Topological Quantum Field Theory

Ronen Brilleslijper

July 12, 2017

Combined bachelor’s thesis mathematics and physics

Supervisors: dr. Hessel Posthuma and dr. Miranda Cheng

Korteweg-de Vries Instituut voor Wiskunde

Institute of Physics

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

Universiteit van Amsterdam

Abstract

This thesis aims to provide an overview of the concepts needed to comprehend thesimplest examples of Topological Quantum Field Theories (TQFT’s). After briefly cov-ering the essential mathematical preliminaries, the category nCob of n-dimensionalcobordisms is described in detail, leading to the definition of a TQFT as a functornCob → VectK that satisfies some additional properties. Next, the focus is turnedto the definition of G-coverings and a description will be given of two ways to classifyG-coverings over a given base space X. The first one uses the concept of Cech cocycles toclassify G-coverings that are trivial over a specified open covering of X. The classifyingspace of a group, denoted BG, is introduced in order to look at a second way of classify-ing G-coverings, by considering homotopy classes of maps X → BG. This classificationuses Cech cocycles to construct a map X → BG from a G-covering. After covering all ofthese mathematical structures, Dijkgraaf-Witten theory is introduced. First, the focusis layed on the untwisted version, which heuristically counts the number of G-coveringover a manifold with a weight corresponding to the number of automorphisms of thecovering. The twisted version of Dijkgraaf-Witten theory generalizes this concept byincluding a second weight factor coming from a fixed cohomology class α ∈ Hn(BG,T).The untwisted version then corresponds to the case where α is trivial. Finally, Dijkgraaf-Witten theory is discussed within the framework of Quantum Field Theory (QFT). Boththe untwisted and twisted version are covered from this point of view, checking theircompatibility with the axioms of QFT.

Title: Topological Quantum Field TheoryAuthor: Ronen Brilleslijper, 10855041Supervisors: dr. Hessel Posthuma and dr. Miranda ChengSecond graders: dr. Raf Bocklandt and dr. Vladimir GritsevDate: July 12, 2017

Korteweg-de Vries Instituut voor WiskundeUniversiteit van AmsterdamScience Park 904, 1098 XH Amsterdamhttp://www.science.uva.nl/math

Institute of PhysicsUniversiteit van AmsterdamScience Park 904, 1098 XH Amsterdamhttp://iop.uva.nl

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Contents

1 Introduction 51.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Physical relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Preliminaries 82.1 Differential Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Morse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 (Co)homology groups . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Homology of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Axioms for TQFT 113.1 The category nCob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.2 Identity morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Definition of a TQFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Relationship with quantum mechanics . . . . . . . . . . . . . . . . . . . . 16

3.3.1 Operators and probabilities . . . . . . . . . . . . . . . . . . . . . . 16

4 G-coverings and classification 184.1 G-coverings and isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.1 From G-covering to Cech cocycle . . . . . . . . . . . . . . . . . . . 214.2.2 From Cech cocycle to G-covering . . . . . . . . . . . . . . . . . . . 21

4.3 Homotopy description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3.1 (Semi)simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3.2 Classifying space of a group . . . . . . . . . . . . . . . . . . . . . . 244.3.3 Homotopy classification . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Dijkgraaf-Witten theory 295.1 Untwisted Dijkgraaf-Witten theory . . . . . . . . . . . . . . . . . . . . . . 295.2 Twisted Dijkgraaf-Witten theory . . . . . . . . . . . . . . . . . . . . . . . 32

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6 Dijkgraaf-Witten as QFT 346.1 Field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.1.1 Classical field theory . . . . . . . . . . . . . . . . . . . . . . . . . . 346.1.2 From classical to quantum . . . . . . . . . . . . . . . . . . . . . . . 35

6.2 Untwisted Dijkgraaf-Witten theory . . . . . . . . . . . . . . . . . . . . . . 356.3 Twisted Dijkgraaf-Witten theory . . . . . . . . . . . . . . . . . . . . . . . 36

Popular summary 38

Bibliography 40

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1 Introduction

”That is easy, in one sentence, at long distance Topological Quantum Field The-ory is the relevant approximation, and why it’s so important for, for instance, con-densed matter physics.” - Prof. Greg Moore on the importance of Topological QuantumField Theory

Geometry plays a large role in theoretical physics. A particular area where geometryis heavily used is, for example, General Relativity. When describing global propertiesof physical systems, the topology of the systems involved becomes important. Oneremarkable interplay between physics and topology is described by Topological QuantumField Theories (TQFT’s) [A88]. The concept of TQFT was mathematically formalizedin [A88] with the formulation of a set of axioms. Intuitively, a TQFT assigns a vectorspace AΣ to each (n − 1)-dimensional space Σ and a linear map AΣ0 → AΣ1 to eachn-dimensional space-time that interpolates between the spaces Σ0 and Σ1. TQFT’s areof interest to both mathematicians and physicists. In mathematics, they are studiedbecause they assign topological invariants to manifolds of certain dimensions [K03].An example of a mathematical theorem regarding TQFT’s is given in [K03], where 2-dimensional TQFT’s are related to Frobenius algebras. The physical interst in TQFTcomes mainly from the fact that it is simple enough to do calculations and gives insightinto more complicated quantum field theories (QFT’s). At low energies, QFT’s canbe approximated by TQFT’s, thus making computations easier (prof. Greg Moore,personal communication, June 7th 2017). This is the main application of TQFT to,amongst others, condensed matter physics (see for example [KTu15]). Another area ofphysics where TQFT plays a role is quantum computation. The importance of TQFTto quantum computation is described in [Na08].

1.1 Objective

The goal of this thesis is to provide an introduction to the concepts that are needed tounderstand TQFT and to formulate simple examples. Also, the analogy with operatorsand probabilities from quantum physics will be discussed. The example of a TQFTthat will be covered is Dijkgraaf-Witten theory. This ”toy-model” of a TQFT was firstintroduced in [DW90] and is widely accepted as one of the simplest TQFT’s that stillgives insight into the way the theory works (see for example [FQ93]). Dijkgraaf-Wittentheory is a simplification of the more involved Chern-Simons theory. Whereas Chern-Simons theory uses bundles over a Lie Group, Dijkgraaf-Witten theory considers onlyfinite groups. This reduces the complicated path integrals to finite sums, thus eliminatingmost of the analytical difficulty, while preserving the algebraic properties.

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While writing this thesis it became clear that there is a lot of literature availableon the concept of TQFT as well as on different approaches to Dijkgraaf-Witten theory(see for example [Mu07], [Fr92], [FQ93] and [Q91]). However, most of the literatureeither assumes extensive preknowledge or skips important details, leaving the readerin the dark about some of the necessary steps. Therefore, it is my intention that thisthesis will provide a detailed exposition of the theory, that is accessible for mathematicsstudents in the final phase of their undergraduate degree.

1.2 Physical relevance

As stated in the previous section, Dijkgraaf-Witten (DW) theory is useful as a TQFTbecause of its analytical simplicity. However, DW theory also has actual physical signifi-cance, coming from condensed matter physics. The physics behind phases of matter andphase transitions all comes from the underlying symmetries of the material [W16]. Theso-called Landau-Ginzburg symmetry breaking formalism describes the physics of thesesymmetries and stands at the basis of condensed matter physics [HZK17]. Symmetriesare described by a group G and phases of matter can be described at low-energy by aTQFT with symmetry group G. Thus, to understand these phases of matter, an under-standing of such TQFT’s is needed. In the case where G is finite, DW theory describesall possible TQFT’s with symmetry group G. As will be seen in chapter 5, DW theoriesare classified by cocycles in Hn(BG;T).1 This implies that these special kind of phasesare in bijection with the cohomology of the space BG [HZK17]. Unfortunately though,it turns out that this classification does not give the full picture [KTh15]. However, DWtheory still provides a lot of insight into the physics governing the phases of matter ofmaterials with finite symmetry group.

1.3 Overview

This thesis is structured as follows. Chapter 2 starts by briefly covering the conceptsrequired to understand this thesis. The relevant constructions and theorems from dif-ferential and algebraic topology are stated and references are given to readers that wishto read a more detailed discussion of these concepts. This chapter is primarily intendedfor readers that have a basic knowledge of differential and algebraic topology, but whomight not have an active knowledge of all the concepts needed. Readers with extensiveknowledge of these subjects can choose to skip this chapter, while readers with verylittle understanding of the topics are adviced to turn to the references given at the startof the chapter. In chapter 3, first the concept of cobordisms is introduced. A detaileddescription is given of the category nCob that has closed oriented (n− 1)-manifolds asits objects and equivalence classes of cobordisms as its morphisms. The main challengein the definition of this category lies in the description of the composition of two cobor-disms and the proof that this is well-defined. The chapter continues with the definition

1The notation BG for the classifying space of G will be introduced in chapter 4.

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of a TQFT as a functor nCob → VectK satisfying two aditional properties, whereVectK is the category of vector spaces over some ground field K. Finally, chapter 3makes a connection between the language of quantum mechanics and that of TQFT.In order to comprehend Dijkraaf-Witten theory, one must be familiar with the conceptof G-coverings. Therefore, chapter 4 starts by introducing the concept of G-coveringsfor a discrete group G. Afterwards, this chapter will cover two ways of classifying allG-coverings over a given base space up to isomorphisms. Using the concept of Cechcocycles a bijection is constructed from the set of isomorphism classes of G-coveringsthat are trivial over an open covering of the base space to the set of cohomology classesof Cech cocycles over the open covering. Next, the classifying space of G, denoted BG, isconstructed using the geometric realization of (semi)simplicial sets. Using this space andthe Cech cocycle classification, the homotopy classification of G-coverings is described.This construction associates a G-covering to every homotopy class of maps from the basespace to BG using the so-called pull back. After having covered all of these concepts,chapter 5 turns to Dijkgraaf-Witten theory. This theory has a simple version, called un-twisted Dijkgraaf-Witten theory, and a general version, called twisted Dijkgraaf-Wittentheory. First, the untwisted version is covered, which assigns the space C[Σ, BG] toevery closed (n − 1)-manifold Σ. Then linear maps between two such vector spaces in-duced by n-manifolds M are constructed by counting the number of G-coverings over Mtimes the inverse of the size of their automorphism group. A detailed discussion is givenof the proof that this defines a TQFT. Following this paragraph is the explanation oftwisted Dijkgraaf-Witten theory. This version associates an extra weight factor to everyG-covering corresponding to a fixed cohomology class α ∈ Hn(BG;T), where T ⊆ C∗ isthe group consisitng of all elements of C∗ with unit norm. This cohomology class alsoalters the vector spaces that are assigned to closed (n− 1)-manifolds. The final chapterof this thesis again covers Dijkgraaf-Witten theory, but this time as a QFT. First, thischapter covers the axioms of QFT. Then untwisted and twisted Dijkgraaf-Witten theoryare treated within this framework, checking their compatibility with these axioms.

1.4 Acknowledgement

First of all, I want to thank dr. Hessel Posthuma and dr. Miranda Cheng for theirsupervision during the process of writing this thesis. A thank you also goes to dr. ChrisZaal for pushing me to arrange everything in time before going on exchange. Finally, Iwould like to thank my family for supporting me and reviewing my presentations, eventhough they probably did not understand a word of what I said.

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2 Preliminaries

This chapter will briefly cover some of the concepts that will be used in the rest ofthis thesis. For a more detailed introduction to differential topology, see [Le03], and foralgebraic topology, see [Br93].

2.1 Differential Topology

Recall that an n-dimensional topological manifold is a second countable Hausdorff spacethat is locally homeomorphic to Rn. In this thesis the coordinate maps are assumed to behomeomorphisms onto Rn, in contrast to the usual convention of using homeomorphismsonto open subsets of Rn. However, these definitions are equivalent and our approachwill avoid some of the technical details in the next chapter. A (smooth) manifold is atopological manifold where all coordinate charts are smoothly compatible. If M is anorientable manifold, then an orientation for M is a smooth choice of positive bases forthe tangent spaces. When a map between oriented manifolds takes positive bases topositive bases, it is called orientation preserving.

For an oriented manifold with boundary, two types of boundary components can bedistinguished: in-boundaries and out-boundaries [K03]. They are defined as follows.

Definition 2.1. Let M be a manifold and ι : Σ → M an orientation preserving dif-feomorphism from an oriented manifold Σ onto a disjoint union of components of theboundary of M . For x ∈ ι(Σ) a positive normal is a vector w ∈ TxM , such that[v1, . . . , vn−1, w] is a positive basis for TxM , where [v1, . . . , vn−1] is a positive basis forTxι(Σ). The vector w can now locally be seen as a vector in Rn that is either pointingin or out of the halfspace Rn+ := {x ∈ Rn : xn ≥ 0}. If it points inwards for all x ∈ ι(Σ)then ι(Σ) is called an in-boundary of M and if it points outwards for all such x thenι(Σ) is called an out-boundary.

With this definition, the boundary of a manifold can be divided into a disjoint unionof in-boundaries and out-boundaries.

2.1.1 Morse functions

In the next chapter, a few results from Morse theory will be used. This subsection,which is based on [K03], will introduce the necessary vocabulary.

Definition 2.2. For a smooth map f : M → I from a manifold to the unit interval, apoint x ∈M is called a critical point if dfx = 0. If x is not a critical point, it is called a

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regular point. The images of critical and regular points under f are called critical andregular values respectively.

For a critical point x ∈ M one could look at the Hessian matrix ( ∂2f∂xi∂xj

)i,j in somechosen coordinate system. If this matrix is invertible, then x is called nondegenerate.

Definition 2.3. A smooth map f : M → I is called a Morse function if all of its criticalpoints are nondegenerate. Also it is assumed that f−1(∂I) = ∂M and that 0, 1 ∈ I areregular values.

When M is compact, it can be shown that a Morse function M → I has finitely manycritical points. In particular, we can therefore find an ε > 0 such that the intervals[0, ε] and [1− ε, 1] contain only regular values. The most important result about Morsefunctions is that they exist for every manifold. This will be used in the construction ofthe composition of two cobordisms in the next chapter.

2.2 Algebraic topology

This section is mostly based on [Br93]. A map Y → X between path connected Hausdorffspaces is called a covering map if each point inX has an open neigbourhood that is evenlycovered. The fiber over x ∈ X of this covering is denoted Yx. An important property ofcoverings is that homotopies can be lifted over them, as the following theorem states.

Theorem 2.4. Let p : Y → X be a covering and H : W × I → X a homotopy, whereW is a locally connected space. Suppose there is a lift h : W → Y of H0 := H|W×{0}.Then we can find a unique H : W × I → Y such that H0 = h and p ◦ H = H.

An important topological invariant of a space is its fundamental group. This conceptcan be generalized to arbitrary homotopy groups. The n-th homotopy group of a pointedspace (X,x0) is defined as

πn(X,x0) = [(Sn, ∗), (X,x0)],

where ∗ ∈ Sn is any point and the brackets denote equivalence classes of maps Sn → Xunder homotopy relative the basepoint. For n = 1, this is just the fundamental groupand for n = 0, it consists of the path components of X.

2.2.1 (Co)homology groups

Other topological invariants are the homology and cohomology of a space. Recall that asingular n-simplex of a space X is a map ∆n → X, where ∆n = {(x0, . . . , xn) ∈ Rn+1 :∑n

i=0 xi = 1, xi ≥ 0 for all i} is the standard n-simplex. The free abelian group on thesingular n-simplices of X is called the singular n-chain group Sn(X). These groups definea chain complex with a boundary map ∂ that maps a singular simplex to the alternating

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sum of its faces. The homology of this complex is called the singular homology of X andis denoted as H∗(X). In other words:

Hn(X) =ker (∂ : Sn(X)→ Sn−1(X))

im (∂ : Sn+1(X)→ Sn(X)).

To define cohomology, first fix an abelian group F and define Sn(X;F ) := Hom(Sn(X), F ),where Hom denotes the group of group homomorphisms. Now the boundary map∂ : Sn(X) → Sn−1(X) induces a map δ : Sn−1(X;F ) → Sn(X;F ) by sending a grouphomomorphism f to f ◦ ∂. This makes S∗(X;F ) into a cochain complex, whose coho-mology H∗(X;F ) is called the singular cohomology of X with coefficients in F . So

Hn(X;F ) =ker(δ : Sn(X;F )→ Sn+1(X;F )

)im (δ : Sn−1(X;F )→ Sn(X;F ))

.

2.2.2 Homology of manifolds

Now we turn to the calculation of the homology of an n-dimensional manifold. See [Ma99]as a reference. Let M be an orientable n-manifold without boundary and take a pointx ∈ M . By definition there is an open subset U ⊆ M with x ∈ U and U ∼= Rn. SinceM\U ⊆ int(M\{x}) excision gives an isomorphism Hi(M,M\{x}) ∼= Hi(U,U\{x}) forall i. Then the long exact sequence for the pair (U,U\{x}) and the fact that the reducedhomology of a point is trivial induces an isomorphism Hi(U,U\{x}) ∼= Hi−1(U\{x}).Finally by homotopy invariance this is again isomorphic to Hi−1(Sn−1), which is Z fori = n and trivial otherwise. So Hn(M,M\{x}) is generated by a single element.1

Definition 2.5. If [M ] ∈ Hn(M) is an element such that (ιx)∗([M ]) generates Hn(M,M\{x})for all x ∈ M , then [M ] is called a fundamental class of M . Here, ιx is the inclusionM → (M,M\{x}).

Note that since Z has two generators (plus and minus one), there are two choices fora fundamental class of M (provided that M is orientable). These correpond to the twoorientations on M .

1For an explanation of the axioms of homology, including exactness, excision and homotopy invariance,see [Br93].

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3 Axioms for TQFT

This chapter will introduce the basic concepts needed to describe a toplogical quantumfield theory (TQFT). First, the notion of cobordisms is introduced and the category ofcobordisms nCob is defined. The main part of this definition lies in the compositionof two cobordisms. Afterwards, we define a TQFT as a functor from nCob to VectK .Finally, the relationship between TQFT and quantum mechanics is discussed. Most ofthis chapter is based on chapter 1 from [K03].

3.1 The category nCob

Recall that to describe a category, we need to define its objects and morphisms, alongwith the composition of morphisms and the identity morphisms. As the objects ofnCob we take the closed oriented (n− 1)-manifolds. Note that ´closed´ in this contextmeans compact without boundary. To define the morphisms, we first need the followingdefinition.

Definition 3.1. Let Σ0 and Σ1 be two objects in nCob and M a compact orientedn-manifold. If ιj : Σj → M (j = 0, 1) are two maps as in definition 2.1, the triple(M, ι0, ι1) is called an (oriented) cobordism from Σ0 to Σ1, provided that ι0(Σ0) is anin-boundary of M , ι1(Σ1) is an out-boundary of M and ∂M = ι0(Σ0)

∐ι1(Σ1).

Intuitively a cobordism can be seen as a ”smooth” way to transform system Σ0 intoΣ1. On the class of cobordims the following relation is defined.

Definition 3.2. For Σ0,Σ1 objects in nCob let M,M ′ : Σ0 ⇒ Σ1 be two cobordims.Denote the embeddings of Σj into M and M ′ by ιj respectively ι′j for j = 0, 1. DefineM ∼M ′ if there is an orientation preserving diffeomorphism ψ : M →M ′ such that thefollowing diagram commutes.

M

Σ0 Σ1

M ′

ψ

ι0

ι′0

ι1

ι′1

(3.1)

Note that this condition says that ψ maps the in-boundary of M to the in-boundary ofM ′ and similarly for the out-boundary.

It turns out that ∼ is an equivalence relation on the cobordisms from Σ0 to Σ1.Reflexivity and transitivity are not hard to prove. To see that it is symmetric, let M

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and M ′ be two cobordisms from Σ0 to Σ1, such that M ∼M ′. This means that there isan orientation preserving diffeomorphism ψ : M →M ′ that makes diagram 3.1 commute.Now ψ−1 : M ′ → M is a diffeomorphism as well. We prove that it is also orientationpreserving. Take p ∈ M ′ and q = ψ−1(p). For a positive basis Bp for TpM

′ we wantto prove that (ψ−1)∗(Bp) is a positive basis for TqM . Assume that this is not the case.Then ψ∗((ψ

−1)∗(Bp)) is a negative basis for TpM′, since ψ preserves orientation. But

ψ∗ ◦ (ψ−1)∗ = (ψ ◦ψ−1)∗ = IdTpM ′ , so Bp is a negative basis for TpM′, which contradicts

our assumption. So ψ−1 is orientation preserving. Also, for j = 0, 1

ψ−1 ◦ ι′j = ψ−1 ◦ ψ ◦ ιj = ιj ,

so ψ−1 maps the in-boundary of M ′ to the in-boundary of M and similarly for theout-boundary. So M ′ ∼M .

This equivalence relation induces equivalence classes of cobordisms from Σ0 to Σ1. Theequivalence class of a cobordism M is denoted [M ]. We define the class of morphismsHomnCob(Σ0,Σ1,) as the class of these equivalence classes.

3.1.1 Composition

Given a morphism in HomnCob(Σ′,Σ,) and one in HomnCob(Σ,Σ′′,), the goal is to definea morphism in HomnCob(Σ′,Σ′′,). That is, from two cobordisms we want to define anew manifold with a smooth structure that is unique up to diffeomorphisms. We startby defining the manifold.

The topological composition

Let M0 : Σ′ ⇒ Σ and M1 : Σ ⇒ Σ′′ be two cobordisms with embeddings ι0 : Σ → M0

and ι1 : Σ→ M1. On the disjoint union M0∐M1, the following equivalence relation is

defined: let every point be equivalent to itself and let m0 ∈M0 be equivalent to m1 ∈M1

if there is an x ∈ Σ such that ι0(x) = m0 and ι1(x) = m1. This is clearly an equivalencerelation and we denote the quotient space as M0

∐ΣM1 (or as M0

∐ι0,ι1Σ M1 if we want

to stress the embeddings used in the equivalence relation). On this space, we take thequotient topology induced by the projection map π : M0

∐M1 →M0

∐ΣM1. From now

on, M0 and M1 will be seen as subspaces of M0∐

ΣM1 where we identify these spaceswith their image under π. We want to make M0

∐ΣM1 into a topological manifold. For

points that are not on the gluing edge ι0(Σ) = ι1(Σ) (seen as subspaces of M0∐

ΣM1) wealready have charts. For a point p on this ”edge”, we have to find an open neighbourhoodU and a homeomorphism with Rn. Pick an open neighbourhood U of this point. Bydefinition, the sets U0 := U ∩M0 and U1 := U ∩M1 are open in M0 and M1 respectively.Note that U = U0

∐Σ U1. By shrinking U if necessary, we may assume that U0 and

U1 are the domains of two coordinate maps ψ0 : U0 → Rn− = {x ∈ Rn : xn ≤ 0} andψ1 : U1 → Rn+ = {x ∈ Rn : xn ≥ 0}. Define Ξ := U ∩ ι0(Σ) = U0 ∩ U1. The mapsp0 := ψ0|Ξ and p1 := ψ1|Ξ give embeddings of Ξ in Rn±, whose images are the boundary ofthese spaces. Therefore, we can define the space Rn−

∐p0,p1

Ξ Rn+ ∼= Rn. When Rn+ and Rn−are seen as subspaces of Rn under this identification, ψ0 and ψ1 can be regarded as maps

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to Rn that give homeomorphisms to their image. Because these images agree on Ξ bydefintion, they induce a continuous map ψ : U → Rn. By the same reasoning, also ψ−1

is continuous and therefore ψ is a homeomorphism. Left to prove is that the structurefound this way is unique. Suppose we had chosen different coordinate maps from U0

and U1, say χ0 and χ1. These can be glued together again to form a homeomorphismχ : U → Rn. The transitionfunctions between the ψj ’s and the χj ’s are homeomorphismsα0 : Rn− → Rn− and α1 : Rn+ → Rn+. Define for j = 0, 1 the embeddings qj : αj ◦ pj . Thenwe can glue the images of the αj ’s again to form Rn−

∐q0,q1Ξ Rn+ ∼= Rn. Now we get a

continuous map α : Rn → Rn that is precisely the transitionfunction between ψ and χ.So both coordinate maps belong to the same maximal atlas, which is therefore uniquelydefined.

Smooth structure

Next we want to assign a smooth structure to the topological manifold formed above.The next theorem shows that when we find a smooth structure, it is unique up todiffeomorphisms. This theorem will not be proven here. See the references given by[K03] for more details.

Theorem 3.3 ([K03]). Let Σ be an out-boundary of M0 and an in-boundary of M1. Ifα and β are two smooth structures on M0

∐ΣM1, such that for every coordinate chart

(U,ϕ) in α or β the coordinate charts ((π|Mj )−1(U), ϕ◦π|Mj ) are in the smooth structure

on Mj for j = 0, 1, then there is a diffeomorphism φ : (M0∐

ΣM1, α)→ (M0∐

ΣM1, β)such that φ|Σ = IdΣ. Here, π : M0

∐M1 →M0

∐ΣM1 is again the quotient map.

First, we will discuss how to glue together cylinders. Take two cobordisms M0 : Σ0 ⇒Σ1 and M1 : Σ1 ⇒ Σ2, such that [M0] = [Σ1 × [0, 1]] and [M1] = [Σ1 × [1, 2]]. DefineS := Σ1×[0, 2]. The standard smooth structure on S matches the structures on Σ1×[0, 1]and Σ1 × [1, 2]. Let

i0 : Σ1 × [0, 1]→ Σ1 × [0, 2]

i1 : Σ1 × [1, 2]→ Σ1 × [0, 2]

be the inclusions and

α0 : M0 → Σ1 × [0, 1]

α1 : M1 → Σ1 × [1, 2]

be orientation preserving diffeomorphisms. Define

α : M0

∐Σ1M1 → S

α(x) =

{i0 ◦ α0(x) x ∈M0

i1 ◦ α1(x) x ∈M1.

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This clearly defines a homeomorphism. The restrictions α|M0 and αM1 are orientationpreserving diffeomorphisms. Define a smooth structure on M0

∐Σ1M1 by pulling back

the smooth structure on S through α. This gives the composition of the two cylinders.The next theorem, coming from Morse theory, gives a criterium for when a cobordism

is diffeomorphic to a cylinder.

Theorem 3.4 ([K03]). Let M : Σ0 ⇒ Σ1 be a cobordism and f : M → [0, 1] a smoothmap without critical points. Suppose that f−1({0}) = Σ0 and f−1({1}) = Σ1. Then thereis a diffeomorphism φ : Σ0×[0, 1]→M such that f ◦φ = π2, where π2 : Σ0×[0, 1]→ [0, 1]is the projection on the second coordinate.

With the tools developed so far, we are now able to glue together arbitrary manifolds.Take two cobordisms M0 : Σ0 ⇒ Σ1 and M1 : Σ1 ⇒ Σ2. Then there exist two Morsefunctions f0 : M0 → [0, 1] and f1 : M1 → [1, 2] that can be glued together to a functionf : M0

∐Σ1M1 → [0, 2]. Choose ε > 0 small enough such that the intervals [1− ε, 1] and

[1, 1 + ε] are regular for f0 and f1 respectively. Then using theorem 3.4, it can be seenthat N0 := f−1

0 ([1− ε, 1]) and N1 := f−11 ([1, 1 + ε]) are diffeomorphic to cylinders. Let

α0 : N0 → Σ1 × [0, 1]

α1 : N1 → Σ1 × [1, 2]

be the corresponding diffeomorphisms. Because we know already how to glue togethercylinders, a homeomorphism α : N0

∐Σ1N1 → S := Σ1 × [0, 2] is found, such that α|N0

and αN1 are diffeomorphisms. Define an atlas on M0∐

Σ1M1 as follows:

• For every chart (U,ψ) in Mj let (U,ψ) be a chart in M0∐

Σ1M1 (where we regard

U ⊆Mj as a subset of M0∐

Σ1M1);

• For every chart (V,Ψ) in S let (α−1(V ),Ψ ◦ α) be a chart in M0∐

Σ1M1.

We now have to prove that the transition maps between these charts are diffeomorphisms.Suppose that (U,ψ) is a chart in Mj and (V,Ψ) in S, such that W := α−1(V ) ∩ U 6= ∅.Then on W

ψ ◦ (Ψ ◦ α)−1 = ψ ◦ α−1 ◦Ψ−1

= ψ ◦ (α|Nj )−1 ◦Ψ−1

is a diffeomorphism, since ψ, Ψ and α|Nj are diffeomorphisms. So this atlas is well de-fined and gives a smooth structure on M0

∐Σ1M1. The resulting cobordism is denoted

M0M1.1 On this cobordism we choose the unique orientation that makes this into acobordism from Σ0 to Σ2 and makes the embeddings Σj → M0M1 (for j = 0, 2) intoorientation preserving maps. By theorem 3.3, the structure found this way is unique up

1Note that the smooth structure depends on the maps f0 and f1. Therefore, a better notation wouldbe M0 ·f0f1 M1. However, we will only be interested in the equivalence class of this cobordism, whichdoes not depend on these maps, according to theorem 3.3.

14

to diffeomorphisms. The corresponding diffeomorphisms clearly have to preserve orien-tation and map in-boundaries to in-boundaries and out-boundaries to out-boundaries.Therefore the equivalence class [M0M1] is uniquely defined in HomnCob(Σ0,Σ2,). Start-ing with two morphisms [M0] and [M1] the composition that we write as [M0][M1] =[M0M1] is hereby well defined, since it does not depend on the chosen representative.

Associativity

The last thing left to check is that the defined composition is associative. Let M0 :Σ0 ⇒ Σ1, M1 : Σ1 ⇒ Σ2 and M2 : Σ2 ⇒ Σ3 be three cobordisms. First note that astopological spaces, the equality (M0

∐Σ1M1)

∐Σ2M2 = M0

∐Σ1

(M1∐

Σ2M2) applies,

when regarding the original spaces as subsets of the formed quotientspace like discussedearlier. Also, observe that we can choose the ε’s in the definition of the compositionsmall enough such that the corresponding ”cylinders” are disjoint. Therefore, the twocompositions do not affect each other. These comments can be combined to see that([M0][M1])[M2] = [M0]([M1][M2]).

3.1.2 Identity morphisms

The last thing to do in order to complete the description of the category of cobordisms, isto show that every object has an identity morphism. Let Σ be a closed oriented (n− 1)-manifold. We claim that the morphism [C] is the identity on Σ, where C := Σ × [0, 1].To show this, take a cobordism M : Σ ⇒ Σ′. As mentioned in the contruction of thecomposition of two cobordisms, M can be seen as the composition of M[0,ε] and M[ε,1],where M[0,ε] is diffeomorphic to a cylinder over Σ. Two cylinders can clearly be gluedtogether to form another cylinder, so there is an orientation preserving diffeomorphismCM[0,ε] →M[0,ε]. So now

[C] [M ] = [C]([M[0,ε]

] [M[ε,1]

])=([C][M[0,ε]

]) [M[ε,1]

]=[M[0,ε]

] [M[ε,1]

]= [M ] .

This proves that [C] is a left unit for Σ. A similar proof shows that it is also a rightunit. In conclusion, [C] is the identity on Σ.

3.2 Definition of a TQFT

Now that all the relevant concepts are introduced, we can finally define what a TQFTis. Let K be some fixed ground field.

Definition 3.5. An n-dimensional TQFT is a functor A from nCob to the categoryVectK of vector spaces over K that satisfies the following two conditions:

15

1. If Σ and Σ′ are two objects in nCob, then A(Σ∐

Σ′) ∼= A(Σ)⊗K A(Σ′).

2. A(∅) = K, where ∅ is the empty (n− 1)-manifold.

Note that the second condition implies that the empty cobordism gets sent to IdK , sincethe empty cobordism is the cylinder over the empty manifold.

Mathematically the two conditions reflect the monoidal structure of the two categoriesinvolved. To be precise, A is a symmetric monoidal functor (see chapter 3 from [K03]).To give some physical intuition, a TQFT can be seen as a rule that assigns to a physicalsystem its state space and to a time evolution between two systems a linear map betweenthe corresponding state spaces. The first condition in the defintion then corresponds tothe fact that in quantum mechanics the state space of two independent systems is thetensor product between the state spaces of the seperate systems.

3.3 Relationship with quantum mechanics

This section will explain the relationship between TQFT and quantum mechanics (QM).In QM, a physical system is associated to its space of states. Operators on the systemcorrespond to linear transformations of the state space. By computing inner products,probabilities can be calculated that measurements give certain outcomes. We will discusshow these operators and probabilities can be seen in the context of TQFT. This sectionis based on [Ba95].

The following notation for a TQFT will be used: a boundary Σ gets mapped to avector space V (Σ) and a cobordism M to a linear map ZM . Note that a cobordismM : ∅⇒ ∂M corresponds to a state ZM ∈ V (∂M), since the map ZM : C→ V (∂M) iscompletely determined by the image of 1. Denoting Σ for the manifold Σ with oppositeorientation, there is a canonical isomorphism V (Σ) ∼= V (Σ)∗, where V (Σ)∗ is the dualspace of V (Σ) consisting of the linear functionals V (Σ) → C [DW90]. This sectionwill assume the existence of a linear map aΣ : V (Σ) → V (Σ) for all boundaries Σ,corresponding to the inner product < σ, τ >:= aΣ(σ)(τ), with σ, τ ∈ V (Σ). We willassume that a∂M (ZM ) = ZM . Also, for closed M , we will assume a(ZM ) = ZM = ZM ∈C, where the last bar denotes complex conjugation in C. Throughout this chapter, the¨bra-ket” notation from QM will be used:

|M〉 := ZM ∈ V (∂M)

〈M | := ZM ∈ V (∂M).

Then if ∂N = ∂M the inner producted between ZM and ZN can be denoted 〈M |N〉 =ZM (ZN ) = a(ZM )(ZN ) =< ZM , ZN >.

3.3.1 Operators and probabilities

As mentioned at the end of section 3.2, a boundary Σ can be identified with a physi-cal system and the vector space V (Σ) with the space of possible states of the system.

16

A manifold C with in-boundary Σ and out-boundary Σ corresponds to a linear mapV (Σ) → V (Σ). If ZC happens to be corresponding to an operator on the system Σ,we can look at the result of applying the operator C to the state ZM ∈ V (Σ), where∂M = Σ. This is done by gluing M and Σ to get the manifold M

∐ΣC corresponding

to a state in V (Σ), which can be denoted C |M〉. Now if there is another operator Don Σ, then the manifold M

∐ΣC

∐ΣD can be constructed. Here, the order of gluing

reflects the time order in which the operators are applied to the system. If operator Cis applied first and then operator D is applied, we must glue the in-boundary of C to Mand the in-boundary of D to the out-boundary of C.

If π : Σ⇒ Σ is an operator corresponding to an orthogonal projection in V (Σ), then inQM π is associated to the proposition of a measurement resulting in a certain outcome.Here π |M〉 asserts the desired outcome and (1 − π) |M〉 asserts the opposite. We canlook at the probabilities of this happening:

pyes =〈M |π|M〉〈M |M〉

pno =〈M |1− π|M〉〈M |M〉

,

where of course pyes + pno = 1. If there are two such projections π1, π2, they can becombined to give:

pyes,yes =〈M |π1π2π1|M〉〈M |M〉

pyes,no =〈M |π1(1− π2)π1|M〉

〈M |M〉

pno,yes =〈M |(1− π1)π2(1− π1)|M〉

〈M |M〉

pno,no =〈M |(1− π1)(1− π2)(1− π1)|M〉

〈M |M〉.

In conclusion, we have shown that many of the fundamental concepts of QM can betranslated to the language of TQFT.

17

4 G-coverings and classification

In this chapter, the notion of G-coverings will be introduced. In sections 4.2 and 4.3,two ways to classify the G-coverings over a certain space will be described. Both clas-sifications will be of importance in the next chapter to understand Dijkgraaf-Wittentheory. In this chapter, the topological group G is always assumed to be equipped withthe discrete topology.

4.1 G-coverings and isomorphisms

The following section is based on chapters 11 through 16 from [Fu95]. The informaldefinition of a G-covering is a covering space that arises from a group action on atopological space. Recall that a (left) group action from G on a topological space Y isa map G× Y → Y denoted (g, y) 7→ g · y, that satisfies:

• Associativity: g · (h · y) = (gh) · y for all g, h ∈ G and y ∈ Y ;

• Neutral element: e ·y = y for all y ∈ Y , where e denotes the identity element of G.

Furthermore, we always assume that for each g ∈ G the map y 7→ g · y is a homeomor-phism of Y . This implies that if U is open in Y , then so is gU . When such a groupaction is defined, the set Y/G denotes the set of orbits with the quotient topology comingfrom the projection map Y → Y/G. In order for this map to be a covering map, wehave to put some extra condition on the group action. Therefore, we make the followingdefinition.

Definition 4.1. We call a group action properly discontinuous if every y ∈ Y has aneighbourhood V such that g · V ∩ h · V = ∅ for all g 6= h in G.

It turns out that this condition is sufficient to make the projection map a coveringmap as stated in the next lemma.

Lemma 4.2. If the group action of G on Y is properly discontinuous, then the projectionp : Y → Y/G is a covering map.

Proof. By definition, p is continous. It is also open, since for an open U ⊆ Y the setp−1◦p(U) =

⋃g∈G gU is the union of open sets and is therefore itself open. By definition

of the quotient topology, this means that p(U) is open which then implies that the mapp is in fact open. Left to prove is that each point y ∈ Y/G has an open neighbourhoodthat is evenly covered by p. If we take y ∈ Y such that p(y) = y, y has an open

18

neighbourhood V that satisfies the condition in definition 4.1. Define V := p(V ). Thisis an open neighbourhood of y since p is open. We want to show that every set inthis disjoint union p−1(V ) =

∐g∈G gV is mapped homeomorphically to V by p. It is

enough to show that p|gV : gV → V is a bijection for every g ∈ G, since p is open andcontinuous. Suppose that p(gy1) = p(gy2) for some y1, y2 ∈ V . Then there is an h ∈ Gsuch that hgy1 = gy2, so this element lies in hgV ∩gV . Since the group action is properlydiscontinuous, this can only be true if h = e and therefore gy1 = gy2. This proves theinjectivity. The surjectivity follows from the fact that V = p(V ) = p(gV ).

Now that we know that properly discontinuous group actions induce covering maps,it makes sense to consider coverings that arise in this way.

Definition 4.3. i) A G-covering is a covering p : Y → X together with a properlydiscontinuous action of G on Y , such that there is a homeomorphism ψ : X → Y/Gthat makes the following diagram commute:

Y

X Y/G.

pπ

ψ

Here, π is just the projection of Y onto its orbits. A G-covering is often denoted asa triple (Y, p,X).

ii) Two G-coverings p : Y → X and p′ : Y ′ → X over the same base space are calledisomorphic if there is a homeomorphism φ : Y → Y ′ such that p′ ◦ φ = p andφ(gy) = gφ(y) for all y ∈ Y and g ∈ G. In this case φ is called an isomorphism ofG-coverings.

iii) The category G(X) is defined to have G-coverings over X as its objects and isomor-phisms of G-coverings as morphisms.

For a given base space X, there is a natural way to create a G-covering by settingY = X × G and defining g(x, h) = (x, gh) for g, h ∈ G and x ∈ X. The projectionX ×G→ X is called the trivial G covering over X. It turns out that every G-coveringlocally looks like a trivial G-covering. More formally, if p : Y → X is a G-covering thenevery point in X has a neigbourhood U such that p|p−1(U) : p−1(U)→ U is isomorphic asa G-covering to the trivial G-covering over U . This can easily be seen by setting U equalto V as in the proof of lemma 4.2 and defining φ : p−1(U)→ U ×G as φ(gv) = (p(v), g).Since G is discrete and p is a homeomorphism when restricted to the appropriate domainand codomain, φ is a homeomorphism. Clearly φ also satisfies the other properties forbeing an ismorphism of G-coverings, which proves the claim.

Next, we want to describe the G-coverings over a given base space up to isomorphisms.There are two ways to do this. The first one uses the concept of so called Cech cocycles.The second method uses homotopy classes of maps to a fixed base space.

19

4.2 Cech cohomology

We know that every G-covering is locally trivial. In this section we will describe a wayto classify the G-coverings that are trivial over a certain open cover of a space. First,we introduce the concept of Cech cocycles.

Definition 4.4. Let X be a topological space with covering U = {Uα : α ∈ A}. Thena Cech cocycle on U with coefficients in G is a collection {gαβ : α, β ∈ A} of locallyconstant functions gαβ : Uα ∩ Uβ → G satisfying:

1. gαα = e for all α ∈ A,

2. gβα = (gαβ)−1 for all α, β ∈ A,

3. gαγ = gαβgβγ on Uα ∩ Uβ ∩ Uγ for all α, β, γ ∈ A.

Two Cech cocycles {gαβ} and {g′αβ} are called cohomologous if for all α, β ∈ A suchthat Uα ∩ Uβ 6= ∅ there are locally constant functions hα : Uα → G such that g′αβ =

(hα)−1gαβhβ on Uα∩Uβ. It is not hard to see that being cohomologous defines an equiva-lence relation on the set of cocycles and we call the equivalence classes Cech cohomologyclasses on U with coefficients in G. The set H1(U , G) is the set of these classes1.

We start with a technical lemma, describing isomorphsims of trivial G-coverings. Theproof is based on [No15].

Lemma 4.5. Consider the trivial G-covering p : Y = X × G → X over X. Then forevery isomorphism of G-coverings φ : Y → Y there is a unique locally constant funcionh : X → G such that φ(x, g) = (x, g · h(x)).

Proof. Let φ : Y → Y be an isomorphism of G-coverings. Suppose that φ sends (x, e)to (x′, h(x)), where h is a function from X to G. Then for g ∈ G

φ(x, g) = gφ(x, e) = g(x′, h(x)) = (x′, gh(x)).

Now x = p(x, g) = p ◦ φ(x, g) = p(x′, gh(x)) = x′, so the first coordinate is preserved.Define s : X → Y , x 7→ (x, e) and π : Y → G, (x, g) 7→ g. Clearly, both are continuous.Then

π ◦ φ ◦ s(x) = π ◦ φ(x, e)

= π(x, h(x))

= h(x),

and since π, φ and s are continuous, so is h. Now for a connected component U ⊆ X,we can write U =

⋃g∈G(h|U )−1({g}), which is a disjoint union of open sets so only

one can be non-empty. Therefore, h is locally constant. Clearly if h′ also satisfiesφ(x, g) = (x, g · h′(x)), then h′ = h.

1Note that G is not necessarily Abelian.

20

4.2.1 From G-covering to Cech cocycle

From a G-covering p : Y → X a Cech cocycle can be constructed by looking at atrivialisation of the covering and applying lemma 4.5. This is done in the following way.Since the G-covering is locally trivial, there is an open covering U = {Uα : α ∈ A} ofX, such that p−1(Uα) → Uα is isomorphic to a trivial covering for every α. Therefore,we can find isomorphisms of G-coverings φα : Uα × G → p−1(Uα). By composing therestrictions of φα with φ−1

β on Uα ∩ Uβ for α, β ∈ A such that Uα ∩ Uβ 6= ∅ we get

isomorphisms of G-coverings φ−1β ◦ φα : (Uα ∩ Uβ) × G → (Uα ∩ Uβ) × G. By lemma

4.5, these isomorphisms can be written as (x, g) 7→ (x, g · gαβ(x)) for unique locallyconstant functions gαβ : Uα ∩Uβ → G. These functions can easily be seen to satisfy theproperties listed in definition 4.4. Therefore, we have constructed a Cech cocycle from theG-covering p. Note that the resulting cocycle depends on the trivialization {φα : α ∈ A}chosen. We claim though, that if we had used a different trivialization {φ′α : α ∈ A} theresulting cocycles are cohomologous. To see this, first fix α ∈ A. Then φα and φ′α definetwo isomorphism of G-coverings Uα ×G→ p−1(Uα). So the composition (φ′α)−1 ◦ φα isan isomorphism of G-coverings from Uα ×G to itself. Therefore by using lemma 4.5 wefind a unique locally constant function hα : Uα → G such that this isomorphism can bewritten as (x, g) 7→ (x, g · hα(x)). Then we get the following relation:

(x, g · gαβ(x)hβ(x)) = ((φ′β)−1 ◦ φβ) ◦ (φ−1β ◦ φα)(x, g)

= (φ′β)−1 ◦ φα(x, g)

= ((φ′β)−1 ◦ φ′α) ◦ ((φ′α)−1 ◦ φα)(x, g)

= (x, g · hα(x)g′αβ(x)),

where {g′αβ} is the cocycle formed from the trivialization {φ′α}. So the two cocycles are

related by g′αβ = (hα)−1gαβhβ, which means that they are cohomologous. In the sameway we can prove that two isomorphic G-coverings induce cohomologous cocycles whenwe take the same sets Uα over which the coverings are trivial.

4.2.2 From Cech cocycle to G-covering

Now we want to reverse this process and use a Cech cocycle {gαβ : α, β ∈ A} on U to makea G-covering that is trivial over all the Uα’s. To do this first define Σ :=

∐α∈A(Uα×G).

Then we define for x ∈ Uα∩Uβ and g ∈ G that (xα, g) ∼ (xβ, g·gαβ(x)). Here, xα denotesx regarded as element of Uα in the disjoint union. From the properties listed in definition4.4, it can easily be seen that this defines an equivalence relation on Σ. We denote q : Σ→Y as the quotient map, where Y is the set of equivalence classes, and equip Y with thequotient topology. A group action on Y can be defined by g′ · [(x, g)] = [(x, g′ · g)] whereg′ ∈ G and [(x, g)] = q((x, g)). This action is well-defined, since the equivalence relationis compatible with left-multiplication by G. Also, it clearly satisfies the requirements ofassociativity and neutral element defined at the start of this chapter. We also need tocheck that for g′ ∈ G the map µ : Y → Y, [(x, g)] 7→ [(x, g′ · g)] is a homeomorphism.If U ⊆ Y is open, then V := q−1(U) ⊆ Σ is open. µ(U) = µ ◦ q(V ) = q(g′V ), so

21

q−1(µ(U)) = q−1 ◦ q(g′V ) = g′V is open. Therefore µ(U) is open. Since µ−1 is just µwith g′ replaced by (g′)−1, µ−1 is also open. The map is clearly bijective and thereforeµ is a homeomorphism. So this multiplication satisfies all the requirements for a groupaction. Left to check is that the action on Y is properly discontinuous. Take [(x, g)] ∈ Yand let V = q(Uα × {g}). For elements in V we can find a unique representation [(y, g)]with (y, g) ∈ Uα × G ⊆ Σ. Let h1, h2 ∈ G and suppose that h1V ∩ h2V 6= ∅. Thenthere exists an element in this intersection that we can represent in the representationdescribed before. Since the element is in h1V it can be denoted as [(y, h1g)] and since itis in h2V as [(y, h2g)]. Since in both notations we have y ∈ Uα in the disjoint union Σ,we must have h1 = h2. This proves that the action is properly discontinuous.

Define p : Y → X as the projection [(x, g)] 7→ x, which clearly is well-defined. Also letψ : X → Y/G be the map x 7→ [(x, e)]G. This is a homeomorphism and ψ ◦ p([(x, g)]) =[(x, e)]G = [(x, g)]G, so ψ ◦ p is just the projection Y → Y/G. By lemma 4.2 thisprojection is a covering. Since ψ is a homeomorphism, p is also a covering and therefore aG-covering. It can easily be seen that p is trivial over each Uα, since p−1(Uα) = q(Uα×G)and q restricts to an isomorphism of G-coverings Uα ×G→ q(Uα ×G).

If we start with two cohomologous cocycles {gαβ} and {g′αβ}, there are locally constant

functions {hα : α ∈ A} such that g′αβ = (hα)−1gαβhβ on Uα∩Uβ. We get two equivalencerelations with corresponding qoutient maps on Σ. The first relation ∼R is given by(xα, g) ∼R (xβ, g · gαβ(x)) and the second, ∼R′ , by (xα, g) ∼R′ (xβ, g · g′αβ(x)). Wedenote the two quotient maps as follows:

q : Σ→ Y := Σ/ ∼R(x, g) 7→ [(x, g)]

q′ : Σ→ Y ′ := Σ/ ∼R′

(x, g) 7→ (x, g).

Now define φ : Y → Y ′ as [(x, g)] 7→ (x, g · hα(x)) for x ∈ Uα. We first show this iswell-defined. If x ∈ Uα ∩ Uβ, then [(x, g)] = [(x, g · gαβ(x))]. We know

φ([(x, g · gαβ(x))]) = (x, g · gαβ(x)hβ(x))

= (x, g · hα(x)g′αβ(x))

= (x, g · hα(x))

= φ([(x, g)]),

so φ is well-defined. It is also a homeomorphism since the hα’s are locally constant. Ifwe take g, g ∈ G then we clearly have φ(g[(x, g)]) = gφ([(x, g)]). Also, if p and p′ arethe G-coverings constructed from the two cocycles, then p′ ◦ φ = p since the coveringsare just projections on X. So φ satisfies all the conditions for being an isomorphism ofG-coverings. Thus cohomologous cocycles induce isomorphic G-coverings.

In summary, in this section we proved the following theorem.

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Theorem 4.6. The constructions from this section give well-defined maps

MG(X;U)→ H1(U , G) and H1(U , G)→MG(X;U).

Here, the set MG(X;U) denotes the set of equivalence classes of G-coverings that aretrivial over each Uα, where two G-coverings are equivalent if they are isomorphic.

These maps turn out to be inverses to each other, which will not be shown here. See[Fu95] for the proof.

4.3 Homotopy description

The main idea of this section is to construct a so-called universal G-covering. This meansthat we will define a covering space EG with corresponding base space BG such that forevery space X there is a bijection from the homotopy classes of maps X → BG to theset of equivalence classes of G-coverings over X. Some parts of this section will sketchthe steps involved rather than going into the details of the process. This is done in orderto improve the legibility and because the detailed process won’t be important for therest of the chapters.

4.3.1 (Semi)simplicial sets

To define the spaces EG and BG, we first need the concept of (semi)simplicial spaces.For a reference for this subsection, see [Mu07].

Definition 4.7. A semisimplicial set (SSS) X? is a collection of spaces Xn, where n ≥ 0,

and face maps ∂(n)i : Xn → Xn−1, where n ≥ 1 and 0 ≤ i ≤ n, such that

∂(n−1)i ◦ ∂(n)

j = ∂(n−1)j−1 ◦ ∂(n)

i , (4.1)

for all i < j. For two SSS’s X? and Y? a collection of maps mn : Xn → Yn, for n ≥ 0 iscalled a morphism of SSS’s if the mn’s commute with the ∂i’s. That is, for n ≥ 1 and0 ≤ i ≤ n the following diagram commutes.

Xn Xn−1

Yn Yn−1

∂X,i

mn mn−1

∂Y,i

(4.2)

This defines a category SSS of SSS’s.

Note that in the diagram, the index above the face maps is omitted. This is a com-mon convention, as it rarely causes confusion. In the following text we will adopt thisconvention as well.

There is a construction that turns a SSS into a topological space, called the geometricrealisation, which we define below. As it turns out, this construction can be extended

23

to a functor SSS → Top. To define the geometric realization, first we briefly lookat maps between standard n-simplices. For 0 ≤ i ≤ n define di : ∆n−1 → ∆n by(x0, . . . , xn−1) 7→ (x0, . . . , xi−1, 0, xi, . . . , xn−1). This maps the standard (n− 1)-simplexto a face of the standard n-simplex. Now we can define the geometric realization.

Definition 4.8. For a SSS X? let all the Xn be equipped with the discrete topology.On the space

∐n≥0Xn ×∆n define the equivalence relation ∼, where ∼ is generated by

(x, di(y)) ∼ (∂i(x), y), for x ∈ Xn and y ∈ ∆n−1. Define the geometric realization ofX? as

|X?| =

∐n≥0

Xn ×∆n

/ ∼,

where we give |X?| the quotient topology.

As the following theorem proves, a morphism of SSS’s induces a continuous mapbetween the geometric realizations of the involved SSS’s, making | · | into a functor.

Theorem 4.9. Let m : X? → Y? be a morphism of SSS’s. Then there is a continuousmap |m| : |X?| → |Y?|.

Proof. First define |m|0 :∐n≥0Xn × ∆n →

∐n≥0 Yn × ∆n by (x, t) 7→ (mn(x), t) for

x ∈ Xn and t ∈ ∆n. Now |m|0(∂i(x), t) = (mn∂i(x), t) = (∂imn(x), t), because m is amorphism of SSS’s. Also |m|0(x, di(t)) = (mn(x), di(t)) ∼ (∂imn(x), t), so |m|0 factorsthrough the equivalence relation to give a function |m| : |X?| → |Y?|. Since |m|0 iscontinuous, the map pY ◦|m|0 is too, where pY :

∐n≥0 Yn×∆n → |Y?| is the quotient map.

So for U ⊆ |Y?| open, (pY ◦ |m|0)−1(U) is open. Then |m|−1(U) = pX((pY ◦ |m|0)−1(U)),where pX :

∐n≥0Xn ×∆n → |X?| is again the quotient map. Since pX is an open map,

|m|−1(U) is open and |m| is continuous.

The constructions above can be extended to simplicial sets. A simplicial set is asemisimplicial set that also has maps σi : Xn → Xn+1 for 0 ≤ i ≤ n, called thedegeneracy maps. The degeneracy maps have to satisfy certain commutation relationswith the face maps, that we won’t discuss in detail, because they are in our case nearlytrivial. The geometric realization of a simplicial set is the same as that of a semisimplicialset with an extra equivalence relation induced by the degeneracy maps and the mapssi : ∆n+1 → ∆n given by si(t0, . . . , tn+1) = (t0, . . . , ti−1, ti + ti+1, ti+2, . . . , tn+1). For amore detailed introduction to simplicial sets, see [Ma99].

4.3.2 Classifying space of a group

The construction in the previous section can be applied to groups. This will form twospaces: EG and BG. We will give two descriptions of BG, namely as the geometricrealization of a simplicial set and as the quotient EG/G. Most of this subsection isbased on [Ma99].

24

Definition 4.10. Define En(G) = Gn+1. This defines a simplicial set with face maps∂i : En(G)→ En−1(G) and degeneracy maps σi : En(G)→ En+1(G) defined as follows:

∂i(g1, . . . , gn+1) =

{(g2, . . . , gn+1) i = 0

(g1, . . . , gi−1, gigi+1, gi+2, . . . , gn+1) 1 ≤ i ≤ nσi(g1, . . . , gn+1) = (g1, . . . , gi−1, e, gi, . . . , gn+1),

for 0 ≤ i ≤ n. The space EG is now defined as the geometric realization of E?(G). SoEG =

∐n≥0(Gn+1 ×∆n)/ ∼, where ∼ is generated by the two relations

(ζ, di(t)) ∼ (∂i(ζ), t) for ζ ∈ Gn+1 and t ∈ ∆n−1

(ζ, si(t′)) ∼ (σi(ζ), t′) for ζ ∈ Gn+1 and t′ ∈ ∆n+1.

We define a group action on EG by g · [((gi)n+1i=1 , (ti)

ni=0)] = [((g · gi)n+1

i=1 , (ti)ni=0)]. Here

[(ζ, t)] denotes the equivalence class of (ζ, t) in EG for ζ ∈ En(G) and t ∈ ∆n. Somultiplication by g ∈ G, amounts to multiplying all the gi ∈ G that represent the elementin En(G) by g, while leaving the t ∈ ∆n unchanged.

To see that we get a G-covering EG→ EG/G, consider the next theorem.

Theorem 4.11. The action of G on EG is properly discontinuous.

Proof. Let p := ((g1, . . . , gn+1), t) ∈ Gn+1 × ∆n. Assume t ∈ ∂∆n. Then there is at′ ∈ int(∆n+1) such that si(t

′) = t and therefore p ∼ (σi(g1, . . . , gn+1), t′). So withoutloss of generality we may take p such that t ∈ int(∆n). Let V = {(g1, . . . , gn+1)}×int(∆n)and V be the corresponding set in EG. Then [p] ∈ V . First we prove that V is open.Observe that V is open, since G is discreet. Now in the proof of theorem 4.9, we sawthat the quotient map from a (semi)simplicial set to its geometric realization is open andtherefore V is open. Now we prove that V satisfies the condition from definition 4.1. Note

that EG =⋃q≥0

(∐0≤m≤q(G

m+1 ×∆m)/ ∼)

. Clearly in∐

0≤m≤n−1(Gm+1 ×∆m)/ ∼we have gV ∩ hV = ∅ if g, h ∈ G with g 6= h, because p is not equivalent to anything inthis set. So also in EG, we must have gV ∩ hV = ∅ if g 6= h. It follows that the actionis properly discontinuous.

As a corollary of this theorem, we see that the map πG : EG→ EG/G is a G-covering.The base space EG/G will be denoted by BG and is called the classifying space of G.We define ξG = (EG, πG, BG) as this G-covering. It turns out that BG can also bedefined as the geometric realization of a simplicial set. Since this description will be ofuse later, this way of constructing BG will also be covered here.

Definition 4.12. Define Bn(G) = Gn, together with face maps ∂i : Bn(G)→ Bn−1(G)

25

and degeneracy maps σi : Bn(G)→ Bn+1(G) defined by:

∂i(g1, . . . , gn) =

(g2, . . . , gn) i = 0

(g1, . . . , gi−1, gigi+1, gi+2, . . . , gn) 1 ≤ i ≤ n− 1

(g1, . . . , gn−1) i = n

σi(g1, . . . , gn) = (g1, . . . , gi−1, e, gi, . . . , gn),

for 0 ≤ i ≤ n. Then BG = |B?(G)| is the geometric realization of this simplicial set.

4.3.3 Homotopy classification

Now that the classifying space of a group has been defined, this space will be used toconstruct a bijection [X,BG]→MG(X), where [X,BG] is the set of homotopy classesof maps X → BG andMG(X) are the isomorphism classes of G-coverings over X. Theconstruction that will be used for this bijection is called the pull back. See [Hus94] as areference.

Definition 4.13. Let p : E → B be a G-covering and f : X → B a map. Then wedefine the pull back of f as (f∗(E), f∗(p), X), where

f∗(E) = {(x, e) ∈ X × E : f(x) = p(e)}

and f∗(p) : f∗(E) → X is the projection (x, e) 7→ x. The action of G on f∗(E) isdefined as multiplication with the second coordinate. This G-covering is often denotedf∗(ξ), where ξ = (E, p,B). There is a canonical map fξ : f∗(E) → E that satisfiesp ◦ fξ = f ◦ f∗(p), namely the projection on the second coordinate.

As the reader can check, the pull back indeed defines a G-covering. Note that forx ∈ X, we can identify the fiber of f∗(ξ) over x with the fiber of p over f(x), sincethe former is given by (f∗(E))x = {(x, e) ∈ X × E : f(x) = p(e)} and the latter byEf(x) = {e ∈ E : p(e) = f(x)}. Moreover, two homotopic maps define isomorphicG-coverings as the next theorem states. The proof of this theorem is based on [C98].

Theorem 4.14. Let ξ = (E, p,B) be a G-covering. If two maps f, g : X → B arehomotopic, then their induced G-coverings f∗(ξ) and g∗(ξ) are isomorphic.

Proof. Since f and g are homotopic, there exists a homotopy H : X×I → B, with H0 =f and H1 = g. Then F := H ◦(f∗(p)×IdI) : f∗(E)×I → B is also a homotopy and fξ isa lifting of F0, so by theorem 2.42 there is a lifting of F to a homotopy F : f∗(E)×I → E,with F0 = fξ. Now for (x, t) ∈ X × I on the level of fibers F maps (f∗(E) × I)(x,t) to

EH(x,t) = (H∗(E))(x,t). So F gives rise to a map H : f∗(E)× I → H∗(E) that satisfies

2Note that here a slightly more general version of the homotopy lifting lemma is used than the onedescribed in the preliminaries. As a reference see [Hus94] or [C98]. This version imposes extraconditions on the spaces involved, which we will not ellaborate on as the spaces we are concernedwith will always satisfy these conditions.

26

H∗(p) ◦ H = f∗(p)× IdI . Now by restricting this map to f∗(E)× {1} ⊂ f∗(E)× I, weget the following commutative diagram:

f∗(E)× {1} g∗(E)× {1}

X × {1} .

H

f∗(p) g∗(p)

The reader is invited to check that this map indeed defines an isomorphism of G-coverings.

Recall that we defined a G-covering ξG = (EG, πG, BG). For a topological space X,a map X → BG then induces a G-covering over X. By the previous theorem, the mapψ : [X,BG] → MG(X) with ψ([f ]) = f∗(ξG) is well defined, where [f ] denotes thehomotopy class of f . The main result of this section will be that ψ is a bijection. Inthe following paragraphs an inverse map will be constructed to sketch this proof. Thismethod is based on [S68].

Definition 4.15. Let X be a topological space and U = {Ui : i ∈ I} an open cover.Construct the following simplicial set:

X0 =∐i∈I

Ui

Xn =∐

i0,...,in∈I(Ui0 ∩ · · · ∩ Uin),

with face maps ∂j : Xn → Xn−1 defined by ∂j(xi0,...,in) = xi0,...,ij−1,ij+1,...,in and degener-acy maps σj : Xn → Xn+1 defined by σj(xi0,...,in) = xi0,...,ij ,ij ,...,in. Here, xi0,...,in denotesthe element x seen as an element of Ui0 ∩ · · · ∩ Uin in the disjoint union.

As the reader may check, these maps satsify the properties for a simplicial space.From a G-covering we will construct a morphism of simplicial sets. As shown before,a G-covering induces a Cech cocycle. So for a G-covering over X, let U be an opencovering such that the covering is trivial over U . Then, let X? be the simplicial setdefined above and {gij}i,j∈I the cocycle over U induced by the covering. Now let m0 :X0 → B0(G) = {∗} be the unique map to a point space and mn : Xn → Bn(G) be givenby mn(xi0,...,in) = (gi0i1 , . . . , gin−1in). An easy check confirms that these maps commutewith the face and degeneracy maps and therefore define a morphism of simplicial spaces.Now by the extension of theorem 4.9 to simplicial spaces, this induces a continuous mapfrom |X?| to |B?(G)| = BG. Since |X?| is homotopy equivalent to X (see [S68]), thisdefines a map X → BG up to homotopy. It turns out that this construction factors to amapMG(X)→ [X,BG], which we won’t prove here. Also, this map will be the inverseof the map ψ that was defined above. For a different approach to the proof that ψ is abijection, see chapter 16 of [Ma99].

27

In summary, this chapter has introduced the concept of G-coverings and has giventwo ways of classifying the equivalence classes of G-coverings over a given space. In thenext chapter these concepts will be used to define a TQFT.

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5 Dijkgraaf-Witten theory

This chapter will cover the details of Dijkgraaf-Witten (DW) theory. We fix a finitegroup G throughout this chapter and write M(X) :=MG(X) for the G-coverings overa topological space X. First, the simplest version of DW theory is introduced, whichis called untwisted DW theory. Afterwards, a generalization is introduced using a fixedcohomology class in the cohomology of BG with coefficients in the circle group T = {z ∈C : |z| = 1} ⊆ C∗. The main concept behind DW theory is the counting of G-coveringsover manifolds. To each G-covering a certain ”mass” is associated, by counting thenumber of automorphisms of the covering. This brings us to the following defininition.

Definition 5.1. Let M and N be manifolds (either with or without boundary) such thatN ⊆M . This induces a functor r : G(M)→ G(N) by restriction of G-coverings over Mto the subspace N . For a G-covering ξ in G(M) this functor maps automorphisms of ξto automorphisms of rξ. Define

Aut(ξ; r) := {ϕ ∈ HomG(M)(ξ, ξ) : rϕ = Idrξ}.

If for example N = ∂M in the previous definition, then Aut(ξ; r) consists of all theautomorphisms of ξ that are trivial on the boundary of M . With this definition in mind,we start with the untwisted version of DW theory.

5.1 Untwisted Dijkgraaf-Witten theory

This section is largely based on [Lu12]. For any manifold N with (possibly empty)boundary, we define the complex vector space

V (N) = C[N,BG] = {f :M(N)→ C}. (5.1)

First, we want to see that when N is compact, this defines a finite-dimensional space asthe following lemma states.

Lemma 5.2. The complex vector space V (N) defined in equation 5.1 is finite dimen-sional for every compact manifold N .

Proof. First, suppose N is connected. Then a G-covering M → N induces a homomor-phism π1(N,n)→ G, where n is a basepoint of N , in the following way. Fix a basepointm ∈M . A loop at n can be lifted to a path in M starting at m in a unique way. Sincethe endpoint m′ of this path sits in the fiber over n there is a unique g ∈ G such thatm′ = g ·m and this group element is the same for homotopic loops. Changing the base-point n or m amounts to a conjugation of the constructed homomorphism. If we let G act

29

on Hom(π1(N), G) by conjugation, we get an isomorphism M(N) ∼= Hom(π1(N), G)/G[Fr92]. The fundamental group of a compact manifold is finitely generated1, so thereare only finitely many homomorphisms π1(N) → G. This means that V (N) is a finite-dimensional vector space. If N is not connected then N =

∐i∈I Ni, where the Ni are

the components of N and V (N) = V (∐i∈I Ni) ∼=

⊗i∈I V (Ni) as will be shown in theo-

rem 5.4. Here, the tensor product is taken over C. This shows that V (N) is still finitedimensional.

Note that a functor r : G(M)→ G(N) induces a function r :M(M)→M(N), since afunctor preserves ismorphisms. This in turn defines a linear map r∗ : V (N)→ V (M) byf 7→ f ◦ r, called the pull-back. It will be needed to also have a map r! : V (M)→ V (N)going in the other direction. There is a natural way of defining this map, sometimesreferred to as integration over fibers. In what follows, we will not make a notationaldifference between a G-covering in G(N) and its isomorphism class in M(N), in orderto make the formulas look somewhat more clear. For f : M(M) → C in V (M) andξ ∈M(N) define

r!(f)(ξ) =∑

ν∈M(M)rν=ξ

f(ν)1

#Aut(ν; r).

This means that r!(f) sums the values of f on all isomorphism classes of G-coveringsthat restrict to the G-covering of its input with the appropriate mass as discussed indefinition 5.1.

Now that we have access to these different construction of maps, we can associate alinear map to a cobordism M : Σ0 → Σ1. Recall from the definition of a cobordism(definition 3.1) that M comes with inclusions Σi ↪→ M for i = 0, 1. As before, theseinclusions give rise to functors ri : G(M) → G(Σi). Define ZM : V (Σ0) → V (Σ1) asZM = (r1)! ◦ r∗0. This clearly defines a linear map, since both (r1)! and r∗0 are linear.The formula for ZM is:

ZM (f)(σ) =∑

ν∈M(M)r1ν=σ

f ◦ r0(ν)1

#Aut(ν; r1), (5.2)

where f ∈ V (Σ0) and σ ∈M(Σ1).

Theorem 5.3. Define A : nCob→ VectC as

Σ 7→ V (Σ);

M 7→ ZM ,

for Σ an object in nCob and M a morphism. Then A is a functor.

1See exercise 4 on page 500 of [Mu00]. The statement follows from the fact that every manifold issemilocally simply connected.

30

Proof. First note that when M ∼ M ′ : Σ0 → Σ1 as in definition 3.2, then ZM = ZM ′ .So A is well-defined. Now we check that A behaves well under composition. Take twocobordisms M0; Σ′ → Σ and M1 : Σ→ Σ′′. This gives rise to the following diagram:

G(M0∐

ΣM1)

G(M0) G(M1)

G(Σ′) G(Σ) G(Σ′′).

r0 r1

p′ p q q′′

Here, all the functors are restriction to a subspace. Note that the diagram commutes:p ◦ r0 = q ◦ r1. Computing ZM0

∐ΣM1

amounts to pulling back along p′ and r0 and thenintegrating over fibers along r1 and q′′, whereas ZM1 ◦ ZM0 = q′′! ◦ q∗ ◦ p! ◦ (p′)∗. So tocheck that ZM0

∐ΣM1

= ZM1 ◦ZM0 , we have to verify q∗◦ p! = (r1)!◦ r∗0. Take f ∈ V (M0)and ξ ∈M(M1). Computing both sides of the equation, we get:

q∗ ◦ p!(f)(ξ) =∑

µ∈M(M0)pµ=qξ

f(µ)1

#Aut(µ; p)(5.3)

(r1)! ◦ r∗0(f)(ξ) =∑

η∈M(M0∐

ΣM1)r1η=ξ

f ◦ r0(η)1

#Aut(η; r1)(5.4)

=∑

µ∈M(M0)pµ=qξ

f(µ)∑

η∈M(M0∐

ΣM1)r1η=ξr0η=µ

1

#Aut(η; r1).

Note that when pµ = qξ, we can glue µ and ξ to give an element η ∈M(M0∐

ΣM1) inexactly one way up to isomorphism, so the second sum in the last line of equation 5.4 onlycontains one term. So to check that equations 5.3 and 5.4 are equal, we must prove that#Aut(µ; p) = #Aut(η; r1). To see this, note the map Aut(η; r1)→ Aut(µ; p) : ϕ 7→ r0ϕis a bijection with inverse given by gluing an automorphism of µ that leaves pµ intactwith the identity on ξ to give an automorphism of η that leaves ξ intact. This provesq∗ ◦ p! = (r1)! ◦ r∗0, which in turn shows ZM0

∐ΣM1

= ZM1 ◦ ZM0 . Left to check is thatZΣ×I = IdV (Σ). To see this, take a manifold M with Σ as its in-boundary. Then by theprevious arguments ZM ◦ ZΣ×I = Z(Σ×I)

∐ΣM

= ZM , where the last equality followsfrom the fact that M and (Σ× I)

∐ΣM are equivalent. The same reasoning shows that

ZΣ×I ◦ZM = ZM , when Σ is an out-boundary of M . This proves that A is a functor.

As a final theorem of this section we prove that A defines a TQFT.

Theorem 5.4. The functor A : nCob→ VectC from theorem 5.3 is a TQFT.

31

Proof. The two conditions from definition 3.5 need to be checked. The second one istrivial, since there is only one map ∅ → BG, so V (∅) = C. For the first condition,note that [Σ

∐Σ′, BG] = [Σ, BG] × [Σ′, BG]. Then the map C([Σ, BG] × [Σ′, BG]) →

C[Σ, BG]⊗C[Σ′, BG] given by (f, f ′) 7→ f ⊗f ′ is easily seen to be a linear bijection (see[Q91]). This proves the theorem.

This concludes the exposition of untwisted DW theory. As a remark, note that thisTQFT does not ¨see¨ orientation. The orientation of the manifold M is not used in thedefinition of ZM , when M is closed. This will change when the twist is added in thenext section.

5.2 Twisted Dijkgraaf-Witten theory

After having explained untwisted DW theory in much detail, the twisted version willmerely be sketched. This section is mainly for the purpose of describing the conceptsinvolved and not to give rigorous proofs. References will be given for more detailed texts.

To define twisted DW theory, one fixes a cohomology class [α] ∈ Hn(BG;T). For amanifold M , a G-covering over M is given by a map ν : M → BG up to homotopyas covered in chapter 4. Therefore, we can pull back the cohomology class [α] to acohomology class ν∗[α] ∈ Hn(M ;T), where v∗[α]([z]) = [α][ν#z] for a cycle z in M .The main difference between untwisted and twisted DW theory is that formula 5.2 getsan extra wait factor of W (ν) for every G-covering ν. For a closed manifold M , thisweight is defined as ν∗[α]([M ]) [DW90], where [M ] is the fundamental class of M (seedefinition 2.5). For manifolds with boundary, this definition is more complicated, sincethe fundamental class becomes a relative class in Hn(M,∂M), so it is not in the domainof ν∗[α]. There are different approaches to solving this problem. In [DW90] this problemis solved for a four-dimensional TQFT by seeing M as a union of oriented tetrahedraand defining the weight W as the product of the weights of the tetrahedra. Anotherapproach is given in [Q91], where representatives α for the cohomology class, ν forthe classifying map and m ∈ Cn(M) are chosen, such that i(m) represents [M ], wherei : Cn(M)→ Cn(M,∂M). Then ν∗α(m) is a well-defined complex number. Lemma 5.3from [Q91] states that for a different choice ν ′ such that ν ' ν ′ rel ∂M and m′ such that∂m′ = ∂m the outcome stays the same. For a more detailed explanation of these twosolutions we refer to the original articles.

The addition of this weight factor complicates the definition of the vector spaces thatget assigned to closed (n− 1)-manifolds. Several different solutions can be given for thisproblem. In [T16], a complex number χα(ϕ) is defined for every automorphism ϕ of aG-covering. Then V (Σ) is taken to be the direct sum of C over all coverings that haveχα(ϕ) = 1 for all automorphisms ϕ:

V (Σ) :=⊕

ν∈M(Σ)χα(Aut(ν))={1}

C.

32

Note that this is the same space as defined in equation 5.1, except for the fact that thedirect sum runs over a smaller set. For different approaches to define the vector spaces,see for example [Q91].

In conclusion, we have seen that DW theories come from cocycles in Hn(BG;T), wherethe untwisted version corresponds to the case where the cocycle is trivial. Referring backto section 1.2, these cocycles therefore provide information about the phases of certainkinds of matter. In the next chapter we will see how DW theory fits into a slightlydifferent framework, namely that of quantum field theory, comparing the results withthose obtained in this chapter.

33

6 Dijkgraaf-Witten as QFT

Up to this point, we solely discussed topological quantum field theory, mostly from amathematical point of view. However, in physics one is mostly interested in quantumfield theory, which is usually more complicated. This chapter will cover axioms of bothclassical and quantum field theory and discuss how Dijkgraaf-Witten theory fits into thispicture. Most of this chapter is based on [Fr92].

6.1 Field theory

6.1.1 Classical field theory

To describe an n-dimensional field theory, two ingredients are needed. To every n-manifold M (the space-time), a space of fields CM is assigned. In addition, there is afunction SM : CM → R for every M , called the action. The space-times can be equippedwith some extra structure. For example they can be oriented or have a metric. Thefields and actions are required to satisfy the following axioms.

1. Let f : M ′ →M be a diffeomorphism between two space-times that preserves theextra structure. Then there is a map f∗ : CM → CM ′ such that

SM ′(f∗ν) = SM (ν),

for all ν ∈ CM .

2. If M denotes M with opposite orientation1, then CM = CM and SM (ν) = −SM (ν),for all ν ∈ CM . This axiom reflects the unitarity of the theory. There are alsononunitary field theories in which this axiom does not have to be true.

3. If M and M ′ are two space-times then CM∐M ′∼= CM × CM ′ and furthermore

SM∐M ′(ν t ν ′) = SM (ν) + SM ′(ν

′),

for all ν ∈ CM and ν ′ ∈ CM ′ .

4. If Σ ⊆M is a submanifold of codimension one, M can be ¨cut¨ along Σ to obtaina new manifold M cut. For every ν ∈ CM with corresponding νcut ∈ CMcut

SM (ν) = SMcut(νcut).

1Note that not all field theories must have oriented space-times. This axiom is only applicable fortheories that do.

34

6.1.2 From classical to quantum

To make a classical field theory into a quantum field theory one extra ingredient isneeded: a measure µM on the space of fields CM . Then for a closed space-time M , onecan define the partition function

ZM :=

∫CM

eiSM (ν) dµM (ν).

For space-times M with boundary ∂M , ZM becomes a function of the fields on ∂M .Define CM (σ) = {ν ∈ CM : ∂ν = σ} for fields σ ∈ C∂M . By abuse of notation, themeasure on this new space of fields is still denoted as µM . Then ZM is defined as

ZM (σ) :=

∫CM (σ)

eiSM (ν) dµM (ν).

6.2 Untwisted Dijkgraaf-Witten theory

Now that field theories have been introduced, we will describe DW theory as a fieldtheory, starting with untwisted DW theory. For a space-time M the space of fieldsis defined as CM = MG(M) = [M ;BG]. The definition of the action is very simple.We simply take SM : CM → R to be the zero function. This action clearly satisfiesall the axioms given in subsection 6.1.1. Left to check is that the space of field doestoo. Axioms 2 and 4 are trivially checked. If f : M ′ → M is a diffeomorphism ofspace-times then f∗ : CM → CM ′ is defined as f∗([ν]) := [ν ◦ f ] for ν : M → BG.This is clearly well-defined, proving axiom 1. Finally, for two space-times M and M ′

[M∐M ′;BG] ∼= [M ;BG]× [M ′;BG], which proves axiom 3. So these definitions make

untwisted DW theory into a classical field theory.Now, a measure on the space of fields is needed, in order to define a quantum field

theory. We start by taking M to be a closed space-time. Define µM (ν) = 1#Aut(ν) as

defined in the previous chapter. Then by definition

ZM =

∫MG(M)

dµM =∑

ν∈MG(M)

1

#Aut(ν),

where the integral turns into a finite sum. For space-times M with boundary ∂M definer : G(M) → G(∂M) to be the functor that restricts G-coverings to the boundary asin the previous chapter. Then CM (σ) = {ν ∈ CM : rν = σ} for σ ∈ MG(∂M) andµM (ν) = 1

#Aut(ν;r) . Now

ZM (σ) =

∫CM (σ)

dµM =∑

ν∈MG(M)rν=σ

1

#Aut(ν; r).

Note the similarity in both notation as in formulas between the ZM from this chapterand from the previous chapter (equation 5.2). Of course, this similarity is no coincidence.They both are essentially the same object, written in a different way.

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6.3 Twisted Dijkgraaf-Witten theory

In untwisted DW theory, the action was trivial. This changes when moving to twistedDW theory. Fix a cohomology class [α] ∈ Hn(BG;T). Instead of looking at the action,it is easier in this case to consider the function WM = eiSM : CM → T. For twistedDW theory, this is defined as WM (ν) = ν∗[α]([M ]) for closed M and ν ∈ CM , just likein the previous chapter. Again we face the problem of defining WM for space-timeswith boundary, which is solved in the same way as described there. For the case ofclosed space-times we will check that WM satisfies axioms 1, 2 and 3. Axiom 4 and thecase where the space-time has a boundary will not be covered here, as a more detailedsolutions to defining WM for these space-times is needed for that.

Let M and M ′ be two closed space-times and let f : M ′ → M be an orientation-preserving diffeomorphism. This entails that f sends the fundamental class of M ′ tothat of M . In the untwisted version of DW theory, we saw that f∗ sends a field [ν] to[ν ◦f ]. For notational simplicity, from now on we will write ν for both the map M → BGand the homotopy class of this map. The first axiom states that WM ′(f

∗ν) = WM (ν).This can be seen in the following way.

WM ′(f∗ν) = WM ′(ν ◦ f)

= (ν ◦ f)∗[α]([M ′])

= [α][(ν ◦ f)#M′]

= [α][ν#M ]

= ν∗[α]([M ])

= WM (ν).

For the second axiom, note that the fundamental class of M equals −[M ]. So WM (ν) =ν∗[α](−[M ]) = (ν∗[α]([M ]))−1, since α is multiplicative. So WM (ν) = (WM (ν))−1,which is exactly the statement of axiom 2 after taking the complex exponent. Left toprove is axiom 3, which turns into the statement that WM

∐M ′(νtν ′) = WM (ν)WM ′(ν

′),for closed space-times M and M ′ and fields ν : M → BG and ν ′ : M ′ → BG. First,note that the fundamental class of M

∐M ′ is [M ] + [M ′]. So

WM∐M ′(ν t ν ′) = (ν t ν ′)∗[α]([M ] + [M ′])

= [α][(ν t ν ′)#(M +M ′)]

= [α][ν#M + ν ′#M′]

= ([α][ν#M ]) ([α][ν ′#M′])

= WM (ν)WM ′(ν′).

This shows that at least for closed space-times, WM satisfies the axioms. Now for such

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M , the partition function can be calculated. By definition:

ZM :=

∫CM

WM dµM

=∑

ν∈MG(M)

ν∗[α]([M ])

#Aut(ν).

This concludes the description of DW theory as a QFT.

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Popular summary

When you are walking on the surface of the earth, you get the impression that you arewalking on a flat surface. However, as everybody knows, the earth is actually round.Therefore, the surface of the earth is an object that is not flat when looked at as a whole,but seems flat when you zoom in enough. In mathematics there is a name for these kindof objects that are ”locally flat”: manifolds. Other examples of manifolds are a circleand a donut (see figure 6.1). A circle, for example, is obviously not flat, but if you zoom

Figure 6.1: Three examples of manifolds: a circle, a sphere and a donut.

in enough on its edge, it will almost look like a straight line. A donut will look just likea flat surface if you zoom in enough. Manifolds have a certain dimension. For example,the dimension of the surface of the earth is two and that of the edge of a circle is one.You can see this the following way. If you are walking on the surface of the earth, youcan walk forwards and backwards as well as sidewards, whereas when you’re walkingalong the edge of a circle you can only walk forwards and backwards.

In physics, manifolds are used to model real-world systems. For example, you couldhave an experiment where a particle can move only along the edge of a circle. Asmany people know, the space we live in is three-dimensional and time is viewed as thefourth dimension. A physical model of our space would therefore be a three-dimensionalmanifold and a 4-dimensional manifold (which might be hard to envision) could be amodel of the so-called space-time. Here, space-time means the four-dimensional spacewe live in, consisting of three space-dimensions and one extra dimension for the time.If we look at the example at the beginning of this section, the particle moves in onedimension (along a circle), so the space-time would be two-dimensional.

When looking at a system, physicists want to determine the state the system is in.For the example of the experiment with the particle that moves along a circle, a state ofthe system can be labelled by, amongst others, the position and velocity of the particle.Other states of physical systems can involve the temperature, pressure and accelarationof the system. By considering all of the states a system can be in, you get a wholecollection of different states, that together we call the state space. Now, because of our

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identification of manifolds with physical systems, we can couple a manifold to the statespace of the corresponding system. This is what a Topological Quantum Field Theory(TQFT) does. It takes a manifold as input and produces a state space as output. For thespace we live in, 4-dimensional TQFT’s are most useful, since we live in 4 dimensions,including the time dimension. What happens, is that this 4-dimensional TQFT actuallytakes 3-dimensional spaces as input (for example the space we live in) to give the statespace as output. If we put a 4-dimensional space-time as input, it gives as output howthe state space changes over time.

The construction of TQFT’s is very important in physics, because it gives a lot ofinsight into the processes of determining the state space of a system. Also, mathemati-cians are interested in TQFT’s, since they give information about the manifolds thatyou give as input. In short, TQFT’s are very important to multiple branches of science,providing information about both mathematical and physical concepts.

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