symmetry, defects, and gauging of topological phases · b. obstruction to fractionalization 20 c....
TRANSCRIPT
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Symmetry, Defects, and Gauging of Topological Phases
Maissam Barkeshli,1 Parsa Bonderson,1 Meng Cheng,1 and Zhenghan Wang1, 2
1Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA2Department of Mathematics, University of California, Santa Barbara, California 93106, USA
(Dated: November 13, 2014)
We examine the interplay of symmetry and topological order in2+1 dimensional topological quantum phasesof matter. We present a precise definition of thetopological symmetrygroup Aut(C), which characterizes thesymmetry of the emergent topological quantum numbers of a topological phaseC, and we describe its relationwith the microscopic symmetry of the underlying physical system. This allows us to derive a general frame-work to classify symmetry fractionalization in topological phases, including phases that are non-Abelian andsymmetries that permute the quasiparticle types and/or areanti-unitary. We develop a theory of extrinsic defects(fluxes) associated with elements of the symmetry group, extending previous results in the literature. This pro-vides a general classification of2+ 1 dimensional symmetry-enriched topological (SET) phases derived from atopological phase of matterC with symmetry groupG. We derive a set of data and consistency conditions, solu-tions of which define the algebraic theory of the defects, known as aG-crossed braided tensor categoryC×
G . Thisallows us to systematically compute many properties of these theories, such as the number of topologically dis-tinct types of defects associated with each group element, their fusion rules, quantum dimensions, zero modes,braiding exchange transformations, a generalized Verlinde formula for the defects, and modular transformationsof theG-crossed extensions of topological phases. We also examinethe promotion of the global symmetry toa local gauge invariance (“gauging the symmetry”), whereinthe extrinsicG-defects are turned into deconfinedquasiparticle excitations, which results in a different topological phaseC/G. We present systematic methods tocompute the properties ofC/G whenG is a finite group. The quantum phase transition between the topologicalphasesC/G andC can be understood to be a “gauge symmetry breaking” transition, thus shedding light on theuniversality class of a wide variety of topological quantumphase transitions. A number of instructive and/orphysically relevant examples are studied in detail.
Contents
I. Introduction 2A. Summary of Main Results 3B. Relation to Prior Work 5
II. Review of Algebraic Theory of Anyons 6A. Fusion 6B. Braiding 9C. Gauge Transformations 11
III. Symmetry of Topological Phases 12A. Topological Symmetry 12B. Global Symmetry 14C. H3
[ρ](G,A) Invariance Class of the SymmetryAction 15
IV. Symmetry Fractionalization 17A. Physical Manifestation of On-Site Global
Symmetry 18B. Obstruction to Fractionalization 20C. Gauge Transformations 23D. Classification of Symmetry Fractionalization 23E. Projective Representations of the Global
Symmetry 24F. Quasi-On-Site Global Symmetry 25
V. Extrinsic Defects 28A. Physical Realization ofg-Defects 28B. Topologically Distinct Types ofg-Defects 31
VI. Algebraic Theory of Defects 32
A. G-graded Fusion 32B. G-Crossed Braiding 33C. Gauge Transformations 39D. G-Crossed Invariants, Twists, andS-Matrix 39
VII. G-Crossed Modularity 45A. G-Crossed Verlinde Formula andωa-Loops 45B. Torus Degeneracy andG-Crossed Modular
Transformations 46C. Higher Genus Surfaces 50
VIII. Gauging the Symmetry 52A. Microscopic Models 53B. Topological Charges and Fusion Rules ofC/G 53C. Modular Data of the Gauged Theory 56D. Genusg Ground State Degeneracy 59E. Universality Classes of Topological Phase
Transitions 59
IX. Classification of Symmetry Enriched TopologicalPhases 59A. Classification ofG-Crossed Extensions 60B. Relation to PSG Framework 62C. Continuous, Spatial, and Anti-Unitary
Symmetries 62
X. Examples 63A. Trivial Bosonic State withG symmetry 63B. Trivial Fermionic StateZ(1)
2 with G symmetry 64
C. SemionsZ(± 1
2 )2 with Z2 symmetry 66
D. SemionsZ(± 1
2 )2 with Z2 × Z2 symmetry 67
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E. Z(p)N Anyons withN odd andZ2 symmetry 68
F. Z(p+ 1
2 )
N Anyons withN even andZ2 symmetry 69G. ZN -Toric Code D(ZN ) with Z2 symmetry 70H. Double-Layer SystemsB × B with Z2 symmetry 72I. S3-Gauge Theory D(S3) with Z2 symmetry 73J. 3-Fermion Model SO(8)1, a.k.a. D′(Z2), with S3
symmetry 74K. Rep(D10) with Z2 symmetry: AnH3
[ρ](G,A)
Obstruction 78
Acknowledgments 79
A. Review of Group Cohomology 79
B. Projective Representations of Finite Groups 80
C. Z(w)N and Ising(ν) Anyon Models 81
D. Categorical Formulation of Symmetry, Defects,and Gauging 821. Categorical Topological and Global Symmetry 822. Symmetry Defects 823. Gauging Categorical Global Symmetry 844. General Properties of Gauging 845. New Mathematical Results 84
References 85
I. INTRODUCTION
The last two decades of research in condensed matterphysics has yielded remarkable progress in the understand-ing of gapped quantum states of matter. In the absence ofany symmetry, gapped quantum systems at zero temperaturemay still form distinct phases of matter that exhibittopolog-ical order, which is a new kind of order characterized bypatterns of long range entanglements [1, 2]. Topologicallyordered phases possess numerous remarkable properties, in-cluding quasiparticle excitations with exotic, possibly non-Abelian, exchange transformations (statistics), robust pat-terns of long range quantum entanglement, robust topology-dependent ground state degeneracies, and protected gaplessedge modes.
Recently, a number of exciting new directions haveemerged in the study of topological phases of matter, oneof which is the study of extrinsic defects [3–21]. This in-cludes the study of extrinsically imposed point-like defects,which are not finite-energy quasiparticle excitations, butnev-ertheless have a nontrivial interplay with the topologicalorder.These point-like defects can themselves give rise to topologi-cally protected degeneracies, non-Abelian braiding exchangetransformations, and exotic localized zero modes. From apractical standpoint, they might be useful in enhancing thecomputational power of a topological phase used for topo-logically protected quantum information processing [2, 22–27]. For example, one may engineer non-Abelian defects inan Abelian topological phase, or even defects that realize a
computationally universal braiding gate set in a non-Abelianphase that otherwise would not have computationally univer-sal braiding [12]. Several microscopic realizations of such de-fects have been proposed in the past few years, ranging fromlattice dislocations in certain microscopic models [4, 6–8, 16–19] to unconventional methods of coupling fractional quan-tum Hall (FQH) edge states [6, 9–14]. In addition to point-likeextrinsic defects, topological phases also support a rich vari-ety of extrinsic line-like defects. These may either be gappedor gapless, and in both cases there is necessarily a nontriv-ial interplay with the topological order. In particular, gappedline-like defects, such as gapped boundaries [5, 13, 14, 28–33], have recently been proposed to be used for robust experi-mental signatures of certain topologically ordered states, suchas fractionalization in spin liquids and topological degeneracyin FQH states [34–37].
A second direction that has generated intense research is theinterplay of symmetry with topological order. In the presenceof symmetries, gapped quantum systems acquire a finer clas-sification [38–60]. Specifically, it is possible for two phasesof matter to be equivalent in the absence of the symmetry, butdistinct in the presence of symmetry. These are referred toas symmetry-protected topological (SPT) states if the gappedphase is trivial in the absence of symmetry, and as symmetry-enriched topological (SET) states if the gapped phase is topo-logically nontrivial, even when all symmetries are broken.One-dimensional Haldane phases in spin chains [61, 62],two-dimensional quantum spin Hall insulators [63–65], andthree-dimensional time-reversal-invariant topologicalinsula-tors [66–68] are all well-known examples of SPT states. Incontrast, FQH states and gapped quantum spin liquids are ex-amples of SET states, because they possess symmetries (par-ticle number conservation or spin rotational invariance) to-gether with topological order.
In the presence of symmetries, quasiparticles of a topolog-ical phase of matter can acquire fractional quantum numbersof the global symmetry. For example, in theν = 1
3 Laugh-lin FQH state [69], the quasiparticles carry charge in unitsofe/3; in gappedZ2 quantum spin liquids [70], the quasipar-ticles can carry unit charge and no spin (chargeons/holons),or zero charge and spin-1
2 (spinons). With symmetry, an evenlarger class of extrinsic defects are possible, as one can alwaysconsider a deformation of the Hamiltonian that forces a fluxassociated with the symmetry into a region of the system, evenif this flux is not associated with any deconfined quasiparticleexcitation.
When a Hamiltonian that realizes a topological phase ofmatter possesses a global symmetry, it is natural to considerthe topological order that is obtained when this global sym-metry is promoted to a local gauge invariance, i.e. “gaugingthe symmetry.” This is useful for a number of reasons: (1)The properties of the resulting gauged theory can be used asa diagnostic to understand the properties of the original, un-gauged system [71–74]. (2) Gauging the symmetry providesa relation between two different topological phases of matter,and can give insight into the nature of the quantum phase tran-sition between them [75–78]. (3) Understanding the relationbetween such phases may aid in the development of micro-
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scopic Hamiltonians for exotic topological phases (describedby the gauged theory), by starting with known models of sim-pler topological phases (described by the ungauged theory).
Although a remarkable amount of progress has been madeon these deeply interrelated topics, a completely general un-derstanding is lacking, and many questions remain. For ex-ample, although there are many partial results, the currentun-derstanding of fractionalization of quantum numbers, alongwith the classification and characterization of SETs is incom-plete. Moreover, while there have been many results towardsunderstanding the properties of extrinsic defects in topologi-cal phases, there has been no general systematic understand-ing and, in particular, no concrete method of computing allthe rich topological properties of the defects for an arbitrarytopological phase. The study of topological phase transitionsbetween different topological phases is also missing a generaltheory.
In this paper, we develop a general systematic frameworkto understand these problems. We develop a way to char-acterize the interplay of symmetry and topological order in2 + 1 dimensions, thus leading us to a general understand-ing of how symmetries can be consistently fractionalized ina given topological phase. Subsequently, we develop a math-ematical framework to describe and compute the propertiesof extrinsic point-like defects associated with symmetries ofthe topological phase. Our construction utilizes results andideas from recent mathematical literature [79–82]. However,since our focus is on concrete applications to physics, our ap-proach and formalism is quite different from the more abstractcategorical formalism that has been presented in the mathe-matical literature. Our framework for understanding the topo-logical properties of extrinsic defects then provides us with away to systematically classify and characterize SETs (includ-ing SPTs) in2 + 1 dimensions. Finally, we again build onresults from the mathematics literature [80, 83] to provideasystematic prescription for gauging the symmetry of a systemin a topological phase of matter.
A. Summary of Main Results
Due to the length of this paper, we will briefly summarizethe main results of our work here. Before we proceed, we notethat our starting framework to describe a topological phasewithout symmetry is in terms of an anyon modelC, for whichwe provide a detailed review of the general theory in Sec. II.Mathematically,C is referred to as a unitary modular tensorcategory (UMTC). Physically, it can be thought of as the setof topological charges, which label the topologically distincttypes of quasiparticles (anyons), together with data that self-consistently specifies their fusion, associativity, and braidingexchange transformations. As this paper draws upon a numberof technical mathematical concepts, we have made an effort toinclude precise definitions and explanations of most of theseconcepts, in order to make it as self-contained as possible.
1. Symmetry and Fractionalization
Symmetry fractionalization refers to the manner in whichtopologically nontrivial quasiparticles carry quantum numbersthat are (in a sense) fractions of the quantum numbers of theunderlying local constituents of the system, such as electronsor spins. We show that for a symmetryG (continuous or dis-crete, unitary or anti-unitary), symmetry fractionalization isclassified by two objects,[ρ] and[t], which we briefly describehere. The square brackets here indicate equivalence classes,as there are non-physical redundancies, i.e. a sort of gaugefreedom, associated with both objects that should be factoredout.
We first define the group oftopological symmetries, de-noted Aut(C), of a topological phase of matter described byC. Roughly speaking, this corresponds to all of the differentways the theoryC can be mapped back onto itself, includ-ing permutations of topological charges, in such a way thatthe topological properties are left invariant. A subset of suchauto-equivalence maps called “natural isomorphisms,” whichdo not permute topological charges and leave all the basic dataunchanged, provide the redundancy under which one equatesthe auto-equivalence maps to form the group Aut(C). Sim-ple examples of auto-equivalence maps include layer permu-tations in multi-layer systems that consist of multiple identicalcopies of a topological phase, or electric-magnetic duality inphases described by aZn gauge theory.
We next consider a physical system in a topological phasedescribed byC, which also has a global symmetry describedby the groupG. One must specify howG acts upon the topo-logical degrees of freedom and thus interplays with the topo-logical symmetry. This is characterized by a group action
[ρ] : G→ Aut(C). (1)
The notation means that we assign an auto-equivalence mapρg to each group elementg ∈ G and take the equivalenceclasses of these maps under natural isomorphism. (It is usefulto work with a specific choiceρ ∈ [ρ] when deriving results,and then demonstrate invariance within the equivalence classfor certain quantities at the end.)
Once[ρ] is specified, we examine the symmetry action inan underlying physical system described by a microscopicHamiltonian. We show that symmetry fractionalization is pos-sible only when a certain obstruction class[O] ∈ H3
[ρ](G,A)
vanishes. HereA corresponds to the group of Abelian topo-logical charges inC, where group multiplication is defined byfusion. H3
[ρ](G,A) is the 3rd cohomology group ofG withcoefficients in the groupA, where the subscript[ρ] indicatesthe inclusion of the symmetry action in the definition of thecohomology, which, in this context, is a potential permutationof the topological charge values inA (and, hence, is indepen-dent of the choice ofρ ∈ [ρ]). When the obstruction vanishes,it is possible to consistently fractionalize the symmetry in thesystem, meaning one can specify a local projective symme-try action that is compatible with the symmetry action on thetopological degrees of freedom. The different ways in whichthe symmetry can be fractionalized is classified by the 2nd co-
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homology groupH2[ρ](G,A), with there being a distinct frac-
tionalization class for each element[t] ∈ H2[ρ](G,A). More
precisely, the set of symmetry fractionalization classes formanH2
[ρ](G,A) torsor, which means the classes are not them-
selves elements ofH2[ρ](G,A), but rather the distinct fraction-
alization classes are related to each other by an action of dis-tinct elementsH2
[ρ](G,A). The precise definitions of thesemathematical objects will appear in the main text and appen-dices.
2. Extrinsic Defects
When the physical system has a symmetryG, one canconsider the possibility of point-like defects associatedwithgroup elementsg ∈ G, which may be thought of as fluxes.In many ways, a defect behaves like quasiparticle. However,an important distinction is that when a quasiparticle is trans-ported around ag-defect, it is acted upon by the correspond-ing symmetry actionρg, possibly permuting the quasiparti-cle’s topological charge value. Another important distinctionis that, sinceG describes a global symmetry and not a localgauge invariance in this context, these defects do not corre-spond to finite-energy excitations of the system. Thus, theymust beextrinsically imposedby modifying the Hamiltonianin a manner that forces theg-flux into the system. If the posi-tion of the defects are allowed to fluctuate quantum mechan-ically, the energy cost of separating such defects will groweither logarithmically or linearly in their separation. There-fore they may also be viewed asconfinedexcitations of thesystem.
The extrinsic defects of a topological phase have many richtopological properties, and one purpose of this paper is to de-velop a concrete algebraic formalism, analogous to the alge-braic theory of anyons, that can be used to characterize andsystematically compute the many topological properties ofsuch defects. For this, we begin by generalizing the notionof topological charge to apply to defects, with distinct typesof g-defects carrying distinct values of topological charge. Wethen extend the description of the original anyon modelC, de-scribing the topological phase, to aG-graded fusion theory
CG =⊕
g∈GCg, (2)
where each sectorCg describes the topologically distinct typesof g-defects and the fusion and associativity relations respectthe group multiplication ofG, i.e. ag-defect and anh-defectfuse to agh-defect. In this way, the quasiparticles of the origi-nal topological phase correspond to the0-defects, i.e.C0 = C.
Subsequently, we introduce a generalized notion of braid-ing transformations that incorporates the symmetry actionρgas a quasiparticle or defect passes around ag-defect. This isreferred to as “G-crossed braiding” and defines aG-crossedbraided tensor category (BTC), which we denote asC×
G . Sim-ilar to anyon models, we provide a diagrammatic represen-tation of the states and operators of the theory and identify
the basic data that fully characterizes the theory. We intro-duce consistency conditions on the basic data, which gener-alize the famous hexagon equations for braiding consistencyto “heptagon equations” forG-crossed braiding, and imposeconsistency of the incorporation of the symmetry action andits fractionalization within the theory.
Given the basic data of theG-crossed theory, we are ableto compute all properties of the defects, including their fusionrules, quantum dimensions, localized zero modes, and braid-ing statistics. We find that topological twists, which charac-terize the braiding statistics of objects, is not a gauge invari-ant quantity for defects, which meshes well with the notionthat the defects are associated with confined objects. Anotherimportant property that we derive is that the total quantum di-mensionDg of the sectorCg is the same for allg ∈ G, i.e.Dg = D0 (this holds generally for aG-graded fusion cate-gory). We also find that the number of topologically distinctg-defects,|Cg|, is equal to the number ofg-invariant topolog-ical charges [i.e. those for whichρg(a) = a] in the originalUMTC C0.
We describe the notion ofG-crossed modular transforma-tions when the system inhabits a torus or surfaces of arbitrarygenus. These extend the usual definition of modular trans-formations, generated byS andT matrices, to cases wherethere are defect branch lines wrapping the cycles of the torusor higher genus surface. We derive aG-crossed generaliza-tion of the Verlinde formula, which relate the fusion rules ofdefects (and quasiparticles) to theG-crossedS-matrix.
For every2 + 1 dimensional SET phase, one can constructa correspondingG-crossed theoryC×
G describing the defectsin the topological phase. Therefore, theG-crossed defecttheoriesC×
G provide both a classification and a characteriza-tion of SET phases in2 + 1 dimensions. In this way, onecan classify SETs by solving theG-crossed consistency rela-tions. However, it has recently been proven [81] that, for afinite groupG, which describes unitary on-site symmetries,the distinctG-crossed extensionsC×
G of a topological phasedescribed byC are fully classified by three objects: the sym-metry action[ρ] : G → Aut(C) describing how the globalsymmetry acts on the topological degrees of freedom, an ele-ment[t] ∈ H2
[ρ](G,A) that classifies the symmetry fractional-
ization, and an element[α] ∈ H3(G,U(1)) that classifies thedefects’ associativity.
Importantly, not every fractionalization class inH2[ρ](G,A)
corresponds to a well-defined SET in2 + 1 dimensions. Insome cases there can be an additional obstruction that preventsthe existence of a solution of theG-crossed consistency rela-tions (such as the heptagon equations). The inability to solvethese consistency conditions and thus to construct a consistentextended theoryC×
G indicates that the symmetry fractionaliza-tion class is anomalous. A number of recent examples haveshown that anomalous realizations of symmetry fractionaliza-tion, while they cannot exist in 2+1 dimensions, can insteadexist as a surface termination state of a3+1 dimensional SPTstate [84–94].
Similar to theH2[ρ](G,A) classification of fractionaliza-
tion classes, the set of defect associativity classes formsa H3(G,U(1)) torsor, meaning distinctG-crossed theories
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(with the sameC0, symmetry action, and fractionalizationclass) are related to each other by an action of the distinct ele-ments ofH3(G,U(1)). This action by[α] ∈ H3(G,U(1)) isessentially factoring in a groupG SPT state with associativitydefined by[α] to possibly produce anotherG-crossed theory.Whether factoring in this groupG SPT actually provides adistinct SET can be determined in our framework by seeingwhether factoring in the SPT has the equivalent effect of rela-belling differentg defects. These results together then suggestthat([ρ], [t], [α]) parametrize SETs in 2+1 dimensions, at leastfor finite on-site unitary symmetries.
3. Gauging the Symmetry
Given a topological phase of matterC, together with itssymmetry-enriched class, i.e. itsG-crossed defect theoryC×
G ,one can promote the symmetryG to a local gauge invariance(“gauging the symmetry”). This results in a different topo-logical order, which we denoteC/G, in which theg-defectsbecome deconfined quasiparticle excitations. Importantly, thegauged theoryC/G depends on the particularG-crossed ex-tensionC×
G of C, which thus forms the input data necessary toconstruct the gauged theory. The topological properties ofthegauged theoryC/G can alternatively be viewed from a dif-ferent perspective as topological invariants of the associatedSET, which is described byC×
G .We first examine the question of how one may obtain a
microscopic Hamiltonian that realizes the topological phaseC/G, given a Hamiltonian that realizes a topological phaseC.Along this line, we provide a concrete model demonstratinghow this may be done in the case whereG is an Abelian finitegroup.
Next, we provide a review of some known results fromthe mathematics literature for obtaining the properties ofC/Gfrom those ofC×
G , in particular the topological charge con-tent, quantum dimensions, and fusion rules. It follows fromthese results that the total quantum dimension of the gaugedtheory C/G is always related to the total quantum dimen-sion of the original theoryC and itsG-crossed extension byDC/G = |G| 12DCG = |G|DC . We further conjecture, based onphysical considerations, a formula for the topological twistsof quasiparticles inC/G. Based on this conjecture we showthat the chiral central charge (mod 8) is the same in these the-ories. We also derive a formula for the modularS-matrix ofC/G in terms of the data ofC×
G . Finally, we discuss how tocompute the ground state degeneracy ofC/G on higher genussurfaces in terms of the properties ofC×
G , without needing toderive the full fusion rules ofC/G. This is useful for practicalcomputations of the number of topological charge types andtheir quantum dimensions.
Finally, we observe that, sinceC/G andC are related toeach other by gaugingG, the topological quantum phase tran-sition between them can be understood as a discreteG “gaugesymmetry breaking” transition. This point of view providesinsight into the universality class of the topological phase tran-sitions between a wide variety of distinct topological phases.
4. Examples
After developing the general theory, summarized above, westudy many concrete examples. We focus on examples thatare physically relevant and/or which illustrate differenttech-nical aspects and subtleties of using the theory and methodsdeveloped in this paper to derive the various properties ofG-crossed extensions and gauged theories. A particularly inter-esting example that we examine is the “three-fermion theory,”also known as SO(8)1, with the non-Abelian symmetry groupG = S3 acting nontrivially. Gauging theS3-symmetry ofthe three-fermion theory results in a rank12 (weakly integral)UMTC that has not been previously described elsewhere.
B. Relation to Prior Work
The background context of our work is closely related to alarge number of works spanning many different fields. Herewe briefly comment on the relation to some of the most closelyrelated works.
A framework, called the projective symmetry group (PSG),to address the problem of classifying SETs was originally in-troduced in Ref. [38]. As we discuss in Sec. IX B, the PSGframework only captures a subset of possible types of sym-metry fractionalization and, thus, misses a large class of pos-sible SETs for a given topological phase. Our results on thegeneral classification of symmetry fractionalization in termsof H2
[ρ](G,A) extends the previous result of Ref. [51], whichspecifically applies to Abelian topological phases where thesymmetries do not permute the topological charge values. Apreliminary consideration of some of these ideas can also befound at a more abstract level in the discussion in Appendix Fof Ref. [95].
The notion of aG-crossed braided tensor category (BTC)was originally introduced in the mathematics literature inRefs. [79, 82]. Similarly, the full classification ofG-crossedextensions in terms of the objects([ρ], [t], [α]) and the possi-ble obstructions, summarized in the previous subsection, haspreviously appeared in the mathematics literature [81] in theproblem of extending a fusion category or a braided fusioncategory by a finite groupG.
With respect to these prior mathematical results, our resultscan be viewed as both providing (1) a new and detailed con-crete formulation of the theory ofG-crossed BTC, and (2) pro-viding the physical context and interpretations of the abstractmathematical results by directly linking them to their physicalrealizations. In particular, we provide a physical interpreta-tion of these mathematical objects in terms of the fusion andbraiding properties of extrinsic defects associated with groupelementsg ∈ G. Moreover, since the mathematical construc-tions are highly abstract, they may obscure many of the impor-tant details that are of interest for physical applications. Forexample, we provide concrete definitions of the[O] obstruc-tion and [t] classification objects in terms of the symmetryaction on the states of quasiparticles. The mathematical treat-ment that we utilize in this paper, working directly with thetopologically distinct classes of simple objects (quasiparticles
6
and defects), their basic data (F -symbols,R-symbols, etc.),and their consistency conditions, is referred to in mathemat-ical parlance as a “skeletonization” of a category. Our workmay, thus, be viewed as a new mathematical result that intro-duces the skeletonization ofG-crossed BTCs and provides anew definition of the theory ofG-crossed BTCs.
Extrinsic defects in topological phases of matter have beenincreasingly studied in various examples in the condensedmatter physics literature [4–21]. One purpose of our work isto provide a totally general treatment of extrinsic twist defectsthat captures all of their topologically nontrivial properties,provides a framework for computing them, and can be appliedto arbitrary topological phases of matter. In recent years,suchdefects have also been studied in the mathematical physicsliterature, both for conformal field theory (CFT) [96] and fortopological quantum field theory (TQFT) [97, 98]. While ourwork has some overlap with these, our approach is quite dif-ferent. Our emphasis is on developing concrete methods thatcan be used to compute various topological properties of thedefects and direct physical interpretations that apply in thecondensed matter physics setting.
The idea of “gauging” a discrete symmetry of a topologicalphase of matter is closely related to the concept of “orbifold-ing” in rational CFT [99, 100]. However, while there are oftenclose relations between CFTs and topological phases of mat-ter, they are distinct physical systems, and so they each requiretheir own physical understanding. Many of our general resultsand examples go beyond the analogous problem that has beenstudied in the CFT literature, for which the general resultsarelimited. For example, much of the CFT work on orbifolding istypically focused on holomorphic CFTs, which correspond toonly a small class of possible topological phases. The impor-tant classifying objects([ρ], [t], [α]) summarized above alsohave not, to our knowledge, been generally discussed in theCFT literature on orbifolding.
Our work on gauging topological phases of matter is closelyrelated to work of Refs. [80, 101], which sets out to find amathematical formulation in terms of MTCs of the conceptof orbifolding in CFTs. For example, Ref. [80] also containsresults on the extended Verlinde algebra. Again, our resultsextend some of these mathematical results and put them intomore concrete terms with direct physical context.
In recent years, the notion of gauging symmetries of atopological phase has been increasingly studied in the con-densed matter literature. The resulting non-Abelian topo-logical phases that are obtained by gauging either the layerexchange symmetry of bilayer Abelian FQH states, or theelectric-magnetic duality ofZN toric code models were stud-ied in Refs. [77, 102, 103]. In studies of SPT phases, the no-tion of gauging the symmetry of the system has been power-ful in developing an understanding of the distinction betweenSPT states [71, 74]. While those were isolated classes of ex-amples, our work provides a concrete prescription to derivethe properties obtained when any topological phase of matterC is gauged by any finite groupG.
While gauging a discrete global symmetryG of a topo-logical phaseC gives rise to a new topological phaseC/G,there is an inverse process, known as topological Bose con-
densation [75], which takesC/G to C. The quantum phasetransition betweenC/G and C corresponds to a confine-ment/deconfinement transition or, in other words, a “gaugesymmetry breaking” transition. The notion of condensa-tion was discussed mathematically in Refs. [104, 105]. Thishas been studied in the context of topological phases inRefs. [28, 75, 106, 107]. In the topological Bose condensa-tion picture, there is an intermediate stage betweenC/G andC, referred to as theT -theory in Ref. [75], which includes theobjects that are confined by the condensate. These confinedobjects areg-defects with aG-crossed theoryC×
G that pro-vides the complete description of the topological properties oftheT -theory, including their braiding transformations, whichhas not been previously identified. Most of the prior workalong these lines has focused on the nature of the topologicalphase that is obtained when topologically non-trivial bosonsof a topological phase are condensed. However, Refs. [76–78] focused on the nature of the universality class of quan-tum phase transitions associated with topological Bose con-densation by studying some simple classes of examples whenG = Z2. We generalize these results to an understanding ofthe universality class of topological Bose condensation transi-tions betweenC/G to C for general finiteG.
II. REVIEW OF ALGEBRAIC THEORY OF ANYONS
This section provides a summary review of anyon models,known in mathematical terminology as unitary braided ten-sor categories (UBTC) [108, 109]. We use a diagrammaticrepresentation of anyonic states and operators acting on them,following Refs. [95, 110–112]. (Many relations in this re-view section are stated without proof. For additional detailsand proofs, we refer the reader to the references listed hereor, in some cases, to Secs. VI and VII where one may findthe generalized versions.) This formalism encodes the purelytopological properties of anyons, i.e. quasiparticle excitationsof topological phases of matter, independent of any particularphysical realization.
A. Fusion
In this section, we describe the properties of fusion tensorcategories, and will introduce braiding in the next. We beginwith a setC of superselection sector labels called topologicalor anyonic chargesa, b, c . . . ∈ C. [172] (We will often alsouse the symbolC to refer the category itself.) These conservedcharges obey an associative fusion algebra
a× b =∑
c∈CN cabc (3)
where the fusion multiplicitiesN cab are non-negative integers
which indicate the number of different ways the chargesa andb can be combined to produce the chargec. We require thatfusion is finite, meaning
∑cN
cab is a finite integer for any
7
fixeda andb. Associativity requires these to satisfy
∑
e
NeabN
dec =
∑
f
NdafN
fbc. (4)
In the diagrammatic formalism, each line segment is ori-ented (indicated with an arrow) and ascribed a value of topo-logical charge. Each fusion product has an associated vectorspaceV cab with dimV cab = N c
ab, and its dual (splitting) spaceV abc . The states in these fusion and splitting spaces are as-signed to trivalent vertices with the appropriately correspond-ing anyonic charges, with basis states written as
(dc/dadb)1/4
c
ba
µ = 〈a, b; c, µ| ∈ V cab, (5)
(dc/dadb)1/4
c
ba
µ = |a, b; c, µ〉 ∈ V abc , (6)
whereµ = 1, . . . , N cab. (Many anyon models of interest have
no fusion multiplicities, i.e.N cab = 0 or 1 only, in which
case the trivial vertex labelsµ will usually be left implicit.)The bra/ket basis vectors are orthonormal. The normalizationfactors(dc/dadb)
1/4 are included so that diagrams will be inthe isotopy invariant convention, as will be explained in thefollowing. Isotopy invariance means that the value of a (la-beled) diagram is not changed by continuous deformations,so long as open endpoints are held fixed and lines are notpassed through each other or around open endpoints. Openendpoints should be thought of as ending on some boundary(e.g. a timeslice or an edge of the system) through which iso-topy is not permitted. We note that the diagrammatic expres-sions of states and operators are, by design, reminiscent ofparticle worldlines, but there is not a strict identification be-tween the two. The anyonic charge lines are only a diagram-matic expression of the algebraic encoding of the topologicalproperties of anyons, and interpreting them as worldlines isnot always correct.
Diagrammatically, inner products are formed by stackingvertices so the fusing/splitting lines connect
a b
c
c′
µ
µ′= δcc′δµµ′
√dadbdc
c
, (7)
which can be applied inside more complicated diagrams.Note that this diagrammatically encodes charge conservation.
Since we want to use this to describe the states associated withanyonic quasiparticles (in a topological phase of matter),werequire the inner product to be positive definite, i.e.da arerequired to be real and positive.
With this inner product, the identity operator on a pair ofanyons with chargesa andb is written (diagrammatically) asthe partition of unity
11ab =
ba
=∑
c,µ
√dcdadb
c
ba
ba
µ
µ. (8)
A similar decomposition applies for an arbitrary number ofanyons.
More complicated diagrams can be constructed by connect-ing lines of matching charge. The resulting vector spaces obeya notion of associativity given by isomorphisms, which can bereduced using the expression of three anyon splitting/fusionspaces in terms of two anyon splitting/fusion
V abcd∼=⊕
e
V abe ⊗ V ecd∼=⊕
f
V bcf ⊗ V afd , (9)
to isomorphisms calledF -moves, which are written diagram-matically as
a b c
e
d
α
β=∑
f,µ,ν
[F abcd
](e,α,β)(f,µ,ν)
a b c
f
d
µ
ν. (10)
The F -moves can be viewed as changes of bases for thestates associated with quasiparticles. To describe topologicalphases, these are required to be unitary transformations, i.e.
[(F abcd
)−1]
(f,µ,ν)(e,α,β)=[(F abcd
)†]
(f,µ,ν)(e,α,β)
=[F abcd
]∗(e,α,β)(f,µ,ν)
. (11)
In order for this notion of associativity to be self-consistent,any two sequences ofF -moves applied within an arbitrarydiagram which start from the same state space and end inthe same state space must be equivalent. MacLane’s coher-ence theorem [113] establishes that this consistency can beachieved by imposing the constraint called the Pentagon equa-tion
∑
δ
[F fcde
](g,β,γ)(l,δ,ν)
[F able
](f,α,δ)(k,λ,µ)
=∑
h,σ,ψ,ρ
[F abcg
](f,α,β)(h,σ,ψ)
[F ahde
](g,σ,γ)(k,λ,ρ)
[F bcdk
](h,ψ,ρ)(l,µ,ν)
(12)
8
e
g
c d
e
f
c db
e
g
a c db
Fk
a dbe
c dbFbal
F
f
e
F Fc
a
a
kl
hh
FIG. 1: The Pentagon equation enforces the condition that differentsequences ofF -moves from the same starting fusion basis decom-position to the same ending decomposition gives the same result.Eq. (12) is obtained by imposing the condition that the abovedia-gram commutes.
which equates the two sequences ofF -moves shown in Fig. 1.In other words, given a set of fusion rules, one can find allconsistent fusion categories by solving the Pentagon equationsfor all consistent sets ofF -symbols.
We require the existence of a unique “vacuum” charge0 ∈ C for which fusion (and braiding) is trivial. In particular,the fusion coefficients must satisfyN c
a0 = N c0a = δac, charge
lines can be added and removed from diagram at will (in otherwords, there are canonical isomorphisms betweenV a0a , V 0a
a ,andC), and the associativity relations must obey
[F abcd
]= 11
if any one ofa, b, or c equals0 when the involved fusionsare allowed (this enforces the compatibility ofF -moves withthe previously mentioned canonical isomorphism and corre-sponds to choosing the basis vectors ofV a0a andV 0a
a suchthat they map to1 in the canonical isomorphisms mentionedabove). Note that it is not required that
[F abcd
]= 11 when
d = 0, nor is this even generally possible. We often speciallydenote vacuum lines as dotted lines.
For eacha ∈ C, we require the existence of a conjugatecharge, or “antiparticle,”a ∈ C, for which [F aaaa ](0,α)(0,µ) 6=0. It follows thatN0
ab = δba, i.e. a is unique anddimV 0aa = 1.
Also, 0 = 0 and¯a = a. Thus, we can write
[F aaaa ]00 =κada, (13)
where we have defined the quantum dimensionda of chargeato be
da =∣∣[F aaaa ]00
∣∣−1(14)
andκa is a phase. It follows thatd0 = 1,
da = da = a . (15)
Here we have introduced the convention of smoothing out thechargea line at |a, a; 0〉 vertices to form a “cup” when we re-move the vacuum charge0 line, and similarly forming a “cap”from 〈a, a; 0|.
We also define the total quantum dimension ofC to be
D =
√∑
a∈Cd2a. (16)
In the diagrammatic formalism, reversing the orientation ofa line is equivalent to conjugating the charge labeling it, i.e.
a=
a. (17)
Isotopy invariance is essentially the ability to introduceandremove bends in a line. Bending a line horizontally (so thatthe line always flows upward) is trivial (in that it utilizes thecanonical isomorphisms of adding/removing vacuum lines),but a complication arises when a line is bent vertically. Tounderstand this, consider theF -move associated with this typeof bending
a a a
0
0
= κa a . (18)
(Notice the vertex normalization comes into play here.) Ingeneral, the phaseκa = κ∗
a is not equal to1, but for a 6=a, it is gauge dependent and can be fixed to1 by a gaugechoice. Fora = a, κa = ±1 is a gauge invariant quantity,known as the Frobenius-Schur indicator. Thus, we see thatone needs more than just diagrammatic vertex normalizationto produce isotopy invariance for this kind of bending. Thiscan be dealt with using flags that keep track of nontrivialκaphases and unitary transformations (which can be defined interms of theF -symbols) when the legs of a vertex are bent upor down, which can be used, for example, to prove the pivotalproperty. (We refer the reader to Refs. [95, 111] for details.)It follows that the dimension of fusion/splitting spaces relatedby bending lines are equal, so
N cab = N b
ac = Nacb = N a
bc = N bca = N c
ba. (19)
We can also define a diagrammatic trace of operators(known as the “quantum trace”) by closing the diagram withloops that match the outgoing lines with the respective incom-ing lines at the same position
TrX = Tr
X
. . .
. . .
=
∑
a1,...,an
X
. . .
. . .
. . .
a1 an
. (20)
Connecting the endpoints of two lines labeled by differenttopological charge values violates charge conservation, sosuch diagrams evaluate to zero. One can equivalently takethe trace either by looping the lines around to the right (as
9
shown above) or to the left (with their equality following fromda = da).
By taking the trace of11ab and using isotopy, together withEqs. (7) and (8), we obtain the important relation
dadb =∑
c
N cabdc. (21)
Let us define fusion matricesNa using the fusion coeffi-cients to be[Na]bc = N c
ba. We note that the bending rela-tions indicate thatNT
a = Na. From Eq. (21), we see thatthe vectorv with componentsvc = dc/D is a normalizedeigenvector of each matrixNa with corresponding eigenvalueda. Moreover, the Perron-Frobenius theorem assures us thatv is the only eigenvector (up to overall multiplicative fac-tors) ofNa with all positive components and thatda is thelargest (in absolute value) eigenvalue ofNa. Thus, the di-mension of the state space asymptotically grows as powersof da as one increases the numbern of a quasiparticles, i.e.∑
c dimV a...ac =∑
c [Nna ]0c ∼ dna asn → ∞. If da = 1,
we call chargea Abelian, which is equivalent to saying it hasunique fusion with all other charges (
∑cN
cab = 1 for all b).
Otherwise,da > 1 and we call it non-Abelian.Given fusion rules specified byN c
ab, we can define the cor-responding Verlinde algebra spanned by elementsva whichsatisfyva = v†
a and
vavb =∑
c
N cabvc. (22)
Notice thatva may be (faithfully) represented byNa.
B. Braiding
The theory described in the previous subsection defineda unitary fusion tensor category with positive-definite innerproduct. We now wish to introduce braiding. For this, werequire the fusion algebra to also be commutative, i.e.
N cab = N c
ba, (23)
so that the dimension of the state space is unaltered when thepositions of anyons are interchanged.
We note that this, together with associativity, impliesNaNb = NbNa, i.e. all of the fusion matrices commute witheach other. Hence, the fusion matrices are also normal andsimultaneously diagonalizable by a unitary matrixP. Specif-ically, Na = PΛ(a)P−1, where[Λ(a)]bc = λ
(a)b δbc and the
eigenvalues areλ(a)b = Pab/P0b. The eigenvalues form thefusion characters of the Verlinde algebra, i.e. for eachb themapλb : a 7→ λ
(a)b is a fusion character satisfying the rela-
tions
λ(a)e λ(b)e =∑
c
N cabλ
(c)e , (24)
∑
a
λ(a)b λ(a)∗c = δbc |P0b|−2
. (25)
Moreover, we have the relation
N cab =
∑
x
PaxPbxP∗cx
P0x. (26)
The counterclockwise braiding exchange operator of twoanyons is represented diagrammatically by
Rab =
a b
=∑
c,µ,ν
√dcdadb
[Rabc
]µν
c
ba
ab
ν
µ , (27)
where theR-symbols are the mapsRabc : V bac → V abc thatresult from exchanging two anyons of chargesb anda, re-spectively, which are in the chargec fusion channel. This canbe written as
c
ba
µ =∑
ν
[Rabc
]µν
c
ba
ν . (28)
Similarly, the clockwise braiding exchange operator is
(Rab
)−1=
b a
. (29)
In order for braiding to be compatible with fusion, werequire that the two operations commute. Diagrammati-cally, this means we can freely slide lines over or under fu-sion/splitting vertices
x
c
ba
µ
=x
c
ba
µ
(30)
x
c
ba
µ
=x
c
ba
µ
. (31)
These relations imply the Yang-Baxter equations for braid-ing operators,Rj,j+1Rj−1,jRj,j+1 = Rj−1,jRj,j+1Rj−1,j ,whereRj,j+1 is the operator that braids the strands in thejthand(j + 1)th positions in the counterclockwise sense, whichare equivalent to the property that lines can slide over braids,since the ability to freely slide lines over/under verticesal-lows lines to slide over/under braiding operators. Diagram-matically, this is written as
= . (32)
Requiring consistency between fusion and braiding, we findconditions that must be satisfied by theF -symbols andR-symbols, which may be expressed as the Hexagon equations
10
F
R
F
R
F
d
e
a cb
d
e
a b c d
c
g
a cb
g
d
f
a cb
d
a cb
R
a b
f
d
R−1R−1
F
F Fd
ca b d
a cb
d
a cb
ge
e
d
a cb
fR−1
d
f
a cb
g
d
a cb
FIG. 2: The Hexagon equations enforce the condition that braiding is compatible with fusion, in the sense that differentsequences ofF -movesandR-moves from the same starting configuration to the same ending configuration give the same result. Eqs. (33) and (34) are obtained byimposing the condition that the above diagram commutes.
∑
λ,γ
[Race ]αλ[F acbd
](e,λ,β)(g,γ,ν)
[Rbcg]γµ
=∑
f,σ,δ,ψ
[F cabd
](e,α,β)(f,δ,σ)
[Rfcd
]
σψ
[F abcd
](f,δ,ψ)(g,µ,ν)
, (33)
∑
λ,γ
[(Rcae )
−1]
αλ
[F acbd
](e,λ,β)(g,γ,ν)
[(Rcbg)−1]
γµ=
∑
f,σ,δ,ψ
[F cabd
](e,α,β)(f,δ,σ)
[(Rcfd
)−1]
σψ
[F abcd
](f,δ,ψ)(g,µ,ν)
.(34)
These relations are represented diagrammatically in Fig. 2.MacLane’s coherence theorem [113] establishes that if thePentagon equation and Hexagon equations are satisfied, thenany two sequences ofF -moves andR-moves (braiding) ap-plied within an arbitrary diagram which start from the samestate space and end in the same state space are equivalent,which is to say that fusion and braiding are consistent. TheF -symbols andR-symbols completely specify a braided ten-sor category (BTC).
Given the trivial associativity of the vacuum charge0(F abcd = 11 whena, b, or c = 0), the Hexagon equations implythat braiding with the vacuum is trivial, i.e.Ra0a = R0a
a =(Ra0a
)−1=(R0aa
)−1= 1.
If we further require unitarity of the theory, then(Rab
)−1=(Rab
)†, which can be expressed in terms of
R-symbols as[(Rabc
)−1]
µν=[Rabc
]∗νµ
(which are simply
phases whenN cab = 1).
An important quantity derived from braiding is the topolog-ical twist (or topological spin) of chargea
θa = θa =∑
c,µ
dcda
[Raac ]µµ =1
da a
, (35)
which is a root of unity [114]. This can be used to show thattheR-symbols satisfy the “ribbon property”
∑
λ
[Rabc
]µλ
[Rbac
]λν
=θcθaθb
δµν . (36)
Another important quantity is the topologicalS-matrix
Sab = D−1∑
c
N cab
θcθaθb
dc =1
D a b . (37)
It is clear thatSab = Sba = S∗ab andS0a = da/D. A related
invariant quantity
Mab =S∗abS00
S0aS0b(38)
is the monodromy scalar component, which plays an impor-tant role in anyonic interferometry [112, 115, 116] and whichwill show up later in the classification of symmetry fractional-izations and group extensions of categories. IfMab = eiφ(a,b)
is a phase, then the braiding ofa with b is Abelian in the sensethat
a b
= eiφ(a,b)
ba
. (39)
Moreover, when this is true, it follows thatMabMac = Mae
wheneverNebc 6= 0.
An important property that follows from the definition oftheS-matrix is the ability to remove closed loops that encir-cle other line, which is done by acquiring an amplitude deter-
11
mined by theS-matrix. In particular, we have
a
b
=SabS0b
b(40)
which can be verified by taking the trace of both sides, closingtheb charge line into a loop.
Using Eq. (40) for a diagram with two loops of topologicalchargea andb, respectively, linked on a line of topologicalchargex, together with Eqs. (7) and (8) and isotopy, we obtainthe important relation
SaxS0x
SbxS0x
=∑
c
N cab
ScxS0x
. (41)
This relation shows thatλ(a)[x] = Sax/S0x is a character of theVerlinde algebra. Here, we wrote[x] to indicate an equiva-lence class of topological charges that correspond to the samecharacter, reflecting the fact that theS-matrix may be degen-erate.
When theS-matrix is non-degenerate it is unitary, and thisis equivalent to the condition that braiding is non-degenerate,which means that for each topological chargea 6= 0 there issome chargeb such thatRabRba 6= 11ab.
Indeed, when theS-matrix is unitary, the equivalenceclasses[x] of topological charges corresponding to the sameVerlinde algebra character are singletons and all the fusioncharacters of the Verlinde algebra are specified by theS-matrix and given byλ(a)x = Sax/S0x. In this case, we canalso writePab = Sab, which is often phrased as “theS-matrix diagonalizes the fusion rules.” In this case, we canuse the inverse of theS-matrix with Eq. (41) to determine thefusion rules from theS-matrix, as specified by the Verlindeformula [117]
N cab =
∑
x∈C
SaxSbxS∗cx
S0x. (42)
When theS-matrix is unitary, the braided tensor categoryis called a modular tensor category (MTC). Such theories canbe consistently defined for 2D manifolds of arbitrary genusand are related to(2 + 1)D TQFTs. In this case, theS-matrixtogether with theT -matrix,Tab = θaδab, and the charge con-jugation matrixCab = δab obey the modular relations
(ST )3 = ΘC, S2 = C, C2 = 11 (43)
where
Θ =1
D∑
a∈Cd2aθa = ei
2π8 c− (44)
is a root of unity andc− ≡ c − c is the chiral central charge.These correspond to the TQFT’s projective representation ofthe respective modular transformations on a torus.
Another useful property of a UMTC is that, if a given topo-logical chargea has Abelian braiding with all other charges,i.e. if Mab = eiφ(a,b) is a phase for all chargesb ∈ C, thenais Abelian in the sense that it hasda = 1 (and hence Abelianfusion and associativity). This follows from unitarity of theS-matrix, which implies that
1 =∑
b
|Sab|2 =∑
b
∣∣∣∣S0aS0b
S00Mab
∣∣∣∣2
=∑
b
∣∣∣∣dadbD eiφ(a,b)
∣∣∣∣2
= d2a. (45)
In other words, non-Abelian topological charges (those withda > 1) necessarily have non-Abelian braiding in a UMTC.
Finally, we establish the following property for MTCs,which will be useful for establishing the classification of sym-metry fractionalization. If there are phase factorseiφa (de-fined for all charge values) that satisfy the relation
eiφaeiφb = eiφc (46)
wheneverN cab 6= 0, then it must be the case that
eiφa =M∗ae (47)
for some Abelian topological chargee. To verify this claim,we writeλ(a) = dae
iφa and notice that
λ(a)λ(b) =∑
c
N cabλ
(c). (48)
Hence, it is a fusion character and must be given byλ(a) =Sae/S0e for some topological chargee. Thus, we have
eiφa =λ(a)
da=SaeS00
S0eS0a=M∗
ae, (49)
and since this makesMae a phase for all values ofa, it followsthat e must be an Abelian topological charge. In this case,M∗ae = Sae/S0a.
C. Gauge Transformations
Distinct sets ofF -symbols andR-symbols describe equiv-alent theories if they can be related by a gauge transformationgiven by unitary transformations acting on the fusion/splittingstate spacesV abc andV cab, which can be though of as a redefi-nition of the basis states as
˜|a, b; c, µ〉 =∑
µ′
[Γabc]µµ′ |a, b; c, µ′〉 (50)
whereΓabc is the unitary transformation. Such gauge transfor-mations modify theF -symbols as
12
[F abcd
]
(e,α,β)(f,µ,ν)=
∑
α′,β′,µ′,ν′
[Γabe]αα′ [Γ
ecd ]ββ′
[F abcd
](e,α′,β′)(f,µ′,ν′)
[(Γbcf)−1]
µ′µ
[(Γafd )−1
]
ν′ν(51)
and theR-symbols as[Rabc
]
µν=∑
µ′,ν′
[Γbac]µµ′
[Rabc
]µ′ν′
[(Γabc)−1]
ν′ν. (52)
One must be careful not to use the gauge freedom associatedwith Γa0a andΓ0b
b to ensure that fusion and braiding with thevacuum0 remain trivial. More specifically, one should fixΓa0a = Γ0b
b = Γ000 . (One can think of this as respecting the
canonical isomorphisms that allow one to freely add and re-move vacuum lines. Alternatively, one could allow the useof these gauge factors and compensate by similarly modify-ing the canonical isomorphisms.) It is often useful to con-sider quantities of the anyon model that are invariant undersuch gauge transformation. The most relevant gauge invari-ant quantities are the quantum dimensionsda and topologicaltwist factorsθa, since these, together with the fusion coeffi-cientsN c
ab, usually uniquely specify the theory (there are noknown counterexamples).
III. SYMMETRY OF TOPOLOGICAL PHASES
We would like to consider a system that realizes a topolog-ical phase described by a UMTCC and which has a globalunitary or anti-unitary symmetry of the microscopic Hamilto-nian described by a groupG. In this section, we do not requireG to be discrete, nor do we assume that the symmetry is on-site. In order to characterize the interplay of symmetry andtopological order, we first define the notion of the “topolog-ical symmetry” ofC, which is independent of the groupG.We then consider the action of the global symmetry on thetopological properties through its relation to the topologicalsymmetry (via a homomorphism from the global symmetrygroup to the topological symmetry group).
A. Topological Symmetry
The symmetries of a categoryC are described by invert-ible mapsϕ : C → C from the category to itself. Each suchmapϕ can be classified according to whether it is unitary oranti-unitary, and whether it preserves or reverses the spatialparity. We will first consider unitary, parity-preserving sym-metries. Such maps are called auto-equivalences, or braidedauto-equivalences for a BTC, and may permute the topologi-cal charge labels
ϕ(a) = a′, (53)
in such a way that all of the topological properties are left in-variant. In particular, the vacuum must always be left invari-ant under symmetry, so0′ = 0, and gauge invariant quantities
will be left invariant under these permutations of topologicalcharge, so that
N c′a′b′ = N c
ab (54)
da′ = da (55)
θa′ = θa (56)
Sa′b′ = Sab (57)
under auto-equivalence maps.Quantities in the theory that are not gauge invariant must
be left invariant by auto-equivalence maps, up to some gaugetransformation. At a more detailed level, an auto-equivalenceϕ maps basis state vectors of fusion/splitting spaces to (pos-sibly different) basis state vectors of the corresponding fu-sion/splitting spaces
ϕ (|a, b; c, µ〉) = ˜|a′, b′; c′, µ〉=∑
µ′
[ua
′b′c′
]
µµ′|a′, b′; c′, µ′〉 , (58)
where[ua
′b′c′
]is a unitary transformation that is included so
that the map will leave the basic data exactly invariant, ratherthan just gauge equivalent to their original values. Noticethatthis mapping to new basis states is generally the same as ap-plying a permutation of labels together with a gauge trans-formation, so we have used a similar notation to that of theprevious section describing fusion/splitting vertex basis gaugetransformations.
Under such mappings of the fusion/splitting basis states, thebasic data map to
ϕ (N cab) = N c′
a′b′ = N cab (59)
ϕ([F abcd
](e,α,β)(f,µ,ν)
)=[F a
′b′c′d′
]
(e′,α,β)(f ′,µ,ν)
=[F abcd
](e,α,β)(f,µ,ν)
(60)
ϕ([Rabc
]µν
)=[Ra
′b′c′
]
µν=[Rabc
]µν. (61)
We see that this would generally result in gauge equivalentvalues of theF -symbols andR-symbols without the factorsua
′b′c′ , but including these factors in the definition of symmetry
maps gives the stronger condition that theF -symbols andR-symbols are left exactly invariant.
The collection of all such mapsϕ that leave all propertiesof C invariant form the set of braided auto-equivalences ofC. However, there is redundancy in these maps given by the“natural isomorphisms,” which, in this context, are the braidedauto-equivalence maps of the form
Υ(a) = a (62)
Υ(|a, b; c, µ〉) =γaγbγc
|a, b; c, µ〉 , (63)
13
for some phasesγa. It is straightforward to see that such mapsalways leave all the basic data exactly invariant. Hence, onecan think of these natural isomorphisms as vertex basis gaugetransformations of the form[Γabc ]µν = γaγb
γcδµν , which leave
the basic data unchanged. [173]Consequently, we wish to consider braided auto-
equivalence maps as equivalent if they are related by anatural isomorphism, and doing so defines a group, whichwe denote as Aut0,0(C). (The0, 0 here indicates unitary andparity preserving, as we will further explain.) In particular,if ϕ = Υ ϕ for a natural isomorphismΥ, then the braidedauto-equivalence mapsϕ andϕ represent the same equiva-lence class[ϕ] = [ϕ]. In this way, group multiplication inAut0,0(C) is defined by composition up to natural isomor-phism[ϕ1]·[ϕ2] = [ϕ1ϕ2]. In other words,[ϕ3] = [ϕ1]·[ϕ2]if for any representativesϕ1, ϕ2, andϕ3 of the correspondingequivalence classes, there are natural isomorphismsΥ1,Υ2, and Υ3 such thatΥ3 ϕ3 = Υ1 ϕ1 Υ2 ϕ2,or, equivalently, if there is a natural isomorphismκ suchthat ϕ3 = κ ϕ1 ϕ2. (These definitions are related byκ = Υ−1
3 Υ1 ϕ1 Υ2 ϕ−11 .)
There is yet another level of redundancy that arises in thedecomposition of the natural isomorphisms into topologicalcharge dependent phase factors, as in Eq. (63). Specifically,there is freedom to equivalently choose
Υ(|a, b; c, µ〉) =γaγbγc
|a, b; c, µ〉 (64)
γa = ζaγa, (65)
for phasesζa that satisfyζaζb = ζc wheneverN cab 6= 0. In
other words, the phase factorsζa that obey this condition pro-vide a way of decomposing the completely trivial natural iso-morphismΥ = 11 into topological charge dependent phasefactors. As explained at the end of Sec. II B, phase factorsthat obey this condition are related to some Abelian topologi-cal chargez through the relation
ζa =M∗az. (66)
As such, this redundancy of natural isomorphisms be-tween braided auto-equivalence maps (the natural isomor-phisms themselves being a redundancy of the braided auto-equivalences) is classified by the subsetA ⊂ C of Abeliantopological charges of the UMTCC, which can also be con-sidered an Abelian group where multiplication in this groupisgiven by the fusion rules. [174]
We may also consider anti-unitary symmetries of the BTCC, which we called braided anti-auto-equivalences. Thesewere previously examined in the context of time-reversal sym-metries in Refs. [89, 118]. For anti-unitary symmetries, themapϕ is anti-unitary, which means it is a bijective, anti-linearmap, i.e.
ϕ (Cα|α〉+ Cβ |β〉) = C∗α ϕ (|α〉) + C∗
β ϕ (|β〉) , (67)
for any states|α〉 and|β〉 and complex numbersCα, Cβ ∈ C,that also obeys the condition
〈ϕ(α)|ϕ(β)〉 = 〈α|β〉∗. (68)
Any anti-unitary operatorA can be written asA = UK,whereU is a unitary operator andK is the complex conju-gation operator. Its inverse isA−1 = A† = KU−1 = KU †.
The vertex basis states transform as in Eq. (58) whenϕ isanti-unitary, though any (complex-valued)coefficients infrontof such states would be complex conjugated. Under such anti-auto-equivalence mappings of the fusion/splitting basis states,the basic data map to
ϕ (N cab) = N c′
a′b′ = N cab (69)
ϕ([F abcd
](e,α,β)(f,µ,ν)
)=[F a
′b′c′d′
]
(e′,α,β)(f ′,µ,ν)
=[F abcd
]∗(e,α,β)(f,µ,ν)
(70)
ϕ([Rabc
]µν
)=[Ra
′b′c′
]
µν=[Rabc
]∗µν. (71)
Anti-unitarity similarly introduces complex conjugationforthe gauge invariant quantities, so that
θa′ = θ∗a (72)
Sa′b′ = S∗ab. (73)
As mentioned above, when including both unitary and anti-unitary topological symmetries (braided auto-equivalences), itis useful to define a function
q (ϕ) =
0 if ϕ is unitary1 if ϕ is anti-unitary , (74)
which specifies when a braided auto-equivalence map is uni-tary or anti-unitary. When we form equivalence classes ofmaps related by natural isomorphism, the combined set of uni-tary and anti-unitary topological symmetries is again a group.The functionq provides a homomorphism from this group toZ2, i.e. q ([ϕ1 ϕ2]) = q ([ϕ1]) q ([ϕ2]), since the compo-sition of a unitary transformation and an anti-unitary trans-formation is anti-unitary and the composition between twoanti-unitary transformations is unitary. This homomorphismdefines aZ2-grading of the group of unitary and anti-unitaryauto-equivalences.
We can also include spatial parity symmetry, which is a uni-tary symmetry, by introducing an additionalZ2 grading struc-ture. The action of spatial parity on the topological state spaceand basic data is a somewhat complicated matter, because thequasiparticles may, in principle, exist in a 2D surface of arbi-trary topology, and the action of parity depends on both howone chooses to linearly order the quasiparticles for the pur-poses of writing a fusion tree decomposition of the states, andwhat is the line across which one performs the parity reflec-tion. The full details of such parity transformations will not beused in this paper, so we will not present them here. However,it is simple to state the transformation of the gauge invariantquantities
ϕ (N cab) = N c′
a′b′ = N cab (75)
ϕ (θa) = θa′ = θ∗a (76)
ϕ (Sab) = Sa′b′ = S∗ab, (77)
14
which holds for any parity reflection transformation, regard-less of the details of quasiparticle ordering or reflection line.
With this in mind, we introduce the function
p (ϕ) =
0 if ϕ is spatial parity even1 if ϕ is spatial parity odd . (78)
Forming equivalence classes of symmetry transformations un-der natural isomorphisms, this provides anotherZ2-grading ofthe resulting group, since the composition of two parity re-versing (odd) transformations is obviously parity preserving(even), and thusp ([ϕ1 ϕ2]) = p ([ϕ1]) p ([ϕ2]).
We write the full group of quantum symmetries of the topo-logical theory as
Aut(C) =⊔
q,p∈0,1Autq,p(C), (79)
where Autq,p(C) is the set of equivalence classes (under nat-ural isomorphisms) of braided auto-equivalence maps that areunitary forq = 0 or anti-unitary forq = 1, and parity preserv-ing for p = 0 or parity reversing forp = 1.
We consider Aut(C) to be thetopological symmetrygroupof C, because it describes the symmetry of the emergent topo-logical quantum numbers of the topological phase, as de-scribed byC. This is in contrast to and independent of anyglobal symmetry of the underlying physical system, as de-scribed by the microscopic Hamiltonian.
B. Global Symmetry
We now consider the case where a physical system that re-alizes a topological phase described by the UMTCC, has aglobal symmetry groupG of the microscopic Hamiltonian.We restrict our attention to the case where the elements ofG correspond to symmetries that preserve the orientation of
space, i.e. those withp = 0. Since the elements ofG actas symmetries onC, their action must correspond to a grouphomomorphism
[ρ] : G→ Aut(C), (80)
to the topological symmetry group Aut(C), which is to say that[ρg] · [ρh] = [ρgh]. In other words, for each elementg ∈ G,the action ofg can be described by a (unitary or anti-unitary)braided auto-equivalence mapρg, which is a topological sym-metry ofC, that respects group multiplication by satisfying
κg,h ρg ρh = ρgh, (81)
whereκg,h is the corresponding natural isomorphism neces-sary to equateρg ρh with ρgh. We denote the identity el-ement ofG as0 and letρ0 = 11 be the completely trivialtransformation. Clearly, this givesκg,0 = κ0,h = 11.
The group action on topological charge labels is simply per-mutation [withρgh(0) = 0], and so must satisfyρg ρh(a) =ρgh(a). Consequently,κg,h is trivial with respect to the ac-tion on topological charge labels, i.e.κg,h(a) = a. It will beconvenient to introduce the shorthand notations
ga = ρg(a) (82)
g = g−1 (83)
q(g) = q(ρg). (84)
We emphasize that the transformation factorsua′b′c′ associ-
ated withρg acting on vertices need not be the same for differ-entg, and, in general, may require nontrivial action of the nat-ural isomorphismκg,h in order to respect the group multipli-cation. We denote the transformation factorsua
′b′c′ for a given
ρg that leaves the basic data invariant asUg (ga, gb; gc).
Thus, with this symmetry action, we have
ρg (|a, b; c, µ〉) =∑
µ′
[Ug(ga, gb; gc)]µµ′ | ga, gb; gc, µ′〉 , (85)
ρg (Ncab) = N
gcga gb = N c
ab, (86)
ρg
([F abcd
](e,α,β)(f,µ,ν)
)=
∑
α′,β′,µ′ν′
[Ug(ga, gb; ge)]αα′ [Ug(
ge, gc; gd)]ββ′
[F
ga gb gcgd
]
(ge,α′,β′)(gf,µ′,ν′)
×[Ug(
gb, gc; gf)−1]µ′µ
[Ug(
ga, gf ; gd)−1]ν′ν
= Kq(g)[F abcd
](e,α,β)(f,µ,ν)
Kq(g) (87)
ρg
([Rabc
]µν
)=∑
µ′,ν′
[Ug(gb, ga; gc)]µµ′
[R
ga gbgc
]
µ′ν′
[Ug(
ga, gb; gc)−1]ν′ν
= Kq(g)[Rabc
]µνKq(g), (88)
κg,h (|a, b; c, µ〉) =∑
ν
[κg,h(a, b; c)]µν |a, b; c, ν〉 , (89)
[κg,h(a, b; c)]µν =∑
α,β
[Ug(a, b; c)
−1]µαKq(g)
[Uh(
ga, gb; gc)−1]αβKq(g) [Ugh(a, b; c)]βν . (90)
We note that, to account for the possibility of anti-unitary symmetries, we have inserted the complex conjugation oper-
15
atorsK in such a way that has the effect of complex conju-gating theF -symbol,R-symbol, orUh-symbol that is sand-wiched between a pair ofK operators wheng corresponds toan anti-unitary symmetry, which hasq(g) = 1.
Sinceκg,h is a natural isomorphism, its action on verticestakes the form
[κg,h(a, b; c)]µν =βa(g,h)βb(g,h)
βc(g,h)δµν , (91)
whereβa(g,h) are phases that only depend on the topologicalchargea and the group elementsg andh.
As discussed in the previous subsection, there is redun-dancy due to the freedom of choosing how one decomposesa natural isomorphism into the topological charge dependentphase factors. Specifically, it is always possible to transformtheβa(g,h) phases into
βa(g,h) = νa(g,h)βa(g,h), (92)
while leavingκg,h(a, b; c) = κg,h(a, b; c) unchanged, if thephasesνa(g,h) satisfyνa(g,h)νb(g,h) = νc(g,h) when-everN c
ab 6= 0. Moreover, it is clear that whenever twosets of phase factorsβa(g,h) and βa(g,h) give the sameκg,h(a, b; c), they must be related byνa(g,h) of this form.Therefore, the derived properties ofβa(g,h) andβa(g,h) re-lated in this manner should be considered equivalent, and thisredundancy should be viewed as a sort of gauge freedom.
Requiring the symmetry action on vacuum to be trivial im-poses the conditions
Ug(0, 0; 0) = Ug(a, 0; a) = Ug(0, a; a) = 1, (93)
which makes the symmetry action compatible with introduc-ing and removing vacuum lines at will. Clearly,ρ0 = 11 re-quiresU0(a, b; c) = 11.
Eq. (93) requires
κg,h(0, 0; 0) = β0(g,h) = 1. (94)
Sinceκg,0 = κ0,h = 11, it follows that
βa(g,0)βb(g,0) = βc(g,0) (95)
βa(0,h)βb(0,h) = βc(0,h) (96)
wheneverN cab 6= 0. Given the gauge freedom described in
Eq. (92), it is always possible to freely modify such terms tobe trivial, so we will always impose on them the simplifyingcondition
βa(0,0) = βa(g,0) = βa(0,h) = 1, (97)
as a choice of gauge.We can use Eq. (81) to write the decomposition ofρghk in
the two equivalent ways related by associativity (leaving thesymbols implicit from now on)
ρghk = κg,hkρgρhk
= κg,hkρgκh,kρhρk
= κg,hkρgκh,kρ−1g ρgρhρk
= κgh,kρghρk
= κgh,kκg,hρgρhρk. (98)
This gives the consistency condition onκg,h
κg,hkρgκh,kρ−1g = κgh,kκg,h. (99)
We emphasize that theρg transformation here may be anti-unitary, so that it applies complex conjugation (as well as thetopological charge permutation) to theκh,k which it conju-gates.
Since we consider braided auto-equivalence maps to beequivalent when they are related by natural isomorphisms,we may equivalently choose to use the auto-equivalence mapsρg = Υgρg for the global symmetry action. With this choiceof action, we have the redefined quantities
[Ug(a, b; c)
]µµ′ =
γa(g)γb(g)
γc(g)[Ug(a, b; c)]µµ′ . (100)
These result in a correspondingly redefinedκg,h, for whichwe may choose the redefined vertex decomposition factors
βa(g,h) =γa(gh)
Kq(g)γ ga(h)Kq(g)γa(g)βa(g,h). (101)
We emphasize that the transformation of theF -symbols andR-symbols are precisely the same forρg andρg, since theyare related by a natural isomorphism. In order to preservethe trivial action on the vacuum charge and the triviality ofthe factorβa(0,0) = 1, we must fixγ0(g) = γa(0) = 1.We may think of the relation between auto-equivalence mapsby natural isomorphisms as a sort of gauge transformation forthe symmetry action, which is a notion that will be made moreclear in Sec. VI C.
C. H3[ρ](G,A) Invariance Class of the Symmetry Action
Given the global symmetry action[ρ] described inSec. III B, we wish to find an invariant that would allow us todetermine whether or not it would be possible to fractionalizethe symmetry action. In this subsection, we will define suchan invariant[O] ∈ H3
[ρ](G,A), and in the following section,we will demonstrate that the symmetry can be fractionalizedwhen[O] = [0], whereas[O] 6= [0] indicates that there is anobstruction to fractionalizing the symmetry. (See Appendix Afor a review of group cohomology.)
We begin by defining (for a particular choice ofρ ∈ [ρ]) thequantity
Ωa(g,h,k) =Kq(g)βρ−1
g (a)(h,k)Kq(g)βa(g,hk)
βa(gh,k)βa(g,h), (102)
which is a phase from which we will obtain the desired invari-ant. From this definition, it immediately follows that
Kq(g)Ωρ−1g (a)(h,k, l)K
q(g)Ωa(g,hk, l)Ωa(g,h,k)
Ωa(gh,k, l)Ωa(g,h,kl)= 1.
(103)By using Eqs. (91) and (99), we see that
Ωa(g,h,k)Ωb(g,h,k) = Ωc(g,h,k) (104)
16
wheneverN cab 6= 0. As explained in the end of Sec. II B, this
implies
Ωa(g,h,k) =M∗aO(g,h,k) (105)
for someO(g,h,k) ∈ A, whereA ⊂ C is the subset oftopological charges inC that are Abelian. (One can alsothink of A ⊂ C as a subcategeory ofC.) More precisely,
O(g,h,k) ∈ C3(G,A) is a 3-cochain, since it is a func-tion of three group elementsg,h,k ∈ G to A, which we cannow consider to be the Abelian group whose elements are theAbelian topological charges ofC with group multiplicationgiven by their corresponding fusion rules. Moreover, throughthis relation, Eq. (103) maps to the condition
1 = Kq(g)Mρ−1g (a)O(h,k,l)K
q(g)M∗aO(gh,k,l)MaO(g,hk,l)M
∗aO(g,h,kl)MaO(g,h,k)
= Maρg[O(h,k,l)]M∗aO(gh,k,l)MaO(g,hk,l)M
∗aO(g,h,kl)MaO(g,h,k)
= Maρg[O(h,k,l)]MaO(gh,k,l)MaO(g,hk,l)MaO(g,h,kl)MaO(g,h,k)
= Ma,ρg[O(h,k,l)]×O(gh,k,l)×O(g,hk,l)×O(g,h,kl)×O(g,h,k), (106)
Here, we used the symmetry propertySρg(a)ρg(b) = Kq(g)SabKq(g), the relationS∗
ab = Sab, and the fact that ifMab is a phase,thenMabMac =Mae wheneverNe
bc 6= 0. Since this condition holds for alla, the non-degeneracy of braiding implies that
dO(g,h,k, l) = ρg[O(h,k, l)]× O(gh,k, l)× O(g,hk, l)× O(g,h,kl)× O(g,h,k) = 0. (107)
In other words,O(g,h,k) satisfies the3-cocycle condition,when treated as a 3-cochain. Thus, there is an invertible mapbetween the phaseΩa(g,h,k) and the3-cocycleO(g,h,k) ∈Z3ρ(G,A).As explained in the discussion around Eq. (92), there is
gauge freedom to modify the phasesβa(g,h) to βa(g,h) =νa(g,h)βa(g,h), for phase factorsνa(g,h) that satisfyνa(g,h)νb(g,h) = νc(g,h) wheneverN c
ab 6= 0. The cor-respondingly modified
Ωa(g,h,k) =Kq(g)βρ−1
g (a)(h,k)Kq(g)βa(g,hk)
βa(gh,k)βa(g,h)
=Kq(g)νρ−1
g (a)(h,k)Kq(g)νa(g,hk)
νa(gh,k)νa(g,h)Ωa(g,h,k) (108)
is to be considered in the same equivalence class asΩa(g,h,k) and obeys the same properties asΩa(g,h,k), ex-ceptΩa(g,h,k) =M∗
aO(g,h,k)maps to a potentially different
O(g,h,k), which should therefore be considered to be in thesame equivalence class asO(g,h,k). To find the relation be-tween these, we note that we similarly have the condition that
νa(g,h) =M∗av(g,h), (109)
wherev(g,h) ∈ C2(G,A) is a2-cochain taking values in theset of Abelian topological charges. Using this in Eq. (108)and employing the same properties utilized in Eq. (106), weobtain the corresponding relation
O(g,h,k) = ρg[v(h,k)]× v(gh,k)×v(g,hk)× v(g,h)× O(g,h,k)
= dv(g,h,k)× O(g,h,k), (110)
which shows thatO(g,h,k) and O(g,h,k) in the sameequivalence class are related by fusion with a3-coboundarydv(g,h,k) ∈ B3
ρ(G,A). Thus, the equivalence classes[O]are elements of the3rd cohomology group given by taking thequotient of3-cocycles by3-coboundaries
[O] ∈ H3ρ(G,A) =
Z3ρ(G,A)
B3ρ(G,A)
. (111)
We emphasize that the equivalence class[O] is defined entirelyin terms ofρ (which definesκg,h). We further emphasize that[O] = [0] does not necessarily imply thatβa(g,h)βb(g,h) =βc(g,h) wheneverN c
ab 6= 0 nor, equivalently, thatκg,h = 11.We can also see from the definitions that the equivalence
class[O] is actually an invariant of the equivalence class[ρ]of symmetry actions that are related by natural isomorphisms.In particular, if we instead used the actionρg = Υgρg, whereΥg is a natural isomorphism, and the corresponding modifiedvertex decomposition factorsβa(g) as given in Eq. (101), thenwe would find that the corresponding quantityΩa(g,h,k) =Ωa(g,h,k) is unchanged. Thus, any such symmetry actionsrelated by natural isomorphisms define the same equivalenceclass[O] = [O], so we actually have
[O] ∈ H3[ρ](G,A). (112)
We note that if the symmetry action does not permutetopological charges, i.e.ρg(a) = a for all a andg, thenit is always the case that[O] = [0]. To demonstrate thisproperty, we observe thatρg are actually natural isomor-phisms when this is the case. It follows that we can write[Ug(a, b; c)]µν = χa(g)χb(g)
χc(g)δµν , whereχa(g) are phases, and
that we can make a choice within the equivalence class for
17
whichβa(g,h) =χa(gh)
Kq(g)χρ−1g (a)
(h)Kq(g)χa(g). Using this with
the definition, we findΩa(g,h,k) = 1 and hence[O] = [0].(Alternatively, we could have used a gauge transformation tosetρ = 11, Ug(a, b; c)µν = δµν , andβa(g,h) = 1, whichobviously givesΩa(g,h,k) = 1.)
GivenC andG, there are many different possible choicesof ρ. These different choices correspond to different waysthat the global symmetry (of the microscopic Hamiltonian)and the topological order can interplay with each other. Fromthe above discussion, we see that clearly the first importantchoice is howρg permutes the various anyons. The next im-portant choice depends on more subtle properties of the gaugetransformations that are required when implementingρg. Inthe next section, we examine how these properties lead to aconcept known as symmetry fractionalization, whereby thequasiparticles have the ability to form a sort of projectiverep-resentation of the symmetry group. We will classify the waysin which the symmetry can fractionalize and, in doing so, findthat[O] 6= [0] indicates that there is an obstruction to fraction-alizing the symmetry.
IV. SYMMETRY FRACTIONALIZATION
Before carrying out the detailed derivation, we will state theresult of this section and provide a summary overview of thearguments (and direct the reader to Appendix A, if a review ofgroup cohomology is needed):
For a system that realizes a topological phase described bythe UMTC C and which has the global symmetry groupGwith corresponding group action[ρ] : G→ Aut(C):
1. There is an obstruction to symmetry fractionalization if[O] 6= [0], where[O] ∈ H3
[ρ](G,A) was the invariant of[ρ] defined in Sec. III C.
2. When[O] = [0], symmetry fractionalization may occurand is classified by the cohomology groupH2
[ρ](G,A),whereA is defined to be the finite group whose ele-ments are the Abelian topological charges ofC withgroup multiplication given by their corresponding fu-sion rules. More precisely, the set of distinct symmetryfractionalization classes is anH2
[ρ](G,A) torsor. [175]
In this section, we assume that the global symmetry acts inan on-site or “quasi-on-site” fashion on the underlying phys-ical system, where quasi-on-site is a generalization of thenotion of on-site that may include symmetries that act non-locally, such as anti-unitary, time-reversal, translation, rota-tion, and other spacetime symmetries. The on-site and quasi-on-site properties of symmetry actions are fundamental re-quirements for symmetry fractionalization, so we will defineprecisely what we mean when we use these terms. We do notrestrict the symmetry groupG to be discrete.
In order to explain the above mathematical statement ofsymmetry fractionalization, we begin by examining the ac-tion of a unitary on-site symmetry on the physical Hilbertspace of the underlying physical system and its microscopic
Hamiltonian. We argue that the action of the global symmetryoperatorRg on the physical states|Ψaj〉, corresponding tothe system withn quasiparticles carrying topological chargesa1, . . . , an, respectively, can always be written as
Rg|Ψaj〉 =n∏
j=1
U (j)g ρg|Ψaj〉. (113)
Here we have separated local unitary transformationsU(j)g
from the non-local unitary transformationρg that acts as thesymmetry action on the topological quantum numbers.
Since Rg are the physical symmetry transformations,RgRh = Rgh (at least projectively). Writing out the local-ized forms explicitly leads to the relation
n∏
j=1
U (j)g ρgU
(j)h ρ−1
g =n∏
j=1
U(j)gh κg,h. (114)
We can also argue that the local operators satisfy the pro-jective multiplication relation acting on quasiparticle states
U (j)g ρgU
(j)h ρ−1
g |Ψaj〉 = ηaj (g,h)U(j)gh |Ψaj〉 (115)
for some phase factorsηaj (g,h) that only depend on the topo-logical chargeaj and group elementsg andh. Then the con-ditionRgRh = Rgh yields
n∏
j=1
βaj (g,h)
ηaj (g,h)= 1, (116)
whereβa(g,h) are the phase factors that decompose the nat-ural isomorphismκg,h, as in Sec. III B.
The associativity of the local operators leads to the cocycle-like relation
ηρ−1g (a)(h,k)ηa(g,hk)
ηa(gh,k)ηa(g,h)= 1. (117)
This imposes a condition onβa(g,h) factors, which definesan obstruction given by the previously described invarianceclass[O] ∈ H3
[ρ](G,A).When the obstruction class is trivial, one is guaranteed
to have at least one set ofηa(g,h) which can satisfy bothEq. (116) and Eq. (117). It follows that there are actuallymany solutions, since, given one solution with phase factorsηa(g,h), another solutionηa(g,h) = τa(g,h)
−1ηa(g,h) isobtained from it by dividing by phasesτa(g,h) that satisfythe conditions
τρ−1g (a)(h,k)τa(g,hk)
τa(gh,k)τa(g,h)= 1, (118)
τa(g,h)τb(g,h) = τc(g,h), if N cab 6= 0. (119)
However, there is some redundancy in these solutions that isdue to the freedom to redefine the operatorsU
(j)g by local op-
eratorsZ(j)g that do not affectRg, which means
∏j Z
(j)g = 11.
This property requires that the action on quasiparticle state
18
Z(j)g |Ψaj〉 = ζaj (g)|Ψaj〉, whereζa(g) are phase that
satisfyζa(g)ζb(g) = ζc(g) wheneverN cab 6= 0. This redef-
inition of local operators changes the phasesηa(g,h) in thefollowing way
ηa(g,h) =ζa(gh)
ζρ−1g (a)(h)ζa(g)
ηa(g,h). (120)
Thus, if two sets solutions are related by such a transforma-tion, they should be considered physically indistinguishable,so they belong to to a single equivalence class of solutions.
SinceC is modular, the factorsτa(g,h) uniquely define a2-cocyclet ∈ Z2
ρ(G,A) and the factorsζa(g) uniquely de-fine a 1-cochainz ∈ C1(G,A), which makes the equivalenceclasses related by2-coboundaries dz ∈ B2
ρ(G,A). Taking thequotient (and noting the invariance of the results under thechoice ofρ ∈ [ρ]) results in the classification of solutions byH2
[ρ](G,A).After these arguments, we will generalize the results to the
case where the global symmetry action is a projective rep-resentation. Finally, we will introduce the notion of quasi-on-site symmetry and explain how the on-site symmetry argu-ments and results are generalized to apply to such symmetries.
A. Physical Manifestation of On-Site Global Symmetry
We wish to examine the quantum states of the underlyingphysical system in which there are quasiparticles present.Ini-tially, let us consider the case when there are two quasiparti-cles, and we will subsequently generalize to an arbitrary num-ber. We assume the two quasiparticles possess topologicalchargesa and a, respectively, and that they are respectivelylocalized within the well-separated, simply-connected regionsR1 andR2. Well-separated means that the minimum distancer12 ≡ minrj∈Rj
|r1 − r2| between any two points of the dis-tinct regions is much larger than the correlation lengthξ ofthe system, i.e.r12 ≫ ξ. (We typically think ofRj as a diskcentered at the quasiparticle coordinaterj with a radius that isa few correlations lengths.)
For concreteness, we consider the system to be defined ona sphere (or any genus zero surface) and assume that there areno other quasiparticles nor boundaries that carry topologicalcharge, so this pair must fuse to vacuum. (The analysis canbe generalized to surfaces of arbitrary genus with any num-ber of boundaries, but we will not do so in this paper.) Ingeneral, sinceN0
aa = 1, there is a single topological sectorin such a setup, which is described by|a, a; 0〉 in the topo-logical state space. However, this topological state representsa universality class of many microscopic states that share itstopological properties and which differ by the applicationoflocal operators. Such a state in this universality class canbeobtained by starting from the uniform HamiltonianH0 of thesystem in the topological phase, adiabatically creating a pairof quasiparticles with chargesa anda from vacuum by tuningthe Hamiltonian to locally favor the existence of such quasi-particles that are not well-separated, and then subsequentlymoving the quasiparticles individually to regionsR1 andR2,
respectively, through a sequence of similar modifications ofthe Hamiltonian (which return the Hamiltonian to its originalform in the regions away from the quasiparticles positions).
The corresponding Hamiltonian resulting after this processis of the form
Hαa,a;0 = H0 + h(1)a;α + h
(2)a;α, (121)
whereh(j)a;α is a modification of the Hamiltonian whose non-trivial action is localized withinRj and which favors the lo-calization of a quasiparticle of chargea in this region. Thelabelα is a parameter which simply identifies these terms asone of many that favors localization of a quasiparticle of thistype. We write the ground state of this HamiltonianHα
a,a;0
as |Ψαa,a;0〉 (which is in the|a, a; 0〉 universality class). Weemphasize that|Ψαa,a;0〉 with different values of the parame-ter α are not necessarily orthogonal; in fact, we expect thatthey may have very high overlaps for some different valuesof α. However, such states with different values of topologi-cal charge will be orthogonal, up to exponentially suppressedcorrections, i.e.〈Ψαa,a;0|Ψβb,b;0〉 ≈ 0 whenevera 6= b.
Let us now assume that the symmetry acts on the system inan on-site manner, withRg being the unitary operator repre-senting the action ofg. By on-site, we mean that if we decom-pose the space manifoldM =
⋃k∈I Mk into a collection of
simply connected disjoint regionsMk (a subset of which canbe taken to be the regionsRj) with index setI, the symmetryoperators take the form
Rg =∏
k∈IR(k)
g , (122)
whereR(k)g is a unitary operator that has nontrivial action lo-
calized in regionMk. Sinceg is a symmetry of the systemthat acts onC by ρg, the Hamiltonians should transform as
RgH0R−1g = H0 (123)
RgHαa,a;0R
−1g = H
g(α)ga, g a;0 (124)
where
h(j)ga;g(α) = Rgh
(j)a;αR
−1g (125)
of the new Hamiltonian remains an operator that is localizedin the regionRj , but now favors the localization of a quasi-particle of chargega = ρg(a). Indeed, since the symmetry ison-site, any operatorO(j) whose nontrivial action is localizedin a regionRj remains localized in this region when actedupon by the symmetry transformation, i.e.
gO(j) ≡ RgO(j)R−1g = R(j)
g O(j)R(j)−1g (126)
is localized inRj . We stress that the labelg(α) of the Hamil-
tonian defined withh(j)ga;g(α) obtained from the symmetrytransformation indicates that this Hamiltonian need not equalthe Hamiltonian defined with the modificationh(j)ga;α for local-izing a chargega quasiparticle, to which we already ascribed
19
the labelα. In other words, while the universality class ofstates transforms as
|a, a; 0〉 7→ ρg|a, a; 0〉 = Ug(ga, ga; 0)| ga, ga; 0〉 (127)
under the action ofg, the ground state of the Hamiltoniantransforms as
|Ψαa,a;0〉 7→ Rg|Ψαa,a;0〉 = |Ψg(α)ga, ga;0〉, (128)
where|Ψg(α)ga, g a;0〉 is not necessarily equal (nor proportional)
to |Ψαga, ga;0〉.In fact, we have not yet made clear what it even means to
have states|Ψαa,a;0〉 and |Ψαga, ga;0〉 in different topologicalcharge sectors with the same labelα. For this, we make achoice of complete orthonormal basis states|ϕsa,a;0〉 for eachtopological charge sector. Then, given a state
|Ψαa,a;0〉 =∑
s
As|ϕsa,a;0〉 (129)
we identify the corresponding state in the different topologicalcharge sector to be
|Ψαga, ga;0〉 =∑
s
As|ϕsga, g a;0〉. (130)
We can now define the unitary operatorUg via the basisstates of each subspace (with respect to which the operator isblock diagonal)
〈ϕrga, ga;0|Ug|ϕsga, g a;0〉= 〈ϕrga, ga;0|Rg|ϕsa,a;0〉. (131)
This gives the relation
Rg|Ψαa,a;0〉 = Ug|Ψαga, ga;0〉 (132)
for any state|Ψαa,a;0〉 in the |a, a; 0〉 universality class. Weemphasize thatUg is independent ofα, but it does depend onthe choice of basis, and simply provides the relation betweenthe orthonormal basis given by the states|ϕsga, ga;0〉 and theorthonormal basis given by the statesRg|ϕsa,a;0〉.
Since the quasiparticles are localized at well-separated po-sitions, the system has exponentially decaying correlations,and the system is locally uniform and symmetric away fromthe quasiparticles, the states in the|a, a; 0〉 universality classwill be locally indistinguishable from the ground state|Ψ0〉of H0 in (simply-connected) regions well-separated from thequasiparticles’R1 andR2. More specifically, we expect thatany two such states|Ψαa,a;0〉 and|Ψβa,a;0〉 in this universalityclass can be related by unitary operators acting independentlyin regionsR1 andR2, i.e. there exist unitary operatorsV (j)
whose nontrivial action is localized withinRj such that
|Ψβa,a;0〉 ≈ V (1)V (2)|Ψαa,a;0〉. (133)
The approximation in this expression is up toO(e−r12/ξ) cor-rections, which we will leave implicit in the following. [176]
Rg
R1
R2
R3R4
a1
a3
a2
a4
ga1
ga2
ga3
ga4
U(1)g
U(3)g
U(4)g
U(2)g
FIG. 3: The global on-site symmetry action on states containingquasiparticles takes the form given in Eq. (139), where the globalactionRg factorizes into the global symmetry action operatorρg,which acts only on the topological quantum numbers, and local trans-formationsU (j)
g , each of which only acts nontrivially within a regionRj well-localized around thejth quasiparticle carrying topologicalchargeaj .
Thus, it follows that we can write the symmetry action as
Rg|Ψa,a;0〉 = U (1)g U (2)
g Ug(ga, ga; 0)|Ψ ga, ga;0〉, (134)
for any state|Ψa,a;0〉 in the|a, a; 0〉 universality class (we now
drop the inconsequential labelα). In this expression,U (1)g and
U(2)g are unitary operators whose nontrivial action is localized
within R1 andR2, respectively. The quantityUg(ga, ga; 0)
is precisely the transformation on the topological state spacefrom Eqs. (85)-(90) that leaves the basic data invariant. Inpar-ticular, Ug(
ga, ga; 0) is an overall phase that depends onlyon the universality class of the state. Normally, one wouldsafely ignore such an overall phase, but we include it here tomatch with the symmetry action on the topological degrees offreedom, as this will play an essential role in the subsequentgeneralization ton quasiparticles. In this way, we have de-composedUg = U
(1)g U
(2)g Ug(
ga, ga; 0) into terms that actlocally around the quasiparticles and the term that acts on thetopological state space. Clearly,U (1)
g andU (2)g commute with
each other, since their respective nontrivial actions are in twowell-separated regions.
Given Eq. (134), we can define the operator
ρg = U (1)−1g U (2)−1
g Rg (135)
acting on the physical Hilbert space that has the same actionon states|Ψa,a;0〉 in the |a, a; 0〉 universality class as does thepreviously defined symmetry operatorρg ( see Sec. III B ) act-ing on|a, a; 0〉 in the topological state space, i.e.
ρg|Ψa,a;0〉 = Ug(ga, ga; 0)|Ψ ga, ga;0〉. (136)
We note that, similar toRg, this operator also has the form
ρg =∏k∈I ρ
(k)g . [177]
We now generalize to consider the system in a configurationwith n quasiparticles with corresponding topological chargesaj localized in well-separated regionsRj (for j = 1, . . . , n),with corresponding Hamiltonians
Hαa1,...,an;0 = H0 +
n∑
j=1
h(j)aj ;α. (137)
20
The same steps can be followed as above, though one mustbe more careful to properly account for fusion degeneracies.In particular, there will beN c
ab distinct ways to create twoquasiparticles with respective chargesa andb from a singlequasiparticle of topological chargec, and this will be reflectedin the corresponding states and Hamiltonians. For a systemwith n quasiparticles, the topological state space may be de-generate, with the dimensionality given by
N0a1...an =
∑
c12,c123,...,c1...n−1
N c12a1a2N
c123c12a3 . . .N
0c1...n−1an ,
(138)where here we use the standard basis decomposition of thetopological state space where topological charges are fusedtogether successively in increasing order ofj, andc1...k is thecollective topological charge of quasiparticles1, . . . , k. Thestates will correspondingly carry the labelsc12, . . . , c1...n−1,as well as the fusion space basis labelsµ12, . . . , µ1...n−1. (Wecan, of course, write the states in a different basis relatedbyF -moves.) We write all these topological charges and fusionbasis labels of the state collectively asa; c, µ, with the un-derstanding that the overall fusion channel of then quasipar-
ticles is vacuum (i.e.c1...n = 0), so we can more compactlywrite a state in this universality class as|Ψa;c,µ〉. Followingthe same arguments given above, we find that the symmetryaction on such states will take the form
Rg|Ψa;c,µ〉 = U (1)g . . . U (n)
g ρg|Ψa;c,µ〉, (139)
where the unitary operatorU (j)g has its nontrivial action local-
ized withinRj . This is shown schematically in Fig. 3. (Again,
the operatorsU (j)g depend on a choice of basis within the uni-
versality class, but not on the particular state it is actingupon.)Here, we use the generalized definition of the operator (in thephysical Hilbert space)
ρg =
n∏
j=1
U (j)−1g Rg (140)
which acts on physical states|Ψa;c,µ〉 in the universalityclass |a; c, µ〉 precisely as the operatorρg acts on states|a; c, µ〉 in the topological state space. Explicitly, this isgiven by
ρg|Ψa;c,µ〉 =∑
µ′12,...,µ
′1...n−1
[Ug(ga1,
ga2;gc12)]µ12µ′
12[Ug(
gc12,ga3;
gc123)]µ123µ′123
× . . .
. . .× [Ug(gc1...n−2,
gan−1;gc1...n−1)]µ1...n−1µ′
1...n−1Ug(
gc1...n−1,gan; 0)|Ψ ga; gc,µ′〉.(141)
Given the physical states containing quasiparticles and thesymmetry transformationsRg acting upon them, one may usethese expressions as a means of determining the global sym-metry actionρg on the topological state space.
We can consider symmetry transformations taking the formin Eq. (139) when acting on states in the physical Hilbertspace containing quasiparticles in a topological phase to bethe fundamental condition from which the symmetry fraction-alization arguments follow, regardless of the particular formof the Hamiltonian.
B. Obstruction to Fractionalization
We will allow the global symmetry action to form either lin-ear or projective representations of the symmetry group whenacting on the physical Hilbert space, but first consider the caseof linear representations of the global symmetry, and then re-turn to the case of projective representations in Sec. IV E.For linear representations, the symmetry operators will sat-isfy Rgh = RgRh. However, the local operatorsU (j)
g cannonetheless take a projective form, and we wish to classifythe types of projective forms that they can realize. We com-
pare the action ofgh, which is given by
Rgh|Ψa;c,µ〉 =
n∏
j=1
U(j)gh ρgh|Ψa;c,µ〉
=
n∏
j=1
U(j)gh κg,hρgρh|Ψa;c,µ〉, (142)
whereκg,h = ρghρ−1h ρ−1
g (as in Sec. III B), and the succes-sive actions ofg andh, which is given by
RgRh|Ψa;c,µ〉 = Rg
n∏
j=1
U(j)h ρh|Ψa;c,µ〉
= Rg
n∏
j=1
U(j)h R−1
g Rgρh|Ψa;c,µ〉
= Rg
n∏
j=1
U(j)h R−1
g
n∏
k=1
U (k)g ρgρh|Ψa;c,µ〉
=n∏
j=1
gU(j)h U (j)
g ρgρh|Ψa;c,µ〉, (143)
wheregU(j)h = RgU
(j)h R−1
g = R(j)g U
(j)h R
(j)−1g has its non-
trivial action localized within the regionRj , and we used the
21
fact that operators whose nontrivial actions are localizedindifferent regions commute with each other. Comparing theseexpressions, we see that
n∏
j=1
U (j)−1g
gU(j)−1h U
(j)gh κg,h = 11 (144)
when acting in the subspace of states of the form|Ψa;c,µ〉corresponding to the system withn quasiparticles. We notethat
gO(j)U (j)g = U (j)
g ρgO(j)ρ−1g , (145)
for any operatorO(j) localized inRj , so we could rewrite
these expressions usingU (j)g ρgU
(j)h ρ−1
g instead ofgU (j)h U
(j)g ,
if desired.Since the action ofρg on the physical states of the form
|Ψa;c,µ〉 is precisely the same as the action ofρg on thestates|a; c, µ〉 in the topological state space, we know thatthe action ofκg,h on physical states of the form|Ψa;c,µ〉also matches the action ofκg,h in the topological state space,and thus takes the form
κg,h|Ψa;c,µ〉 =n∏
j=1
βaj (g,h)|Ψa;c,µ〉, (146)
whereβa(g,h) are the phases defined in Sec. III B that de-pends only on the topological charge valuea, and group ele-mentsg andh. Let us define a unitary operatorB(j)
g,h localizedin regionRj whose action on a quasiparticle state producesthe phaseβaj (g,h) of the topological charge contained in theregionRj , that is [178]
B(j)g,h|Ψa;c,µ〉 = βaj (g,h)|Ψa;c,µ〉. (147)
We can now define the unitary operators
W(j)g,h = U (j)−1
ggU
(j)−1h U
(j)ghB
(j)g,h
= ρgU(j)−1h ρ−1
g U (j)−1g U
(j)ghB
(j)g,h. (148)
Since theU (j)g andB(j)
g,h are all unitary operators with nontriv-ial action localized within the regionRj , this is also true for
W(j)g,h. From the above relations, we see that
n∏
j=1
W(j)g,h = 11 (149)
when acting in the subspace ofn quasiparticles states of theform |Ψa;c,µ〉, for any values ofa; c, µ.
Since the respective regionsRj whereW (j)g,h act nontriv-
ially are well-separated from each other, each one of theseoperators can, at most, change a state of the form|Ψa;c,µ〉by an overall phase factor. Hence, we have
〈Ψa;c,µ|W (j)g,h|Ψb;e,ν〉 = ωaj (g,h)δa;c,µb;e,ν, (150)
where the phaseωaj (g,h) only depends on the topologicalchargeaj contained in the regionRj .
In order to see that the phasesωaj (g,h) do not dependon anything else, we first note that the phase factor can ob-viously depend, at most, on the group elementsg and h,and the properties of the state|Ψa;c,µ〉 that are local tothe regionRj . In order to see that the only property of thestate that the phase depends on is the topological charge con-tained in the regionRj , we must show that the phase is actu-ally independent of the specific state|Ψa;c,µ〉 taken fromthe |a; c, µ〉 universality class. For this, assume that thephase may depend on the specific state, which we indicateby writing it asω(g,h; Ψa;c,µ). Then consider any two or-
thonormal states|Ψαa;c,µ〉 and |Ψβa;c,µ〉 from this univer-
sality class, and their normalized superposition|Ψγa;c,µ〉 =
Cα|Ψαa;c,µ〉 + Cβ |Ψβa;c,µ〉. The above expression yieldsthe relation
ω(g,h; Ψγa;c,µ) = |Cα|2ω(g,h; Ψαa;c,µ)+|Cβ |2ω(g,h; Ψβa;c,µ) (151)
which can only be true for arbitraryCα andCβ if
ω(g,h; Ψγa;c,µ) = ω(g,h; Ψαa;c,µ) = ω(g,h; Ψβa;c,µ)(152)
which shows that the phase is the same for all states in the uni-versality class. Since the only universal property of the statethat is local to the regionRj is the topological chargeaj con-tained in that region, this establishes the claimed dependenceof the phase.
It follows that, within the subspace of states of the form|Ψa;c,µ〉, the operatorsW (j)
g,h,W (j)k,l ,B(j)
g,h, andB(j)k,l all com-
mute with each other. It also follows that
ηaj (g,h)U(j)gh |Ψa;c,µ〉 = U (j)
g ρgU(j)h ρ−1
g |Ψa;c,µ〉= gU
(j)h U (j)
g |Ψa;c,µ〉, (153)
where the projective phases are given by
ηa(g,h) =βa(g,h)
ωa(g,h). (154)
Eq. (153) exhibits a characteristic property of symmetry frac-tionalization, which is that the action of the symmetry can bebroken up into topological and local actions, where the lo-cal actions are locally consistent in a projective fashion.Ofcourse, the topological action is topologically consistent, andthe local and topological actions must also be consistent witheach other. For this, we have already decomposed the consis-tency of the topological action into termsβaj (g,h) that onlydepend on the localized topological charge values, and mustnow examine the phasesωaj (g,h) to analyze the consistencyof the interplay between the local and topological actions ofthe symmetry.
It is clear that we should have
η0(g,h) = 1, (155)
22
since the symmetry action on the ground state is trivial (andany regionRj containing total topological chargeaj = 0 canbe locally transformed into the ground state). Additionally,we will always fix
ηa(0,0) = ηa(g,0) = ηa(0,h) = 1, (156)
since we can always freely setU (j)0 = 11 as a gauge choice,
which we will describe in more detail in Sec. IV C. It followsthat we also haveωa(0,0) = ωa(g,0) = ωa(0,h) = 1.
Given Eq. (149), the phasesωaj (g,h) must obey the con-straint
n∏
j=1
ωaj (g,h) = 1. (157)
We emphasize that this does not mean that the product ofthe phases
∏nj=1 ηaj (g,h) is equal to 1, nor that the product
of the phases∏nj=1 βaj (g,h) is equal to 1. These products
would only individually equal 1 whenκg,h = 11, which isnot generally true (though, this condition is often satisfied byexamples of physical interest).
Considering the case ofn = 2 quasiparticles with respec-tive topological chargesa anda, we find the relation
ωa(g,h) = ωa(g,h)−1. (158)
Considering the case ofn = 3 quasiparticles, with respectivetopological chargesa, b, andc, for whichN c
ab 6= 0, and usingthe result from then = 2 case, we find the relation
ωa(g,h)ωb(g,h) = ωc(g,h) (159)
for any chargesa, b, andc with N cab 6= 0. Thus, as explained
at the end of Sec. II B, the phase factors are given by
ωa(g,h) =M∗aw(g,h), (160)
for some Abelian topological charge valuew (g,h) ∈ A ⊂ C.
It also follows from Eqs. (154) and (159) that
ηa(g,h)ηb(g,h)
ηc(g,h)=βa(g,h)βb(g,h)
βc(g,h)= κg,h(a, b; c)
(161)wheneverN c
ab 6= 0.
Next, we consider the product of three symmetry operationsand apply the relationU (j)
gh = gU(j)h U
(j)g W
(j)g,hB
(j)−1g,h in the
two distinct, but equivalent orders to obtain
U(j)ghk = ghU
(j)k U
(j)ghW
(j)gh,kB
(j)−1gh,k
= ghU(j)k
gU(j)h U (j)
g W(j)g,hB
(j)−1g,h W
(j)gh,kB
(j)−1gh,k
= gU(j)hkUgW
(j)g,hkB
(j)−1g,hk
= ghU(j)k
gU(j)h
gW(j)h,k
gB(j)−1h,k U (j)
g W(j)g,hkB
(j)−1g,hk
= ghU(j)k
gU(j)h U (j)
g ρgW(j)h,kρ
−1g ρgB
(j)−1h,k ρ−1
g W(j)g,hkB
(j)−1g,hk . (162)
This gives the relation
ρgW(j)h,kρ
−1g ρgB
(j)−1h,k ρ−1
g W(j)g,hkB
(j)−1g,hk =W
(j)g,hB
(j)−1g,h W
(j)gh,kB
(j)−1gh,k , (163)
which, when applied to a state|Ψa;c,µ〉, yields the crucial relation
Ωa(g,h,k) = βρ−1g (a)(h,k)βa(gh,k)
−1βa(g,hk)βa(g,h)−1
= ωρ−1g (a)(h,k)ωa(gh,k)
−1ωa(g,hk)ωa(g,h)−1 (164)
where we use the definition ofΩa(g,h,k) from Sec. III C. This relation is equivalent to the condition
ηρ−1g (a)(h,k)ηa(gh,k)
−1ηa(g,hk)ηa(g,h)−1 = 1 (165)
on the projective phases of the local terms, which is a sort oftwisted2-cocycle condition. From this, one might naıvely expect aclassification of fractionalization byH2(G,U(1)), however the relation toβa(g,h) andωa(g,h) impose additional constraintsthat further restrict the classification, as we will now describe.
UsingΩa(g,h,k) = M∗aO(g,h,k) andωa(g,h) = M∗
aw(g,h), whereO (g,h,k) ∈ Z3ρ(G,A) andw (g,h) ∈ C2(G,A) are
Abelian topological charges, together with the relationS∗ab = Sab and the symmetry propertySρg(a)ρg(b) = Sab, this becomes
MaO(g,h,k) = Maρg[w(h,k)]M∗aw(gh,k)Maw(g,hk)M
∗aw(g,h)
= Maρg[w(h,k)]Maw(gh,k)Maw(g,hk)Maw(g,h)
= Ma,ρg[w(h,k)]×w(gh,k)×w(g,hk)×w(g,h). (166)
23
In the last line, we used the fact that ifMab is a phase andNebc 6= 0, then it follows thatMabMac = Mae. Finally, the
non-degeneracy of braiding in a MTC makes this equivalent tothe condition
O (g,h,k) = ρg[w (h,k)]× w (gh,k)× w (g,hk)× w (g,h) = dw (g,h,k) . (167)
Thus, we have found that consistency between the local andtopological portions of the symmetry action requires thatO (g,h,k) is necessarily a 3-coboundary, which is to say thatO ∈ B3
ρ(G,A) and its equivalence class is[O] = [0]. Thisestablishes the first statement regarding symmetry fractional-ization, which was that[O] 6= [0] indicates that there is anobstruction to fractionalizing the symmetry, since this wouldcontradict the result in Eq. (167). In particular, such an ob-struction implies that it is not actually possible for the symme-try of the system to take the assumed on-site form of Eq. (122)with the corresponding action on quasiparticle states given inEq. (139), as the symmetry action cannot be consistently splitinto local and topological components.
We can also viewηa(g,h) as a particular choice of theκg,h(a, b; c) decomposition factors, i.e.ηa(g,h) = βa(g,h)with νa(g,h) = ωa(g,h), such that Eq. (165) is satis-fied. From this perspective, the obstruction class[O] indicateswhether or not such a choice is possible.
When the symmetry action does not permute topologicalcharge values, one can interpret Eq. (153) as indicating thatthe local operatorsU (j)
g provide projective representations ofthe groupG.
C. Gauge Transformations
There is gauge freedom to redefine the local operatorsU(j)g
by the local transformations
U (j)g = U (j)
g Y (j)−1g (168)
whereY (j)g are unitary operators whose nontrivial action is
localized in regionRj . In order to leave the global operatorRg unchanged, there must be a corresponding transformationof the symmetry action operator
ρg =
n∏
j=1
Y (j)g ρg. (169)
In order for this operator to again act on the physical stateswith quasiparticles as does a symmetry action on the topolog-ical state space, we require it to only depend on the topologicalquantum numbers (and the group elementg). SinceY (j)
g actslocally in region in the regionRj , the only topological quan-tum number it can depend upon is the topological chargeajin that region. Thus, we must have
〈Ψa;c,µ|Y (j)g |Ψb;e,ν〉 = γaj (g)δa;c,µb;e,ν, (170)
whereγaj (g) is some phase factor that depends only on thetopological chargeaj and the group elementg. Of course, the
notation we used here anticipated the fact that these gaugetransformations have precisely the form of natural isomor-phisms, as described in Sec. III by
ρg = Υgρg, (171)
with corresponding decomposition into the phase factorsγa(g) when acting on fusion vertex states.
We notice that, under these transformations, the projectivephases transform as
ηa(g,h) =γa(gh)
γ ga(h)γa(g)ηa(g,h). (172)
For the choice of
βa(g,h) =γa(gh)
γ ga(h)γa(g)βa(g,h), (173)
as in Eq. (101), this exactly cancels to leaveωa(g,h) =ωa(g,h) unchanged. As previously mentioned, it alsoleavesΩa(g,h,k) = Ωa(g,h,k) and henceO (g,h,k) =O (g,h,k) unchanged.
In this way, the nontrivial transformations of these quanti-ties are relegated to the transformations
βa(g,h) = νa(g,h)βa(g,h), (174)
where νa(g,h)νb(g,h) = νc(g,h) wheneverN cab 6= 0,
corresponding to the freedom of decomposing the action ofκg,h on vertices into factorsβa(g,h). These transformationsgive ωa(g,h) = νa(g,h)ωa(g,h), while Ωa(g,h,k) andO (g,h,k) are given in Eqs. (108) and (110).
D. Classification of Symmetry Fractionalization
We now wish to classify the different ways in which thesymmetry can fractionalize, when there is no obstruction. Forthis, we must analyze the solutions of Eq. (167) for a givenρandO.
Since[O] = [0], there must exist somev(g,h) ∈ C2(G,A)such thatO = dv. This is just the equivalence class statementthat one can use the gauge transformation in Eq. (174) forsomeνa(g,h) = M∗
av(g,h) which results inΩa(g,h,k) = 1
and O = 0. Thus, we are guaranteed to have at least onesolution of Eq. (167) given byw = v.
Given a solutionw(g,h) of Eq. (167), it is straightforwardto see that another solution
w′(g,h) = t(g,h)× w(g,h) (175)
24
can be obtained from it by multiplying by a2-cocyclet(g,h) ∈ Z2
ρ(G,A). In fact, it should be clear that all so-lutions of Eq. (167) may be obtained from any given solutionin this way.
This way of obtaining different solutions can be thought ofas utilizing transformations like those in Eq. (174), givenbyβ′a(g,h) = τa(g,h)βa(g,h), whereτa(g,h) are phases that
satisfy the condition thatτa(g,h)τb(g,h) = τc(g,h) when-everN c
ab 6= 0, but which are also required to satisfy the addi-tional condition
τga(h,k)τa(g,hk) = τa(g,h)τa(gh,k). (176)
Alternatively, one may think of this as similarly modifyingthelocal phase factorsη′a(g,h) = τa(g,h)
−1ηa(g,h), as the twonotions are not really distinguishable in this context.
There is, however, a sense in which different solutionsw (g,h) should be considered equivalent. In particular, if we
locally redefine the operatorsU (j)g by a transformation
U (j)g = U (j)
g Z(j)−1g , (177)
whereZ(j)g are unitary operators whose nontrivial action is lo-
calized withinRj , this redefinition will not change the globalactionRg on states as long as these operators satisfy
n∏
j=1
Z(j)g = 11 (178)
when acting in the subspace of quasiparticle states of the form|Ψa;c,µ〉. These are gauge transformations, and so they
should be treated as trivial modifications of the operatorsU(j)g ,
i.e. all operators related by such a transformation are in thesame equivalence class.
By similar arguments as used forW (j)g,h, it follows from
Eq. (178) that
〈Ψa;c,µ|Z(j)g |Ψb;e,ν〉 = ζaj (g)δa;c,µb;e,ν, (179)
whereζaj (g) is a phase that only depends on the topologicalchargeaj contained in the regionRj and that these phasesobey the constraint
n∏
j=1
ζaj (g) = 1. (180)
This similarly leads to the property thatζa(g)ζb(g) =ζc(g) wheneverN c
ab 6= 0, which, in turn, gives the relation
ζa(g) =M∗az(g), (181)
for some Abelian topological chargez(g) ∈ C1(G,A). Theseare precisely the same redundancies that arose due to thefreedom to decompose the trivial natural isomorphism intotopological charge dependent phase factors, as described inSec. III.
Under such transformations, the operatorsW(j)g,h transform
into
W(j)g,h = Z(j)
g U (j)−1g
gZ(j)h
gU(j)−1h U
(j)ghZ
(j)−1gh B
(j)g,h
= Z(j)g ρgZ
(j)h ρ−1
g W(j)g,hZ
(j)−1gh . (182)
Acting on states of the form|Ψa;c,µ〉, this produces theequivalent relations
ωa(g,h) =ζρ−1
g (a)(h)ζa(g)
ζa(gh)ωa(g,h), (183)
Maw(g,h) = Mρg[z(h)]M∗az(gh)Maz(g)Maw(g,h)
= Ma,ρg[z(h)]×z(gh)×z(g)×w(g,h) (184)
from which we obtain
w(g,h) = ρg[z(h)]× z(gh)× z(g)× w(g,h)= dz(g,h)× w(g,h), (185)
showing thatw(g,h) andw(g,h) that are related by fusionwith a 2-coboundary dz(g,h) ∈ B2
ρ(G,A) correspond pre-
cisely to operatorsW (i)g,h andW (i)
g,h that are related by gaugetransformations, and so should be considered equivalent, i.e.one should take the quotient byB2
ρ(G,A).Thus, the solutions of Eq. (167) for the[O] = [0] equiva-
lence class are classified by
[t] ∈ H2ρ(G,A) =
Z2ρ(G,A)
B2ρ(G,A)
. (186)
One should not, however, think of the set of solutions itselfas being equal toH2
ρ(G,A), but rather anH2ρ(G,A) torsor.
In particular, the distinct cohomology classes[t] ∈ H2ρ(G,A)
relate distinct equivalence classes of solutions[w], with differ-ent solutions being related byw′(g,h) = t(g,h) × w(g,h).The number of distinct symmetry fractionalization classesisthus equal to|H2
ρ(G,A)|. In this sense, there is no notionof an identity element of the set of solutions (as might havenaıvely seemed to be the case had one chosen to use the rep-resentativeO = 0 of the [O] = [0] equivalence class). WhenO = 0, Eq. (167) becomes a cocycle condition onw(g,h), so,in this case, the equivalence classes of solutions are actuallycohomology classes[w] ∈ H2
ρ(G,A), though this is not aninvariant statement.
Once again, symmetry actions in the same equivalenceclass related by natural isomorphisms lead to the same resultshere, so this classification of solutions is actually independentof the choiceρ ∈ [ρ]. Thus, the symmetry fractionalization isclassified by
[t] ∈ H2[ρ](G,A). (187)
E. Projective Representations of the Global Symmetry
In the above discussion, we assumed that the local Hilbertspace on each site transforms in a linear representation of the
25
global symmetryG. However this is not fully general, andit is possible that instead the local Hilbert space on each sitetransforms according to a projective representation ofG. Thecanonical example is a spin-1
2 system. While the global sym-metry of spin rotation isG = SO(3), each site contains a spin-12 which transforms in a projective representation of SO(3).Describing symmetry fractionalization when the local Hilbertspace already forms a projective representation ofG requiressome minor modifications of the previous arguments. In par-ticular, the action of a projective symmetry representation onthe ground state will take the form
Rgh|Ψ0〉 = eiΦg,hRgRh|Ψ0〉, (188)
whereeiΦg,h are the projective representation phase factors.The projective representations are classified byH2 (G,U(1)).In particular, the phaseseiΦg,h must satisfy the 2-cocycle con-dition
eiΦh,ke−iΦgh,keiΦg,hke−iΦg,h = 1 (189)
in order for the two different, but equivalent ways of relat-ingRghk andRgRhRk to be consistent. Additionally, differ-
ent projective phase factorseiΦg,h andeiΦg,h are consideredequivalent if they are related by a 2-coboundary
eiΦg,h = eifhe−ifgheifgeiΦg,h (190)
for some phase functioneifg of the group elements ofG, sincetheir difference could simply be absorbed into the operatorRg by the trivial redefinitionRg = eifgRg. The equivalenceclass[eiΦg,h ] ∈ H2 (G,U(1)) of the projective representationis a global property of the system that does not change un-der application of local operations, such as those that createquasiparticles. Thus, we also have
Rgh|Ψa;c,µ〉 = eiΦg,hRgRh|Ψa;c,µ〉 (191)
with the sameeiΦg,h for any state of the form|Ψa;c,µ〉 ob-tainable from the ground state|Ψ0〉 through adiabatic creationand manipulation of quasiparticles. We now defineW
(j)g,h as
before forj = 2, . . . , n, while for j = 1 we slightly modifythe definition to be
W(1)g,h = e−iΦg,hU (j)−1
ggU
(j)−1h U
(j)ghB
(j)g,h. (192)
With this definition, we retain the properties thatW (j)g,h is lo-
calized in regionRj , and that theW (j)g,h satisfy Eqs. (149) and
(150). This allows the argument relating the eigenvalues ofW
(j)g,h to Abelian topological charges to go through unaltered.
To see that the cocycle relations are unchanged, we only needto check that Eq. (163) remains the same forW
(1)g,h. This fol-
lows from the previous argument, together with the fact thateiΦg,h itself satisfies the 2-cocycle condition of Eq. (189).Thus, the same cohomological relations hold and all the ar-guments go through as before to give the same results for ob-struction and classification of symmetry fractionalization.
F. Quasi-On-Site Global Symmetry
There are a number of symmetries, such as time-reversalsymmetry and translation symmetry, that do not act in an on-site fashion, but which may nonetheless be fractionalized.Inorder to understand fractionalization of such symmetries,wemust generalize the notion of symmetries acting in an on-sitefashion so as to include the possibility of anti-unitary symme-tries and other nonlocal symmetries.
Let us again decompose the space manifoldM =⋃k∈I Mk into a collection of simply connected disjoint re-
gionsMk (a subset of which can be taken to be the regionsRj) with index setI.
We call a symmetry operator “quasi-on-site” if it takes theform
Rg =∏
k∈IR(k)
g Pg, (193)
whereR(k)g is a unitary operator that has nontrivial action lo-
calized in regionMk, andPg is a unitary or anti-unitary op-erator that preserves locality in the following sense: For anyoperatorO(j) localized in the simply connected regionRj ,the operator
Og(j) ≡ PgO(j)P−1g (194)
is localized in the (possibly distinct) simply connected regionthat we denote asgRj , and two such simply connected regionsRj andRk are disjoint, i.e.Rj ∩ Rk = ∅, if and only if thecorresponding regionsgRj andgRk are disjoint, i.e.gRj ∩gRk = ∅.
Specific examples that we have in mind for the operatorPg
are the complex conjugation operatorK, in which casegRj =Rj , or a translation operatorT~x (in a translationally invariantsystem), in which casegRj is the regionRj translated by thevector~x. Clearly, on-site symmetries havePg = 11.
We can now repeat the entire analysis of this section witha few small, but important modifications to account for thequasi-on-site generalization. We note that our treatment hererequires that the symmetries also leave the spatial orientationof the fusion/splitting spaces invariant. Consequently, we omitspatial symmetries involving rotations or parity.
The first modification is to the conjugation of local opera-tors byRg. In particular, given the above locality preservingproperty ofPg, we generalize the definition in Eq. (126) to
gO(j) ≡ RgOg(j)R−1g = R(j)
g PgOg(j)P−1g R(j)−1
g , (195)
which is thus an operator whose nontrivial action is localizedin the regionRj .
The next modification is that when thejth quasiparticle ofthe state|Ψa;c,µ〉 is localized in regionRj , it follows thatthe jth quasiparticle of the stateRg|Ψa;c,µ〉 is localized inthe regiongRj . Consequently, the action ofRg on statesin the physical Hilbert space containing quasiparticles, as inEq. (139), is modified to
Rg|Ψa;c,µ〉 = Ug(1)g . . . Ug(n)
g ρg|Ψa;c,µ〉, (196)
26
R1
R2R3
a1
a2a3
ga2
ga3
ga1
Rg
gR1
gR2
gR3
Ug(1)g
Ug(2)g
Ug(3)g
FIG. 4: The action of a global quasi-on-site symmetry operator Rg
on a state with quasiparticles may move the locations where thequasiparticles are localized, from the regionsRj to the regionsgRj .The quasi-on-site property ensures that the regionsgRj are mutuallydisjoint for distinctj whenever the regionsRj are mutually disjointfor distinct j. Additionally, the quasi-on-site symmetry action in-duces unitary transformationsUg(j)
g that are, respectively, localizedin the regionsgRj , together with a global transformationρg whichstrictly acts on the topological quantum numbers.
whereUg(j)g is a unitary operator whose nontrivial action is
localized in the regiongRj , and we have definedρg exactly asbefore, which now makes it a quasi-on-site operator, in accordwith Rg. In particular,
ρg =
n∏
j=1
Ug(j)−1g Rg, (197)
and it also follows thatρg =∏k∈I ρ
(k)g Pg. We can leave
Eq. (141) unmodified, with the understanding that if thejthquasiparticle of the state|Ψa;c,µ〉 is localized is regionRj ,then thejth quasiparticle of the state|Ψ ga; gc,µ′〉 (and thestateρg|Ψa;c,µ〉) is localized in the regiongRj . It is alsoimportant to emphasize that nowρg includes the action ofPg,so, in addition to potentially modifying the localization re-gions, it will complex conjugate coefficients in front of thestate wheneverg corresponds to an anti-unitary symmetry.
With these modifications, one must be careful to modify thelocalization regions of the operators appropriately in allstepsof the arguments of the previous sections, but, in the end, thisdependence drops out entirely. In particular, we note that weshould modify Eq. (145) to
gO(j)U (j)g = U (j)
g ρgOg(j)ρ−1g , (198)
the definition of the operatorW (j)g,h, which has its nontrivial
action localized in the regionRj , to
W(j)g,h = U (j)−1
ggU
(j)−1h U
(j)ghB
(j)g,h = ρgU
g(j)−1h ρ−1
g U (j)−1g U
(j)ghB
(j)g,h, (199)
and the relation of Eq. (153) to
ηaj (g,h)U(j)gh |Ψa;c,µ〉 = gU
(j)h U (j)
g |Ψa;c,µ〉 = U (j)g ρgU
g(j)h ρ−1
g |Ψa;c,µ〉. (200)
The final relation in terms of operators, given in Eq. (163), is modified to
ρgWg(j)h,k ρ−1
g ρgBg(j)−1h,k ρ−1
g W(j)g,hkB
(j)−1g,hk =W
(j)g,hB
(j)−1g,h W
(j)gh,kB
(j)−1gh,k . (201)
Applying this relation to a state|Ψa;c,µ〉, we find that the dependence on localization regions drops out of the resulting relationin terms of (eigenvalue) phases, and the only modification that we must now account for is the potential complex conjugationdue tog being an anti-unitary symmetry (which was encoded in the operatorρg). Specifically, this yields the modification ofEq. (164) to the relation
Ωa(g,h,k) = Kq(g)βρ−1g (a)(h,k)K
q(g)βa(gh,k)−1βa(g,hk)βa(g,h)
−1
= Kq(g)ωρ−1g (a)(h,k)K
q(g)ωa(gh,k)−1ωa(g,hk)ωa(g,h)
−1, (202)
and the modification of Eq. (165) to
Kq(g)ηρ−1g (a)(h,k)K
q(g)ηa(gh,k)−1ηa(g,hk)ηa(g,h)
−1 = 1. (203)
UsingΩa(g,h,k) = M∗aO(g,h,k) andωa(g,h) = M∗
aw(g,h)
exactly as before, though with the relationSρg(a)ρg(b) =
Kq(g)SabKq(g) that applies for unitary and anti-unitary sym-
27
metries, we obtain precisely the same consistency condition
O (g,h,k) = dw (g,h,k) (204)
of Eq. (167). We emphasize that the complex conjugationsdue to symmetries being anti-unitary dropped out in the pro-cess of mapping the relation of phases into the relation ofCn(G,A) cochains.
The remaining arguments that lead to the classification re-sults are similarly modified. Similar to the steps describedabove, the localization region dependence drops out when theoperator relations are converted into phase relations by apply-ing them to states of the form|Ψa;c,µ〉, and the complexconjugations that occur for anti-unitary symmetries drop outwhen these phase relations are converted into cochain rela-tions. Thus, the obstruction of fractionalization by nontrivial[O] ∈ H3
[ρ](G,A) and the classification of symmetry fraction-alization (when the obstruction vanishes) in terms of the co-homology classH2
[ρ](G,A) is precisely the same for unitaryand anti-unitary quasi-on-site symmetries as it was for unitaryon-site symmetries.
We note that the projective representation analysis ofSec. IV E must include the complex conjugation of anti-unitary symmetries, so they are classified byH2
q (G,U(1)),which includes complex conjugation from anti-unitary sym-metry action. In particular, the boundary operator includes thecomplex conjugation through theρg action, so the2-cocyclecondition on the projective phases becomes
eiq(g)Φ(h,k)e−iΦ(gh,k)eiΦ(g,hk)e−iΦ(g,h) = 1, (205)
and the projective phase is a2-coboundary when
eiΦ(g,h) = eiq(g)f(h)e−if(gh)eif(g) (206)
for some phaseeif(g) ∈ C(G,U(1)). These modifications donot affect the symmetry fractionalization results.
When we specify a fusion basis decomposition of the topo-logical state space ofn quasiparticles, we first specify an or-der in which to place the quasiparticles from left to right atthe top of a fusion tree. Specifying an order in which onelists the quasiparticles is equivalent to specifying a lineinthe 2D manifold that passes through the quasiparticles in thatorder. The inclusion of rotational and spatial parity sym-metry is complicated by the fact that these symmetry op-erations generally change the positions of the quasiparticleswith respect to their ordering line. For spatial parity symme-tries, we note that one can repeat the analysis above, with themodification that when phases in the analysis are mapped toCn(G,A) cochains, the action ofρg on the group elementsAis modified to include topological charge conjugation when-everp(ρg) = 1. This modification follows from the relationSρg(a)ρg(b) = Kq(g)+p(g)SabK
q(g)+p(g), which modifies theρ action in the cohomology structure, i.e. in the coboundaryoperator and the groupsHn
[ρ](G,A), wheneverρg correspondsto a non-trivial parity symmetry, which changes the orienta-tion of space.
Before concluding this section, we note that the above con-siderations provide a framework to classify the different pos-sible types of symmetry fractionalization. However, not all
elements of theH2[ρ](G,A) classes will be allowed in gen-
eral. WhenG corresponds to a spatial symmetry, there can beadditional constraints that rule out certain types of fractional-ization [119–123]. Even for on-site symmetries, as we willsee, some of the fractionalization classes are anomalous andcannot be realized in a purely2 + 1 dimensional system.
1. Time Reversal Symmetry Fractionalization and Local KramersDegeneracy
It is worth considering fractionalization in more detail forthe case of time reversal symmetry, or, rather, a group elementT ∈ G such thatT2 = 0 andq(T) = 1, i.e. it is an anti-unitary symmetry.
We first note that the state of the system can either forma linear representation withR2
T|Ψ〉 = |Ψ〉 or a projectiverepresentation withR2
T|Ψ〉 = −|Ψ〉. This follows fromtheH2
q (ZT2 ,U(1)) classification of projective representations.
In particular, the modified2-cocycle condition of Eq. (205)is simply the conditionei2ΦT,T = 1, and the modified2-coboundary condition of Eq. (206) iseiΦT,T = 1. The projec-tive representationeiΦT,T = −1 gives the usual degeneracyfrom Kramers theorem, where|Ψ〉 andRT|Ψ〉 are necessarilyorthogonal and degenerate in energy for any state|Ψ〉 whenRT commutes with the Hamiltonian. Physically, this corre-sponds to the case where the system has half-integer angularmomentum, i.e. an odd number of electrons in the system.
The symmetry action on the topological state space is spec-ified by the action on fusion vertex states
ρT|a, b; c, µ〉 =∑
ν
[UT(Ta, Tb; Tc)]µν |Ta, Tb; Tc, ν〉.
(207)Since this is an anti-unitary symmetry, it follows that
[κT,T(a, b; c)]µν =∑
λ
[UT(Ta, Tb; Tc)]∗µλ[UT(a, b; c)]λν
=βa(T,T)βb(T,T)
βc(T,T)δµν
=ηa(T,T)ηb(T,T)
ηc(T,T)δµν . (208)
The obstruction class is defined by
Ωa(T,T,T) =1
βTa(T,T)βa(T,T). (209)
The condition that the obstruction vanishes is equivalent tothere being someωa(T,T) such that
βTa(T,T)βa(T,T) = ωTa(T,T)ωa(T,T), (210)
ωa(T,T)ωb(T,T) = ωc(T,T), if N cab 6= 0. (211)
We assume that the obstruction vanishes and that this anti-unitary symmetry acts in the quasi-on-site fashion withRT =∏k∈I R
(k)T K. From the quasi-on-site symmetry fractionaliza-
tion analysis, we have
RT|Ψa;c,µ〉 = U(1)T . . . U
(n)T ρT|Ψa;c,µ〉. (212)
28
The localized symmetry action operatorsU (j)T have the pro-
jective consistency relation
ηaj (T,T)|Ψa;c,µ〉 = TU(j)T U
(j)T |Ψa;c,µ〉
= RTU(j)T R−1
T U(j)T |Ψa;c,µ〉
= U(j)T ρTU
(j)T ρ−1
T |Ψa;c,µ〉 (213)
where the projective phasesηa(T,T) satisfy Eq. (203),which, in this case, is simply the condition that
ηTa(T,T) = ηa(T,T)∗. (214)
WhenTa = a, this condition implies
ηa(T,T) = ±1, (215)
and we interpretηa(T,T) as the “localT2” value ascribed tothe topological chargea. Whenηa(T,T) = −1, there is alsoa local Kramers degeneracy [49] associated with the topolog-ical chargea. In other words, quasiparticles that carry topo-logical chargea also carry a local degenerate state space inphysical systems that possess this symmetry. We also empha-size thatθTa = θ∗a, so, whenTa = a, we also haveθa = ±1.However, we stress that it is not necessarily the case thatθaequalsηa(T,T), as one might have naıvely expected from theusual understanding of Kramers degeneracy in terms of spinand fermionic parity.
WhenTa = a, Tb = b, andTc = c, (andN cab 6= 0), we
have
ηa(T,T)ηb(T,T)
ηc(T,T)δµν =
βa(T,T)βb(T,T)
βc(T,T)δµν
=∑
λ
[UT(a, b; c)]∗µλ[UT(a, b; c)]λν . (216)
ForN cab = 1, the second line of this relation is simply equal
to 1, which implies that
ηa(T,T)ηb(T,T) = ηc(T,T). (217)
WhenTc = c andN caTa = 1, the ribbon property gives
RaTa
c RTaac = θc, (218)
and the transformation of theR-symbols underT gives
ρT
(Ra
Tac
)=(Ra
Tac
)∗=UT(a,
Ta; c)
UT(Ta, a; c)R
Taac
= κT,T(a,Ta; c)R
Taac . (219)
It follows that
ηc(T,T) = θc = ±1 (220)
whenTc = c andN caTa = 1.
The properties given in Eqs. (217) and (220) are useful fordetermining the localT2 values of quasiparticle excitationsin typical time-reversal invariant topological phases, see e.g.Refs. [89, 90, 118].
The analysis of fractionalization of time reversal symmetrypresented in this section precisely matches that of Ref. [89]. Incontrast with Ref. [49], our definition of localT2 for thejthquasiparticle of a state|Ψa;c,µ〉 (which carries topologicalchargeaj) is the corresponding eigenvalueηaj (T,T) of the
operatorRTU(j)T R−1
T U(j)T whose nontrivial action is localized
in the regionRj containing thejth quasiparticle. In particular,this definition applies to the general case where there are anarbitrary number of regions/quasiparticles that transform non-trivially underT and where the entire system may transformprojectively withT2 = −1. In considering the case wherethere are only two regions that transform nontrivially underT and where and the entire system transforms asT2 = 1,Ref. [49] interprets the operatorRTU
(1)T as the “localT” op-
erator for regionR2 andRTU(2)T as the “localT” operator
for regionR1. We avoid interpreting the operatorRTU(j)T as
a “localT” operator (of some complementary region), as it isnot a local operator and even its action on a quasiparticle state,which is given by
RTU(j)T |Ψa;c,µ〉 = ηaj (T,T)
∏
k 6=jU
(k)T ρT|Ψa;c,µ〉,
(221)is generally not localized in one region (even when all thetopological charges involved areT-invariant).
V. EXTRINSIC DEFECTS
Given the existence of a global symmetryG, we can in-troduce point-like defects that carry flux associated with thegroup elementsg ∈ G. In this section, we will describe a wayto create such defects and some of their basic properties. Wefirst give a prescription for creatingg-defects in some simplelattice model systems, and subsequently generalize this con-struction to an arbitrary system in a topological phase. Atthe end of this discussion, we will briefly discuss the casewhere there is no global symmetry, which still allows non-trivial point-like defects as long as Aut(C) is nontrivial. In thefollowing section (Sec. VI), we will build upon the physicalmotivation of this section and provide a detailed presentationof the algebraic theory of extrinsic defects, which is knowninthe mathematical literature asG-crossed braided tensor cate-gory theory [79, 82].
A. Physical Realization ofg-Defects
1. Simple lattice model
We begin by considering a concrete model system, in whichwe can precisely describe the general idea we wish to abstract.In particular, we consider a system with a local Hilbert spacedefined on the sites of a square lattice, whose HamiltonianH0
has a local on-site unitary symmetryG. For simplicity, werestrict to the case where the interactions inH0 are just near-est neighbor or plaquette interactions, so that the Hamiltonian
29
takes the form
H0 =∑
i
hi +∑
〈ij〉hij +
∑
[ijkl]
hijkl, (222)
wherehi consists of local operators that act on sitei, hij con-sists of local operators that act on a pair of neighboring sitesi andj connected by the link〈ij〉, andhijkl consists of localoperators that act on a plaquette[ijkl] defined by the sitesi,j, k, andl.
A pair of defects carrying fluxesg andg−1, respectively,can be created and localized at a well-separated pair of pla-quettes by modifying the Hamiltonian as follows. Imagine alineC emanating from the center of one of the defect’s corre-sponding plaquette, cutting across a set of links of the lattice,and terminating at the center of the other defect’s plaquette,as shown in Fig. 5. We modify the original Hamiltonian byreplacing each term inH0 that straddles the lineC with thecorresponding operator obtained from that term by acting withthe symmetry locally on the sites only on one side of the lineC.
In order to make this procedure well-defined, we first as-cribe an orientation of the lineC, indicated by an arrow point-ing from theg−1-defect endpoint towards theg-defect end-point. (If g = g−1, it will not matter which orientation wechoose.) This provides a well-defined notion of sites beingimmediately to the left or to the right of the lineC. Specifi-cally, the sitei is immediately to the left ofC and the sitejis immediately to the right of the lineC, if C crosses the link〈ij〉 of the lattice connecting sitesi andj with i to the left andj to the right, with respect to the orientation of the lineC. Wedenote the set of all sites immediately to the left ofC asCl andthe set of all sites immediately to the right asCr. We can nowdefine a term in the Hamiltonian to be straddling the lineC ifit only acts nontrivially on sites in the unionCl∪Cr and it hasnontrivial action on sites in bothCl andCr. [179] Finally, weconjugate such terms by the operatorR
(Cr)g =
∏j∈Cr R
(j)g ,
whereR(j)g represents the local action ofg ∈ G acting on site
j. (Recall that the global on-site symmetry action can be writ-ten as the product of local operatorsRg =
∏k∈I R
(k)g , where
I in this example is simply the set of all sites.)Thus, the modified Hamiltonian is given by
Hg,g−1 = H0 +∑
〈ij〉:i∈Cl;j∈Cr
[R(j)g hijR
(j)−1g − hij ]
+∑
[ijkl]:i,l∈Cl;j,k∈Cr
[R(j)g R(k)
g hijklR(j)−1g R(k)−1
g − hijkl]. (223)
Here, we have assumed that the lineC is straight for simplic-ity. If C was not a straight line, the last line in this Hamil-tonian would include plaquette terms with one site on oneside ofC and three sites on the other side ofC, correspond-ing to the plaquettes whereC makes turns. This HamiltonianHg,g−1 defines a line defect associated with the lineC. Thetwo end points ofC are codimension-2 point defects whichcarry fluxg andg−1, respectively. We refer to the lineC as ag-defect branch line.
(a) (b)
gC g-1
i
j k
l
(c)g g-1
a ρ (a)g (d)
C
C
wr
l w
FIG. 5: (a) When the system is cut along a lineC, quasiparticles can-not propagate across the cut. (b) The system can be reglued togetheralongC in a manner that conjugates bond/plaquette operators strad-dling the cut by a localg-symmetry action on one side of the cut, asindicated by red dots. The result is ag andg−1 pair of defects atthe end-points of the cut. (c) Such a construction effectively imple-ments ag-symmetry transformation on quasiparticles that propagateacross the cut around the defects. For example, a quasiparticlea willbe transformed intoρg(a) when it encircles theg-defect in a coun-terclockwise fashion. For symmetries that are not on-site,such astranslational or rotational symmetries,g-defects correspond to lat-tice dislocations or disclinations, respectively.
2. g-conjugation of quasiparticles across defect line
When a quasiparticle is adiabatically transported around ag-defect, it will be transformed by the symmetry action of thegroup elementg, as a consequence of crossing theg-defectbranch line. When the actionρg on topological charges isnon-trivial, as a quasiparticle with topological chargea en-circles the point-likeg-defect at the end of the defect lineC,the quasiparticle is transformed into one that carries topologi-cal chargeρg(a). Defects that permute the topological chargevalues of quasiparticles are sometimes referred to as “twistdefects.”
In order to understand this property, it is useful to firstconsider starting from the uniform system with HamiltonianH0, and introducing some quasiparticles using local poten-tials of the formh
(j)aj , as described in Sec. IV A, with the
corresponding HamiltonianHa1,...,an;0. We now consider anoperatorTak(k, k
′) that moves the quasiparticle of chargeakfrom sitek on one side of the lineC (which at this point issimply an imaginary line drawn on the system) to the sitek′ on the other side ofC in a manner that crosses the lineC. Such an operator annihilates a quasiparticle of topolog-ical chargeak at sitek, creates a quasiparticle of chargeakat sitek′, and commutes with the Hamiltonian away from thesitesk andk′. (One may think of this as a “string operator.”)Thus, if |Ψa;c,µ〉 were the ground states of the Hamiltonian
Ha1,...,an;0 with h(j)aj localizing the quasiparticle at sitej, then|Ψ′
a;c,µ〉 = Tak(k, k′)|Ψa;c,µ〉 are the ground states of the
HamiltonianH ′a1,...,an;0 with kth term changed toh(k
′)ak lo-
calizing the quasiparticle at sitek′ (perhaps up to some addi-tional unitary transformations localized around the sitesk andk′). Consequently, it is possible to adiabatically change the
30
Hamiltonian between these configurations and, in doing so,adiabatically move the quasiparticle of chargeak from sitekto sitek′.
We next imagine cutting all bonds of the system along thelineC, as indicated in Fig. 5(a). The corresponding Hamilto-nian is
Hcut(C) = H0 −∑
〈ij〉:i∈Cl;j∈Cr
hij −∑
[ijkl]:i,l∈Cl;j,k∈Cr
hijkl, (224)
where we have again assumedC is a straight line for simplic-ity. In this system, it is no longer possible to adiabaticallymove a quasiparticle across the lineC (without reintroducingthe excised terms in the Hamiltonian), because there are noterms in the Hamiltonian that connect the system acrossC. Ifwe introduce quasiparticles away from the lineC using localpotentials to similarly produce a HamiltonianHa1,...,an;cut(C),we would find that the operatorTak(k, k
′) does not commutewith the HamiltonianHcut(C) in the vicinity ofC (nor in thevicinity of the sitesk andk′), hence it will create quasiparti-cles there. Consequently, this operator would now correspondto moving the quasiparticle from sitek to its nearer side thecut lineC, pair creating quasiparticles of chargeak andak onthe other side of the cut lineC, and moving the chargeak ofthat pair to sitek′, while leaving the chargeak quasiparticlenext to the cut lineC on the opposite side from the originalquasiparticle. Such a process involves more than just adia-batically transporting the quasiparticle, since one must eitherintroduce additional local potentials for the extra quasiparti-cles, or cost energy above the gap for creating the additionalquasiparticles.
We now imagine reintroducing the bond/plaquette opera-tors that connect the system across the cut lineC with a con-jugation of these operators by the symmetry action ofg actinglocally only on the sites on one side of the cut, to obtain theHamiltonianHg,g−1 . Then we introduce quasiparticles awayfrom C using local potentials to similarly produce a Hamil-tonianHa1,...,an;g,g−1. We similarly find that the operatorTak(k, k
′) will, in general, not commute with the HamiltonianHcut(C) in the vicinity ofC (nor in the vicinity of the siteskandk′), and, therefore, must create extra quasiparticles there.
However, in this case, the lineC is not an untraversablecut line, and one can actually construct an operator that cor-responds to adiabatically transporting a quasiparticle acrossC (without creating extra quasiparticles). For this, we startfrom the operatorTak(k, k
′), which can be written as a prod-uct of local operators, and modify it in the following way. Thelocal terms in the product whose nontrivial action is entirelyon the left side ofC are left unaltered, the local terms in theproduct whose nontrivial action is entirely on the right side ofC are conjugated byRg, and the local terms in the product
that straddleC are conjugated byR(Cr)g . The resulting oper-
ator, which we denoteTak;g(k, k′), annihilates a quasiparti-
cle of topological chargeak at sitek, creates a quasiparticleof chargegak at sitek′, and commutes with the HamiltonianHg,g−1 away from the sitesk andk′. (Note that if the un-modified operatorTak(k, k
′) commutes withH0 away fromthe sitesk and k′, then so doesRgTak(k, k
′)R−1g .) Thus,
if |Ψa;c,µ;g,g−1〉 were the ground states of the Hamiltonian
Ha1,...,an;g,g−1 with h(j)aj localizing the quasiparticle at sitej,then|Ψ′
a′;c′,µ′;g,g−1〉 = Tak;g(k, k′)|Ψa;c,µ;g,g−1〉 are the
ground states of the HamiltonianH ′a1,..., gak,...,an;g,g−1 with
the kth term changed toh(k′)
gak localizing a quasiparticle ofchargegak at sitek′ (perhaps up to some additional unitarytransformations localized around the sitesk andk′). Conse-quently, it is possible to adiabatically change the Hamiltonianbetween these configurations (without creating extra quasipar-ticles), and, in doing so, adiabatically move the quasiparticlefrom sitek to sitek′, while also transforming its topologicalcharge fromak to ρg(ak) as it crosses theg-defect branchline.
3. General construction ofg-defects
We can generalize the above discussion and prescriptionfor creating defects to a general topologically ordered systemwith a local HamiltonianH0. Again, we first draw an ori-ented lineC in the system. We then define regionsCl andCr,which are “immediately” to the left and right of the lineC, re-spectively. These regions should have widthw such that anyterm in the Hamiltonian that straddles the lineC has nontrivialaction that is localized (perhaps up to exponentially dampedtails) in the unionCl ∪ Cr. Typically, this will require thewidthw to be a few correlation lengthsξ. The precise detailsof how these regions,Cl andCr, terminate near the endpointsof the lineC is unimportant for establishing that there is ag-defect (though it may play a role in determining which typeof g-defect is preferred, as we will explain below). We nextidentify the terms inH0 whose nontrivial action is localizedentirely withinCl ∪ Cr, and denote the sum of these termsasH0(C). We define the operatorR(Cr)
g =∏j:Mj⊂Cr R
(j)g ,
where we decompose the space manifoldM = ∪k∈IMj intoa collection of simply connected disjoint regionsMj , none ofwhich straddle the lineC, i.e. C ∩ int(Mj) = ∅ for all j.Finally, we define the defect Hamiltonian
Hg,g−1 = H0 + [R(Cr)g H0(C)R
(Cr)−1g −H0(C)]. (225)
It should be clear that these constructions can also be gen-eralized to describe the system with an arbitrary numbern ofdefects which carry group elementsg1, . . . ,gn whose productis identity
∏nj=1 gj = 0.
4. Point-like nature and confinement ofg-defects
WhenG is continuous or is physically obtained by sponta-neously breaking a larger continuous symmetry, theg-defectscan be created gradually. This property is familiar in the caseof superfluid vortices, where the phase of the order parameterrotates continuously by2π. For symmetries that are not on-site, such as translational or rotational symmetries, the defectscorrespond to lattice dislocations or disclinations. In all ofthese cases, theg-defects are well-defined even though there
31
is no specificg-branch line across which theg-action takesplace. In other words, theg-defects are truly point-like ob-jects.
In fact, from the perspective of the topological order andquantum numbers, the defect branch lines are completely in-visible in general. There are no local measurements one canperform using topological properties and operations, suchasquasiparticle braiding, that can identify the location of ade-fect branch line. Only the end-points of the branch lines,where theg-defects are localized, are locally detectable bytopological objects or operations. We stress that this doesnot necessarily mean that the branch lines are invisible toall forms of local measurements. Depending on the physi-cal realization, the branch lines may or may not be a phys-ically well-localized and measurable object. For example,in superconductor-semiconductor heterostructure-basedreal-izations of Majorana and parafendleyon wires [9–11, 124–127], the defect branch lines are the segments of nanowires inthe topological phase, and are clearly locally measurable andidentifiable. On the other hand, for multi-layer systems withgenons [6, 12], which are defects whose group action trans-fers quasiparticles from one layer to another, abstractly theremay be no precise, well-defined location of the branch lines,whereas there may be in some experimental realizations [34].
Theg-defects defined above areextrinsic defectsin the sys-tem, in the sense that they are imposed by deforming the uni-form Hamiltonian to the defect HamiltonianHg,g−1. The lo-cations of theg-defects are classical parameters inHg,g−1
and thus do not fluctuate quantum mechanically. However,if we allow the defects to become dynamical objects, whosepositions do fluctuate quantum mechanically, then there is aquestion of whether they are confined or deconfined. If theyare confined, then the energy cost to separating the dynamicalg-defects will grow with their separation. If they are decon-fined, then the energy cost for separating theg-defects willbe finite and independent of their separation, up to exponen-tially small corrections. Given the Hamiltonian of the sys-tem, diagnosing whether theg-defects correspond to confinedor deconfined excitations may be a non-trivial task. We ex-pect that one possible way to do this would be to obtain theground state|Ψg,g−1〉 ofHg,g−1, and then to compute the av-erage energy of this ground state with respect to the originalHamiltonian:E0
g,g−1 = 〈Ψg,g−1|H0|Ψg,g−1〉. The confine-ment/deconfinement of the defects would then correspond towhetherE0
g,g−1 diverges with the separation between the de-fects or is bounded by a finite value, respectively, in the limitof large separations.
If the g-defects are deconfined, as described above, thenthey correspond to quasiparticle excitations of the phaseC. Insuch a case, the globalG symmetry effectively becomes anemergent local gauge invariance with gauge groupG at longwavelengths. In what follows, we focus on the case where theg-defects correspond toconfinedobjects, and in fact we willreserve the termg-defect for this case. The case whereG ispromoted to a local gauge invariance is described in Sec. VIII.
5. Twist defects without global symmetry
It is important to note that even when the underlying phys-ical system has no exact global symmetry of its microscopicHamiltonian (i.e. G is trivial), the existence of topologicalsymmetry Aut(C) of the emergent topological phaseC im-plies the possibility of twist defects. In particular, one canpotentially have point-like twist defects associated withnon-trivial group elements in Aut(C). However without any globalsymmetries, creating such twist defects with a generic micro-scopic Hamiltonian is a more complicated issue, which we donot address here. [180]
As a simple example of the realization of twist defects di-rectly from Aut(C), without a global symmetry, consider thetwist defects associated with layer exchange in a double-layertopological phase [6]. These defects are well-defined evenin the absence of an exact layer-exchange symmetry. There-fore the concept of a twist defect is not logically dependenton the global symmetry of the microscopic Hamiltonian. Inwhat follows we focus on extrinsic point-like defects associ-ated with elements of a global symmetryG. This is because(a) twist defects without global symmetries can still be con-sidered in the same formalism by takingG = Aut(C), (b)we wish to develop a complete characterization of symmetry-enriched phases associated with a global symmetryG, and (c)we also wish to study the mechanism of gauging the globalsymmetryG, which requires us to start with a system whereG is an exact microscopic global symmetry.
B. Topologically Distinct Types ofg-Defects
In the previous subsection, we provided an example of howto modify the Hamiltonian to realizeg-defects. However, it isnot necessarily the case that there is a unique type ofg-defectthat may be physically realized in a given topological phase.In principle, a topological phase may support multiple typesof g-defects that cannot be transformed into one another bythe application of a local operator. In these cases, there wouldbe topologically distinct types ofg-defects.
As a simple example, we may consider a Hamiltonianwhich makes it locally preferable for a quasiparticle withtopological chargeb to be bound to theg-defect. Under cer-tain circumstances, this composite object might correspond toa topologically distinct type ofg-defect as compared to theoriginal one. Indeed, as we will explain in the next subsec-tion, two topologically distinct types ofg-defects can alwaysbe obtained from each other by fusion with a quasiparticlecarrying an appropriate value of topological charge. This canbe understood intuitively, since topologically distinct typesof g-defects can only differ by topological properties of thetopological phase that can be point-like localized at the defect(endpoint of ag-branch line). While there is no preference be-tween topologically distinctg-defects when considered in thetopological context, it will generically be the case that therewill be an energetic preference between distinctg-defects, asthey will have different energy costs for a given physical real-ization.
32
g g=? b
e
e
FIG. 6: A g-defect can possibly be altered into a topologically dis-tinct type ofg-defect by fusing it with a quasiparticle carrying non-trivial topological chargeb ∈ C. Whether the originalg-defectand theb-g composite object are topologically distinct depends onwhether there is some topological chargee ∈ C, whose Wilson looparound the defects can distinguish them. Such a topologicalchargeemust beg-invariant,ρg(e) = e, otherwise its Wilson loop could notclose upon itself after crossing theg-defect branch line.
If two g-defects are topologically distinct, then there mustbe a topological process that can distinguish them. This pro-cess corresponds to the Wilson loop operatorWe associatedwith a ρg-invariant topological chargee encircling theg-defect, as shown schematically in Fig. 6. Different possibleeigenvalues ofWe can be used to distinguish topologicallydistinct types of defects. In fact, we will later show that thisstatement can be made more precise. In particular, for a modu-lar theoryC, we will show that one can write a linear combina-tions of such Wilson loop operators which acts as orthogonalprojectors on the enclosed area onto each topologically dis-tinct type ofg-defect. (We will also show that the number oftopologically distinct types ofg-defects is equal to the numberof ρg invariant topological charges in the original topologicalphaseC.)
In order to refer to topologically distinct types ofg-defects,we must use a more refined labeling system than simply as-signing them the group elementg. We give each topologicallydistinct type of defect its own labela, which, in accord withprior terminology, we call topological charge. We write theset of topological charges corresponding to distinct typesofg-defects asCg. We will often use the notationag as a short-hand to indicate thata ∈ Cg. We emphasize that this doesnot meanag is a composite object formed by ag-defect and atopological chargea ∈ C from the original topological phase.In this notation, the topological charge set labeled by the iden-tity group element0 is equal to the original set of topologicalcharges of the topological phase, i.e.C0 = C. We write theset of all topological charges asCG.
VI. ALGEBRAIC THEORY OF DEFECTS
We now wish to develop a mathematical description ofthe topological properties, such as fusion and braiding, ofg-defects in a topological phaseC with global symmetryG,that generalizes (and includes) the UBTC theory used to de-scribe (deconfined) quasiparticle excitations of the topologicalphase. The proper mathematical description of such defectsisknown as aG-crossed braided tensor category [79, 82]. Inthis section, we present theG-crossed theory, starting withG-graded fusion and then introducingG-crossed braiding. We
derive the consistency conditions and a number of importantproperties for such theories. In Appendix D, we provide aconcise presentation ofG-crossed categories more properlyusing the abstract formalism of category theory.
A. G-graded Fusion
It is clear that combining ag-defect with anh-defect shouldyield agh-defect. Hence, the fusion of defects must respectthe group multiplication structure ofG, leading to the notionof G-graded fusion.
A fusion categoryCG isG-graded if it can be written as
CG =⊕
g∈GCg. (226)
In particular, this means each topological chargea ∈ CG isassigned a unique group elementg ∈ G and correspondingcharge subsetCg to which it belongs, such that fusion respectsthe group multiplication ofG, i.e. if a ∈ Cg andb ∈ Ch, thenN cab can only be nonzero ifc ∈ Cgh.We recall the shorthand notationag used to indicate that
a ∈ Cg. With this, we can write the fusion rules [of Eq. (3)]as
ag × bh =∑
c∈CGN cabc =
∑
c∈Cgh
N cabc =
∑
c
N cabcgh. (227)
All the properties and constraints of fusion categories fromSec. II A carry over directly toG-graded fusion categories.Clearly, the vacuum charge0 ∈ C0, where we write the iden-tity element of the groupG as0. It should be clear thatC0 is it-self a fusion category, since it is closed under fusion. As such,we consider aG-graded categoryCG to be a “G-extension” ofits subcategoryC0.
The unique charge conjugate of a topological chargeag isag ∈ Cg−1 . Sinceag is the unique topological charge withwhich ag can fuse into vacuum, it follows that for any twodistinct topological chargesag, cg ∈ Cg, there must existsome nontrivial topological chargesb0, b′0 ∈ C0 such thatcgis one of the fusion outcomes obtained from fusingag withb0 or fusing b′0 with ag, i.e. N
cgagb0
= N b0agcg
6= 0 and
Ncgb′0ag
= Nb′0cgag
6= 0. Physically, this means that differenttypes ofg-defects in (aG-extension of) a topological phasedescribed byC0 can indeed be obtained from each other byfusing quasiparticles, which carry topological charges inC0,with theg-defects. [181]
As before, the quantum dimensions (which are defined inthe same way) obey the relation
dagdbh =∑
c
N cabdcgh . (228)
We define the (total) quantum dimension ofCg to be
Dg =
√∑
a∈Cg
d2ag . (229)
33
Using Eq. (228) and the fact thatN cab = Na
cb, we see, by
picking some arbitraryc ∈ Cg, that
D20 =
∑
a∈C0
d2a0 =∑
a,b∈C0
x∈Cg
da0d−1cg N
xacgd
−1cg N
bxcgdb0
=∑
a,b∈C0
x∈Cg
d−2cg N
axcgda0N
bxcgdb0
=∑
x∈Cg
d−2cg
(dxg
dcg)2
=∑
x∈Cg
d2xg= D2
g (230)
for any g ∈ G with nonemptyCg 6= ∅. In particular, thequantum dimension of every nonemptyCg is
Dg = D0 = |H |− 12DCG , (231)
whereDCG is the total quantum dimension ofCG and we de-fine the subgroup
H = h ∈ G | Ch 6= ∅ ≤ G. (232)
That H forms a subgroup ofG follows from the fact thatCg, Ch 6= ∅ implies thatCgh 6= ∅, together with the existenceof a vacuum charge and charge conjugates.
In this paper, we will focus our attention to faithfullyG-graded categories, i.e. those withH = G, so that there isno g ∈ G with Cg = ∅. In other words, we study the fulldefect theory associated with all group elementsg ∈ G. Wenote that one could instead choose to study the defect theoryassociated with a subgroupH ≤ G. In this case, one canleaveCg for g /∈ H empty and then study the resulting non-faithfully G-graded category. Such a non-faithfullyG-gradedcategory would just be a faithfullyH-graded category, withthe empty setsCg for g /∈ H included formally. This is onlynontrivial once we also include the symmetry action of suchg /∈ H .
B. G-Crossed Braiding
We can consider a continuous family of HamiltoniansH(λ)of the physical system containing defects (possibly includingquasiparticles, which we consider to be0-defects), where thelocations of the defects and their corresponding branch linesare changed adiabatically as a function of the parameterλ.This allows us to implement physical operations that exchangethe positions of defects.
With this in mind, we wish to define a notion of braiding ofdefects, called “G-crossed braiding,” that includes group ac-tion and which is compatible with aG-graded fusion categoryCG. We denote such aG-crossed braided tensor category asC×G . This requires some modification of the usual definition of
braiding. In fact, whenG is a non-Abelian group, fusion inaG-graded fusion category is not commutative, so the usualnotion of braiding cannot even be applied. In particular, theremust be a group action when the positions of objects (carry-ing nontrivial group elements) are exchanged. (Of course, the
g h
a b
(a)
g
h
ab
(b)
gh
gb a
g
(c)
FIG. 7: (a) Each symmetry defect is labeled by a topological chargeand has a corresponding defect branch line emanating from itchar-acterized by a symmetry group element. Here we show ag-defectwith chargea and anh-defect with chargeb, and their correspondingbranch lines in a 2D system. (b) As ag-defect is braided with anh-defect in the counterclockwise sense, one can imagine deformingthe corresponding branch lines, so that no objects cross them. (c) Inorder to return to the original configuration of branch lines, one mustpass theg branch line across theh-defect and its branch line. As theh-defect of topological chargeb passes through theg branch line, thetopological chargeb is transformed toρg(b) and theh branch line istransformed into agh = ghg−1 branch line. This corresponds totheG-crossed braiding operatorR
gbhag , as defined in Eq. (233).
usual definition of braiding still applies within the subcategoryC0, which is a BTC.)
As the mathematical formalism is developed, it will be-come clear that one can also physically implement braidingtransformations for non-Abelian defects by using topologi-cal charge measurements and/or tunable interactions, follow-ing the “measurement-only” methods of Refs. [128–130]. Asthese methods remove the need to physically move the de-fects, they may provide a more preferable physical implemen-tation of braiding transformations, depending on the details ofthe physical system.
When the objects carry non-trivial group elements, they areconsidered symmetry defects, which one can think of as hav-ing a branch cut line emanating from the otherwise point-likeobject. These branch cuts are oriented and are labeled by thegroup element of the object at which they terminate, so thattaking an object through ag-branch in the counterclockwisesense around the branch point at the correspondingg-defectgivesg-action on that object, as shown in Fig. 7. In order todescribe this using diagrammatics, we choose the conventionwhere the branch lines, which form worldsheets that end onthe worldlines of the defects, go into the page, and then weleave the branch line worldsheets implicit in the diagrammat-ics. This does not impose any restriction on how the defectbranch lines must be physically configured in the actual sys-tem. Rather, it is merely a bookkeeping tool that allows usto consistently keep track of the effects of the branch linesinthe diagrammatics, while only drawing the worldlines of thedefects and not the branch line worldsheets. With this con-vention, ag-defect worldline applies group action on objectswhen it crosses over their worldlines. In particular, we define
34
G-crossed braiding by
Ragbh =
ag bh
bhhag
=∑
c,µ,ν
√dcdadb
[Ragbhcgh
]
µν
cgh
bhag
hagbh
ν
µ,(233)
where theR-symbols for aG-crossed theory are the maps
Rabc : Vbh
hagcgh → V
agbhcgh that result from exchanging (in a
counterclockwise manner) two objects of chargesbh andhag,respectively, which are in the chargecgh fusion channel. Werecall that
gbh = ρg(bh) (234)
g = g−1 (235)gh = ghg−1 (236)
in the shorthand notation introduced in Sec. III B for the sym-metry group action on topological charges. [182]
The symmetry action[ρ] : G →Aut(C0) on the originaltheory must now be self-consistently extended to an action ofthe symmetry group
[ρ] : G→ Aut(C×G) (237)
that is incorporated within the structure of the extended the-ory. Notice, for example, that compatibility with theG-gradedfusion rules required thatgbh ∈ Cghg−1 , i.e.
ρg : Ch → Cghg−1 . (238)
More generally, compatibility with the fusion algebra requires
Ncghagbh
= Ncghgbhag
. (239)
From this, together with the properties of charge conjugates,it follows thatN0
agbh= N0
gbhag= δbhag , and hence any topo-
logical charge inCg will be invariant under the action of thecorrespondingg, i.e.
gag = gnag = ag, (240)
for all n ∈ Z.For some theories (this may occur also in FTCs or BTCs),
it may be possible for a topological chargeag to remain un-changed after fusion/splitting with another nontrivial topo-logical chargeb0. In particular, this occurs whenNag
agb0=
Nagb0ag
6= 0. In this case,b0 quasiparticles can be absorbed oremitted at theag-defect without changing the localized topo-logical charge or localization energy of the defect. As such,we say that defects (or quasiparticles) that carry chargeag lo-calize a “b0 zero mode.” It is clear from
Nagagb0
= Nag
agb0= N
agb0ag
= N b0agag
= Nagagb0
= Naggb0ag
(241)
that if ag localizes ab0 zero mode, then: (1)ag also localizesa b0 zero mode, (2)ag andag localize b0 zero modes andalso zero modes associated with the entireg-orbit of chargesgnb0, and (3)b0 is one of the fusion channels ofag with itsconjugateag, as isb0 and gnb0.
TheG-crossedR-symbols can equivalently be written interms of the relation
cgh
bhag
µ =∑
ν
[Ragbhcgh
]
µνcgh
bhag
ν . (242)
Similarly, the clockwiseG-crossed braiding exchange op-erator is
(Ragbh
)−1=
bhhag
ag bh
=∑
c,µ,ν
√dcdadb
[(Ragbhcgh
)−1]
µν
cgh
hagbh
bhag
ν
µ.(243)
In order forG-crossed braiding to be compatible with fu-sion, we again wish to have the ability to slide lines over orunder fusion vertices. However, we may no longer assumethat such operations are completely trivial, since one mustatleast account for the group action on a vertex. The appropriaterelations are given by the unitary transformations
xkkb
kcgh
bhag
µ
=∑
ν
[Uk (a, b; c)]µν xk
kcgh
cgh
bhag
ν
(244)
xk
gx
hgxk
cgh
bhag
µ
= ηx (g,h)xk
hgxk
cgh
bhag
µ
. (245)
We have used the same notation[Uk (a, b; c)]µµ′ andηx (g,h)that we previously introduced for the global symmetry actionon the topological degrees of freedom in Sec. III B and thefractionalized (projective) local symmetry action in Sec.IV B,because, as we will see, these are precisely the same quantitiesextended to the entireG-crossed theory. Intuitively, it shouldbe clear why this is the case, since aag line in theG-crossedbraided diagrammatics has an implicitg branch sheet behindit that applies ag action to anything that passes through it,i.e. anything that theag line passes over. Hence, sliding axkline over a vertex, as in Eq. (244), passes the vertex throughthek branch sheet, and should result in thek action on thatvertex. Similarly, passing a|ag, bh; cgh, µ〉 vertex over axkline, as in Eq. (245), should capture the local projective re-lation of equatinggh action on chargex with successively
35
appliedg andh actions on chargex, as the vertex indicateswhere thegh branch sheet splits into ag branch sheet and anh branch sheet. The validity of this claim will be establishedthrough the following consistency arguments and conditions.The quantity[Uk (a, b; c)]µµ′ here corresponds to a specificchoice ofρ ∈ [ρ], and we will see that the relation betweenchoices within a symmetry action equivalence class (relatedby natural isomorphisms) will take the form of a gauge trans-formation in this theory.
We begin by arguing that the factors in these expressionsmust have the given dependence on the various topologicaland group quantities. In particular, in Eq. (244), we see thatthe nontrivial interaction is between thek-branch line and thevertex, hence there may be dependence onk, but not the spe-cific x within Ck, and the transformation on the fusion statespace may be nontrivial, so it may depend on all the vertexlabels. For Eq. (245), we see that the nontrivial interactionis between theg, h, andgh branch lines and the topologicalchargex, so this expression should not depend on the specifictopological charge valuesa, b, or c, (just their correspondinggroup elements), nor should it have any effect within the fu-sion state space (on the vertex).
Sliding a line over a vertex, as in Eq. (244) is a unitary
transformation betweenVka kbkc
andV abc , as specified by theunitary operatorsUk (a, b; c). This requires the dimensional-ity of the fusion spaces to be preserved under the correspond-ing symmetry action, giving
Nkcghkag kbh
= Ncghagbh
(246)
for anyk acting on a vertex. It follows that the quantum di-mensions are also invariant
dag = d kag . (247)
Clearly, if the sliding line has vacuum chargexk = 0, thesliding transformations should be trivial, so
[U0 (a, b; c)]µν = δµν (248)
η0 (g,h) = 1. (249)
We require that the sliding rules are compatible with theproperty that vacuum lines can be freely added or removedfrom a diagram, i.e. sliding over/under a vertex|a, b; c〉 witha = 0 or b = 0 should be trivial, since it is equivalent tosimply sliding over a line. This imposes the conditions
Uk (0, 0; 0) = Uk (a, 0; a) = Uk (0, b; b) = 1 (250)
ηx (0,0) = ηx (g,0) = ηx (0,h) = 1. (251)
Combining Eqs. (244) and (245) with trivial braidings, suchas
ba
=
a b
, (252)
we see that sliding lines over or under vertices with the oppo-site braiding are given by
xk
kcgh
c
bhag
µ
=∑
ν
[Uk
(ka, kb; kc
)]µν xk
kcgh
bhag
kakbν
(253)
xk
hgxk
cgh
bhag
µ
= ηx (g,h)
xk
hgxk
gx
cgh
bhag
µ
. (254)
Compatibility with the inner product Eq. (7) gives the cor-responding relations for sliding over and under fusion (ratherthan splitting) vertices
xk
cgh
ab
kbhkag
µ
=∑
ν
[Uk (a, b; c)]νµ xk
cgh
kc
kbhkag
ν
(255)
xk
cgh
kc
kbhkag
µ
=∑
ν
[Uk
(ka, kb; kc
)]νµ xk
cgh
ab
kbhkag
ν
(256)
hgxk
xk gxk
cgh
bhag
µ
= ηx (g,h)
hgxk
xk
cgh
bhag
µ
(257)
xk
hgxk
cgh
bhag
µ
= ηx (g,h)gxk
xk
hgxk
cgh
bhag
µ
(258)
We require that the sliding moves are consistent with eachother by requiring that any two sequences of sliding movesthat start in the same configuration and end in the same con-figuration are equivalent. This can be achieved by equat-ing the two different sequences of sliding moving shown inFig. 8, which results in the consistency conditions betweentheUk (a, b; c) and theηx (g,h) factors given by
36
ag bh xk yl
zkl lkcgh
U
ag bh xk yl
zkl lkcgh
U
ag bh xk yl
zkl lkcgh
η
ag bh xk yl
zkl lkcgh
η
ag bh xk yl
zkl lkcgh
η
ag bh xk yl
zkl lkcgh
U
FIG. 8: TheG-crossed symmetry action consistency equation provides consistency between the sliding moves, which implement theU andη transformations associated with the global and fractionalized (local projective) symmetry action. Eq. (260) is obtained by imposing thecondition that the above diagram commutes.
ηb (k, l) ηa (k, l)∑
λ
[Ul
(ka, kb; kc
)]
µλ[Uk (a, b; c)]λν = [Ukl (a, b; c)]µν ηc (k, l) . (259)
If we defineκk,l = ρklρ−1l ρ−1
k andκk,l|ag, bh; cgh, µ〉 =∑ν [κk,l(a, b; c)]µν |ag, bh; cgh, ν〉, we see that this condition can
be rewritten as the symmetry action consistency equation
[κk,l(a, b; c)]µν =∑
α,β
[Uk (a, b; c)
−1]
µα
[Ul
(ka, kb; kc
)−1]
αβ
[Ukl (a, b; c)]βν =ηa (k, l) ηb (k, l)
ηc (k, l)δµν . (260)
Using this condition to decomposeUklm (a, b; c) in the two equivalent ways related by associativity, one obtains the followingconsistency condition on theκk,l
κl,m( ka, kb; kc)κk,lm(a, b; c) = κk,l(a, b; c)κkl,m(a, b; c). (261)
Thus, we see that sliding anxk line over a vertex or operator can indeed be thought of as implementing theG-crossed exten-sion of the symmetry actionρk, with Uk (a, b; c) playing the same role as in Sec. III B. Similarly, sliding anxk line under a|ag, bh; cgh, µ〉 vertex can be thought of as implementing theG-crossed extension of the projective phasesηx (g,h) relating thelocal symmetry action ofg andh to gh.
We continue expounding the relation of the sliding moves to the symmetry action by next requiring consistency between thesliding moves and theF -moves. Sliding a line over a fusion tree before or after application of anF -move gives
xk
ka kb kc
a b c
e
d
α
β
=∑
α′,β′,f,µ′,ν′
[Uk
(ka, kb; ke
)]αα′
[Uk
(ke, kc; kd
)]ββ′
[F
ka kb kckd
]
( ke,α′,β′)( kf,µ′,ν′)xk
ka kb kc
kf
kd
d
µ′
ν′
=∑
f,µ,ν,µ′,ν′
[F abcd
](e,α,β)(f,µ,ν)
[Uk
(kb, kc; kf
)]µµ′
[Uk
(ka, kf ; kd
)]νν′
xk
ka kb kc
kf
kd
d
µ′
ν′
, (262)
which yields the consistency condition∑
α′,β′,µ′ν′
[Uk(
ka, kb; ke)]αα′
[Uk(
ke, kc; kd)]ββ′
[F
ka kb kckd
]
(ke,α′,β′)(kf,µ′,ν′)
×[Uk(
kb, kc; kf)−1]µ′µ
[Uk(
ka, kf ; kd)−1]ν′ν
=[F abcd
](e,α,β)(f,µ,ν)
.(263)
37
This condition is the statement of invariance of theF -symbols (of theG-crossed theory) under the symmetry action.Similarly, sliding a line under a fusion tree before or afterapplication of anF -move gives
x
ag bh ck
egh
dghk
α
β
= ηx (g,h) ηx (gh,k)∑
f,µ,ν
[F abcd
](e,α,β)(f,µ,ν)
x
ag bh ck
fhk
dghk
µ
ν
=∑
f,µ,ν
[F abcd
](e,α,β)(f,µ,ν)
ηgx (h,k) ηx (g,hk)
x
ag bh ck
fhk
dghk
µ
ν, (264)
which yields the consistency condition
ηgx (h,k) ηx (g,hk) = ηx (g,h) ηx (gh,k) . (265)
This is the statement of fractionalization being consistent in theG-crossed theory. Recall from Sec. IV that this relation translatesinto the condition that the obstruction to fractionalization vanishes, so here we see a direct way in which a nontrivial obstructionwould make it impossible to consistently extend the original theoryC0 to aG-crossed theoryC×
G .Sliding a line under aG-crossed braiding operation gives theG-crossed Yang-Baxter equation
bh hagxk
kagkbh xk
=η ka(khk,k)
η ka(k,h)
bh hagxk
kagkbh xk
. (266)
Here, we slid thea line under theRbx braiding operator and obtained theηa factors by expanding theRbx braiding operator interms of fusion and splitting vertices.
Alternatively, we can obtain a similar relation by sliding thex line over theRab braiding operator, but this case there will besymmetry action applied to the braiding operation, so we must explicitly expand it, giving
bh hagxk
kagkbh xk
=∑
c,µ,ν
√dcdadb
[Rabc
]µν
bh hagxk
kagkbh xk
cghν
µ
=∑
c,µ,ν,µ′,ν′
√dcdadb
[Uk(
kb, kha; kc)−1]
µ′µ
[Rabc
]µν
[Uk(
ka, kb; kc)]νν′
bh hagxk
kagkbh xk
kcν′
µ′ . (267)
Comparing this relation with the G-crossed Yang-Baxter equation by expanding theRab braiding operator in Eq. (266), weobtain the consistency condition between braiding and sliding moves
η ka(khk,k)
η ka(k,h)
∑
µ′,ν′
[Uk(
kb, kha; kc)]
µµ′
[R
ka kbkc
]
µ′ν′
[Uk(
ka, kb; kc)−1]ν′ν
=[Rabc
]µν
(268)
38
a cb
e
d
R
a cb
e
d
F
a cb
d
g Ra cb
d
g
Fa cb
d
kf
U
a cb
d
R
a cb
f
d
F
a hgcb
e
d
R−1
a hgcb
e
d
F
a hgcb
d
g R−1
a hgcb
d
g
Fa hgcb
d
f
η
a hgcb
d
R−1
a hgcb
f
d
F
FIG. 9: TheG-crossed Heptagon equations provide consistency conditions betweenG-crossed braiding, fusion, and sliding moves. Eqs. (272)and (273) are obtained by imposing the conditions that the above diagrams commute.
This is theG-crossed generalization of the statement that theR-symbols are invariant under the symmetry action. Notice thepresence of theη factors, as compared to Eq. (88), to which this expression reduces whena, b, c ∈ C0.
We reemphasize the fact that imposing consistency on the sliding moves has resulted in consistency conditions that preciselyreplicate the symmetry action constraints and properties described in Secs. III and IV, and extend them from acting on theC0 the-ory to itsG-crossed extensions. This justifies our use of the same symbols [Uk (a, b; c)]µµ′ andηx (g,h) for the transformationsassociated with the sliding moves.
We note, for future use, that sliding a line under and anotherline over a vertex gives the relation
ηkx
(kg, kh
)=ηgx
(k,khk
)
ηgx
(h, k
) ηx(gh, k
)
ηx(k,kghk
) ηx(k,kgk
)
ηx(g, k
) ηx (g,h) (269)
for howηx (g,h) transforms underk-action. This can be obtained from
x
yk
cgh
kbhkag
bhag
µ
= ηx (g,h)[Uk(
ka, kb; kc)−1]
µν
ηx(gh, k
)
ηx(k,kghk
) kx
yk
cgh
kcgh
kbhkag
ν
(270)
=ηgx
(h, k
)
ηgx
(k,khk
) ηx(g, k
)
ηx(k,kgk
)[Uk(
ka, kb; kc)−1]
µνηkx
(kg, kh
) kx
yk
cgh
kcgh
kbhkag
ν
(271)
where the two lines in this expression correspond to the two orders in which one can slide thex andy lines.Finally, we require consistency betweenG-crossed braiding and fusion, as well as the sliding moves, so that any two sequences
of moves that start from the same configuration and end in the same configuration must be equivalent. This is achieved byimposing the followingG-crossed Heptagon equations, which are analogous to the Hexagon equations of BTCs, a diagrammaticrepresentation of which is shown in Fig. 9. The Heptagon equation for counterclockwise braiding exchanges is
∑
λ,γ
[Race ]αλ
[F ac
kbd
]
(e,λ,β)(g,γ,ν)
[Rbcg]γµ
=∑
f,σ,δ,η,ψ
[F c
ka kbd
]
(e,α,β)( kf,δ,σ)[Uk (a, b; f)]δη
[Rfcd
]
σψ
[F abcd
](f,η,ψ)(g,µ,ν)
, (272)
in which we left the group labels forag, bh, ck, dghk, egk, fgh, andghk implicit. Similarly, the Heptagon equation for clockwise
39
braiding exchanges is
∑
λ,γ
[(Rcae )−1
]
αλ
[F a
gcbd
]
(e,λ,β)(g,γ,ν)
[(R
gcbg
)−1]
γµ
=∑
f,σ,δ,ψ
[F cabd
](e,α,β)(f,δ,σ)
ηc (g,h)
[(Rcfd
)−1]
σψ
[F ab
hgcd
]
(f,δ,ψ)(g,µ,ν), (273)
in which we left the group labels forag, bh, ck, dkgh, ekg,fgh, and ggkgh implicit (the differences being due to howthe group action enters braiding in the counterclockwise vs.clockwise braiding operators).
Given the trivial associativity of the vacuum charge0(F abcd = 11 whena, b, or c = 0), the Heptagon equationsimply that braiding with the vacuum is trivial, i.e.Ra0a =
R0aa =
(Ra0a
)−1=(R0aa
)−1= 1 for any value ofa ∈ CG.
If we further require unitarity of the theory, then(Rab
)−1=(Rab
)†, which can be expressed in terms ofR-
symbols as[(Rabc
)−1]
µν=[Rabc
]∗νµ
.
C. Gauge Transformations
The basic data given byN cab, F
abcd , Rabc , ρk [which in-
cludesUk(a, b; c)], andηa(g,h) that satisfy the consistencyconditions described in the previous subsections defines aG-crossed braided tensor category, which we can consider to beageneralized anyon and defect model. There is, however, someredundancy between different collections of basic data duetogauge freedom, similar to the case of BTCs. Thus, we againwish to characterize theories as equivalent when they are re-lated by gauge transformations. ForG-crossed BTCs, it isuseful to separate gauge transformations into two classes.
The first type of gauge transformation is familiar fromBTCs. In particular, these gauge transformations derive fromthe redundancy of redefining the fusion/splitting vertex basisstates
˜|a, b; c, µ〉 =∑
µ′
[Γabc]µµ′ |a, b; c, µ′〉 (274)
whereΓabc is a unitary transformation. Such gauge transfor-mations modify theF -symbols in precisely the same way wehave previously seen in Eq. (51). The transformation ofG-crossedR-symbols is slightly modified from that of BTCs toaccommodate the symmetry actions that are incorporated inbraiding, and is given by
[Ragbhcgh
]
µν=∑
µ′,ν′
[Γb
hac
]
µµ′
[Ragbhcgh
]
µ′ν′
[(Γabc)−1]
ν′ν.
(275)
The symmetry action transformation become
[Uk (a, b; c)
]
µν= (276)
∑
µ′,ν′
[Γ
ka kbkc
]
µµ′[Uk (a, b; c)]µ′ν′
[(Γabc)−1]
ν′ν.
These gauge transformations leaveηx(g,h) = ηx(g,h) un-changed, and consequentlyκg,h = κg,h is also unchanged.
The second type of gauge transformation is derived fromthe equivalence of symmetry actions by natural isomorphisms,i.e. ρg = Υgρg, which we discussed in Secs. III and IV.In particular, these gauge transformations enact the followingmodifications of the basic data
[F abcd
](e,α,β)(f,µ,ν)
=[F abcd
](e,α,β)(f,µ,ν)
, (277)[Ragbhcgh
]
µν= γa(h)
[Ragbhcgh
]
µν, (278)
[Uk (a, b; c)
]µν
=γa(k)γb(k)
γc(k)[Uk (a, b; c)]µν ,(279)
ηx(g,h) =γx(gh)
γ gx(h)γx(g)ηx(g,h), (280)
which leave theF -symbols unchanged, since the symme-try action is incorporated through braiding. [The symmetryaction on topological charge labels is unchangedρg(a) =ρg(a).] Thus, theories with different choices ofρ ∈ [ρ] areequivalent under this type of gauge transformation.
We refer to these two types of gauge transformations as ver-tex basis gauge transformations and symmetry action gaugetransformations, respectively. It is straightforward to checkthat all the consistency conditions are left invariant under bothtypes of gauge transformations.
As before, one must be careful not to use the gauge freedomassociated with the canonical gauge choices associated withmaking fusion, braiding, and sliding with the vacuum trivial,and respecting the canonical isomorphisms that allow one tofreely add and remove vacuum lines. In particular, one mustfix Γa0a = Γ0b
b = Γ000 , as in the case of BTCs, and also fix
γ0(h) = γa(0) = 1.
D. G-Crossed Invariants, Twists, andS-Matrix
It is useful to consider quantities of aG-crossed theorythat are invariant under gauge transformations, as we did forBTCs. (In this section, we will discuss invariants that are
40
straightforward to obtain in theG-crossed theory, e.g. us-ing diagrammatics, but we will later see that another classof invariants can be constructed by gauging the symmetryof the theory.) Clearly, invariants derived from fusion andF -symbols alone are the same in both BTCs andG-crossedBTCs, since the new symmetry action gauge transformationsdo not affect theF -symbols. In particular, the quantum di-mensionsda = da = d ka are invariants.
Eq. (263) withe = f = 0 yields the relation
κ ka
κa=
[F
ka ka kaka
]
00
[F aaaa ]00=Uk(
ka, ka; 0)
Uk( ka, ka; 0). (281)
Whena = a, the Frobenius-Shur indicatorκa = ±1 is agauge invariant quantity and it follows from Eq. (281) thatκa = κ ka. (We recall that, more generally,κa = κ−1
a .)Whenka = a is k-invariant, it follows from Eq. (281) that
Uk(a, a; 0) = Uk(a, a; 0). (282)
On the other hand, we must be more careful when tryingto carry over gauge invariant quantities that are derived frombraiding operations, such as the twist factors andS-matrix, asthese may no longer be gauge invariant in aG-crossed theory.Consequently, we will examine these in more detail.
The topological twists are defined the same way as beforeby taking the quantum trace of a counterclockwise braid of atopological charge with itself
θa =1
da a
=∑
c,µ
dcda
[Raac ]µµ . (283)
We immediately see thatθag is always invariant under the ver-tex basis gauge transformations, but is only invariant under thesymmetry action gauge transformations ifg = 0, since
θag = γag(g)θag . (284)
This corroborates the interpretation of topological chargesagwith g 6= 0 as describing extrinsic defects, for which oneshould not expect invariant braiding or exchange statistics inthe usual sense, since they are not true quasiparticles (decon-fined topological excitations) of the system. We will examinethis matter in more detail.
We can immediately notice that∑µ[Raac ]µµ
∑µ′
[Raac′ ]µ′µ′(285)
is gauge invariant under both types of gauge transformations.Using Eq. (266) with the definition of the twist, we find the
general relation betweenθa andθ ka is
θag =η kag(kgk,k)
η kag(k,g)θ kag =
ηag(k,kgk)
ηag(g, k)θ kag . (286)
Whenka = a, it follows that
ηag(g,k) = ηag(k,g). (287)
We also note that Eq. (265) givesηkx(k, k) = ηx(k,k) foranyx andk, so we also have
ηag(k, k) = ηag(k,k) (288)
whenka = a.The definition of topological twists can also be written in
the form
ag
ag
= θa
ag
ag
=
ag
ag
, (289)
as is the case with BTCs. It is clear that the inverse topologicaltwists are similarly obtained from clockwise braidings
ag
ag
= θ−1a
ag
ag
=
ag
ag
. (290)
For unitary theories, it is straightforward to see thatθ−1a = θ∗a,
and hence the topological twist factors must be phases.Unlike a BTC, it is not necessarily the case thatθag andθag
are equal in aG-crossed BTC. In particular, from the follow-ing diagrammatic manipulations
ag
= Ug(ag, ag; 0) ag (291)
= Ug(ag, ag; 0)ηag(g,g)
ag
, (292)
we find that they are related through the following expression
θag = Ug(ag, ag; 0)ηag(g,g)θag . (293)
By evaluating the diagrams on the first line of this sequence,one also finds the relations
θag = Ug(ag, ag; 0)κag
(Ragag0
)−1
(294)
= ηag(g, g)−1κ−1
ag
(Ragag0
)−1
. (295)
41
We can now derive theG-crossed generalization of the rib-bon property
∑
λ
[Rbh
hagcgh
]
µλ
[Ragbhcgh
]
λν=
θcθaθb
[Ugh(a, b; c)]µν
ηa(g,h)ηb(h, hg),
(296)
which is obtained using the following diagrammatic steps
ag bh
cgh
µ
=∑
ν
[Ugh(a, b; c)]µν
ag bh
cgh
ν
= θc∑
ν
[Ugh(a, b; c)]µνcgh
bhag
ν (297)
= ηb(hg, hgh)ηa(
hg, hgh)
ag bh
cghµ
= ηa(g,h)ηb(h, hgh)
cgh
ag bh
µ
= θaθbηa(g,h)ηb(h, hgh)∑
ν,λ
[Rbh
hagcgh
]
µλ
[Ragbhcgh
]
λνcgh
bhag
ν (298)
Notice that the first and second lines are related using the piv-otal property and we used the Yang-Baxter relation and thefact that lines can slide freely under a twist.
Clearly, the operatorRagbhRbhhag is not gauge invariant,
unlessg = h = 0. However, whenhag = ag andgbh = bh,the quantities
∑µ,ν
[Rbhagcgh
]
µν
[Ragbhcgh
]
νµ
∑µ′,ν′
[Rbhagc′gh
]
µν
[Ragbhc′gh
]
νµ
(299)
are invariant under both types of gauge transformations.
Once again, we define the topologicalS-matrix by
Sagbh =1
D0a b
=1
D0
∑
c,µ,ν
dc[Rbac
]µν
[Rabc
]νµ
=1
D0
∑
c,µ
dcθcθaθb
[Ugh(a, b; c)]µµηa(g,h)ηb(h, g)
. (300)
We emphasize that, whena ∈ Cg andb ∈ Ch, theS-matrixis only well-defined ifha = a andgb = b, and consequentlygh = hg. Otherwise, the topological charge values wouldchange in the braiding and one would not be able to closethe lines back upon themselves. We note that we have usedD0 = Dg, the total quantum dimension of each subsectorCg,rather than the total quantum dimensionDCG = |G| 12D0 ofthe entireG-crossed theoryC×
G for reasons that will be madeclear later.
The elements of theS-matrix do not obey all the same re-
42
lations as that of a BTC, nor are they gauge invariant, unlessg = h = 0, or unless eithera = 0 or b = 0 (in which caseSab = dadb/D0), since
Sagbh = γa(h)γb(g)Sagbh . (301)
Nonetheless, theS-matrix will be an important quantity thatagain plays an important role in defining the system and mod-ular transformations on higher genus surfaces, so we will ex-amine its properties in detail.
We first note that
S kag kbh =η ka(k,h)η kb(k, g)
η ka(khk,k)η kb(kgk,k)Sagbh , (302)
which follows from the definition and Eq. (266). It followsthat, whenkag = hag = ag andkbh = gbh = bh, we have
ηa(k,h)ηb(k, g)
ηa(h,k)ηb(g,k)= 1. (303)
It is straightforward to see that
S∗agbh =
1
D0a b (304)
for a unitary theory. It also follows immediately from the def-inition (and the cyclic property of the trace) that
Sagbh = Sbhag . (305)
While theseS-matrix relations are the same as for UBTCs, wemust be more careful with properties obtained by deforminglines, because of the nontrivial sliding rules of aG-crossedtheory.
Whenhag = ag andgbh = bh (and hencegh = hg), sothat the correspondingS-matrix element is well-defined, wehave the loop-removal relation
ag
bh
=SabS0b
bh
, (306)
which can be verified by closing theb line upon itself in thisexpression. In fact, if eitherhag 6= ag or gbh 6= bh, then lefthand side of the equation evaluates to zero, so, for these pur-poses, we can considerSab = 0 when it is not well-defined.
In writing this relation, we must be more careful than ina BTC to indicate clearly where the lines are drawn with re-spect to vertices, including local minima and maxima (cupsand caps). Recall that the minima/maxima of the cups/capscorrespond to splitting/fusion vertices, respectively, betweena topological charge, its conjugate, and the vacuum. There-fore, we see that
ag
bh
=Uh(a, a; 0)
ηb(g, g) ag
bh
. (307)
Since one can equivalently take the trace of Eq. (307) byclosing theb-line on itself into a loop to the left or right, itleads to the relation
Sagbh =Uh(a, a; 0)
ηb(g, g)S∗bhag
. (308)
Combining Eqs. (305) and (308) yields a relation between theS-matrix and its transpose
Sagbh =Uh(a, a; 0)ηa(h,h)
Ug(b, b; 0)ηb(g, g)Sbhag . (309)
Another useful relation allows us to flip the tilt of a loopencircling another line, as follows
ag
bh
= θa
agbh
= θaag
bh
=
ag bh
, (310)
in which we used Eqs. (266) and (287).
An important diagrammatic relation, which is the precursorof theG-crossed Verlinde formula, is obtained by putting two
43
loops on a line and using a partition of identity to relate it to a single loop on the line
ag bh
xk
=∑
c,µ
√dcdadb
µ
µ
bac
xk
=∑
c,µ,ν
√dcdadb
[Uk(b, a; c)]µν
ηx(h, g)
µ
ν
cb a
xk
=∑
c,µ
[Uk(b, a; c)]µµ
ηx(h, g)cgh
xk
(311)
Combining Eqs. (311) and (306), we find that whenkag =ag, kbh = bh, andgxk = hxk = xk, we have the importantrelation
Sagxk
S0xk
Sbhxk
S0xk
=∑
cgh,µ
[Uk(b, a; c)]µµ
ηx(h, g)
Scghxk
S0xk
. (312)
We can similarly obtain
Sxkag
Sxk0
Sxkbh
Sxk0=∑
cgh,µ
[Uk(a, b; c)]µµηx(g,h)
Sxkcgh
Sxk0. (313)
If we takex ∈ C0, these expressions become
Sagx0Sbhx0
S0x0
ηx(h, g) =∑
cgh
N cabScghx0
(314)
Sx0agSx0bh
Sx00ηx(g,h) =
∑
cgh
N cabSx0cgh , (315)
which show that one may think ofSagx0/S0x0
(or, equiva-lently, Sx0ag/Sx00) as projective characters of the extended(non-commutative) Verlinde algebra.
We will now establish several interesting relations that wewill find particularly useful for the discussion of modularity.The first is
∑
a∈Cg
daθaUg(a, a; 0) ag
bg
=D0Θ0
ηb(g, g)θbbg
, (316)
where
Θ0 =1
D0
∑
c∈C0
d2cθc (317)
is the normalized Gauss sum of theC0 BTC. In order to obtainthis relation, we use the fact that whena, b ∈ Cg, theS-matrixtakes the form
Sagbg =1
ηa(g,g)ηb(g, g)
1
D0
∑
c∈C0
N cabdc
θcθaθb
(318)
and therefore obeys the property
∑
ag
daθaUg(a, a; 0)
Sagbg =1
ηb(g, g)θb
1
D0
∑
ag,c0
N cabdadcθc
=dbΘ0
ηb(g, g)θb, (319)
which is established using Eqs. (228) and (293).
The next relation (which holds even whenhag 6= ag orgbh 6= bh) is
44
∑
x∈Cgh
dxθxηx(hg, hgh)
Ugh(x, x; 0) xgh
ag bh
hgbhhag
=∑
x,c∈Cghµ,ν
dxθxUgh(x, x; 0)
√dcdadb
[Uhg(
ha, hgb; c)]
µν
ν
µ
c
xgh
ag bh
hgbhhag
=∑
cgh,µ,ν
√dcdadb
[Uhg(
ha, hgb; c)]
µν
D0Θ0
ηc(gh, hg)θccgh
hgbhhag
bhag
ν
µ
=D0Θ0
θaθbηa(gh, hg)ηb(gh, hg)ηa(g,h)ηb(h, hg)
ag bh
hgbhhag
, (320)
which is obtained by using Eq. (316), the relation
[Uk(
ka, kb; kc)]
µν=
ηc(k, k)
ηa(k, k)ηb(k, k)
[Uk(a, b; c)
−1]µν, (321)
[which is the sliding move consistency Eq. (260) withl = k,] and the (inverse of the) the ribbon property given in Eq. (296).Finally, whenhag = ag (which requiresgh = hg), we have the relation
∑
x∈Cgh
dxθxηx(g,h)
Ugh(x, x; 0) xgh
ag
bh
gbh
=∑
x∈Cgh
dxθxηx(g,h)
Ugh(x, x; 0)
Ug(x, x; 0)
ηa(gh, hg)
ηx(g, g)
Ugh(a, a; 0) xgh
ag
bh
gbh
=D0Θ0ηb(g, g)
θaθbUh(a, a; 0)ηa(gh, hg)ηb(gh, hg)ηa(g,h)ηb(h,g) ag
gbh
bh
. (322)
To obtain these relations, we used Eqs. (307) and (316) in both lines, though, in the second line, we first applied Eq. (320). Weemphasize that the individual diagrams in this equation evaluate to zero, unlesshag = ag, gbh = bh, andgxgh = hxgh = xgh.In particular, the sum here can be taken to be overxgh ∈ Cg,h
gh = Cggh ∩ Ch
gh, the topological charges inCgh that are bothg-invariant andh-invariant, where we define the invariant topological charge subsets
Chg = a ∈ Cg | ha = a . (323)
Taking the trace of Eq. (322), i.e. closing theb-line back on itself (which requiresgbh = bh), we finally obtain the importantrelation
∑
x∈Cgh
ηa(g,h)θagSagxgh
Ugh(a, a; 0)ηx(gh, g)θxgh
Sxghbh
Uh(x, x; 0)ηb(h, hg)θbh = Θ0
SagbhUh(a, a; 0)
. (324)
45
In order to manipulate the trace of Eq. (322) into this form,we have used the relations
ηa(k, l)ηa(k, l) =Ukl(a, a; 0)
Uk(a, a; 0)Ul( ka, ka; 0), (325)
ηx(gh, g) =ηx(g, g)
ηgx(g,h), (326)
ηgb(h, hg) =ηb(gh, hg)ηb(g,h)
ηb(g, g), (327)
the first of which is the sliding move consistency Eq. (260)with c = 0, while the second and third are special cases ofEq. (265).
We conclude this section by noting that a number of addi-tionalG-crossed gauge invariant quantities will naturally arisein the context of modular transformations of theG-crossedtheory and gauging the symmetry of theory. As these quanti-ties would be somewhat out of context and mysterious here,we leave their discussion for the subsequent Secs. VII andVIII.
VII. G-CROSSED MODULARITY
An important property of a topological phase of matter isthe ground state degeneracy when the system inhabits mani-folds with different topologies. For a2 + 1 dimensional topo-logical phase, the ground state degeneracy will depend on thegenusg of the surface inhabited by the system and the topo-logical charge values of the quasiparticles (and boundaries) ofthe system. More generally, it is important that the theory de-scribing a topological phase is well-defined and consistentforthe system on arbitrary topologies. In other words, the topo-logical properties of the system are described by a TQFT. Interms of the BTC theory, this is achieved by requiring the the-ory to be a modular tensor category (MTC), i.e. to have uni-taryS-matrix. In this case, theS-matrix andT -matrix providea projective representation of the modular transformations forthe system on the torus. (More general modular transforma-tions for the system on a manifold of arbitrary topology andquasiparticle content can similarly be defined in terms of theMTC properties.)
We wish to establish a similar notion of modularity forG-crossed BTCs, which allows one to relate the theory to aG-crossed TQFT that describes the topological phase with de-fects on arbitrary 2D surfaces. TheG-crossed extended defecttheoryC×
G admits a richer set of possibilities, as defect branchlines can wrap the nontrivial cycles of surfaces with genusg > 0, thus giving rise to “twisted” sectors. ForG-crossedmodularity, we will require that the set ofg-defect topologi-cal chargesCg is finite for eachg ∈ G (though not necessarilythatG is finite or even discrete). Some special cases ofG-crossed modular transformations have been studied recentlyin Refs. [131, 132].
In this section, we will develop an understanding of thetwisted sectors and their associated topological ground statedegeneracies. We also establish the notion ofG-crossed mod-ularity and the corresponding modular transformations forthe
system when it includes twisted sectors. The topologicalground state degeneracies of the defect sectors, together withtheG-crossed modular transformations, can provide valuableinformation about the symmetry-enriched topological order.
A. G-Crossed Verlinde Formula andωa-Loops
Before considering theG-crossed theory and modulartransformations for a system on surfaces with genusg > 0,we first investigate some properties that are closely related tomodularity, namely the Verlinde formula andωa-loops. Forthis, we begin with the minimal assumption that the originaltheoryC0 is a MTC, which is to say that itsS-matrix is unitary.From this assumption and Eqs. (314) and (315), we obtain theformula
N c0agbg
=∑
x0∈Cg0
Sagx0Sbgx0
S∗c0x0
S0x0
ηx(g, g) (328)
=∑
x0∈Cg0
Sx0agSx0bgS∗x0c0
Sx00ηx(g, g), (329)
where the sums in these expressions are over the subsetCg0 of
g-invariant topological charges inC0. (Actually, we could letthe sums go over the entireC0 if we consider theS-matricesto be equal to zero whengx 6= x.)
Settingc = 0 in these expressions and using Eqs. (305) and(308), we obtain
δaga′g =∑
x0∈Cg0
Sagx0S∗a′gx0
=∑
x0∈Cg0
Sx0agS∗x0a′g
. (330)
Now, we can use Eq. (330) with Eqs. (314) and (315) toobtain theG-crossed Verlinde formula
Ncghagbh
=∑
x0∈Cg,h0
Sagx0Sbhx0
S∗cghx0
S0x0
ηx(h, g) (331)
=∑
x0∈Cg,h0
Sx0agSx0bhS∗x0cgh
Sx00ηx(g,h), (332)
whereCg,h0 = Cg
0 ∩ Ch0 is the subset of topological charges in
C0 that are bothg-invariant andh-invariant.Moreover, we may use these properties to defineωag -loops,
which are linear combinations of loops of topological chargelines that act as topological charge projectors on the collec-tion of topological charge lines passing through them. [Theseshould not to be confused with theωa(g,h) phase factorsassociated with symmetry fractionalization in Sec. IV, norshouldωag be confused with an element ofCg.] Similar totheir definition in MTCs, theωag -loops in aG-crossed theoryare defined by
ωag
bg
=∑
x0∈Cg0
S0agS∗x0ag
x0
bg
= δagbg
bg
,
(333)
46
a
b
FIG. 10: A topological phase described by the MTCC on a torushas ground state degeneracy equal to the number of distinct topolog-ical charge types|C|. A basis for the degenerate ground state sub-space is provided by the states|a〉l which have definite topologicalcharge valuea ascribed to the charge line passing through the inte-rior of the torus around the longitudinal cycle. Alternatively, a basisis provided by the states|b〉m which have definite topological chargevalueb ascribed to the charge line passing through the exterior of thetorus around the meridional cycle. These two bases are related bythe modularS transformation, which is represented in a MTC by thetopologicalS-matrix, giving|a〉l =
∑
b∈CSab|b〉m.
where the first equality is a definition, and the last step usedEqs. (306) and (330) to show that act ong-defects as projec-tors that distinguish between the different topological chargevalues ofg-defects. Eq. (333) establishes our previous claimin Sec. V B that, when the original theoryC0 is modular, thereare physical processes involving theg-invariant topologicalcharges inC0 which are able to distinguish between the dis-tinct types ofg-defects.
It is worth re-emphasizing that, so far, we have only as-sumed thatC0 is modular (i.e. has unitaryS-matrix), andmade no further assumption about theS-matrix of the ex-tendedG-crossed theory. The results here seem to suggestthat it may be the case that requiringC0 to be modular wouldbe sufficient to obtain a notion of modularity of theG-crossedtheory. Indeed, by combining theorems from Refs. [80, 133]that relateC0 andC×
G to the theoryC/G obtained by gaug-ing the symmetry, one has the property thatC0 is a MTC ifand only ifC×
G is G-crossed modular (both of which are trueif and only if C/G is a MTC). We now define the notion ofaG-crossed BTC beingG-crossed modular in the followingsubsection.
B. Torus Degeneracy andG-Crossed ModularTransformations
When a topological phase of matter characterized by aUMTC C inhabits a torus, it possesses a topologically pro-tected ground state degeneracy equal to the number of distincttopological charges inC. More specifically, a basis for thisdegenerate ground state subspace on the torus is given by thestates|a〉l, which are, respectively, identified as having thedefinite topological charge valuea ∈ C ascribed to the chargeline passing through the interior of the torus around the lon-gitudinal cycle, as indicated in Fig. 10. In other words, if onewere to perform a topological charge measurement along aloop around the meridian of the torus, the resulting measure-
a
aa
FIG. 11: A topological charge measurement around a meridionalloop for the state|a〉l yields the measurement resulta. Similarly,if one cuts open the torus along the meridian for the state|a〉l, thetwo resulting boundaries will have chargesa anda, respectively.
ment value would bea for the state|a〉l. This is equivalent tocutting open the torus along the meridian and inspecting theresulting topological charge value on the resulting boundaries,which would be found to bea anda for the basis state|a〉l, asshown in Fig. 11. Along these lines, the ground state|a〉l canbe obtained from the ground state|0〉l by adiabatically creat-ing a pair of quasiparticles with topological chargesa anda,respectively, from vacuum, transporting the chargea quasi-particle around the longitudinal cycle (in the positive sense),and then pair annihilating the quasiparticles (fusing thembackinto vacuum).
Alternatively, one may interchange the roles of the longi-tudinal and meridional cycles of the torus. In this way, wecan equivalently define a basis for the ground state subspaceby |b〉m, which are, respectively, identified as having the defi-nite topological charge valuea ∈ C ascribed to the charge linepassing through the exterior of the torus around the meridionalcycle, as indicated in Fig. 10. These two bases are related bythe modularS transformation, which interchanges the cyclesof the torus (and flips the direction of one of them). As men-tioned in Sec. II B, theS-matrix of a MTC provides a (projec-tive) representation of the modularS transformation, wherethe bases are related by
|a〉l =∑
b∈CSab|b〉m. (334)
This relation is motivated by the observation of thea andbtopological charge lines passing around the complementarycycles of the torus forming linked loops, as in the topologicalS-matrix.
In order to generate all modular transformations on thetorus, we additionally consider the modularT transforma-tions, known as Dehn twists. This transformation replacesthe longitudinal cycle around the torus with one that wrapsonce (in the positive direction) around the longitude and oncearound the meridian (in the positive direction), as shown inFig. 12. We can, alternatively, think of this transformationas being obtained by cutting open the torus along a meridian,twisting the torus by2π around the longitudinal axis, and glu-ing it back together, so that a2π twist has been introduced.Providing the topological charge linea with a framing, which
47
a
FIG. 12: The modularT transformation, known as a Dehn twist, re-places the longitudinal cycle with a cycle (shown here in black) thatwraps once around the longitude and once around the meridian. Suchtransformations are represented in a MTC by the topologicaltwists,i.e. Tab = θbδab, and relate the basis states|a〉l to the basis de-fined with respect to the longitude plus meridian basis states |a〉l+m
through the relation|a〉l =∑
b∈CTab|b〉l+m.
is equivalent to drawing a line on the surface of the torus run-ning parallel to thea line around the longitudinal cycle, wesee that this transformation puts a twist in the framing ribbonof the charge line. This ribbon twist, which one can equate tothe topological twist, motivates the definitionTab = θbδab inthe transformation
|a〉l =∑
b∈CTab|b〉l+m, (335)
since a2π twist must be introduced to go from the basisstates|a〉l+m to the basis states|a〉l, and such a twist doesnot change the topological charge value that wraps around thelongitudinal cycle.
As mentioned in Sec. II B, when theS-matrix of a UBTCis unitary, the theory is considered modular, as theS andTmatrices provide a projective representation of SL(2,Z), themodular transformations on a torus, i.e.
(ST )3 = ΘC, S2 = C, C2 = 11, (336)
whereCab = δab is the topological charge conjugation opera-tor. In this case, one may also define the corresponding mod-ular transformations for punctured torii, and consequently, thetheory can be consistently defined on arbitrary surfaces.
In the defect theory described by aG-crossed BTCC×G , the
situation becomes more complicated. Clearly, theC0 subcat-egory, which describes the original topological phase withoutdefects, must behave exactly the same as described above.In other words, whenSa0b0 is a unitary matrix (when re-stricted to topological charge labelsa, b ∈ C0), so thatC0 is aUMTC, the ground states on a torus without defect branchesare described exactly as above and the operatorsSa0b0 andTa0b0 = θb0δa0b0 provide a projective representation of themodular transformations in the subtheory without defects.Wecall this restriction to the defect-free theory on the torusthe(0,0)-sector and denote the corresponding modular transfor-mations defined in this way asS(0,0) andT (0,0).
When we allow for the inclusion of defects in the theory,we can produce twisted sectors on the torus, each of which islabeled by two group elementsg,h ∈ G, which correspondto defect branch lines that, respectively, wind around the two
g
ag
bh
h
FIG. 13: The(g,h)-twisted sector on a torus, where a closedg-defect branch line wraps around the longitudinal cycle of the torusand ah-defect branch line wraps around the meridional cycle of thetorus. A basis for the degenerate ground state subspace of the(g,h)-twisted sector is given by the states|a(g,h)g 〉l corresponding to defi-nite topological charge valueag ∈ Ch
g ascribed to a charge line pass-ing through the interior of the torus around the longitudinal cycle.Alternatively, one may consider this to be a(h, g)-twisted sector ona torus by interchanging the roles of the longitudinal and meridionalcycles. In this case, a basis for the ground state subspace isgivenby the states|b(h,g)
h 〉m corresponding to definite topological chargebh ∈ Cg
h ascribed to the charge line passing through the exterior ofthe torus around the meridional cycle. These two bases are relatedby the modularS transformation|a(g,h)
g 〉l =∑
b∈Cg
h
S(g,h)agbh
|b(h,g)h 〉m.
non-contractible cycles of the torus, as shown in Fig. 13.The original UMTCC0 is described by the trivial twist sec-tor (g,h) = (0,0). One can obtain a state in the(g,h)-twisted sector from the(0,0)-sector by adiabatically creatinga h,h−1 defect pair from vacuum, transporting theh-defectaround the meridional cycle (in the positive sense), pair anni-hilating the defect pair, and then doing the same process withang,g−1 defect pair winding around the longitudinal cycle.This is only possible when
gh = hg, (337)
since otherwise the group element ascribed to the defectswould necessarily change type as they crossed the other de-fect branch line wrapping around the complementary cycle,making it impossible to pair-annihilate the defects or close thebranch line on itself. In this way, we see that the topologicalcharge line that runs through the interior of the torus aroundthe longitudinal cycle can only take values inCg, since it mustbe created by ag-defect encircling the cycle. Moreover, thistopological charge must beh-invariant, since the charge linescross theh-branch. Similarly, the topological charge line thatruns through the exterior of the torus around the meridionalcycle can only takeg-invariant topological charge values inCh. It is clear that states from different(g,h)-sectors can-not be superposed, since the defects are extrinsic, confinedobjects, which can be thought of as defining distinct superse-lection sectors.
We label the ground state subspace associated with the(g,h)-sector of the system on a torus asV(g,h). Similar to
UMTCs, a basis forV(g,h) is given by states|a(g,h)g 〉l cor-responding to definite topological charge valueag ∈ Ch
g as-cribed to a charge line passing through the interior of the torus
48
g
ag
gh
g
ag
h
=
FIG. 14: The modularT transformation (Dehn twist) maps be-tween the(g,h)-sector on a torus to the(g, gh)-sector shown here.This transformation acts diagonally (i.e. with relative phases) be-tween bases for the(g,h) and (gh,h) sectors, both of which arelabeled by topological charge valuesag ascribed to the topologi-cal charge line passing through the interior of the torus around thelongitudinal cycle. These two bases are related by|a(g,h)
g 〉l =∑
b∈Chg
T (g,h)agbg
|b(g,gh)g 〉l+m
around the longitudinal cycle. For aG-crossed theory, if weinterchange the roles of the longitudinal and meridional cycles(and flip one of their directions), corresponding to a modularS transformation, then the system from this perspective is inthe (h, g)-twisted sector on a torus. In this case, a basis for
the ground state subspace is given by the states|b(h,g)h 〉m cor-responding to definite topological chargebh ∈ Cg
h ascribed tothe charge line passing through the exterior of the torus aroundthe meridional cycle. Thus, there must be a unitary operatorrelating these two bases which represents the modularS trans-formation between the(g,h) and(h, g) sectors. In particular,this takes the form
|a(g,h)g 〉l =∑
b∈Cg
h
S(g,h)agbh
|b(h,g)h 〉m. (338)
Similarly, the modularT transformation (Dehn twist) takesthe system between the(g,h) and (g,gh) sectors, as indi-cated in Fig. 14, with basis states related by
|a(g,h)g 〉l =∑
b∈Chg
T (g,h)agbg
|b(g,gh)g 〉l+m. (339)
Thus, we can write the modularS andT transformationsfor aG-crossed theory in the form
S =⊕
(g,h) | gh=hgS(g,h) (340)
T =⊕
(g,h) | gh=hgT (g,h), (341)
where these transformations map from one twisted sector toanother (without mixing sectors)
S(g,h) : V(g,h) → V(h,g) (342)
T (g,h) : V(g,h) → V(g,gh). (343)
For example, theG-crossed modular transformations for
G = Z2 = 0, 1 take the block form
S =
S(0,0) 0 0 0
0 0 S(0,1) 0
0 S(1,0) 0 0
0 0 0 S(1,1)
, (344)
T =
T (0,0) 0 0 0
0 T (0,1) 0 0
0 0 0 T (1,0)
0 0 T (1,1) 0
, (345)
where the rows and columns are separated into(0, 0), (0, 1),(1, 0), and(1, 1) sectors, in that order.
Thus, imposing unitarity on the representations of the mod-ular S andT transformations amounts to imposing unitarityon their restricted actionsS(g,h) andT (g,h) for each(g,h)-sector individually. Since the system in the(g,h)-sector hasa ground state degeneracy
N(g,h) = dimV(g,h) = |Chg | (346)
equal to the number ofh-invariant topological charges inCg,it follows that requiring the modular transformations to beuni-tary gives the condition that
|Cgh| = |Ch
g |, (347)
whenevergh = hg. In particular, forh = 0, this gives us theimportant property
|Cg| = |Cg0 |, (348)
which says the number of topologically distinct types ofg-defects (i.e. the topological charge types inCg) is equal to thenumber ofg-invariant topological charges inC0.
We now wish to provide a projective representation of themodular transformations that are defined by theG-crossedUBTC data. Let us take the representation of the modulartransformations defined by
S(g,h)agbh
=Sagbh
Uh(a, a; 0)(349)
T (g,h)agbg
= ηa(g,h)θagδagbg . (350)
Recall thatSagbh is the topologicalS-matrix defined in (300).It is convenient for us to also define theG-crossed “chargeconjugation” transformation
C =⊕
(g,h):gh=hgC(g,h) (351)
C(g,h) : V(g,h) → V(g,h) (352)
|a(g,h)g 〉l =∑
b∈Chg
C(g,h)agbg
|b(g,h)g 〉−l (353)
C(g,h)agbg
=1
Uh(b, b; 0)ηb(h, h)δagbg . (354)
49
Given the properties derived for a generalG-crossed UBTCin Sec. VI, we can obtain the relation∑
w,x,y,z
T (g,h)agwg
S(g,gh)wgxgh
T (gh,g)xghygh
S(gh,h)yghzh
T (h,hg)zhbh
= Θ0S(g,h)agbh
(355)from Eq. (324), whereΘ0 = 1
D0
∑c∈C0
d2cθc, the relation
S(g,h)agbh
=∑
x
[S(h,g)xhag
]∗C
(h,g)xhbh
(356)
from Eq. (308), and the relation∑
x
C(g,h)agxg
C(g,h)xgbg
= δagbg , (357)
from Eq. (260). Thus, without imposing unitarity of the topo-logicalS-matrix nor any other extra conditions on aG-crossedUBTC, the transformations defined by Eqs. (349), (350), and(354) obey the relations
(ST )3 = Θ0S2 (358)
S = S†C (359)
C2 = 11. (360)
We can also show that
CS = SC (361)
using Eqs. (305) and (309), and that
CT = T C (362)
using Eqs. (293) and (325)-(327).It is clear from these relations that all that is needed for
these operators to provide a projective representation of themodular transformations is to impose a condition on the topo-logical S-matrix that makes the modular operatorS definedby Eq. (349) unitary, in which case Eq. (359) would become
S2 = C. (363)
We can see that requiringS to be unitary is equivalent tothe condition that the topologicalS-matrix of theG-crossedUBTC gives unitary matrices when it isG-graded, by whichwe mean that for any fixed pair of group elementsg andh,the matrix defined bySagbh with indicesa ∈ Ch
g andb ∈ Cgh
is a unitary matrix. Thus, when the topologicalS-matrix of aG-crossed UBTCC×
G is G-graded unitary (in the fashion de-scribed here), we say thatC×
G isG-crossed modular or that itis aG-crossed modular tensor category (MTC).
We note that, for a modular theory, the quantity
Θ0 =1
D0
∑
c∈C0
d2cθc = ei2π8 c− (364)
is a phase related toc− which is the chiral central charge ofthe topological phase described by the UMTCC0. Thus, wecan ascribe the same chiral central charge to theG-crossedextensions of a topological phase.
It follows from the definition ofG-crossed modularity thattheC0 subcategory of aG-crossed MTC is a MTC. As previ-ously mentioned, the converse is also true, as can be shown bycombining highly nontrivial theorems from Refs. [80, 133].Thus, the conditions of modularity of a UBTC and itsG-crossed extensions are equivalent, i.e.C×
G is a G-crossedUMTC if and only if C0 is a UMTC.
We note that, just as in the case of a MTC, we could actuallyobtain a linear (rather than projective) representation ofthemodular transformations on the torus if we instead defined theDehn twist transformation to be given by
T (g,h)agbg
= e−i2π24 c−ηa(g,h)θagδagbg , (365)
as this would give the relation(ST )3= S2. This convention
may be more useful when performing concrete calculations orphysical simulations on the torus. However, it is not generallypossible to trivialize the projective phases for the representa-tions of modular transformations for higher genus surfaces,so we will not generally include the central charge dependentphase.
We also note that the quantities representing theG-crossedmodularS andT transformations defined here are not gaugeinvariant, except in the(0,0)-sector (which was also the casewith the topological twists andS-matrix in theG-crossed the-ory). In particular, while they are invariant under vertex basisgauge transformations, they transform under symmetry actiongauge transformations as
S(g,h)agbh
=γb(g)
γa(h)S(g,h)agbh
(366)
T (g,h)agbg
=γb(gh)
γa(h)T (g,h)agbg
. (367)
This is not unexpected, since these two modular transfor-mations map the(g,h)-sector to the(h, g)-sector and the(g,gh)-sector, respectively, and there is no well-definedgauge invariant notion of comparing distinct superselectionsectors, i.e. there is no canonical map between different sec-tors. (This is related to the fact that the defects are extrinsicobjects which define different superselection sectors for dif-ferent group elements and for which one should not expectoverall phases to be well-defined.) As such, it is important tobe careful with the details of how one sets up configurationsand analyzes their modular transformations when working ona torus or higher genus system.
On the other hand, we may expect some modular trans-formations to be gauge invariant [in addition to the(0,0)-sectors]. From Eqs. (366) and (367), and the fact thatS andT generate the modular transformations on the torus, it fol-lows that a general modular transformationQ that maps the(g,h)-sector to the(g′,h′)-sector, i.e.
Q(g,h) : V(g,h) → V(g′,h′), (368)
transforms under symmetry action gauge transformations as
Q(g,h)agbg′ =
γb(h′)
γa(h)Q(g,h)agbg′ . (369)
50
From this expression, it is easy to see that (a) if a modulartransformationQ maps a(g,h)-sector to itself, thenQ(g,h)
agag isa gauge invariant quantity and (b) ifQ maps a(g,0)-sector to
itself, thenQ(g,0)agbg
is a gauge invariant quantity.For example, ifgn = 0, thenT n will map a (g,h)-sector
to itself, and the coefficients
[T n](g,h)agag= θnag
n−1∏
j=0
ηag(g,gjh) (370)
provide gauge invariant quantities of theG-crossed theory (foranyh that commutes withg). If g2 = 0, we see that
S(g,g)agag =
SagagUg(a, a; 0)
(371)
[ST S](g,0)agbg=∑
x0
Sagx0θx0
Sx0bg
Ug(x, x; 0)(372)
[T ST ](g,0)agbg
=θagSagbgθbgηb(g,g)
Ug(a, a; 0)(373)
are also gauge invariant quantities [the last two are, of course,not independent, given Eq. (358)].
C. Higher Genus Surfaces
When the system is on a genusg surface, the topologi-cal ground state degeneracy is more complicated. In gen-eral, it can be obtained by summing over the possible statesassociated with a fusion tree of topological charge lines thatpass through either the interior or exterior of the surface,andwhich encircle independent non-contractible cycles, as shownin Fig. 15. For a UMTC (without defects), this leads to theground state degeneracy
Ng =∑
b,z,c∈CN c12z1z2N
c123c12z3 · · ·N0
c1...g−1zg
g∏
j=1
Nzjajaj
= D2g−20
∑
x∈Cd−(2g−2)x , (374)
where the evaluation may be carried out using the Verlindeformula.
For a topological phase with defects, described by aG-crossed UMTCC×
G , the system on a genusg surface may havedefect branch lines around any non-contractible loop, similarto the case of the torus. In this case, we can label the distincttwisted sectors of a genusg surface by2g group elements,gj,hj, j = 1, . . . , g, each of which corresponds to a de-fect branch line wrapping around an independent generatingcycle. We write the corresponding ground state subspace asVgj ,hj. The group elementsgj ,hj must satisfy relationsto ensure that the corresponding defect branch lines can closeconsistently upon themselves. In this case, we do not requirethatgj andhj necessarily commute. When they do not, one ofthe branch lines at a given handle may have its group elementlabel change as it crosses the other branch line. If we pick
h1h2 h3 h4
g1 g2 g3 g4
a1 a2 ag−1 ag
z1 z2 zg−1 zg
h1 h2hg−1 hg
c12c1...g−1
· · ·
· · ·
FIG. 15: The twisted sectors on a genusg surface can be labeled by2g group elementsgj ,hj for j = 1, . . . , g, which are ascribed tothe defect branch lines around two independent non-contractible cy-cles associated with thejth handle. In this case, one does not requirethatgj andhj commute, so one of the branch lines at a given han-dle may change as it crosses the complementary branch line atthathandle. We pick thehj-branch lines to close around their cycles un-changed, while thegj branch lines transform intoh−1
j gjhj branchlines when they cross thehj-branch. This requires akj-branch line,wherekj = gjh
−1j g−1
j hj , to enter the handle to cancel the left-over branch. Similarly, theaj ∈ Cgj charge lines used to define
basis state may also transform nontrivially ashjaj when it crossesthehj -branch loop. This requires a line of chargezj ∈ Ckj withN
zj
ajhj aj
6= 0 to enter the handle to cancel the leftover topological
charge. Thezj charge lines from different handles form a fusiontree. These charge line configurations, together with the fusion ver-tex state labels, provide a basis of states for the genusg surface inthegj ,hj-sector.
thehj-branch lines to close around their cycles unchanged,then thegj branch lines transform intoh−1
j gjhj branch lineswhen they cross thehj-branch. When this branch line loopsback on itself, we are left with a nontrivial branch line, whichrequires akj-branch line, where
kj = gjh−1j g−1
j hj , (375)
to enter the handle and cancel this off, as shown in Fig. 15.Thus, while we do not requiregj andhj to commute, we
do, however, require that the product of their commutatorskjequals the identity group element, that is
g∏
j=1
kj =
g∏
j=1
gjh−1j g−1
j hj = 0, (376)
as this condition is necessary for a consistent configuration ofbranch lines that do not contain any free endpoints, as can beseen from Fig. 15.
A basis for the ground state subspaceVgj,hj of thegj,hj-sector can be given in terms of fusion trees of topo-logical charge lines passing through the interior of the sur-face, as shown in Fig. 15. Using the choice where thehj-branch lines loop around their cycles unchanged, we may havea charge lineaj ∈ Cgj that winds around the complemen-
tary cycle of thejth handle and transforms intohjaj when
51
it crosses thehj -branch loop. In closing back on itself, thistopological charge loop must fuse with a possibly nontrivialline of chargezj ∈ Ckj such thatNzj
ajhjaj
6= 0. The charge
zj lines from the different handles then form a fusion tree thatmust terminate in the trivial topological charge.
In particular, the basis states described in this way can bewritten as
g⊗
j=1
|aj , hjaj ; zj, µj〉|c1...j−1, zj; c1...j , ν1...j〉, (377)
for all possible values (allowed by fusion) of topological
chargesaj ∈ Cgj , zj ∈ Ckj , andc1...j ∈ Clj for lj =j∏i=1
ki,
and fusion vertex basis labelsµj = 1, . . . , Nzj
ajhjaj
, and
ν1...j = 1, . . . , Nc1...jc1...j−1zj . We setc∅ = c1...g = 0 (which
givesc1 = z1) andlg = 0, in order to letj = 1, . . . , g for allthese quantities.
We note that the states in Eq. (377) may transform non-trivially under the symmetry action ofq ∈ G. In particular,
ρq : Vgj ,hj → Vqgjq−1,qhjq−1 (378)
|ψ〉 7→ ρq(|ψ〉) (379)
This symmetry action will play a crucial role whenG is pro-moted to a local gauge invariance.
In order to obtain the number of ground states in thegj,hj-sector
Ngj,hj = dim Vgj,hj, (380)
we can sum over the fusion channels
Ngj ,hj =∑
aj∈Cgj
zj∈Ckj
N0z1z2···zg
g∏
j=1
Nzj
ajhjaj
, (381)
where
N0z1z2···zg =
∑
c1...j∈Clj
N c12z1z2N
c123c12z3 · · ·N0
c1...g−1zg (382)
is the number of ways the topological chargesz1, . . . , zg canfuse to0. We can evaluate these expressions using theG-crossed Verlinde formula Eq. (332), together withG-gradedmodularity and other properties that we derived for theS-matrix in Sec. VI D, which yields
Ngj ,hj = D2g−20
∑
x∈Cgj,hj0
d−(2g−2)x
×g∏
j=1
ηx(hj , gj)ηx(gj , hjgjhj)
ηx(gj , gj)ηx(hj gjhj , hj)ηx(lj−1,kj), (383)
whereCgj,hj0 is the set of all topological charges inC0 that
aregj-invariant andhj-invariant for allj = 1, . . . , g. Whenhj = 0 for all j, this expression simplifies to
Ngj ,hj=0 = D2g−20
∑
x∈Cgj0
d−(2g−2)x , (384)
g1 g2 g3
h1h2 h3
h
FIG. 16: WhenG = Zp for p prime, any twisted sector can bemapped via Dehn twists to the sector with a single twist correspond-ing to a elementh ∈ Zp, which generates the group. Thus, allgj ,hj-sectors that are not completely trivial must have the sameground state degeneracy.
which clearly satisfiesD2g−20 ≤ Ngj ,hj=0 ≤ D2g
0 . FromEq. (383), we see that, in general, the genusg degeneracyNgj ,hj ≤ N0j ,0j, and generally scales asNgj ,hj ∼D2g
0 in the largeg limit, regardless of the twisted sector. Thisprovides a physical interpretation of the total quantum dimen-sionD0 = Dg of eachCg subsector.
We note that another physical interpretation of the totalquantum dimensionD0 is given by the topological entangle-ment entropy [134, 135]. One can use the properties ofG-crossed modularity to compute the topological entanglemententropy of a region, following the arguments of Ref. [134].Unsurprisingly, this yields the same result as for MTCs thatStopo = −n logD0, wheren is the number of connected com-ponents of the boundary of the region in question, regard-less of the number of branch lines passing through the region.There are also anyonic contributionsSa = log da to the en-tanglement entropy when there are quasiparticles org-defectswithin the region whose collective topological charge isa (seealso Ref. [18]).
1. Dehn twists on high genus surfaces
Another powerful method of computingNgj,hj on agenusg surface is to make use of modular transformations.Similar to the case of the torus, we can define operators usingthe data of aG-crossed UMTCC×
G , that provide a projectiverepresentation of the modular transformations of the genusgsurface. We will not go into these details here, but, instead,will simply utilize the property that the modular transforma-tions can be used to interchange, combine, and twist the var-ious non-contractible cycles of the surface, as we saw for thetorus. Unitarity of the modular transformations implies thatwhen two different twisted sectorsgj,hj andg′
j,h′j can
be related by such modular transformations, they must havethe same ground state degeneracy.
As a simple example, let us considerG = Zp and takegto be a generator of this group. Whenp is prime, any elementh ∈ Zp generates the group. In this case, every nontrivialgj,hj-sector can be related by Dehn twists to the sectorwith only a singleh-defect branch line wrapped around a sin-gle cycle (see Fig. 16). The proof of this statement, and somegeneralizations, is given below.
Specifically, in the following we show that for a genusgsurface, whenG = ZN , the (N2g − 1) non-trivial twistedsectors can all be obtained by Dehn twists from a small “gen-
52
B3
A3A2
A1
B1 B2C1 C2
FIG. 17: Non-contractible cocycles on ag = 3 surface.
erating” set of generating sectors (the caseN = 2 wasproven in [102]). We start by examining a torus (g = 1).Since we are considering a cyclic group, group multiplica-tion will be denoted additively. We label the twisted sectorby(g,h) = (m,n) wherem,n = 0, 1, . . . , N − 1. There areN2 − 1 nontrivial sectors in total.
An arbitrary modular transformation acts on a twisted sec-tor (m,n) by a SL(2,Z) matrix
[a b
c d
](m
n
)=
(am+ bn
cm+ dn
). (385)
Heread − bc = 1, a, b, c, d ∈ Z. Letting r = gcd(m,n),we now show that(m,n) can be obtained from(r, 0) by amodular transformation. To see this, we seta = m
r , c = nr
in the SL(2,Z) matrix. We then need to findb, d such thatmr d − n
r b = 1. Since gcd(mr ,
nr
)= 1, this equation has
integral solutions.Next, we show that for arbitrarym, (m, 0) can be obtained
from (s, 0) wheres = gcd(m,N). From Eq. Eq. (385), wesee that we need to find an SL(2,Z) matrix with a = m
s andc = N
s . We need to find integersb, d such thatad−bc = ms d−
Ns b = 1, which is solvable since gcd(m,N) = s. Therefore
we have established thatthe twist sectors(r, 0), wherer isa divisor ofN , is a generating set.That is, the number ofgenerating twist sectors is equal to the number of divisors ofN .
We now consider a genusg surface. A similar reduction of ageneral twisted sector to a small number of generating twistedsectors is also possible. The inequivalent cycles associatedwith each handle are labeled byAi, Bi wherei = 1, . . . , g(see Fig. 17). The twisted sectors are now labeled by2g inte-gers (modN ) (m1, n1), . . . , (mg, ng). We note that apply-ing a Dehn twist alongC1 has the following effect:
A1 → A1
B1 → B1 + C1 = B1 −A1 +A2
A2 → A2
B2 → B2 −A1 +A2.
(386)
The configuration then becomes(m1, n1 − m1 +m2), (m2, n2 −m1 +m2), . . . .
The arguments from the genusg = 1 case above im-ply that by applying Dehn twists alongAi or Bi, wecan always map any general twisted sector to the form(m1, 0), (m2, 0), . . . , (mg, 0). If at least one of themi’s iscoprime withN , we can further perform Dehn twists to reducethe configuration to a twist along a single cycle. To see this,let
us assume gcd(m1, N) = 1. We can do an S transformation tomap to the configuration(m1, 0), (0,m2), . . . . After apply-ing k Dehn twists along−C1, we get(m1, km1), (0,m2 +km1), . . . . Since gcd(m1, N) = 1, there exists aksuch thatm2 + km1 ≡ 0 (modN), resulting in the sec-tor (m1, km1), (0, 0), . . . . This can be further reduced to(m1, 0), (0, 0), . . . by Dehn twists. A similar argument canbe applied in the case whenm1 = m2 = · · · = mg, withoutthe need to assume gcd(mi, N) = 1.
In particular, the above arguments imply that whenN isprime, then the general twisted sector can always be mappedto a sector with a single elementary twist along only one cycleof the genusg surface.
VIII. GAUGING THE SYMMETRY
We have, so far, studied the properties of the defects, whichcorrespond to extrinsically imposed (confined) fluxes of thesymmetry groupG, as described by aG-crossed theoryC×
G .In this section, we consider the nature of the resulting phasewhen the global symmetryG is promoted to a local gaugeinvariance – “gauging the symmetry.” This is also referredto as “equivariantization” in the mathematical literature. Aphysical consequence of our investigation is that the confinedg-defects become deconfined quasiparticle excitations of thegauged phase. We would like to understand how to obtain theproperties and basic data of the gauged theory, which is de-noted asC/G, from theG-crossed extensionC×
G of the UMTCC describing the original topological phase. We will see that,given the complete data of theG-crossed UMTCC×
G , we canobtain the quasiparticle content, fusion rules, and modulardata of the corresponding UMTCC/G.
On the other hand, we can consider the inverse of thisconstruction. Starting from the gauged theoryC/G, we cantune the interactions so that the “charged” matter, whichtransforms under irreducible representations ofG, condenses,and the gauge theory undergoes a continuous confinement-deconfinement transition into the Higgs phase. The resultingtopological order can be analyzed using the theory of topolog-ical Bose condensation [75], where the subcategory consistingof gauge charges ofG (referred to as Rep(G)) condenses. Inshort, all gauge fluxes inC/G become confined, while the de-confined remnants give rise toC. The algebraic theory of topo-logical defects that we have developed in this paper provides acomplete description, in particular providing the braiding andmodular transformations, of the sectors that are confined inthe condensation ofRep(G), which is called theT -theory inRef. [75].
We summarize the relation betweenC, C×G , andC/G by the
following diagram:
C C×G C/G
Defectification
Confinement
Gauging
Condensation
In this section, we consider only finite symmetry groups
53
G. We will first examine the problem of how to modifya microscopic Hamiltonian that realizes a topological phaseC and has an on-site symmetryG in a manner that gaugesthe symmetry and realizes the topological phaseC/G. Thenwe will study how to derive the mathematical properties ofthe gauged phase’s UMTCC/G from the correspondingG-crossed UMTCC×
G .
A. Microscopic Models
Gauging a symmetry of a microscopic Hamiltonian is awell-known notion in physics. However, a gauge theory doesnot, in general, have a local Hilbert space. Suppose we aregiven aG-symmetric microscopic HamiltonianH that (1)is defined on a Hilbert space that decomposes into a tensorproduct of local Hilbert spaces on each site, (2) has localinteractions, and (3) realizes a topological phaseC at long-wavelengths. Here, we address the question of whether or notwe can produce a new HamiltonianHG that also satisfies (1)and (2) above, but realizesC/G at long-wavelengths.
We will briefly describe the case whereG = Z2. Supposethat the Hamiltonian consists of nearest neighbor interactionson a two-dimensional lattice. We assume that there is a finite-dimensional bosonic Hilbert space at each site of the lattice,and there is a global on-siteZ2 symmetry withRg =
∏j R
(j)g .
Such aZ2 symmetric Hamiltonian can generically be writtenas
H =∑
〈ij〉Jαβ+,ijO+,α
i O+,βj + Jαβ−,ijO−,α
i O−,βj
+∑
i
mαi O+,α
i + H.c., (387)
where O±,αj are a complete set ofZ2 even/odd local
operators at sitej. In particular, these operators satisfyR
(j)g O±
j R(j)−1g = ±O±
j .Now, let us introduce a two-dimensional Hilbert space on
each bond〈ij〉 of the lattice. The gauged Hamiltonian is de-fined as
HZ2 =∑
〈ij〉Jαβ+,ijO+,α
i O+,βj +
∑
i
mαi O+,α
i
+∑
〈ij〉Jαβ−,ijO−,α
i O−,βj σzij + H.c.
−K∑
∏
〈ij〉∈
σzij − Γ∑
〈ij〉σxij − U
∑
+
R(i)g
∏
〈ij〉∈+
σxij .
(388)
We always assume thatU is the largest energy scale, whicheffectively imposes aZ2 analog of Gauss’s law in the low-energy Hilbert space:
∏〈ij〉∈+ σ
xij = R
(i)g . It is straight-
forward to extend the construction to Hamiltonians involvinglonger-range interactions.
We notice that the full gauged Hamiltonian (not just thelow-energy subspace) still preserves theZ2 symmetryRg. Inthe low-energy subspaceU → ∞ where the dynamics can be
described by aZ2 gauge theory with matter, the global sym-metry is enhanced to a local gauge symmetry generated byprecisely the local conserved quantityR(i)
g
∏〈ij〉∈+ σ
xij . The
gauged Hamiltonian has the feature that whenΓ = 0 andK,U are both much larger than any energy scale inH , thelow-energy spectrum without anyZ2 fluxes is identical to thatof H . However, the states must be projected to the gauge-invariant Hilbert space.
We now review the phase diagram of the gauge the-ory [136], focusing on the three parametersJ−,K and Γ.Three limiting cases can be easily identified. WhenJ−,Γ ≪K, the gauge field is in the deconfined phase. WhenJ− ≫K,Γ, the gauge theory is in the Higgs phase and theZ2
fluxes (i.e. visons) are linearly confined. IfΓ is dominant,Z2 charges are linearly confined. It is however well-knownthat the Higgs and the confinement phases are smoothly con-nected. Hence there are only two phases which are separatedby a second-order phase transition belonging to the 3D Isinguniversality class.
The above construction can straightforwardly be general-ized to the caseG = ZN . The generalization to a generalfinite groupG is technically more involved and will be left forfuture work.
B. Topological Charges and Fusion Rules ofC/G
We now turn to the derivation of the topological proper-ties of the gauged theory. The simplest information about thegauged theoryC/G that we can read off fromC×
G is the topo-logical charge content. The mathematical description of thiswas provided in Ref. [83].
For each topological charge (simple object)a ∈ C×G of the
G-crossed theory, we define its orbit underG to be the set ofcharges
[a] = ga, ∀g ∈ G. (389)
Heuristically, the reason for consideringG orbits is that, un-der theG action, all topological charges within an orbit mustcombine into a single object by “quantum superposition” oncethe global symmetry is promoted to a local gauge invariance.In this way, the original topological charges inC×
G becomeinternal degrees of freedom. In particular, if we ignore thetopological charge labels within eachCg and only focus on thegroup elements, the orbit would simply be a conjugacy classof G, which is what labels gauge fluxes in a discrete gaugetheory. Keeping track of the topological charge labels, it isclear that there can be multiple orbits associated with a givenconjugacy class ofG.
Additionally, we need to take into account the different rep-resentations of the symmetry, which thus allows us to includethe gauge charges and flux-charge composites. For this, wedo not consider the full symmetry groupG, but rather the sub-groups that keep the relevant topological charge labels invari-ant. More precisely, for a given[a], we choose a representativeelementa ∈ [a], and define its stabilizer subgroup
Ga = g ∈ G | ga = a . (390)
54
The topological charges ofC/G are then defined to be thepairs
([a], πa), (391)
whereπa is an irreducible projective representation ofGawith the factor set given byηa, i.e.
πa(g)πa(h) = ηa(g,h)πa(gh), g,h ∈ Ga. (392)
We will refer to such an irreducible projectiveηa-representation as anηa-irrep. The phasesηa(g,h) here areprecisely the projective symmetry fractionalization phases oftheG-crossed theory, defined in Sec. VI. Thus, we see thatthe dataηa are essential in defining the quasiparticles of thegauged theory.
In order for this definition of topological charge to be well-defined, the specific choice ofa within the conjugacy class[a]should not lead to essential differences in the correspondingprojective representations. To make this notion more precise,we first notice that conjugation byk ∈ G provides a canonicalisomorphism betweenGa andGka
k : Ga → Gka
g 7→ kg. (393)
Next, from Eq. (265), we see that, for group elementsg,h ∈Ga, we have the cocycle condition
ηa (h,k) ηa (g,hk)
ηa (g,h) ηa (gh,k)= 1, (394)
so ηa ∈ Z2(Ga,U(1)). From Eq. (269), we see that, forg,h ∈ Ga, we have the relation
ηka(kg,kh) =
ηa(k,kh)
ηa(h, k)
ηa(gh, k)
ηa(k, kgkh)
ηa(k,kg)
ηa(g, k)ηa(g,h)
= dεa,k(g,h)ηa(g,h), (395)
where we have defined the1-cochainεa,k ∈ C1(Ga,U(1))
to beεa,k(g) = ηa(k,kg)
ηa(g,k). Thus, when viewed in terms of
cohomology, we see that thek-action does not change the co-homology class ofηa, i.e.
[ηka(kg,kh)] = [ηa(g,h)] ∈ H2(Ga,U(1)). (396)
Moreover, it is clear that we then also have
[ηka(kg,kh)] = [ηa(g,h)] ∈ H2(Gka,U(1)). (397)
As discussed in Appendix B, this implies that there is a canon-ical one-to-one correspondence between the set ofηa-irreps ofGa and the set ofηka-irreps ofGka. We will write
kπa(kg) = εa,k(g)πa(g) (398)
to denote theηka-irrep ofGka which is canonically isomor-phic to theηa-irrepπa of Ga under this mapping.
In this way, the topological charges ofC/G are essentially“flux-charge” composites, very much like the dyonic excita-tions in discrete gauge theories, but which also take into ac-count the possibility of having distinct types ofg-flux defects,corresponding to distinct topological charge valuesa ∈ Cg.
With this definition of the topological charges ofC/G, it isstraightforward to determine the corresponding quantum di-mensions. In particular, we just sum over the quantum di-mensions of all the charges in the orbit and multiply by thedimension of the attachedηa-irrep, so that([a], πa) has quan-tum dimension given by
d([a],πa) = da ·∣∣[a]∣∣ · dim(πa), (399)
whereda is the quantum dimension ofa (which is the samefor all a ∈ [a]),
∣∣[a]∣∣ the number of elements in the orbit[a],
and dim(πa) the dimension of theηa-irrepπa.Having specified the topological charges ofC/G and their
quantum dimensions, it is straightforward to prove that thetotal quantum dimension is
DC/G = |G| 12DCG = |G|D0. (400)
For this, we first consider the differentηa-irreps of the stabi-lizer subgroupGa of a in a given orbit[a]. It is known that∑πa
|dim(πa)|2 = |Ga| for suchηa-irreps, as shown in Ap-pendix B. With this, and the fact that
∣∣[a]∣∣|Ga| = |G|, we
obtain the result
D2C/G =
∑
([a],πa)∈C/Gd2([a],πa)
=∑
([a],πa)
d2a∣∣[a]∣∣2|dim(πa)|2 =
∑
[a]
d2a∣∣[a]∣∣2|Ga|
= |G|∑
[a]
d2a∣∣[a]∣∣ = |G|
∑
a∈C×G
d2a
= |G|D2CG = |G|2D2
C0. (401)
The fusion rules for the topological charges ofC/G havealso been recently described in the mathematical litera-ture [83]. To obtain these, we need to understand both howto fuse twoG-orbits and how to fuse twoηa-irreps. For ped-agogical reasons, we will give a heuristic discussion to justifythe fusion rules ofC/G before presenting the actual expres-sion.
We first consider a very coarse version of the problem. Inparticular, we suppress the topological charge label associatedwith an orbit and multiply two conjugacy classesC1 andC2
of G. For this, we first form the product set
g1g2 |g1 ∈ C1,g2 ∈ C2,
which can be equivalently expressed using representative ele-mentsg1 ∈ C1 andg2 ∈ C2 as
hg1h−1kg2k
−1 |h ∈ G/Ng1 ,k ∈ G/Ng2,
where
Ng = h ∈ G |gh = hg, (402)
55
denotes the centralizer ofg in G. Now the problem isto decompose the product set into conjugacy classes. Tothis end, we observe that ifh′ = lh and k′ = lk, thenh′g1h
′−1k′g′
2k′−1 = l(hg1h
−1kg2k−1)l−1, i.e. the two
elements are in the same coset. Hence, we are naturally led toconclude that the conjugacy classes contained in the productset are given by the coset of diagonal left multiplication onG/Ng1 ×G/Ng2 , which is the double cosetNg1\G/Ng2 .
We now return to the problem of the fusion of two orbits[a] and [b], neglecting for the moment theηa-irreps attachedto them. Selecting representative elementsa ∈ [a] andb ∈[b], the fusion of the two orbits give a direct sum of all theelements in the set
ρg(a)× ρh(b) |g ∈ G/Ga,h ∈ G/Gb,
where we take the coset overGa andGb here, since thesesubgroups do not modify the corresponding labels. We nowneed to decompose this set further intoG-orbits. For this, wehave the similar property that ifg′ = kg andh′ = kg, then
ρg′(a)× ρh′(b) = ρk(ρg(a)
)× ρk
(ρh(b)
)
= ρk(ρg(a)× ρh(b)
). (403)
This essentially says that the fusion channels ofρg′(a) ×ρh′(b) are exactly the image of those ofρg(a) × ρh(b) un-der the action ofk. Therefore, fusion of orbits correspond tothe equivalence classes ofG/Ga ×G/Gb under diagonal left(or right) multiplication, which is known to be isomorphic tothe double cosetGa\G/Gb.
Next, we consider how theηa-irreps attached to the defectsshould be combined. Naıvely, one would expect that we justtake the tensor product of the representations and decomposeit as a direct sum of irreps. However, an important subtlety
in this case is that the fusion/splitting spaces of the defectscan transform nontrivially under the symmetry group action,and this should also be taken into account in the fusion. Moreexplicitly, we consider the fusion/splitting vertex statespacesVcghagbh
andV agbhcgh , and we define the stablizer subgroup forthis space asH(a,b;c) = Ga ∩Gb ∩Gc. The symmetry action(sliding moves) consistency Eq. (259) tells us that
∑
λ,δ
[Ul(a, b; c)]µλ[Uk(a, b; c)]λν
=ηc(k, l)
ηa(k, l)ηb(k, l)[Ukl(a, b; c)]µν (404)
for k, l ∈ H(a,b;c). We notice that theU transformations canbe thought of as being associated with the action on the split-ting spacesV abc , while the transposeUT corresponds to theaction on the fusion spacesV cab, as seen in Eqs. (255) and(256). The symmetry action consistency implies thatUT forma projective representation ofH(a,b;c), with a factor set givenby
κk,l(a, b; c)−1 =
ηc(k, l)
ηa(k, l)ηb(k, l)(405)
restricted tok, l ∈ H(a,b;c). We will denote this projectiverepresentation ofH(a,b;c) byUT asπ(a,b;c) and its character isgiven by
χπ(a,b;c)(k) =
∑
µ
[Uk(a, b; c)
]µµ. (406)
With the above discussion as justification, we present theformula for the fusion coefficients of theC/G MTC [83]
N([c],πc)([a],πa)([b],πb)
=∑
(t,s)∈Ga\G/Gbm(πc∣∣H(ta, sb;c)
, tπa∣∣H(ta, sb;c)
⊗ sπb∣∣H(ta, sb;c)
⊗ π(ta, sb;c)
), (407)
where H(ta, sb;c) = Gta ∩ Gsb ∩ Gc and the notationπ∣∣H(ta, sb;c)
means the restriction of the irrepπ to the sub-
groupH(ta, sb;c). As we discussed above, the tensor producttπa∣∣H(ta, sb;c)
⊗ sπb∣∣H(ta, sb;c)
⊗ π(ta, sb;c) has the factor set
given by
ηa(k, l)ηb(k, l)κk,l(a, b; c)−1 = ηc(k, l) (408)
for k, l ∈ H(ta, sb;c), which is precisely the same factor setasπc
∣∣H(ta, sb;c)
. We note that the restriction of an irrep to a
subgroup is not necessarily an irrep of the subgroup. Finally,m(·, ·) is a sort of integer-valued inner product that, in somesense, measures the multiplicity of the entries with respect toeach other. If one of the entries is an irrep, then this multi-plicity function simply counts the number of times this irrep
occurs in the other entry’s irrep decomposition. However, thegeneral description of the multiplicity function is more com-plicated than the statement that it counts the number of timesone entry occurs in the other. The precise definition of thismultiplicity functionm is given in Appendix B. For practi-cal purposes, it may be computed in terms of the projectivecharacters of the projective representations, as in Eq. (B8).
The formula in Eq. (407) may appear obtuse without someexperience in using it for concrete computations. For this,werefer the reader to Sec. X, where this formula is utilized toderive the fusion rules of the gauged theory for several exam-ples.
As the first application of this formula, we determinethe topological charge conjugate (antiparticle) of([a], πa).It should be clear that if([b], πb) is the charge conjugate
56
([a], πa), then[b] = [a], since, for eacha ∈ [a], there must bean elementb ∈ [b] such thatN0
ab 6= 0. Regarding theηa-irrepof the conjugate charge, a natural guess would be the conju-gate irrepπ∗
a, sinceπa ⊗ π∗a = 11⊕ · · · . However, the factor
set ofπ∗a is η∗a, which is in general only gauge-equivalent to
ηa. In fact, from the symmetry action consistency Eq. (259),we have the relation
ηa(k, l)ηa(k, l) =Ukl(a, a; 0)
Uk(a, a; 0)Ul(a, a; 0), (409)
for k, l ∈ Ga. It follows that we should define the chargeconjugate’s irrep to be
πa(k) = Uk(a, a; 0)−1π∗
a(k). (410)
This is, indeed, anηa-irrep ofGa, i.e. it has the factor setηa.Thus, the topological charge conjugate of([a], πa) ∈ C/G is
([a], πa) = ([a], πa) (411)
with a the charge conjugate ofa ∈ C×G andπa the ηa-irrep
of Ga defined in Eq. (410). We can verify this by plugging([a], πa) and([a], πa) into Eq. (407), where we would find thatthe tensor product in the second entry ofm simply becomesπa ⊗ π∗
a which contains the trivial representation11 preciselyonce.
C. Modular Data of the Gauged Theory
The basic data of a MTC can be conveniently organizedinto the modularS andT matrices or, equivalently, the fusionmultiplicities, quantum dimensions, and topological twists ofthe topological charges. In fact, it is a widely believed conjec-ture that this topological data uniquely characterizes theMTCdescribing a topological phase, i.e. that it uniquely specifiestheF -symbols andR-symbols, up to gauge equivalence. Assuch, we will simply focus on these quantities here.
First, we want to find the topological twists of the topolog-ical charges inC/G. As we have discussed above, a topo-logical charge inC/G has the form of a generalized “flux-charge” composite, the “flux” being aG-orbit of defects andthe “charge” being a projectiveη-irrep. Thus, we expect thatthe topological twist of such objects will receive a contributionfrom the defect’s twist (carrying over from theC×
G theory),as well as an Aharonov-Bohm type phase from the (internal)braiding of the object’s flux and charge around each other. Thelatter contribution is roughly given by the character of thepro-jective irreps, as it is in discrete gauge theories. Therefore, weconjecture the following formula for the topological twists oftopological charges inC/G
θ([a],πa) = θagχπa(g)
χπa(0). (412)
In this expression,θag is the topological twist ofag ∈ C×G and
χπa(g) = Tr[πa(g)
](413)
is the projective character of theηa-irrepπa (see Appendix B).χπa(0) = dim(πa) is equal to the dimension ofπa.
It is straightforward to see that this expression forθ([a],πa)is indeed equal to a phase. Specifically, sinceηag(g,h) =ηag(h,g) for all h ∈ Ga, it follows that πa(g)πa(h) =πa(h)πa(g). Using Schur’s lemma, we deduce thatπa(g) ∝11. Since the representations are unitary, it follows thatχπa (g)
χπa (0)
is aU(1) phase.We stress that the projective character depends on the par-
ticular factor setηa, not just the equivalence class to which itbelongs. While neitherθag norχπa(g) is individually invari-ant under the symmetry action gauge transformations, theirproduct actually is invariant under such gauge transforma-tions. More explicitly, under a symmetry action gauge trans-formation, as in Eq. (280), the projective character transformsas
χπa(g) = γ−1a (g)χπa(g) (414)
andθag = γa(g)θag . Thus,
χπa(g)θag = χπa(g)θag . (415)
We also notice that vertex basis transformations leave bothθagandχπa , and henceθ([a],πa) invariant.
We must also check thatθ([ag],πa) does not depend uponthe choice ofag ∈ [a]. Consider a different representativeelementkag with k ∈ G/Ga. In Eq. (286), we saw that
θkag =ηa(g, k)
ηa(k, kg)θag . (416)
As shown in the previous subsection, there is a canonical cor-respondence between the projective representations ofGka
andGa. Thus, we choose the projective representation forka to bekπa. According to Eq. (398), we have
χkπa(kg) =
ηa(k,kg)
ηa(g, k)χπa(g). (417)
This results in the relation
θkagχkπa(kg) = θagχπa(g), (418)
which demonstrates that the expression for the topologicaltwist is indeed independent of the choice ofa ∈ [a].
As a special case, we notice that if[a] ∈ C0, thenθ[a] = θa,which is expected from the theory of topological Bose con-densation.
Given our formula in Eq. (412) for the topological twists ofC/G, we now prove that the chiral central chargec− (mod8)of the gauged theory is the same as that ofC0. To see this, wefirst evaluate the Gauss sum for a specificG-orbit [a], sum-ming overηa-irreps∑
πa
d2([a],πa)θ([a],πa) =∑
πa
d2a∣∣[a]∣∣2|χπa(0)|2θag
χπa(g)
χπa(0)
= d2a∣∣[a]∣∣2θag
∑
πa
χπa(g)χπa(0)
= d2a∣∣[a]∣∣2θag |Ga|δg,0
= d2a|G| ·∣∣[a]∣∣θagδg,0.
(419)
57
From this result, we obtain
ΘC/G =1
DC/G
∑
([ag],πa)∈C/Gd2([a],πa)θ([a],πa)
=1
DC/G
∑
[ag]
d2a|G| ·∣∣[a]∣∣θagδg,0
=1
D0
∑
[a0]
d2a∣∣[a]∣∣θa =
1
D0
∑
a∈C0
d2aθa
= Θ0 = ΘC . (420)
ThusC/G has the same chiral central charge mod8 asC andC×G .
Given the topological twists and fusion rules, we essentiallyhave the complete modular data of the UMTCC/G, since thequantum dimensions can be obtained from the fusion rules andtheS-matrix is defined in terms of these quantities by Eq. (37).We now derive an expression forS
([a],πa)([b],πb)in terms of
theG-crossedS-matrix of C×G [where we use the topological
charge conjugate of([a], πa) to simplify the subsequent ex-pressions]
S([a],πa)([b],πb)=
1
DC/G
∑
([c],πc)
N([c],πc)([a],πa)([b],πb)
d([c],πc)θ([c],πc)
θ([a],πa)θ([b],πb)
=1
|G|D0
∑
(t,s)∈Ga\G/Gb
∑
[c]
∑
πc
dc∣∣[c]∣∣ θc tg sh
θtagθsbh
χπc(tg sh)
χtπa(tg)χsπb(
sh)
× n tπan sπb
|H( ta, sb;c)|∑
k∈H( ta, sb;c)
χ∗πc(k)χtπa(k)χsπb(k)
∑
µ
[Uk(tag,
sbh; c tg sh)]µµ. (421)
Herenπ ≡ χπ(0) = dim π, and we used Eqs. (399), (406), (407), (412), and (B8). We maychose to use any representatives ofthe topological charge orbits in this expression, but we have specifically chosen to usec ∈ [c] such thatc ∈ C tg sh (correspondingto the choicesag ∈ [a] andbh ∈ [b]) in order to make the evaluation more direct. The sum breaks into three parts: (1) a sumover(t, s) ∈ Ga\G/Gb, (2) a sum overG-orbits[c], and (3) a sum over irreducibleηc-representationsπc. We first carry out thesum overπc. In order to apply the orthogonality relation Eq. (B14), we notice that, inGc, tg sh by itself forms anηc-regularconjugacy class and its centralizer isGc. Thus, we can apply Eq. (B14) to evaluate the sum
∑
πc
χπc(tg sh)χ∗
πc(k) = |Gc|δtg sh,k. (422)
Sincek ∈ H( ta, sb;c), we conclude that in order to havek = tg sh (so that the sum is non-vanishing), we must havetg ∈ Gsb
andsh ∈ Gta. Using these properties to evaluate the sums overπc andk, we obtain
S([a],πa)([b],πb)=
1
D0
∑
(t,s)∈Ga\G/Gb
∑
[c]c∈Ctgsh
dcn tπan sπb
|H( ta, sb;c)|θc
θtaθsb
χtπa(tg sh)χsπb(
tg sh)
χtπa(tg)χsπb(
sh)
∑
µ
[Utg sh(ta, sb; c)]µµ, (423)
where we indicate the choicec ∈ Ctgsh on the[c] sum in order to reduce clutter. We further notice that
χtπa(tg sh) = Tr
[πta(
tg sh)]= Tr
[ηta(
tg, sh)−1πta(tg)πta(
sh)]= ηta(
tg, sh)−1χtπa(tg)
ntπa
χtπa(sh), (424)
where we have used the fact thatπta(tg) ∝ 11. There is a similar relation forχsπb(
tg sh). From these relations, we obtain
S([a],πa)([b],πb)=
1
D0
∑
(t,s)∈Ga\G/Gb
∑
[c]c∈Ctgsh
dc|H( ta, sb;c)|
θcθtaθsb
∑µ[Utgsh(
ta, sb; c)]µµ
ηta(tg, sh)ηsb(tg, sh)χtπa(
sh)χsπb(tg)
=1
D0
∑
(t,s)∈Ga\G/Gb
∑
[c]
dc|H( ta, sb;c)|
∑
µ,ν
[Rta sbc ]µν [R
sb tac ]νµχtπa(
sh)χsπb(tg), (425)
where we used theG-crossed ribbon identity Eq. (296) in the last step. (We can now drop thec ∈ Ctgsh, since this condition isimplicitly enforced by theR-symbols.)
Thus, we have found
S([a],πa),([b],πb)=
1
D0
∑
(t,s)∈Ga\G/Gb
∑
[c]
dc|H( ta, sb;c)|
Tr[R
ta sbc R
sb tac
]χtπa(
sh)χsπb(tg). (426)
58
By sliding a line over a double braid and applying Eq. (266), we can show (whenha = a andgb = b) that
Tr[R
kagkbh
kcR
kbhkag
kc
]=
ηb(g, k)ηa(h, k)
ηb(k, kg)ηa(k, kh)Tr[Ragbhc Rbhagc
]. (427)
Using Eq. (398), we also have
χkπa(kh) =
ηa(k,kh)
ηa(h, k)χπa(h), χkπb(
kg) =ηb(k,
kg)
ηb(g, k)χπb(g). (428)
Putting these together, we find the relation
Tr[R
kagkbh
kcR
kbhkag
kc
]χkπa(
kh)χkπb(kg) = Tr
[Ragbhc Rbhagc
]χπa(h)χπb(g), (429)
which shows that this quantity is invariant underG action.Finally, we carry out the sum over the orbits[c], replacing it with a sum over the actual topological chargesc ∈ C×
G to obtain
S([ag],πa)([bh],πb)=
1
D0
∑
(t,s)∈Ga\G/Gb
∑
[c]
1
|G|∑
k∈G
dkc
|H( ta, sb; c)|Tr[R
kta ksbkc R
ksb ktakc
]χktπa(
ksh)χksπb(ktg)
=1
|G|D0
∑
(t,s)∈Ga\G/Gb
∑
[c]
∑
k∈G/H( ta, sb; c)
dkcTr[R
kta ksbkc R
ksb ktakc
]χktπa(
ksh)χksπb(ktg). (430)
We write k ∈ G/H( ta, sb; c) as k = lk1 where k1 ∈ Gta ∩ Gsb ∩ G/H( ta, sb; c) ≡ M(ta,sb;c) and l ∈[G/H( ta, sb; c)]/M(ta,sb;c) ≡ L(ta,sb). We purposefully drop the indexc in the definition ofL, sinceL contains cosets ofelements that at least change one ofta andsb, without referencing toc. In other words,k1 are all elements inG/H( ta, sb; c) thatkeep bothta andsb invariant and by definition necessarily transformsc nontrivially within the same orbit. Once we sum overthosek1 and[c], we actually have a sum over allc in ta× sb:
S([ag],πa)([bh],πb)=
1
|G|D0
∑
(t,s)∈Ga\G/Gb
∑
l∈L( ta, sb)
∑
[c]
∑
k1∈M(ta,sb;c)
dlk1cTr[R
lta lsblk1c R
lsb ltalk1c
]χltπa(
lsh)χlsπb(ltg)
=1
|G|∑
(t,s)∈Ga\G/Gb
∑
l∈L( ta, sb)
Slta lsb χltπa(lsh)χlsπb(
ltg)
(431)
Now recall that the double cosetGa\G/Gb is defined as the equivalence classes of elements inG/Ga × G/Gb, under thediagonal left multiplication. Therefore carrying out the sum overl is equivalent to lift the double coset back toG/Ga ×G/Gb.Finally we arrive at the following expression:
S([ag],πa)([bh],πb)=
1
|G|∑
t∈G/Gas∈G/Gb
Sta sb χtπa(sh)χsπb(
tg).(432)
We can now use Eqs. (308), (410), and (B5) to rewrite this finalexpression as
S([ag],πa)([bh],πb) =1
|G|∑
t∈G/Gas∈G/Gb
Stag sbh χtπa(sh)χsπb
(tg). (433)
Thus, we have found that theS-matrix of the gauged UMTCC/G can be obtained from theS-matrix of the correspondingG-crossed UMTCC×
G by taking a linear combination ofS-matrix elements that is weighted by the projective charactersof the corresponding projective irreps.
In general, gauge-inequivalentG-crossed extensions al-ways lead to distinctC/G as topological gauge theories. How-
ever, when viewed as UMTCs (so that we neglect the origin ofthe charge and flux labels of the quasiparticles inC/G), dif-ferentG-crossed extensions can potentially result in the sameC/G. Examples of such phenomena have been noticed forgauging bosonic SPT phases in Refs. [52, 74].
59
D. Genusg Ground State Degeneracy
An alternative way of computing a number of propertiesof C/G is by computing the ground state degeneracyNg ofthe theory on a genusg surface. It is well-known that thisis related to the quantum dimensions ofC/G via the formula(which can be derived using the Verlinde formula)
Ng = D2g−2∑
A∈C/Gd−(2g−2)A . (434)
Therefore, knowledge ofNg for enough values ofg can beused to extract the quantum dimensionsdj for every topolog-ical chargeA ∈ C/G.
The ground state degeneracyNg of C/G can also be ob-tained from the genusg ground state degeneracy ofC×
G ,which was discussed in Sec. VII B, by projecting onto theG-invariant subspace of states. In other words, we consider everystate|ψ〉 ∈ Vgj ,hj for everygj ,hj-sector. As discussedin Sec. VII C, these states transform under theG action. Theprojection keeps only the subspace of states that are invariantunder thisG-action. That is, one takes
|ψG〉 =∑
g∈Gρg(|ψ〉), (435)
for each state|ψ〉, belonging to anygj ,hj-sector of theG-crossed theory. The ground state degeneracyNg is then thedimension of the space spanned by suchG-invariant states|ψG〉.
E. Universality Classes of Topological Phase Transitions
A wide class of quantum phase transitions between topo-logically distinct phases of matter can be understood in termsof the condensation of some set of “bosonic” quasiparti-cles [75, 106], i.e. those whose topological chargea has triv-ial topological twistθa = 1. In these cases, the topologicalproperties of the resulting phase can be derived from thoseof the parent phase – some of the topological charge values(quasiparticle types) become confined due to the new conden-sate, some are equated with other topological charges, relatedto each other by fusion with the condensed bosons, but other-wise go through the transition essentially unmodified, and oth-ers may split into multiple distinct types of topological chargewhen going through the transition. We note that the mathe-matics underlying these transitions was initially developed inRefs. [104, 105].
Most of the current understanding of such topological phasetransitions focuses on the formal mathematical structure,suchas the nature of the topological order on the two sides ofthe transition. However, another very important property ofa phase transition is its universality class. For the simplestcases, where only one bosona with fusion a × a = 0 con-denses, it has been shown that the resulting phase transitionscan be understood asZ2 gauge-symmetry breaking transi-tions [76, 78]. Here, we will extend these results to a more
general understanding of the universality classes of topologi-cal bose condensation transitions.
Let us consider a topological phase of matter described bya UMTC M that contains a subtheoryB, which is itself aUBTC (i.e. it contains topological charges that are closedunder fusion) with the following properties: (a) all the topo-logical twists are trivial, i.e. θa = 1, ∀a ∈ B, and (b)DSab = dadb, ∀a, b ∈ B. In other words, the subcategoryB is symmetric, which means braiding is completely trivial,i.e.Raa = RabRba = 11 for all a, b ∈ B.
When these conditions are satisfied, a theorem due toDeligne [137] guarantees thatB is gauge-equivalent to the cat-egory Rep(G) for some finite groupG. This category Rep(G)has its topological charges given by all irreducible linearrep-resentations ofG, with the fusion rules being precisely givenby the tensor product of the irreducible representations andtheF -symbols being given by the corresponding Wigner6j-symbols. The braiding of Rep(G) is symmetric, i.e. com-pletely trivial. We notice, however, that one generally doesneed the full knowledge ofF -symbols andR-symbols ofB tounambiguously recover the groupG from the representationcategory [138].
In such a case, one can always condense the quasiparticlesbelonging toB following the formal rules given in Refs. [75,106, 107]. LetC denote the phase obtained by condensingtheB quasiparticles inM. It was proven in Ref. [139] thatM can always be obtained by starting fromC, and gauginga symmetry groupG. This implies the following property ofthe topological phase transition:The universality class of topological quantum phase transi-tions corresponding to the condensation of a Rep(G) subsetof a UMTC can be understood as discrete gauge symmetrybreaking transitions associated with the finite groupG.
Since discrete gauge symmetry breaking transitions arewell-understood and can be simulated easily using numeri-cal methods or, in simple cases, through analytical methods,this means that we can immediately understand the criticalexponents for local correlations of this much wider class oftopological quantum phase transitions. [183]
IX. CLASSIFICATION OF SYMMETRY ENRICHEDTOPOLOGICAL PHASES
We have developed a general framework to understand theinterplay of symmetry and topological order in2 + 1 dimen-sions. Our work leads to a systematic classification and char-acterization of SET phases in2 + 1 dimensions, for unitarysymmetry groupsG, which describe on-site and/or transla-tion symmetries, based on inequivalent solutions of the de-fect theoryC×
G . Our formalism forC×G encapsulates in detail
the properties of the extrinsicg-defects and the way in whichsymmetries relate to the topological order. Below we will de-scribe the classification ofC×
G and discuss the relation to thePSG framework for classifying SET phases. The extension tocontinuous, other spatial (non-on-site), or anti-unitarysym-metries will also be be briefly discussed below.
60
A. Classification ofG-Crossed Extensions
One can, in principle, obtain allG-crossed BTCs by solv-ing the consistency equations. In practice, this can quicklybecome computationally intractable. Fortunately, addressingthis problem is aided by the theorems of Ref. [81], which clas-sify theG-crossed extensions of a BTCC0 for finite groupsG(and also extensions of fusion categories). In our paper, werestrict our attention to the case whereC0 is a UMTC.
We have already examined part of this classification in de-tail in our paper. The most basic part of the classification,discussed in Sec. III, is the choice of the symmetry action[ρ] : G → Aut(C0), which is incorporated as a fundamen-tal property of the defects of the extended theory.
The next part of the classification was discussed in Sec. IV,where we showed that, given a specific symmetry action[ρ],the symmetry fractionalization is classified byH2
[ρ](G,A).
This required that the obstruction class[O] ∈ H3[ρ](G,A) be
trivial [O] = [0], since, otherwise, there would be no solutions.More precisely, the symmetry fractionalization classes weregiven by the equivalence classes of solutions[w], which areelements of anH2
[ρ](G,A) torsor. This means distinct classesof solutions are obtained from each other by action of dis-tinct elements ofH2
[ρ](G,A). In particular, theUg(a, b; c) and
ηx(g,h) transformations of aG-crossed MTCC×G (or, rather,
their restriction to theC0 sector) are precisely the symme-try action transformations of fusion vertex states and symme-try fractionalization projective phases, respectively, that en-coded symmetry fractionalization. Similarly, theG-crossedconsistency relations of theUg(a, b; c) and ηx(g,h) trans-formations are precisely the corresponding consistency rela-tions that arose in the fractionalization analysis. Thus, theH2
[ρ](G,A) classification of symmetry fractionalization car-ries over to theG-crossed extensions ofC0, where the de-fects in the extended theory incorporate the symmetry actionthrough the braiding operations.
In this sense, the set of gauge inequivalentG-crossed MTCsthat are extensions of a MTCC0 with specified[ρ] is anH2
[ρ](G,A) torsor. By this, we mean that, given aG-crossed
MTC C×G , each element[t] ∈ H2
[ρ](G,A) specifies a potential
way of modifyingC×G to obtain a distinct, gauge inequivalent
G-crossed MTCC×G , with a different fractionalization class
[w] = [t × w]. From the above discussion, it is clear that animportant property of aG-crossed extension that is modifiedby [t] in this way is the symmetry action and fractionalizationthat is encoded in the defects, particularly their action withrespect to theC0 sector.
We can also see that, for a choice oft ∈ [t], theG-gradedfusion rules of the defects inC×
G are modified to become
ag × bh = t(g,h)×∑
cgh
Ncghagbh
cgh, (436)
so that the fusion coefficients of the modified theoryC×G are
given by
Ncghagbh
= Nt(g,h)×cghagbh
. (437)It follows from the2-cocycle condition ont that these mod-ified fusion coefficients automatically provide an associativefusion algebra. We note that such a modification may or maynot actually give a distinct fusion algebra. Clearly, the rest ofthe basic data ofC×
G will also be modified by[t], but we willnot go into these details here.
Importantly, while there is a different symmetry fraction-alization class[w] for each element[t] ∈ H2
[ρ](G,A), it isnot guaranteed that each[w] can be consistently extended todefine a fullG-crossed defect theoryC×
G . In fact, the symme-try fractionalization class defines a new obstruction classinH4(G,U(1)) [81]. Only when this obstruction class is triv-ial can aG-crossed BTCC×
G be constructed, as there wouldotherwise be no solutions to theG-crossed consistency condi-tions. When this obstruction vanishes, the classification the-orem established in Ref. [81] says that the remaining multi-plicity of G-crossed extensions (after specifiying[ρ] and[w])is classified byH3(G,U(1)). The set ofG-crossed exten-sions (with specified symmetry action and symmetry fraction-alization class) is anH3(G,U(1)) torsor in a similar sense asabove. In particular, given aG-crossed MTCC×
G , each ele-ment[α] ∈ H3(G,U(1)) specifies a way of modifyingC×
G toobtain a distinct, gauge inequivalentG-crossed MTCC×
G withthe same symmetry action and fractionalization class.
We now describe how one may modify a particularG-crossed theoryC×
G to obtain anotherG-crossed theoryC×G , for
a given[α] ∈ H3(G,U(1)). We first note that the bosonicSPT states for symmetry groupG are completely classified bythe elements[α] ∈ H3(G,U(1)), as discussed in Sec. X A.
We will denote these states as SPT[α]G . Then it is easy to see
that, for each[α], we can produce anotherG-crossed theoryby factoring in SPT states in such a way that the group elementlabels match up with those ofC×
G , i.e. we take the restrictedproduct
C×G = SPT[α]G × C×
G
∣∣∣(g,ag)
, (438)
where topological charges inCg from theG-crossed theory arepaired up withg-defects from the SPT. To be more explicit,for a choice ofα ∈ [α], that is, a3-cocycleα ∈ Z3(G,U(1)),and the choice of gauge, given in Sec. X A, that makes all thebraiding phases trivial for SPT[α]G , the basic data ofC×
G can bemodified as
61
Ncghagbh
= Ncghagbh
(439)[Fagbhckdghk
]
(e,α,β)(f,µ,ν)= α(g,h,k)
[Fagbhckdghk
]
(e,α,β)(f,µ,ν)(440)
[Ragbhcgh
]
µν=[Ragbhcgh
]
µν(441)
[Uk(ag, bh; cgh)
]
µν=
α(g,k, kh)
α(g,h,k)α(k, kg, kh)Uk(ag, bh; cgh) (442)
ηxk(g,h) =
α(g, gk,h)
α(g,h, hgk)α(k,g,h)ηxk
(g,h) (443)
to give the basic data ofC×G , which automatically satis-
fiesG-crossed consistency conditions. We note that, sinceα(g,h,k) = 1 if g, h, or k = 0, the transformationsUk(a0, b0; c0) = Uk(a0, b0; c0) and ηx0
(g,h) = ηx0(g,h)
with respect to theC0 sector are unchanged by the abovemodification. Thus, such modifications of aG-crossed theoryleaves the symmetry action[ρ] : G→ Aut(C0) and symmetryfractionalization class[w] fixed.
We believe modifications of this type precisely give theH3(G,U(1)) classification, or, in other words, they generateall gauge inequivalentG-crossed MTCs for a specified sym-metry action and symmetry fractionalization class.We refer tosuch distinctG-crossed theories with the same symmetry ac-tion and fractionalization class as having different defect as-sociativity classes.
It is straightforward to check that whenα ∈ B3(G,U(1))is a3-coboundary, i.e. when
α(g,h,k) = dε(g,h,k) =ε(h,k)ε(g,hk)
ε(gh,k)ε(g,h)(444)
for someε ∈ C2(G,U(1)), that the above modification of theG-crossed theory byα produces aC×
G that is gauge equivalentto C×
G through the vertex basis and symmetry action gaugetransformations
[Γagbhcgh
]
µν= ε(g,h)δµν (445)
γag(k) =ε(g,k)
ε(k, kgk). (446)
This establishes the fact that one should take a quotient byB3(G,U(1)), since such modifications are just gauge trans-formations. What remains to be shown is that every pair ofG-crossed extensions ofC0 with the same symmetry action[ρ] and fractionalization class[w] is related by such a modifi-cation for some3-cocycleα ∈ Z3(G,U(1)) and that distinctcohomology classes[α] give gauge inequivalent solutions. Wedo not establish this here, but note that it may be partially ver-ified (or wholly verified for simple enough examples) usinginvariants of theG-crossed or gauged theory, and it is true forall the examples we study in Sec. X. The classification is es-tablished in Ref. [81] by working at a higher category level,
with the subsectorsCg (which areC0 bimodules) playing therole of objects.
In summary, theG-crossed extensions of a MTCC0 for fi-nite groupG are classified by the symmetry action[ρ], thesymmetry fractionalization class[w], which is an element ofaH2
[ρ](G,A) torsor, and the defect associativity class, which
is an element of aH3(G,U(1)) torsor. This yields the clas-sification of2 + 1 dimensional SET phases for a system in atopological phase described by a UMTCC0 and an (on-site)global unitary symmetry described by a finite groupG. Basedon the classification theorem of Ref. [81], we believe that allof the inequivalentG-crossed extensions can be parameter-ized in this way.
1. On relabeling topological charges
Before we conclude this section, we emphasize one ad-ditional point regarding the equivalence of distinct solutionsof the consistency equations forC×
G . In some cases, differ-ent fractionalization classes[w] can be related to each otherby a relabeling of the topological charges inC0, i.e. thequasiparticle types. In these cases, one might naıvely thinkthat fractionalization classes that are related in this mannershould be identified as the same SET phase, but this is notcorrect. The different fractionalization classes classified byH2
[ρ](G,A) should be considered to be associated with dis-tinct SET phases, even if they are related by a relabeling ofthe topological charges inC0.
As a simple example that illustrates the main idea involvedhere, consider the case of theZ2 toric code model, for whichthe topological charges areI, e,m, ψ, and letG = Z2 withρ acting trivially on the topological charges (i.e. no permuta-tions). This example is examined in Sec.X G. In this case, thefractionalization is classified byH2(Z2,Z2×Z2) = Z2×Z2,which physically corresponds to whether thee andm quasi-particles carry fractionalZ2 charges. The fractionalizationclass wheree carries half-integerZ2 charge andm carries in-tegerZ2 charge is therefore related to the one wheree carriesintegerZ2 charge andm carries half-integerZ2 charge by therelabelinge ↔ m. Nevertheless, these are distinct phasesof matter. To understand why this is the case, first break the
62
globalG = Z2 symmetry and pick a single point in phasediagram (i.e. a single point in the space of Hamiltonians thatrealize theZ2 toric code model). At this point in the param-eter space, we can make a choice of labelingI, e,m, ψ foreach of the different types of topological excitations. Next,we can adiabatically vary the parameters of the Hamiltonianaway from this point, while staying in the same topologicalphase. Doing this allows us to extend the labelingI, e,m, ψof topological excitations to all other points in the phase di-agram that can be reached adiabatically without closing thegap. In other words, this provides a way of fixing the labelingof quasiparticle types, i.e. topological charges, in a consis-tent way throughout the (adiabatically connected) toric codephase.
Once the labeling of topological charges has been fixed ev-erywhere in the toric code phase space, we can then considerthe subset of the toric code phase space that isZ2-symmetric.TheZ2-symmetric phase diagram will break up into disjointregions, where different SETs cannot be continuously con-nected without either breaking theZ2 symmetry or closingthe energy gap and thus passing through a phase transition.The region wheree andm respectively have half-integer andintegerZ2 charge cannot be adiabatically connected to the re-gion wheree andm respectively have integer and half-integerZ2 charge.
From this simple example, we see that two SETs associ-ated with different fractionalization classes cannot be adiabat-ically connected, even if they can be related by relabeling thetopological charges inC0. It is also worth remarking that thephases are only distinct relative to each other, since the data oftheir associated defect theoriesC×
G can be related to each otherby relabeling the topological charges inC0. This is at the heartof the statement that the set of fractionalization classes is anH2
[ρ](G,A) torsor.In light of this discussion, we do not allow the relabeling of
the topological charges inC0 when determining whether twoG-crossed theories should be considered equivalent. How-ever, the argument given above does not apply to the possiblerelabeling ofG-defects. Specifically, when theG symmetryis broken, theg-defects are no longer well-defined objects.Consequently, we expect that distinctG-crossed theories thatcan be related by relabeling elements withinCg, for g 6= 0,should be considered physically equivalent. In Sec. X G 1, wediscuss an example where two solutions forC×
G related by dis-
tinct SPT[α]G are equivalent under such a relabeling of theg-defects. In other words, allowing the relabeling of defect typeswithin the sameCg sector (withg 6= 0) can relate solutionsassociated with distinct classes inH3(G,U(1)). This partic-ular example was discussed previously from a different per-spective, using Chern-Simons field theory, in Refs. [52, 140].Another example of this kind is discussed in Sec. X F.
B. Relation to PSG Framework
At this stage, it is worth understanding how the frame-work that we have developed for classifying and character-izing SETs relates to the projective symmetry group (PSG)
classification proposed in Ref. [38]. A complementary discus-sion can also be found in Ref. [51]. In the PSG formulation,a topological phase is considered with a low-energy descrip-tion in terms of a gauge theory with gauge groupH and aglobal symmetryG. Different PSGs are classified by differentmean-field solutions within a slave-particle framework [1]. Acrucial role is played by group extensions, labelledPSG, ofH by G, which mathematically meansPSG/H = G. It isnot clear whether the classification of different slave-particlemean-field solutions, as originally formulated in Ref. [38], isequivalent to classifying the group extensionsPSG such thatPSG/H = G. Nevertheless, each such mean-field solutionmust be described by such a group extension, even if the cor-respondence is not one-to-one. Here, we will briefly discussthe problem of classification of such group extensions, andcompare the results to our approach.
WhenG andH are both finite, the mathematical problem ofclassifying group extensions has the following solution [141].One first picks a mapρ : G → Out(H), where Out(H) isthe group of outer automorphisms ofH . Different group ex-tensions are then classified byH2
ρ(G,Z(H)), whereZ(H) de-notes the center ofH . [184] WhenH is finite and the topo-logical phase is fully described by a discreteH gauge the-ory, i.e. C0 is the (untwisted) quantum double ofH , then wesee that the classification of distinct group extensionsPSGforms only a subset of the classification that we have de-veloped in this paper. This is becauseZ(H) is only a sub-set of the Abelian anyonsA. [185] Furthermore, even whenone specifies the symmetry fractionalization class accordingtoH2
ρ(G,Z(H)), there are still additional possibilities for dis-tinct SETs, as indicated by theH3(G,U(1)) part of the clas-sification ofG-crossed extensionsC×
G . Through the simpleexample of a topological phase described by pure discreteHgauge theory, we see that these are also not captured by clas-sifying the different group extensionsPSG.
Another important distinction between the PSG approachand our approach is that the former requires knowledge ofH , which is, in general, not unique and difficult to determinewhen given a generic topological phase in terms of an UMTCC0.
Ref. [53] has proposed an alternative framework, besidesPSG, to classify the SET phases of quantum doubles of a dis-crete group (i.e. discrete gauge field theories). This classifica-tion is also incomplete for those classes of states, as it missesthe full set of symmetry fractionalization classesH2
[ρ](G,A)
described in this paper.
C. Continuous, Spatial, and Anti-Unitary Symmetries
The theory that we have developed in this paper is mostcomplete whenC is a UMTC, and the symmetryG is a finite,on-site unitary symmetry. However, much of the frameworkwe have developed applies more generally as well.
Our general discussion of the symmetry of topologicalphases in Sec. III and symmetry fractionalization in Sec. IVis valid for any general symmetryG. However, when spa-tial symmeries are considered, there may be additional con-
63
straints on what types of symmetry fractionalization are al-lowed [119–123]. We leave a systematic study of this for fu-ture work.
WhenG is continuous, one also requires additional condi-tions that the mapsρ : G → Aut(C) respect the continuity ofG by mapping all group elements in a single connected com-ponent ofG to the same element of Aut(C). The cochainsvalued inA, such as theO andw described in Secs. III andIV, should similarly respect the continuity ofG.
Similarly, the definition ofg-defects and the notion of topo-logically distinct types ofg-defects is valid (or can be straight-forwardly generalized) for any unobstructed unitary symme-try G, even if it is not discrete and on-site. It is unclear howto generalize the constructions and formalism of defects toinclude anti-unitary symmetries, as the complex conjugationoperation is inherently nonlocal (except when acting on prod-uct states and operators).
WhenG is not a finite group, our formalism forG-crossedUBTCs described in Sec. VI may still be applied as long asfusion is finite, meaning there are only a finite number of fu-sion outcomes when fusing two topological charges. The dis-cussion ofG-crossed modularity for generalG requires thefurther restriction that|Cg| be finite for allg, but again doesnot requireG to be finite.
WhenG is a continuous group, the consistency conditionsthat we have described in Sec. VI are not complete. In partic-ular, the basic data ofC×
G , such as theF , R, U , andη sym-bols, must somehow reflect the topology and continuity of thegroupG. For SPT states, which consist of the case wherethe original categoryC is trivial, it was argued in Ref. [47]that whenG is continuous the classification is given in termsof Borel cohomologyH3
B(G,U(1)). In our language, thisamounts to the condition that theF -symbols ofC×
G be Borelmeasurable functions on the group manifold. Therefore a nat-ural assumption is that SETs with continuous symmetryG areclassified by distinctC×
G , with the additional condition thatF , R, U , andη be Borel measurable functions on the groupmanifold. However, a detailed study ofG-crossed extensionsfor continuousG, in addition to the framework for gaugingcontinuousG, will be left for future work.
In the case whereG contains spatial symmetries, such astranslations, rotations, and reflections, we expect that the ba-sic data and consistency conditions forC×
G will be valid, al-though as mentioned above, additional constraints may needto be included that would impose additional important rela-tions on the data ofC×
G . A systematic study of these will alsobe left for future work.
Finally, we note that the classification theorems of Ref. [81]for C×
G and, in particular, the statement that distinctC×G are
fully classified by([ρ], [t], [α]) require thatG is a finite, on-site unitary symmetry.
X. EXAMPLES
In this section, we consider a number of examples, whichwe label by the initial anyon model (UBTC)C0 and the sym-metry groupG. We obtain the data of the correspondingG-
crossed UBTCsC×G by solving the consistency equations, us-
ing various properties and classification theorems when use-ful, and present as much of the basic data as is reasonable.We also present explicit derivations of the fusion rules andthemodular data of the corresponding gauged theoriesC/G.
The purpose of these examples is twofold: (1) to providethe basic data ofC×
G and C/G for some of the more inter-esting and perhaps more physically relevant models, and (2)to illustrate the different types of nontrivial issues and struc-tures that may arise when concrete calculations are performed.Most of the examples examined here have Abelian symme-try groupG, but we thoroughly consider an example with anon-Abelian symmetryG = S3. We use the final exampleto briefly present an example of a non-vanishingH3
[ρ](G,A)
obstruction. Partial results from some of the examples thatwe examine have also been obtained in previous works [6, 9–12, 47, 77, 92, 100, 102, 103, 142–144], though mostly usingdifferent methods.
In the following, we adopt the convention that the vacuumtopological charge is always referred to as either0 or I andthe identity element ofG is referred to as either0 or 11.
A. Trivial Bosonic State with G symmetry
Consider the case where the starting topological phaseC0is trivial in the sense that it only contains topologically trivialbosonic excitations, i.e.C0 = 0, but possesses a symmetrygroupG. This would describe a bosonic symmetry-protectedtopological (SPT) phase with symmetry groupG. In this casethe construction of the extended categoryC×
G is straightfor-ward. EachCg contains a single defect type, which will bedenoted byg. Fusion of defects is given by group multipli-cation, i.e.g × h = gh. Since the fusion categoryCG willappear elsewhere, we will refer to it asVecG. Mathemati-cally, this is the category ofG-graded vector spaces. It is awell-known result that the equivalence classes ofF -symbolsunder vertex basis gauge transformations are determined bythe3rd group cohomologyH3(G,U(1)) [95, 143]. Given a3-cocycleα ∈ Z3(G,U(1)), we define theF -symbols as
[F g,h,kghk ]gh,hk = α(g,h,k). (447)
As usual, we requireF g,h,k = 1 whenever any ofg,h,k is0, so we always impose this condition on the3-cocycleα. Wecan also always apply the symmetry action gauge transforma-tion to setRg,h
gh = 1 for all values ofg andh. (If these werenontrivial in this example, we would apply the symmetry ac-tion gauge transformationγg(h) = [Rg,h
gh ]−1 to remove anynontrivial braiding phases.) The corresponding braiding op-erators simply involve theG-action of group elements actingby conjugation. In this case, the correspondingUk andηk areuniquely determined by theG-crossed consistency equations
64
to be
Uk(g,h;gh) =α(g,k, kh)
α(g,h,k)α(k, kg, kh), (448)
ηk(g,h) =α(g, gk,h)
α(g,h, hgk)α(k,g,h). (449)
TheH3(G,U(1)) classification of theF -symbols of theG-crossed extensionsC×
G for generalC is therefore in one-to-one correspondence with the classification of 2D bosonic SPTstates with symmetry groupG developed in Ref. [47]. There-fore, we see thatclassifyingC×
G reproduces the classificationof bosonic SPT states.
TheG action onC×G (for C0 trivial) is obviously given sim-
ply by conjugation. Therefore, we immediately obtain thequasiparticle labels in the gauged theory as a pair([g], πg)
where[g] = hgh−1, ∀h ∈ G is a conjugacy class ofG(i.e. an orbit underG action) andπg is an irreducible projec-tive representation of the stabilizer group, i.e. the centralizerof a representative elementg ∈ [g]. If we consider trivialF -symbols on VecG, we see that allU andη can be set to1. Theanyon content thus agrees exactly with the well-known quan-tum double construction D(G) of G. In general, the gaugedtheory is a twisted quantum double Dα(G) [144–146].
For illustration, let us consider theG-crossed braiding forG = ZN . SinceG is Abelian, in the following we denotegroup multiplication inG by addition+. TheG-extension issimply VecαG equipped with a3-cocycle
α(a, b, c) = e2πip
N2 a(b+c−[b+c]N ), (450)
where we use the notation[a]N ≡ a modN . When there is noambiguity we will also sometimes drop the subscript and write[a] instead of[a]N . In this case we find it more illustrativeto choose a gauge in whichη ≡ 1. We find the followingsolutions to theG-crossed heptagon equations
Raba+b = e−2πip
N2 [a]N [b]N e2πimaN
b. (451)
We can get theU symbols
Uc(a, b; a+ b) = e−4πip
N2 c(a+b−[a+b]N )e2πiNc(ma+mb−ma+b),
(452)wherema ∈ Z. Clearly all thema’s are symmetry actiongauge redundancy. The topological twists and double braidare given by
θa = e−2πip[a]2NN2 e
2πiamaN , (453)
Raba+bRbaa+b = e−
4πip
N2 [a]N [b]N e2πi(mba+mab)
N . (454)
It is evident from these expressions thatma can be understoodas the number ofZN charges attached to the defecta, due tonon-universal local energetics. This explains why these solu-tions should be considered as being gauge equivalent, sinceinthe extended theoryZN charges are still part of the vacuumsector.
Let us consider the gauged theory. SinceG is Abelian, eacha ∈ C×
G is anG-orbit. They can also carry gauge charges la-beled again byn ∈ ZN . We therefore obtain|G|2 quasiparti-cles labeled by(a, n). Their fusion rules are
(a, n)× (a′, n′) (455)
=
([a+ a′]N ,
[n+ n′ − 2p
N(a+ a′ − [a+ a′]N )
]
N
),
where the additional gauge charges come from the nontrivialsymmetry action on the fusion state of the defects. The topo-logical twist of(a, n) is then
θ(a,n) = e−2πip[a]2NN2 e
2πinaN . (456)
These results agree exactly with the twisted quantum doubleDα(ZN ) [144–147].
B. Trivial Fermionic State Z(1)2 with G symmetry
Our next example is to consider a trivial fermionic topo-logical phase. Even though the fermion is a local excitationin such a case, it turns out to be useful to view it as a non-trivial element of the category and therefore to treat it as atopological charge. To describe such a situation, we use theUBTC with C0 = I, ψ whereI is the vacuum charge,ψ isthe fermion, andψ × ψ = I. C0 should be viewed thereforeas a topological abstraction of gapped fermionic systems withonly short-range entanglement.
TheF -symbols andR-symbols are
[Fψψψψ ]ψψ = 1, RψψI = −1. (457)
Notice that the braiding is not modular, since theS matrix
S =1√2
[1 1
1 1
](458)
is singular. We note that our results on classification of sym-metry fractionalization do not directly apply, since modularitywas an essential part of the argument. Using the notation ofAppendix C, we will denote this category asC0 = Z(1)
2 .Let us consider the extension ofC0 by G = Z2. The ex-
tended category isZ2-graded:CG = C0⊕ Cg, whereg is
the non-trivial element ofZ2. As explained in Sec. VI A, wemust haveDg = D0 = 2. This leaves two physically distinctways of constructingC×
G : (1) there is a single non-Abelian de-fect in Cg with quantum dimension
√2, denoted byσ, or (2)
there are two Abelian defects inCg, denoted byσ± and thusσ± × ψ = σ∓.
Below we will study these two cases in turn. We willfind that each case admits four distinctZ2 extensionsC×
Z2,
for a total of 8 possibleZ2 extensions ofC0. This repro-duces the known result for the classification of fermionic SPTstates with unitary on-siteZ2 symmetry, which is known to be
65
Z8 [148–151]. In fact, this approach can be further general-ized to fermionic SPT phases with an arbitrary finite symme-try groupG, and the classification is given by three cohomol-ogy groupsH1(G,Z2), H
2(G,Z2) andH3(G,U(1)) whichclassifies bosonic SPT phases [152].
1. Non-Abelian extensions
We first consider case (1), where there is a single defecttype Cg = σ. Since we must haveσ = σ and the quan-tum dimensions must satisfy the formuladadb =
∑cN
cabdc,
there fusion rules forσ must be: σ × σ = I + ψ. Weconclude that the extension as a fusion category must beidentical to the Ising theory. TheF -symbols of the Isingcategory are completely classified, see Eq. (C4). Thereare two gauge-inequivalentF -symbols, distinguished by theFrobenius-Schur indicatorκσ = ±1.
In this simple case (G = Z2 with an action that does notpermute any topological charges), we can use a symmetry ac-tion gauge transformation to pick a gauge in whichηa = 1.Solving the consistency equations forG-crossed braiding, weobtain
Rψσσ = iα,Rσψσ = iβ
RσσI = ±√κσe
iπ8 α, Rσσψ = −iαRσσIUg(σ, σ; I) = Ug(σ, σ;ψ) = αβ
Ug(ψ, σ;σ) = Ug(σ, ψ;σ) = αβ
Ug(ψ, ψ; I) = 1.
(459)
Here,α2 = β2 = 1. Notice thatβ = +1 and−1 give equiv-alent solutions under symmetry action gauge transformations,as do the± in RσσI . Thus, we find that, for this case, there arefour distinctZ2 extensions, distinguished byα andκσ.
Let us now gauge theZ2 symmetry. The topologicalcharges inC/G are(a,±) wherea = I, ψ, or σ and where±indicates the trivial and alternating irrep, respectively. Sincethe mutual braiding phase betweenψ andσ is−αβ, the topo-logical charge(ψ,−αβ) has trivial braiding with(σ,±), sincethe irrep± results in a phase±1 when it braids around the de-fectσ.
Thus, the four distinctC×G yield four distinctC/G, which
we can identify with Ising(n) × Z(1)2 wheren = 1, 3, 5, 7.
More explicitly, we have
α = 1,κσ = 1 −→ Ising(1) × Z(1)2
α = 1,κσ = −1 −→ Ising(5) × Z(1)2
α = −1,κσ = 1 −→ Ising(7) × Z(1)2
α = −1,κσ = −1 −→ Ising(3) × Z(1)2 .
(460)
Physically, all these fermionic phases can be realized innon-interacting spin-12 superconductors, where the spin upor down fermions form a class D topological superconduc-tor with Chern numberν or −ν, respectively, whereν =1, 3, 5, 7. In other words, this is|ν| copies ofpx ± ipysuperconductors. TheZ2 symmetry is the fermion parity
of spin up fermions. Gauging theZ2 symmetry results inIsing(ν) × Z(1)
2 , where theZ(1)2 corresponds to the spin down
fermions. We will refer to such a fermionic SPT phase as(px + ipy)
ν × (px − ipy)ν .
2. Abelian extensions
We now consider case (2), where there are two Abelian de-fect typesCg = σ+, σ−. In this case, there are two sets offusion rules possible: (a)Z2 × Z2 fusion, i.e.σ+ × σ+ = Ior (b)Z4 fusion, i.e.σ+ × σ+ = ψ.
In case 2(a), the extended theories can be written as prod-ucts of Z(1)
2 with the Z2 SPT theories, which have beenworked out in Sec. X A. This gives two distinct extensions forthis case, corresponding to the twoG-crossed theories SPT[α]Z2
with α(g,g,g) = ±1.In case 2(b), we haveCG = VecZ4 , and we will again find
two distinctZ2 extensions. Explicitly, we relate the topologi-cal charges byI ≡ [0]4, σ+ ≡ [1]4, ψ ≡ [2]4, andσ− ≡ [3]4.The F -symbols and theG-crossed braiding ofVecZ4 havebeen completely solved in Sec. X A. We thus continue usingthe same notationp to label different3-cocyclesα andma
which enters theG-crossedR-symbols, see Eqs. (450) and(451).
However, there are additional constraints now onp andma
in order to match theC0 sector. SinceFψψψψ = 1, p must be
an even integer. In order to matchRψψI = −1, we should have(−i)p(−1)m2 = −1. One more constraint can be obtained bydemandingUc(a, b; [a + b]4) to only depend on[c]2, whichmeans thatma +mb −ma+b is always even. An immediateconsequence is thatm2 is even, sop = 2. In addition,m1
andm3 must have the same parity. Hence in summary we canparameterizem1,2,3 as
ma = 2na + [a]2m′, a = 1, 2, 3. (461)
Herem′ = 0 or 1, while na ∈ Z. TheR-symbols andU -symbols become
Raba+b = e−πi4 [a]4[b]4(−1)nabim
′b[a]2
Uc(a, b; a+ b) = (−1)c(na+nb−na+b)im′([a]2+[b]2−[a+b]2).
(462)One can easily see the factor(−1)nab in Raba+b can be re-moved by a symmetry action gauge transformation. There-fore, as previously mentioned, there are two distinct typesofG-crossed braiding given bym′ = 0, 1. From theR-symbols,we can compute the topological twists and braiding statisticsto be
θσ+ = Rσ+σ+
ψ = e−πi4 (−1)n1im
′
θσ− = Rσ−σ−ψ = e−
πi4 (−1)n3(−i)m′
Rσ+ψσ− Rψσ+
σ− = −(−1)n2
Rσ−ψσ+
Rψσ−σ+
= −(−1)n2
Rσ−σ+
I Rσ+σ−I = −i(−1)n1+n3+m
′
(463)
66
Next, we can gauge the symmetry of these theories. We la-bel the topological charges inC/G by (a, α) wherea ∈ Z4
andα = ±1. We can setna = 0 and assumem′ = 0 forsimplicity, then allU symbols are1. The fusion rules of thegauged theory are then simply given by:(a, α) × (b, β) =([a + b]4, αβ). We observe that(2,−) has trivial full braid-ing with all other topological charge types and that we canwrite the topological charges with− irreps as(a,−) = ([a+2]4,+) × (2,−). In this manner, we can write the theory as
the productZ(1)2 ×Z(−1/2)
4 , where theZ(1)2 corresponds to the
topological charges(0,+), ([2],−) and theZ(−1/2)4 corre-
sponds to the topological charges(a,+)|a = 0, 1, 2, 3. For
m′ = 1, we similarly obtain the gauged theoryZ(1)2 × Z(1/2)
4 .Thus, we have two distinct gauged theories for case 2(b), onefor each distinctG-crossed extension.
The Z2-crossed extensions discussed here should not beconfused with a different concept, which is referred to as amodular extension of fermions. Physically, a modular exten-sion of fermions corresponds to gauging theZ2 symmetry offermion parity conservation, such that there are no indepen-dent (bosonic)Z2 charges other than theψ fermions them-selves. The extended category in such a modular extensionwill, by definition, be a modular one and will be braided inthe usual sense (as opposed toG-crossed braided). In con-trast,Z2-crossed extensions considered in this section are dif-ferent, as bosonicZ2 charges are allowed, and the braiding isG-crossed.
In summary, we have found four AbelianZ2-crossed exten-sions ofZ(1)
2 (one of which is trivial), all of which can be real-ized again in non-interacting superconductors as(p + ip)ν ×(p−ip)ν as described previously, but with even Chern numberν = 0, 2, 4, 6. We notice thatν = 0 is the trivial extension,ν = 4 corresponds to taking the product of the trivial fermiontheory with a (nontrivial) bosonicZ2 SPT [153], and the othertwo are nontrivial fermionic SPT phases [74, 148].
C. SemionsZ(± 1
2)
2 with Z2 symmetry
In this section, we examine the semion theory forG = Z2.
The semion theoryZ(± 1
2 )2 consists of only two topological
chargesC0 = I, s, wheres is a semion with fusions×s = Iand topological twistθs = ±i. Such a theory would describethe topological properties of the bosonicν = 1/2 Laugh-lin FQH state. The nontrivialF -symbols andR-symbols are
F ssss = −1 andRs,sI = ±i. We will focus on theZ( 12 )
2 theoryin this section.
Since there is only one nontrivial topological charge type,the symmetry action does not permute topological charge val-ues. As discussed at the end of Sec. III C, this automati-cally means the obstruction vanishes and we can setO = I.The symmetry fractionalization is classified byH2(Z2,Z2) =Z2, which gives two equivalence classes corresponding tow(g,g) = I ands, respectively. Physically, these two co-homology classes correspond to whether the semion carries aZ2 charge of0 or 1
2 . The two fractionalization classes corre-
spond to distinct fusion rules for theZ2 defects, which are,respectively, given by:g×g = I org×g = s. In the follow-ing, we focus on the latter case and systematically work outthe gauging procedure (although there are other simpler waysto get the gauged theory). We will use a choice of gauge inwhichηs(g,g) = 1.
First, we construct theCZ2 theory. For the nontrivialH2(Z2,Z2) classw(g,g) = s, we haveCg = σ+, σ−,and σ+ × σ+ = s. The extended category has the samefusion rules asVecZ4 , similar to one of theZ2 extension offermions discussed in Sec. X B. As such, we again identifyI ≡ [0], s ≡ [2], σ+ ≡ [1], σ− ≡ [3]. TheF -symbols aregiven in Eq. (450). In order to match theC0 sector, we musthaveα([2], [2], [2]) = F ssss = −1, and, hence,p must be anodd integer.
TheG-crossed braiding ofVecZ4 were found to be
Rab[a+b] = e−πip8 [a]4[b]4e
2πima4 b, (464)
Uc(a, b) = e−πip4 c(a+b−[a+b]4)e
πi2 c(ma+mb−ma+b).(465)
Herema are arbitrary integers. The analysis is very similar tothe AbelianZ2 extension ofZ(1)
2 except that now theC0 sectoris different. We will not repeat the steps, but just give the finalresult:p = −1, and
ma = 2na + [a]2m′, a = 1, 2, 3, (466)
wherem′ = 0 or 1, andna ∈ Z.From theR-symbols, we can compute the topological
twists and braiding statistics
θ+ = Rσ+σ+s = e
πi8 (−1)n1 im
′
θ− = Rσ−σ−s = −e πi8 (−1)n3(−i)m′
Rσ+sσ− Rsσ+
σ− = i(−1)m′+n2
Rσ−sσ+
Rsσ−σ+
= −i(−1)m′+n2
Rσ−σ+
I Rσ+σ−I = e
3πi4 (−1)n1+n3
(467)
The integerm′ taking the values0 or 1 corresponds to twodistinct solutions forC×
G . This corresponds to the two dis-tinct possible values ofH3(Z2,U(1)) = Z2. Therefore, wefind that the distinctZ2-crossed extensions of the semion the-ory are indeed in one-to-one correspondence with the differentchoices ofH2(G,A) = Z2 andH3(G,U(1)) = Z2.
We now describe the gauged theory. The topologicalcharges in the gauged theory are parameterized by([a], α)wherea ∈ Z4 andα = ±1 labels theZ2 irreps (trivial andalternating, respectively). The fusion rules are given by
(a, α) × (b, β) = ([a+ b]4, αβUg(a, b)). (468)
We can verify that the fusion algebra Eq. (468) is isomorphicto those of aZ8 theory. For the solution withm′ = 0, thegauged theory isZ(1/2)
8 , which is equivalent to a U(1)8 Chern-
Simons theory. Form′ = 1, the gauged theory isZ(5/2)8 .
The original semion theory can then be obtained from thesegauged theories by condensing the bosonic quasiparticle[4]8in theZ8 theories.
67
Physically, the nontrivialZ2-crossed extension of thesemion model withm′ = 0 can be constructed by startingfrom a Kalmeyer-Laughlin chiral spin liquid with U(1) Sz ro-tational symmetry [154], where the semions carry half U(1)charges, and then breaking the U(1) down to aZ2 subgroupto obtain aZ2 symmetry-enriched semion theory.
D. SemionsZ(± 1
2)
2 with Z2 × Z2 symmetry
We consider the semion theoryC0 = I, s with symmetrygroupG = Z2 × Z2 ≡ 11, X, Y, Z. Again because there isno permutation, the obstructionO is identicallyI. The sym-metry fractionalization classes are given byH2(G,Z2) = Z3
2.Among the7 nontrivial 2nd cohomology classes, three are justthe nontrivial cocycle inH2(Z2,Z2) for the threeZ2 sub-groups, which has been considered in the previous section.Thus, we will focus on other four cohomology classes, whichare distinguished byw(g,g),g = X,Y, Z subject to the con-straintw(X,X)w(Y, Y )w(Z,Z) = s.
1. w(g,g) = s, ∀g 6= 11
First we consider the case where the2-cocyclew(g,g) = sfor g = X,Y, Z. The extended theory hasCg = σ+
g , σ−g
whereg = X,Y, Z andσ±g = s×σ∓
g . The fusion rules of theextended theory can be easily obtained from the2-cocycles:
σ+g × σ+
g = σ−g × σ−
g = s
σ+X × σ+
Y = σ+Z , σ
+Y × σ+
X = σ−Z
(469)
The other fusion rules can be obtained by cyclic permutationcombined with the relation betweenσ+
g andσ−g . We notice
that the fusion rules match exactly with the multiplicationta-ble of the quaternion groupQ8. In other words,CG ≃ VecQ8 .TheF -symbols are then classified byH3(Q8,U(1)) = Z8.Four of them can match theF -symbols ofC11.
In order to make sense ofG-crossed braiding, we need toextend the symmetry action onto theCg sector. For example,we must have
X : σ±X → σ±
X , σ±Y → σ∓
Y , σ±Z → σ∓
Z . (470)
The action ofY andZ can be obtained by cyclic permutations.We have checked that theG-crossed hexagon equations havesolutions with the help of Mathematica. Interestingly, we canset all theηa(g,h) to 1 excepta = s andηs is a nontrivial2-cocycle inH2(G,U(1)), which is the expected result.
Let us now consider the gauged theory. With the symme-try action given in Eq. (470), forg 6= 11 eachCg forms anorbit underG. For g = X,Y, Z, the stabilizer group of theorbit σ+
g , σ−g is theZ2 subgroup generated byg. So we get
3 × 2 = 6 anyons with quantum dimensiond = 2. Theirtopological twists can be computed from theG-crossedR-symbols, and all of them have±eiπ8 twist factor. For theC11sector, the stabilizer group is the wholeZ2 ×Z2. The vacuum
simply splits according to the linear irreducible representa-tions ofG, which gives fourd = 1 Abelian anyons. However,the semion carries a projective representation ofZ2×Z2, sincethe factor setηs belongs to the nontrivial cohomology class inH2(Z2 × Z2,U(1)) = Z2. It is well-known that there is aunique two-dimensional irreducible representation with thisfactor setηs up to similarity, essentially given by Pauli ma-trices. According to Sec. VIII B, this implies that the semionbecomes ad = 2 non-Abelian quasiparticle in the gauged the-ory with the topological twist still beingi. Therefore, we findthere are7 non-Abelian anyons withd = 2 and four Abelianones, for a total of11 anyons.
Another way to obtain the number of anyons is to computethe ground state degeneracy of the gauged theory on the torus,as described in Sec. VIII D. To do this, note that the sys-tem can have twisted sectors with group elements inZ2 ×Z2.This implies1 untwisted sector and15 twisted sectors. Theuntwisted sector has two invariant states. The twisted sec-tors can be understood as follows. There are3 twisted sectorswhere there is a twist only along the longitudinal cycle and notwist along the meridian. In each such twisted sector inC×
G ,there are 2 states. However from theZ2 × Z2 action it is easyto see that only one state is invariant under the symmetry andsurvives the gauging. Therefore, the sectors with twists onlyalong the longitudinal and not the meridian yields3 states,and similarly for sectors with twists only along the meridianand not the longitudinal cycles. For sectors with twists alongboth cycles, it can be easily verified that the sectors(g,h)with g 6= h contain no states, while sectors of the form(g,g)contain one state. Therefore the twisted sectors in total yield9 states. Combined with the untwisted sector, we find a torusdegeneracy of11 for the gauged theory. This implies that thereare11 distinct quasiparticles, as explained above.
One can obtain the fusion rules of the gauged theory usingthe solutions of theG-crossed consistency equations, but wechoose to use a different method, namely gauging the symme-try sequentially. Without loss of generality, we first gaugetheZ2 subgroup11, Z. Sincew(Z,Z) = s, from the previoussubsection we have learnt that gauging this subgroup resultsin a U(1)8 theory. The otherZ2 group then has a nontrivial ac-tion in the U(1)8 theory, namely it acts as a charge conjugationsymmetry. We will discuss the charge conjugation symmetryin U(1)8 theory in more detail in Section X F. Here we notethat the gauged theory can be understood in terms of theZ2
orbifold of a U(1)8 CFT, which was analyzed in Ref. [100].Interestingly, it can also be understood as theD2 ≡ Z2 × Z2
orbifold of the SU(2)1 CFT, which fits naturally within ourapproach. The gauged theory can be identified withSO(8)2.
To have a physical realization of the semion theory with thisZ2 × Z2 symmetry, one can start from a chiral spin liquid inspin-12 systems, which has SO(3) spin rotational symmetry.The semion carries spin-1/2, i.e. a projective representationof the SO(3) symmetry. One can then break the SO(3) sym-metry down to theZ2 × Z2 subgroup, i.e.π rotations aroundthree orthogonal axes. We can explicitly write down the localunitary operators on a semion:
UX = iσx, UY = iσy, UZ = iσz. (471)
68
ApparentlyU2g = −1 soeiφs(g,g) = −1. This is the familiar
fact that spin-1/2 acquires aπ phase after a2π rotation.
2. Other cocycles and obstruction to definingC×G
We now turn to the other three cocycles, which has twos and oneI amongw(g,g),g = X,Y, Z. Without loss ofgenerality we considerw(X,X) = w(Y, Y ) = I,w(Z,Z) =s. The fusion rules of the defects can be worked out as before:
σ±X × σ±
X = σ±Y × σ±
Y = I, σ±Z × σ±
Z = s
σ+X × σ+
Y = σ+Z , σ
+Y × σ+
X = σ−Z
σ+Y × σ+
Z = σ−X , σ
+Z × σ+
Y = σ+Z
σ+Z × σ+
X = σ−Y , σ
+X × σ+
Z = σ+Y
(472)
The other fusion rules can be obtained using the relation be-tweenσ+
g andσ−g . With these fusion rules, we see thatCG ≃
VecD8 . HereD8 = Z4 ⋊ Z2 is the dihedral group of order8.TheF -symbols are classified byH3(D8,U(1)) = Z2
2 × Z4.Among the16 classes,8 of them can match theC0 sector.
Therefore, the extended category as a usual fusion categorydoes exist. However, we find that theG-crossed consistencyequations admit no solutions. Therefore the extended cate-gory has noG-crossed braiding. This implies that the symme-try action can not be consistently extended to the wholeCGand therefore we cannot continue our gauging procedure. Inother words, there is an obstruction to gauging the symmetry.This is in full agreement with the result obtained in Ref. [92]using a different method. The physical meaning of this isthat the semion theory with such a symmetry fractionaliza-tion class cannot exist in 2+1 dimensions, but could possiblyexist at the surface of a 3+1 dimensional system, as discussedin Ref. [92].
E. Z(p)N Anyons withN odd andZ2 symmetry
In this subsection we considerZ2 symmetry in theZ(p)N
anyon model whereN is odd andp is integer. Z(p)N hasN
distinct topological charges, labeled bya = 0, 1, . . . , N − 1.The fusion rules are just given by addition moduloN : a×b =[a+ b]N . TheF -symbols are all trivial and theR-symbols aregiven by
Rab[a+b]N = e2πipabN . (473)
Physically, these models show up in the mathematical descrip-tion of ν = 1
m Laughlin FQH states withm odd, which
have the same topological order asZ(2)m × Z(1)
2 . These the-
ories have aAut(Z(p)N ) = Z2 topological symmetry associ-
ated with charge-conjugation, which is given by the action[a] → [−a] [12, 155]. A twist defect associated with the per-mutation of quasiparticles into quasiholes can be engineeredin the Laughlin state by creating a superconducting trench inthe bulk, or as a superconducting/magnetic domain wall onthe edge of a fractional topological insulator [9–12].
We have two choices for the mapρ : Z2 → Aut(Z(p)N ). It
can either map both elements ofZ2 to the identity element ofAut(Z(p)
N ), or it can map the non-trivial element ofZ2 to the
non-trivial element ofAut(Z(p)N ). Here we will consider the
latter case.First we calculate the symmetry fractionalization classes
H2ρ(Z2,ZN ). The cocycle condition simplifies to
gw(g,g) = w(g,g). (474)
Thereforew(g,g) must be ag-invariant charge. In fact, thisholds for anyg of order2, i.e. g2 = 0. SinceN is odd, thereis no fixed point under theG action, henceH2
ρ (Z2,ZN ) = 0.As discussed in Sec. VII B, since there are nog-invariant
topological charges inC0, we have|Cg| = 1. In other words,there is exactly one type of defect, which will be denoted byσ.We can also prove this statement directly from associativity ofthe fusion rules without using modularity. If there is anotherdefectσ′, it must be related toσ by fusing with somea ∈C0 ≡ Z(p)
N . Assumeσ′ = σ × a. Whena is taken aroundthe defect it becomes[−a]N , which implies thatσ × a =σ× [−a]N = σ′. We then concludeσ′×a = σ′× [−a]N = σand thusσ × a2 = σ. Using this relation, we calculateσ′ =σ × [−a]N again to be
σ× [−a]N = σ× a2 × a2 × · · · a2︸ ︷︷ ︸N−1
2
= (σ× a2)× · · ·a2 = σ,
(475)which provesσ′ = σ.
The fusion rules ofσ can be easily obtained to be
σ × a = a× σ = σ (476)
σ × σ =∑
a∈C0
a. (477)
The fusion categoryCG = C0 ⊕ Cg is known as the Tambara-Yamagami category [156]. TheF -symbols ofCG are com-pletely classified in Ref. [156] and are given by theF -symbolsof the original categoryC0, together with:
[F aσbσ ]σσ = [F σaσb ]σσ = χ(a, b),
[F σσσσ ]ab =κσ√Nχ−1(a, b), (478)
Hereχ is a U(1)-valued function onZN × ZN , satisfying
χ(a, b) = χ(b, a)
χ(ab, c) = χ(a, b)χ(b, c),
χ(a, bc) = χ(a, b)χ(a, c),
(479)
together with normalization conditionχ(0, a) = χ(a, 0) =1. Such aχ is called a symmetric bi-character.κσ = ±1is the Frobenius-Schur indicator of theg defect. It is worthmentioning that the two solutions ofF -symbols distinguishedbyκσ is directly related toH3(Z2,U(1)) = Z2. Interestingly,this fusion categoryCG does not admit braiding in the usualsense.
69
We now considerG-crossed braiding. First we gauge fixη.It is easy to see that using symmetry-action gauge transforma-tions we can setησ(g,g) all to 1. Forηa(g,g), they transformunder the symmetry-action gauge transformations as follows:
ηa(g,g) = γ[−a](g)γa(g)ηe(g,g) (480)
In addition, the associavitity constraint onη yields
η[−a](g,g) = ηa(g,g). (481)
Therefore we can also chooseγa(g) andγ[−a](g) to set allηa(g,g) to 1. We notice that the remaining symmetry-basedgauge transformations areγσ(g) being±1.
With this gauge fixing, we find the following solutions totheG-crossed braiding consistency equations:
χ(a, b) = Raba+b = e2πipabN ,
Rσaσ = s(a)(−1)pae−πipa2
N , Raσσ = (−1)pae−πipa2
N
Rσσa = γ(−1)paeπipa2
N
γ2 =κσ√N
N−1∑
n=0
(−1)pne−πipn2
N .
(482)
We have chosenη ≡ 1, which is possible sinceG = Z2. Heres(a) ≡ Ug(σ, σ, a) = ±1 is an arbitrary sign with a constraints(a) = s(−a). We notice that none of theUg are intrinsic inthe sense that they are all essentially maps between differentsplitting spaces, exceptUg(σ, σ; 0) = 1.
Having obtained theR-symbols, we can formally calculatethe topological spin:
θσ =∑
a
dadσRσσa = γ∗. (483)
We can also calculate theG-crossed modular matrices:
S =
S0 0 0 0
0 0 1 0
0 1 0 0
0 0 0 γ2eπi4 c−
,
T =
T0 0 0 0
0 1 0 0
0 0 0 γ∗
0 0 γ∗ 0
.
(484)
HereS0, T0 are the topologicalS andT matrices for theZ(p)N
theory. The basis states of the twisted sectors are chosen tobe|0(0,g)〉, |σ(g,0)〉, |σ(g,g)〉 (in this order). It is straightforwardto check thatS, T form a representation ofSL(2,Z).
We now proceed to derive the properties of the gauged the-ory C/G. Under the group action the extended category isdivided into N+1
2 orbits: [0], [a], [−a], σ. The stabilizersubgroups areZ2 for [0] andσ, and trivial for theN−1
2 or-bits [a], [−a]. Therefore the vacuum should split into aZ2
even charge([0],+) ≡ I (i.e. the vacuum inC/G) and aZ2
odd charge([0],−) ≡ e (i.e. theZ2 charge) which satisfiese2 = I. The orbits[a], [−a], 1 ≤ a ≤ N−1
2 becomeN−12
non-Abelian anyons withd = 2, which we label byφa. Theirfusion rules are given by:
φa × φb =
φmin(a+b,N−a−b) + φ|a−b| a 6= b
I + e+ φmin(2a,N−2a) a = b(485)
We turn to the defect sectors. As we have argued, there isone defect in the extended theory, which should split into two(σ,±) in C/G. They have the same quantum dimensions
√N .
Their fusion rules are
(σ,+)× (σ,+) = I +∑
a
φa,
(σ,+)× (σ,−) = e +∑
a
φa. (486)
Therefore,C/G has2+ N−12 +2 = N+7
2 topological charges.To further identify the gauged theory, we calculate the topo-
logical twists. The twist factors ofφa are identical toa (or
[−a]N), so we haveθφa = e2πipa2
N . We find that whenp = N−1
2 , i.e. C0 = SU(N)1, the gauged theory is equiv-alent to the SO(N)2 category. We can confirm this by com-puting the twist factor of the defects. The Gauss sum inthe expression ofγ2 evaluates toe−
2πi8 (N−1) and we obtain
θ2σ = (γ∗)2 = κiN−1
2 , consistent with SO(N)2 whenκ = 1.
The gauged theories forκ = −1 as well as the otherZ(p)N are
Galois conjugates of SO(N)2.We note that the relation between theZ3 theory and the
gauged theory SO(3)2 = SU(2)4 was previously observed inRefs. [75, 102, 157].
F. Z(p+ 1
2)
N Anyons with N even andZ2 symmetry
Here we considerZ2 symmetry in theZ(p+ 1
2 )
N anyon modelwhereN is even andp is integer. This model hasN topo-logical charges labeled bya = 0, . . . , N − 1 whose fusionrule is again given by addition modN . TheF -symbols andR-symbols are given by
F abca+b+c = eiπNa([b]+[c]−[b+c]),
Raba+b = e2πiN
(p+ 12 )[a][b]. (487)
p = 0 represents the topological order of the well-knownU(1)N Chern-Simons theory, which describes the bosonicν = 1/N Laughlin FQH states.
This theory has a topological symmetryAut(Z(p+ 1
2 )
N
)=
Z2 associated with charge conjugation, which is given by theaction[a] → [−a]. As in the previous section, we consider the
case where theρ : Z2 → Aut(Z(p+ 1
2 )
N
)maps the non-trivial
element in the symmetry groupZ2 to the non-trivial element
of Aut(Z(p+ 1
2 )
N ). BecauseN is even, the anyonN2 is a fixedpoint underg. This implies that the second cohomology group
70
H2ρ(Z2,ZN ) = Z2 whenN is even. To see this, note that
the cocycle condition has a solutionw(g,g) = [N/2]. Sincea coboundary is alwaysga × a = [0], this solution indeedrepresents a nontrivial cohomology class.
Due to the existence of oneZ2 invariant anyon inC0 ≡Z(p+ 1
2 )
N , there are two species of twist defects, labeled byσ±.Their fusion rules are
σ+ × σ+ =∑
a even
[a]N
σ+ × σ− =∑
a odd
[a]N
σ± × [a]N =
σ± a even
σ∓ a odd
(488)
The quantum dimensions of the defects aredσ± =√
N2 .
Let us first considerN = 2m wherem is an odd integer.
We have a decompositionZ(p+ 1
2 )2m = Z(2p+1)
m ×Z(mp+m
2 )2 . The
mapping is given by
[a]2m ↔([
a+m[a]22
]
m
, [a]2
). (489)
We can easily see that theZ2 action induces the charge con-jugation symmetry on theZm theory. The anyon[m]2m isactually a semion, and the symmetry fractionalization classesH2(Z2,ZN ) = Z2 only affect the semion theory, which wasconsidered in Sec. X C.
Next we consider a simple but nontrivial example withN = 4. The nontrivial cocylew(g,g) = [2]. By solvingthe pentagon equations forCG, we find there are four gauge-inequivalent solutions [158], distinguished by the Frobenius-Schur indicators of the defects:κσ+ = ±1,κσ− = ±1.This should be contrasted with theZ2-extended toric code dis-cussed in the next subsection, where similarly there are twotypes ofZ2 defects, but with identical Frobenius-Schur indi-cators. In fact, the appearance of solutions whereσ± havedifferent Frobenius-Schur indicator is closely related totheexistence of a nontrivial2-cocycle inH2
ρ(Z2,Z4).Physically, the nontrivial2-cocycle implies that both[1] and
[3] carry a “half”Z2 charge, while being permuted under theZ2 action. To see this explicitly, we can calculate the symme-try fractionalization phasesηa(g,g). The cocycle conditionimpliesη[1](g,g) = η[3](g,g). We also have
ωa(g,g) = βa(g,g)η−1a (g,g) (490)
being a character onZ4, soωa(g,g) = eiπ2 an wheren ∈ Z4.
See the discussions around Eq. (153) for more details. Bydirectly solving theF -symbol invariance condition Eq. (87),we findβ[1](g,g) = β[3](g,g) = 1, β[2](g,g) = −1. Thesetogether impliesn = 0 or n = 2. Apparentlyn = 2 is thenontrivial solution, withη[1](g,g) = η[3](g,g) = −1. Thisconfirms that[1] and[3] indeed carry halfZ2 charges.
Sinceσ+ × [1] = σ+ × [3] = σ−, one expects that eitherσ+ or σ− has to carry a halfZ2 charge, while the other has a
trivial Z2 charge. The halfZ2 charge changes the Frobenius-Schur indicator of one of the two defects. In addition to this,we can change theF -symbols by a nontrivial3-cocycle inH3(Z2,U(1)), which actually amounts to gluing aZ2 SPTstate, changing their Frobenius-Schur indicators simultane-ously. However, the solutions withκσ± = ±1 and the onewith κσ± = ∓1 are clearly related by relabeling the two de-fects. Therefore, allowing relabeling defects reduces thenum-ber of distinct solutions solutions from four to three.
After gauging, we find that the solution(κσ+ , κσ−) =(1, 1) becomes Ising×Ising, and the solution(κσ+ , κσ−) =
(−1,−1) becomes Ising(5) × Ising(3)
. The other two solu-
tions collapse to Ising(3) × Ising(1)
.
G. ZN -Toric Code D(ZN ) with Z2 symmetry
The anyon modelD(ZN ) arises as the quantum double ofZN [159]. Physically they can be realized byZN general-izations of Kitaev’s toric code model, or asZN lattice gaugetheories. The anyons are gauge charges (the unit of which isdenoted bye), gauge fluxes (the unit of which is denoted bym) and their bound states, the dyons. So there areN2 anyonlabelsa = (a1, a2) ≡ ea1ma2 , a1, a2 = 0, 1, . . . , N − 1, withe ≡ (1, 0),m ≡ (0, 1). Their fusion rules are straithforward:(a1, a2)× (b1, b2) = ([a1 + a2]N , [b1 + b2]N ), i.e. they formaZN × ZN fusion algebra. TheF -symbols of the theory areall trivial and theR-symbols are given byRa,b = e
2πiNa2b1 .
Aut(D(ZN )) is generally a complicated group. Besidesthe trivial action, we can easily identify twoZ2 symmetries:(1) ρg(e) = m, ρg(m) = e. This is known as the electric-magnetic duality symmetry, and can be realized in a slightlydifferent formulation of the toric code model by Wen [160] aslattice translations [4, 7]. Alternatively, one can also realizethis type of symmetry of aZN toric code in an on-site fash-ion [161]. (2)ρg(e) = eN−1, ρg(m) = mN−1, this is sort ofa charge conjugation symmetry. In this section we considerthe trivial action and the electric-magnetic duality symmetry.
We first calculate the symmetry fractionalization class.Since theF symbols of D(ZN ) are all trivial, the obstruc-tion can be set toI identically. The Abelian anyons form agroupZN × ZN . (1) The action on the charge label setρ istrivial. We generally havew(g,g) = en1mn2 . It is easy tosee that whenN is odd, this is a coboundaryw = dz wherez(g) = e2
−1n1m2−1n2 , with 2−1 being the inverse of2 inZN . For evenN , if either ofn1, n2 is odd, it is a nontrivialcocycle. (2) The action on the charge label set is the electric-magnetic duality symmetry. As we have shown in Eq. (474),w(g,g) must be ag-invariant anyon, sow(g,g) = enmn
wheren = 0, 1, . . . , N − 1. However, this is a coboundarysinceenmn = ρg(e
n) × en. Therefore we have shown thatH2ρ(Z2,ZN × ZN ) = 0.
71
1. Trivial action
We now consider theZ2 toric code with trivial symme-try action on the label set. We denote the four anyons byI, e,m, ψ whereψ = e × m. Using our previous labeling,e = (1, 0),m = (0, 1) andψ = (1, 1). There are four frac-tionalization classes, given byw(g,g) = I, e,m, ψ.
There are four defects which are labeled byσI , σe, σm, σψwhereσa = σI × a for a = e,m, ψ. They all satisfyσ2
a =w(g,g).
We will not show the complete set ofG-crossed braideddata of the extended category, but just give theG-crossed in-variantsT 2 (with all η set to1). Forw(g,g) = m, we find
θ2σI = θ2σm = (−1)p, θ2σe = θ2σψ = −(−1)p. (491)
Herep = 0, 1. We notice that the two solutions are related bygluing aZ2 SPT state. Forw(g,g) = ψ, we have
θ2σI = θ2σm = i(−1)p, θ2σe = θ2σψ = −i(−1)p. (492)
However, we must keep in mind that the topological chargelabels of the defects are arbitrary. For example, which of thedefect is defined asσI is an arbitrary choice. We can redefineσ′I = σe, σ
′e = σI , σ
′m = σψ , σ
′ψ = σm and the two so-
lutions become identical. In other words, gluing a nontrivialH3(Z2,U(1)) solution does not give distinctG-crossed ex-tensions for the three nontrivial fractionalization classes. Thisphenomena was observed in Refs. [52, 140] using a Chern-Simons field theory approach.
2. Electric-magnetic duality symmetry
To understand the extended category and the gauged the-ory, we notice that forN odd we have the following decom-position ofD(ZN ): D(ZN ) = Z(1)
N × Z(−1)N . To make it
explicit, theZ(±1)N subcategory are formed by(k,±k), k =
0, 1, . . . , N − 1. In the decomposition, the symmetry onlyacts on theZ(−1)
N sector and the extension problem has beenthouroughly analyzed in the previous subection. Thereforethegauged theory isZ(1)
N ×(Z(−1)N /Z2). Similar results have been
obtained previously in Ref. [103].For evenN , such a decomposition no longer exists. Let
us considerN = 2 as an illustrating example [17, 52, 103].Becauseψ is a fixed point under theG action, there are twospecies of twist defects labeled byσ+ andσ−. They differ byfusion with ane orm anyon:
σ+ × σ+ = σ− × σ− = I + ψ
σ+ × σ− = e+m
σ+ × e = σ+ ×m = σ−.
(493)
TheZ2 action can be straightforwardly extended to defects:ρg(σ±) = σ±.
TheF -symbols of the extended category can be found:
F aσbσ = F σaσb = (−1)a1b1 , [F σσσσ ]ab =κ√2(−1)a1b1 .
(494)
We then solve for theR-symbols from theG-crossed con-sistency equations. Similar to the gauge fixing done in Sec.X E, we can set allη to 1 and solve theG-crossed heptagonequations to get
Rσ−,a = (−1)a1Rσ+,a, Rσ+,a = saia1
Ra,σ− = Ra,σ+ , Ra,σ+Rga,σ+ = (−1)a1a2
Rσ+,σ+
I = s+√κeisψ
π8 , R
σ−,σ−I = s−
√κe−isψ
π8
Rσ+,σ+
ψ = s+√κe−isψ
3π8 , R
σ−,σ−ψ = s−
√κeisψ
3π8
Rσ+,σ−e = −iseRσ+,σ+
I , Rσ+,σ−m = smR
σ+,σ+
I
Rσ−,σ+e = iseR
σ−,σ−I , Rσ−,σ+
m = smRσ−,σ−I
(495)
Heresa is aZ2 character on the fusion group, i.e.sasb = sab,ands± = ±1. Notice that(Rψσ+)2 = −1, soRψσ+
σ+ = s′ψiwheres′ψ = ±1. Notice thats± and s′ψ are all symmetryaction gauge degrees of freedoms.
We also find all theU symbols which are invariant undervertex-based gauge transformations:
Ug(σ+, σ+; I) = 1, Ug(σ+, σ+, ψ) = sψs′ψ
Ug(σ−, σ−; I) = 1, Ug(σ−, σ−, ψ) = −sψs′ψUg(σ+, ψ, σ+) = Ug(ψ, σ+, σ+) = sψs
′ψ
Ug(σ−, ψ, σ−) = Ug(ψ, σ−, σ−) = −sψs′ψUg(ψ, ψ; I) = 1
(496)
We can recognize that two subcategoriesI, ψ, σ+ andI, ψ, σ− closely resemble theZ2-crossed braided Ising cat-egory discussed in Section X B. The topological twists of thedefects are given by
θσ± = s±√κe±isψ
π8 . (497)
So the two Ising subcategories are conjugate to each other andthus flipping the value ofsψ = ±1 is equivalent to relabelingσ+ andσ−, and the same is true for the remaining sign ambi-guity se. Therefore, we have found that there are two gauge-inequivalentG-crossed extensions labeled byκ = ±1. In thefollowing we sets′ψ = s± = 1.
72
TheG-crossed modularS andT matrices are given by:
S =
S0 0 0 0 0 0 0
0 0 0 1√2
1√2
0 0
0 0 0 − sψ√2
sψ√2
0 0
0 1√2
− sψ√2
0 0 0 0
0 1√2
sψ√2
0 0 0 0
0 0 0 0 0 0 κ
0 0 0 0 0 κ 0
,
T =
T0 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 −1 0 0 0 0
0 0 0 0 0 κ12 e
iπ8 0
0 0 0 0 0 0 κ12 e−
iπ8
0 0 0 κ12 e
iπ8 0 0 0
0 0 0 0 κ12 e−
iπ8 0 0
.
(498)Here the basis states for the twisted sectors are chosen to be|I(0,g)〉, |ψ(0,g)〉, |σ(g,0)
+ 〉, |σ(g,0)− 〉, |σ(g,g)
+ 〉, |σ(g,g)− 〉.
Now let us consider the gauged theory. There are nine topo-logical charges, labeled by(I, s), (ψ, s), (σ±, s), Y wheres = ±1 andY corresponds to the orbite,m so dY = 2.A straightforward application of Eq. (407) yields the follow-ing fusion rules:
(ψ, s)× (σ+, s′) = (σ+, ss
′)
(ψ, s)× (σ−, s′) = (σ−,−ss′)
(σ+, s)× (σ+, s′) = (I, ss′) + (ψ, ss′)
(σ−, s)× (σ−, s′) = (I, ss′) + (ψ,−ss′)
(σ+, s)× (σ−, s′) = Y
(σ±, s)× Y = (σ∓,+) + (σ∓,−)
(499)
The fusion rules are identical to those of two copies of Isingcategories:I, σ1, ψ1 × I, σ2, ψ2 once we make the fol-lowing identification:
I ↔ (I,+), ψ1ψ2 ↔ (I,−)
ψ1 ↔ (ψ,+), ψ2 ↔ (ψ,−)
σ1 ↔ (σ+,+), σ1ψ2 ↔ (σ+,−)
σ2 ↔ (σ−,+), σ2ψ1 ↔ (σ−,−)
σ1σ2 ↔ Y
(500)
The topological twists of(σ±,±) are±√κe±i
π8 . Depending
on whetherκ = 1 or −1, the gauged theory is identified with
Ising(1) × Ising(1)
or Ising(3) × Ising(3)
.As aforementioned the electric-magnetic duality inZ2 toric
codes can be realized as an on-site symmetry. We now brieflydescribe a concrete model. We start from a spin-1/2 fermionicsuperconductor with the pairing(p+ ip)ν↑ × (p− ip)ν↓ whereν is an odd integer. This is a model of theZ2 fermionic SPTphase discussed in Sec. X B. Now we gauge theZ2 fermionparity of the whole system, i.e. coupling all fermions to aZ2
gauge field, and we obtain aZ2 toric code, wherem is theπ
flux in the original superconductor ande is the bound state oftheπ flux and a fermion. TheZ2 symmetry that protects theSPT phase, namely the fermion parity of the spin↑ fermions,now becomes thee ↔ m symmetry of the toric code. To seethis, we first notice that before theZ2 total fermion parity isgauged, aπ flux localizes two Majorana zero modesγ↑ andγ↓since it penetrates twop± ip superconductors. Under theZ2
symmetryγ↑ → −γ↑, γ↓ → γ↓, so the local fermion parityiγ↑γ↓ on theπ flux changes sign under the on-siteZ2 symme-try, which interchangese andm after the total fermion parityis fully gauged. This provides the desired on-site realization.We can turn this model of a fermionic superconductor coupledto aZ2 gauge field into a Kitaev-type spin model.
In this model, gauging theZ2 symmetry becomes particu-larly easy: we simply gauge the fermion parities of the spin↑ and ↓ fermions separately, and the result is nothing but
Ising(ν) × Ising(ν). However,ν andν + 8, as well asν and−ν, lead to exactly the same topological gauge theories, sowe only obtain two distinct gauge theories corresponding toν = 1, 3, in agreement with our previous analysis.
H. Double-Layer SystemsB × B with Z2 symmetry
Let us consider two identical, non-interacting layers of atopological phase described by the UMTCB. The topologicalorder of the double-layer system is described byB×B. Thereis aZ2 symmetry which exchanges the two layers. LetA de-note the subcategory of Abelian anyons inB. We can gener-ally prove that the symmetry fractionalizationH2
ρ(Z2,A×A)is trivial: as shown in previous examples, the2-cocycle con-dition leads tow(g,g) = (a, a) wherea ∈ A. However, thisis again a coboundary, since(a, a) = g(a, 0) × (a, 0). SoH2ρ(Z2,A×A) = 0.The number of defects is obviously equal to|B|. There is a
“bare” defectX0 which has the following fusion rule:
X0 ×X0 =∑
a∈B(a, a). (501)
The quantum dimensiondX0 =√∑
a d2a = D. The other
defects areXa wherea ∈ B:
X0 × (a, 0) = X0 × (0, a) = Xa. (502)
The fusion rules ofXa can be easily deduced. For example,
Xa × (b, c) = X0 × (a, 0)× (b, 0)× (c, 0) =∑
e
NeabcXe
Xa ×Xb =∑
c
(c, c)× (a, 0)× (b, 0) =∑
c,e
Neabc(e, c)
(503)Here we introduceNe
abc = dimV eabc =∑f N
fabN
efc.
Let us consider the gauged theory. For eacha ∈ B, (a, a)is an orbit which splits into two topological charges. For eachpair a, b ∈ B, a 6= b, (a, b), (b, a) is an orbit which be-comes a topological charge with dimensions2dadb. Each de-fect splits into two topological charges. All together we have|B|(|B|+7)
2 topological charges.
73
Let us consider the ground state degeneracy on a genusgsurface. This consists of22g different sectors, depending onwhether there is aZ2 twist along any of the2g independentnon-contractible cycles. As discussed in Section VII B, whenG = Z2 all twisted sectors can be mapped onto each otherusing Dehn twists, and therefore have the same number ofground states. It follows that the ground state degeneracy ona genusg surface,Ng, is given by
Ng =NBg (NB
g + 1)
2+ (22g − 1)
NB2g−1 + IB
2g−1
2, (504)
whereNBg is the ground state degeneracy ofB on a genusg
surface. The first term on the RHS is the number ofZ2 invari-
ant states in the untwisted sector. The factorNB
2g−1+IB2g−1
2 isthe number ofZ2 invariant states in each twisted sector, andthere are22g−1 twisted sectors total. Since all twisted sectorscan be mapped to each other with Dehn twists, it is sufficientto consider the sector with a singleZ2 twist along a single cy-cle. The system is then equivalent to a single copy ofB on agenus2g−1 surface, with a non-trivialZ2 action.IB
2g−1 is thenumber ofZ2 invariant states, which is given by the formula:
IB2g−1 =
g−1∏
i=1
(N biaiai
)2Nbgagag
Ib1,b2,...,bg , (505)
with
Ib1,b2,...,bg =∑
ci∈C(N c1
b1b2N c2c1b3
...Ncg−2
cg−3bg−1)2N
cg−2
cg−2bg(506)
To give an interesting example, letB be the Fibonnaci cate-gory. The quantum dimensions of the gauged theory agree ex-actly with theSU(2)8 theory. One can also check that by con-densing the highest spin boson inSU(2)8 the resulting phaseis indeed Fib× Fib [75].
As another example, consider the case whereB = D(G),whereB is the quantum double of a discrete groupG. Then,the gauged theory is(B × B)/Z2 = D((G × G) ⋊ Z2). Forexample, whenG = Z2, B is theZ2 toric code phase, andgauging theZ2 symmetry of two layers of toric code givesthe quantum double of(Z2 ×Z2)⋊Z2, which is the dihedralgroup of order8, D8. This theory has, for example,22 stateson a torus.
I. S3-Gauge Theory D(S3) with Z2 symmetry
First we briefly review the anyon model of D(S3). Thetopological charges are labeled by a pair([a], πa) wherea ∈ S3, [a] is the corresponding conjugacy class andπa isan irreducible representation of the centralizer ofa in S3.There are three conjugacy classes inS3: C1 = e, C2 =(12), (23), (13), C3 = (123), (132). For C1, the central-izer of the identity elemente is just S3 which has three ir-reducible representations, the trivial one, the sign one and the2-dimensional one. The corresponding anyon labels are de-noted byI, B,C, whereI is the vacuum. ForC2, we pick
a = (12) and the centralizer isZ2, so we have two anyonlabelsD andE whereD corresponds to the trivial representa-tion of Z2. ForC3, we pick(123) whose centralizer isZ3, sowe get three anyon labelsF,G,H whereF corresponds to thetrivial representation ofZ3. Altogether we have8 topologicalcharges. For a complete list of the fusion rules,F,R symbolsand the modular data we refer the readers to Ref. [162].
There is an interestingZ2 symmetry in this theory, namelywe can exchangeC andF , a kind of “electromagnetic dual-ity”, since we exchange a charge (representation) and a flux (anontrivial conjugacy class).
We start from the fusion rules of the twist defects. Naıvely,one might write down the following fusion rules
σ × σ = I +G+H. (507)
This is however incorrect. To see the inconsistency let us con-sider the fusionC × C × σ × σ. Consistency between asso-ciativity andG-crossed action requires that
C × C × σ × σ = C × F × σ × σ. (508)
SinceC × σ = σ ×gC = σ × F . The left-hand side gives
(I+B+C)×(I+G+H) = I+B+C+2F+3G+3H, (509)
while the right-hand side
(G+H)×(I+G+H) = 2(I+B+C+F+G+H). (510)
This proves that Eq. (507) does not yield a consistent fusiontheory.
In fact, we can check thatσ×σ = I+G orσ×σ = I+Hboth satisfy Eq. (508). We therefore postulate that both fusionrules are realized inCg. In fact, the resulting gauged theory isSU(2)4×SU(2)4. Condensing the self-dual boson (the boundstate of the highest spin particle in SU(2)4 and the correspond-ing one in the conjugate) indeed gives back D(S3) [78].
To further verify this, and to give an example of some non-trivial features of the ground state degeneracy calculations, letus consider the ground state degeneracy of the gauged the-ory on a torus,N1. D(S3) has 8 states on a torus, but only7 of them are invariant under theZ2 transformation. Further-more, the three twisted sectors each contain the same degener-acy, because they can be mapped into each other under Dehntwists. In the presence of a single twist along one cycle, thereare 6 states, because there are 6Z2 invariant anyons in D(S3).Summing all these states we obtain
N1 = 7 + 3× 6 = 25, (511)
which agrees with the degeneracy of SU(2)4 × SU(2)4 on atorus, as expected.
Now let us consider the genus2 degeneracy,N2. This canbe written as
N2 = I02 + 15Itw
2 . (512)
I02 is the number ofZ2 invariant states in the untwisted sector,
which is the number ofZ2 invariant states of the D(S3) theory
74
on a genus2 surface.Itw2 is the number ofZ2 invariant states
in the presence of a singleZ2 twist along one cycle of thegenus2 surface; the factor15 is the total number of twistedsectors.
Let us begin by computingI02 . To do so, we first pick a
basis for the states in the ungauged theory, which we denote|a1, a2, b〉, and a diagrammatic representation is given by
|a1, a2, b〉 ≡a1
a2
b
And the number of such states is denoted byN 02 . Let us con-
sider theZ2 actionρg on these states. If any of the three la-bels is changed underρg, the state is clearly non-invariant. Allsuch states can be grouped into pairs which are permuted intoeach other under theZ2 action.
The rest of the states all haveZ2-invariant anyon labels, butthis alone does not mean that they areZ2 invariant. Let usdenote the number ofZ2-odd states (but with invariant anyonlabels) byA0
2. TheZ2 action on the fusion states is given by
eiθ(a1,a2,b) =Ug(a1, a1; b)Ug(a2, a2; 0)
Ug(a2, a2; b)Ug(a1, a1; 0). (513)
For simplicity we assume here that all fusion vertices have nomultiplicities.
We considera1 = G, a2 = H, b = 0 orB. The relevantfusion rules are
G×G = I +B +G
G×H = I +B +H
G×H = C + F
(514)
In order to determine theU symbols, we need the followingF -symbols [162]
FGGHH =1√2
[1 1
1 −1
]. (515)
Here the rows of the matrix are indexed byI, B and thecolumns are indexed byC,F . The invariance of theF -symbolunder theZ2 action gives
U(G,G; I) = U(G,H ;C)U(G,C;H)
U(G,G; I) = U(G,H ;F )U(G,F ;H)
U(G,G;B)U(B,H ;H) = −U(G,H ;C)U(G,C;H)
U(G,G;B)U(B,H ;H) = −U(G,H ;F )U(G,F ;H),(516)
from which we conclude that
U(G,G; I) = −U(G,G;B)U(B,H ;H). (517)
Using bending moves one can derive thatU(B,H ;H) =U(H,H ; I)U−1(H,H ;B). Therefore
eiθ(G,H,B) =U(G,G;B)U(H,H ; I)
U(H,H ;B)U(G,G; I)
=U(G,G;B)U(B,H ;H)
U(G,G; I)= −1
(518)
Similarly, we haveeiθ(H,G,B) = −1. One can show that allthe other states areZ2-invariant. Therefore we conclude thatA0
2 = 2.Now we can count the number ofZ2-invariant states, de-
noted byD02. It is easy to see that
D02 =
′∑
a1,a2,b6=C,FN ba1a1N
a2ba2
(519)
where the prime on the sum indicates that the terms where(a1, a2, b) = (G,H,B) or (H,G,B) should be left out. Per-forming the sum, we find
D02 = 56. (520)
The remaining states in the ungauged theory are permutedinto each other under theZ2 action. It follows that
I02 =
N 02 −A0
2 +D02
2(521)
Thus we findI02 = 85.
Now let us computeItw2 . Here, we follow the same strat-
egy as before. In the ungauged theory, there areN 0;tw2 =∑
a1,a2,bN ba1,a1
Nρ(a2)b,a2
= 98 states with a singleZ2 twistalong one cycle. Of these,2 of them acquire a minus signunder theZ2 action,D0;tw
2 = D02 = 56 are already invariant
under theZ2 action, and the rest are permuted into each otherunder theZ2 action. The number ofZ2 invariant states in thetwisted sector is thereforeI0;tw
2 = (98 − 2 + 56)/2 = 76.Therefore we find a total of
N2 = 85 + 15× 76 = 1225, (522)
which agrees exactly with the genus 2 degeneracy of SU(2)4×SU(2)4, as expected.
J. 3-Fermion Model SO(8)1, a.k.a. D′(Z2), with S3 symmetry
In this section, we consider a theory with a non-Abeliansymmetry: the SO(8)1 theory, which has three mutuallysemionic fermionic anyons. We denote the three fermions byψi, i = 1, 2, 3. They form aZ2 × Z2 fusion algebra, similarto that of theZ2 toric code. TheF -symbols are all trivial andtheR-symbols are
Rψiψi1 = −1, Rψiψj = −Rψjψi for i 6= j
Rψ1ψ2
ψ3= Rψ2ψ3
ψ1= Rψ3ψ1
ψ2= −1.
(523)
75
The three fermions can be arbitrarily permuted and the modu-lar data are completely invariant, so the theory has aS3 sym-metry. Recently, this topological phase has been proposedto exist at the surface of a bosonic 3D time-reversal-invarianttopological superconductor [86, 163]. We also notice that thistheory can arise in the following physical way: consider threeidentical layers of semionsI, s1 × I, s2 × I, s3, e.g.three layers ofν = 1
2 bosonic Laughlin states. We identifya subtheoryI, s1s2, s2s3, s1s3 as the SO(8)1 theory withc− = 4 and the rest is the conjugate of semionsI, s1s2s3with c− = −1. So theS3 symmetry is just the permuta-tion symmetry of the three layers, which only acts on thethree-fermion sector. In fact, this type of layer permutationsymmetry and the associated defects have been considered inRef. [12].
As before, first one can calculate the symmetry fractional-ization classH2
ρ(S3,Z2 × Z2) = 0 [164]. So there are nonontrivial symmetry fractionalizations.
Let us set up some notations for theS3 group. Werepresent S3 as the permutation group of three ob-jects and the five non-identity elements are denoted by(12), (23), (13), (123), (132). SinceS3 = Z3 ⋊ Z2, we startour preparatory analysis from the two subgroups.
1. Z2 symmetry
Without loss of generality, we consider theZ2 symmetryexchangesψ1 andψ2, i.e. the sectorC(12). The analysis isalmost the same as what we have done for the toric code, andone can easily see that the two defects should have topological
twists ±e iπn8 and±e iπ(8−n)8 . The distinct choices aren =
1, 3, due toH3(Z2,U(1)) = Z2. For completeness we list thefusion rules:
z± × z± = I + ψ3
z+ × z− = ψ1 + ψ2
z± × ψ1 = z∓, z± × ψ2 = z∓z± × ψ3 = z±
(524)
Here z± denotes the twoZ2 defects. If theZ2 symmetryis gauged, we end up with Ising(1) × Ising(7) or Ising(3) ×Ising(5).
The other two sectorsC(23) andC(13) are similar.
2. Z3 symmetry
UnderG = Z3 action the three fermions are cyclically per-muted:ρg(ψi) =g ψi = ψ[i+1]3 . A simple calculation showsH2ρ(Z3,Z2 × Z2) = 0. There are two defect sectors, corre-
sponding to the two nonzero group elements inG, dual to eachother. Since there is no fixed point, each defect sector has onlyone type of defect, which we denote byw andw respectively.Their fusion rules are easily obtained:
w × ψi = w,w × ψi = w
w × w = I + ψ1 + ψ2 + ψ3,(525)
which impliesdw = dw = 2.Sincew is aZ3 defect, fusingw with itself should givew.
In order to match the quantum dimensions, we have to allowmultiplicity here:
w × w = 2w,w × w = 2w. (526)
Physically, the multiplicity can be understood from the twoanti-commuting Wilson loop operators around twow defects.
To obtain theF symbols we solve the pentagon equa-tions [16, 158]. First we giveF -symbols where no fusionmultiplicities are involved:
F abww = χ(b, ga)
Fwabw = χ(a, gb)
F awbw = χ(b, ga)χ(a, gb)
Fwwab = χ(a, gb)
F awwb = χ(a− b, ga)
Fwawb = χ(b, ga)χ(a, g(a+ b))
[Fwwww ]ab =1
2χ(b, g(a+ b))χ(a, gb)
(527)
Hereχ is a symmetric bi-character on the fusion algebra. Inthe above equations, we can exchangew and w, and at thesame time replaceg with g = g−1 on the right sides. Thebi-characterχ is fixed by theG-crossed heptagon equations:
χ(a, b) = Rba. (528)
We list the rest of the nontrivialF -symbols, which involvefusion multiplicities, in Table I. TheZ3 phaseα, being a 3rdroot of unity, is closely related to the 3rd Frobenius-Schurin-dicator related to the trivalent vertexw × w → w, which isdetermined by the choice inH3(Z3,U(1)) = Z3.
SolvingG-crossed hexagon equations with a gauge fixingη ≡ 1 yields theR-symbols
Rwa = Rwa = −1
Rwww = θwe− πi
3√
3(1,1,1)·σ
, Rwww = θweπi
3√
3(1,1,1)·σ
RwwI = Rwwa = θ−1w
RwwI = θ−1w , Rwwa = −θ−1
w
(529)
Herea = ψ1, ψ2, ψ3 andθ3w = θ3w = α−1. Obviously,θw andθw are the topological twist factors of thew andw defects, re-spectively. As expected, they are determined up to 3rd rootsofunity. Physically this uncertainty can be attributed to possibleZ3 charges attached to the defects.
Notice that theG-crossed hexagon equations can not com-pletely fixRaww andRaww , but subject to the following condi-tions:
∏
a
Raww = 1, Raww Rga,ww = 1. (530)
We also extract the relevantU symbols:
Ug(w,w;w) = − θwαθw
e− iπ
3√
3(1,1,1)·σ
,
Ug(w,w; I) =θwθw.
(531)
76
a I ψ1 ψ2 ψ3
[F awww ](w,0,µ),(w,ν,0) σ0 −iσ1 −iσ3 −iσ2
[Fwaww ](w,0,µ),(w,0,ν) σ0 iσ2 iσ1 iσ3
[Fwwaw ](w,µ,0),(w,0,ν) σ0 iσ3 iσ2 iσ1
[F awww ](w,0,µ),(w,ν,0) σ0 −iσ2 −iσ1 −iσ3
[Fwaww ](w,0,µ),(w,0,ν) σ0 −iσ1 −iσ3 −iσ2
[Fwwaw ](w,µ,0),(w,0,ν) σ0 iσ3 iσ2 iσ1
[Fwwwa ](w,µ,0),(w,ν,0) −αe−
πi
3√
3(1,1,1)·σ
αeπi
3√
3(1,−1,1)·σ
αeπi
3√
3(−1,1,1)·σ
αeπi
3√
3(1,1,−1)·σ
[Fwwwa ](w,µ,0),(w,ν,0) −αe
πi
3√
3(1,1,1)·σ −αe
πi
3√
3(1,−1,−1)·σ −αe
πi
3√
3(−1,−1,1)·σ −αe
πi
3√
3(−1,1,−1)·σ
[Fwwww ](w,µ,ν),(a,0,0)
1√2eπi
3√
3(−1,−1,1)·σ − 1√
2eπi
3√
3(−1,1,−1)·σ − 1√
2eπi
3√
3(1,1,1)·σ 1√
2eπi
3√
3(1,−1,−1)·σ
[Fwwww ](w,µ,ν),(a,0,0)
1√2eπi
3√
3(−1,1,1)·σ 1√
2eπi
3√
3(1,1,−1)·σ − 1√
2e− πi
3√
3(1,1,1)·σ − 1√
2eπi
3√
3(1,−1,1)·σ
[Fwwww ](a,0,0),(w,ν,µ) − i√
2ασ2 − 1√
2ασ0
i√2ασ3 − i√
2ασ1
[Fwwww ](a,0,0),(w,ν,µ)
i√2ασ2 − i√
2ασ3
i√2ασ1
1√2ασ0
TABLE I: F -symbols that involve fusion multiplicities.µ, ν index the fusion states. Theµν matrix element of each entry gives the value ofthe correspondingF -symbol.
In other words, theZ3 symmetry action on theV www space isnontrivial.
We now consider the gauged theory. TheG orbits of C×G
areI, ψ1, ψ2, ψ3, w, w. I, w, w each splits into3, labeled ase.g.(I, n) wheren = 0, 1, 2 labels irreducible representationsof Z3. ψ1, ψ2, ψ3 becomes ad = 3 anyon, which will bedenoted byY . Altogether we have3 + 6 + 1 = 10 anyons.Assumingα = e
2πik3 , θw = θw, the fusion rules between non-
Abelian topological charges can be computed using Eq. (407):
(w, n)× (w,m) = (w, [n+m−k+1]3)+(w, [n+m−k−1]3)
(w, n)× (w,m) = (I, [n+m]3) + Y
(w, n) × Y =∑
m
(w,m)
(w, n)× Y =∑
m
(w,m)
Y × Y =∑
n
(I, n) + 2Y.
(532)Notice that in computing the fusion of(w, n) and (w,m),there are two terms on the right hand side becauseV www carriesa nontrivial reducible2-dimensional representation ofZ3, seeEq. (531).
This fusion category as given in Eq. (532) is identified withthat of SU(3)3 for k = ±1. For later convenience, we labelthe three gauge charges by(I, 0) ≡ 1, (I, 1) ≡ a, (I, 2) ≡ a.The six fluxes are labeled bywi ≡ (w, i), wi = (w, i), i =0, 1, 2.
3. S3 Symmetry
Now we consider theS3 extension. SinceH4(S3,U(1)) =0, there is no obstruction to extension. There are six sectorsCg whereg ∈ S3. From our previous discussions we canwrite down all fusion rules within each sector. The fusionrules between the(12), (23), (13) sectors can also be written
down rather straightforwardly:
xα × zβ = w, zβ × xα = w
xα × yβ = w, yβ × xα = w
yα × zβ = w, zβ × yα = w.
(533)
Let us consider the fusion rules betweenx, y, z andw,w:
x+ × w = x− × w = y+ + y−w × x+ = w × x− = z+ + z−
(534)
The other fusion rules can be obtained by cyclically permutingx, y, z. One may wonder whey we exclude the other possibil-ity x+×w = 2y+(or y−). If this is the case, we can fuse bothsides withψ2 and we getx− × w = 2y−. On the other hand,we can consider fusingψ3 × x+ × w. We first fuseψ3 withx+ and we get backx+ × w. We can also takeψ3 aroundwfirst and it becomesψ1, then fuse it withx+, which leaves uswith x− ×w. So we conclude thatx+ ×w = x− ×w, whichexcludes thex± × w = 2y± fusion rule.
In addition, we need to understand the symmetry actionson the defect sectors. In general,g-action takesCh to Cghg−1.SinceC(123) andC(132) each contain one defect, the nontrivialZ2 action is obviously given by:
(12), (23), (13) : w ↔ w. (535)
The Z3 symmetry has nontrivial actions onC(12),(23),(13).Since each of these sectors contains two defects, we need todetermine the specific actions of(123). Let us consider theaction onC(12). The two defectsz± are distinguished by theeigenvalue of the localψ3 Wilson loop around the defect. Un-der (123), C(12) is mapped toC(23) andψ3 is mapped toψ1.So it is natural to associate defects with the same eigenvaluesof the invariant Wilson loops:
(123) : z± → x±. (536)
The actions on the other two sectors can be obtained similarly.
77
4. Sequentially gauging theS3 symmetry
We are now ready to gauge the wholeS3 symmetry. Ourstrategy is to break theS3 symmetry into theZ3 which is anormal subgroup and theZ2 subgroup and gauge them se-quentially [165]. We have learnt that gauging theZ3 sym-metry gives a SU(3)3-type theory. So we just need to gaugethe remainingZ2 symmetry, which has a nontrivial action onSU(3)3:
ga = a,gwi = wi,gY = Y. (537)
Although not necessary in our following discussion, wewould like to remark that the symmetry action on the two-dimensionalY × Y → Y fusion space is nontrivial. Thiscan be seen by imposing the invariance ofF -symbols undersymmetry action. In this case, the relevantF -symbols are
FY Y Ya =
[− 1
2 −√32√
32 − 1
2
], FY Y Ya =
[− 1
2
√32
−√32 − 1
2
]. (538)
Assume that the symmetry action
˜|Y, Y ;Y, µ〉 =∑
ν
uµν |Y, Y ;Y, ν〉, (539)
where u is a unitary matrix and the conditionFY Y Ya =uFY Y Ya u−1 gives−σy = uσyu
−1. In other words, the sym-metry action on the|Y, Y, Y 〉 space is nontrivial.
Let us consider theZ2 defects. There are two of them dueto the existence of a fixed-point anyonY and we denote themby σ±. We first state the conjectured fusion rules:
σ± × σ± = I + a+ a+ Y +∑
i
(wi + wi)
σ± × σ∓ = 2Y +∑
i
(wi + wi)
w × σ± = σ+ + σ−Y × σ+ = 2σ− + σ+
Y × σ− = 2σ+ + σ−a× σ± = σ±
(540)
To justify these fusion rules, it is useful to revert to theS3
extended category. TheZ2-extended SU(3)3 category shouldbe equivalent to theS3-extended category, but “gauging” theZ3 subgroup. Armed with this perspective, we immediatelysee that theZ2 defects in SU(3)3 are the “equivariantized”orbit of theZ2 defects inC(12),(23),(13). Schematically, wecan write
σ± ≃ x± + y± + z±. (541)
To actually use the general formula Eq. (407), we will haveto solve the entire extended category to obtain theU symbols.However we will just use this expression to make heuristicderivation of the fusion rules. For example,
σ+ × σ+ = (x+ + y+ + z+)× (x+ + y+ + z+)
= (I + ψ1) + (I + ψ2) + (I + ψ3)+
(w + w) + (w + w) + (w + w)
(542)
The multiple occurrence of the vacuumI should be inter-preted asI+a+a (a is theZ3 charge) and similar forw+w.ψ1 + ψ2 + ψ3 is identified withY . We therefore have
σ+ × σ+ = I + a+ a+ Y +∑
i
(wi + wi), (543)
which givesdσ± = 3√2. The other fusion rules can be “de-
rived” in a similar fashion. We have checked that the fusionrules are associative and satisfy all the symmetry properties.
In addition, without solving the crossed hexagon equationsfor the complicated SU(3)3 theory, we can directly read offthe topological twists of theσ± defects, since their twists arethe same as theZ2 defects in the SO(8)1 theory, as suggestedby Eq. (412).
We are now ready to attack our final goal, the gauged the-ory. First we count the number of topological charges. Thevacuum1 and theG-invariant topological chargeY each splitinto two. The twoZ2 defectsσ± split into fourZ2 fluxes.a and a are symmetrized into ad = 2 boson. wi andwi are symmetrized into threed = 4 topological charges,and the twist factors are unchanged. So altogether, we have4+1+3+2+2 = 12 topological charges. Their quantum di-mensions are1, 1, 2, 3, 3, 4, 4, 4, 3
√2, 3
√2, 3
√2, 3
√2. Their
topological twists factors are also known and are listed in Ta-ble II. To get the fusion rule of the gauged theory in principleone needs the full data of theG-crossed braiding, especiallytheUg symbols. Fortunately, in this case we find that merelyrequiring associativity is enough to constrain the fusion rulesobtained by equivariantization. With the fusion rules and thetopological twist factors, we can compute theS-matrix. Thereare6 possibilities for the topological twists in accordance withH3(S3,U(1)) = Z6. Choosingα = e
4πi3 , ν = 1, the result-
ing S-matrix is [165]
DS = (544)
1 1 2 3 3 4 4 4 3√2 3
√2 3
√2 3
√2
1 1 2 3 3 4 4 4 −3√2 −3√2 −3√2 −3√22 2 4 6 6 −4 −4 −4 0 0 0 0
3 3 6 −3 −3 0 0 0 −3√2 −3√2 3√2 3
√2
3 3 6 −3 −3 0 0 0 3√2 3
√2 −3√2 −3√2
4 4 −4 0 0 b c a 0 0 0 04 4 −4 0 0 c a b 0 0 0 04 4 −4 0 0 a b c 0 0 0 0
3√2 −3√2 0 −3√2 3
√2 0 0 0 0 0 6 −6
3√2 −3√2 0 −3√2 3
√2 0 0 0 0 0 −6 6
3√2 −3√2 0 3
√2 −3√2 0 0 0 6 −6 0 0
3√2 −3√2 0 3
√2 −3√2 0 0 0 −6 6 0 0
Here a = −8 cos 2π9 , b = −8 sin π
9 , c = 8 cos π9 . Thecolumns(rows) are ordered as in Table II. We will not writethe fusion rules explicitly, since they can be obtained easilyfrom theS-matrix using the Verlinde formula. To the bestof our knowledge, this12-particle MTC was not previouslyknown.
5. Ground state degeneracy of the gauged theory
Let us also illustrate the ground state degeneracy computa-tions for genus1 and2, by using information obtained only
78
Label d θ
(I,+) 1 1
(I,−) 1 1
a, a 2 1
(Y,+) 3 −1
(Y,−) 3 −1
w,w 4 α−1/3
wa,wa 4 ωα−1/3
wa,wa 4 ω2α−1/3
(σ+,+) 3√2 e
iπν8
(σ−,+) 3√2 −e− iπν
8
(σ+,−) 3√2 −e iπν8
(σ−,−) 3√2 e−
iπν8
TABLE II: Topological charges in the gauged theoryC/S3
from the SU(3)3 theory, before the finalZ2 gauging. To be-gin, the ground state degeneracy on a genus one surface is
N1 = I1 + 3Itw1 , (545)
whereI1 consists of those states of theSU(3)3 theory on atorus that areZ2 invariant, whileItw
1 is the number ofZ2
invariant states in the sector where there is aZ2 twist alongone cycle of the torus. TheSU(3)3 theory has 10 states ona torus; of these, 6 of them areZ2 invariant, soI1 = 6. Thenumber of states in the presence of a singleZ2 twist along onecycle of the torus is 2, and both areZ2 invariant, soItw1 = 2.Therefore, we find
N1 = 12, (546)
in agreement with the results found through other methods.Now let us compute the genus2 degeneracy,N2, which
takes the form:
N2 = I2 + 15Itw2 , (547)
where nowI2 is the number of states in theSU(3)3 theory ona genus2 surface after projecting to theZ2 invariant subspace.Itw2 is the number ofZ2 invariant states in theSU(3)3 theory
on a genus2 surface with aZ2 twist along one cycle. TocomputeI2, we pick a basis|a1, a2, b〉µ for the states in theSU(3)3 theory on a torus. Here the indexµ labels states inthe fusion space.I2 can be obtained by computing the totalnumber of statesN 0
2 =∑
a1,a2,bN ba1,a1
Na2ba2
= 166 and thenumber of diagonal statesD0
2 which satisfy
ρ(|a1, a2, b〉µ) = |a1, a2, b〉µ. (548)
In order to determine the number of such states, it is impor-tant to recall that the two-dimensional fusion/splitting space|Y, Y ;Y 〉 transforms non-trivially under theZ2 action. There-fore,
D02 =
′∑
a1,a2,b∈1,Y N ba1,a1N
a2ba2
= 4. (549)
The prime on the sum indicates that vertices wherea1 = b =Y , or a2 = b = Y , must be omitted as these transform non-trivially under theZ2 action. Therefore,
I2 =166 + 4
2= 85. (550)
Now let us computeItw2 . First, we compute the to-
tal number of states in the presence of aZ2 twist alonga single cycle inSU(3)3, which is given byN 0;tw
2 =∑a1,a2,b
N ba1,a1
Nρ(a2)ba2
= 40. The number of diagonal statesare
D0;tw2 =
′∑
a1,a2,b∈1,Y N ba1,a1N
ρ(a2)ba2
= 4, (551)
where again the prime on the sum indicates that vertices wherea1 = b = Y , or a2 = b = Y , must be omitted as thesetransform non-trivially under theZ2 action. Therefore,Itw
2 =(40 + 4)/2 = 22. Thus we find
N2 = 85 + 15× 22 = 415. (552)
The above calculation agrees exactly with the formulaN2 =
D2(g−1)∑i d
−2(g−1)i , for the quantum dimensions listed
above.
K. Rep(D10) with Z2 symmetry: An H3[ρ](G,A) Obstruction
We provide an example of theH3[ρ](G,A) obstruction (i.e.
obstruction to symmetry fractionalization) in a pre-modularcategory [166]. Consider the dihedral groupD10 = Z10 ⋊ Z2
generated from two elementsr, s with r10 = s2 = 1, srs =r−1. It has8 irreducible representations, four of which are1-dimensional and the others2-dimensional. We will considerthe BTCC = Rep(D10). The fusion rules ofC can be easilydeduced from the character table ofD10, which we spell outexplicitly here: There are four Abelian topological charges,I, A,B,C = AB which form aZ2 × Z2 fusion subalgebraand4 non-Abelian onesX1,2,3,4 of dimension2, such that
A×Xi = Xi
B ×Xi = C ×Xi = X5−i
Xi ×Xj =
Xi+j +X|i−j| i+ j ≤ 5
X10−i−j +X|i−j| i+ j > 5
(553)
where we defineX0 = I +A,X5 = B + C. TheF -symbols(or Wigner 6j-symbols) of this category can be computedfrom the Clebsch-Gordon coefficients.
In addition, this category also admits braiding. In fact, therepresentation category of any finite group can be endowedwith symmetric braiding, i.e. all topological charges havetwist factors1 andSab = dadb
D for all topological chargesa, b, which is apparently non-modular. Therefore a represen-tation category Rep(G) for any finite groupG can not existalone physically, but one can always embed it into the quan-tum double D(G) as the charge sector. It is further known that
79
the quantum double D(G) is the minimal modular extensionof Rep(G) [167].
We now define an obstructedZ2 symmetry on Rep(D10).We first define an automorphismρ on the groupD10 as fol-lowing: ρ(r) = r7, ρ(s) = r5s. We can easily check(ρ ρ)(r) = r−1 = srs, (ρ ρ)(s) = s = sss, soρ ρ isthe conjugation bys. Therefore, althoughρ is not an exactZ2
automorphism on the group (only aZ2 outer automorphism),it still induces aZ2 action on the representations, since rep-resentations are defined up to similarity transformations.Theexplicit action on the label set can be easily found:
ρg(B) = C, ρg(X1) = X3, ρg(X2) = X4. (554)
One can check that the fusion rules and modular data are allinvariant under this symmetry.
However, by directly checking the definition of thesymmetry action, we find that thisZ2 symmetry is notfractionalizable. In other words, it is impossible to frac-tionalize the symmetry in a manner as described in Sec. IV.Therefore, the symmetry is obstructed. Notice that becausethe Rep(D10) category is not modular, we can not directlyrelate the obstruction to an obstruction class inH3
ρ(G,A).However, the group automorphism can actually be turnedinto a topologicalZ2 symmetry in the quantum doubleD(G), which restricts to the obstructed symmetry in the Repcategory and therefore is also obstructed.
Acknowledgments
We would like to thank M. Hermele, A. Kitaev, M. Metlit-ski, V. Ostrik, J. Slingerland, M. Titsworth, and K. Walker forenlightening discussions and especially X. Cui, C. Galindo,and J. Plavnik for sharing unpublished work. M.C. would liketo thank Y.-Z. You for help on figures.
Note added: During the preparation of this manuscript, welearned of related unpublished works [168, 169].
Appendix A: Review of Group Cohomology
In this appendix, we provide a brief review of group coho-mology.
Given a finite groupG, letM be an Abelian group equippedwith aG actionρ : G → M , which is compatible with groupmultiplication. In particular, for anyg ∈ G anda, b ∈M , wehave
ρg(ab) = ρg(a)ρg(b). (A1)
(We leave the group multiplication symbols implicit.) Suchan Abelian groupM with G actionρ is called aG-module.
Let ω(g1, . . . ,gn) ∈ M be a function ofn group elementsgj ∈ G for j = 1, . . . , n. Such a function is called an-cochain and the set of alln-cochains is denoted asCn(G,M).They naturally form a group under multiplication,
(ω · ω′)(g1, . . . ,gn) = ω(g1, . . . ,gn)ω′(g1, . . . ,gn), (A2)
and the identity element is the trivial cochainω(g1, . . . ,gn) = 1.
We now define the “coboundary” mapd : Cn(G,M) →Cn+1(G,M) acting on cochains to be
dω(g1, . . . ,gn+1) = ρg1 [ω(g2, . . . ,gn+1)]
×n∏
j=1
ω(−1)j (g1, . . . ,gj−1,gjgj+1,gj+1, . . . ,gn+1)
× ω(−1)n+1
(g1, . . . ,gn).
(A3)
One can directly verify thatddω = 1 for anyω ∈ Cn(G,M),where1 is the trivial cochain inCn+2(G,M). This is whydis considered a “boundary operator.”
With the coboundary map, we next defineω ∈ Cn(G,M)to be ann-cocycle if it satisfies the conditiondω = 1. Wedenote the set of alln-cocycles by
Znρ (G,M) = ker[d : Cn(G,M) → Cn+1(G,M)]
= ω ∈ Cn(G,M) | dω = 1 .(A4)
We also defineω ∈ Cn(G,M) to be ann-coboundary if itsatisfies the conditionω = dµ for some(n − 1)-cochainµ ∈Cn−1(G,M). We denote the set of alln-coboundaries byAlso we have
Bnρ (G,M) = im[d : Cn−1(G,M) → Cn(G,M)]
= ω ∈ Cn(G,M) | ∃µ ∈ Cn−1(G,M) : ω = dµ .(A5)
Clearly,Bnρ (G,M) ⊂ Znρ (G,M) ⊂ Cn(G,M). In fact,Cn, Zn, andBn are all groups and the co-boundary maps arehomomorphisms. It is easy to see thatBnρ (G,M) is a normalsubgroup ofZnρ (G,M). Since d is a boundary map, we thinkof the n-coboundaries as being trivialn-cocycles, and it isnatural to consider the quotient group
Hnρ (G,M) =
Znρ (G,M)
Bnρ (G,M), (A6)
which is called then-th cohomology group. In other words,Hnρ (G,M) collects the equivalence classes ofn-cocycles that
only differ byn-coboundaries.It is instructive to look at the lowest several cohomology
groups. Let us first considerH1ρ(G,M):
Z1ρ(G,M) = ω | ω(g1)ρg[ω(g2)] = ω(g1g2)
B1ρ(G,M) = ω | ω(g) = ρg(µ)µ
−1 .(A7)
If the G-action onM is trivial, then B1ρ(G,M) = 1
andZ1ρ(G,M) is the group homomorphisms fromG to M .
In general,H1ρ(G,M) classifies “crossed group homomor-
phisms” fromG toM .For the second cohomology, we have
Z2ρ(G,M) = ω | ρg1 [ω(g2,g3)]ω(g1,g2g3)
= ω(g1,g2)ω(g1g2,g3) B2ρ(G,M) = ω | ω(g1,g2) = ε(g1)ρg1 [ε(g2)]ε
−1(g1g2) .(A8)
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If M = U(1), it is well-known thatZ2(G,U(1)) is exactly thefactor sets (also known as the Schur multipliers) of projectiverepresentations ofG, with the cocycle condition coming fromthe requirement of associativity.H2(G,U(1)) classifies allinequivalent projective representations ofG.
For the third cohomology, we have
Z3ρ(G,M) = ω | ω(g1g2,g3,g4)ω(g1,g2,g3g4)
= ρg1 [ω(g2,g3,g4)]ω(g1,g2g3,g4)ω(g1,g2,g3) (A9)
ForM = U(1) and trivialG action,Z3(G,U(1)) is the set ofF -symbols for the fusion category VecG, with the3-cocyclecondition being the Pentagon identity.B3(G,U(1)) is iden-tified with all theF -symbols that are gauge-equivalent to thetrivial one.H3(G,U(1)) then classifies the gauge-equivalentclasses ofF -symbols on VecG.
Appendix B: Projective Representations of Finite Groups
We briefly summarize some basic results of the theory ofprojective representations of finite groups over the complexnumbersC and discuss the unitary case without loss of gener-ality. For proofs, we refer the readers to Ref. [170].
Consider a finite groupG and a normalized2-cocycleω ∈Z2(G,U(1)). SupposeV is a non-zero vector space overC.A ω-representation ofG over the vector spaceV is a mapπ : G→ GL(V ) such that
π(g)π(h) = ω(g,h)π(gh), ∀g,h ∈ G
π(0) = 11.(B1)
We denote theω-projective representative by a triple(ω, π, V ), or for brevity (π, V ) or simply π below. Alsonπ ≡ dimV .
Two ω-representations(π1, V1) and (π2, V2) are ω-isomorphic, denoted asπ1 ∼ω π2, if and only if there exits anisomorphismS betweenV1 andV2 such thatSπ1(g)S−1 =π2(g), ∀g ∈ G.
Given twoω-representations(π1, V1) and(π2, V2), we canform their direct sum, which is aω-representation ofG overV1 ⊕ V2. In matrix form, we have
(π1 ⊕ π2)(g) ≡[π1(g) 0
0 π2(g)
]. (B2)
Clearly π1 ⊕ π2 also has the same factor setω. How-ever, there is no natural way of defining a direct sum of aω-representation and aω′-representation whenω 6= ω′.
One can also define a tensor product of two projec-tive representations. Given two projective representations(ω1, π1, V1) and(ω2, π2, V2), their tensor productπ1 ⊗ π2 isdefined as(π1 ⊗ π2)(g) = π1(g) ⊗ π2(g) over the vectorspaceV1 ⊗ V2. The factor set of the tensor productπ1 ⊗ π2 isω1ω2.
Similar to linear representations, one can define reducibleand irreducible projective representations. A projectiverepre-sentation(ω, π, V ) is called irreducible if the vector spaceV
has no invariant subspace under the mapπ other than0 or V .A projective representation is reducible if it is not irreducible.A reducible projective representation always decomposes intoa direct sum of irreducible projective representations with thesame factor set.
Given a projective representationπ of G, its characterχπ :G→ C is defined to be
χπ(g) = Tr[π(g)
]. (B3)
It follows that
χπ(0) = nπ (B4)
χπ(g−1) = ω(g,g−1)χ∗
π(g) (B5)
where we use the identityω(g,g−1) = ω(g−1,g).Another more nontrivial relation is
χπ(hgh−1) =
ω(h−1,hgh−1)
ω(g,h−1)χπ(g), (B6)
which reveals an important difference between projective andregular characters, because regular characters depend only onthe conjugacy classes.
Given twoω-representationsπ1 andπ2, obviously, one hasχπ1⊕π2 = χπ1 + χπ2 andχπ1⊗χ2 = χπ1χπ2 .
As in the theory of linear representations, characters are im-portant because they distinguish the isomorphism classes ofirreducible projective representations:
Twoω-representations areω-isomorphic if and only if theyhave the same character.
Analogous to the familiar character theory of linear repre-sentations, one can show that the projective characters satisfysome orthogonality relations. We give the first orthogonalityrelation here and discuss the second one later. For two irre-ducibleω-representationsπ1 andπ2, we have
1
|G|∑
g∈Gχπ1(g)χ
∗π2(g) =
1 if π1 ∼ω π20 otherwise
(B7)
One can use the characters to decompose projective repre-sentations. Namely, fix a factor setω, let π be a projectiverepresentation (not necessarily irreducible) ofG andπ′ an ir-reducible projective representation. The multiplicity ofπ′ inπ can be computed by
m(π′, π) =1
|G|∑
g∈Gχπ′(g)χ∗
π(g). (B8)
In general, given twoω-representationsπ andπ′ (neitherof which is necessarily irreducible), we define the multiplicitym(π′, π) as
m(π′, π) = dim HomG(Vπ′ , Vπ). (B9)
HereHomG(Vπ′ , Vπ) is the space of intertwining operators,i.e. linear maps betweenVπ′ andVπ which commute with theG actions. Note that theG action onVπ is given exactly by therepresentationπ. Schur’s lemma implies that ifπ is an irrep,
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thenHomG(Vπ, Vπ) = C11Vπ , i.e. all intertwiners are scalarmultiplications. If π andπ′ are irreducible representationsthat are not isomorphic, thenHomG(Vπ , Vπ′) = 0. There-fore, given twoω-representationsπ andπ′, we can decom-pose them into the direct sum ofω-irreps:π = ⊕iNiπi, π′ =⊕iN ′
iπi, whereπi is the complete set ofω-irreps andNi, N ′i
are multiplicities, respectively. Then a general intertwinerΦ ∈ HomG(Vπ′ , Vπ) is of the form
Φ =⊕
i
(Mi ⊗ 11Vπi ). (B10)
HereMi is a linear map betweenCNi andCN′i , i.e. anNi×N ′
i
complex matrix, which can be thought as a vector in anNiN′i -
dimensional complex vector space. It follows that
dimHomG(Vπ′ , Vπ) =∑
i
NiN′i . (B11)
For applications, we can show that Eq. (B8) applies to thegeneral case, too.
A special projective representation, theω-regular represen-tation, is defined asR(g)eh = ω(g,h)egh, whereeg|g ∈G is a basis for a|G|-dimensional vector space. Its characterχR(g) = |G|δg0. Using Eq. (B8), we see that theω-regularrepresentation is reducible and each irreducible projective rep-resentationπ appears exactlynπ times in its decomposition.Consequently, we have the following two relations
∑
π
n2π = |G|,
∑
π
nπχπ(g) = |G|δg0. (B12)
The sum is over all irreducibleω-projective representationsπ.An elementg ∈ G is called anω-regular element if and
only if ω(g,h) = ω(h,g) for all h ∈ Ng, whereNg is thecentralizer ofg in G. Moreover,g is ω-regular if and onlyif all elements in its conjugacy class[g] areω-regular. Thisproperty follows from the2-cocycle condition.
Now considerh ∈ Ng, so
χπ(g) = χπ(h−1gh) =
ω(h,g)
ω(g,h)χπ(g) (B13)
Therefore, ifg is notω-regular,χπ(g) = 0. In fact, one canshow that an elementg is ω-regular if and only ifχπ(g) 6= 0for some irreducible representationπ.
We then have the following important result:Fix a factor setω. The number of non-isomorphic irre-
ducible projectiveω-representations ofG is equal to the num-ber ofω-regular conjugacy classes ofG.
We can now state the second orthogonality relation: Letg1,g2, . . . be a complete set of representatives forω-regularclasses ofG. For any twoω-regular elementsgi andgj,
∑
π
χπ(gi)χ∗π(gj) = |Ngi |δij . (B14)
The sum is over all irreducibleω-projective representationsπ.
If two factor setsω andω′ belong to the same equivalenceclass inH2(G,U(1)), then we have
ω′(g,h) =µ(g)µ(h)
µ(gh)ω(g,h) (B15)
for someµ(g) : G→ U(1) with µ(0) = 11.Given aµ as above and an irreducibleω-projective repre-
sentationπ, we can then construct anotherω′-projective rep-resentationπ′(g) = µ(g)π(g). Clearly, the two proceduresabove define a one-to-one correspondence. Their charactersalso differ byµ, that isχπ′(g) = µ(g)χπ(g).
Appendix C: Z(w)N and Ising(ν) Anyon Models
TheZ(w)N anyon model forN a positive integer can have
w ∈ Z for all N andw ∈ Z + 12 for N even. It hasN
topological charges labeled bya = 0, 1, . . . , N − 1 whichobey the fusion rulesa× b = [a+ b]N . TheF -symbols are
[F abc[a+b+c]N][a+b]N ,[b+c]N = e
2πiwN
a(b+c−[b+c]N). (C1)
Notice that they are all1 whenw ∈ Z. Forw ∈ Z+ 12 , some
of theF -symbols are equal to−1 which can not be gaugedaway in general.
TheR-symbols are
Rab[a+b]N = e2πiwN
ab. (C2)
The twist factors areθa = e2πiwN
a2 .Notice thatw is periodic inN . For oddN , Z(w)
N are mod-
ular exceptw = 0. For evenN , Z(w)N are modular only for
w ∈ Z+ 12 .
The anyon model Ising(ν) whereν is an odd integer hasthree topological chargesI, σ, ψ, where the vacuum chargehere is denotedI, and the nontrivial fusion rules are given by
ψ × ψ = I, σ × ψ = σ, σ × σ = I + ψ. (C3)
The nontrivialF -symbols are
Fψσψσ = F σψσψ = −1
[F σσσσ ]ab =κσ√2
[1 1
1 −1
]
ab
.(C4)
Here the column and row values of the matrix take valueI
andψ (in this order).κσ = (−1)ν2−1
8 is the Frobenius-Schurindicator ofσ.
TheR-symbols are
Rψσσ = Rσψσ = (−i)ν
RσσI = κσe−iπν8 , Rσσψ = κσe
i 3πν8 .(C5)
The twist factorθσ = eiπν8 uniquely distinguishes the eight
distinct Ising(ν) anyon models, as does the chiral centralchargec−mod8 = ν
2 .The Ising TQFT corresponds toν = 1, SU(2)2 corresponds
to ν = 3, andν ≥ 5 can be realized in SO(ν)1 Chern-Simonsfield theory.
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Appendix D: Categorical Formulation of Symmetry, Defects,and Gauging
In this appendix,G will always denote a finite group andCa unitary modular tensor category (UMTC) unless otherwisestated explicitly. AlsoAut(C) below isAut0,0(C) in the maintext. For a categoryC, x ∈ C means thatx is an object ofC,andC is the complex conjugate category ofC. The materials inthis appendix are distilled from Refs. [80–82, 101, 139, 165].
1. Categorical Topological and Global Symmetry
A categorical-groupG is a monoidal categoryG whose ob-jects and morphisms are all invertible. The complete invariantof a categorical-groupG is the triple(π1(G), π2(G), φ(G)),whereπ1(G) is the group of the isomorphism classes of ob-jects ofG, π2(G) the abelian group of the automorphisms ofthe tensor unit1 of G, andφ(G) ∈ H3(π1(G);π2(G)) thegroup3-cocyle that represents the associativity of the tensorproduct⊗ of G (π1(G) acts onπ2(G) and they form a crossmodule as the notation suggests) [171].
A groupG can be promoted to a categorical-groupG asfollows: the objects ofG are the group elements ofG, andthe morphism set Hom(g,h) of two objectsg,h is empty ifg 6= h and contains only the identity ifg = h. We willuseAut(C) to denote the categorical-group of braided ten-sor autoequivalences ofC. The tensor product of two braidedtensor autoequivalences is their composition. The morphismbetween two braided tensor autoequivalences are the naturalisomorphisms between the two functors. We will callAut(C)the categorical topological symmetry group ofC.
Given a UMTCC, π2(C), i.e. π2(G) for G = Aut(C), isisomorphic to the group of the invertible object classes ofCas an abstract finite abelian group, which we denote byA inthe main text, but the finite groupπ1(C), i.e. π1(G) for G =Aut(C), is difficult to determine in general except for abelianmodular categories.
We will also useAut(C) to denoteπ1(C): the group ofequivalence classes of braided tensor autoequivalences ofC.This ordinary group is the demotion (or decategorification)of the categorical-groupAut(C) and is called the topologicalsymmetry ofC.
Definition 1. Given a groupG, a monoidal functorρ : G →Aut(C) is called a categorical global symmetry ofC.
We will denote the categorical global symmetry as(ρ,G)or simplyρ and say thatG acts categorically onC.
A categorical global symmetry can be demoted to a grouphomomorphismρ : G → Aut(C), which is called a globalsymmetry ofC.
To understand a categorical-group actionG on a UMTCC,we will start with a global symmetryρ : G → Aut(C). It isnot true that we can always lift such a group homomorphismto a categorical-group functorρ. The obstruction for the exis-tence of such a lifting is the pull-back group cohomology classρ∗(φ(C)) ∈ H3(G;π2(C)) of φ(C) ∈ H3(π1(C), π2(C)) byρ.
If this obstruction class does not vanish, thenG cannot act cat-egorically onC so that the decategorified homomorphism isρ.If this obstruction does vanish, then there are liftings ofρ tocategorical-group actions, but such liftings are not necessar-ily unique. The equivalence classes of all liftings form a tor-sor overH2
ρ(G, π2(C)). We will denote the categorical globalsymmetryρ also by a pair(ρ, t), whereρ : G → Aut(C) andt ∈ H2
ρ(G, π2(C)).
2. Symmetry Defects
A module categoryM over a UMTCC is a categorical rep-resentation ofC. A left module categoryM overC is a semi-simple category with a bi-functorαM : C × M → M thatsatisfies the analogues of pentagons and the unit axiom. Sim-ilarly for a right module category. A bi-module category isa simultaneously left and right module category such that theleft and right actions are compatible. Bi-module categoriescan be tensored together just like bi-modules over algebras.WhenC is braided, a left module category naturally becomesa bi-module category by using the braiding. A bi-module cat-egoryM overC is invertible if there is another bi-module cat-egoryN such thatM⊠N andN ⊠M are both equivalent toC—the trivial bi-module category overC. The invertible (left)module categories over a modular categoryC form the Picardcategorical-group Pic(C) of C. The Picard categorical-groupPic(C) of a modular categoryC is monoidally equivalent tothe categorical-groupAut(C) [81]. This one-one correspon-dence between braided auto-equivalences and invertible mod-ule categories is an important relation between symmetry andextrinsic topological defects.
Given a categorical global symmetry(ρ,G) of a UMTCC and an isomorphism of categorical groupsPic(C) withAut(C), then eachρg ∈ Aut(C) corresponds to an invertiblebi-module categoryCg ∈ Pic(C).
Definition 2. An extrinsic topological defect of fluxg ∈ G isa simple object in the invertible module categoryCg ∈ Pic(C)overC corresponding to the braided tensor autoequivalenceρg ∈ Aut(C).
The analogue of the Picard categorical-group of a mod-ular category for a fusion categoryC is the Brauer-Picardcategorical-group of invertible bi-module categories over C.But invertible bi-module categories over a fusion categoryCis in one-one correspondence with braided auto-equivalencesof the Drinfeld centerD(C) of C (also known as the quan-tum double ofC in physics literature) [81], not tensor auto-equivalences ofC itself. WhenC is modular, thenD(C) ∼= C⊗C. Note thatPic(C) is naturally included in the Brauer-Picardgroup ofC andAut(C) included naturally in the categorical-group of braided tensor auto-equivalences ofD(C). The im-ages of the two inclusions intersect trivially.
The topological defects in theg-flux sector form an in-vertible bi-module categoryCg over the UMTCC. Defectscan be fused and their fusion corresponds to the tensor prod-uct of bi-module categories. Since all defects arise from the
83
same physics, fusions of defects for all flux sectors shouldbe consistent. Such a consistency is encoded as the collec-tion Cg,g ∈ G of flux sectors gives rise to an extensionof C to a unitaryG-crossed modular category. Given a cat-egorical global symmetry(ρ, t), it is not always possible todefine defect fusions so that we could obtain such an exten-sion. Given fluxesg,h, we need to choose an identificationMgh : Cg ⊠ Ch ∼= Cgh. For four fluxesg,h,k, l ∈ G,the two paths of the pentagon using theMgh’s to identify((Cg⊠Ch)⊠Ck)⊠Cl with Cg⊠(Ch⊠(Ck⊠Cl)) could differ bya phase. The collection of those phases forms a cohomologyclass inH4(G; U(1)), which is the obstruction class to con-sistent pentagons for the flux sectors. If this obstruction classvanishes, then we need to choose a group cohomology classα ∈ H3(G; U(1)) to specify the associativity of the flux sec-tors. A subtle point here is that the consistency requirementvia pentagons for flux sectorsCg is strictly stronger than thatfor all defects separately.
Given a triple(ρ, t, α) as above when the obstruction classin H4(G,U(1)) vanishes, where(ρ, t) is a categorical globalsymmetry andα ∈ H3(G,U(1)) specifies associtivity of theflux sectors, we can construct aG-crossed modular extensionof C, which describes the extrinsic topological defects ofC.In the following, we will call such a triple(ρ, t, α) a gaugingdata. The extensionC×
G = C(ρ,t,α) =⊕
g∈G Cg of C = C0is a unitaryG-crossed modular category—a unitaryG-crossedfusion category with a compatible non-degenerateG-braiding.
A G-grading of a fusion categoryC is a decomposition ofC into
⊕g∈G Cg. We will consider only faithfulG-gradings
so that none of the componentsCg = 0. The tensor productrespects the grading in the senseCg ⊠ Ch ⊂ Cgh. SinceCg−1
is the inverse ofCg, Cg is naturally an invertible bi-modulecategory overC0, where0 ∈ G is the identity element. Acategorical actionρ of G onC is compatible with the gradingif ρ(g)Ch ⊂ Cghg−1 . A G-graded fusion categoryC with acompatibleG-action is called aG-crossed fusion category.
SupposeC×G =
⊕g∈G Cg is an extension of a unitary fu-
sion categoryC0, i.e. C×G is a unitaryG-crossed fusion cat-
egory. LetIg,g ∈ G be the set of isomorphism classes ofsimple objects inCg andIrr(Cg) = Xii∈Ig be a set of rep-resentatives of simple objects ofCg. The cardinality ofIg is
called the rank of the componentCg, andDg =√∑
i∈Ig d2i
is the total quantum dimension of componentCg, wheredi isthe quantum dimension ofXi ∈ Irr(Cg).
Theorem D.1([80, 82]). Let C =⊕
g∈G Cg be an extensionof a unitary fusion categoryC0. Then
1. The rank ofCg is the number of fixed points of the actionof g onI0.
2. D2g = D2
h for all g,h ∈ G.
The extensionC×G =
⊕g∈G Cg of a UMTCC0 for the sym-
metry (ρ, t), while not braided in general, has aG-crossedbraiding. Given aG-crossed fusion categoryC with categor-ical G-actionρ, we will denoteρg(Y ) for an objectY of Cby gY . A G-braiding is a collection of natural isomorphisms
cX,Y : X ⊗ Y → gY ⊗ X for all X ∈ Cg, Y ∈ C, whichsatisfies a generalization of the Hexagon equations.
A UMTC is a unitary fusion category with a non-degeneratebraiding. A unitaryG-crossed modular category is a unitaryG-crossed fusion category with a non-degenerateG-braiding.An easy way to define non-degeneracy of braiding is throughthe non-degeneracy of the modularS-matrix. To define thenon-degeneracy of theG-crossed braiding, we will introducethe extendedG-crossedS andT operators on an extendedVerlinde algebra. Likewise, the extendedS andT operatorswill give rise to a projective representation ofSL(2,Z). Webelieve that theS andT operators will determine the unitaryG-crossed modular categoryC×
G .
Theorem D.2([81]). The unitaryG-crossed fusion categoryextensionC×
G of a UMTCC has a canonicalG-braiding andcategoricalG-action that makeC×
G into a unitaryG-crossedmodular category.
Given a categorical global symmetryρ : G → Aut(C) ofa UMTC C, an extension ofC to a non-degenerate braidedfusion category corresponds to a lifting ofρ to a categorical2-group functorρ : G → Aut(C). The existence of such lift-
ings has an obstruction inH4(G; U(1)), which is the same asthe obstruction for solving pentagons of flux sectors. Whenthe obstruction class vanishes, the choices correspond to co-homology classes inH3(G; U(1)). If we choose a cohomol-ogy classα ∈ H3(G; U(1)), then we have a lifting to a cate-gorical2-group morphism. Since all other higher obstructionclasses vanish, the categorical global symmetry can be liftedto a morphism of any higher categorical number. As extendedG-action andG-braiding are higher categorical-number mor-phisms, so they can always be lifted. Furthermore, since allhigher obstruction classes vanish, the liftings are unique.
To see theG-action andG-crossed braiding concretely,consider the functor category Fun(Cg, Cgh). On one hand, thiscategory can be identified asCh by Cg ⊠ Ch ∼= Cgh, and onthe other hand, asCghg−1 by Cghg−1 ⊠ Cg ∼= Cgh. There-fore, we have an isomorphismCg ∼= Cghg−1 . This defines anextended action ofG on C×
G . By the same consideration, wehaveCg ⊠ Ch ∼= Cghg−1 ⊠ Cg. This defines theG-crossedbraiding ofC×
G .To define the extendedS, T -operators, we first define an
extended Verlinde algebra. For each pairg,h of commutingelements ofG, we define the following extended Verlinde al-gebra component:Vg,h(C) =
⊕i∈Ih Hom(Xi,
gXi).Then the extended Verline algebra is
V(C) =⊕
(g,h)|gh=hgVg,h(C).
Note that V0,0 is the Verlinde algebra ofC, whichhas a canonical basis given by the identity morphisms ofHom(Xi, Xi), i ∈ I0. Unlike the usual Verlinde algebra ofC, the extended Verlinde algebra does not have such canon-ical basis. One choice of basis isρg : Xi → gXi, and theywill give rise to extendedG-crossedS andT transformations.However, this depends on the choice of cocycle representa-tive of α. Therefore, the extendedS andT operators are not
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canonically matrices. We call aG-crossed braided sphericalfusion categoryG-crossed modular if the extendedS operatoris invertible.
3. Gauging Categorical Global Symmetry
Let G be the promotion of a groupG to a categorical2-group, andAut(C) be the categorical2-group of braided ten-
sor auto-equivalences.
Definition 3. A categorical global symmetryρ : G→ Aut(C)can be gauged ifρ can be lifted to a categorical2-group func-tor ρ : G→ Aut(C).
Given a categorical global symmetry(ρ,G) of a UMTCC,gaugingG is possible only when the obstruction as above inH4(G; U(1)) vanishes. Then the gauging result in generaldepends on a gauging data(ρ, t, α). Given a gauging data(ρ, t, α), gauging is defined as the following two-step pro-cess: first extendC to a unitaryG-crossed modular categoryC×G with a categoricalG action; Then perform the equivari-
antization of the categoricalG action onC×G , which results in
a UMTC (C×G)
G, also simply denoted asC/G. The “bosonic”
symmetric categoryRep(G) is always contained inC/G as aTannakian subcategory. Therefore, gauging actually leadstoa pairRep(G) ⊂ C/G.
SupposeC is a fusion category with aG action. The equiv-ariantization ofC, denoted asCG, is also called orbifolding.The result of equivariantization of aG-action on a fusion cat-egoryC is a fusion category whose objects are(X, φgg∈G),whereX is an object ofC andφg : gX → X an isomorphismsuch thatφ0 = id andφg · ρg(φh) = φgh · κg,h, whereκg,hidentifiesρh · ρg with ρgh. Morphisms between two objects(X, φgg∈G) and(Y, ψgg∈G) are morphismsf : X → Ysuch thatf · φg = φh · ρg(f).
The simple objects ofCG are parameterized by pairs([X ], πX), where[X ] is an orbit of theG-action on simple ob-jects ofC, andπX is an irreducible projective representationof GX—the stabilizer group ofX . The quantum dimensionof ([X ], πX) is dim(πX) · N[X] · dX , whereN[X] is the sizeof the orbit[X ]. Fusion rules can be similarly described usingalgebraic data [83].
In general, gauging is difficult to perform explicitly. Thefirst extension step is very difficult. The second equivarianti-zation step is easier if the6j-symbols of the gauged UMTCC/G are not required explicitly. Different triples of gaugingdata might lead to the same gauged UMTC.
Gauging has an inverse process, which is the condensationof anyons in the Tannakian subcategoryRep(G). This con-densation process is mathematically called taking the coreofthe pairRep(G) ⊂ C/G [139]. Taking a core is a powerfulmethod to verify a guess for gauging because anyon conden-sation is sometimes easier to carry out than gauging.
When C is a G-crossed modular category with faithfulgrading, then its equivariantization is also modular and viseversa [80]. There is the forgetful functorF : CG →
C by F (X, φgg∈G) = X and its adjointG(X) =⊕g∈G(
gX, (µX)g), where(µX)g = κg,h. They inter-twine the extendedS, T operators.
Our equivariantization in gauging is applied to aG-crossedextensionC×
G of a modular categoryC. WhenC×G has a faith-
ful grading, then the non-degeneracy of the braiding ofC isequivalent to the non-degeneracy of the braiding ofC×
G [139].
4. General Properties of Gauging
Gauging and its inverse – condensation of anyons – are in-teresting constructions of new modular categories from oldones. The resulted new modular categories have many inter-esting relations with the old ones.
Theorem D.3 ([139]). Let C be a UMTC with a categoricalglobal symmetry(ρ,G). ThenC ⊗ C/G ∼= D(C×
G).It follows that
1. Chiral topological central charge is invariant undergauging (mod8).
2. The total quantum dimensionDC/G = |G|DC .
The following theorem says that gauging a quantum doubleresults in a quantum double.
Theorem D.4. SupposeG acts categorically onD(C). ThenD(C)/G = D(C×
G).
When the symmetry groupG has a normal subgroupN ,then we can first gaugeN , and then gauge their quotientH =G/N . This sequentially gauging is useful for gauging non-abelian groupsG such asS3.
Theorem D.5. Let ρ : G −→ Pic(C), then there existρ1 :
N −→ Pic(C), ρ2 : H −→ Pic(C/N), such that(C/N)/H
is braided equivalent toC/G.
Proofs of Theorems D.4 and D.5 will appear in Ref. [165].The construction from a modular categoryC with a G-
action to a modular categoryC/G with a Tannakian subcate-goryRep(G) by gauging can be regarded as a new way to con-struct interesting modular categories in the same Witt class.WhenC is weakly integral, then the gauged categoryC/G isalso weakly integral. The inverse process of condensation im-plies that pairs(C, ρ) and(C,Rep(G)) are in one-one corre-spondence.
5. New Mathematical Results
In higher category theory, it is common practice to stric-tify categories as much as possible by turning natural isomor-phisms into identities. This is desirable because strictificationsimplifies many computations and does not lose any generalitywhen we are interested in gauge invariant quantities in classi-fication problems. The drawback is that we have to work withmany objects. In this paper, our preference is the opposite,
85
in the sense that we would like to work with as few objectsas possible. Hence, our goal is to have a skeletal formulationwith full computational power, so that we can calculate nu-merical quantities, such as amplitudes of quantum processes,which are not necessarily gauge invariant. A category is skele-tal if there is only one object in each isomorphism class, andingeneral strictness and skeletalness cannot be obtained simulta-neously, as may be demonstrated, for example, by the semiontheoryZ(1/2)
2 . Therefore, we need to skeletonize the exist-ing mathematical theories. The situation is analogous to theone of a connection or gauge field: mathematically it is goodto define a connection as a horizontal distribution, while, inpractice, it is better to work with Christoffel symbols, espe-cially in physics.
Our first mathematical result is a skeletonization ofG-crossed braided fusion category in Sec. VI. We pro-vide a definition of aG-crossed braided category usinga collection of quantities organized into a basic data set:N cab, F
abcd , Rabc , Uk (a, b; c), andηx(g,h) that satisfy certain
consistency polynomial equations. The fusion coefficientsN cab and associativityF -symbolsF abcd are as usual, but theR-
symbolsRabc are extended to incorporate theG-crossed struc-ture. The new dataUk (a, b; c) andηx(g,h), respectively en-code the categorical symmetry: monoidal functors and naturalidentificationsρgh with ρgρh. A good example of new con-sistency equations are ourG-crossed Heptagon Eqs. (272) and
(273), which generalize the usual Hexagon equations.Our numericalization of aG-crossed braided fusion cate-
gory provides the full computational power for any theory us-ing diagrammatical recouplings, though care has to be takenwhen strands pass over/under local extremals. This com-putational tool is especially useful for dealing with gauge-dependent quantities, which, in theG-crossed theory, includethe important extendedG-crossed modularS andT transfor-mations. As an application, we generalize the Verlinde formu-las to theG-crossed Verlinde formulas Eqs. (331) and (332).The diagrammatical recouplings also allow us to prove thetheorems mentioned above in an elementary way. In partic-ular, we prove that the extendedG-crossed modularS andT transformations indeed give rise to projective representa-tions of SL(2,Z). We also conjecture the topological twistsfor the gauged (equivariantized) theory and derive the modu-lar S-matrix of the gauged theory.
In Sec. X, we catalog many examples. Those examples il-lustrate our theory and also potentially lead to new modularcategories. An interesting example is the gauging of theS3-symmetry of the three-fermion theory SO(8)1. The resultingrank12 weakly integral modular tensor category has not pre-viously appeared in the literature. It would be interestingtosee if the triality of the Dynkin diagramD4 would provide in-sight into the construction of this new modular category fromSO(8)× S3.
[1] X.-G. Wen, Quantum Field Theory of Many-body Systems:From the Origin of Sound to an Origin of Light and Elec-trons, Oxford Graduate Texts (Oxford University Press, Ox-ford, 2004).
[2] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D.Sarma, Rev. Mod. Phys.80, 1083 (2008), arXiv:0707.1889.
[3] S. B. Bravyi and A. Kitaev, quant-ph/9811052.[4] H. Bombin, Phys. Rev. Lett.105, 030403 (2010),
arXiv:1004.1838.[5] A. Kitaev and L. Kong, Comm. Math. Phys.313, 351 (2012).[6] M. Barkeshli and X.-L. Qi, Phys. Rev. X2, 031013 (2012),
arXiv:1112.3311.[7] Y.-Z. You and X.-G. Wen, Phys. Rev. B86, 161107 (2012),
arXiv:1204.0113.[8] Y.-Z. You, C.-M. Jian, and X.-G. Wen, Phys. Rev. B87,
045106 (2013), arXiv:1208.4109.[9] N. H. Lindner, E. Berg, G. Refael, and A. Stern, Phys. Rev. X
2, 041002 (2012), arXiv:1204.5733.[10] D. J. Clarke, J. Alicea, and K. Shtengel, Nature Comm.4,
1348 (2013), arXiv:1204.5479.[11] M. Cheng, Phys. Rev. B86, 195126 (2012), arXiv:1204.6084.[12] M. Barkeshli, C.-M. Jian, and X.-L. Qi, Phys. Rev. B87,
045130 (2013), arXiv:1208.4834.[13] M. Barkeshli, C.-M. Jian, and X.-L. Qi, Phys. Rev. B88,
241103(R) (2013), arXiv:1304.7579.[14] M. Barkeshli, C.-M. Jian, and X.-L. Qi, Phys. Rev. B88,
235103 (2013), arXiv:1305.7203.[15] A. Vaezi, Phys. Rev. B87, 035132 (2013).[16] J. C. Teo, A. Roy, and X. Chen, arXiv:1306.1538.[17] J. C. Y. Teo, A. Roy, and X. Chen, Phys. Rev. B90, 155111
(2014), arXiv:1308.5984.
[18] B. J. Brown, S. D. Bartlett, A. C. Doherty, and S. D. Barrett,Phys. Rev. Lett.111, 220402 (2013).
[19] O. Petrova, P. Mellado, and O. Tchernyshyov, Phys. Rev. B88, 140405 (2013).
[20] A. Kapustin, arXiv:1306.4254.[21] M. N. Khan, J. C. Y. Teo, and T. L. Hughes, arXiv:1403.6478.[22] A. Y. Kitaev, Annals Phys.303, 2 (2003), quant-ph/9707021.[23] M. H. Freedman, Proc. Natl. Acad. Sci. USA95, 98 (1998).[24] J. Preskill, inIntroduction to Quantum Computation, edited
by H.-K. Lo, S. Popescu, and T. P. Spiller (World Scientific,1998), quant-ph/9712048.
[25] M. H. Freedman, M. J. Larsen, and Z. Wang, Commun. Math.Phys.227, 605 (2002), quant-ph/0001108.
[26] M. H. Freedman, M. J. Larsen, and Z. Wang, Commun. Math.Phys.228, 177 (2002), math/0103200.
[27] M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang, Bull.Amer. Math. Soc. (N.S.)40, 31 (2003), quant-ph/0101025.
[28] F. A. Bais, J. K. Slingerland, and S. M. Haaker, Physical Re-view Letters102, 220403 (2009), arXiv:0812.4596.
[29] S. Beigi, P. W. Shor, and D. Whalen, Comm. Math. Phys.306,663 (2011).
[30] J. Wang and X.-G. Wen, arXiv:1212.4863.[31] M. Levin, Phys. Rev. X3, 021009 (2013).[32] T. Lan, J. Wang, and X.-G. Wen, arXiv:1408.6514.[33] L.-Y. Hung and Y. Wan, arXiv:1408.0014.[34] M. Barkeshli and X.-L. Qi, arXiv:1302.2673.[35] M. Barkeshli, E. Berg, and S. Kivelson, Science346, 722
(2014), arXiv:1402.6321.[36] M. Barkeshli, Y. Oreg, and X.-L. Qi, arXiv:1401.3750.[37] T. Iadecola, T. Neupert, C. Chamon, and C. Mudry,
arXiv:1407.4129.
86
[38] X.-G. Wen, Phys. Rev. B65, 165113 (2002).[39] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig,
Phys. Rev. B78, 195125 (2008).[40] A. Kitaev, AIP Conf. Proc.1134, 22 (2009).[41] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys.82, 3045 (2010).[42] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys.83, 1057 (2011).[43] L. Fidkowski and A. Kitaev, Phys. Rev. B81, 134509 (2010).[44] A. M. Turner, F. Pollmann, and E. Berg, Phys. Rev. B83,
075102 (2011).[45] X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B83, 035107
(2011).[46] L. Fidkowski and A. Kitaev, Phys. Rev. B83, 075103 (2011).[47] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B
87, 155114 (2013), arXiv:1106.4772.[48] Y.-M. Lu and A. Vishwanath, Phys. Rev. B86, 125119 (2012).[49] M. Levin and A. Stern, Phys. Rev. B86, 115131 (2012),
arXiv:1205.1244.[50] D. Freed and G. Moore, Annales Henri Poincare14, 1927
(2013).[51] A. M. Essin and M. Hermele, Phys. Rev. B87, 104406 (2013).[52] Y.-M. Lu and A. Vishwanath, arXiv:1302.2634.[53] A. Mesaros and Y. Ran, Phys. Rev. B87, 155115 (2013).[54] Z. Bi, A. Rasmussen, and C. Xu, arXiv:1309.0515.[55] A. Kapustin, arXiv:1403.1467.[56] C. Wang, A. C. Potter, and T. Senthil, Science343, 629 (2014).[57] T. Neupert, C. Chamon, C. Mudry, and R. Thomale (2014),
arXiv:1403.0953.[58] M. A. Metlitski, L. Fidkowski, X. Chen, and A. Vishwanath,
arXiv:1406.3032.[59] D. S. Freed, arXiv:1406.7278.[60] D. V. Else and C. Nayak, arXiv:1409.5436.[61] F. D. M. Haldane, Phys. Lett. A93, 464 (1983).[62] F. D. M. Haldane, Phys. Rev. Lett.50, 1153 (1983).[63] C. L. Kane and E. J. Mele, Phys. Rev. Lett.95, 146802 (2005).[64] C. L. Kane and E. J. Mele, Phys. Rev. Lett.95, 226801 (2005).[65] B. A. Bernevig, T. Hughes, and S. Zhang, Science314, 1757
(2006).[66] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett.98, 106803
(2007).[67] J. E. Moore and L. Balents, Phys. Rev. B75, 121306 (2007).[68] R. Roy, Phys. Rev. B79, 195322 (2009).[69] R. B. Laughlin, Phys. Rev. Lett.50, 1395 (1983).[70] L. Balents, Nature464, 199 (2010).[71] M. Levin and Z.-C. Gu, Phys. Rev. B86, 115109 (2012).[72] L.-Y. Hung and X.-G. Wen, arXiv:1211.2767.[73] B. Swingle, Phys. Rev. B90, 035451 (2014).[74] M. Cheng and Z.-C. Gu, Phys. Rev. Lett.112, 141602 (2014).[75] F. A. Bais and J. K. Slingerland, Phys. Rev. B79, 045316
(2009), arXiv:0808.0627.[76] M. Barkeshli and X.-G. Wen, Phys. Rev. Lett.105, 216804
(2010), arXiv:1007.2030.[77] M. Barkeshli and X.-G. Wen, Phys. Rev. B84, 115121 (2011).[78] F. J. Burnell, S. H. Simon, and J. K. Slingerland, New J. Phys.
14, 015004 (2012).[79] V. Turaev, math/0005291.[80] A. J. Kirillov, math/0401119.[81] P. Etingof, D. Nikshych, and V. Ostrik, Quantum Topology1,
209 (2010), arXiv:0909.3140.[82] V. Turaev,Homotopy Quantum Field Theory(European Math-
ematical Society, 2010).[83] S. Burciu and S. Natale, J. Math. Phys.54, 013511 (2013).[84] K. Walker and Z. Wang, Frontiers of Physics7, 150 (2012),
arXiv:1104.2632.[85] C. W. von Keyserlingk, F. J. Burnell, and S. H. Simon, Phys.
Rev. B 87, 045107 (2013), arXiv:1208.5128.[86] A. Vishwanath and T. Senthil, Phys. Rev. X3, 011016 (2013),
arXiv:1209.3058.[87] C. Wang and T. Senthil, Phys. Rev. B87, 235122 (2013),
arXiv:1302.6234.[88] C. Wang, A. C. Potter, and T. Senthil, Phys. Rev. B88, 115137
(2013), arXiv:1306.3223.[89] P. Bonderson, C. Nayak, and X.-L. Qi, Journal of Statis-
tical Mechanics: Theory and Experiment9, 09016 (2013),arXiv:1306.3230.
[90] X. Chen, L. Fidkowski, and A. Vishwanath, Phys. Rev. B89,165132 (2014), arXiv:1306.3250.
[91] M. A. Metlitski, C. L. Kane, and M. P. A. Fisher,arXiv:1306.3286.
[92] X. Chen, F. J. Burnell, A. Vishwanath, and L. Fidkowski,arXiv:1403.6491.
[93] G. Y. Cho, J. C. Y. Teo, and S. Ryu, Phys. Rev. B89, 235103(2014).
[94] A. Kapustin and R. Thorngren, arXiv:1404.3230.[95] A. Kitaev, Annals Phys.321, 2 (2006), cond-mat/0506438.[96] J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert, Nucl. Phys.
B 763, 354 (2007).[97] A. Kapustin, arXiv:1004.2307.[98] J. Fuchs, C. Schweigert, and A. Valentino, Commun. Math.
Phys.321, 543 (2013), arXiv:1203.4568.[99] P. D. Francesco, P. Mathieu, and D. Senechal,Conformal Field
Theory(Springer, 1997).[100] R. Dijkgraaf, C. Vafa, E. Verlinde, and H. Verlinde, Comm.
Math. Phys.123, 485 (1989).[101] M. Muger, Journal of Algebra277, 256 (2004).[102] M. Barkeshli and X.-G. Wen, Phys. Rev. B81, 045323 (2010),
arXiv:0909.4882.[103] M. Barkeshli and X.-G. Wen, Phys. Rev. B86, 085114 (2012),
arXiv:1012.2417.[104] A. Bruguieres, Math. Ann316, 215 (2000).[105] M. Muger, Advances in Mathematics150, 151 (2000).[106] L. Kong, Nucl. Phys. B886, 436 (2014), arXiv:1307.8244.[107] I. Eliens, J. Romers, and F. Bais, arxiv:1310.6001.[108] V. G. Turaev,Quantum Invariants of Knots and 3-Manifolds
(Walter de Gruyter, Berlin, New York, 1994).[109] B. Bakalov and A. Kirillov, Lectures on Tensor Categories
and Modular Functors, vol. 21 of University Lecture Series(American Mathematical Society, 2001).
[110] J. Preskill (2004), lecture notes, URLhttp://www.theory.caltech.edu/ ˜ preskill/ph219/topological.ps .
[111] P. H. Bonderson, Ph.D. thesis, California Institute of Technol-ogy (2007).
[112] P. Bonderson, K. Shtengel, and J. K. Slingerland, Annals ofPhysics323, 2709 (2008), arXiv:0707.4206.
[113] S. Mac Lane,Categories for the Working Mathematician,Graduate Texts in Mathematics (Springer-Verlag, New York,1998), 2nd ed.
[114] C. Vafa, Phys. Lett. B206, 421 (1988).[115] P. Bonderson, K. Shtengel, and J. K. Slingerland, Phys. Rev.
Lett. 97, 016401 (2006), cond-mat/0601242.[116] P. Bonderson, K. Shtengel, and J. K. Slingerland, Phys. Rev.
Lett. 98, 070401 (2007), quant-ph/0608119.[117] E. Verlinde, Nucl. Phys. B300, 360 (1988).[118] L. Fidkowski, X. Chen, and A. Vishwanath, Phys. Rev. X3,
041016 (2013), arXiv:1305.5851.[119] M. Oshikawa, Phys. Rev. Lett.84, 1535 (2000).[120] A. Paramekanti and A. Vishwanath, Phys. Rev. B70, 245118
(2004).
87
[121] M. B. Hastings, Phys. Rev. B69, 104431 (2004).[122] M. B. Hastings, EPL (Europhysics Letters)70, 824 (2005).[123] M. P. Zaletel and A. Vishwanath, arXiv:1410.2894.[124] A. Y. Kitaev, Physics-Uspekhi44, 131 (2001), cond-
mat/0010440.[125] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett.
105, 077001 (2010), arXiv:1002.4033.[126] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett.105,
177002 (2010), arXiv:1003.1145.[127] P. Fendley, J. Stat. Mech. p. P11020 (2012), arXiv:1209.0472.[128] P. Bonderson, M. Freedman, and C. Nayak, Phys. Rev. Lett.
101, 010501 (2008), arXiv:0802.0279.[129] P. Bonderson, M. Freedman, and C. Nayak, Annals Phys.324,
787 (2009), arXiv:0808.1933.[130] P. Bonderson, Phys. Rev. B78, 035113 (2013),
arXiv:1210.7929.[131] M. Zaletel, arXiv:1309.7387.[132] L.-Y. Hung and X.-G. Wen, Phys. Rev. B89, 075121 (2014),
arXiv:1311.5539.[133] P. Etingof, D. Nikshych, and V. Ostrik, Ann. Math.162, 581
(2005).[134] A. Kitaev and J. Preskill, Phys. Rev. Lett.96, 110404 (2006),
hep-th/0510092.[135] M. Levin and X.-G. Wen, Phys. Rev. Lett.96, 110405
(pages 4) (2006), cond-mat/0510613.[136] E. Fradkin and S. H. Shenker, Phys. Rev. D19, 3682 (1979).[137] P. Deligne, Moscow Math. Journal2, 227 (2002).[138] P. Etingof and S. Gelaki, arXiv:math/0007196.[139] V. Drinfeld, S. Gelaki, D. Nikshych, and V. Ostrik, Selecta
Mathematica16, 1 (2010).[140] M. Metlitski, unpublished.[141] K. S. Brown,Cohomology of Groups(Springer-Verlag, 1982).[142] G. Moore and N. Seiberg, Physics Letters B220, 422 (1989),
ISSN 0370-2693.[143] R. Dijkgraaf and E. Witten, Commun. Math. Phys.129, 393
(1990).[144] M. de Wild Propitius, Ph.D. thesis, University of Amsterdam
(1995).[145] R. Dijkgraaf, V. Pasquier, and P. Roche, Proceedings of the
International Colloquium on Modern Quantum Field Theory(1991).
[146] Y. Hu, Y. Wan, and Y.-S. Wu, Phys. Rev. B87, 125114 (2013),arXiv:1211.3695.
[147] C.-H. Lin and M. Levin, Phys. Rev. B89, 195130 (2014),arXiv:1402.4081.
[148] Z.-C. Gu and M. Levin, Phys. Rev. B89, 201113 (2014).[149] X.-L. Qi, New J. Phys.15, 065002 (2013).[150] S. Ryu and S.-C. Zhang, Phys. Rev. B85, 245132 (2012).[151] H. Yao and S. Ryu, Phys. Rev. B88, 064507 (2013).[152] Z. Bi, M. Cheng, and Z.-C. Gu, unpublished.[153] Z. Bi, A. Rasmussen, Y.-Z. You, M. Cheng, and C. Xu,
arXiv:1404.6256.[154] V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett.59, 2095
(1987).[155] Y.-M. Lu and L. Fidkowski, Phys. Rev. B89, 115321 (2014).[156] D. Tambara and S. Yamagami, Journal of Algebra209, 692
(1998).[157] D. Clarke, private communication.[158] M. Titsworth, private communications.[159] A. Y. Kitaev, Annals Phys.303, 2 (2003), quant-ph/9707021.[160] X.-G. Wen, Phys. Rev. Lett.90, 016803 (2003), quant-
ph/0205004.[161] M. Cheng, unpublished.[162] S. X. Cui, S.-M. Hong, and Z. Wang, arXiv:1401.7096.
[163] F. J. Burnell, X. Chen, L. Fidkowski, and A. Vishwanath,arXiv:1302.7072.
[164] M. Hartl and S. Leroy, arXiv:0707.0291.[165] S. X. Cui, C. Galindo, J. Plavnik, and Z. Wang, unpublished.[166] A. M. Cegarra, A. R. Garon, and J. A. Ortega, Journal of Al-
gebra241, 620 (2001).[167] M. Muger, Proc. Lond. Math. Soc.87, 291 (2003),
arXiv:math/0201017.[168] L. Fidkowski, N. Lindner, and A. Kitaev, unpublished.[169] J. Teo, T. Hughes, and E. Fradkin, unpublished.[170] G. Karpilovsky,Projective Representation of Finite Groups,
vol. 94 of Pure and Applied Mathematics(Marcel Dekker,New York/Basel, 1985).
[171] J. C. Baez and A. D. Lauda, Theory and Applications of Cat-egories12, 423 (2004), arXiv:math/0307200.
[172] It is often assumed that the set of topological chargesC is fi-nite, but we may allow it to be infinite in the definition of afusion tensor category or a braided tensor category, as longasfusion is finite. However, for a modular tensor category, wewill requireC to be a finite set.
[173] In fact, we believe that all gauge transformations of a BTCthat leave the basic data unchanged must take this form andthus they are actually natural isomorphisms, so that the twoconcepts are synonymous for BTCs.
[174] In mathematical parlance, the braided auto-equivalence mapsϕ are1-automorphism functors, the natural isomorphismsΥare2-isomorphisms between the auto-equivalence functor, andthe decomposition freedom of natural isomorphisms (given bythe phase factorsζa) are the automorphisms of the identityfunctor.
[175] Given a groupG, aG torsor is a non-empty setX upon whichG acts freely and transitively. In other words, it is what youget if the groupG had lost its identity. In the context of clas-sification, this means that distinct symmetry fractionalizationclasses are related to each other by the action of distinct ele-ments ofH2
[ρ](G,A).[176] We note that, givenHα
a,a;0 and its ground state|Ψαa,a;0〉, it is
always possible to construct a Hamiltonian for which anotherstate|Ψβ
a,a;0〉 in this universality class is the ground state. In
particular, one can useh(j)a;β = V (j)
[
h(j)a;α +H
(j)0
]
V (j)−1 −H
(j)0 , whereH(j)
0 is the sum of the terms inH0 that act non-trivially in Rj .
[177] From this, we can see that|Ψαρg(a),ρg(a);0
〉 is the ground state
of the HamiltonianHαρg(a),ρg(a);0
= ρgHαa,a;0ρ
−1g , for which
the correspondingh(j)ρg(a);α
= ρg[
h(j)a;α +H
(j)0
]
ρ−1g −H
(j)0
is again localized withinRj , whereH(j)0 is the sum of the
terms inH0 that act nontrivially inRj .[178] Such an operator localized inRj can be defined, for example,
by takingB(j)g,h =
∑
a,b βa(g,h)S0aSabWb(∂Rj), whereWb(∂Rj) is a Wilson loop of topological chargebwhose pathfollows the loop delineated by the boundary∂Rj (or just in-side the boundary) of the region in a counterclockwise fashion.
[179] We could modify this definition slightly to include also theplaquettes that contain the end-points of the lineC. Such amodification corresponds to a local change in the Hamiltonianand would also describe ag andg−1 pair of defects.
[180] This requires a detailed understanding of gapped line defects;see, e.g., Refs. [5, 14].
[181] After introducingG-crossed braiding in the next section, wewill see that the same chargeb0 can always be used for eitherleft or right fusion withag to obtaincg, i.e. there exists some
88
b0 such thatNcgagb0
= Nb0agcg
= Nb0cgag
= Ncgb0ag
6= 0.[182] Had we allowed theG-crossed braiding action to depend on
the topological charge value, rather than only depending onthe corresponding group element, i.e. if we replacedρg witha more general mapρag , compatibility with fusion would re-quire thatρag ρbh = ρcgh wheneverNc
ab 6= 0. Combin-ing this this an axiom thatρb0 act trivially on all topologicalcharges for allb0 ∈ C0, i.e. ρb0(e) = e for any e ∈ C×
G ,would lead back toρag = ρg being independent of the partic-ular topological chargea within Cg. In particular, for any twodistinct chargesag 6= cg in Cg , there is always someb0 ∈ C0
with Ncab 6= 0, and henceρag = ρag ρb0 = ρcg . This ax-
iom is physically natural, because the topological chargesinC0 correspond to quasiparticles, which are truly point-like lo-calized (they do not have defect branch cut lines) and henceshould be unable to alter operators or topological charges lo-calized in a distant region, unless it enters that region.
[183] We note that while critical exponents for local correlationfunctions can be captured in this way, non-local correlationsand the topological structure of the critical points may havesubtle differences depending on the rest of the structure ofthe
topological phases in question.[184] There can also be an obstruction to the group extension, which
is classified byH3ρ(G,Z(H)).
[185] In fact, one can show that every element ofZ(H) correspondsto an Abelian anyon in the quantum double D(H). To see this,recall that elements of D(H) are labelled by([h], π(Ch)),whereh labels a conjugacy class andπ(Ch) is an irreduciblerepresentation of the centralizer ofh. Every element inZ(H)corresponds to a distinct conjugacy class inH with Ch ≃ H .The Abelian anyons in this conjugacy class are in one-to-onecorrespondence with group homomorphisms ofH to U(1).In particular, there is a trivial homomorphism 11 where eachgroup element is mapped to1, corresponding to an Abeliananyon (h, 11). This shows thatZ(H) is a true subgroup ofthe Abelian anyons in D(H) as long asH1(H,U(1)) is notempty. This statement does not hold in general for twistedquantum doubles, where theπ(Ch) may be a projective ir-reducible representation, which, in general, may not be one-dimensional, and therefore(h, π(Ch)) may not be Abelianfor anyπ(Ch).