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Electronic Theses, Treatises and Dissertations The Graduate School

2012

Periods and Motives: Applications inMathematical PhysicsDan Li

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THE FLORIDA STATE UNIVERSITY

COLLEGE OF ARTS AND SCIENCES

PERIODS AND MOTIVES: APPLICATIONS IN MATHEMATICAL PHYSICS

By

DAN LI

A Dissertation submitted to theDepartment of Mathematicsin partial fulfillment of the

requirements for the degree ofDoctor of Philosophy

Degree Awarded:Fall Semester, 2012

Dan Li defended this dissertation on August 23, 2012.

The members of the supervisory committee were:

Matilde MarcolliProfessor Directing Dissertation

Laura ReinaUniversity Representative

Paolo AluffiCommittee Member

Amod AgasheCommittee Member

Ettore AldrovandiCommittee Member

The Graduate School has verified and approved the above-named committee mem-bers, and certifies that the dissertation has been approved in accordance with theuniversity requirements.

ii

To my wife, my parents and my younger brother

iii

ACKNOWLEDGMENTS

This dissertation would not have been possible without the help of several in-dividuals who in one way or another contributed their valuable assistance in thepreparation and completion.

First and foremost, I owe my deepest gratitude to my advisor, Dr. Matilde Mar-colli, who has been an inspiration to me for the last five years. Throughout my re-search, she provided encouragement, sound advice and her insights into the specificsof my work. She taught me on various subjects, guided me through the difficultiesand confusions, and sets a moral example as a mathematician. One simply could notwish for a better mentor and friend.

I would like to thank professors in the algebraic geometry group: Dr. Paolo Aluffi,Dr. Ettore Aldrovandi, Dr. Amod Agashe and Dr. Mark van Hoeij. From theirclasses and seminars, I have learned a lot. What I value most was the opportunitiesto discuss with them and absorb their ideas from different perspectives. Due to mylimited time and energy, I only learned a small part, I really hope I can learn morefrom them. Special thanks to Dr. Aluffi for his suggestions to improve my research,I enjoy reading his papers and notes.

I next thank professors in the geometry and topology group: Dr. Eric Klassen,Dr. Wolfgang Heil, Dr. Philip Bowers, Dr. Eriko Hironaka, Dr. Kate Petersen, theyoffered very interesting and helpful classes. With their enthusiasm and their greatefforts to explain things clearly and simply, they made me fall in love with geometryand topology.

iv

TABLE OF CONTENTS

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1 Introduction 1

2 Periods 6

2.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Fuchsian equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Families of elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.6 Residues and Griffiths-Dwork method . . . . . . . . . . . . . . . . . . 212.7 Schwarzian equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Motives 30

3.1 Weil cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Algebraic cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Tannakian category . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Pure motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.5 Mixed Hodge structure . . . . . . . . . . . . . . . . . . . . . . . . . . 483.6 Mixed motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.7 Mixed Tate motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.8 Grothendieck ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Algebraic geometry of Harper Operators 62

4.1 Bloch wave and Fermi surface . . . . . . . . . . . . . . . . . . . . . . 634.2 Harper operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3 Geometry of the Bloch variety . . . . . . . . . . . . . . . . . . . . . . 684.4 Monodromy of Fermi curves . . . . . . . . . . . . . . . . . . . . . . . 734.5 Blow-ups and compactification . . . . . . . . . . . . . . . . . . . . . . 754.6 Density of states as periods . . . . . . . . . . . . . . . . . . . . . . . 814.7 Picard-Fuchs equation of density of states . . . . . . . . . . . . . . . 874.8 Spectral functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.9 Almost Mathieu operator . . . . . . . . . . . . . . . . . . . . . . . . . 94

v

4.10 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . 97

5 Ponzano-Regge model and Feynman motives 100

5.1 Loop quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2 Ponzano-Regge model . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.3 Star product and noncommutative field theory . . . . . . . . . . . . . 1055.4 3d gravity coupled with matter . . . . . . . . . . . . . . . . . . . . . 1065.5 Parametric Feynman integrals . . . . . . . . . . . . . . . . . . . . . . 1095.6 Kirchhoff polynomial with κ-correction . . . . . . . . . . . . . . . . . 1115.7 Hamiltonian action at vertices . . . . . . . . . . . . . . . . . . . . . . 1145.8 Graph hypersurfaces and motives . . . . . . . . . . . . . . . . . . . . 1165.9 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . 119

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

vi

LIST OF FIGURES

4.1 Hofstadter’s butterfly . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1 Tetrahedron with vertices and edges labeled . . . . . . . . . . . . . . . 117

vii

ABSTRACT

The study of periods arose in number theory and algebraic geometry. Periods areinteresting numbers like multiple zeta values, on the other hand periods are integralsof algebraic differential forms over domains described by algebraic relations. Viewedas abstract periods, we also consider their relations with motives. In this work, weconsider two problems in mathematical physics as applications of the ideas and toolsfrom periods and motives.

We first consider the algebro-geometric approach to the spectral theory of Harperoperators in solid state physics. When the parameters are irrational, the compacti-fication of the associated Bloch variety is an ind-pro-variety, which is a Cantor-likegeometric space and it is compatible with the picture of Hofstadter butterfly. Oneach approximating component the density of states of the electronic model can beexpressed in terms of period integrals over Fermi curves, which can be explicitlycomputed as elliptic integrals or periods of elliptic curves.

The above density of states satisfies a Picard-Fuchs equation, whose solutions aregenerally given by hypergeometric functions. We use the idea of mirror maps as inmirror symmetry of elliptic curves to derive a q-expansion for the energy level basedon the Picard-Fuchs equation. In addition, formal spectral functions such as thepartition function are derived as new period integrals.

Secondly, we consider generalized Feynman diagram evaluations of an effectivenoncommutative field theory of the Ponzano-Regge model coupled with matter in loopquantum gravity. We present a parametric representation in a linear κ-approximationof the effective field theory derived from a κ-deformation of the Ponzano-Regge modeland define a generalized Kirchhoff polynomial with κ-correction terms. Setting κequal to 1, we verify that the number of points of the corresponding hypersurface ofthe tetrahedron over finite fields does not fit polynomials with integer coefficients bycomputer calculations. We then conclude that the hypersurface of the tetrahedronis not polynomially countable, which possibly implies that the hypersurface of thetetrahedron as a motive is not mixed Tate.

viii

CHAPTER 1

INTRODUCTION

In algebraic geometry, a period is defined as the integral of an algebraic differentialform on an algebraic variety over some domain in Rn described by algebraic equalitiesor inequalities. In general, the set of periods, denoted by P, is a special class of(typically transcendental) numbers determined by algebraic information.

For example, the transcendental number π is a period defined as the ratio of a cir-cle’s circumference to its diameter, and obviously the circumference can be expressedas an integral. Because of the Cauchy’s integral formula, we always extend periods bymultiplying 1/2πi to include residues from contour integrals, that is P = P[(2πi)−1].From the point of view of number theory, P contains the most interesting numbers inour world. For example, multiple zeta values are on the top of the watch list, whichare periods of mixed Tate motives and also occur in Feynman integral computations.

Elliptic functions and elliptic curves play important roles in algebraic geometryand physics. Elliptic integrals are the periods (in the usual sense) of elliptic functions.For an elliptic curve, its periods can be calculated as integrals of the canonical one-form over a standard basis of homological cycles. Given a family of elliptic curvesE → P1 parametrized by P1, the varying periods of the fibers satisfy a second orderdifferential equation called the Picard-Fuchs equation. The ratio of the periods keepstrack of how the complex structure changes in the family, which is characterized bythe Schwarzian equation derived from the Picard-Fuchs equation. Period mappingsand period domains were introduced by Griffiths as higher dimensional generalizationof periods of elliptic curves, and the (higher dimensional) geometric picture of Picard-Fuchs equations is encoded in Gauss-Manin connections.

In the context of mirror symmetry, the B-model correlation function only dependson the complex structure modulus of the mirror partner, which is described by aPicard-Fuchs equation. The A-model Yukawa coupling, which can be defined byGromov-Witten invariants, only depends on the Kahler structure modulus of theCalabi-Yau manifold. The mirror map between local coordinates of the complexmoduli space and the Kahler moduli space is then characterized by the correspondingSchwarzian equation.

In general, instead of numerical periods P one considers the algebra of abstractperiods P. Let X be a nonsingular algebraic variety over Q, D ⊂ X a divisor

1

with normal crossings, ω ∈ Ωdim(X)(X) an algebraic differential form of top degree,and γ ∈ Hdim(X)(X(C), D(C);Q) a relative homology class on the complex manifoldX(C) with boundary in D(C). Abstract periods are equivalence classes of sym-bols [(X,D, ω, γ)] defined as quadruples (X,D, ω, γ) modulo the relations: linearity,change of variables and Stokes’ formula, and they form an algebra P. Kontsevichconjectured that the evaluation homomorphism ev : P→ P; [(X,D, ω, γ)] 7→

∫γω is

an isomorphism. Now we can talk about periods of motives.As before, we extend the abstract periods P by formally inverting the element

whose evaluation is 2πi, denote it by P = P[(2πi)−1]. One important reason toconsider P instead of P lies in that one would like to count the pure Tate motives asperiods, whose prototype is the Tate Hodge structure Z(1) or H2

B(P2) = 1

2πiQ = Q(1)

in the Betti cohomology.There exist important comparison isomorphisms connecting different classical co-

homology theories in algebraic geometry. A Weil cohomology is a desirable goodcohomology theory satisfying Poincare duality, Kunneth formula and Lefschetz the-orems. Grothendieck’s motive theory was originally introduced as a universal coho-mology theory with realizations to be Weil cohomologies. As a “linearization” of thecategory of smooth projective varieties, the category of pure motives is an abeliancategory (through standard conjectures on algebraic cycles). Moreover, it is a neutralTannakian category with a fiber functor mapping to the category of graded finitedimensional vector spaces GrVect.

For possibly non-compact or singular varieties, we do not have an abelian categoryof mixed motives but we do have a triangulated category of mixed motives DM(k)over a field k, which is expected to be the bounded derived category of the abeliancategory of mixed motives. One may think of mixed motives as objects endowed withfiltrations whose graded pieces are pure motives, and the extensions between thosegraded pieces are highly non-trivial.

The triangulated category of mixed Tate motives DMT(k) is a full triangulatedsubcategory of DM(k) generated by the Tate motives Q(n). Over a number field K,as the heart of a specific t-structure in DMT(K) the category of mixed Tate motivesMTM(K) is abelian, just as expected it is a neutral Tannakian category with a fiberfunctor given by the Hodge realization. It is remarkable that multiple zeta values areperiods of mixed Tate motives MTM(Z) over Z.

Numerical computations on Feynman diagrams show that multiple zeta values alsooccur in perturbative quantum field theory. For example, the wheel with n+1 spokesdelivers ζ(2n− 1). Under the parametric representation of Feynman integrals, eachFeynman graph Γ gives rise to a graph hypersurface variety XΓ, and its correspondingmotive is called a Feynman motive. One would like to know whether [XΓ] has valuein the (mixed Tate) subring Z[L] of the Grothendieck ring of algebraic varieties andgives a mixed Tate period.

In this work we apply algebro-geometric tools in periods and motives to studyphysical problems from solid state physics and loop quantum gravity. More precisely,we study the geometry of Harper operators acting on a single electron in a magnetic

2

field from solid state physics and the parametric representation of the Ponzano-Reggemodel coupled with matter in loop quantum gravity.

In a perfect crystal, the static lattice approximation under low temperature isassumed to fix the ions at lattice points. Furthermore, the independent electronapproximation is used to reduce the many-body electronic system to a single electronmodel. Classically, the electronic state can be described by a Bloch wave, whosegeometry is encoded in a finite dimensional algebraic variety called Bloch variety. Inphysical terms, the Bloch variety describes the complex energy-crystal momentumdispersion relation. In addition, the electrical properties of a metal are determinedby the shape of the Fermi surface, which is an abstract boundary of the occupiedelectronic orbitals at absolute zero.

When a uniform magnetic field is turned on, the Bloch theory breaks down, thatis the Hamiltonian no longer commutes with the lattice translation operator. In-stead the magnetic translation operators are introduced, meanwhile, the randomwalk operator is replaced by Harper operators as the discretized Laplacian. Fromthe noncommutative geometry point of view, the presence of a magnetic field leadsto a noncommutative space, i.e. the noncommutative torus.

In the setting of the noncommutative torus, the integer quantum Hall effect canbe explained as an index of a Fredholm operator based on the knowledge of Kubo’sformula and Chern-Connes character.

Another classical result by functional analysis is that the spectrum of the almostMathieu operator (i.e. the one dimensional Harper operator) has a Cantor struc-ture for irrational frequencies of zero Lebesgue measure, which is manifested in theHofstadter butterfly.

In this work we consider the algebraic geometry of Harper operators to generalizethe spectral theory of electrons developed for discrete periodic Schrodinger operatorsby Gieseker, Knorrer and Trubowitz [44]. In particular, we study the geometry of theBloch variety and Fermi curves. After building up the geometric space, we derive thedensity of states as periods and its Picard-Fuchs equation.

When the magnetic fluxes are irrational real numbers, the Bloch variety associ-ated to the Harper operator is an inductive limit of algebraic varieties, call it theBloch ind-variety. For each approximating component, the Fermi curve induces anelliptic fibration. Resolving the singularities by blow-ups delivers its compactificationas an ind-pro-variety which is a Cantor-like space compatible with the Hofstadterbutterfly. More precisely, according to a result of [67] by Kapranov, the category oflocally compact Hausdorff totally disconnected spaces can be identified with a fullsubcategory of IndPro(Set0).

The density of states can be expressed as a period over approximating componentsof Fermi curves, which can be explicitly computed by elliptic integrals. It turns outthat the density of states is a composition of Landen transformation and the completeelliptic integral of the first kind. It is easy to derive the Picard-Fuchs equation of thedensity of states, as a by-product we obtain a q-expansion of the energy level in thespirit of the mirror map as in mirror symmetry of elliptic curves.

3

With the density of states in hand, the partition function of the electronic modelcan be formally written as a spectral function. Finally, we compare the density ofstates between the Harper operator and the almost Mathieu operator.

The second problem considered in this work is an effective noncommutative fieldtheory arose from the Ponzano-Regge model in loop quantum gravity. We derive theparametric representation of its Feynman diagram evaluations and look into the mo-tivic properties of graph hypersurfaces defined by a generalized Kirchhoff polynomial.

A noncommutative star product was introduced in [40] to construct an effectivefield theory of the Ponzano-Regge model coupled with matter in 3d quantum gravity.For a trivalent graph in a triangulation of the spacetime manifold, the spin foammodel is identified with generalized Feynman diagram evaluations in the perturbativeexpansion of this noncommutative field theory.

As an analogy of the classical parametric representation of Feynman integrals,we derive the parametric representation in a linear κ-approximation under the newstar product and define a generalized Kirchhoff graph polynomial with κ-corrections.In the no-gravity limit, this new Kirchhoff polynomial agrees with the usual firstKirchhoff-Symanzik graph polynomials. With the help of the generalized Kirchhoffpolynomial we can define graph hypersurface varieties as usual.

For the simplest spherical graph, the tetrahedron, we use computer calculationsto test the motivic complexity of its hypersurface by counting rational points overfinite fields. It turns out that under the assumption κ = 1 the graph hypersurfaceof the tetrahedron does not fit into a polynomial with integer coefficients, i.e. is notpolynomially countable. From the relation between polynomially countability andbeing mixed Tate motive, this may imply that (the motive of) the hypersurface ofthe tetrahedron is not mixed Tate.

The organization of this work is the following.In the second chapter we go over the definition of periods with plenty of examples

and explain every aspect of Picard-Fuchs equations in details. In order to understanda Picard-Fuchs equation, one first has to know its regular singularities and its mon-odromy. Different examples of Picard-Fuchs equations mainly come from family ofelliptic curves (or K3 fibration), generalized hypergeometric functions and modularforms. Basic methods to derive Picard-Fuchs equations include direct computation,hypergeometric functions and Griffiths-Dwork reduction. One important applicationcan be found in mirror symmetry, especially the derivation of mirror maps.

The third chapter is devoted to a brief introduction to motives. One importantpoint in the theory of motives is to construct a universal cohomology theory. Thecategory of pure motives is abelian so one has homological algebra available, mean-while in the triangulated category of mixed motives the distinguished triangles alsogenerate long exact sequences of cohomology groups. The Tannakian formalism isvery important for motives, on the one hand the fiber functor associates a Tannakiancategory to the category of graded vector spaces, on the other hand it presents onewith a motivic Galois group. Mixed Hodge structures provide concrete examples inthe study of quite abstract mixed motives. We are mostly interested in mixed Tate

4

motives, as mentioned before multiple zeta values are periods of mixed Tate motivesover Z. There are two useful computational tools to test the motivic complexity:Grothendieck ring of algebraic varieties and Chern-Schwartz-MacPherson class in anambient projective space, and we only give a short introduction to Grothendieck ring.

The fourth chapter will discuss about the algebraic geometry of Harper operators.As discretized Laplacian operators acting on electronic states, we consider the Harperoperator and the almost Mathieu operator. The geometry of the Bloch variety andFermi curves is described in a categorical language, since it is convenient to capturethe geometric picture of the compactification as an ind-pro-variety. The density ofstates is the key to understanding an electronic system in solid state physics, whichis studied as a period integral in our case. Based on the Picard-Fuchs equationof the density of states, a q-expansion of the energy level is obtained with rationalcoefficients. Formal spectral functions such as the partition function are new periodsin terms of the density of states over approximating components.

In the last chapter the Feynman motive of a new noncommutative field theoryresulted from the Ponzano-Regge model is discussed along with a short introductionto three dimensional loop quantum gravity. Introducing Schwinger parameters givesrise to the parametric representation of Feynman integrals, which is useful in studyingthe renormalizability and analytic properties of Feynman graphs. Kirchhoff polyno-mials can be used to define graph hypersurface varieties and then one considers themotivic picture of Feynman graph hypersurfaces. In our case, a generalized Kirchhoffpolynomial with κ-correction is derived and the hypersurface of the tetrahedron isverified to be not polynomially countable by computer calculations. Then it is highlypossible that the hypersurface of the tetrahedron is not mixed Tate.

5

CHAPTER 2

PERIODS

The study of periods arose both in number theory and algebraic geometry, on the onehand periods are interesting numbers such as special values of Riemann zeta function,on the other hand periods are integrals of algebraic differential forms over domainsdescribed by algebraic relations.

The history of this subject can be traced back to Euler’s work on elliptic integrals.In the 18th century, Fagnano and Euler first studied elliptic integrals, which wasinspired by figuring out the circumference of an ellipse. Later on elliptic integralsand elliptic functions were further studied by other famous mathematicians. Oneimportant problem, in the language of algebraic geometry, was to find out how thecomplex structure of a complex torus varies in a family of elliptic curves.

In this chapter we go over the basic concepts and tools related to periods forlater physical application, mostly following the references [25], [28], [73]. We focuson the relation between periods and Picard-Fuchs equations. Geometric intuitionbehind important concepts will be explained with motivating examples. The relationbetween periods and motives will be treated in the next chapter. We will not discussspecial values of L-functions, which is also very important for the study of periods.The choice of topics reflects the limited knowledge, particular interests and bias ofthe author.

2.1 Definition and examples

This section is closely following §1.1 in [73], and the reader can find more inter-esting examples in the same survey by Kontsevich and Zagier.

Example 1 (Pi). The first example might be periods of trigonometric functions, thatis, the period of the sine function is 2π and that of the tangent function is π. We canexpress π as integrals

π =

∫ 1

−1

dx√1− x2

=

∫ ∞

−∞

dx

1 + x2,

which are evaluations of the arcsine or arctangent function by the fundamental theo-rem of calculus.

6

Example 2 (elliptic integrals and elliptic functions). Doubly periodic elliptic func-tions generalize trigonometric functions in that elliptic functions were discovered asinverse functions of elliptic integrals.

Periods of Jacobi elliptic functions are given by special elliptic integrals. Moreprecisely, it is the complete elliptic integral of the first kind K(k), also called thequarter period, which can be defined as

K(k) =

∫ 1

0

dt√(1− t2)(1− k2t2)

If k is the elliptic modulus then k′ =√1− k2 is called the complementary modulus,

and the complementary quarter period is defined as K ′(k) = K(k′). For example, theJacobi elliptic function sn(u, k) is periodic in K(k) and K ′(k) as

sn(u+ 2mK + 2niK ′, k) = (−1)msn(u, k).

We will see later that K(k) and K ′(k) are linearly independent period solutions to thecorresponding Picard-Fuchs equation.

The complete elliptic integral of the second kind E(k), which can be used to com-pute the circumference of an ellipse, is defined as

E(k) =

∫ 1

0

√1− k2t2√1− t2

dt

For any non-zero complex numbers ω1, ω2 ∈ C, one forms a lattice of periodsΛ = mω1 + nω2 : m,n ∈ Z. Then God created a doubly periodic function withperiods ω1 and ω2: the famous Weierstrass’s elliptic function

℘(z;ω1, ω2) =1

z2+

∑

n2+m2 6=0

1

(z +mω1 + nω2)2− 1

(mω1 + nω2)2

This function plays an important role in the theory of elliptic curves since the field ofelliptic functions with respect to the lattice Λ is generated by ℘ and its derivative ℘′.

The above two examples are periods of periodic functions in the usual sense, thegeneral definition of a period is as follows.

Definition 1 (Definition in §1.1 [73]). A period is a complex number whose real andimaginary parts are values of absolutely convergent integrals of rational functions withrational coefficients, over domains in Rn given by polynomial inequalities or equalitieswith rational coefficients.

In general, periods are usually transcendental numbers described by at most acountable amount of information, and the integral forms expressing the same periodare usually not unique. Such as the Feynman periods, they sometimes show up atthe intersection of different disciplines in mathematics and physics.

7

Example 3 (logarithmic Mahler measure). Given a Laurent polynomial P (x1, · · · , xn)usually with integer coefficients, the logarithmic Mahler measure is defined as

m(P ) =

∫

|x1|=1

· · ·∫

|xn|=1

log|P (x1, · · · , xn)|dx1x1· · · dxn

xn

Suppose one could expand the integrand log|P | in Taylor series, then it would giverise to a period.

Example 4 (Beta function). The beta function is an Euler integral of the first kindand it has close relation with the gamma function.

B(a, b) =

∫ 1

0

ta−1(1− t)b−1 dt =Γ(a)Γ(b)

Γ(a + b)

for Re(a), Re(b) > 0, where the gamma function is as usual

Γ(z) =

∫ ∞

0

tz−1 e−t dt

However, a generic value of the gamma function is not necessarily a period.

Example 5 (hypergeometric functions). The Gaussian hypergeometric function isformally defined by

2F1(a, b; c; z) =

∞∑

n=0

(a)n(b)n(c)n

zn

n!

for |z| < 1 with a, b ∈ C and c ∈ C \ 0,−1,−2, · · · , where the Pochhammer symbolis the following

(a)n =

1 if n = 0a(a+ 1) · · · (a + n− 1) if n > 0

By analytic continuation, 2F1(a, b; c; z) can be defined for any z ∈ C, and its Eulerintegral form is given by

2F1(a, b; c; z) =Γ(c)

Γ(b)Γ(c− b)

∫ 1

0

tb−1(1− t)c−b−1

(1− zt)a dt with Re(c) > Re(b) > 0

In particular, the above elliptic integrals K and E are special cases

K(k) = π2 2F1

(12, 12; 1; k2

), E(k) = π

2 2F1

(−1

2, 12; 1; k2

).

A generalized hypergeometric function can be defined similarly as

pFq(a1, . . . , ap; b1, . . . , bq; z) =

∞∑

n=0

(a1)n . . . (ap)n(b1)n . . . (bq)n

zn

n!

8

For instance, the dilogarithm can be represented by

Li2(z) =∞∑

n=1

zn

n2= z 3F2(1, 1, 1; 2, 2; z).

Generalized hypergeometric functions play an important role in studying Calabi-Yaumanifolds and mirror symmetry.

Example 6 (Riemann zeta function and multiple zeta values). The Riemann zetafunction was originally defined for s ∈ C with Re(s) > 1

ζ(s) =

∞∑

k=1

1

ks,

and we are mostly interested in the Riemann zeta value ζ(n) at integers n ≥ 2 whichhas a multiple integral representation

ζ(n) =

∫ 1

0

· · ·∫ 1

0

∏ni=1 dxi

1−∏n

i=1 xi.

As a generalization, multiple zeta values (MZV) of weight n and depth r are definedsimilarly by

ζ(n1, . . . , nr) =∑

k1>k2>···>kr>0

1

kn11 · · · knr

r

with ni ∈ N and n1 ≥ 2. By the sum theorem, such as ζ(2, 1) = ζ(3), multiple zetavalues are also periods.

Multiple zeta values surprisingly appear in calculating Feynman graphs in pertur-bative quantum field theory, which is the motivation to study Feynman periods (orFeynman motives), we will give more details in later chapters.

In algebraic geometry, periods are just integrals of algebraic differential forms oversome cycles embedded in ambient varieties. Suppose these forms or cycles could varywith respect to extra parameters, such as the quarter period K(k) depending on theelliptic modulus k, then the resulting period integrals would satisfy some Picard-Fuchstype ordinary differential equation.

2.2 Fuchsian equation

Based on any given ordinary differential equation, the points of P1 are classifiedinto ordinary points and singular points with respect to this ODE. Amongst singularpoints, we are mostly interested in regular singular points. In this section, we willrecall the definition of regular singular points and Fuchsian equations following thenotes [15], one can also find the subject in any standard textbook on ODE.

9

Consider the linear differential equation

y(n) + p1(z)y(n−1) + · · ·+ pn(z)y = 0, pi(z) ∈ C(z) (2.1)

In a small neighborhood around any point P ∈ P1, one usually chooses t = z − Pfor P ∈ C or t = 1/z for P = ∞ as a local parameter. In this work, we are mainlyconcerned with second order ordinary differential equations.

P ∈ P1 is an ordinary point if pi(t) has no pole at P for i = 1, · · · , n. And P ∈ P1

is a regular singular point if pi(t) has a pole of order at most i. In other words, Pis an ordinary point or a regular singular point if limt→0 t

ipi(t) exists for all i ≤ n.Otherwise, we call other points irregular singularities.

Example 7 (Bessel equation). In three dimensional cylindrical coordinates, Bessel’sequation is obtained in solving Laplace’s equation for the electrostatic potential byseparation of variables:

d2f

dx2+

1

x

df

dx+

(1− k2

x2

)f = 0,

where k ∈ Z is the order of the Bessel function.Bessel’s equation has a regular singular point at P = 0, because p1(x) = 1/x has

a pole of first order at x = 0, when k 6= 0, p2(x) = (1 − k2/x2) has a pole of secondorder at x = 0.

However, it has an irregular singularity at P = ∞. Under the transformationx = 1/w, one rewrites the above equation as

d2f

dw2+

1

w

df

dw+

[1

w4− k2

w2

]f = 0.

Now p1(w) = 1/w has a pole of first order at w = 0, but p2(w) has a pole of fourthorder at w = 0.

Definition 2. The differential equation 2.1 is called a Fuchsian equation if its singularpoints, including the point at infinity, are all regular singular points.

If x = a is a regular singular point of the second-order equation

y′′ + p(x)y′ + q(x)y = 0,

then one has the Fuchs’s theorem to find possible local solutions.

Theorem 1 (Fuchs). Given a second-order Fuchsian equation, applying the Frobeniusmethod based on the indicial equation, any solution around x = a can be written as

y = (x− a)r∞∑

n=0

an(x− a)n, a0 6= 0

10

for some real r, or

y = y0 ln(x− a) + (x− a)s∞∑

n=1

bn(x− a)n

for some real s, where y0 is a solution of the first kind. In addition, r and s aredetermined by the indicial equation.

The indicial polynomial is the coefficient of the lowest power in x obtained byplugging in the Frobenius series. Setting the indicial polynomial equal zero, one getsthe indicial equation. The solutions to the indicial equation are sometimes called theexponents of the singularity.

Example 8 (Bessel functions). Even though Bessel’s equation is not Fuchsian, onestill can apply the Frobenius method to find local solutions around the regular singularpoint at x = 0, which are the famous Bessel functions.

Around x = 0, take the Frobenius series y =∑∞

n=0 anxn+r and plug into the Bessel

equation

x2d2f

dx2+ x

df

dx+ (x2 − k2)f = 0,

one immediately gets

∞∑

n=0

(n+ r)(n+ r − 1)anxn+r +

∞∑

n=0

(n+ r)anxn+r +

∞∑

n=2

an−2xn+r − k2

∞∑

n=0

anxn+r = 0.

The indicial equation obtained by setting n = 0 is then

r(r − 1) + r − k2 = 0,

so the exponents of x = 0 are r = ±k.Look at the first case r = k, comparing the coefficients of the infinite series gives

(2k + 1)a1 = 0, i.e. a1 = 0

and

(n2 + 2kn)an + an−2 = 0, i.e an = − 1

n(2k + n)an−2, n ≥ 2

This delivers the Bessel function of the first kind Jk(x),

Jk(x) = a0

∞∑

m=0

(−1)mm! Γ(m+ k + 1)

(x2

)2m+k

.

For the second case r = −k, one proceeds similarly and gets the other solution

J−k(x) = (−1)kJk(x).

11

Example 9 (hypergeometric equation §1 in [15]). Hypergeometric equation

d2f

dz2+c− (a+ b+ 1)z

z(1− z)df

dz− ab

z(1 − z)f = 0.

has regular singular points at 0, 1 and ∞, so it is Fuchsian, whose solutions are builtout of Gaussian hypergeometric function 2F1(a, b; c; z) using the Frobenius method.

If c /∈ Z , then two independent solutions around z = 0 are given by

2F1(a, b; c; z), z1−c2F1(1 + a− c, 1 + b− c; 2− c; z).

Similarly if (c− a− b) /∈ Z, then two independent solutions around z = 1 are

2F1(a, b; 1 + a+ b− c; 1− z), (1− z)c−a−b2F1(c− a, c− b; 1 + c− a− b; 1− z).

Finally if (a− b) /∈ Z, then two independent solutions around z =∞ are

z−a2F1(a, 1 + a− c; 1 + a− b; z−1), z−b

2F1(b, 1 + b− c; 1 + b− a; z−1).

If either c, c−a−b or a−b is an integer, the second solution can be constructed byFuchs’s theorem using the logarithmic function. In general, Kummer gave 24 solutionsto the hypergeometric solution, and the relations between them are called connectionformulas, see all the solutions in §1 of [15].

There are important results about the Riemann-Hilbert correspondence, histori-cally called the Hilbert’s twenty-first problem, related to Fuchsian equations, but wewon’t discuss it in this work.

Given a Fuchsian equation, taking their monodromy into account, we are mainlyinterested in Picard-Fuchs equations.

Definition 3 (p.9 in [73]). The differential equation 2.1 is called of Picard-Fuchs typeif it is Fuchsian and its monodromy group is contained in SL(n, Q), where n is theorder of the equation and Q is the algebraic closure of Q.

Monodromy will be discussed in a later section in this chapter. Other alternativedefinitions of Picard-Fuchs equations are also discussed in the introduction of chapter2 of [73].

Higher dimensional generalization of Picard-Fuchs equation is the Gauss-Maninconnection, which is important in algebraic geometry and mirror symmetry.

In the following sections, the Picard-Fuchs equation of certain family of ellipticcurves will be derived with no background knowledge, and modular forms also giveanother class of important examples of Picard-Fuchs equation.

12

2.3 Families of elliptic curves

Elliptic curves are Riemann surfaces of genus 1, and families of elliptic curveshave different Picard-Fuchs equations for different realizations in weighted projectivespaces. We recall the Picard-Fuchs equations of the Legendre family (closely following§1.1 in [25]) and the Weierstrass family in this section. The detailed derivation ofthe Picard-Fuchs equation of the Legendre family can be found in §1.1 of [25] andthe Picard-Fuchs equation related to the j-invariant is covered in many textbooks onelliptic curves such as [107].

Let us first consider the Legendre family

Eλ : y2 = x(x− 1)(x− λ), where λ 6= 0, 1 (2.2)

Taking the square root on both sides gives the multivalued function

y =√x(x− 1)(x− λ),

which has branch points at x = 0, 1, λ,∞ ∈ P1. In order to get a single valuedfunction, one has to extend its domain and do some surgery. Indeed, by the cut-and-paste method, one assembles a complex torus. More precisely, if the cuts are from 0to 1 and from λ to∞ on the Riemann sphere, then the Riemann surface of y consistsof two copies of the Riemann sphere with two holes (resulted from the above cuts)cross-pasted. As usual, we take the standard basis δ, γ ∈ H1(Eλ,Z) of the resultingcomplex torus such that the intersection number of the basis is one, i.e. δ · γ = 1.

There exists a canonical holomorphic one form on the elliptic curve Eλ,

ω =dx√

x(x− 1)(x− λ). (2.3)

If we call integrals (∫δω,

∫γω) the period vector of Eλ, then the τ -invariant is defined

as the ratio of the periods

τ(δ, γ) =

∫γω∫

δω,

which describes the complex structures of elliptic curves Eλ. It is easy to prove thatthe imaginary part of τ is positive, i.e. Im(τ) > 0 or τ ∈ H, where H is the upperhalf-plane of complex numbers.

However, by definition this τ -invariant depends on the choice of the cycles δ and γwith δ ·γ = 1. If we choose a different pair of cycles δ′, γ′ ∈ H1(Eλ,Z) with δ′ ·γ′ = 1,then δ′ = aδ + bγ and γ′ = cδ + dγ since (δ, γ) is the standard basis of the homologygroup. The condition δ′ · γ′ = 1 ensures that the transformation matrix belongs tothe full modular group (

a bc d

)∈ Γ = SL(2,Z).

13

Further the corresponding τ ′ relates to τ by a fractional linear transformation

τ ′(δ′, γ′) =dτ + c

bτ + a, with ad− bc = 1

With this remark, we can just assume τ ∈ H/Γ, meaning that τ is independent ofthe choice of homology basis on the elliptic curve Eλ.

Now let the parameter λ vary, we will derive the Picard-Fuchs equation of theLegendre family closely following (pp. 15-16) §1.1 of [25]. We first show a straightfor-ward method based on derivatives of ω(λ), a modern general method using residueswill be introduced in a later section.

Rewrite the canonical one form as

ω(λ) = x−12 (x− 1)−

12 (x− λ)− 1

2 dx.

Take the first and second derivatives of ω with respect to λ, one has

ω′ = 12x−

12 (x− 1)−

12 (x− λ)− 3

2 dx

ω′′ = 34x−

12 (x− 1)−

12 (x− λ)− 5

2 dx

Introduce an auxiliary function f(x) = x12 (x− 1)

12 (x− λ)− 3

2 , and its differential is

df = (x− 1)ω′ + xω′ − 2x(x− 1)ω′′

Reorganize the coefficients and cancel out the x’s, one obtains

− 1

2df =

1

4ω + (2λ− 1)ω′ + λ(λ− 1)ω′′ (2.4)

Thus for any cycle ξ ∈ H1(Eλ,Z), the integral π(λ) =∫ξω(λ) satisfies the second

order differential equation

λ(λ− 1)d2π

dλ2+ (2λ− 1)

dπ

dλ+

1

4π = 0, (2.5)

which is called the Picard-Fuchs equation of the Legendre family. Obviously, theperiods

∫δω,

∫γω are linearly independent solutions of the above equation.

Next we recall the Picard-Fuchs equation of a Weierstrass family. The Weierstrassform of elliptic curves is given by

E : y2 = 4x3 − g2x− g3. (2.6)

The modular discriminant is defined as

∆ = g32 − 27g23,

and the modular j-invariant is

j(τ) = 1728g32∆,

14

which is an isomorphism from the one-point compactification H/Γ to the Riemannsphere P1.

Again we take the canonical one form

ω =dx

y=

dx√4x3 − g2x− g3

and consider π(j) =∫ξω as a function of j for any cycle ξ ∈ H1(E,Z). The corre-

sponding Picard-Fuchs equation is then given by

d2π

dj2+

1

j

dπ

dj+

31j − 4

144j2(1− j)2π = 0. (2.7)

Since every modular function can be expressed as a rational function of j(τ), theabove equation sometimes is called the universal Picard-Fuchs equation.

One way to obtain the Picard-Fuchs equation of the Weierstrass family is to deriveit from that of the Legendre family. The relation between the modular functions λ(τ)and j(τ) listed below can be found in any textbook on elliptic curves.

g2 =3√43(λ2 − λ+ 1), g3 =

127(λ+ 1)(2λ2 − 5λ+ 2)

∆ = λ2(λ− 1)2, j(λ) = 256(λ2 − λ+ 1)3/λ2(λ− 1)2

2.4 Modular forms

Elliptic modular functions can be thought of as functions on the moduli space ofisomorphism classes of elliptic curves, in this section we will discuss about generalmodular forms and modular functions. Certain modular forms in connection withhypergeometric functions and modular functions satisfy Picard-Fuchs equations aswell. This short review is closely following §2.3 in [73]. Besides standard textbookssuch as [36], a good introduction to modular forms written by Zagier can be found inthe book [23], and there is also a nice chapter of modular stuff in [43].

Recall that a modular form of weight k ∈ Z is a holomorphic function f(z) on Hsatisfying

f

(az + b

cz + d

)= (cz + d)kf(z),

where the matrix γ =(a bc d

)belongs to the full modular group Γ or some subgroup of

finite index in Γ, further f(z) is required to be holomorphic at the cusps. Relax therequirements of the above definition, a modular function can be defined similarly. Amodular function is a meromorphic function f(z) on H which transforms as

f

(az + b

cz + d

)= f(z),

and f(z) is required to be meromorphic at the cusps.We first recall some basic modular forms and the relations between them.

15

Example 10 (Eisenstein series). Given a lattice Λτ = m+nτ : m,n ∈ Z generatedby τ ∈ H, the classical Eisenstein series G2k(τ) with integer k ≥ 2 is defined by

G2k(τ) =∑

λ∈Λτ\0

1

λ2k,

which is a modular form of weight 2k for the full modular group Γ. Alternatively, ifone expands the Eisenstein series in Fourier series (i.e. q-expansions with q = e2πiτ),then one has a power series form (still call it the Eisenstein series)

E2k(q) =G2k(τ)

2ζ(2k)= 1− 4k

B2k

∞∑

n=1

σ2k−1(n)qn,

where B2k are Bernoulli numbers, ζ(z) is Riemann zeta function and σp(n) is thedivisor sum function.

For instance, the first two Eisenstein series have q-expansions as follows

E4(q) = 1 + 240∑∞

n=1n3qn

1−qn= 1 + 240q + 2160q2 + 6720q3 + · · ·

E6(q) = 1− 504∑∞

n=1n5qn

1−qn= 1− 504q − 16632q2 − 122976q3 + · · ·

As an important fact, the ring of all modular forms for the full modular group Γ isfreely generated by E4 and E6. In particular, the modular discriminant of weight 12may be expressed as

∆(q) =1

1728(E3

4 − E26) =

∞∑

n=1

τ(n)qn = q − 24q2 + 252q3 − 6048q4 + · · ·

where the multiplicative τ(n) is the Ramanujan tau function.

Example 11 (Jacobi theta function). The classical Jacobi theta function is definedfor z ∈ C and τ ∈ H,

ϑ(z; τ) =∞∑

n=−∞exp(πin2τ + 2πinz) =

∞∑

n=−∞qn

2

ω2n,

where q = exp(πiτ) (the so-called nome) and ω = exp(πiz). As mentioned in theabove example, we also use the notation q = exp(2πiτ) = q2, which may be slightlydifferent from the literature.

In particular,

ϑ(τ) = ϑ(0; τ) =

∞∑

n=−∞qn

2

= 1 + 2q + 2q4 + 2q9 + · · ·

ϑ(τ) is a modular form of weight 12with a nontrivial multiplier for Γϑ,

Γϑ =

⟨(0 −11 0

),

(1 20 1

)⟩=

(a bc d

)∈ Γ | ac ≡ bd ≡ 0 (mod 2)

(2.8)

Sometimes ϑ(τ) is also called θ3(τ) in the literature.

16

Example 12 (Dedekind eta function). The Dedekind eta function is defined as aq-expansion by

η(τ) = q1/24∞∏

n=1

(1− qn) = q1/24(1− q − q2 + q5 + q7 − q12 − · · · )

which is a modular form of weight 12with nontrivial multiplier for the full modular

group Γ.Dedekind eta function is another important ingredient in defining other modular

forms. For instance, the relation between Jacobi theta function and Dedekind etafunction is

ϑ(τ) =η2((τ + 1)/2)

η(τ + 1).

As another example, the modular discriminant may be defined as

∆(τ) = (2π)12η(τ)24.

Eta quotients further give an special class of modular forms.

Example 13 (modular lambda function). As an example of eta quotient, the ellipticmodular lambda function may be defined as

λ(τ) = 16η8(τ/2)η16(2τ)

η24(τ)= 16q − 128q2 + 704q3 − 3072q4 + · · · ,

which is a modular function for Γ(2). Recall that the principal congruence subgroupof level N , denoted by Γ(N), is defined as

Γ(N) =

(a bc d

)≡

(1 00 1

)(mod N)

.

In addition, λ(τ) generates the function field of the quotient space H/Γ(2), i.e. it isa Hauptmodul for the modular curve X(2).

Jacobi theta function and the modular lambda function is connected by the quar-ter period K(k),

ϑ2(τ) =2

πK(k) = 2F1(

12, 12; 1;λ(τ)). (2.9)

Then the modular form ϑ2 of weight 1 as a function of λ satisfies the Picard-Fuchsequation of the Legendre family 2.5.

This example is a special case of a general theorem on modular forms.

Theorem 2 (Zagier). Let Γ′ ⊆ Γ = SL2(Z) be a subgroup of finite index. Let f(τ)be a modular form of weight k for Γ′ and t(τ) a modular function for Γ′. Then themany-valued function F (t) defined by F (t(τ)) = f(τ) satisfies a linear differentialequation of order k+1 with algebraic coefficients. In addition, the monodromy of thedifferential equation is the image of Γ′ under the k-th symmetric power representationSL2(R)→ SLk+1(R).

17

Thus there exists a differential equation satisfied by F (t),

a0(t)dk+1F

dtk+1+ · · ·+ ak(t)

dF

dt+ ak+1(t)F = 0

with ai(t) in the function field C(Γ′) ≡ C(XΓ′).

Example 14 (Fricke-Klein). The classical identity

4√E4 = 2F1(

1

12,5

12; 1;

1728

j(τ)), with t(τ) =

1728

j(τ), f(τ) = 4

√E4

is an interesting example connecting the modular function j(τ) and the 4-th root of E4

by Gaussian hypergeometric function, and the Picard-Fuchs equation is easily obtainedby plugging the parameters into the hypergeometric equation.

Example 15 (Γ1(6) Stienstra-Beukers [112]). Consider a family of elliptic curves:

Et : t(x+ y + z)(xy + yz + zx)− xyz = 0.

One takes the modular form of weight 1,

f(τ) =η(2τ)6η(3τ)

η(τ)3η(6τ)2,

and the modular function

t(τ) =η(6τ)8η(τ)4

η(2τ)8η(3τ)4,

which is a generator of the function field C(Γ1(6)). Recall that the definition of themodular group Γ1(N) is the following:

Γ1(N) =

(a bc d

)∈ Γ : a ≡ d ≡ 1, c ≡ 0 (mod N)

.

Then the function F (t) defined by F (t(τ)) = f(τ) satisfies the second order differentialequation,

t(t− 1)(9t− 1)d2F

dt2+ (27t2 − 20t+ 1)

dF

dt+ (9t− 3)F = 0,

which happens to be the Picard-Fuchs equation of the family Et. More details aboutthis example can be found in Stienstra and Beukers’s paper [112].

18

2.5 Monodromy

In order to identify a Fuchsian equation as a Picard-Fuchs equation, one has toknow its monodromy as well. In this section, we go over the concept of monodromy,show how to compute the monodromy of the Legendre family, and briefly introducethe monodromy of hypergeometric equation. One can find a systematic treatment ofmonodromy in [121] and more applications of Picard-Lefschetz formula in [117].

Monodromy was first introduced in differential topology as a generalization of thedeck transformation of covering spaces. Assume f :M → N is a smooth proper mapbetween two smooth manifolds with k = dimM − dimN > 0. Let K ⊂ N be the setof critical values of f , so N \K is the set of regular values. For any point q ∈ N \K,the fiber f−1(q) =Mq is a k-dimensional smooth submanifold.

Every loop ℓ in N \K with base point q defines a diffeomorphism ρ(ℓ) :Mq → Mq

which induces a representation ρ : π1(N \ K, q) → Aut(Hn(Mq)) = GL(dn,C) for0 ≤ n ≤ k, where dn is the dimension of Hn(Mq) as a vector space. ρ is called thelocal monodromy and its image is the global monodromy group.

The Picard-Lefschetz formula provides an easy way to compute the local mon-odromy in complex geometry. In general, we consider a family of nonsingular complexprojective varieties parametrized by the projective line f : X → P1, such as an ellipticsurface, and let S ⊂ P1 be the set of singular points of f . For each point si ∈ S,there exists a vanishing cycle δi in Hm−1(Xq) with dimX = m. The Picard-Lefschetztransformation is then given by

T (γ) = γ + (−1)m(m+1)/2(γ · δi)δi (2.10)

where γ ∈ Hm−1(Xq) is any homology cycle and γ · δi is the intersection number.In other words, Picard-Lefschetz formula describes how the monodromy around thesingular fiber acts on the top homology group of fibers in connection with all possiblevanishing cycles.

The global monodromy is the image of the fundamental group taking into accountthe relations between those local monodromy representations, which is supposed tobe a subgroup of SL(n, Q) for the Picard-Fuchs equations.

In the theory of differential equations, a local solution may give rise to otherlinearly independent solutions by analytic continuation. The transformation connectsthese two sets of solutions is called the monodromy transformation, which is basicallythe monodromy representation in the context of ODE. The key observation is thatthe monodromy representation for period solutions of the Picard-Fuchs equation isthe same as the geometric monodromy discussed above.

Example 16 (Legendre family §1.1 in [25] ). Let us consider the Picard-Fuchs equa-tion of the Legendre family again

λ(λ− 1)d2π

dλ2+ (2λ− 1)

dπ

dλ+

1

4π = 0.

19

Fix a point λ0 6= 0, 1 on the complex plane, then the fundamental group π1(P1 \

0, 1,∞, λ0) ≃ Zℓ0 ∗ Zℓ1 is a free group of rank 2 generated by one loop around 0,call it ℓ0, and the other around 1, call it ℓ1.

The homology group H1(Eλ,Z) ≃ Zδ × Zγ is generated by a standard basis δ, γwith δ · γ = 1. Suppose that we have a cut from 0 to λ and a cut from 1 to ∞ onP1, then δ is a loop around the cut from 0 to λ, and γ is a loop through the two cuts.In addition, letting λ → 0 forces δ to be a vanishing cycle and similarly if λ → 1, γbecomes a vanishing cycle.

Picard-Lefschetz formula details the action of ρ(ℓ0), ρ(ℓ1) on δ, γ. The transfor-mation matrices change the local solutions are sometimes called monodromy matrices,corresponding to the action of ρ(ℓ0), ρ(ℓ1) in this case.

First, let us consider the vanishing cycle δ when λ → 0, λ has to go around 0by 360 degrees along ℓ0 to form a loop. If we denote a 180-degree turn around 0by transformation T , then the action by ρ(ℓ0) corresponds to T 2. Apply the Picard-Lefschetz formula to find out T directly

T (δ) = δ, T (γ) = γ − (γ · δ)δ = γ + δ

then

ρ(ℓ0)

(δγ

)=

(1 02 1

)(δγ

)(2.11)

Similarly, when λ → 1, γ will be the vanishing cycle. Denote a 180-degree turnaround 1 by S, then ρ(ℓB) is given by S2 because λ goes around 1 by 360 degrees alongℓB. S is given by the Picard-Lefschetz formula

S(δ) = δ − (δ · γ)γ = δ − γ, S(γ) = γ

then

ρ(ℓ1)

(δγ

)=

(1 −20 1

)(δγ

)(2.12)

We emphasize that the role of vanishing cycle is interchanged between δ and γ in theabove local monodromy representations.

Finally, the monodromy group is generated by the monodromy matrices withoutany relations in Aut(H1(Eλ)) = GL(2,C), that is the global monodromy is given by

Γ(2) =

⟨(1 02 1

),

(1 −20 1

)⟩(2.13)

as a subgroup of the full modular group Γ of index 6, i.e. |Γ : Γ(2)| = 6. More detailsabout this example can be found in §1.1 of [25].

For a Fuchsian equation, the monodromy matrices Msi corresponding to loopsaround regular singularities si are usually chosen so that the last M∞ goes around

20

(usually) the point at infinity clockwisely and each other Msi goes around si coun-terclockwisely. In this case, the only relation of the monodromy group is given byMs1 · · ·MskM∞ = I with increasing order for s1, · · · , sk.

In the above example, the monodromy matrix corresponding to the clockwise looparound ∞ is

M∞ = (M0M1)−1 =

(−3 2−2 1

).

Example 17 (hypergeometric equation §3.8 in [15]). Gaussian hypergeometric func-tion satisfies the hypergeometric equation

z(1− z)d2y

dz2+ [c− (a+ b+ 1)z]

dy

dz− aby = 0

with three regular singular points at 0, 1 and ∞, the corresponding monodromy ma-trices are M0,M1,M∞ with M0M1M∞ = I. Let us assume a, b, c ∈ R, from theRiemann scheme, we know the exponents at 0 are 0, 1 − c, those at ∞ are a, b andthose at 1 are 0, c− a− b. Further based on the exponents, the eigenvalues of M0 are1, e2πi(1−c), those of M∞ are e2πia, e2πib, and those of M1 are 1, e2πi(c−a−b). Then themonodromy group is generated by M0 and M∞ and the eigenvalue of M∞M0 = M−1

1

is 1.The hypergeometric equation is called irreducible if its global monodromy as a group

of matrices is irreducible, i.e. it has no nontrivial invariant subspaces. For irreduciblehypergeometric equation, its monodromy group is determined up to conjugation by thevalues of a, b, c modulo Z.

A curvilinear triangle is defined as a connected open subset of the Riemann sphereP1 whose boundary is the union of three open segments (edges) of a circle or straightline and three vertices. A curvilinear triangle is uniquely determined by its verticesand angles.

The Schwarz map is defined by D : H → P1; z 7→ f(z)/g(z), where f, g are twoindependent local solutions of the hypergeometric equation in the upper half-plane H.

Theorem 3 (Schwarz). Let λ = |1−c|, µ = |c−a−b|, ν = |a−b| with 0 ≤ λ, µ, ν < 1.Then the Schwarz map D maps H ∪ R one-to-one onto a curvilinear triangle in theRiemann sphere. The vertices correspond to the points D(0), D(1), D(∞) and thecorresponding angles are given by λπ, µπ, νπ.

Next step is to study the analytic continuation of the Schwarz map D, to this end,one has to apply the Schwarz reflection principle. More details about monodromy ofhypergeometric equation can be found in Beukers’ lecture notes [15].

2.6 Residues and Griffiths-Dwork method

We give some examples of periods by contour integrals, then generalize the con-cept of residue to higher dimensions. A method for getting Picard-Fuchs equations,

21

which was first introduced for projective hypersurfaces due to Griffiths and Dwork, isexplained with an example closely following Schnell’s notes [102], the Griffiths-Dworkmethod has been covered in [25], [28].

A class of periods is provided by contour integrals or residues in complex analysis.

Example 18. The simplest example is the contour integral of f(z) = z−1,

2πi =

∮

C

dz

z

with the unit circle C : |z| = 1 as the contour.

Example 19 (Pochhammer contour). The beta function B(a, b) is basically a contourintegral over the Pochhammer contour C,

(1− e2πia)(1− e2πib)B(a, b) =

∮

C

ta−1(1− t)b−1 dt.

The above contour integral converges for all a, b ∈ C, thus it defines the analyticcontinuation of the usual beta function only for Re(a), Re(b) > 0. Applying the samemethod gives the analytic continuation of the Gaussian hypergeometric function as acontour integral over the Pochhammer contour C,

2F1(a, b; c|z) =Γ(c)

Γ(b)Γ(c− b)

∮

C

xb−1(1− x)c−b−1(1− zx)−a dx.

The residue of a meromorphic function f(z) at an isolated pole P , denoted byResPf(z), is the normalized contour integral along a counterclockwise simple closedpath γ enclosing P . More precisely, the simplest form of the Cauchy’s residue formulais given by

ResPf(z) =1

2πi

∮

γ

f(z)dz.

In general, assume that X ⊂ Pn is a nonsingular projective hypersurface, onecan define the residue map ResX : Hn(Pn \X,C)→ Hn−1(X,C) as a generalizationof the classical residue in one-variable complex analysis. Let ω be a rational n-form with poles along X representing an element of Hn(Pn \ X,C). For any cycleγ ∈ Hn−1(X,Z), let T (γ) be a small tube around γ such that T (γ) ∈ Hn(Pn \X,Z).Definition 4 (p.5 in [102]). The residue map ResX : Hn(Pn\X,C)→ Hn−1(X,C);ω 7→ResXω is defined by

ResX ω : Hn−1(X,Z) −→ Cγ 7→

∫T (γ)

ω(2.14)

In other words, ResX ω ∈ Hom(Hn−1(X,Z),C) ≃ Hn−1(X,C) satisfies∫

γ

ResX ω =

∫

T (γ)

ω (2.15)

22

In particular, we can rephrase the definition of the residue of f(z) at P to fit intothis general setting. Instead of a meromorphic function f(z), we view it as a rational1-form ω = f(z)dz, then Res is the map that assigns a 0-form on P (the numberResPf(z)) to ω with a pole at P (plus the normalization (2πi)−1). Of course, thispoint P is the hypersurface in P1, so the general residue map is a generalization ofthe classical residue.

If the hypersurface X is defined locally by F = 0 and ω is a form with a pole oforder one on X , then one has the following formula to compute the residue,

ResX ω = α|X, if ω =dF

F∧ α+ β (2.16)

As a special example, holomorphic 1-forms on a nonsingular curve X ⊂ P2 are theresidues of rational 2-forms with a simple pole along X . In general, one has to reducethe order of the pole to one, then use the above formula.

Let H be the hyperplane class in Pn, recall that the primitive part of the coho-mology (or simply the primitive cohomology) is defined by

PHn−1(X,C) = ξ ∈ Hn−1(X,C)|ξ ∩H = 0.

From the point of view of intersection theory, [X ] ∼ degF · H , so ResXω ∩ H = 0for all ω ∈ Hn(Pn \ X,C). Thus the residue map actually defines a map into theprimitive cohomology,

ResX : Hn(Pn \X,C)→ PHn−1(X,C). (2.17)

By Hodge theory, one has the Hodge filtration of the primitive cohomology,

PHn−1(X,C) = F 0PHn−1(X,C) ⊇ F 1PHn−1(X,C) ⊇ · · · ⊇ F n−1PHn−1(X,C) ⊇ 0,

where

F pPHn−1(X,C) ≃ PHn−1,0(X,C) + PHn−2,1(X,C) + · · ·+ PHp,n−1−p(X,C).

Theorem 4 (Theorem 3.2.12 in [25]). Let X ⊆ Pn be a nonsingular hypersurface de-fined by F = 0 of homogeneous degree d, and R(F ) be the graded ring C[z0, · · · , zn]/J(F ),as usual the Jacobian ideal is J(F ) =

(∂F∂z0, · · · , ∂F

∂zn

). Then one has the isomorphisms

between graded pieces

R(F )d(n−p)−(n+1) ≃ PHp,n−1−p(X,C)P 7→ ResX(PΩ/F

n−p)(2.18)

where Ω is the canonical n-form on Pn,

Ω =n∑

j=0

(−1)jxjdx0 ∧ · · · ∧ dxj ∧ · · · ∧ dxn.

23

One important fact is that when the dimension of X is odd, the primitive coho-mology is the same as the full cohomology. The reader can find more details aboutthis theorem in §3.2 of [25] or Schnell’s lecture notes [102].

Example 20 (P8 [102]). Consider a family of elliptic curves given by

Et : t(x3 + y3 + z3)− 3xyz = 0 (2.19)

and call the defining equation F (t, x, y, z). For each t, F (t) = 0 defines a hypersurface(an elliptic curve) Et in P2. We will apply the Griffiths-Dwork method to find out thePicard-Fuchs equation of this family.

From the above theorem, if we take P = 1 ∈ R(F )0, then p = 1 since n = 2, d = 3,so Res(Ω/F ) ∈ H1,0(Et) ≃ C with the canonical 2-form

Ω = xdy ∧ dz − ydx ∧ dz + zdx ∧ dy.

In a neighborhood around a point such that say ∂xF 6= 0, the residue of Ω0(t) = Ω/Fcan be computed explicitly

ResΩ0(t) =ydz − zdy

∂xF=

ydz − zdy3(tx2 − yz) (2.20)

by rewriting

Ω0 =dF

F∧ ydz − zdy

∂xF+

3dy ∧ dz∂xF

(2.21)

Take the derivatives with respect to t and define

Ω1(t) =dΩ0

dt, Ω2(t) =

d2Ω0

dt2

There must be a relation between Ω0,Ω1 and Ω2 of the form

Ω2(t) +B(t)Ω1(t) + C(t)Ω0(t) ≡ 0 (2.22)

at the level of cohomology classes.If we define

π(t) =

∫

γ

ResΩ0(t) =

∫

T (γ)

Ω0(t) (2.23)

as a period for any cycle γ ∈ H1(Et,Z), then the desired Picard-Fuchs equation hasthe same coefficients as in (2.22)

d2π

dt2+B

dπ

dt+ Cπ = 0.

We will use the following formula repeatedly in the computation

Ω

F k(A∂xF +B∂yF + C∂zF ) ≡

Ω

(k − 1)F k−1(∂xA+ ∂yB + ∂zC), (2.24)

24

which is a trick first developed by Griffiths.Let us take a look at

Ω2(t) =2(x3 + y3 + z3)2

F 3Ω

A Grobner basis calculation gives

(x3+y3+z3)2 =

(x3 + y3 + z3

t+

3xyz

t2

)F+

3x2yz

t2(t3 − 1)∂xF+

3x3y

t(t3 − 1)∂yF+

3x2y2

t3 − 1∂zF

(2.25)With the help of (2.24), we reduce the order of the pole by one

Ω2(t) ≡(2(x3 + y3 + z3)2

t+

6txyz

t3 − 1+

3x3

t(t3 − 1)

)Ω

F 2(2.26)

modulo exact forms. Since H0,1(Et,C) (when p = 0) is one dimensional, there existssome B(t) such that

Ω2(t) +B(t)Ω1(t) ≡P

F 2Ω, with P ∈ J(F )

Form the requirement that P falls into the Jacobian ideal J(F ), we get

B(t) =4t3 − 1

t(t3 − 1)(2.27)

With this choice of B, we have

Ω2(t) +BΩ1(t) ≡ −2t3 + 1

t2(t3 − 1)

Ω

F+

x

t2(t3 − 1)∂xF

Ω

F 2

Then use the formula (2.24) again, we finally obtain

Ω2(t) +BΩ1(t) ≡ −2t

t3 − 1Ω0(t)

that is, C is just given by

C(t) =2t

t3 − 1(2.28)

Thus the Picard-Fuchs equation is the following

d2π

dt2+

4t3 − 1

t(t3 − 1)

dπ

dt+

2t

t3 − 1π = 0. (2.29)

More details about the computations of this example can be found in Schnell’slecture notes [102]. The interested reader is referred to [28] for more examples, includ-ing the quintic mirror, about applying the Griffiths-Dwork method to compute thePicard-Fuchs equation. In general, this method can be generalized to hypersurfacesin weighted projective spaces or even toric varieties [28].

25

2.7 Schwarzian equation

We have seen several families of elliptic curves and their Picard-Fuchs equations.As complex manifolds, the fibers in a family of elliptic curves are complex tori ratherthan simply topological tori, which means we are also concerned with the informationof complex structures. As a matter of fact, the complex structure always changes fromfiber to fiber in the family. In order to catch such information we study the periodsassociated to each elliptic curve, and it turns out that the change of periods manifestsitself in the Picard-Fuchs equation.

More precisely, let us consider a family of elliptic curves π : E → P1, take thecanonical 1-form as before ω = dx/y, and let δ, γ be the standard basis of H1(Et,C),then the period lattice is generated by periods

∫

δ

ω(t),

∫

γ

ω(t)

depending on the parameter t. The subspace H1,0(Et) ⊂ H1(Et,C) ≃ C2 is a line inthe direction of ω(t), and obviously its slope is

τ(t) =

∫γω(t)∫

δω(t)

.

We know that the Picard-Fuchs equation describes how the periods change, and wewill introduce the Schwarzian equation which directly characterizes how the complexstructure (i.e. τ(t)) changes in the family. We only talk about second order Picard-Fuchs equation and its associated Schwarzian equation in this work, more generalhigher order equations are discussed for example in [72].

We give a short introduction to Schwarzain equation following the first part of thesurvey [93]. The Schwarzian derivative of a holomorphic function f(z) is defined by

(Sf)(z) = f, z =(f ′′(z)

f ′(z)

)′− 1

2

(f ′′(z)

f ′(z)

)2

=f ′′′(z)

f ′(z)− 3

2

(f ′′(z)

f ′(z)

)2

.

Recall that fractional linear transformations (also called Mobius transformations) map(generalized) circles to circles and they are the only bijective conformal maps of theRiemann sphere P1 to itself. The Schwarzian derivative vanishes for fractional lineartransformations, conversely fractional linear transformations are the only functionshaving this property.

The Schwarzian derivative has an important inversion formula,

(Sw)(v) = −(dw

dv

)2

(Sv)(w)

derived from the inverse function theorem.

26

If f is any holomorphic function and set u = (f ′)−1/2 then it is easy to see that

u′′ +1

2(Sf)u = 0.

Conversely, consider a second order equation (in the Q-form)

u′′ +Qu = 0,

and let u1 and u2 be two linearly independent solutions. One can easily reconstructthe potential by Q = 1

2Sf from the quotient f = u1/u2. More generally, if u1 and u2

are independent solutions of an equation

u′′ + Pu′ +Qu = 0,

then f = u1/u2 satisfies

Sf = 2Q− 1

2P 2 − P ′

and we call it the Schwarzian equation from now on.

Example 21 (hypergeometric equation §2 in [93]). The hypergeometric equation

z(1− z)d2y

dz2+ [c− (a+ b+ 1)z]

dy

dz− aby = 0

may be turned into the Q-form

d2u

dz2+Q(z)u(z) = 0

by making the substitution y = uv and eliminating the middle term. One finds that

Q =z2[1− (a− b)2] + z[2c(a + b− 1)− 4ab] + c(2− c)

4z2(1− z)2

and v is determined by

d

dzlog v(z) = −c− z(a + b+ 1)

2z(1 − z) .

Suppose we use another set of parameters

λ = 1− c, µ = c− a− b, ν = a− b

then the corresponding Schwarzian equation is

Sf(z) =1− λ22z2

+1− µ2

2(1− z)2 +1− λ2 − µ2 + ν2

2z(1 − z) . (2.30)

As mentioned before, a solution to the above Schwarzian equation (the Schwarztriangle map D) maps the upper half-plane onto the interior of a curvilinear trianglewith angles πλ, πµ, πν.

27

Example 22 (Universal Picard-Fuchs equation §3.1 in [72]). Recall that the Picard-Fuchs equation of the Weierstrass family is given by

d2y

dj2+

1

j

dy

dj+

31j − 4

144j2(1− j)2 y = 0, (2.31)

and its Q-form can be easily obtained

d2f

dj2+

1− 1968j + 2654208j2

4j2(1− 1728j)2f = 0.

The corresponding Schwarzian equation is then

(Sτ)(j) =3

8(1− j) +4

9j2+

23

72j(1− j) (2.32)

Applying the inversion formula, we get the more important Schwarzian equation interms of τ

j(τ), τ+[

3

8(1− j(τ)) +4

9j(τ)2+

23

72j(τ)(1− j(τ))

](dj(τ)

dτ

)2

= 0. (2.33)

We change the variable ξ = 1/j in order to look into the behavior of the equationaround j =∞, the Picard-Fuchs equation 2.31 can be rewritten as

d2y

dξ2+

1

ξ

dy

dξ+

31ξ − 4ξ2

144ξ2(1− ξ)2y = 0. (2.34)

For holomorphic functions near ξ = 0 (i.e. j =∞)

P = 1, Q =31ξ − 4ξ2

144(1− ξ)2

the above equation is of the form

d2y

dξ2+P

ξ

dy

dξ+Q

ξ2y = 0,

which means the universal Picard-Fuchs equation has a regular singular point at j =∞.

One computes the indicial equation, which is

r(r − 1) + P (0)r +Q(0) = r2,

it has a double root at r = 0. Thus the two dimensional solution space is spannedby y1(ξ), which is holomorphic near ξ = 0, and y2(ξ) = y1(ξ)logξ + ρ(ξ) for someholomorphic function ρ(ξ).

28

Thus the solution to the Schwarzian equation 2.32 has the form

τ(ξ) =y2(ξ)

y1(ξ)=y1(ξ)logξ + ρ(ξ)

y1(ξ)= log(ξ) +

ρ

y1. (2.35)

One step further, we define q as the exponentiation

q(ξ) = eτ = ξ · eρ/y1 , (2.36)

which is an invertible holomorphic map from a disk |ξ| < S to another disk |q| < R.The inverse map ξ(q) therefore gives an analytic solution to 2.33, we call it the mirrormap.

One can apply the Frobenius method to find out y1, y2 and get

ξ(q) = 1− 744q2 + 356652q3 − 140361152q4 + · · · , (2.37)

furthermore the famous q-expansion of the j-invariant

j(q) =1

q+ 744 + 196884q + 21493760q2 + 864299970q3 + · · · . (2.38)

For a mirror pair (V, ω) and (V , ω), the B-model correlation function over themirror dual V is only depending on the complex structure modulus of V and suchdependence is encoded in the Picard-Fuchs equation.

In order to identify the mirror symmetry, one has to determine the map from thecomplex structure modulus of V to the Kahler structure modulus of V , the so-calledmirror map. Given a local coordinate system of the complex structure moduli space(call it τ) and that of the Kahler structure moduli space (call it q), the mirror mapgives rise to the expansion of q in terms of τ , sometimes called the q-expansion, whichis characterized by the Schwarzian equation. In particular, the mirror symmetry forelliptic curves consists of interchanging the complex structure modulus τ of V withthe Kahler structure modulus t of V . For the universal Picard-Fuchs equation, (2.37)(or its inverse) gives the mirror map and q-expansion, for more examples see [72],[86], and a systematic treatment of mathematical mirror symmetry is offered by thebook [28].

29

CHAPTER 3

MOTIVES

The theory of motives was first introduced by Grothendieck as a universal coho-mology for algebraic varieties. In his own words: [53] “ Contrary to what occurs inordinary topology, one finds oneself confronting a disconcerting abundance of differentcohomological theories. One has the distinct impression that each of these theories‘amount to the same thing.’ In order to express this intuition, of the kinship of thesedifferent cohomological theories, I formulated the notion of ‘motive’ associated to analgebraic variety. By this term, I want to suggest that it is the ‘common motive’behind this multitude of cohomological invariants attached to an algebraic variety, orindeed, behind all cohomological invariants that are a priori possible.”

Grothendieck’s speculations roughly sketch the picture:

MotRH

%%LLLLLLLLLL

Var

i;;vvvvvvvvv

H∗// GrVect

(3.1)

In plain language, an abelian category of motives Mot is expected to exist at theuniversal level together with a motivic cohomology, such that the category of algebraicvarieties Var is embedded intoMot. Then any classical cohomology theory H∗ wouldbe in the image of the realization functor RH from the universal motivic cohomology.

In this chapter, we will see this picture can be built up at least for nonsingularprojective varieties and pure motives. For general (possibly singular) varieties, wehave a triangulated category of mixed motives, and we are mostly interested in theabelian category of mixed Tate motives. General introductions to the theory of mo-tives are the book by Andre [5] and notes from the AMS Seattle conference [65], andthe material reviewed in this chapter is mostly following the references [18], [27], [35],[65], [92].

3.1 Weil cohomology

In this section, we will briefly go over some classical cohomology theories of alge-braic varieties with amazing common features. Abstracting these common properties

30

gives the notion of a Weil cohomology, which is a desirable good cohomology as arealization of the motivic cohomology. We are closely following §8.1 in [27] by Connesand Marcolli and §4 in [98] by Rej, and more details can be found in Kleiman’s notesin [65].

Let us assume X is a nonsingular projective algebraic variety over a field k inthis section, there are different classical cohomology theories associated to X , moresignificantly, these classical cohomology groups are related by canonical comparisonisomorphisms.

Example 23 (de Rham cohomology). If the field k is of characteristic zero, one hasthe algebraic de Rham cohomology defined as the hypercohomology of the de Rhamcomplex Ω•

X/k of algebraic differentials forms on X,

H idR(X, k) = Hi(X,Ω•

X/k).

Example 24 (Betti cohomology). If σ : k → C is an embedding, one has the Betticohomology associated to σ,

H iB(X,Q) = H i(X(C),Q),

which is the singular cohomology of X(C) viewed as a complex variety. For example,the only non-trivial Betti cohomology group of P1 is H2

B(P1,Q) = 1

2πiQ.

The period isomorphism identifies Betti and de Rham cohomology over C,

H iB(X,Q)⊗Q C ≃ H i

dR(X, k)⊗k C.

By taking the dualH i

B(X(C),C) ≃ Hom(Hi(X(C),C)), (3.2)

a Betti cohomology class could be represented by some homology cycle. A periodintegral is then given by the pairing

H idR(X,C)⊗Hi(X(C),C) → C

ω ⊗ γ 7→∫γω

(3.3)

We call it the period isomorphism since the isomorphism is literally induced by aperiod matrix.

Example 25 (etale cohomology). If the field k is of characteristic p (some prime),one has etale cohomology defined as the sheaf cohomology with respect to the etalesites on X. Furthermore, one has ℓ-adic (etale) cohomology as the inverse limit ofetale cohomology,

H iet(X,Zℓ) = lim←−H

iet(X,Z/ℓ

kZ),

where ℓ is a prime different from the characteristic p. In practice, one has to considerthe action of the absolute Galois group Gal(ksep/k) on etale cohomolgoy as well, whereksep is a separable closure of k, so sometimes it is convenient to use

H iet(X,Qℓ) with X = X ⊗k k

sep.

31

Similar to the period isomorphism, there exists a canonical comparison isomor-phism relating etale and Betti cohomology by comparing the etale topology with thecomplex topology,

H iet(X,Zℓ)⊗Zℓ

Qℓ ≃ H iB(X(C),Q)⊗Q Qℓ.

Etale cohomology is a deep theory related to Weil conjectures on zeta functionand the Riemann hypothesis over a finite field, more details can be found in [91] andoriginal papers by Artin, Deligne and Grothendieck.

Example 26 (crystalline cohomology). Suppose k is a perfect field of characteristic p,letW (k) be the ring of Witt vectors of k. For example, if k = Fp, thenW (Fp) = Zp thering of p-adic integers. Crystalline cohomology is the sheaf cohomology with respectto the crystalline site on X,

H icrys(X, k) = H i(Crys(X/k),OX/k).

If X/k can be lifted to a nonsingular variety X/W (k), then the crystalline cohomologyof X/k can be calculated in terms of the algebraic de Rham cohomology of X/W (k),and this construction is independent of the choice of liftings. In other words, thereexists a canonical isomorphism

H icrys(X, k) ≃ H i

dR(X,W (k)).

Let k be the ground field and K be the coefficient field of characteristic zero,consider the category of smooth projective varieties over k, denoted by SmProjk,and the category of graded vector spaces over K, denoted by GrVectK .

Generally speaking, a Weil cohomology theory is a contravariant functor fromSmProjk to GrVectK satisfying Poincare duality, Kunneth formula, algebraic cycleaxioms, and Lefschetz theorems. As expected, the classical cohomology theories: deRham, Betti, etale and crystalline cohomology are all Weil cohomologies.

Definition 5. A Weil cohomology theory is a contravariant functor:

H∗ : SmProjk → GrVectKX 7→ H∗(X)

(3.4)

SupposeX is a nonsingular projective variety of dimension n, then H∗(X) =⊕

i∈ZHi(X)

is subject to the following properties:(1) Finiteness H i(X) are finite-dimensional vector spaces and H i(X) vanish for

i < 0 or i > 2n.(2) Trace map The trace map Tr gives rise to an isomorphism

Tr : H2n(X)∼−→ K. (3.5)

32

(3) Poincare duality There exists a non-degenerate pairing

H i(X)×H2n−i(X)∼−→ H2n(X) ≃ K

(α, β) 7→ 〈α, β〉 = Tr(α ∪ β) (3.6)

If we introduce the notation of dual vector space H i(X)∨ = Hom(H i(X), K), thenthe pairing gives rise to an isomorphism

H i(X)∨ ≃ H2n−i(X). (3.7)

(4) Kunneth formula The projections from the fiber product X×kY to the subspaces

X ×k Y

prXzzvvvvvvvvv

prY$$H

HHHHHHHH

X Y

induce the isomorphism

H∗(X)⊗K H∗(Y ) ≃ H∗(X ×k Y )α⊗ β 7→ pr∗X(α) ∪ pr∗Y (β)

(3.8)

(5) Cycle map Let C i(X) denote the free abelian group of algebraic cycles in Xof codimension i. There exist group homomorphisms for 0 ≤ i ≤ n, called the cyclemaps,

γiX : C i(X)→ H2i(X) (3.9)

satisfying the following:(i) Normalization If X = P is a point, then the cycle map γP : C0(P )→ H0(P )

is just the inclusion Z ⊂ K.(ii) Multiplicativity One has the commutative diagram by the Kunneth formula,

C i(X)⊗ Cj(Y )

γiX⊗γj

Y

// C i+j(X ×k Y )

γi+jX×kY

H2i(X)⊗K H2j(Y )∼

// H2(i+j)(X ×k Y )

in terms of elements,

pr∗X(γiX(Z)) ∪ pr∗Y (γiY (W )) = γi+j

X×kY(Z ×k W ). (3.10)

In other words, the cycle map is multiplicative

γiX(Z)⊗K γiY (W ) = γi+jX×kY

(Z ×k W ). (3.11)

33

(iii) Pullback Let f : X → Y be a morphism and C i(X), C i(Y ) be the groups ofalgebraic cycles, then functoriality requires a commutative diagram for each i,

C i(Y )

γiY

f∗

// C i(X)

γiX

H2i(Y )f∗

// H2i(X)

(3.12)

(iv) Pushforward We first recall the definitions of push-forward for algebraic cyclesand cohomolgy groups since they are not so obvious compared to pull-backs.

Fix the dimensions as usual, dimX = n and dimY = m. Assume Z ⊆ X is aclosed subvariety of dimension i, the push-forward on cycles is defined by

f∗ : Ci(X) → Ci(Y )Z 7→ deg(f) · f(Z) (3.13)

one has to take into account the degree of the morphism, which is the degree of theextension of function fields [k(Z) : k(f(Z))] under mild conditions. Change that intocodimension, one gets

f∗ : Cn−i(X)→ Cm−i(Y ) (3.14)

Given the pull-backf ∗ : H2i(Y )→ H2i(X),

one defines the push-forward as

f∗ : H2n−2i(X) → H2m−2i(Y )α 7→ f∗(α)

(3.15)

such that for any β ∈ H2i(Y ),

TrX(α ∪ f ∗(β)) = TrY (f∗(α) ∪ β). (3.16)

For push-forwards functoriality requires a commutative diagram for each i,

Cn−i(X)

γn−iX

f∗// Cm−i(Y )

γm−iY

H2n−2i(X)f∗

// H2m−2i(Y )

(3.17)

(6) Lefschetz theorems Let ι :W → X be the inclusion of a nonsingular hyperplanesection W in X.

(i) Weak The weak Lefschetz theorem states that the restriction

ι∗ : H i(X)→ H i(W )

is an isomorphism if 1 ≤ i ≤ n− 2is an injection if i = n− 1

(3.18)

34

(ii) Hard Recall that the Lefschetz operator L is defined by

L : H i(X) → H i+2(X)α 7→ α ∪ γ1X(W ),

(3.19)

the hard Lefschetz theorem claims that

Ln−i : H i(X)∼−→ H2n−i(X) (3.20)

is an isomorphism for each i, where 1 ≤ i ≤ n.

3.2 Algebraic cycles

The main difficulty of studying algebraic varieties is due to the highly non-linearstructure of the objects. A practical strategy is to attach a linear structure, such as avector space, to a given variety. We will see in a later section that a “linearization” ofSmProjk is achieved by the abelian category of pure motives. Roughly speaking, anabelian category behaves like the category of modules over a ring, which is amenableto the techniques from homological algebra. In this new category, smooth projectivevarieties are kept as (part of) the objects and morphisms are given by correspondencesrealized by algebraic cycles in the product.

In this section, we will first give the definition of correspondences [88], which isthe key to constructing morphisms in the desired category of pure motives. Aftera short review of the adequate equivalence relations on algebraic cycles, we will listGrothendieck’s standard conjectures on algebraic cycles [52]. We are closely following§8.1 of [27] and §4,§5 of [98] in this section.

First let us recall the definition of algebraic cycles, sometimes called cycles. Weassume thatX and Y are smooth projective varieties with dimX = n and dimY = m.

Definition 6. An algebraic cycle W in X of codimension i is a formal linear combi-nation of irreducible closed subvarieties Wα of codimension i, that is,

W =∑

α

nαWα, nα ∈ Z. (3.21)

As before, C i(X) denotes the free abelian group of cycles in X of codimension i.

Thanks to the cycle map γiX : C i(X) → H2i(X), a cohomology class in the Weilcohomology could be the image of certain algebraic cycle, though we do not knowhow to trace it back. Consider a morphism f : X → Y and its graph is defined asusual

Γf = (x, y) | y = f(x) ⊆ X × Y,which is a closed connected subvariety of the product. It is easy to see codimΓf =dimY = m, the composition of Kunneth formula and Poincare duality then induces

35

a “correspondence”:

γmX×Y (Γf ) ∈ H2m(X × Y ) ≃ Hm(X)⊗Hm(Y ) ≃ Hom(H2n−m(X), K)⊗Hm(Y ).(3.22)

Namely, under the cycle map the image of the graph of a morphism can be viewedas a homomorphism from H2n−m(X) to Hm(Y ), where m = codimΓf . This suggeststhat cycles in X×Y with right codimension generalize morphisms between X and Y ,which is the key observation for defining morphisms by correspondences in motives.

Definition 7. For smooth projective varieties X and Y , a degree zero correspondenceZ is formally defined as the linear combination Z =

∑α nαZα with Zα irreducible

closed subvarieties in the product X × Y . We denote the abelian group of degree zerocorrespondences by Corr0(X, Y )Z, similarly Corr0(X, Y )Q for the coefficient field Q.

Based on the observation (3.22), for any morphism f : X → Y its graph Γf is acorrespondence from Y to X since the functor from SmProjk to motives is supposedto be contravariant. Furthermore, Γf ∈ Corr0(Y,X)Z has codimension dimY = m,and the transpose of the graph Γt

f ∈ Corr0(X, Y )Z as a correspondence from X to Yhas codimension dimY = m as well. We have to emphasize that any correspondenceZ ∈ Corr0(X, Y )Z has codimension dimX = n.

As a simple example, let us look at correspondences of smooth curves.

Example 27 (correspondence of curves). (§2.5 in [51]) A correspondence of degreed between two curves C1 and C2 is a holomorphic map

T : C1 → C2; p 7→ T (p), (3.23)

where T (p) is a divisor of degree d on C2. Its graph ΓT , called the curve of correspon-dence, is as usual

ΓT = (p, q) | q ∈ T (p) ⊆ C1 × C2. (3.24)

Conversely, for any curve C ⊆ C1 × C2 an associated correspondence can be definedby

T (p) = ι∗p(C) ∈ Div(C2), (3.25)

where ι : C2 → C1 × C2 is the inclusion map.An interesting example of this type is the Hecke operators of modular forms may

be viewed as correspondences of modular curves.

The group of correspondences Corr0(X, Y )Q is too big to work with, in practice wealways trim it into a smaller space of morphisms by considering equivalence relations.There are different choices of adequate equivalence relations on algebraic cycles [92].Let us use algebraic cycles C i(X) instead of Corr0(X, Y )Q for the moment, we willturn back to the group of correspondences later when we talk about pure motives. Welist below three commonly used equivalence relations on cycles from weak to strong:numerical equivalence, homological equivalence and rational equivalence.

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(1) Numerical equivalence Two cycles Z1, Z2 ∈ C i(X) are numerically equivalent,denoted by Z1 ∽num Z2, if and only if for any cycle Z ∈ Cn−i(X), the intersectionnumbers

♯(Z1 ∩ Z) = ♯(Z2 ∩ Z) (3.26)

are the same, where the intersection is just a linear combination of points by theabove assumption.

(2) Homological equivalence For a given Weil cohomology H∗, two cycles Z1, Z2 ∈C i(X) are said to be homologically equivalent if and only if their images under thecycle map γiX : C i(X)→ H2i(X) are the same at the cohomology level

[γiX(Z1)] = [γiX(Z2)], (3.27)

denoted by Z1 ∽hom Z2 when the choice of H∗ is clear from the context.(3) Rational equivalence Two cycles Z1, Z2 ∈ C i(X) are rationally equivalent, de-

noted by Z1 ∽rat Z2, if Z1 and Z2 are connected through a flat family of cycles inX × P1 parameterized by P1.

Modulo a certain equivalence relation ∽, we obtain a class group, denoted by

C i∽(X) = C i(X)/ ∽ . (3.28)

In particular, the class group of algebraic cycles modulo rational equivalence is theChow group CH i(X) = C i

∽rat(X). Since the numerical equivalence is the weakestequivalence relation and the rational equivalence is the strongest one, we have thegeneral relation between class groups

C i∽rat(X) ⊆ C i

∽hom(X) ⊆ C i∽num(X). (3.29)

In order to construct the abelian category of pure motives, one also needs thestandard conjectures on algebraic cycles, which are open conjectures relating alge-braic cycles and Weil cohomology theories originally stated by Grothendieck. Thestandard conjectures include conjecture B (Lefschetz type), conjecture C (Kunnethtype), conjecture D and the Hodge type standard conjecture. More details can befound in Grothendieck’s original paper [52] or Kleiman’s notes in [65]. In the follow-ing, we assume that a Weil cohomology H∗ is chosen for convenience.

(1) Lefschetz type The Lefschetz type standard conjecture is also called Grothendieck’sconjecture B. Recall that the Lefschetz operator L : H i(X) → H i+2(X) (3.19) in-duces an isomorphism Ln−i : H i(X)

∼−→ H2n−i(X) for each i ≤ n = dimX , which isthe so-called hard Lefschetz theorem.

Define the second Lefschetz operator

Λ : H i(X)→ H i−2(X) for i ≤ n (3.30)

by the composition (Ln−i+2)−1 L Ln−i, i.e.

Λ : H i(X)Ln−i

// H2n−i(X)L

// H2n−i+2(X)(Ln−i+2)−1

// H i−2(X). (3.31)

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Secondly, switch the positions of L and Λ and reverse the arrows in the commutativediagram, one defines the other part

Λ : H2n−i+2(X)→ H2n−i(X) for i ≤ n (3.32)

similarly by the composition Ln−i L (Ln−i+2)−1 , i.e.

Λ : H2n−i+2(X)(Ln−i+2)−1

// H i−2(X)L

// H i(X)Ln−i

// H2n−i(X). (3.33)

Conjecture 1 (B(X)). The Lefschetz conjecture states that the Lefschetz operator Λcan be realized by an algebraic cycle. More precisely, there exists an algebraic cycleZ ∈ C i(X ×X) such that the correspondence

Λn−i = γiX×X(Z) : H2n−i(X)→ H i(X) (3.34)

is the inverse of the Lefschetz isomorphism Ln−i.

Conjecture B has been proved for curves, surfaces and abelian varieties.

(2) Kunneth type The Kunneth type standard conjecture is also called Grothendieck’sconjecture C, which is weaker than the Lefschetz conjecture B(X).

Conjecture 2 (C(X)). The Kunneth projectors πi : H∗(X)→ H i(X) can be realizedby algebraic cycles Zi ∈ C i(X ×X).

Conjecture C holds for curves, surfaces and abelian varieties based on the knowl-edge of the Lefschetz conjecture.

(3) Conjecture D

Conjecture 3 (D(X)). Conjecture D states that numerical equivalence and homolog-ical equivalence are the same.

If conjecture D holds, then the homological equivalence would not depend on thechoice of Weil cohomology theories.

(4) Hodge type Recall that the primitive cohomology is defined by

PH i(X) = KerLn−i+1 : H i(H)→ H2n−i+2(X), (3.35)

we define the primitive algebraic cycles as

P j(X) = Z ∈ Cj(X) | γjX(Z) ∈ PH2j(X), with j ≤ n/2. (3.36)

Consider the non-degenerate pairing on H2i(X) with rational coefficients

< , >: H2i(X)×H2i(X) → H2n(X) ≃ Q(α, β) 7→ (−1)i〈Ln−2iα, β〉 = (−1)iTr(Ln−2iα ∪ β) (3.37)

then restrict this pairing to the primitive part P j(X) (the hidden cycle map is un-derstood), denoted by < , > |P .

38

Conjecture 4 (Hdg(X)). The Hodge standard conjecture claims the positive def-initeness of the above pairing on primitive algebraic classes of codimension j forj ≤ n/2, that is,

< , > |P : P j(X)× P j(X)→ Q (3.38)

is positive definite.

In characteristic zero, the Hodge type standard conjecture holds as a consequenceof Hodge theory (Weil). In nonzero characteristic, the Hodge type standard conjecturewas proved for surfaces (Segre, Grothendieck) and abelian varieties (Weil).

We must stress that the Hodge conjecture in complex geometry is not the sameas the Grothendieck’s Hodge type standard conjecture discussed above.

Based on the Hodge decomposition, for a nonsingular projective variety X over Cone defines the group of Hodge classes as

Hdgp(X) = H2p(X,Q) ∩Hp,p(X). (3.39)

Conjecture 5 (Hodge). The Hodge conjecture states that for a nonsingular projectivevariety X over C, every Hodge class on X can be realized by an algebraic cycle.

3.3 Tannakian category

In this section, we will remind the reader of some basic definitions in categorytheory and give a short introduction to Tannakian category. In a neutral Tannakiancategory, the existence of a fiber functor makes it possible to connect a rigid abeliantensor category with the category of vector spaces. The Tannakian formalism, es-pecially the Tannakian Galois group, will be of great importance in the theory ofmotives. We are closely following §3 in [98] for this topic, standard references areSaavedra-Rivano’s thesis under Grothendieck [99] and the lecture notes by Deligneand Milne [35].

Let us get started with the definition of an additive category.

Definition 8 (additive category). A category is additive if(1) it has a zero object;(2) it has both finite products and finite coproducts;(3) every Hom(A,B) is endowed with the structure of an abelian group such that thecomposition of morphisms is bilinear.

If only the third condition is satisfied, we call it a preadditive category.

Definition 9 (pseudo-abelian category). A pseudo-abelian category is a preadditivecategory such that every idempotent morphism has a kernel. Sometimes it is alsocalled Karoubian category.

39

The construction of effective pure motives in the next section will need the pseudo-abelian completion of a preadditive category, sometimes called the Karoubian enve-lope.

Given a preadditive category C, there exists an additive functor from C to itspseudo-abelian completion Kar(C),

F : C → Kar(C)X 7→ (X, idX)

such that F (p) splits in Kar(C) for any idempotent morphism p in C. Recall thatan idempotent q : X → X splits if there exists another object Y and morphismsf : X → Y, g : Y → X such that q = g f and idY = f g.

In general, objects in Kar(C) are pairs (X, p), where X ∈ Obj(C) and p : X → Xis an idempotent; morphisms f : (X, p) → (Y, q) in Kar(C) are f : X → Y as in C

such that f = q f p.

Definition 10 (abelian category). An abelian category is an additive category satis-fying the following:(1) every morphism has both a kernel and a cokernel;(2) every monomorphism is the kernel of some morphism;(3) every epimorphism is the cokernel of some morphism.

Example 28.

• The prototype of an abelian category is the category of abelian groups Ab.• The category of finite dimensional vector spaces over a field k is an abelian category,denoted by Vectk.• If R is a ring, then the category of left modules over R is an abelian category, de-noted by R-Mod.• (Freyd-Mitchell theorem) Every small abelian category is equivalent to a full subcat-egory of R-Mod over a ring R.

Assume the reader knows what a monoidal category looks like, which is also calledtensor category. For example, Ab, Vectk and R-Mod are all tensor categories.

Definition 11 (internal hom). Let (C,⊗) be a tensor category. For X, Y ∈ Obj(C),if the functor

F : C → Set

T 7→ Hom(T ⊗X, Y )

is representable by an object, then we call it the internal hom Hom(X, Y ), i.e.

F (T ) = Hom(T,Hom(X, Y )) = Hom(T ⊗X, Y ) (3.40)

In particular, when T = Hom(X, Y ),

Hom(Hom(X, Y ), Hom(X, Y )) = Hom(Hom(X, Y )⊗X, Y )

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then one defines evX,Y : Hom(X, Y ) × X → Y as the morphism corresponding to

the identity map idHom(X,Y ). Further, one defines the dual X by the internal homHom(X, 1).

Example 29 (R-Mod). In the category R-Mod,• the internal hom Hom(X, Y ) = HomR(X, Y ) as a module over R,• evX,Y : HomR(X, Y )⊗X → Y is the usual evaluation f ⊗ x 7→ f(x),

• the dual X = HomR(X,R).

Definition 12 (reflexive category). For any X ∈ Obj(C), since Hom(T, X) ≃Hom(T ⊗X, 1), we have

Hom(X,ˆX) ≃ Hom(X ⊗ X, 1). (3.41)

If the composition

X ⊗ X t// X ⊗X Hom(X, 1)⊗X evX,1

// 1 (3.42)

is an isomorphism, then X is reflexive, i.e. X ≃ ˆX. In general, if every object

X ∈ Obj(C) is reflexive, then we call (C,⊗) a reflexive category.

Example 30.

• The category of finitely generated projective modules over a ring R is a reflexivecategory, denoted by ProjR.• R-Mod is not reflexive in general. For instance, in Z-Mod = Ab, take Z/2Z ∈Ab, Z/2Z = Hom(Z/2Z,Z) = 0, hence

Z/2Z = 0 6= Z/2Z.

Definition 13 (rigid tensor category). A rigid tensor category is a reflexive category(C,⊗) such that(1) Hom(X, Y ) exists for all X, Y ∈ Obj(C),(2) all natural maps

⊗

i∈IHom(Xi, Yi)→ Hom(

⊗

i∈IXi,

⊗

i∈IYi)

are isomorphisms.

With the above preparations, we can give the definition of a neutral Tannakiancategory now.

Definition 14 (Neutral Tannakian category). A neutral Tannakian category overa field k is a rigid abelian tensor category C equipped with a k-linear exact faithfultensor functor, called the fiber functor, ω : C→ Vectk.

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An important example of neutral Tannakian category is given by the category offinite dimensional k-linear representations of an affine group scheme over a field k.

For a commutative k-algebra A, recall that an affine group scheme is defined byG = SpecA together with morphisms µ : G×G→ G, e : Spec k → G, and ι : G→ Gsatisfying the product, unit and inverse axioms respectively. By reserving the arrows,one gets k-algebra homomorphisms, ∆ : A → A ⊗k A (coproduct) corresponding toµ, ε : A → k (counit) corresponding to e, and S : A → A (antipode) correspondingto ι. This spells out the fact that the category of affine group schemes is dual to thecategory of commutative Hopf algebras. Thus a commutative Hopf algebra determinesan affine group scheme, and vice versa.

Example 31 (Ga and Gm). The simplest examples of affine group schemes are theadditive group and the multiplicative group.

(1) The additive group Ga = Spec k[t], and its dual Hopf algebra H = k[t] hascoproduct ∆(t) = t⊗ 1 + 1⊗ t, counit ε(t) = 0 and antipode S(t) = −t.

(2) The multiplicative group Gm = Spec k[t, t−1], and its dual Hopf algebra H =k[t, t−1] has coproduct ∆(t) = t⊗ t, counit ε(t) = 1 and antipode S(t) = t−1.

Let V be a finite dimensional vector space over k, a finite dimensional linearrepresentation of an affine group scheme G is a pair (ρ, V ) such that ρ : G→ GL(V )is a homomorphism, sometimes also written as ρ : G× V → V . Let Repk(G) denotethe category of finite representations of G endowed with a natural tensor structuregiven by ρ1 ⊗ ρ2 : G→ GL(V1 ⊗k V2).

Theorem 5 (Tannaka-Krein). Repk(G) is a neutral Tannakian category with thefiber functor ω : Repk(G) → Vectk given by the forgetful functor. Furthermore,there exists an isomorphism

Aut⊗(ω) ≃ G, (3.43)

where Aut⊗(ω) is the group of natural transformations of ω to itself preserving thetensor structure.

The Tannaka-Krein theorem is a special case of the main theorem in Tannakiancategories below, which characterizes a neutral Tannakian category as the categoryof finite representations.

Theorem 6 (Deligne, Grothendieck, Saavedra-Rivano). Let (T,⊗, ω) be a neutralTannakian category over a field k, then Aut⊗(ω) is an affine group scheme over kand we have the equivalence of categories

T ≃ Repk(Aut⊗(ω)). (3.44)

where Repk(−) is the category of finite dimensional k-linear representations.Conversely, if a category T is equivalent to Repk(G) for some affine group scheme

G, then there exists an exact faithful k-linear tensor functor ω : T→ Vectk such that

G ≃ Aut⊗(ω). (3.45)

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In the literature, the affine group scheme G ≃ Aut⊗(ω) is also called the (Tan-nakian) Galois group of the Tannakian category T, which will be very useful inmotives.

3.4 Pure motives

In this section, we will first set up the composition law of correspondences based onintersection products, then the category of effective pure motives will be constructed.After introducing the Tate motives, we obtain the category of (virtual) pure motives,which is a neutral Tannakian category through standard conjectures. We are closelyfollowing §8.2 in [27] and §5 in [98], more details about pure motives can be foundin Murre’s lecture [92] and the textbook [5], other short introductions include [32],[103].

In order to define the composition law of morphisms in motives, let us recall thedefinition of intersection product in intersection theory.

In his book [42], Fulton writes: “ · · · if A and B are subvarieties of a non-singular variety X , the intersection product A · B should be an equivalence class ofalgebraic cycles closely related to the geometry of how A ∩ B, A and B are situatedin X . Two extreme cases have been most familiar. If the intersection is proper, i.e.dim(A ∩ B) = dimA + dimB − dimX , then A · B is a linear combination of theirreducible components of A ∩ B, with coefficients the intersection multiplicities. Atthe other extreme, if A = B is a non-singular subvariety, the self-intersection formulasays that A · B is represented by the top Chern class of the normal bundle of A inX .”

Definition 15 (intersection product). The intersection product of two properly in-tersecting subvarieties A,B of a nonsingular variety X is then defined as

A ·B =∑

i

µ(Zi;A,B)Zi. (3.46)

with intersection multiplicities µ(Zi;A,B) and A ∩B = ∪iZi is the decomposition ofthe set-theoretic intersection into irreducible components.

The intersection multiplicity is defined by Serre’s Tor-formula: Suppose A and Bare given by ideals I and J in the coordinate ring OX . The multiplicity of Zi in theintersection product A · B is defined by

µ(Zi;A,B) =

∞∑

j=0

(−1)jℓOX,z[TorjOZi,z

(OX,z/I,OX,z/J)], (3.47)

i.e. the alternating sum of the length of torsion groups.

From now on, let Corrr(X, Y )Q denote the group of algebraic cycles of codi-mension dimX + r with rational coefficients. Similarly, Corrr

∽(X, Y )Q denotes the

corresponding class group for a given equivalence relation ∽.

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Definition 16 (composition law). For nonsingular projective varieties Xi, i = 1, 2, 3,consider the product varieties and the projections πij : X1×X2×X3 → Xi×Xj. Thecomposition of two correspondences is defined based on the intersection product:

: Corrr∽(X1, X2)Q × Corrs∽(X2, X3)Q → Corrr+s

∽(X1, X3)Q

(Z , W ) 7→ W Z = (π13)∗(π∗12Z · π∗

23W )(3.48)

Recall that a projector p ∈ Corr0∽(X,X)Q is an algebraic cycle (modulo the

equivalence relation) such that the self-intersection is itself, i.e. p · p ≡ p (mod ∽).One only considers degree zero correspondences in effective pure motives.

Definition 17 (Mot+∽(k)). The category of effective pure motives over a field k, de-

noted byMot+∽(k), is a category whose objects are pairs (X, p) with X ∈ Obj(SmProjk)

and a projector p ∈ Corr0∽(X,X)Q. The morphisms between pairs (X, p) and (Y, q)

are given by correspondences q · Corr0∽(X, Y ) · p.

Example 32 (Mot+num(k)). If we take the numerical equivalence, then Mot+num(k)is the pseudo-abelian completion of the preadditive category generated by objects X ∈SmProjk and morphisms Corr0num(X, Y )Q.

Recall that a k-linear abelian category is semisimple if every object is a directsum of finitely many simple objects. If we call the set of simple objects S, thenHom(X,X) = k for any X ∈ S and Hom(X, Y ) = 0 for X, Y ∈ S but X 6= Y .

Theorem 7 (Jannsen [64]). Mot+num(k) is a semisimple abelian category.

Furthermore, Mot+∽(k) is a tensor category with the natural monoidal structure

induced by the Cartesian product of varieties. A desirable property for a tensorcategory is a duality that makes it into a rigid tensor category, this can be achievedby introducing the Tate motives for effective pure motives. Before we define the Tatemotives, let us look at a basic example from Hodge theory.

Recall that a pure Hodge structure of weight n ∈ Z on a (finite dimensionalreal) vector space HR is given by a lattice HZ ⊂ HR and a Hodge decomposition ofHC = HR ⊗R C such that

HR = HZ ⊗Z R, HC =⊕

p+q=n

Hp,q, Hp,q = Hq,p. (3.49)

Example 33 (Tate Hodge structure 1.2.17 in [25]).• For a point P , its cohomology is trivial: H0(P,R) ≃ R. The Hodge structure on

H0(P ) is given by Z ⊂ R and the Hodge decomposition C = C0,0.• Next let us consider the Hodge structure on H2(CP1,R) ≃ R ≃ H1

dR(C∗). Since

H1dR(C

∗) is generated by dz/z and its contour integral is 2πi, we take the lattice(2πi)−1Z (as a normalization). On the other hand, since dz/z is holomorphic on C∗

and lies in H1,0(CP1) = F 1, the Hodge decomposition is just C = H1,1 = F 1 ∩ F 1.

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Thus the Hodge structure on H2(CP1,R), usually denoted by Z(−1), is given by thelattice (2πi)−1Z ⊂ R and the decomposition C = C1,1 with type (1, 1).• In general, the Tate twist Z(−n) is defined as the Hodge structure (onH2n(Pn,R) ≃

R if n > 0) of weight 2n given by the lattice (2πi)−nZ and the decomposition C = Cn,n.• We must stress that the Tate Hodge structure Z(1) is of weight −2 with lattice

2πiZ and the decompostion C = H−1,−1, which corresponds to the pure Tate motiveQ(1) in the theory of motives.

With the Tate Hodge structure in mind, corresponding to Z(−1) we have theLefschetz motive L as a pure motive, which would have the cohomological realizationH2(P1) in a Weil cohomology. The main idea behind this is the analogue as intopology (for algebraic varieties)

[ projective line ] = [ point ] + [ line ], i.e. [P1] = 1 + L (3.50)

that is, the Lefschetz motive L represents a line element in motives.Formally, one defines the Tate motive Q(1), corresponding to Z(1), as the inverse

of the Lefschetz motive L. As mentioned before, the purpose of defining the pureTate motive

Q(1) = L−1, with Q(n) = Q(1)⊗n and Q(0) = 1 (3.51)

is to shift the codimensions of correspondences, in particular introduce formal dualsinto effective pure motives Mot+

∽(k).

In the following we give the definition of virtual pure motives, sometimes justcalled pure motives for short.

Definition 18 (Mot∽(k)). The category of (virtual) pure motives over a field k,

denoted by Mot∽(k), is a category whose objects are triples (X, p, n) with X ∈

Obj(SmProjk), a projector p ∈ Corr0∽(X,X)Q and an integer n called the Tate twist.

The morphisms between triples (X, p, n) and (Y, q,m) are given by correspondencesq · Corrm−n

∽(X, Y ) · p with codimension dimX +m− n.

Example 34. We have the simplest pure motives:• 1 = (Spec k, id, 0), the motive of a point,• the Lefschetz motive L = (Spec k, id,−1), the motive of a line,• the Tate motive Q(1) = (Spec k, id, 1), formal dual of L.

Now we can easily embed the category of smooth projective varieties into thecategory of pure motives.

Definition 19. The embedding functor

i∽: SmProjk →Mot

∽(k) (3.52)

is defined by sending X to (X, idX , 0) and f : X → Y to Γtf ∈ Corr0∽(X, Y )Q.

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The category of pure motives Mot∽(k) is a pseudoabelian category with the nat-

ural tensor structure

(X, p, n)⊗ (Y, q,m) = (X × Y, p× q, n+m). (3.53)

Moreover, Mot∽(k) is a rigid tensor category with the duality

(X, p, n)∨ = (X, pt, dimX − n), (3.54)

where pt is the transpose.Thus with the numerical equivalence, Motnum(k) is a semisimple abelian category

with a rigid tensor structure, one expects a fiber functor to make it into a Tannakiancategory, which would deliver us a realization functor from Motnum(k) to GrVectk.

For a rigid tensor category C, Deligne introduced the definition of the dimensionof an object in C and proved an important theorem in this direction. As usual, weassume K is a field of characteristic zero below, for example we just take K = Q.

Definition 20 (dimension §1 in [35]). Let 1 be the unit object and M be any objectin a rigid tensor category C. If f : M → M is a morphism, then the trace of f onM , denoted by tr(fM), is defined by the composition

tr fM : 1∼

// M ⊗Mf⊗f

// M ⊗M ev// 1 (3.55)

In particular, the dimension of M is defined as dim(M) = tr(idM).

Theorem 8 (Deligne). Let K be a field of characteristic zero and C be an abeliancategory over K with a rigid tensor structure satisfying End(1) = K. Then C is aTannakian category if and only if every M ∈ Obj(C) has an integer dimension, i.e.dim(M) ∈ N.

However, the tensor structure (3.53) would give dim(M) ∈ Z in general. So inorder to get dim(M) ∈ N, one has to modify the tensor structure. Indeed, for any Mof weight i and N of weight j, twisting the usual isomorphism between M ⊗ N andN ⊗M by a sign (−1)ij will do the job.

Assuming conjecture D, i.e∽num=∽hom, one obtains thatMotnum(k) = Mothom(k)is independent of the choice of Weil cohomology theories, denoted by Mot(k) whenthe equivalence relation is understood.

Theorem 9. Let K is a field of characteristic zero. If the standard conjecture D holds,then Mot(K) is a neutral Tannakian category with the modified tensor structure.Thus Mot(K) can be realized for all Weil cohomologies H∗ with the fiber functor R:

R : Mot(K) → GrVectKM = i(X) 7→ H∗(X,K)

(3.56)

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Therefore Grothendieck’s idea can be realized for smooth projective varieties andpure motives in the following commutative diagram:

Mot(K)R

&&NNNNNNNNNNN

SmProjK

i77ppppppppppp

H∗// GrVectK

(3.57)

As a by-product, the Tannakian Galois group gives rise to the so-called motivicGalois group

Gmot(Mot(K)) ≃ Aut⊗(R). (3.58)

Recall that an algebraic group over K is said to be reductive if every finite di-mensional linear representation is a direct sum of irreducible representations. Andan affine group scheme is called proreductive if it is a projective limit of reductivealgebraic groups.

Theorem 10 (Remark 1.85 in [27]). Let K be a field of characteristic zero and Gbe a connected affine group scheme, then G is proreductive if and only if RepK(G) isa semisimple category. If G ≃ Aut⊗(ω) is the Galois group of a neutral Tannakiancategory (C,⊗, ω), then G is proreductive if and only if C is semisimple.

Since we know Mot(K) is a semisimple category (through standard conjectures),(at least the identity component of) its motivic Galois group Gmot(Mot(K)) is aproreductive group from the above theorem.

If T ⊂Mot(K) is a Tannakian subcategory, then there exists a group homomor-phism

Gmot(Mot(K))→ Gmot(T) (3.59)

Example 35 (TMot(K) p.148 in [27]). Let TMot(K) ⊂Mot(K) be the Tannakiansubcategory of pure Tate motives. It is easy to see that the motivic Galois group ofTMot(K) is the multiplicative group, Gmot(TMot(K)) ≃ Gm. Then there exists ahomomorphism

t : Gmot(Mot(K))→ Gm (3.60)

In fact, we also have a homomorphism (embedding)

i : Gm → Gmot(Mot(k)) (3.61)

such that t i = −2 ∈ End(Gm) = Z because of weight(Q(1)) = −2.Example 36 (AMot(K) 8.2.2 in [27]). The Artin motives, denoted by AMot(K),is the Tannakian subcategory of Mot(K) generated by triples (X, p, 0) with X a zerodimensional variety.

The corresponding motivic Galois group is given by

Gmot(AMot(K)) = Gal(Ksep/K) (3.62)

together with a group homomorphism

a : Gmot(Mot(K))→ Gal(Ksep/K). (3.63)

47

There are different important realization functors, here we mention the Tate real-ization and its relation with the Tate conjecture. The Tate conjecture may be recastby the so-called Tate realization:

Conjecture 6 (Tate). the Tate conjecture is equivalent to the following statement:The Tate realization T given by ℓ-adic cohomology H∗

et(−,Qℓ) is a faithful functor,

T : Mot(k)→ RepQℓ(Gal(ksep/k)) (3.64)

with values in semisimple representations.

3.5 Mixed Hodge structure

In this section, we briefly go over the category of mixed Hodge structures, whichgives a concrete example before the introduction of abstract mixed motives. We areclosely following §8.4 in [27], a systematic treatment of this topic can be found inDeligne’s original works, for example [31], [33] or in the book [95].

In the last section, we already mentioned pure Hodge structure (3.49), an alter-native definition is obtained by replacing the Hodge decomposition by the Hodgefiltration.

Definition 21. A pure Hodge structure of weight n on a (finite dimensional) vectorspace H is given by a finite descending filtration of HC by subspaces F pHC :

HC = F 0HC ⊇ F 1HC ⊇ · · · ⊇ 0, (3.65)

satisfying

HC = F rHC ⊕ F sHC and F rHC ∩ F sHC = 0 ∀ r + s = n+ 1.

It is convenient to translate from the Hodge decomposition to the Hodge filtration,and vice versa: for i+ j = p+ q = n,

F pHC =⊕

i≥p

H i,j, Hp,q = F pHC ∩ F qHC.

Example 37. If X is a compact Kahler manifold, then one has a pure Hodge structureof weight n on HR = Hn(X,R). Indeed, HZ = Hn(X,Z) is the lattice and HC =Hn(X,C) has a Hodge decomposition by Hodge theory.

Definition 22 (HS). The category of (pure) Hodge structures, denote by HS, hasHodge structures as objects. The morphisms are those linear transformations preserv-ing the lattice and the decomposition (equivalently the filtration):

φ : A→ B such that φ(Ap,q) ⊆ Bp,q (or φ(F pA) ⊆ F pB)

48

The category HS is a rigid abelian tensor category equipped with the naturaltensor structure and the dual Hodge structure.

Given two Hodge structures A = ⊕p+q=nAp,q of weight n and B = ⊕r+s=mB

r,s ofweight m. The tensor product is given by:

(A⊗ B)i,j =⊕

p,q

Ap,q ⊗Bi−p,j−q, (3.66)

then i + j = (p + q) + (i − p + j − q) = m + n, i.e. A ⊗ B is a Hodge structure ofweight m+ n. The dual object is defined by A = Hom(A,C) of weight −n such thatA−p,−q = Hom(Ap,q,C) as dual vector spaces.

In the language of motives, the Hodge conjecture may be recast as:

Conjecture 7 (Hodge 5). The Hodge conjecture is equivalent to the statement: theHodge realization H is a full functor,

H : Mot(k)→ HSQ. (3.67)

The cohomology groups of a compact Kahler manifold carry pure Hodge struc-tures, in general Deligne proved that the cohomology groups of a complex algebraicvariety carry mixed Hodge structures.

Definition 23. A mixed Hodge structure consists of a triple (V,W•, F •) where(1) V is a finite dimensional vector space;(2) W• is a finite increasing filtration on V , called the weight filtration;(3) F • is a finite decreasing filtration on VC, called the Hodge filtrationsuch that the graded quotient grWn VC = (Wn/Wn−1)⊗ C is a pure Hodge structure ofweight n, i.e.,

grWn VC = F rgrWn VC⊕F sgrWn VC, F rgrWn VC ∩F sgrWn VC = 0 ∀ r+ s = n+1 (3.68)

where the induced filtration on grWn VC is given by

F pgrWn VC = (F pVC ∩Wn ⊗ C)/(F pVC ∩Wn−1 ⊗ C). (3.69)

Mixed Hodge structures with morphisms compatible with the Hodge and weightfiltrations form a category MHS. One similarly defines the category MHSK for anysubfield K of C with characteristic zero. It is easy to see that the category MHSK

is a rigid abelian tensor category by generalizing the results on the category HS.There are two natural choices of a fiber functor on MHSK if we are expecting

a neutral Tannakian category. One is the forgetful functor ω(V,W•, F •) = V andthe other is ωW (V,W•, F •) = grW (V ) = ⊕ngr

Wn (V ). These two fiber functors are

actually isomorphic to each other if we set up an isomorphism between mixed Hodgestructures (V,W•, F •) and (grW (V ),W•, F •

W ) by an induced filtration, more detailscan be found at p. 158 in [27].

49

Theorem 11 (Deligne). If K is a field of characteristic zero, then the category ofmixed Hodge structures MHSK is a neutral Tannakian category and the TannakianGalois group is given by

GMHS = UMHS ⋊ (Gm ×Gm). (3.70)

Recall that given an affine algebraic group G, x ∈ G is called a unipotent elementif its associated right translation Rx on the affine coordinate ring of G is unipotent,i.e. there exists some n ∈ N such that (Rx − Id)n = 0. If every element x ∈ G isunipotent, then G is called a unipotent group. And a pro-unipotent group is just aprojective limit of unipotent groups.

In the above Galois group GMHS, UMHS is a pro-unipotent affine group schemedual to the commutative Hopf algebra HUMHS

= U(LMHS), the universal envelopingalgebra of LMHS, where LMHS is a bigraded Lie algebra with generators e−r,−s ofbidegree (r, s) for r, s ∈ N. The action of Gm ×Gm on UMHS can be described by anaction on LMHS:

(Gm ×Gm)×Lr,sMHS → Lr,s

MHS

( (u, v) , ℓ ) 7→ u−rv−sℓ

3.6 Mixed motives

For smooth projective varieties, we introduced the category of pure motives, whichis a good category in the sense that we have homological algebra techniques in thepackage. For general varieties, we want to construct a category of mixed motivesstill with computational tools such as long exact sequences of cohomology groups.As the relation between pure Hodge structure and mixed Hodge structure suggests,one would expect that mixed motives have filtrations with pure motives as gradedpieces. In order to understand mixed motives, it is important to know the extensionsbetween graded pieces, which turns out to be the main difficulty in studying mixedmotives.

In this section, we will give the description of mixed motives as a triangulatedcategory, from which one could extract an abelian category based on the heart of t-structures. One concrete example of mixed motives is given by Deligne’s one motives.The main references for this mysterious subject are [18], [55], [81], [118].

Let Vark denote the category of algebraic varieties over a field k, the categoryof mixed motives, denoted by MM(k), is a conjectural abelian category with a con-travariant functor

Vark →MM(k). (3.71)

As a universal cohomology theory attached to MM(k), one expects good propertiessuch as homotopy invariance and the existence of Mayer-Vietoris sequences.

The speculated abelian category MM(k) has not been constructed up to now.However, there exists a triangulated category DM(k) independently developed by

50

Hanamura [55], Levine [81], and Voevodsky [118], which is the bounded derived cat-egory of MM(k), i.e.

DM(k) = Db(MM(k)). (3.72)

Definition 24. A triangulated category D is an additive category together with theshift functor (also called the translation functor) T : D → D and a family of distin-guished triangles

X → Y → Z → T (X) = X [1] (3.73)

satisfying(TR 1)(distinguished triangles)

• X id−→ X → 0→ T (X) is a distinguished triangle for any X ∈ Obj(D).• For any f ∈ HomD(X, Y ), there exists an object C(f) ∈ Obj(D), called the

mapping cone, such that Xf−→ Y → C(f)→ T (X) is a distinguished triangle.

• A triangle isomorphic to a distinguished triangle is distinguished.(TR 2) (rotations) Y → Z → T (X) → T (Y ) and T−1(Z) → X → Y → Z of adistinguished triangle X → Y → Z → T (X) are distinguished.(TR 3) Given distinguished triangles X → Y → Z → T (X), X ′ → Y ′ → Z ′ → T (X ′)and morphisms f : X → X ′, g : Y → Y ′ such that the first square commutes, thenthere exists a morphism h : Z → Z ′ such that all squares commute.

X //

f

Y //

g

Z //

h

T (X)

T (f)

X ′ // Y ′ // Z ′ // T (X ′)

(TR 4) (octahedral axiom) Given Xf−→ Y → C(f) → T (X), Y

g−→ Z → C(g) →T (Y ), and the composition X

gf−−→ Z → C(g f)→ T (X), then these three trianglescan be placed on the vertices and edges of an octahedron such that all the faces arecommutative diagrams.

Example 38.

• If A is an abelian category and Kom(A) is the category of chain complexes of A,then the homotopy category K(A) is a triangulated category. The objects are objectsin Kom(A) and morphisms are the homotopy classes of morphisms in Kom(A).• The derived category of A, denoted by D(A), is the prototype of a triangu-

lated category, which can be constructed from K(A) by localizing at the set of quasi-isomorphisms. With boundedness conditions, the bounded derived category Db(A) isalso a triangulated category.

A triangulated category is a good machine to generate natural long exact se-quences. For instance, given a commutative diagram of distinguished triangles as

X //

Y //

Zv

// X [1]

X ′ // Y ′ u// Z // X ′[1]

51

then we have the distinguished triangle

X // X ′ ⊕ Y // Y ′ vu//X [1],

and applying the contravariant functor Hom(−, A) for any A ∈ Obj(D) gives rise toa long exact sequence.

Now we give the definition of a t-structure, it is the heart (also called core) of at-structure that gives an abelian category extracted from a triangulated category.

Definition 25. A t-structure on a triangulated category D is a pair of full subcate-gories (D≤0,D≥0), with general notations D≤n = D≤0[−n], and D≥n = D≥0[−n] forany n ∈ Z. These subcategories have the following properties:

(1)D≤0 ⊂ D≤1 and D≥1 ⊂ D≥0 (3.74)

(2)HomD(X, Y ) = 0 ∀ X ∈ Obj(D≤0), Y ∈ Obj(D≥1) (3.75)

(3) For any Y ∈ Obj(D), there exists a distinguished triangle

X → Y → Z → X [1]

with X ∈ Obj(D≤0) and Z ∈ Obj(D≥1).

Theorem 12 (Beilinson-Bernstein-Deligne [11], Kashiwara-Schapira [68]). The heartof a t-structure on D defined by the full subcategory D0 = D≤0 ∩D≥0 is an abeliancategory.

The simplest concrete example of mixed motives is given by the 1-motives firstintroduced by Deligne [31], and this part closely follows an introduction of 1-motivesin [9] and §8.3.2 in [27].

Example 39 (1-motives §1 in [9]). Let K be an algebraically closed field of char-acteristic zero. First recall that in the category of algebraic groups, a semi-abelianvariety G is defined as an extension of an abelian variety A by an algebraic torus T .According to Deligne, a 1-motiveM over K is defined by the data M = (L,A, T,G, u)where L is a finitely generated torsion free abelian group, G is a semi-abelian variety,and u : L→ G is a homomorphism.

If we expand out the extension G at the bottom, then a 1-motiveM = (L,A, T,G, u)can be represented as:

L

u

1 // T // G // A // 1

(3.76)

or vertically it is a complexM = [Lu−→ G] with L at degree −1 and G at degree 0. The

category of 1-motives, denoted by M1(K), has morphisms just as morphisms of the

52

corresponding complexes. In particular, a lattice L determines a 1-motive [L → 0],sometimes also denoted by L[1]. In other words, the category of Artin motives can beembedded in the category of 1-motives: AMot(K) ⊂M1(K).

Any 1-motive M = (L,A, T,G, u) has a canonical increasing filtration by sub-1-motives:

Wi(M) =

M i = 0[ 0→ G] i = −1[ 0→ T ] i = −20 i = −3

(3.77)

It is easy to see the graded quotients are given by sub-1-motives: grW−2(M) = [0→ T ],grW−1(M) = [0→ A] and grW0 (M) = [L→ 0].

Recall that the Hodge realization H(M) of 1-motives M over C (§10.1.3 in [31])is the mixed Hodge structure (TZ(M),W•, F •). The lattice TZ(M) is obtained by thepull-back of u : L → G along the exponential map exp : Lie(G) → G. The weightfiltration is given by

WiH(M) =

TZ(M) i = 0H1(G) i = −1H1(T ) i = −20 i = −3

(3.78)

The Hodge filtration is defined by

F 0(TZ(M)⊗ C) = ker(TZ(M)⊗ C→ Lie(G)). (3.79)

In addition, grW−1H(M) ≃ H1(A,Z) is a polarized Hodge structure of weight −1.The Hodge realization

H : M1(C) → MHSZ

M 7→ H(M)(3.80)

gives rise to an equivalence of categories between M1(C) and the full subcategory oftorsion free Z-mixed Hodge structures of type

(0, 0), (0,−1), (−1, 0), (−1,−1)

such that grW−1(−) is a polarized pure Hodge structure of weight −1. More detailsabout the Hodge realization, and other realizations can be found in §1 in [9].

3.7 Mixed Tate motives

In this section, we focus on the theory of mixed Tate motives, which is still anactively developing subject. Given the triangulated subcategory of mixed Tate mo-tives generated by Tate motives, one obtains a neutral Tannakian category of mixedTate motives over a number field using the machinery of t-structures. Surprisingly,

53

the periods of mixed Tate motives over Z can be expresses as multiple zeta values.We are closely following Levine’s IHES lecture notes [82] especially chapter 3.

Let DM(k) be the triangulated category of mixed motives over a field k as before.Recall that a full triangulated subcategory is called thick if it is closed under directsums.

Definition 26 (DMT(k)). The triangulated category of mixed Tate motives over k,denoted by DMT(k), is defined as the full triangulated thick subcategory generated bythe (pure) Tate motives Q(n), n ∈ Z.

Let W≤nDMT(k) be the full triangulated subcategory of DMT(k) generated bythe Tate motives Q(−m) withm ≤ n. Similarly,W≥nDMT(k) is the full triangulatedsubcategory generated byQ(−m) withm ≥ n. For each n, (W≤nDMT(k),W≥nDMT(k))is a t-structure on DMT(k) and there exists a tower of triangulated subcategories:

· · · ⊆W≤nDMT(k) ⊆ W≤n+1DMT(k) ⊆ · · · ⊆ DMT(k). (3.81)

Definition 27 (12.10-12.11 in [82]). The truncation functor W≤n is defined as

W≤n : DMT(k) → W≤nDMT(k)M 7→ lim−→N

Hom(Q(N − n),M(N))(−n) (3.82)

and the second truncation functor W≥n can be defined by duality:

W≥n : DMT(k) → W≥nDMT(k)

M 7→ Hom(W≤−nM,Z)(3.83)

where M = Hom(M,Z).

These truncation functors are exact functors, and for M ∈ DMT(k), there exista canonical distinguished triangle

W≤nM →M →W≥n+1M →W≤nM [1] (3.84)

and a canonical weight filtration on M

0 =W≤m′−1M →W≤m′M → · · · →W≤m−1M →W≤mM =M. (3.85)

for some m and m′.

Definition 28. LetWnDMT(k) be the heart of the t-structure (W≤nDMT(k),W≥nDMT(k))generated by Q(−n), i.e.

WnDMT(k) =W≤nDMT(k) ∩W≥nDMT(k). (3.86)

Define the graded quotient functor as

grWn : DMT(k) → WnDMT(k)M 7→ W≤nW≥nM

(3.87)

54

Based on the fact that

Hom(Q(m),Q(m)[p]) = Homp(Q(m),Q(m)) =

0 p 6= 0Q p = 0

(3.88)

we have the equivalence of categories

WnDMT(k) ≃ Db(VectQ), (3.89)

thus it makes sense to talk about the cohomology functor

Hp grWn : DMT(k)→ VectQ. (3.90)

for p ∈ Z.Let DMT(k)≤0 be the full subcategory of DMT(k) such that for every M ∈

Obj(DMT(k)≤0), Hp(grWn M) = 0 for all p ≥ 1 and all n, similar construction gives

DMT(k)≥0.

Conjecture 8 (Beilinson-Soule). The Beilinson-Soule vanishing conjecture statesthat for a field k, the weight q subspace of (Quillen’s algebraic K-group) K2q−p(k)

(q)

equals zero for q ≤ 0, except for the case p = q = 0.

If H is the motivic cohomology defined by

Hp(−,Z(q)) = Hom(−,Z(q)[p]), (3.91)

then this conjecture is equivalent to the statement,

Hp(k,Q(q)) = Hom(Z,Z(q)[p])⊗Q = 0 (3.92)

for p ≤ 0, except for p = q = 0. Indeed, if X is a nonsingular variety over k, then

Hp(X,Q(q)) ≃ K2q−p(X)(q). (3.93)

The weight ℓ subspace of Ki(X)Q = Ki(X)⊗Z Q is defined by

Ki(X)(ℓ) = x ∈ Ki(X)Q |ψk(x) = kℓ.x ∀ k ≥ 2, (3.94)

where ψk is the k-th Adams operator acting on the algebraic K-groups.

Proposition 1 (Prop. 12.15 in [82]). If the Beilinson-Soule vanishing conjectureholds for the field k, then (DMT(k)≤0,DMT(k)≥0) is a t-structure on DMT(k). Inaddition, the heart

MTM(k) = DMT(k)≤0 ∩DMT(k)≥0 (3.95)

is a rigid abelian tensor category containing the Tate motives Q(n), called the (abelian)category of mixed Tate motives over k.

55

MTM(k) is closed under extensions in DMT(k), that is if A→ B → C → A[1]is a distinguished triangle in DMT(k) with A,C ∈ MTM(k), then B is also inMTM(k). Moreover, it is the smallest abelian subcategory of DMT(k) containingthe Tate motives Q(n), n ∈ Z and closed under extensions.

It is easy to see that the category grWn MTM(k) is equivalent toVectQ with Q(−n)corresponding to Q, whose objects are the graded quotients grWn M = WnM/Wn−1Mfrom the canonical weight filtration (3.85).

If k is a finite field or a number field, then it is known that the Beilinson-Soulevanishing conjecture holds. From now on, we assume K is a number field.

Theorem 13 (Borel [80]). The extension groups in the category MTM(K) are

Ext1MTM(K)(Q(0),Q(n)) = K2n−1(K)⊗Q and Ext2

MTM(K)(Q(0),Q(n)) = 0(3.96)

In particular (§2.8 in [89]), when n = 1, the extensions in

Ext1DM(K)(Q(0),Q(1)) = K1(K)⊗Q (3.97)

give rise to the so-called Kummer motives as simplest mixed Tate motives:

K = [Zu−→ Gm] with u(1) = κ ∈ K×. (3.98)

In this case, K has the following period matrix

(1 0

log κ 2πi

)(3.99)

more information about logarithmic motives can be found in [8].

Theorem 14 (Deligne-Goncharov [34]). MTM(K) is a neutral Tannakian categorywith the fiber functor induced by the Hodge realization H : MTM(K) ≃MHS,

ω : MTM(K) ≃ GrVectQM 7→ ω(M)

(3.100)

withω(M) = ⊕nHom(Q(−n), grWn M) (3.101)

The motivic Galois group of MTM(K) is given by

Gmot(MTM(K)) ≃ U ⋊Gm, (3.102)

where U is a pro-unipotent affine group scheme, and its Lie algebra Lie(U) is some-times called the motivic Lie algebra.

For a number field K, the computation ofK∗(K)⊗Q can be related to the cohomol-ogy groups H∗(GLn(K),Q), so one can compute the cohomology groups Hp(K,Q(q))

56

explicitly. More precisely, Hp(K,Q(q)) = 0 unless p = q = 0 or p = 1, q ≥ 1. Andwhen p = 1,

H1(K,Q(q)) =

Qr2 for q ≥ 2 evenQr1+r2 for q ≥ 3 oddK×

Q = ⊕p⊆OK primeQ for q = 1(3.103)

where the rank r1 is the number of real embeddings of K and r2 is the number ofpairs of conjugated complex embeddings of K (Borel, Prop. 12.2 in [19]). Put thesetogether, Lie(UK) is freely generated by homogeneous elements corresponding to abasis of the dual of Ext1(Q(0),Q(n)) (Prop. 2.5 in [34]), i.e. a basis of

⊕

q≥1

H1(K,Q(q))∗ (3.104)

where H1(K,Q(q)) is located at degree −q.

Example 40. We are mostly interested in the case when K = Q, the motivic Liealgebra is freely generated by [p] (p is any prime number) and s2n+1:

Lie(UQ) = LieQ < [2], [3], [5], · · · , s3, s5, · · · > (3.105)

with [p] in degree −1 and s2n+1 in degree −(2n+ 1).If we define the Lie algebra LZ as the quotient Lie(UQ)/[⊕pQ[p], Lie(UQ)], then

the category MTM(Z) ⊆MTM(Q) of mixed Tate motives over Z is given by

MTM(Z) ≃ Rep(U(LZ)⋊Gm) (3.106)

To end this section, we briefly describe how to realize multiple zeta values asperiods of mixed Tate motives over Z. The main references are original papers byBrown [22], Goncharov-Manin [48] and Terasoma [115], and we are closely following§16 in [82].

Recall that multiple zeta values of weight n and depth r can be expressed asperiod integrals (16.4 in [82])

ζ(n1, . . . , nr) =

∫

0≤t1≤t2≤···≤tn≤1

ωn1,··· ,nr (3.107)

where n =∑r

i=1 ni and

ωn1,··· ,nr =dt1

1− t1

n1∏

j=2

dtjtj· · · dtn−nr+1

tn−nr+1

n∏

j=n−nr+2

dtjtj. (3.108)

Definition 29 (framed motives 13.4 in [82]). A framed mixed Tate motive is a triple(M, f, v) with M ∈MTM(k), f : grWn M → Q and v ∈ grWn′M satisfying n′ > n.

57

We first recall the definition of the period of a framed mixed Tate motive (M, f, v).For simplicity, we assumeM = WnM , v ∈ grWn M = Q(−n), and f ∈ Hom(grW0 M,Q)with grW0 M = Q(0) = W0M .

Taking the Hodge realization, (M, f, v) gives a (framed) mixed Hodge structure(HM,Hf,Hv). Since F nHM → grW2nHMC is an isomorphism by the above assumption,we can lift Hv to vC ∈ grW2nHMC. If π : HM → grW0 HM is the projection, then theperiod of (M, f, v) is defined as (Hf π)(vC), which is well-defined modulo periods ofweight < n by the choice of vC.

Consider the moduli spaceM0,n+3 of stable curves of genus zero with n+3 labeledpoints. And the open moduli spaceM0,n+3 can be embedded into (P1)n by sending(x1, x2, xn) to (0, 1,∞) as usual. Moreover, the inclusionM0,n+3 → (P1)n extends toa birational morphism M0,n+3 → (P1)n by a sequence of blow-ups with nonsingularcenters. As a consequence, D =M0,n+3 \M0,n+3 is a divisor with normal crossings.

We are mainly concerned with the real partM0,n+3(R), now take the componentM0,n+3(R)0 such that its coordinates have the standard ordering 0 < t1 < · · · < tn <1. Denote its closure by ∆ =M0,n+3(R)

−0 , which happens to be the integral domain

of multiple zeta values (3.107).Let D∆ be the components of D meeting with ∆, then an important observation

is that[∆] ∈ grW0 H

n(M0,n+3, D∆)∗ (3.109)

Lemma 1 (Lemma 16.3 in [82]). The differential form ωn1,··· ,nr from (3.107) has atworst logarithmic poles along D and is regular at each generic point of D∆.

Thus there exists a divisor D0 ⊆ D such that D0 and D∆ has no common com-ponents and

[ω] ∈ grW2nHn(M0,n+3 \D0) (3.110)

Theorem 15 (Goncharov-Manin). There exists a mixed Tate motive in MTM(Z)with Hodge realization Hn(M0,n+3\D0, D∆\D0), denoted byM(M0,n+3\D0, D∆\D0),such that the period of (M(M0,n+3 \D0, D∆ \D0), [∆], [ω]) is given by ζ(n1, · · · , nr).

The technical details of the proof can be found in [82] or the original paper [48].

3.8 Grothendieck ring

In this section we will briefly recall some basic facts about the Grothendieck ringof varieties, and explain why it offers a useful tool in testing the motivic propertiesof an algebraic variety. We are closely following §2.5 in [89] and §2 in [98].

Definition 30. Let Vark denote the category of algebraic varieties over a field k. TheGrothendieck ring K(Vark) is the abelian group generated by isomorphism classes [X ]of varieties such that for any closed subvariety Y ⊂ X,

[X ] = [Y ] + [X \ Y ]. (3.111)

The fiber product of varieties gives the ring structure on K(Vark): [X ][Y ] = [X×kY ].

58

Roughly speaking, in K(Vark) we look at the decomposition of a variety intopieces by “cut-and-paste”. Let 1 = [A0] be the class of a point and L = [A1] ∈K(Vark) be the class of an affine line (sometimes still called the Lefschetz motive).For example, [Pn] = 1 + L+ · · ·+ Ln ∈ K(Vark) corresponds to the cellular decom-position of Pn into pieces A0 ∪ A1 ∪ · · · ∪ An.

After resolving the singularities by Hironaka’s theorem, K(VarC) can be generatedby nonsingular varieties. More precisely,

Theorem 16 (Bittner [16]). If k is a field of characteristic zero, then the Grothendieckring K(Vark) is generated by isomorphism classes of nonsingular varieties subject tothe blow-up relation [BlYX ]− [E] = [X ]− [Y ],

E

j//

πE

BlYX

π

Y

i// X

(3.112)

where X is a nonsingular variety, Y ⊆ X is a closed nonsingular subvariety, and Eis the exceptional divisor of the blowup BlYX of X along Y .

In the blow-up context, one has the following useful fact:

Proposition 2 (Prop. 2.11, 2.14 in [98]). In the blow-up (3.112), if Y ⊆ X is ofcodimension d, then πE : E → Y is locally a Pd−1-bundle over Y , i.e.

[E] = [Y ][Pd−1] ∈ K(VarC). (3.113)

Furthermore, by the blow-up relation,

[BlYX ]− [X ] = [E]− [Y ] = [Y ]([Pd−1]− 1). (3.114)

Example 41 (banana graph [4]). Banana graph, denoted by Γn, is a graph with twovertices and n parallel edges connecting them. Take the hypersurface XΓn in Pn−1,

XΓn = (t1 : · · · : tn) |ΨΓn = t1 · · · tn(1

t1+ · · ·+ 1

tn) = 0, (3.115)

where ΨΓn is the Kirchoff polynomial of Γn. Then the hypersurface class in theGrothendieck ring is given by

[XΓn ] =Ln − 1

L− 1− (L− 1)n − (−1)n)

L− n(L− 1)n−2 (3.116)

For an algebraic variety, one may think of its class [X ] in the Grothendieck ring asa “universal Euler characteristic” of the variety X , more details see [16]. To explainmore precisely what this means, we recall the following notions.

59

Definition 31. An additive invariant of varieties is a map χ : Vark → R, withvalues in a commutative ring R, satisfying(1) Isomorphism invariance: χ(X) = χ(Y ) if X ∼= Y are isomorphic.(2) Inclusion-exclusion: χ(X) = χ(Y ) + χ(X \ Y ) for Y ⊂ X closed.(3) Multiplicative: χ(X × Y ) = χ(X)χ(Y )

Example 42. (1) Topological Euler characteristic: it is the prototype of additiveinvariants

χtop : VarC → Z; with X 7→∑

i≥0

(−1)idimH ic(X(C),Q) (3.117)

(2) Counting points over Fq:

Nq : VarFq → Z; with X 7→ ♯X(Fq) (3.118)

(3) Hodge polynomial:

PHdg : VarC → Z[u, v]; with X 7→∑

p,q≥0

(−1)p+qhp,q(X)upvq (3.119)

where hp,q(X) = dimHp(X(C),ΩqX) is the (p, q)-Hodge number of X.

Example 43 (Hodge-Deligne invariant [94]). Let K(HS) be the Grothendieck ringof pure Hodge structures. For a Hodge structure V , one defines [V ] ∈ K(HS) by thegraded quotients of the weight filtration: [V ] =

∑n[gr

Wn V ]. Define the Hodge-Deligne

invariant by

χHdg : VarQ → K(HS); with X 7→∑

n

[Hnc (X,Q)] (3.120)

If Y ⊆ X is a closed embedding, there exists the Gysin sequence:

· · · → Hnc (X \ Y,Q)→ Hn

c (X,Q)→ Hnc (Y,Q)→ Hn+1

c (X \ Y,Q)→ · · · (3.121)

Thus the Hodge-Deligne invariant χHdg is an additive invariant,

χHdg(X) = χHdg(Y ) + χHdg(X \ Y ). (3.122)

There exists a ring homomorphism

ψ : K(HS) → Z[u, v]Hn(X,Q) 7→

∑p+q=n(−1)p+qdim(Hp,q(X,Q)) upvq

(3.123)

when combined with χHdg, one obtains the Hodge polynomial PHdg = ψ χHdg.Setting u = v = w, we get another ring homomorphism

ψwt : K(HS) → Z[w]Hn(X,Q) 7→ (−1)ndim(Hn(X,Q))wn (3.124)

The composition Pwt = ψwt χHdg is called the weight polynomial, in particular,Pwt(1) = χtop is the topological Euler characteristic.

60

Every additive invariant must factor through the Grothendieck ring and theninduces a ring homomorphism χ : K(Vark)→ R. Therefore we may think ofK(Vark)as the “universal Euler characteristic”.

Example 44 (Motivic Euler characteristic p.39 in [89]). Let K be a field of char-acteristic zero such as a number field and Mot(K) be the abelian category of puremotives over K. Similarly one can define the Grothendieck ring of pure motives anddenote it by K(Mot(K)). From [45], there exists an additive invariant χ : VarK →K(Mot(K)), and it induces a ring homomorphism as usual

χ : K(VarK)→ K(Mot(K)). (3.125)

In the Grothendieck ring of varieties L = [A1], take into account the formal inverseL−1 as the Tate motive, instead of χ it is natural to consider the ring homomorphism,sometimes called the motivic Euler characteristic,

χmot : K(VarK)[L−1]→ K(Mot(K)). (3.126)

From the discussion on the relation between multiple zeta functions and mixedTate motives over Z, we are mostly interested in the (mixed Tate motive) subringZ[L,L−1] ⊆ K(Mot(K)), or equivalently the subring Z[L] ⊆ K(VarK).

61

CHAPTER 4

ALGEBRAIC GEOMETRY OF

HARPER OPERATORS

In Solid State physics, the Hamiltonian of a lattice electron in a uniform magneticfield, called the discrete magnetic Laplacian, has been studied for years in the tight-binding model [105]. As a special discrete magnetic Laplacian, the Harper operatorcorresponds to a square lattice when the coupling constant is fixed (i.e. λ = 1)[60]. For instance, the Harper operator arises in the study of the integer quantumHall effect [14]. One important spectral property of the Harper operator is that itsspectrum is a Cantor set of zero Lebesgue measure for every irrational frequency [7],[77].

In this chapter, we consider the algebro-geometric approach to the spectral theoryof electrons in solids developed for discrete periodic Schrodinger operators by Gieseker,Knorrer and Trubowitz in [44] and investigate how to extend this approach to thecase of Harper operators and almost Mathieu operators. We show that the differentstructure of the spectrum in the rational and irrational case has an analog in termsof the algebro-geometric properties of the Bloch variety, which is an ordinary varietyin the rational case and an ind-pro-variety in the irrational case.

While this geometric construction is appropriately formulated in algebro-geometricand categorical language where we can appropriately define the necessary double limitoperation, the resulting space we obtain has a simpler heuristic description in moredirectly physical terms. In the original case of [44] of the periodic Schrodinger op-erator without magnetic field, the Bloch variety physically describes the complexenergy-crystal momentum dispersion relation, that is, the set of complex points thatcan be reached via analytic continuation from the band functions. In the case ofHarper and almost Mathieu operators with irrational parameters, one observes thephenomenon that, instead of intervals (band structure), the spectrum takes the formof a Cantor set, which is geometrically a suitable limit of an approximating family in-tervals. Correspondingly, the geometric space describing the complex energy-crystalmomentum dispersion relation, which replaces the Bloch variety, is no longer an al-gebraic variety, but it is obtained in the same way by “analytic continuation” froma Cantor set rather than from bands in the spectrum. It is then natural to expect

62

that such a geometric space will be the “algebro-geometric analog of a Cantor set”,also obtained as a limit, in a way that mirrors the construction of the Cantor set asa limit of intervals. It is then amusing to see that this concrete finding is in completeagreement with a general categorical result of Kapranov [67], [96], which shows thatall ind-pro-varieties in fact really do look like Cantor sets.

In the discrete approximation, we study the geometry of the Bloch variety andFermi curves associated to the spectral theory of Harper operators and we computetheir density of states. The density of states of the two dimensional Harper operatorturns out to be the complete elliptic integral of the first kind combined with theLanden transformation. We find out that this density of states satisfies a Picard-Fuchsequation, then we obtain the fundamental solution of the corresponding Schwarzianequation. This solution can be viewed as a mirror map and gives rise to a q-expansionof the energy level of the electronic system.

When the density of states is available, some spectral functions, such as the zetafunction and the partition function, can be obtained by integrations of special func-tions of the eigenvalues.

A similar strategy is applied to the almost Mathieu operator to study the geometryof the Bloch variety, compute the density of states and calculate the related spectralfunctions. There exists a big difference in their density of states between dimensionone and two. For the (two dimensional) Harper operator, we show that the densityof states turns out to be a period independent of the parameter θ = α − β, whichinvolves the complete elliptic integral of the first kind. However, for the almostMathieu operator (one dimensional case), we show by a similar technique that thedensity of states explicitly depends on the parameter α in a way that recovers theusual phenomenon of the Hofstadter butterfly.

This chapter is based on the author’s results, some of which have appeared in thepapers [83], [85].

4.1 Bloch wave and Fermi surface

In this section, we recall the static lattice approximation, the independent electronapproximation and the classical Bloch theory in a perfect crystal closely following [90].Bloch varieties and Fermi curves are introduced in the algebro-geometric context, andthe density of states shows up as a period in this setting.

In solid state physics, the static lattice approximation is widely used in studyingthe behavior of electrons in a crystal structure of a solid at low temperature. ABravais crystal structure can be described by a lattice Γ ⊆ Rd, we assume Γ ≃ Zd ford = 2 or 3 in our work. At low temperature each atom has almost no thermal energyto move away from its equilibrium position. Then the static lattice approximationsimply states that a single ion is fixed at each lattice point and an electron moves ina periodic potential.

63

Based on a 2-body potential u, the ion-electron interaction can be described by

U(x) =∑

γ∈Γu(x− γ)

for an electron at the position x ∈ Rd. This potential is periodic with respect to thelattice

U(x + γ) = U(x), γ ∈ Γ

And the electron-electron interaction is given by a Coulomb potential W .Ignoring spin, the Hamiltonian for a system of N electrons in a pure, finite sample

of the crystal is given by

H(x1, · · · , xN ) =N∑

i=1

(−∆xi+ U(xi)) +

1

2

∑

i 6=j

W (xi − xj) (4.1)

where xi is the position of the i-th electron.Furthermore, the independent electron approximation is made, in which one does

not consider electron-electron interactions because they are often much weaker thanion-electron interactions. Therefore, the independent electron Hamiltonian providesa good approximation to 4.1,

H ′(x1, · · · , xN ) =N∑

i=1

(−∆xi+ V (xi)) (4.2)

The good thing about the independent electron approximation is that the originalpotential U could be unbounded, but a suitable modification V can be assumed tobe a bounded function.

With this approximation, one reduces a multi-electrons problem to a one-electronproblem. Consider the one-electron Hamiltonian H = −∆ + V , let Tγ denote theunitary operator on L2(Rd) implementing the translation by γ ∈ Γ.

Since the crystalline system and hence the Hamiltonian H is invariant under suchtranslations, Tγ must commute with H , thus they can be simultaneously diagonal-ized. In this way, each eigenfunction of the Hamiltonian is an eigenfunction of thetranslations.

In other words,TγHT

−1γ = H, ∀ γ ∈ Γ

As mentioned above, we simultaneously diagonalize H and Tγ. Suppose Tγψ =c(γ)ψ, γ ∈ Γ, since Tγ1γ2 = Tγ1Tγ2 and these translations are unitaries, we have

c : Γ→ U(1); γ 7→ c(γ) = eik·γ, k ∈ Γ

where Γ is the Pontrjagin dual group of Γ.

64

The Pontrjagin dual Γ of the abelian group Γ ≃ Zd is a compact group isomorphicto Td, obtained by taking (the dual) Rd modulo the reciprocal lattice

Γ♯ = k ∈ Rd | 〈k, γ〉 ∈ 2πZ, ∀ γ ∈ Γ.

Because of the periodicity of crystals, one may consider the Brillouin zones. Thefirst Brillouin zone is the standard fundamental domain for the reciprocal lattice Γ♯.Given any lattice vector k ∈ Γ♯, the hyperplane perpendicularly bisecting k is calleda Bragg hyperplane. In physics, Bragg hyperplanes are where patterns of reflectedX-rays can be observed in a crystal. Then the first Brillouin zone of Γ♯ can be definedas the set of points in the k-space that can be reached from the origin without crossingany Bragg hyperplane.

In general, the n-th Brillouin zone consists of all the points in the reciprocalspace such that the line from that point to the origin crosses exactly (n − 1) Bragghyperplanes. In a periodic medium, the Bloch wave can be completely characterizedby its behavior in the first Brillouin zone and the dispersion relation described byBragg diffractions.

Now it is equivalent to consider a family of self-adjoint elliptic boundary valueproblems, parameterized by the crystal momentum k (also called wave vector) in thereciprocal space,

(−∆+ V )ψ(x) = λψ(x)ψ(x+ γ) = eik·γψ(x)

whose non-trivial solutions ψ(x) are called the Bloch waves.For each value of the momentum k, one has eigenvalues Ej(k) for j ≥ 1, the

so-called band functions describing ranges of energy, and the corresponding normal-ized eigenfunctions ψn(x, k) are called Bloch states. The energy bands satisfy theperiodicity condition:

Ej(k) = Ej(k + ω), ω ∈ Γ♯

As a simple example, if Γ ∼= Z2, then Γ ∼= T2 ∼= R2/Z2 and Ej(k) ∈ C(T2).Based on the Bragg diffractions, one can start to plot E1(k) in the first Brillouin

zone and obtain a mapk → E(k)

by analytic continuation, which is called the energy-crystal momentum dispersionrelation. Similar process gives rise to the Bloch wave ψ(x, k).

Another important geometric concept in condensed matter physics is the Fermisurface of a solid. First recall that the Fermi energy EF is the energy of the highestoccupied quantum state in a system of fermions at absolute zero temperature. Thenthe Fermi surface is an imaginary boundary at the Fermi energy separating the k-space into occupied and non-occupied states at zero temperature, i.e.

F = k ∈ Rd |E(k) = EF.

65

The Fermi energy and hence the Fermi surface are depending on the density of statesof the crystal.

The geometry of the Fermi surface is derived from the periodicity and symmetryof the crystalline lattice and from the occupation of electronic energy bands, i.e.the density of states. The interested reader is referred to 3D VRML Fermi SurfaceDatabase [26] for concrete examples of Fermi surfaces.

In complex geometry, we consider the complex Bloch variety

B(V ) =

(k, λ) ∈ Cd+1 :

∃ψ(x, k) such that(−∆+ V )ψ(x) = λψ(x),ψ(x+ γ) = eik·γψ(x)

With the natural projection π : B(V ) → C onto the last coordinate, one defines thecomplex Fermi hypersurfaces as Fλ(C) := π−1(λ).

In more physical terms, the Bloch variety represents the complex energy-crystalmomentum dispersion relation, that is, the locus of all complex points that can bereached by analytic continuation.

The complex Fermi hypersurface Fλ(C) is a complex geometric hypersurface inCd. Then the real Fermi surface is a cycle Fλ(C) ∩ Rd = Fλ(R) representing a classin the homology group Hd−1(Fλ(C),Z).

Defining the integrated density of states ρ(λ) as the averaged counting function ofthe eigenvalues, the main observation in [44] is that the density of states dρ/dλ canbe expressed as a period integral over the cycle Fλ(R), namely

dρ

dλ=

∫

Fλ(R)

ωλ

where ωλ is a holomorphic form on Fλ(C). This is the period that we will focus onin the rest of this chapter. Namely, we will explicitly compute the period integral ofthe Harper operators, and discuss related physical quantities based on the density ofstates.

4.2 Harper operators

When a magnetic field is present, we replace the discrete Laplacian by Harperoperators. In this section, we introduce the Harper operator and the almost Math-ieu operator in dimension one and two, we also recall their spectral properties infunctional analysis and noncommutative geometry.

In [44], Gieseker, Knorrer and Trubowitz modeled the theory of electron propaga-tion in solids by considering a discrete periodic Schrodinger operator −∆+ V actingon the Hilbert space ℓ2(Z2), with

∆ψ(m,n) = ψ(m+ 1, n) + ψ(m− 1, n) + ψ(m,n+ 1) + ψ(m,n− 1) (4.3)

66

the random walk operator discretizing the Laplacian, and with a real effective poten-tial V periodic with respect to a sublattice aZ + bZ, where a and b are distinct oddprimes. They studied the geometry of the associated Bloch variety

B(V ) = (ξ1, ξ2, λ) ∈ C∗ × C∗ × C | ∃ψ with (−∆+ V )ψ = λψ,such that ψ(m+ a, n) = ξ1ψ(m,n), ψ(m,n+ b) = ξ2ψ(m,n)

(4.4)

In this work we consider a different but closely related spectral problem, whereinstead of the usual Laplacian and its discretization given by the random walk opera-tor, we consider a magnetic Laplacian, whose discretization is a Harper operator. Wealso restrict our attention to the case with trivial potential V ≡ 0. Note that whena magnetic field is present, the commutative relation TγH = HTγ fails and Bloch’stheory breaks down.

The two dimensional Harper operator H acting on ℓ2(Z2) is defined as

Hψ(m,n) = e−2πiαnψ(m+ 1, n) + e2πiαnψ(m− 1, n)+e−2πiβmψ(m,n + 1) + e2πiβmψ(m,n− 1)

(4.5)

where the two unitaries

Uψ(m,n) := e−2πiαnψ(m+ 1, n) and V ψ(m,n) := e−2πiβmψ(m,n + 1)

are the so-called magnetic translation operators with phases α and β respectively, andthe group of magnetic translations Tα,β is generated by U, V . One can then write theHarper operator as H = U +U∗+V +V ∗. Note that our form of the Harper operatoris slightly different from that in the literature, but they are all unitary equivalent bya gauge transformation Tγψ(m,n) = e2πiγmnψ(m,n) on ℓ2(Z2).

Let Tα,β = C∗(Tα,β) be the group C∗-algebra of the group of magnetic translationsTα,β. Recall that the noncommutative torus Aθ is the universal C

∗-algebra generatedby two unitaries u, v subject to the commutation relation uv = e2πiθvu. Settingθ = α − β (called the magnetic flux), we have a representation πθ : Aθ → Tα,β suchthat πθ(u) = U, πθ(v) = V . Thus, the Harper operator H is the image of a boundedself-adjoint element of Aθ.

If the parameters α, β are rational numbers, so is θ, then the rotation algebra Aθ

is isomorphic to the continuous sections of some vector bundle over the two-torus T2.For rational parameter, the one dimensional Harper operator is a periodic operatorand its spectrum consists of energy bands separated by gaps by Bloch-Floquet theory.The Bloch variety associated to a Harper operator with rational phases has onlyfinitely many components, and can be treated similarly to the periodic Schrodingeroperators discussed in [44].

When θ is an irrational real number, the irrational rotation algebra Aθ is a simpleC∗-algebra and it has been studied in noncommutative geometry motivated by Kro-necker foliation, deformation theory etc. For irrational parameter, the spectrum ofthe Harper operator is a Cantor set of zero measure [60], [58].

67

Instead of following the approach to the spectral theory of Harper and almostMathieu operators based on functional analysis and noncommutative geometry, weaim at adapting the algebro-geometric setting developed in [44] in the case of dis-cretized periodic Schrodinger operators without magnetic field. In a sense, what wewant to do is to figure out an algebro-geometric analog of the noncommutative torusAθ.

We assume that α, β, θ are all irrational real numbers in the following sections,which will not be stated otherwise.

A limit case of the Harper operator is the almost Mathieu operator, whose spectraltheory has been widely studied in connection with the famous phenomenon of theHofstadter butterfly, which was first observed by Hofstadter [60], also cf. [13], [78].

Indeed, if we set the parameter β = 0 and let the Harper operator act on ℓ2(Z),we can express the resulting operator in terms of two new unitaries

U ′ϕ(n) := e−2πiαnψ(n) and V ′ϕ(n) := ϕ(n+ 1)

Thus the almost Mathieu operator is defined as H ′ := U ′ + U ′∗ + V ′ + V ′∗, namely

H ′ϕ(n) := 2cos(2παn)ϕ(n) + ϕ(n + 1) + ϕ(n− 1) (4.6)

The physical meaning of this limit process is that one takes the Landau gauge whichforces the vector potential only in one direction, so the almost Mathieu operator issometimes called the Landau Hamiltonian by physicists.

Therefore, we get another representation πα of the noncommutative torus suchthat πα(u) = U ′, πα(v) = V ′. In the literature, the almost Mathieu operator is alsoreferred to as the one dimensional Harper operator. Since the C∗-algebra Aθ is simplefor irrational θ, the almost Mathieu operator has the same spectrum as that of thetwo dimensional Harper operator.

The spectral property of the Harper operator is related to the famous “Ten Martiniproblem”, which was one of Simon’s problems [108]. Now its spectrum is well under-stood and the reader can consult the sources listed as follows. Applying semi-classicalanalysis based on Wilkinson’s renormalization, Hellfer and Sjostrand [58] proved thatthe spectrum of the almost Mathieu operator has a Cantor structure for frequency αhaving continued fraction expansions with big denominators. Last [77] also showedthat it is a zero measure Cantor set for α satisfying a Diophantine condition.

4.3 Geometry of the Bloch variety

In this section, we first describe the geometry of the Bloch variety and algebraicFermi curves associated to the Harper operator in the discrete model. The Blochvariety is actually an inductive limit of ordinary algebraic varieties, call it Blochind-variety. The Fermi curves are a family of elliptic fibers in the Bloch ind-variety,so they are ind-varieties as well. It is easier to adopt the Fourier modes, and thesingularities will be discussed for the approximating components.

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Figure 4.1: Hofstadter’s butterfly

The horizontal axis is the energy E and the vertical axis is the flux parameter α.E(α) as a function of α gives a fractal energy spectrum, where 0 < p/q < 1 are used

to approximate α and −4 ≤ E ≤ 4 [59].

While the rational case is essentially like the original case of [44], in the case ofirrational parameters new phenomena arise, which reflect the different structure ofthe spectrum (Cantor sets instead of bands).

We first show that the analog of the Bloch variety, which describes the complexenergy-crystal momentum dispersion relation, is no longer defined by a finite setof polynomial equations but by a countable family of such equations. Thus, it is nolonger an algebraic variety, but we can still describe it as an inductive limit of algebraicvarieties (an ind-variety) by a procedure that essentially amounts to working with anapproximating family of algebraic varieties obtained by considering only finitely manycomponents at a time, with compatibility conditions.

As in [44], one defines the Bloch variety associated to the Harper operator (4.5)as

B := (ξ1, ξ2, λ) ∈ C∗ × C∗ × C |Hψ = λψ,ψ(m+ a, n) = ξ1ψ(m,n), ψ(m,n+ b) = ξ2ψ(m,n)

(4.7)

This looks very similar to the original case (4.4) with trivial potential V ≡ 0. How-ever, in fact, the locus determined by (4.7) differs significantly from the case withoutmagnetic field, since in the irrational case it defines a countable collection of algebraicvarieties.

To see this, we first describe the spectral problem defining B as a countable col-lection of matrices acting on the vector ψ(m,n) ∈ Cab, when α and β are irrational.This becomes a finite family when both α and β are rational.

69

The correct notion of “limit of algebraic varieties” that we need to employ hereto describe the geometric properties of the resulting space is that of an ind-variety.We first recall the abstract definition and then we give a more heuristic descriptionof its meaning, before we apply it to our concrete and specific problem.

Definition 32. An ind-variety over C is a set X together with a filtration

X0 ⊆ X1 ⊆ X2 ⊆ · · ·

such that ∪n≥0Xn = X and each Xn is a finite dimensional algebraic variety with

the inclusion Xn → Xn+1 being a closed embedding. Such ind-variety X will also bedenoted by lim−→Xn. An ind-variety X = lim−→Xn is said to be affine (resp. projective)if each Xn is affine (resp. projective).

In categorical terms, an ind-variety X is a formal filtered colimit of an inductivesystem Xn of varieties. The notion of an ind-variety was first introduced by Sha-farevich and one can find more examples and properties of ind-varieties for examplein [75].

Let X and Y be two ind-varieties with filtrations Xn and Y n. A map f :X → Y is a morphism if and only if for every n ≥ 0, there exists a number m(n) ≥ 0such that f |Xn : Xn → Y m(n) is a morphism between varieties. Thus we get the Indcategory Ind(V ar) of the category of varieties.

With the notion of ind-variety, in our specific case we have the following fact.

Lemma 2. The Bloch variety (4.7) defined by the spectral problem of the Harperoperator (4.5) is an affine ind-variety that can be written as B = ∪k,ℓ∈ZBk,ℓ, where

Bk,ℓ = (ξ1, ξ2, λ) ∈ C∗ × C∗ × C |Pk,ℓ(ξ1, ξ2, λ) = 0, (4.8)

for polynomialsPk,ℓ(ξ1, ξ2, λ) = det(M (k,ℓ) − λI), (4.9)

with the ab by ab matrix M (k,ℓ) having entries

0 if m′ = m,n′ = ne−2πiα(n+ℓb) if m′ = m+ 1, n′ = ne2πiα(n+ℓb) if m′ = m− 1, n′ = ne−2πiβ(m+ka) if m′ = m,n′ = n+ 1e2πiβ(m+ka) if m′ = m,n′ = n− 1−ξ1 if m = 1, m′ = a, n′ = n−ξ−1

1 if m = a,m′ = 1, n′ = n−ξ2 if m = m′, n = 1, n′ = b−ξ−1

2 if m = m′, n = b, n′ = 10 otherwise

(4.10)

70

Proof. Consider the spectral problem Hψ(m,n) = λψ(m,n). Since we work with theboundary conditions ψ(m + a, n) = ξ1ψ(m,n) and ψ(m,n + b) = ξ2ψ(m,n), we canconsider the range where m = 1, . . . , a and n = 1, . . . , b. If α and β are irrationalnumbers, then the phase factors exp(2πiαn) and exp(2πiβm) are not periodic. Thismeans that, for each (m,n) in the chosen fundamental domain of the aZ⊕ bZ action,we have a collection of problems, parameterized by the choice of an element (k, ℓ)in Z2, which differ only in the presence of the phase factors exp(2πiα(n + ℓb)) andexp(2πiβ(m+ka)). In the case where α is rational, the phase factors exp(2πiα(n+ℓb))repeat periodically, with only finitely many distinct values, and so for exp(2πiβ(m+ka)), when β is rational. Thus, in the case where both α and β are rational, thereare only finitely many different varieties Bk,ℓ to consider, so their union is a genuinealgebraic variety. While in the case where at least one of the two parameters isirrational there are infinitely many components, so their union B is an ind-variety.For each pair (k, ℓ) then one can write the corresponding problem in the form given bythe matrix (4.10). In addition, by the form of the defining matrix, Bk,ℓ is symmetricunder the involution on each fiber, (ξ1, ξ2, λ) 7→ (ξ−1

1 , ξ−12 , λ).

In the following we just rewrite our varieties with a convenient change of coordi-nates coming from Fourier transform, which will be useful later when we deal withthe compactification and singularities problem.

Let the Fourier transform of ψ ∈ ℓ2(Z2) be ψ ∈ L2(R2/Z2), namely

ψ(k1, k2) =∑

(m,n)∈Z2

ψ(m,n)e2πi(mk1+nk2) =∑

(m,n)∈Z2

amnzm1 z

n2

where z1 := e2πik1 , z2 := e2πik2 and amn := ψ(m,n). Furthermore, the Fourier trans-form of Hψ is

Hψ(k1, k2) = e−2πik1ψ(k1, k2 − α) + e2πik1ψ(k1, k2 + α)

+e−2πik2ψ(k1 − β, k2) + e2πik2ψ(k1 + β, k2)(4.11)

i.e.

Hψ(z1, z2) =∑

(m,n)∈Z2

(e−2πiαnz−11 + e2πiαnz1 + e−2πiβmz−1

2 + e2πiβmz2)amnzm1 z

n2 (4.12)

Thus the spectrum of H is given by the Fourier modes in the k-space:

λ = e−2πiαnz−11 + e2πiαnz1 + e−2πiβmz−1

2 + e2πiβmz2

If we consider functions in ℓ2(Z2/aZ+ bZ), then by introducing (k, ℓ) ∈ Z2

λ = e−2πiα(n+ℓb)z−11 + e2πiα(n+ℓb)z1 + e−2πiβ(m+ka)z−1

2 + e2πiβ(m+ka)z2 (4.13)

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For a fixed pair (k, ℓ), we introduce the unramified covering

c : C∗ × C∗ × C→ C∗ × C∗ × C(z1, z2, λ) 7→ (za1 , z

b2, λ)

and the preimage Bk,ℓ := c−1(Bk,ℓ). So the covering c : Bk,ℓ → Bk,ℓ has the structuregroup µa × µb, where by µn we mean the group of roots of unity of order n, with theaction of ρ ∈ µa×µb on the fibers of the form ρ · (z1, z2, λ) = (ρ1z1, ρ2z2, λ). We writeP = P c.

For (m,n) ∈ Z2, zm1 zn2 consists of a basis for the functions in L2(T2). There isan obvious action of µa × µb on this basis ρ · zm1 zn2 := ρm1 ρ

n2z

m1 z

n2 , which is basically a

change of base, and can be naturally extended to an action on L2(T2).

Let ρ act on Hψ(z1, z2). It not only changes the basis from zm1 zn2 to ρm1 ρn2zm1 zn2 ,but it also changes the Fourier modes into

e−2πiα(n+ℓb)ρ−11 z−1

1 + e2πiα(n+ℓb)ρ1z1 + e−2πiβ(m+ka)ρ−12 z−1

2 + e2πiβ(m+ka)ρ2z2

Fix (ρ01, ρ02) = (e2πi/a, e2πi/b), other roots of µa and µb can be written as (ρ1, ρ2) =(ρp01, ρ

q02) for some integers 1 ≤ p ≤ a and 1 ≤ q ≤ b, then rewrite the Fourier modes

asρ−α(n+ℓb)a−p01 z−1

1 + ρα(n+ℓb)a+p01 z1 + ρ

−β(m+ka)b−q02 z−1

2 + ρβ(m+ka)b+q02 z2

Lemma 3. The Harper operator (4.5) determines a family of operators H(k,ℓ), for(k, ℓ) ∈ Z2, which acts as multiplication by the complex number

ρα(n+ℓb)a+p01 z1 + ρ

−α(n+ℓb)a−p01 z−1

1 + ρβ(m+ka)b+q02 z2 + ρ

−β(m+ka)b−q02 z−1

2 (4.14)

Proof. In addition to the above discussion, we also have to take care of the bound-ary conditions. Let us look at one of them ψ(m + a, n) = ξ1ψ(m,n). Taking theFourier transform on both sides gives e−2πiak1ψ(k1, k2) = ξ1ψ(k1, k2), or equivalentlyψ(z1, z2) = za1ξ1ψ(z1, z2). So the boundary conditions are removed naturally in thecovering space Bk,ℓ if we set ξ1 = z−a

1 and ξ2 = z−b2 . In fact, take the involution sym-

metry (ξ1, ξ2, λ) 7→ (ξ−11 , ξ−1

2 , λ) into account, we can just set ξ1 = za1 and ξ2 = zb2.

We write M (k,ℓ) for the diagonal ab× ab matrix with entries

M (k,ℓ)m,n (ρ, z) = ρ

α(n+ℓb)a+p01 z1+ρ

−α(n+ℓb)a−p01 z−1

1 +ρβ(m+ka)b+q02 z2+ρ

−β(m+ka)b−q02 z−1

2 (4.15)

Thus for pairs (k, ℓ), Bk,ℓ is the zero-set of Pk,ℓ(z1, z2, λ) = det(M (k,ℓ) − λI), orequivalently it is given by the zero locus

Bk,ℓ = (z1, z2, λ) |∏

m,n,ρ

(M (k,ℓ)m,n (ρ, z)− λ) = 0. (4.16)

For brevity, hereafter we denote := ((m,n), (ρ1, ρ2)) ∈ Za × Zb × µa × µb.

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Corollary 1. The lifted Bloch ind-variety B =⋃

(k,ℓ)∈Z2 Bk,ℓ =⋃

k,ℓ, Bk,ℓ, is a re-duced affine ind-variety and its components are nonsingular subvarieties

Bk,ℓ, := (z1, z2, λ) ∈ C∗ × C∗ × C | M (k,ℓ)m,n (ρ, z)− λ = 0 (4.17)

where (m,n) ∈ Za × Zb and (ρ1, ρ2) ∈ µa × µb.

Obviously we have a group action of µa×µb on each Bk,ℓ, by acting on each fiberρ · (z1, z2, λ) = (ρ1z1, ρ2z2, λ), in other words the group action changes the powers p, q

in M(k,ℓ)m,n (ρ, z) where ρ = (ρ1, ρ2) = (ρp01, ρ

q02).

We describe explicitly the singularities of the components of the Bloch ind-varietiesthat we need to deal with.

For fixed (k, ℓ) and , consider a typical fiber Eλ = Eλ(k, ℓ, ) of Bk,ℓ, given bythe set

(z1, z2)|λ = ρα(n+ℓb)a+p01 z1+ρ

−α(n+ℓb)a−p01 z−1

1 +ρβ(m+ka)b+q02 z2+ρ

−β(m+ka)b−q02 z−1

2 (4.18)

and take the derivatives formally

∂Eλ

∂z1= ρ

α(n+ℓb)a+p01 − ρ−α(n+ℓb)a−p

01 z−21

∂Eλ

∂z2= ρ

β(m+ka)b+q02 − ρ−β(m+ka)b−q

02 z−22

(4.19)

where we use the same notation Eλ for the variety and for the polynomial

Eλ = ρα(n+ℓb)a+p01 z1 + ρ

−α(n+ℓb)a−p01 z−1

1 + ρβ(m+ka)b+q02 z2 + ρ

−β(m+ka)b−q02 z−1

2 − λ

Then the singular points consist of four points (±ρ−α(n+ℓb)a−p01 ,±ρ−β(m+ka)b−q

02 ).We have an analog of Lemma 5.1 of [44]. When λ = 0, Eλ splits into two compo-

nents(z1, z2)|ρα(n+ℓb)a+p

01 z1 + ρβ(m+ka)b+q02 z2 = 0

(z1, z2)|ρα(n+ℓb)a+p01 z1 + ρ

−β(m+ka)b−q02 z−1

2 = 0(4.20)

When λ = 4 , Eλ is irreducible with (ρ−α(n+ℓb)a−p01 , ρ

−β(m+ka)b−q02 ) as the only singular

point.When λ = −4 , Eλ is irreducible with (−ρ−α(n+ℓb)a−p

01 ,−ρ−β(m+ka)b−q02 ) as its singular

point.Otherwise, when λ ∈ C\0,±4, the typical fiber is a nonsingular complex curve.

4.4 Monodromy of Fermi curves

Now we can identify the components of the Fermi curve associated to the Blochind-variety. It is these countably many components of the Fermi curve that willcontribute to the period computation, after taking care of the compactification andsingularities problem. In this section, we discuss the singularities and monodromiesof the Fermi curve.

73

Recall that affine Fermi curves are defined by Fλ(C) := π−1(λ) with the projectionπ : B → C; (ξ1, ξ2, λ) 7→ λ. Since B = B/µa × µb, Fλ(C) is given by

⋃

k,ℓ,m,n

(ξ1, ξ2)|λ = e2πiα(n+ℓb)ξ1 + e−2πiα(n+ℓb)ξ−11 + e2πiβ(m+ka)ξ2 + e−2πiβ(m+ka)ξ−1

2

(4.21)where 1 ≤ m ≤ a, 1 ≤ n ≤ b and (k, ℓ) running through Z2, so we have countablymany components F k,ℓ,m,n

λ for each λ. Then the Fermi curve itself is, in this case,

an ind-variety. Namely, for fixed integers (k, ℓ) and (m,n), the component F k,ℓ,m,nλ is

given by

(ξ1, ξ2)|λ = e2πiα(n+ℓb)ξ1 + e−2πiα(n+ℓb)ξ−11 + e2πiβ(m+ka)ξ2 + e−2πiβ(m+ka)ξ−1

2 (4.22)

and the singular locus of F k,ℓ,m,nλ is easily derived from that of Eλ.

When λ = 0, F k,ℓ,m,nλ has two components

(ξ1, ξ2)|e2πiα(n+ℓb)ξ1 + e2πiβ(m+ka)ξ2 = 0(ξ1, ξ2)|e2πiα(n+ℓb)ξ1 + e−2πiβ(m+ka)ξ−1

2 = 0 (4.23)

When λ = 4, F k,ℓ,m,nλ is singular only at (e−2πiα(n+ℓb), e−2πiβ(m+ka)), similarly when

λ = −4, F k,ℓ,m,nλ is singular only at (−e−2πiα(n+ℓb),−e−2πiβ(m+ka)) and they are irre-

ducible curves. In addition, these singularities are all ordinary double points.As in Lemma 5.1 of [44], the projective closure of each Fermi curve component

F k,ℓ,m,nλ in P1×P1, call it F k,ℓ,m,n

λ , is an elliptic curve for a generic λ, and the comple-

ment F k,ℓ,m,nλ \ F k,ℓ,m,n

λ is a divisor of type (2, 2). Then we have an elliptic fibrationof the projective closure of each Bk,ℓ,m,n, call it Bk,ℓ,m,n,

π : Bk,ℓ,m,n → P1, π−1(λ) = F k,ℓ,m,nλ

By the same statement as in [44], the above fibration gives rise to a stable family ofelliptic curves with four exceptional fibers, at λ = ±4 type I1, at λ = 0 type I2 andat λ = ∞ type I8, whose global monodromy group is Γ0(8) ∩ Γ0

0(4). The details canbe found in [10].

The local monodromy of the family of elliptic curves has been discussed in Propo-sition 8.1 of [44]. Here we have a similar result for the family of Fermi curve compo-nents.

Let δ be a vanishing cycle, recall that the Picard-Lefschetz formula discussed in§2.5 is

T (x) = x− (x · δ)δfor an arbitrary homology cycle x and x · δ is the intersection number between cycleswith chosen orientation, cf. [117].

Lemma 4. The local monodromies around λ = 4, 0,−4 are given by the Picard-Lefschetz transformations T4, T0, T−4:

T4(γ) = γ + δ1, T−4(γ) = γ − δ2, T0(γ) = γ − δ1 + δ2. (4.24)

where δ1, δ2 are vanishing cycles and γ is an arbitrary homology cycle.

74

Proof. First change the variables by ξ = e2πiα(n+ℓb)ξ1 and η = e2πiβ(m+ka)ξ2. Then theFermi curve components become

F k,ℓ,m,nλ = (e−2πiα(n+ℓb)ξ, e−2πiβ(m+ka)η) ∈ C∗ × C∗|ξ + ξ−1 + η + η−1 = λ

Rewrite the above curve as ξη2+(ξ2−λξ+1)η+ξ = 0, whose discriminant is ∆ = (ξ2−λξ+1)2−4ξ2. Set ∆ = 0, there exist four branch points (λ+2)±

√λ2+4λ

2, (λ−2)±

√λ2−4λ

2

on the ξ-plane.

When λ tends to 4, (λ−2)±√λ2−4λ

2shrink to be an ordinary double point at ξ = 1,

which corresponds to (ξ1, ξ2) = (e−2πiα(n+ℓb), e−2πiβ(m+ka)), the cycle around (λ−2)±√λ2−4λ

2

is a vanishing cycle, call it δ1. Assume γ is a cycle such that δ1·γ = 1, we rotate δ1 once

and interchange the points (λ−2)+√λ2−4λ

2←→ (λ−2)−

√λ2−4λ

2, by the Picard-Lefschetz

transformation T4(γ) = γ + δ1.

When λ tends to −4, (λ+2)±√λ2+4λ

2shrink to be an ordinary double point at ξ =

−1, which corresponds to (ξ1, ξ2) = (−e−2πiα(n+ℓb),−e−2πiβ(m+ka)), the cycle around(λ+2)±

√λ2+4λ

2is a vanishing cycle, call it δ2. The homology cycle γ is the same cycle as

above, so δ2 · γ = −1, we rotate δ2 once and interchange the points (λ+2)+√λ2+4λ

2←→

(λ+2)−√λ2+4λ

2, by the Picard-Lefschetz transformation T−4(γ) = γ − δ2.

When λ tends to 0, (λ+2)±√λ2+4λ

2shrink to be one ordinary double point at

ξ = 1 corresponding to (ξ1, ξ2) = (e−2πiα(n+ℓb),−e−2πiβ(m+ka)) and at the same time(λ−2)±

√λ2−4λ

2shrink to be the other ordinary double point at ξ = −1 corresponding

to (ξ1, ξ2) = (−e−2πiα(n+ℓb), e−2πiβ(m+ka)). The cycles δ1, δ2, γ are the same as above,so by the Picard-Lefschetz transformation T0(γ) = γ − δ1 + δ2.

4.5 Blow-ups and compactification

In this more technical section we explicitly describe the form of the compact-ification of the components of the Bloch ind-variety and the compatibility of thiscompactification operation with the limit procedure.

We describe the blowup procedures that take care of the singularities problem andagain check that these can be carried out compatibly with the operation of passing tothe limit. Performing both of these operations will create a more complicated “doublelimit” procedure, which can be appropriately described, with all the compatibilityconditions directly encoded, by the notion of an ind-pro system of varieties. Theresulting double limit obtained in this way is called an ind-pro-variety and is thenthe geometric space that we need to deal with, which describes the complex energy-crystal momentum dispersion relation in the case of the Harper and almost Mathieuoperators with irrational parameters.

By adding points at infinity, first we have B1 as the projective closure of B inP1×P1×P1. Suppose s, t belong to the set 0,∞, it is easy to see that the intersection

75

B1

⋂(P1 × P1 × P1 \ C∗ × C∗ × C) consists of eight rational curves s × P1 × ∞,

P1 × t × ∞, s × t × P1 as in Lemma 4.1 of [44].B1 is singular at the points Os,t = (s, t,∞), after blowing up these four points, we

define B2 as the strict transform of B1 and the group action of µa × µb can be liftedonto B2 naturally.

Let P s,t be the exceptional divisor over Os,t, the intersection points of P s,t with the

strict transforms of eight rational curves are denoted by ws,t0 ∈ P s,t∩s × t × P1 \Os,t,

ws,t1 ∈ P s,t∩s × P1 × ∞ \Os,t and w

s,t2 ∈ P s,t∩P1 × t × ∞ \Os,t. Obviously

ws,t0 , ws,t

1 , ws,t2 are fixed points of µa × µb.

We then have a direct analog of Lemma 4.2 of [44] adapted to our inductive systemof varieties. The main difference lies in the fact that here we deal with a countablefamily of quadrics arising from the blowups instead of having just ab of them as inthe original case of [44].

Lemma 5. There exist quadrics Qs,tk,ℓ, ⊂ P s,t containing ws,t

0 , ws,t1 , ws,t

2 satisfying

that there are neighborhoods Ui of ws,ti such that Ui ∩ B2 consists of countably many

branches.

Proof. We only consider what happens around the point (0, 0,∞), other cases can betreated similarly.

Using the coordinate system z1, z2, µ = λ−1 in a neighborhood of (0, 0,∞), fixs = t = 0, then w00

0 = (0, 0, 1), w001 = (0, 1, 0), w00

2 = (1, 0, 0).Recall that the closure of the lifted Bloch variety B1 is the zero-set of the deter-

minant of the matrices µz1z2(M(k,ℓ)m,n (ρ, z)− λI) with diagonal entries

ρα(n+ℓb)a+p01 µz21z2+ρ

−α(n+ℓb)a−p01 µz2+ρ

β(m+ka)b+q02 µz1z

22 +ρ

−β(m+ka)b−q02 µz1−z1z2 (4.25)

Define

Qk,ℓ, := (z1 : z2 : µ) ∈ P 00 | ρ−α(n+ℓb)a−p01 µz2 + ρ

−β(m+ka)b−q02 µz1 − z1z2 = 0 (4.26)

therefore P 00 ∩ B2 is the union of infinitely many quadrics Qk,ℓ,. In addition, B2 isnonsingular at every point not being a point of intersection of different quadrics.

Let us look at w000 first, from the tangent part ρ

−α(n+ℓb)a−p01 z2+ρ

−β(m+ka)b−q02 z1, there

are infinitely many transversal branches labeled by (k, ℓ), (m,n) and (p, q) intersectingat L := z1 = z2 = 0.

Then consider w001 , since w00

1 = (0, 1, 0) in the coordinates z1, z2, µ, introduceν 6= 0, u1, u2 such that

z1 = u1ν, z2 = ν, µ = u2ν

And now B2 is given by the determinant of the infinite matrix with diagonal entries

ρα(n+ℓb)a+p01 u21u2ν

2+ρ−α(n+ℓb)a−p01 u2+ρ

β(m+ka)b+q02 u1u2ν

2+ρ−β(m+ka)b−q02 u1u2−u1 (4.27)

So we get a singular line L′ := u1 = u2 = 0 and the tangent cone of B2 at L′ is

the union of infinitely many planes ρ−α(n+ℓb)a−p01 u2 − u1 = 0.

76

Blow up the line L′ by defining new coordinates

u1 = v1v2, u2 = v2

The strict transform of B2 then is given by the determinant of the infinite matrixwith diagonal entries

ρα(n+ℓb)a+p01 v21v

22ν

2 + ρ−α(n+ℓb)a−p01 + ρ

β(m+ka)b+q02 v1v2ν

2 + ρ−β(m+ka)b−q02 v1v2 − v1 (4.28)

In other words,

ρ−α(n+ℓb)a−p01 − v1 + ρ

−β(m+ka)b−q02 ρ

−α(n+ℓb)a−p01 v2

−ρ−β(m+ka)b−q02 (ρ

−α(n+ℓb)a−p01 − v1)v2 + higher corrections

(4.29)

We get countably many singular lines Lℓ,n,p := v1 = ρ−α(n+ℓb)a−p01 , v2 = 0 and the

tangent cone at each Lℓ,n,p is (ρ−α(n+ℓb)a−p01 − v1) + ρ

−β(m+ka)b−q02 ρ

−α(n+ℓb)a−p01 v2, i.e. it

has countably many branches as claimed.Finally consider w00

2 = (1, 0, 0), analyze similarly to w001 and the strict transform

B2 is determined by entries

ρ−β(m+ka)b−q02 − v1 + ρ

−α(n+ℓb)a−p01 ρ

−β(m+ka)b−q02 v2

−ρ−α(n+ℓb)a−p01 (ρ

−β(m+ka)b−q02 − v1)v2 + higher corrections

(4.30)

Countably many singular lines Lk,m,q := v1 = ρ−β(m+ka)b−q02 , v2 = 0 are obtained in

this case as well. Let Lk,ℓ, denote these singular lines Lℓ,n,p and Lk,m,q together.

By Bezout’s theorem Qs,tk,ℓ, and Q

s,tk′,ℓ′,′ have a point of intersection ds,t other than

ws,t0 , ws,t

1 , ws,t2 if and only if (ℓ, n, p) 6= (ℓ′, n′, p′) and (k,m, q) 6= (k′, m′, q′). If such

ds,t does exist, then it can be proved that they are ordinary double points of B2 as in[44].

As in the above lemma (5), we blow up the strict transforms of the rational curvess × t × P1, s × P1 × ∞ and P1 × t × ∞, and let B3 be the stricttransform of B2. Then B3 is nonsingular on the strict transforms of the rationalcurves s × t × P1, since all branches are transversal to each other in this case.On the other hand, it is singular at countably many lines which lie over the stricttransforms of e×P1×∞ and respectively of P1×f×∞. We now take careof these singular lines Lk,ℓ,.

In order to do this, we first need to clarify how we are going to proceed at per-forming the “double limit” that will result when we keep adding more and morecomponents to the Bloch variety (constructing the ind-variety) and at the same timeblowing up all these singular lines, in a compatible way. The double limit procedurewe need is encoded in the notion of an ind-pro-system of algebraic varieties. Again,we first recall the general abstract definition of this procedure, then we give a more

77

heuristic interpretation of its meaning, and then we apply it to our concrete case toobtain explicitly the construction of the compactification and resolution of the Blochvariety in the irrational case.

Definition 33. An ind-pro system of varieties Xni is a double indexed set of vari-

eties such that for each i ≥ 0, Xni is an inductive system while for each n ≥ 0, it

is a projective system and every square of the system is Cartesian, i.e. the horizontalmaps are injections and the vertical ones are surjections, the diagram commutes andthat the top-left corner is the fibered product of the bottom and right map.

Xmi

πmij

// Xni

πnij

Xmj

// Xnj

Using categorical dual notions, we define a pro-object of the category Ind(V ar) asa formal cofiltered limit of a projective system of ind-varieties. Further, we can con-struct the category ProInd(V ar), or Proℵ0Indℵ0(V ar) if the index sets are countable.

Given an ind-pro system of varieties Xni , taking the inductive limit and the pro-

jective limit gives the ind-pro-variety lim←−ilim−→n

Xni , which is an object in Proℵ0Indℵ0(V ar).

What this abstract definition is saying in more heuristic terms is that we are build-ing up a space by performing simultaneously two kinds of operations on a collectionof algebraic varieties. One type of operation is the one we have already seen whendiscussing the ind-limit, namely we organize our varieties in a sequence of inclusionsthat progressively adds more and more components. The other operation takes careof the blowups that progressively remove singularities (these form a projective system,since the smoother blowups map down by projections to the more singular varietiesthey are obtained from). The problem here lies in the fact that the required sequenceof blowups needs to be performed on an infinite number of components that intersecteach other, and the way to make this possible is by compatibly carrying out blowupson the approximating varieties of the inductive system. Notice that, to make sense ofthis double limit, we only need to use the very simple categorical notions of inductiveand projective limit. In fact, the categorical definition above is simply encoding thecompatibility condition that makes this kind of double limit operation possible: it issaying that operations we perform at the level of the approximating algebraic vari-eties Xn

i , as long as they are done in a way that is compatible with the maps betweenthe varieties that define the two limit operations, will carry over to the limit, eventhough the space we obtain as the double limit of this sequence of algebraic varietiesis no longer an algebraic variety but a Cantor-like geometry, which cannot be directlydescribed within classical algebraic geometry.

We now return to our specific case of the Bloch varieties for the Harper operatorin the irrational case.

78

Proposition 3. In the case of irrational parameters, the compactification of the liftedBloch ind-variety, is an ind-pro-object defined by a chain of iterated blowups on theind-variety B.

Proof. For fixed (k, ℓ), (m,n) and (p, q), we can blow up the line Lℓ,n,p or Lk,m,q asusual. Let us look at Lℓ,n,p here. By introducing new coordinates w1, w2 such that

v1 − ρ−α(n+ℓb)a−p01 = w1w2, v2 = w2 (4.31)

the defining equation becomes

ρ−α(n+ℓb)a−p01 ρ

−β(m+ka)b−q02 − w1 + ρ

−α(n+ℓb)a−p01 ρ

β(m+ka)b+q02 ν2 + ρ

−β(m+ka)b−q02 w1w2

+ρ−α(n+ℓb)a−p01 w2ν

2 + 2w1w22ν

2 + ρβ(m+ka)b+q02 w1w2ν

2 + ρα(n+ℓb)a+p01 w2

1w32ν

2

(4.32)First fix (k, ℓ) = (0, 0) and let = (m,n, p, q) vary, we get 2ab singular lines

L0,n,p, L0,m,q, then blow up these 2ab lines, define A0 as the strict transform of B3.We start over this process again, now let |k| ≤ 1, |ℓ| ≤ 1 and vary, blow up these 6absingular lines

⋃|k|≤1,|ℓ|≤1Lk,l,, define A1 as the strict transform of B3. In general,

we have Ai being the strict transform of B3 after blowing up 2(2i+1)ab singular lines⋃|k|≤i,|ℓ|≤iLk,l, for |k| ≤ i, |ℓ| ≤ i and all .Obviously, the projection maps πij : Ai → Aj are surjective for any i ≥ j ≥ 0.

In other words, we have a projective system of ind-varieties since each Ai is an ind-variety derived from the blowup σ : Ai → B3 and B3 being an ind-variety. Hence,the projective limit A = lim←−Ai is an ind-pro-variety in Proℵ0Indℵ0(V ar). We havethe following commutative diagram

Aπi

~~

πj

BBB

BBBB

B

Ai

σ

@@@@

@@@@

πij// // Aj

σ~~

~~~~

~

B3

and it is easy to see our construction satisfies the Cartesian squares.

We call A the compactification of the lifted Bloch ind-variety B and denote it by¯B from now on.

What this result shows is that we can compatibly resolve the compactificationand singularities problem on each individual finite approximation to the Bloch ind-variety, in a way that allows us to pass to the limit. Now, as a consequence ofthis construction, we find that the space we obtained as the double limit of thiscompatible family of algebraic varieties is indeed a Cantor-like space, but one thathas a good approximation by algebraic varieties. In fact, Kapranov showed in [67]

79

(see also [96]) that the category of locally compact Hausdorff totally disconnectedspaces can be identified with a full subcategory of IndPro(Set0), which means thatgeometric spaces obtained by the kind of double limit procedure described above lookCantor-like. One can find a sketch of the proof in [96].

In particular, this means that the compactification of the Bloch variety of theHarper operator with irrational parameters is an ind-pro-variety, which is a Cantor-like geometric space, as one might have expected by thinking of it as the complexenergy-crystal momentum dispersion relation, in a case where the band structure inthe spectrum is replaced by a Cantor set. The important additional information thatthe result above gives us is the fact that this Cantor-like geometry admits a goodapproximation by algebraic varieties: this will be useful later, since it will allow usto reduce the calculation of the density of states for this Cantor-like geometry to asequence of terms that can be computed compatibly over the approximating algebraicvarieties, and that can therefore be identified explicitly with period integrals.

We now rephrase the previous result on the Bloch varieties in terms of Fermicurves, since these will be the curves over which the period calculation for the densityof states will take place.

We have the analog of Theorem 4.2 of [44]. The map π c : B → C extends to a

morphism π : ¯B → P1 and its fibers Fλ := π−1(λ) are called the lifted Fermi curves.

The inclusion map i : B → C∗×C∗×C gives rise to a morphism i : ¯B → P1×P1×P1.

Theorem 17.¯B \ B consists of countable curves

(i) sections of π, Σs,tk,ℓ, with i(Σs,t

k,ℓ,) = s × t × P1

(ii) quadrics Qs,tk,ℓ, with i(Qs,t

k,ℓ,) = s × t × ∞(iii) Hs

ℓ,n,p with i(Hsℓ,n,p) = s×P1×∞ and H t

k,m,q with i(Htk,m,q) = P1×t×∞

These curves meet transversally at only one point, except that ds,t are ordinary

double points of ¯B, every intersection point is a nonsingular point of the compactifi-

cation ¯B.

Taking the quotient by the structure group µa × µb, we get the compactification

B = ¯B/µa×µb of the Bloch ind-variety B. At the same time, we get two morphisms,the inclusion i : B → P1 × P1 × P1 and the projection π : B → P1, whose fibers arethe so-call compactified Fermi curves Fλ. Let Σs,t

k,ℓ,m,n, Qs,tk,ℓ,m,n, H

sℓ,n and H t

k,m in B

be the image under the quotient map, in addition, let ds,t be the image of ds,t in B.

Theorem 18. B \B is the union of countable curves,(i) Σs,t

k,ℓ,m,n with i(Σs,tk,ℓ,m,n) = s × t × P1, nonsingular points of B

(ii) Qs,tk,ℓ,m,n with ordinary double points at ds,t and nonsingular at all points of Qs,t

k,ℓ,m,n\⋃ds,t(iii) Hs

ℓ,n with i(Hsℓ,n) = s × P1 × ∞ and H t

k,m with i(H tk,m) = P1 × t × ∞,

nonsingular points of B.

80

4.6 Density of states as periods

In the above sections we have constructed the desired geometric space, the com-pactification of the Bloch ind-variety, which can be approximated by finite dimen-sional algebraic varieties. In this section, we will proceed to compute the densityof states on those approximating components for the Harper operator. The densityof states is indeed a period integral over Fermi curves, which is also related to theperiods of elliptic curves.

Now we assume |ξ1| = |ξ2| = 1, i.e. (ξ1, ξ2) = (e2πik1, e2πik2) for some (k1, k2) ∈(0, 1]2, then the Bloch ind-variety is given by

B = (e2πik1 , e2πik2 , λ) | Hψ = λψ,ψ(m+ a, n) = e2πik1ψ(m,n),ψ(m,n+ b) = e2πik2ψ(m,n)

(4.33)

Denote its spectrum by σ(H) := Ej(k1, k2), j ∈ N, the function Ej(k1, k2) is theso-called j-th band function.

In order to compute the density of states, we get back to a continuous model bytaking the limit of the lattice model.

Let Hn denote the Harper operatorH acting on ℓ2(Z2/anZ⊕bnZ) for some integern ≥ 1, later we will take the limit as n tends to infinity. It is easy to see that theeigenvalues of Hn are just given by

Ej(m1

n,m2

n)|1 ≤ m1, m2 ≤ n, j ≥ 1

To define the integrated density of states, we first count the number of eigenvaluesless than or equal to λ. Or equivalently, with the step function Θ(x), define

νn(λ) =∞∑

j=1

n∑

m1,m2=1

Θ(λ−Ej(m1

n,m2

n)) (4.34)

The integrated density of states is then defined as the limit

ρ(λ) = limn→∞

1

abn2νn(λ) (4.35)

and the density of states is defined as the derivative dρ/dλ. In the literature, theintegrated density of states can also be defined as the normalized trace on a II1factor in von Neumann algebra theory [105].

Rewrite it as an integration, we get an integral over the continuous variable k ∈[0, 1]2

ρ(λ) = limn→∞1ab

∑∞j=1

1n2

∑nm1,m2=1Θ(λ− Ej(

m1

n, m2

n))

= 1ab

∑∞j=1

∫I2Θ(λ− Ej(k))dk,

(4.36)

Lemma 6. The density of states can be expressed as dρdλ

= 1ab

∫λ∈σ(H)

ωλ, where ωλ is

a differential 1-form.

81

Proof.dρdλ

= 1abΣ∞

j=1

∫I2δ(λ−Ej(k))dk

= 1abΣ∞

j=1

∫Ej(k)=λ

ds|∇kEj |

= 1abΣ∞

j=1

∫Ej(k)=λ

√dk21+dk22√

∂1E2j+∂2E2

j

For fixed λ, from Ej(k1, k2) = λ, we have dEj =∂Ej

∂k1dk1 +

∂Ej

∂k2dk2 = 0,

dk1 = −∂2Ej

∂1Ejdk2 or dk2 = −

∂1Ej

∂2Ejdk1

Then

ds =ö1E

2j + ∂2E

2j

dk1|∂2Ej|

or ds =ö1E

2j + ∂2E

2j

dk2|∂1Ej |

Therefore we can write the density of states as

dρ

dλ=

1

ab

∞∑

j=1

∫

Ej(k)=λ

dk2|∂1Ej|

=1

ab

∞∑

j=1

∫

Ej(k)=λ

dk1|∂2Ej |

=1

ab

∫

λ∈∪jEj(k)

dk1|∂2Ej |

Denote the differential 1-form by ωλ, i.e. ωλ(k1, k2) = dk2|∂1Ej | =

dk1|∂2Ej | , the lemma is

proved.

Let P (ξ1, ξ2, λ) be a general polynomial with ξ1 = e2πik1 and ξ2 = e2πik2 . Thendk1 = dξ1/2πiξ1 and dk2 = dξ2/2πiξ2, we also have dλ = ∂1Ejdk1 + ∂2Ejdk2 fromλ = Ej(k1, k2). Plug into dP = Pξ1dξ1 + Pξ2dξ2 + Pλdλ = 0, we get

(2πiPξ1ξ1 + Pλ∂1Ej) dk1 + (2πiPξ2ξ2 + Pλ∂2Ej) dk2 = 0 (4.37)

Since dk1 and dk2 are independent, so

2πiPξ1ξ1 + Pλ∂1Ej = 02πiPξ2ξ2 + Pλ∂2Ej = 0

(4.38)

Lemma 7. The density of states is a period integral over Fermi curves.

Proof. Recall that the Bloch ind-variety B = ∪(k,ℓ)∈Z2 ∪a,bm,n=1 B(k,ℓ)m,n , where

B(k,ℓ)m,n = e2πiα(n+ℓb)ξ1 + e−2πiα(n+ℓb)ξ−1

1 + e2πiβ(m+ka)ξ2 + e−2πiβ(m+ka)ξ−12 − λ = 0

(4.39)and Fermi curves Fλ = ∪(k,ℓ)∈Z2 ∪a,bm,n=1 F

k,ℓ,m,nλ , where

F k,ℓ,m,nλ = e2πiα(n+ℓb)ξ1+e

−2πiα(n+ℓb)ξ−11 +e2πiβ(m+ka)ξ2+e

−2πiβ(m+ka)ξ−12 = λ (4.40)

P k,ℓ,m,n := e2πiα(n+ℓb)ξ1 + e−2πiα(n+ℓb)ξ−11 + e2πiβ(m+ka)ξ2 + e−2πiβ(m+ka)ξ−1

2 − λ denotethe polynomial, then

P k,ℓ,m,nλ = −1P k,ℓ,m,nξ1

= e2πiα(n+ℓb) − e−2πiα(n+ℓb)ξ−21

P k,ℓ,m,nξ2

= e2πiβ(m+ka) − e−2πiβ(m+ka)ξ−22

(4.41)

82

It follows that on each component F k,ℓ,m,nλ

∂1Ej = 2πiP k,ℓ,m,n

ξ1ξ1

∂2Ej = 2πiP k,ℓ,m,nξ2

ξ2(4.42)

Furthermore

dk2∂1Ej

= dξ2/2πiξ2

2πiP k,ℓ,m,nξ1

ξ1= 1

(2πi)2dξ2

ξ1ξ2Pk,ℓ,m,nξ1

dk1∂2Ej

= dξ1/2πiξ1

2πiP k,ℓ,m,nξ2

ξ2= 1

(2πi)2dξ1

ξ1ξ2Pk,ℓ,m,nξ2

(4.43)

i.e. ωλ(ξ1, ξ2) = dξ14π2|ξ1ξ2P k,ℓ,m,n

ξ2| = dξ2

4π2|ξ1ξ2P k,ℓ,m,nξ1

| . By the change of variables ξ1 =

e2πik1 , ξ2 = e2πik2 , the graph (k1, k2, Ej(k1, k2)) is changed into (ξ1, ξ2, λ). Then

dρ

dλ=

1

ab

∞∑

j=1

∫

Ej(k)=λ

ωλ(k1, k2) =1

ab

∑

(k,ℓ)∈Z2

a,b∑

m,n=1

∫

F k,ℓ,m,nλ

ωλ(ξ1, ξ2) =1

ab

∫

Fλ

ωλ

Notice how in this result we used explicitly the fact that the geometric spacedescribing the complex energy-crystal momentum relation for the Harper operatorwith irrational parameters admits a good approximation by a family of algebraicvarieties. In particular, we see here that the density of states is computed as a periodintegral with compatible contributions from each of the components F k,ℓ,m,n

λ of theFermi curve, as described in Theorem 17 above.

Recall then that the incomplete elliptic integral of the first kind is defined as

F (x, k) :=

∫ x

0

dt√(1− t2)(1− k2t2)

(4.44)

and the complete elliptic integral of the first kind is defined as

K(k) := F (1, k) =

∫ 1

0

dt√(1− t2)(1− k2t2)

(4.45)

The complete elliptic integral of the first kind is also called the quarter period.We rewrite ωλ on F k,ℓ,m,n

λ ∩ |ξ1| = |ξ2| = 1 as

ωλ =dξ1

4π2|ξ1ξ2(e2πiβ(m+ka) − e−2πiβ(m+ka)ξ−22 )| =

dk14π|sin2π(k2 + β(m+ ka))| (4.46)

Let C(k,ℓ)m,n := (k1, k2)|cos2π(k1 + α(n+ ℓb)) + cos2π(k2 + β(m+ ka)) = λ/2 so that

dρ

dλ=

1

4πab

∞∑

k,ℓ=−∞

a,b∑

m,n=1

∫

C(k,ℓ)m,n

dk1|sin2π(k2 + β(m+ ka))| (4.47)

83

Theorem 19. If we define the elliptic modulus as k = 4−|λ|4+|λ| , then the density of states

as a function of k is dρdλ|F k,ℓ,m,n

λ= 1

2π2ab(1+k)K(k) = 1

2π2abK(2

√k

1+k) on each component

of the Fermi curve.

Proof. Denote the definite integral I(k,ℓ)m,n (λ) =

∫C

(k,ℓ)m,n

dk1|sin2π(k2+β(m+ka))| for k1 ∈ I mov-

ing on the curve C(k,ℓ)m,n .

The differential form ωλ is well defined, and k1 6= −α(n + ℓb),−α(n + ℓb) ±1/2 (mod 2π), we can just assume 0 < −α(n+ ℓb) < 1/2.

Let k1 run through a half period of the sine function, i.e. 2π(k1+α(n+ℓb)) ∈ (0, π),

then I(k,ℓ)m,n (λ) is twice of this integral over a half period.

I(k,ℓ)m,n (λ) = 2

∫ 1/2

0d(k1+α(n+ℓb))√

1−(λ/2−cos2π(k1+α(n+ℓb)))2

= 2∫ 1/2

0dcos2π(k1+α(n+ℓb))√

1−(λ/2−cos2π(k1+α(n+ℓb)))2(−2πsin2π(k1+α(n+ℓb)))

= −1π

∫ 1/2

0dcos2π(k1+α(n+ℓb))√

1−(λ/2−cos2π(k1+α(n+ℓb)))2√

1−(cos2π(k1+α(n+ℓb)))2

= 1π

∫ 1

−1dx1√

1−(λ/2−x1)2√

1−x21

in the last line we set x1 := cos2π(k1 + α(n + ℓb)), here −1 ≤ x1 ≤ 1 and −1 ≤x1 − λ/2 ≤ 1, there are two cases:

(i) λ/2− 1 ≤ x1 ≤ 1 when λ ∈ (0, 4);(ii) − 1 ≤ x1 ≤ 1 + λ/2 when λ ∈ (−4, 0).In the first case, λ ∈ (0, 4)

I(k,ℓ)m,n (λ) = 1

π

∫ 1

λ/2−1dx1√

1−(λ/2−x1)2√

1−x21

x := x1 − λ/4 = 1π

∫ 1−λ/4

λ/4−1dx√

1−(λ/4−x)2√

1−(x+λ/4)2

= 1π

∫ 1−λ/4

λ/4−1dx√

((1−λ/4)2−x2)((1+λ/4)2−x2)

= 1π

∫ 1−λ/4

λ/4−1dx

(1−λ/4)(1+λ/4)√

(1−x2/(1−λ/4)2)(1−x2/(1+λ/4)2)

t := x/(1− λ/4) = 4(4+λ)π

∫ 1

−1dt√

(1−t2)(1−k2t2)

k := (4− λ)/(4 + λ) = 1+kπK(k)

In the second case, λ ∈ (−4, 0)

I(k,ℓ)m,n (λ) = 1

π

∫ 1+λ/2

−1dx1√

1−(λ/2−x1)2√

1−x21

x := x1 − λ/4 = 1π

∫ 1+λ/4

−1−λ/4dx√

1−(λ/4−x)2√

1−(x+λ/4)2

t := x/(1 + λ/4) = 4(4−λ)π

∫ 1

−1dt√

(1−t2)(1−k′2t2)

k′ := (4 + λ)/(4− λ) = 1+k′

πK(k′)

84

By identifying k = k′, namely, defining k = 4−|λ|4+|λ| , we conclude that I

(k,l)m,n (λ) =

1+kπK(k) and the density of states on each component F k,ℓ,m,n

λ turns out to be

1 + k

2π2abK(k) =

1

2π2abK(

2√k

1 + k)

In the above, we used some facts about the ascending Landen transformation:

k1 =2√k0

1 + k0

and the corresponding descending Landen transformation:

k−1 =1− k01 + k0

where 0 < k0 < 1 is a modulus of elliptic functions. In our case,

k−1 =|λ|4, k0 = k, k1 =

2√k

1 + k

We have K(k1) = (1 + k0)K(k0) and 2K ′(k1) = (1 + k0)K′(k0). Hence we can

easily see that

2 · K′1

K1

=K ′

K

In other words, the complex structure is changed under the ascending Landen trans-formation by:

τ1 =τ

2where τ =

iK ′

K

By the way, the arithmetic-geometric mean is a useful way to compute the quarterperiod:

K(k) =π/2

agm(1− k, 1 + k).

Interchanging positive and negative λ corresponds to the electron-hole symmetryin solid state physics. We give a representation of the density of states in terms ofelliptic curves.

Corollary 2. On each component of the Fermi curve the density of states is a sumof two half-periods of isomorphic elliptic curves up to a constant, namely dρ

dλ|F k,ℓ,m,n

λ=

18π2ab

(∫γωk +

∫γ′ ω1/k).

85

Proof. First we rewrite the density of states on each component as

dρdλ|F k,ℓ,m,n

λ= 1

4π2ab

∫ 1

−1dt√

(1−t2)(1−k2t2)+ k

4π2ab

∫ 1

−1dt√

(1−t2)(1−k2t2)

u := kt = 14π2ab

∫ k

−kdu√

(k2−u2)(1−u2)+ 1

4π2ab

∫ 1

−1dt√

(1−t2)(1−k2t2)/k2

= 14π2ab

∫ k

−kdt√

(t2−1)(t2−k2)+ 1

4π2ab

∫ 1

−1dt√

(t2−1)(t2−1/k2)

For the elliptic modulus k ∈ (0, 1), consider the elliptic curves given by

Ek := (t, y)|y2 = (t2 − 1)(t2 − k2)E1/k := (t, y)|y2 = (t2 − 1)(t2 − 1/k2)

If we denote the canonical holomorphic form ω = dt/y by ωk for Ek (resp. ω1/k forE1/k), then

dρ

dλ|F k,ℓ,m,n

λ=

1

4π2ab(

∫ k

−k

ωk +

∫ 1

−1

ω1/k)

Recall that for Ek, we cut the Riemann sphere from k to 1 and from −1 to −k,then assemble a Riemann surface homeomorphic to the torus T2. Similarly for E1/k,we cut the Riemann sphere from 1 to 1/k and from −1/k to −1, also assemble aRiemann surface homeomorphic to the torus T2.

A framed elliptic curve (E, δ, γ) is an elliptic curve with an integral basis for thefirst homology such that the intersection number δ · γ = 1. And for the period vector(∫δω,

∫γω), the ratio τ(E, δ, γ) =

∫δω/

∫γω is an invariant for isomorphic complex

tori, which is called the modulus of the isomorphism class.On Ek, we can choose δ as the cycle from k to 1 then from 1 back to k and γ from

−k to k then back to −k, which makes (Ek, δ, γ) a framed elliptic curve. Similarlyfor E1/k, we choose δ′ as the cycle from 1 to 1/k then back to 1 and γ′ from −1 to 1then back to −1.

By deforming the path of integration,∫δωk =

∫ 1

kωk+

∫ k

1ωk = 2

∫ 1

kωk and

∫γωk =∫ k

−kωk +

∫ −k

kωk = 2

∫ k

−kωk. Then

τ(Ek) =∫

δ ωk∫

γ ωk=

∫ 1k ωk

∫ k−k ωk

τ(E 1k) =

∫

δ′ ω1/k∫

γ′ω1/k

=∫ 1/k1 ω1/k∫ 1−1 ω1/k

It is easy to see that τ(Ek) = τ(E1/k), indeed∫ 1

−1ω1/k = k

∫ k

−kωk and

∫ 1k1ω1/k =

k∫ 1

kωk. In other words, we have elliptic curves Ek isomorphic to E1/k and we can

interchange between them by rescaling the fundamental domains.Hence on each component of the Fermi curve

dρ

dλ|F k,ℓ,m,n

λ=

1

8π2ab(

∫

γ

ωk +

∫

γ′

ω1/k) =1

8π2τab(

∫

δ

ωk +

∫

δ′ω1/k)

86

4.7 Picard-Fuchs equation of density of states

In the above section we identified the density of states (DOS) on each componentof the Fermi curve with a period integral independent of the labeling. In this section,we will give the Picard-Fuchs equation of the density of states, and use the Frobeniusmethod to derive a q-expansion of the energy level.

Recall that the density of states is of the form

DOS =1

2π2ab(1 + k)K(k) =

1

2π2abK(

2√k

1 + k)

where k = 4−|λ|4+|λ| is the elliptic modulus.

It is well-known that the quarter period K(k) satisfies the Fuchsian equation

k(1− k2)d2K

dk2+ (1− 3k2)

dK

dk− kK = 0,

with regular singularities at k = −1, 0, 1,∞. If we define

D(k) = 2π2ab DOS = (1 + k)K(k)

then we have the corresponding Picard-Fuchs equation.

Proposition 4. The density of states, or equivalently D(k), satisfies a second orderdifferential equation:

k(1− k)(1 + k)2d2D

dk2+ (1− 2k − k2)(1 + k)

dD

dk+ (k − 1)D = 0. (4.48)

We call (4.48) the Picard-Fuchs equation of the density of states.

Proof. Recall that the relation between K(k) and the complete elliptic integral of thesecond kind E(k) is the following.

dK

dk=

E(k)

k(1− k2) −K(k)

k,

dE

dk=E(k)−K(k)

k

We substitute and compute the first derivative of D(k):

dD

dk= (1 + k)

dK

dk+K(k) =

E(k)

k(1− k) −K(k)

k

In other words,

k(1− k)dDdk

= E(k) + (k − 1)K(k) (4.49)

Thend

dk[k(1− k)dD

dk] =

dE

dk+ (k − 1)

dK

dk+K(k)

87

k(1− k)d2D

dk2+ (1− 2k)

dD

dk=E −Kk

+ (k − 1)(E

k(1− k2) −K

k) +K =

E

1 + k

Here we use (4.49) again to cancel E(k), so

k(1− k)d2D

dk2+ (1− 2k)

dD

dk− k(1− k)

1 + k

dD

dk+

k − 1

(1 + k)2D = 0

And it is easy to see that the equation has regular singularities at k = −1, 0, 1,∞.

Instead of ddk, some authors prefer to use Θ = k d

dk. With this notation, we can

reformulate the Picard-Fuchs equation for the density of states in the following way.

Corollary 3. The Picard-Fuchs equation gives rise to a differential operator in termsof Θ as

Θ2 +2k

k2 − 1Θ− k

(1 + k)2(4.50)

Since the monodromy representation for solutions of the Picard-Fuchs equationis the same as the geometric monodromy representation, we consider the local mon-odromy of D(k) = K(2

√k/(1 + k)), i.e. the monodromy of

y2 = (t2 − 1)(4k

(1 + k)2t2 − 1)

The corresponding canonical holomorphic form is

ω =dt

y=

dt√(t2 − 1)( 4k

(1+k)2t2 − 1)

=ds√

(s2 − 1)(s2 − 4k(1+k)2

)

where s = −1/t.

Lemma 8. The local monodromy representations around k = 0, 1 are given by thePicard-Lefschetz transformations S0, S1,

S0(γ) = γ + δ1, S1(γ) = γ + 2δ2 − 2δ3, (4.51)

Proof. If k approaches 0, then ±2√k/(1 + k) shrink to 0, which gives an ordinary

double point at s = 0. So the cycle around ±2√k/(1 + k) is a vanishing cycle, call it

δ1. Let γ be a cycle intersects with δ1 so that δ1 ·γ = 1. Hence by the Picard-Lefschetzformula S0(γ) = γ + δ1.

If k approaches 1, then ±2√k/(1 + k) go to ±1, which gives two ordinary double

points at s = ±1. So the cycles around 2√k/(1 + k), 1 and −2

√k/(1 + k), − 1

are vanishing cycles, call them δ2, δ3, let γ be a cycle intersect with δ2 and δ3 suchthat δ2 · γ = 1 and δ3 · γ = −1. Similarly, by the Picard-Lefschetz formula S1(γ) =γ + 2δ2 − 2δ3.

88

By the relation k = 4−|λ|4+|λ| , if λ → ±4, then k → 0, and this gives an ordinary

double point as in §4.4; if λ→ 0, then k → 1, and this also gives rise to two ordinarydouble points as before.

Notice that, in the elliptic curve model of two dimensional Ising model, the pointk = 1 corresponds to the critical temperature of the Ising model.

Monodromy of the Picard-Fuchs equation of density of states means that if wego around some singularity then the period solutions (density of states) will changeaccording to the monodromy representation, which forces us to consider the ratioof the period solutions modulo the monodromy group. In the following we changethe Picard-Fuchs equation (4.48) into its Q-form and consider the related Schwarzianequation.

Lemma 9. By change of variable D = UV , the Picard-Fuchs equation is equivalentto

d2U

dk2+

(1 + k2)2

4k2(1− k2)2U = 0 (4.52)

the corresponding differential operator is

Θ2 +k2

(1− k2)2 (4.53)

Proof. Based on (4.48), first define V such that

d

dklnV = −1− 2k − k2

2k(1− k2) , i.e. V ′ =k2 + 2k − 1

2k(1− k2) V

and the second derivative is

V ′′ =7k2 + 2k − 3

2k(1− k2) V ′ +V

k(1− k)

Then by substituting into the Picard-Fuchs equation and canceling out the middleterm and V , it is easy to get the Q-from, where

Q(k) =(1 + k2)2

4k2(1− k2)2 (4.54)

Hence let D1(k), D2(k) be two linearly independent period solutions of the Picard-Fuchs equation (4.48), and let U1(k), U2(k) be the corresponding solutions of theQ-from (4.52), define t = D2/D1 = U2/U1, then t satisfies the Schwarzian equation

t, k = (1 + k2)2

2k2(1− k2)2 . (4.55)

89

This equation characterizes the mirror map k = k(t), now we derive a q-expansion ofthe energy level based on the Frobenius method, where q = et.

The Picard-Fuchs equation (4.48) is equivalent to

d2D

dk2+

1

k

1− 2k − k21− k2

dD

dk+

1

k2−k

(1 + k)2D = 0. (4.56)

which has a regular singularity at k = 0. We want to find out two linearly independentsolutions around 0. If we set

p(x) =1− 2k − k2

1− k2 , q(x) =−k

(1 + k)2

then the indicial equation is given by

r(r − 1) + rp(0) + q(0) = r2 = 0.

It has repeated roots r = 0, which means the local solutions are given by

D1 =∞∑

n=0

dnkn, D2 = D1ln k +

∑

n=1

cnkn

by the Fuchs’s theorem (1).

Lemma 10. Under the normalization D1(0) = 1, the local solutions are

D1 = 1 + k + 14k2 + 1

4k3 + 9

64k4 + 9

64k5 + 25

256k6 + 25

256k7 + · · ·

D2 = D1ln k +14k2 + 1

4k3 + 21

128k4 + 21

128k5 + 185

1536k6 + 185

1536k7 + · · · (4.57)

Proof.

D′1 =

∞∑

n=1

ndn kn−1, D′′

1 =∞∑

n=2

n(n− 1)dn kn−2,

Plug into the Picard-Fuchs equation, we get

∑n=0 dn(k

n+1 − kn) +∑

n=1 ndn(−kn+2 − 3kn+1 − kn + kn−1)+∑n=2 n(n− 1)dn(−kn+2 − kn+1 + kn + kn−1)

It is equivalent to

d1 − d0 + (d0 − d1)k − 3d1k2 − d1k3+∑

n=1[(n+ 1)2dn+1 − dn]kn + [(n2 − 1)dn+1 + dn]kn+1+∑

n=1−(n+ 3)(n+ 1)dn+1kn+2 − (n+ 1)2dn+1k

n+3

From this, we have

d1 = d0, d2 =14d0, d3 =

14d0,

(n+ 1)2dn+1 = [1 + 2n− n2]dn + (n2 − 2)dn−1 + (n− 2)2dn−2

90

It turns out that

d2l = d2l+1 =

[(2l − 1)!!

(2l)!!

]2· d0 (4.58)

If we apply the normalization so that D1(0) = 1, then d0 = 1.Then we look at D2 = D1ln k +

∑n=1 cnk

n,

D′2 = D′

1ln k +1kD1 +

∑n=1 ncn k

n−1,D′′

2 = D′′1 ln k +

2kD′

1 − 1k2D1 +

∑n=2 n(n− 1)cn k

n−2

Plug into the equation,

k(1− k)(1 + k)2[D′′1 ln k +

2kD′

1 − 1k2D1 +

∑n=2 n(n− 1)cn k

n−2]+(1− 2k − k2)(1 + k)[D′

1lnk +1kD1 +

∑n=1 ncn k

n−1]+(k − 1)[D1lnk +

∑n=1 cn k

n]

Cancel the terms with lnk since D1 satisfies the Picard-Fuchs equation, the otherterms can be used to determine cn.

2(1− k)(1 + k)2D′1 +

(k−1)(1+k)2

kD1 +

(1−2k−k2)(1+k)k

D1+(k − 1)

∑n=1 cn k

n + (1− 2k − k2)(1 + k)∑

n=1 ncn kn−1+

k(1− k)(1 + k)2∑

n=2 n(n− 1)cn kn−2

Use the power series of D1, this can be reduced to

∑n=0(2(n+ 1)dn+1 − 2dn + (n + 1)cn+1)k

n+(2(n+ 1)dn+1 − 2dn + (n+ 1)(n+ 2)cn+2 − (n+ 2)cn+1)k

n+1+(−2(n+ 1)dn+1 + (n+ 1)(n+ 2)cn+2 − (3n+ 2)cn+1)k

n+2+(−2(n+ 1)dn+1 − (n+ 1)(n+ 2)cn+2 − (n+ 1)cn+1)k

n+3+−(n + 1)(n+ 2)cn+2k

n+4

Then we have

c1 = 0, c2 =14, c3 =

14,

(n+ 1)2cn+1 = (1 + 2n− n2)cn + (n2 − 2)cn−1 + (n− 2)2cn−2

−2(n + 1)dn+1 − 2(n− 1)dn + 2ndn−1 + 2(n− 2)dn−2

If we do the long division, then∑

n=2 cnkn

∑n=0 dnk

n=

1

4k2 +

13

128k4 +

23

384k6 + · · · (4.59)

Exponentiate the above quotient, we have

exp(∑

n=2

cnkn/∑

n=0

dnkn) = 1 +

1

4k2 +

17

128k4 +

45

512k6 + · · · (4.60)

91

Theorem 20. The energy level has a q-expansion

|λ| = 4− 8q + 8q2 − 6q3 + 4q4 − 39

16q5 +

11

8q6 − · · · (4.61)

where q = eD2/D1.

Proof. Set t = D2/D1, we have the local coordinate q = et as a power series

q = k · exp(∑

n=2

cnkn/∑

n=0

dnkn) = k +

1

4k3 +

17

128k5 +

45

512k7 + · · · (4.62)

This is the mirror map q = q(k), its inverse is given by

k = q − 1

4q3 +

7

128q5 + · · · (4.63)

And the elliptic modulus k is defined by k = 4−|λ|4+|λ| . Hence

|λ| = 41−k1+k

= 41−q+ 1

4q3− 7

128q5+···

1+q− 14q3+ 7

128q5+···

= 4(1− 2q + 2q2 − 32q3 + q4 − 39

64q5 + 11

32q6 + · · · )

(4.64)

4.8 Spectral functions

With the density of states in hand, we can continue to derive some interestingspectral functions. In this section, we will calculate the partition function of thepropagating electron in magnetic field on the components of the Bloch ind-variety.

Fix (k, ℓ,m, n), we consider one irreducible component B(k,ℓ)m,n , other components

can be treated similarly. Recall that

B(k,ℓ)m,n = e2πiα(n+ℓb)ξ1 + e−2πiα(n+ℓb)ξ−1

1 + e2πiβ(m+ka)ξ2 + e−2πiβ(m+ka)ξ−12 − λ = 0

(4.65)For the moment, we omit the superscripts and denote the above defining polynomialas P (ξ1, ξ2, λ).

From dP = Pξ1dξ1 + Pξ2dξ2 + Pλdλ = 0, then on B(k,ℓ)m,n the pullback

π∗(dλ) = −Pξ1dξ1Pλ

− Pξ2dξ2Pλ

= Pξ1dξ1 + Pξ2dξ2

If we wedge this form with

Ωλ =1

(2πi)2dξ1

ξ1ξ2Pξ2

92

then

Ωλ ∧ π∗(dλ) =1

(2πi)2dξ1dξ2ξ1ξ2

(4.66)

This observation was already made in §11 of [44]. Note that Ωλ is slightly differentfrom ωλ. As defined before, ωλ is a volume form so it should be positive and we finallygot a positive period. By contrast, as for Ωλ, we get rid of the absolute value andtake the orientation into account.

Define Ω := Ωλ ∧ π∗dλ over B(k,ℓ)m,n such that Ω|F k,ℓ,m,n

λ= Ωλ, where F

k,ℓ,m,nλ is the

component of the Fermi curve of B(k,ℓ)m,n , Ω is called the density of states form and

then Ωλ is the relative differential form with respect to π∗dλ.

∫B

(k,ℓ)m,n|Ω| =

∫B

(k,ℓ)m,n|Ωλ ∧ π∗dλ|

=∫0<|λ|<4

dλ∫F k,ℓ,m,nλ

ωλ

= 14π2ab

∫0<|λ|<4

dλ(1 + k)K(k)

= 12π2ab

∫ 1

0(1 + k)K(k) 8

(1+k)2dk

= 4π2ab

∫ 1

0K(k)1+k

dk

= 12ab

Then it is possible to construct some interesting spectral functions based on dρfor each component. Let us look at the zeta function of the Harper operator on B

(k,ℓ)m,n ,

ζk,ℓ,m,nH (s) :=

∫

0<|λ|<4

λsdρ =

∫

B(k,ℓ)m,n

λs|Ω| (4.67)

∫0<|λ|<4

λsdρ =∫ 4

0λsdλ

∫F k,ℓ,m,nλ

ωλ +∫ 0

−4λsdλ

∫F k,ℓ,m,nλ

ωλ

= 22s+1

π2ab

∫ 1

0(1−k1+k

)s K(k)1+k

dk + 22s+1

π2ab

∫ 1

0(k−11+k

)sK(k)1+k

dk(4.68)

In particular, when s = 2k + 1 is an odd integer, ζk,ℓ,m,nH (2k + 1) = 0 and when

s = 2k is even, ζk,ℓ,m,nH (2k) = 42k+1

π2ab

∫ 1

0(1−k1+k

)2k K(k)1+k

dk.There is another way to compute the zeta function using two dimensional residue

theorem. Denote

W (ξ1, ξ2) := e2πiα(n+ℓb)ξ1 + e−2πiα(n+ℓb)ξ−11 + e2πiβ(m+ka)ξ2 + e−2πiβ(m+ka)ξ−1

2

namely, P (ξ1, ξ2, λ) = W (ξ1, ξ2)− λ. Since for any function f(ξ1, ξ2, λ),∫T2,0<|λ|<4

f(ξ1, ξ2, λ)δ(λ−W (ξ1, ξ2))dξ1dξ2dλ

=∫T2,0<|λ|<4,λ=W

f(ξ1, ξ2, λ)dσ

|∇P |=

∫T2,0<|λ|<4,λ=W

f(ξ1, ξ2, λ)dξ1dξ2+dξ1dλ+dξ2dλ

√

1+P 2ξ1

+P 2ξ2

=∫B

(k,ℓ)m,n

f(ξ1, ξ2, λ)1+Pξ1

+Pξ2√

1+P 2ξ1

+P 2ξ2

dξ1dξ2

Therefore

93

∫B

(k,ℓ)m,n

λs dξ1dξ24π2ξ1ξ2

=∫T2,0<|λ|<4

λsδ(λ−W )

√

1+P 2ξ1

+P 2ξ2

1+Pξ1+Pξ2

dξ1dξ2dλ4π2ξ1ξ2

=∫T2 W

s

√

1+P 2ξ1

+P 2ξ2

1+Pξ1+Pξ2

dξ1dξ24π2ξ1ξ2

By the residue theorem, only the positive integer powers survive and the contourintegral only depends on terms with 1, ξ1, ξ2, and ξ1ξ2 in the integrand.

Recall that using the zeta regularizatoin, one can define the Laplacian determinantby log det∆ = −ζ ′∆(0). In our context, this gives

log detHB

(k,ℓ)m,n

= − d

dsζk,ℓ,m,nH (s)|s=0 =

1

4π2

∫

T2

logWdξ1dξ2ξ1ξ2

(4.69)

After the change of variables ξ = e2πiα(n+ℓb)ξ1 and η = e2πiβ(m+ka)ξ2, we realize it asa special value of the Mahler measure

log detHB

(k,ℓ)m,n

= −m(ξ + ξ−1 + η + η−1) = 0 (4.70)

In other words, the product of band energies over each component of the Bloch ind-variety is given by a constant. At this point, I would like to thank Dr. Stienstra forpointing out this relation to Mahler measure [110].

We can also have the partition function of the Harper operator on B(k,ℓ)m,n ,

Zk,ℓ,m,nH (t) :=

∫

0<|λ|<4

e−tλdρ =1

2ab+

∞∑

k=1

ζk,ℓ,m,nH (2k)

(2k)!t2k

In order to get the spectral functions on the whole variety B, we should collect allthe contributions from countable components.

4.9 Almost Mathieu operator

We compare in this section the case of the density of states of the two-dimensionalHarper operator analyzed above with the analogous problem for the one-dimensionalalmost Mathieu operator. In particular, we show how to recover in the one-dimensionalcase the familiar picture of the Hofstadter butterfly and the corresponding density ofstates with its explicit dependence on the parameter.

We apply the same process to the almost Mathieu operator, the big difference isthat now its density of states depends on the parameter α. Therefore the derivedspectral functions will have totally different properties compared with those of theHarper operator.

First the Bloch variety of the almost Mathieu operator is now given by

B′ := (ξ, λ) ∈ C∗ × C | H ′ϕ(n) = λϕ(n), ϕ(n+ a) = ξϕ(n) (4.71)

94

It is easy to see that (ξ, λ) belongs to B′ if and only if λ is an eigenvalue of the matrixM ′ with countable components M ′ℓ, ℓ ∈ Z

M ′ℓ =

2cos(2πα(1 + ℓa)) 1 0 . . . −ξ1 2cos(2πα(2 + ℓa)) 1 . . . 0...

.... . .

......

−ξ−1 0 . . . 1 2cos(2πα(ℓ+ 1)a)

(4.72)In other words,

B′ = (ξ, λ) ∈ C∗ × C | det(M ′ − λI) =∏

ℓ∈Zdet(M ′ℓ − λI) = 0 (4.73)

In fact we can expand the determinant as det(M ′ℓ − λI) = pℓ(λ, α)− ξ − ξ−1,

pℓ(λ, α) := (−λ)a + (2

a∑

j=1

cos2πα(j + ℓa))(−λ)a−1 +O(λa−2) (4.74)

Consider a continued fraction expansion cn = pnqn to approximate the irrational

α. Then we let the prime a vary according to qn so that the spectrum of the Blochvariety can be approximated by numerical computation of finite band structures.Indeed, based on semi-classical analysis and renormalization method, the Hofstadterbutterfly is obtained by numerical approximations [13].

Define the unramified covering and B′ := c′−1(B′) as before

c′ : C∗ × C→ C∗ × C(z, λ) 7→ (za, λ)

The structure group µa of the covering c′ : B′ → B′ acts on the fibers as

ρ · (z, λ) = (ρz, λ)

Similarly, we use Fourier transform to convert the spectrum problem into the studyof the Fourier modes, then the almost Mathieu operator is represented by the infinitematrix

M ′ = Diag(ρz + ρ−1z−1 + 2cos2πα(j + ℓa)), 1 ≤ j ≤ a, ℓ ∈ Z (4.75)

So under this representation, the lifted Bloch variety

B′ = (z, λ) |∏

ℓ∈Z

∏

ρ∈µa

a∏

j=1

(ρz + ρ−1z−1 + 2cos(2πα(j + ℓa))− λ) = 0 (4.76)

and the Bloch variety

B′ = (ξ, λ) |∏

ℓ∈Z

a∏

j=1

(ξ + ξ−1 + 2cos2πα(j + ℓa)− λ) = 0 (4.77)

95

In this case, the Fermi curves are degenerate points

F ′λ =

⋃

ℓ∈Z

a⋃

j=1

ξ | ξ + ξ−1 + 2cos2πα(j + ℓa) = λ (4.78)

Let us look at the self-adjoint boundary value problem in dimension one. Fork ∈ I, we now consider

B′ := (e2πik, λ) | H ′ϕ(n) = λϕ(n), ϕ(n+ a) = e2πikϕ(n) (4.79)

Assume its band functions consist of Ei(k).Let H ′

n(n ≥ 1) denote the self-adjoint operator H ′ acting on l2(Z/anZ). Fork ∈ 1

nZ, Ei(k) is an eigenvalue of H ′

n and the spectrum of H ′n is

Ei(m

n)|i ≥ 1, 1 ≤ m ≤ n.

Similarly we have

νn(λ) =

∞∑

i=1

n∑

m=1

Θ(λ− Ei(m

n))

Then the integrated density of states of the almost Mathieu operator is

ρ′(λ) = limn→∞1a

∑∞i=1

1n

∑nm=1Θ(λ− Ei(

mn))

= 1a

∑∞i=1

∫IΘ(λ−Ei(k))dk

(4.80)

And the density of states of the almost Mathieu operator is

dρ′

dλ=

1

a

∞∑

i=1

∫

I

δ(λ− Ei(k))dk =1

a

∞∑

i=1

1

|g′i(ki)|(4.81)

where gi(k) := λ− Ei(k) and ki are the real zeros of gi(k). From the Fermi curve ofthe almost Mathieu operator, we have Ei(k) = Eℓ

j(k) for some j, ℓ

Elj(k) = 2cos2πk + 2cos2πα(j + ℓa) = λ (4.82)

i.e. ki ∈ I satisfies the equation 2cos2πki + 2cos2πα(j + ℓa) = λ. It is easy to seethat

g′i(k) = −d

dkEℓ

j(k) = 4πsin2πk

Hence

dρ′

dλ=

1

a

∞∑

i=1

1

4π|sin2πki|=

1

4πa

∑

ℓ∈Z

a∑

j=1

1√1− (λ/2− cos2πα(j + ℓa))2

Therefore the density of states of the almost Mathieu operator is a function ofthe parameter α. However, as we have seen that the density of states of the Harperoperator is independent of the parameters α and β.

96

Note that, besides the difference in dimension, the generators of the Harper opera-tor have a symmetric form in the variables and parameters, while the almost Mathieuoperator is naturally generated by translation and rotation operators.

In dimension two, the integral variable was absorbed by integrating over one-dimensional Fermi curves and the dependence of the parameters was resolved by thesymmetric form of the magnetic translations. But for the almost Mathieu operator,the Fermi curves are degenerate points and the associated measure is just the countingmeasure, so the dependence of the parameter still remains, as expected, in agreementwith the form of the density described for instance in [57].

Now we can similarly define the zeta function of the almost Mathieu operator oneach component B′

ℓ,j

ζℓ,jH′(s) :=

∫

B′ℓ,j

λsdξ

2πiξ=

∫

T

W ′sℓ,j

dξ

2πiξ=

∫

T

(ξ + ξ−1 + 2cos2πα(j + ℓa))sdξ

2πiξ(4.83)

By the residue theorem, only the positive integer powers survive and the contourintegral only depends on terms with 1 and ξ. So consider powers n ∈ N and expandthe polynomial as

(ξ + ξ−1 + 2cos2πα(j + ℓa))n =∑

k1,k2,k3

Ck1,k2,k3n ξk1−k2(2cos2πα(j + ℓa))k3

There are only two cases

(1) k1 − k2 = 0 and k1 + k2 + k3 = n (2) k1 − k2 = 1 and k1 + k2 + k3 = n

thus ζℓ,jH′(n) can be computed as ‘winding numbers’

ζℓ,jH′(n) =∑

n−2k≥0

n!(2cos2πα(j + ℓa))n−2k

(k!)2(n− 2k)!+

1

2πi

∑

n−2k−1≥0

n!(2cos2πα(j + ℓa))n−2k−1

k!(k + 1)!(n− 2k − 1)!

To get the zeta function of the whole variety B′, we have ζH′(n) =∑

ℓ∈Z∑a

j=1 ζℓ,jH′(n).

Finally, we also have the formal partition function of the almost Mathieu operator onB′

ZH′(t) :=

∫e−tλdρ′ =

∞∑

k=0

ζH′(k)

k!(−t)k

4.10 Conclusions and discussion

In the case of a discretized periodic Schrodinger operator describing electron prop-agation in solids, the complex energy-crystal momentum dispersion relation is de-scribed geometrically by an algebraic variety, the Bloch variety, which consists ofthe set of complex points that can be reached by analytic continuation of the band

97

functions. The density of states can then be computed [44] as a period on the Fermicurve in the Bloch variety.

In this work we consider the case with magnetic field, where the periodic Schrodingeroperator is replaced by the two-dimensional Harper operator, or its degeneration, theone-dimensional almost Mathieu operator. One knows from the spectral theory ofthese operators ([57], [78], [105]) that, in the case of irrational parameters, the bandstructure of the spectrum is replaced by a Cantor set, giving rise to the well knownHofstadter butterfly picture. Thus, one can see that, correspondingly, the geometriclocus describing the complex energy-crystal momentum dispersion relation and re-placing the Bloch variety will no longer be directly described by classical algebraicgeometry. However, as one approximates the Cantor-like spectrum by a family ofintervals, for example by approximating the irrational parameter by rationals via thecontinued fraction algorithm, it should be possible to correspondingly “approximate”this geometric space by ordinary algebraic varieties. The first part of this work con-sists of a geometric result, which shows exactly what this approximation and limitprocedure consists of. In particular, we show that one can obtain the space describ-ing the complex energy-crystal momentum dispersion relation for the Harper operatorwith irrational parameters as a “double limit” of a family of algebraic varieties, ormore precisely as an ind-pro-variety, where one limit takes care of the presence of in-finitely many components and the other limit of the blowups that are needed to dealwith the singularities. The resulting space has indeed a Cantor-like geometry, but onethat admits a good approximation by algebraic varieties, in the sense of this doublelimit procedure, so that one can still use methods from classical algebraic geometry,applied compatibly to the varieties in the approximating family.

We then use this geometric result to show that we can still obtain an explicitcalculation of the density of states for the two-dimensional Harper operator as a periodon the Fermi curve, where the period integral now consists of a sequence of compatiblecontributions from the components of the approximating system of varieties. Thisintegral is then explicitly computed in terms of elliptic integrals and periods of ellipticcurves. On each component, the period is the quarter period combined with theLanden transformation. Because of the electron-hole symmetry, the density of statesof the Harper operator takes the same form for positive and negative energy levels.We then find out the related Picard-Fuchs equation and Schwarzian equation, as ananalog of the mirror symmetry of elliptic curves, we express the energy level as aq-expansion in the spirit of a mirror map. Finally, we obtain explicit formulas for thespectral functions, again in terms of compatible contributions from the approximatingfamily of algebraic varieties.

We apply the same technique to the density of states and spectral functions in thecase of the one-dimensional almost Mathieu operator. The main difference betweenthe two cases is that, in the two-dimensional case the dependence on the parameterdisappears in the density of states, because it is absorbed in the integration over theFermi curve, while in the one-dimensional case the Fermi curves are points with thecounting measure and one recovers the density of states obtained, by different meth-

98

ods, in [57], with its explicit dependence on the parameter. By a residue calculation,we also obtain the zeta function of the almost Mathieu operator as a sum over “wind-ing numbers” associated to the components in the approximating family of algebraicBloch varieties.

Based on the q-expansion of the energy level, a natural question would be what isthe meaning of those coefficients. In moonshine theory, the coefficients of the mirrormap of the j-invariant are identified with character values of the graded representationof the Monster group. It would be very interesting to connect our q-expansion toconformal field theory and explain the coefficients by rational numbers with physicalmeaning.

Another important example in solid state physics is the elliptic curve model ofthe two dimensional Ising model, in which advanced tools (such as modular curves)have been used to study the Ising model, it would be interesting to introduce similartools to study the Fermi curves and the Bloch variety.

Although the algebro-geometric properties of the Bloch variety and Fermi curveshave been studied, there has been no motivic analysis of this family of varieties andof the corresponding periods. It would be interesting to look into the motivic pictureeven with some perturbation (non-zero potential V ) taken into account.

These days graphene is one of the hottest topics in solid state physics, in whicha Dirac equation is used to describe the motion of an electron in the system. Thestudy of Dirac points in graphene also shows similar cone-like singularities as that ofthe Bloch variety. It would be interesting to introduce more algebraic geometry andnoncommutative geometry into the study of graphene.

99

CHAPTER 5

PONZANO-REGGE MODEL AND

FEYNMAN MOTIVES

In perturbative quantum field theory, Feynman diagrams are heavily used to cal-culate probability amplitudes. The parametric representation of Feynman diagramsprovides a powerful tool to study the renormalizability and other analytical propertiesof Feynman graphs. Recently, researchers are also focusing on the relation betweenparametric representation of Feynman integrals and the theory of motives of algebraicvarieties [89].

A new noncommutative field theory is defined by a star product, its perturbativeexpansion gives the same generalized Feynman graph evaluation as the transitionamplitude from the Ponzano-Regge model [40].

We give a parametric representation of the effective noncommutative field theoryderived from a κ-deformation of the Ponzano-Regge model and define a generalizedKirchhoff polynomial with κ-correction terms, obtained in a κ-linear approximation.

Then consider the corresponding graph hypersurfaces and the question of how thepresence of the correction term affects their motivic nature. We look in particular atthe tetrahedron graph, which is the basic case of relevance to quantum gravity. Wecheck the number of Fq-rational points of this hypersurface for several finite fields, itturns out that the number of points does not fit polynomials with integer coefficients,hence the hypersurface of the tetrahedron is not polynomially countable. Since in theclassical case the graph hypersurface of the tetrahedron is known to be polynomiallycountable (and a mixed Tate motive), this shows that the presence of quantum gravitycorrections can significantly alter the motivic structure of the graph hypersurfaces.

The first part of this chapter reviews material related to Ponzano-Regge modelmostly following the book [100] and the paper [40], while the second part is based onthe author’s results, some of which have appeared in the paper [84].

5.1 Loop quantum gravity

Loop quantum gravity (LQG) provides a different approach to constructing aquantum theory of gravity other than string theory.

100

In Einstein-Cartan theory, Cartan recast general relativity based on the mathe-matical language of fiber bundles, where the concept of connection is more essentialthan that of metric (or distance).

Ashtekar first obtained a Hamiltonian formulation of general relativity, in whichthe configuration variable is a spinoral connection, so that the gravitational field isreplaced by a connection field, and the conjugate momentum variable is a coordinateframe at each point in spacetime. It is remarkable that the Ashtekar formulation isbackground independent based on the quantization by Wilson loops. Later, Rovelliand Smolin got an explicit basis of quantum states, i.e. solutions of the Wheeler-DeWitt equation describing loop excitations of the gravitational field, labeled byPenrose’s spin networks (as a generalization of Wilson loops).

In general, a spin network is a graph whose edges (or links) are associated withirreducible representations of a compact Lie groupG and whose vertices (or nodes) areassociated with intertwiners of the edge representations adjacent to it. By definition,an intertwiner between representation spaces V andW of a group G is a G-equivariantlinear transformation from V to W . Basically, a spin network represents a quantumstate in loop quantum gravity, and the set of all possible spin networks modulodiffeomorphisms gives a countable basis of the Hilbert space of quantum states.

Nodes of spin networks represent cells of space, and their volume is given by aquantum number in units of the Planck volume VP = ℓ3P . Recall that the Plancklength is defined as ℓP =

√~G/c3, where ~ is the reduced Planck’s constant, G

is the gravitational constant and c is the speed of light. Two adjacent nodes witha link connecting them are separated by an elementary surface, the area of whichis determined by the quantum number associated with that link. Link quantumnumbers, j, are integers or half-integers, and the area of the elementary surface isA = 8πγℓ2P

√j(j + 1), where γ is the so-called Immirzi parameter. Thus the spectra

of the area and volume operators are quantized in LQG.Dynamics of spacetime (such as curvature and gravitational waves in general rel-

ativity) are replaced by dynamics of spin networks: within a given graph new nodeswith new links can appear (or disappear) according to specific rules. Any evolutionof spin network provides a spin foam of one dimension higher, so a spin foam maybe viewed as the quantum history swept by a spin network. Spin foams are back-ground independent combinatorial objects representing a spacetime, which providean explicit tool to compute the transition amplitudes in quantum gravity.

Once the Immirzi parameter is fixed, one can derive the entropy of a black hole inLQG. This entropy is given by the Bekenstein-Hawking formula, S = AkB/4ℓ

2P , where

A is the area of the event horizon and kB is the Boltzmann’s constant. Moreover, theBig Bang singularity can be resolved in LQG.

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5.2 Ponzano-Regge model

Spin foam models provide a “path integral” formulation of quantum gravity basedon sum-over-paths with representations and intertwiners. The Ponzano-Regge modelis a spin foam model of 3d quantum gravity, whose representations are the unitaryirreducible representations of SU(2), intertwiners are trivial and vertex amplitudesare Wigner 6j symbols. This section closely follows chapter 9 in [100] and the lecturenotes [101] both by Rovelli.

A spin foam σ = (Γ, jf , ie) is a two-complex Γ with irreducible representations jfassociated to faces and intertwiners ie associated to edges. A spin foam model gives asum-over-paths formulation of quantum gravity, one considers the partition function

Z =∑

σ

w(Γ(σ))∏

f

dim(jf )∏

e

Ae(jf , ie)∏

v

Av(jf , ie) (5.1)

The first sum is over spin foams σ = (Γ, jf , ie), and w(Γ) is a weight factor depend-ing on the two complex Γ. The function Av(jf , ie), called vertex amplitude, is anamplitude associated to each vertex v, similarly for the edge amplitude Ae(ff , ie).

Consider Riemannian general relativity in 3d, which is described by a frame field

ei(x) = eia(x)dxa (5.2)

with values in R3 and a spin connection

ωi(x) = ωia(x)dx

a (5.3)

with values in the Lie algebra so(3).The action is defined by

S[e, ω] =

∫ei ∧ Fi[ω] (5.4)

where F [ω] = dω + ω ∧ ω is the curvature two-form.The equations of motion for pure gravity impose that the connection is flat and

the torsion vanishes,F [ω] = 0, T [ω, e] = Dωe = 0. (5.5)

This is actually a topological field theory which has no local degrees of freedom.We introduce the Ponzano-Regge model by discretizing the spacetime manifold,

and the discretization is the exact quantization since the theory is topological. Fixa triangulation ∆ of a 3d spacetime manifold and actually it is more convenient towork with the dual ∆∗ of the triangulation.

Recall some basic facts about the compact Lie group SU(2) first, every elementh ∈ SU(2) can be written as

h = expαiτi = cos(α2

)I2 + i sin

(α2

) αi

ασi (5.6)

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where τi = − i2σi with σi the Pauli matrices and α =

√αiαi < 2π is the rotation

angle of the SO(3) rotation corresponding to h ∈ SU(2). The group manifold ofSU(2) is the three sphere S3 : (x0)

2 + (x1)2 + (x2)

2 + (x3)2 = 1 with x0 = cos(α

2) and

xi = sin(α2)α

i

α. The standard Haar measure on SU(2) can be written as the invariant

measure on S3 : dh = d4x δ(|x|2 − 1).The irreducible unitary representations of SU(2) are labeled by a half-integer spin

number j = 0, 12, 1, 3

2, · · · . The representation space Hj has dimension dj = 2j + 1.

The standard basis that diagonalize τ3 is denoted by vm, m = −j, · · · ,+j. Therepresentation matrices are the Wigner matrices Di(h)mn . The spin-j character isdefined as χj(h) = tr[Dj(h)]. A key property of Wigner matrices is that they areorthogonal with respect to the Haar measure

∫

SU(2)

dh Dj′(h)m′

n′ Dj(h)mn =

1

djδjj

′

δmm′

δnn′. (5.7)

Let L2(SU(2)) be the space of square integrable functions on SU(2) with thestandard Haar measure. In terms of the Dirac notation Dj(h)mn =< h|j,m, n >, theabove orthogonality property can be written as

< j′, m′, n′|j,m, n >= 1

2j + 1δjj

′

δmm′

δnn′, (5.8)

the Wigner matrices form an orthogonal basis of L2(SU(2)). Since Dj : Hj → Hj,i.e. Dj ∈ H∗

j ⊗Hj, we have the decomposition

L2(SU(2)) = ⊕j(H∗j ⊗Hj) (5.9)

Given spins j1, j2, j3, Hj1 ⊗ Hj2 ⊗ Hj3 is the space of tensors vj1j2j3 with indicesin different representations. The invariant subspace of Hj1 ⊗ Hj2 ⊗ Hj3, denotedby Kj1j2j3 = Inv(Hj1 ⊗ Hj2 ⊗ Hj3), is formed by the invariant tensors, also calledintertwiners, satisfying

Dj1(h)m1n1Dj2(h)m2

n2Dj3(h)m3

n3vn1n2n3 = vm1m2m3 . (5.10)

The intertwiner space Kj1j2j3 is always one dimensional, for example, the fully anti-symmetric tensor vijk = εijk are intertwiners in K111 = H1 ⊗H1 ⊗H1.

Let ge be the holonomy of ω along each edge e of ∆∗, that is

ge = Pexp∫

e

ωiτi ∈ SU(2) (5.11)

Let lif ∈ R3 be the line integral of ei along a segment f of ∆, or equivalently a face fof ∆∗. Then the discretized action is

S[lf , ge] =∑

f

lif tr[gfσi] (5.12)

103

wheregf = gef1

· · · gefn (5.13)

is the product of the group elements associated to the edges ef1 , · · · , efn that boundthe face f .

Using this discretization, we define the path integral as

Z =

∫dlif dge e

iS[lf ,ge], (5.14)

integrating over lf gives

Z =

∫dge

∏

f

δ(gf) (5.15)

We can expand the delta function over the group manifold using

δ(g) =∑

j

djχj(g) (5.16)

where the sum is over all unitary irreducible representations of SU(2). Thus we have

Z =∑

j1,···jN

∏

f

djf

∫dge

∏

f

χjf (gf) (5.17)

There is one integral per edge and every edge bounds precisely three faces, eachintegral is then of the form∫

dhDj1(h)m1n1Dj2(h)m2

n2Dj3(h)m3

n3= vm1m2m3vn1n2n3 (5.18)

where vm1m2m3 is the unique normalized intertwiner between the representations ofspin j1, j2, j3.

Each of the two invariant tensors in the right hand side is associated to one of thetwo vertices that bound the edge. Its indices get contracted with the ones comingfrom the other edges at this vertex. At each vertex we have four of these tensorsgiving rise to a function of the six spins associated to the six faces that bound thevertex

6j =j1 j2 j3j4 j5 j6

=

∑

a1···a6va1a5a2va1a4a6va4a2a3va5a3a6 (5.19)

The pattern of the contraction of the indices reproduces the structure of a tetrahedron.The Wigner 6j symbol 6j is a well-defined function in the representation theory ofSU(2).

Bringing all together, we obtain the following form of the partition function of 3dgeneral relativity

ZPR =∑

j1···jN

∏

f

djf∏

v

6jv, (5.20)

which is called the Ponzano-Regge spin foam model. A remarkable result of Ponzanoand Regge is that the partition function ZPR depends on the global topology of the3d spacetime manifold but not on the triangulation ∆.

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5.3 Star product and noncommutative field

theory

In their paper [40], Freidel and Livine proposed an effective field theory to describe3d quantum gravity coupled with matter by introducing a new noncommutative starproduct. In this section, we will recall some properties of this star product and thenoncommutative field theory closely following [40]. More physical background about3d quantum gravity will be given in the next section.

First define the deformation parameter κ = 4πG, where G is the gravitationalconstant. Then the deficit angle θ = κm is resulted from the insertion of a particlewith mass m into spacetime. Viewed as a topological defect, the particle createsa conical singularity with deficit angle θ because a spinless particle is a source ofcurvature

F [ω] = 4πGpδ(x). (5.21)

To define the star product, one needs a new group Fourier transform mappingfunctions on SO(3) to functions on R3 bounded by κ−1,

F : C(SO(3)) → Cκ(R3)

φ(g) 7→ φ(X) =∫SO(3)

dgei2κ

tr(Xg)φ(g)(5.22)

where X ∈ so(3) ≃ R3 and tr(·) is the usual trace function on matrices. Then thedeformed ⋆-product of fields can be defined by Fourier modes

φ ⋆ ψ(X) =

∫dg1dg2e

i2κ

tr(Xg1g2)φ(g1)ψ(g2) (5.23)

In essential, it can be equivalently defined on phases

ei2κ

tr(Xg1) ⋆ ei2κ

tr(Xg2) = ei2κ

tr(Xg1g2) (5.24)

This star product is an associative noncommutative product different from the famousMoyal product. In addition, the inverse group Fourier transform is given by

φ(g) =

∫

so(3)

d3X

8πκ3φ(X) ⋆ e

i2κ

tr(Xg−1) (5.25)

Recall that elements g ∈ SO(3) can be expressed as

g = P4(g) + iκP i(g)σi, s.t. P 24 (g) + κ2P i(g)Pi(g) = 1, P4(g) ≥ 0 (5.26)

where σi are the Pauli matrices. Or equivalently, let ~P (g) = (P1(g), P2(g), P3(g)),then

g =

√1− κ2|~P (g)|2 + κ~P (g) · i~σ (5.27)

105

By the restriction P4 ≥ 0, the 3-vector ~P (g) belongs to a 3-Ball bounded above, i.e.~P (g) ∈ B3

κ = ~P : |~P | ≤ κ−1. Based on the isomorphism between so(3) ∼= su(2) =

spaniσ1, iσ2, iσ3 and the projection ~P (g) = tr(g~σ)/2iκ, one may write the trace asan inner product

ei2κ

tr(Xg) = ei2κ

2X·κP (g) = eiX·P (g) (5.28)

Thus the star product (5.24) can be rewritten in terms of Pi ≡ P (gi):

eiX·P1 ⋆ eiX·P2 = eiX·(P1⊕P2) (5.29)

here one introduces a new addition between 3-vectors in B3κ,

P1 ⊕ P2 = P1 + P2 − κP1 × P2 (5.30)

with × the usual cross product. One remark is in order, notice that the commutator(Lie bracket) in su(2) is the same as the cross product in R3, i.e.

[κP i1σi, κP

j2σj ] = 2iκ2P i

1Pj2 εijkσk, P1 × P2 = εijkP

i1P

j2 (5.31)

in a sense, (5.24) or (5.29) is a natural generalization of the Baker-Campbell-Hausdorffformula.

The action functional (interaction Lagrangian) was introduced in [40]:

S[φ] =

∫d3X

8πκ3[12(∂iφ ⋆ ∂iφ)(X)− 1

2

sin2mκ

κ2(φ ⋆ φ)(X) +

λ

3!(φ ⋆ φ ⋆ φ)(X)

](5.32)

or in the momentum representation

S[φ] =1

2

∫dg

(P 2(g)− sin2mκ

κ2)φ(g)φ(g−1)+

λ

3!

∫dg1dg2dg3δ(g1g2g3)φ(g1)φ(g2)φ(g3)

(5.33)This is our effective field theory describing the dynamics of the matter field afterintegrating out the gravitational sector. This action functional is invariant underthe action of a κ-deformed Poincare group [40]. We then can do the perturbativeexpansion as before and calculate the quantum corrections to the classical theoryusing Feynman diagrams.

5.4 3d gravity coupled with matter

We have seen the Ponzano-Regge model which is a pure quantum gravity theorywithout matter. In this section, follwoing [40] we recall the spin foam model in 3dgravity, still call it Ponzano-Regge model, with matter taken into account.

In order to compare with the usual Feynman diagram amplitude and parametricrepresentation in quantum field theory, we assume the 3d spacetime manifold is givenby S3 with trivial global topology. Fix a triangulation ∆ of S3 and consider an

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arbitrary oriented graph Γ ⊂ ∆ , in this work we only consider spherical graphsΓ ⊆ S2 embedded in a surface S2, for each edge e ∈ E(Γ) we insert a particle withdeficit angle θe.

With matter coupled to 3d quantum gravity, the partition function of the sphericalgraph Γ is a state sum model,

I∆(Γ, θ) =∑

je

∏

e/∈Γdje

∏

e∈Γχje(hθe)

∏

t

je1 je2 je3je4 je5 je6

(5.34)

where the sum is over spin je representations assigned to all edges e ∈ ∆ and theproduct of the 6j symbols is over all tetrahedra t in ∆. In addition, dj = 2j + 1 isthe dimension of the j-representation and the character χj(hθ) = sin djθ/ sin θ.

This amplitude I∆(Γ, θ) is independent of the triangulation ∆ and only dependson the topology of Γ, so we simply denote it as I(Γ, θ) and call it a (generalized)Feynman graph evaluation.

A simplest triangulation of S3 consists of two tetrahedra, one of the tetrahedragives a triangulation of the interior S2, the other one gives a triangulation of theexterior S2, and the three sphere is obtained by gluing the two 3-balls together.

The 6j symbol squared can be written as a group integral. For a tetrahedron,denote its vertices vi, i = 1, · · ·4 and its edges eij. Then in the simplest triangu-lation as above,

je34 je24 je23je12 je13 je14

2

=

∫ ∏

i

gi∏

i<j

χkl(gig−1j )

For a general graph Γ, one considers a more complicated triangulation of S3 andsimilarly replaces the 6j symbols by an integral over the group manifold.

After getting rid of the 6j symbols, we obtain the Feynman graph evaluation:

I(Γ, θ) =∫ ∏

e∈ΓdGe∆(θe)δθe(Ge)

∏

v

δ(Gv) (5.35)

In the above, we changed the variable Ge = gt(e)g−1s(e), that is, the group element

associated to each edge e ∈ Γ with t(e), s(e) being the target and source of e and the

ordered product of the edge group elements meeting at v is defined asGv =−→∏

e⊃vGǫv(e)e

where ǫv(e) = ±1 depends on whether v is the target or source vertex of e. In addition,∆(θ) = sin(θ) and the distribution δθ(g) is defined by

∫

SO(3)

dg f(g)δθ(g) =

∫

SO(3)/U(1)

dxf(xhθx−1)

where hθ =(eiθ 00 e−iθ

)generates the subgroup U(1) and dg, dx are normalized invariant

measures.

107

The delta function on SO(3) can be expanded as

δ(g) =1

8πκ3

∫

so(3)

d3X ei2κ

tr(Xg)

Thus the non-abelian Feynman graph evaluation reads,

I(Γ, θ) =∫ ∏

v∈Γ

d3Xv

8πκ3

∫ ∏

e∈ΓdGe∆(θe)δθe(Ge)

∏

v∈Γe

i2κ

tr(XvGv) (5.36)

One would like to split the exponentials exp i2κtr(Xv

∏eG

ǫv(e)e ) and write the eval-

uation as a standard Feynman amplitude. This can be done by the noncommutativestar product, i.e. we change the usual product by the star product in the last term.

I(Γ, θ) =∫ ∏

v∈Γ

d3Xv

8πκ3

∫ ∏

e∈ΓdGe∆(θe)δθe(Ge)

∏

v∈Γ(⋆e∈∂ve

i2κ

tr(XvGǫv(e)e )) (5.37)

With 2iκ~P (g) = tr(g~σ), or simply ~P ≡ ~P (g), the distribution δθ(g) in the mo-mentum space is

δθ(g) =π

2κ

cos θ

sin θδ(|P |2 − sin2 θ

κ2) (5.38)

The Hadamard propagator δ(|P |2 − sin2 θκ2 ) can be expressed as an improper time in-

tegral

δ(|P |2 − sin2 θ

κ2) =

∫ +∞

−∞

dT

2πeiT (|P |2− sin2 θ

κ2)

If one restricts the integral to be over positive time T ∈ R+ , one obtains the usualFeynman propagator.

With the Feynman propagator, we have the following Feynman graph evaluation:for a spherical graph Γ, the spin foam amplitude spells out as

I(Γ, θ) =∏

e∈Γ(cos θe4κ2

)

∫ ∏

v∈Γ

d3Xv

8πκ3

∫ ∏

e∈ΓdGe

i

|Pe|2 − sin2 θeκ2 + iǫ

∏

v∈Γ

(⋆e∈∂v e

iεveXv·Pe)

(5.39)Any element ge ∈ SU(2) can be written in terms of a Lie algebra element

ge ≡ eκPe·i~σ = cos(κ|~Pe|) + i sin(κ|~Pe|)~n · ~σ (5.40)

where ~n is the direction of the rotation and ~σ the Pauli matrices. Combine this withthe normalization ∫

dge =κ3

π2

∫

B3κ

d3Pe√1− κ2|Pe|2

(5.41)

up to some constant, one obtains the spin foam amplitude

I(Γ, θ) =∫

so(3)

∏

v∈Γd3Xv

∫

B3κ

∏

e∈Γd3Pe

i

|Pe|2 − sin2 meκκ2

∏

v∈Γ

(⋆e∈∂v e

iεveXv·Pe)

(5.42)

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where the vertices Xv are integrated over so(3) ∼= R3 and the momenta Pe are inte-grated over B3

κ, the 3-ball bounded by κ−1. This is the right amplitude that we willuse to derive the parametric representation.

It turns out that these spin foam amplitudes are exactly the generalized Feynmandiagram evaluations of the noncommutative field theory given in the previous section.In other words, the perturbative expansion of the noncommutative field theory (5.32)or (5.33) give the sum of spin foam amplitudes I(Γ, θ) over trivalent diagrams:

∑

Γ trivalent

λV

SΓI(Γ, θ) (5.43)

where λ is a coupling constant, V ≡ ♯V (Γ) is the number of vertices of Γ and SΓ isthe symmetry factor of the graph.

5.5 Parametric Feynman integrals

Consider a quantum scalar field theory based on some interaction Lagrangian, oneobtains the Feynman graphs by expanding out the exponential of the interaction termin the Feynman path integral. Choose an arbitrary Feynman graph Γ, one computesthe amplitude IΓ according to the Feynman rules. The contribution IΓ can be writtenin a parametric form in terms of the Schwinger parameters. In this section, followingthe textbook [63], we will briefly recall how to derive the Kirchhoff-Symanzik graphpolynomials from Feynman diagrams.

The standard notations on oriented graphs will be used. First the incidence matrixεve of Γ is the ♯V (Γ)× ♯E(Γ) matrix defined as

εve =

+1 if v = t(e)−1 if v = s(e)0 if v /∈ ∂(e)

(5.44)

For each vertex v ∈ V (Γ), the conservation of momentum gives rise to the Dirac deltafunction at v

δv(k, p) = δ(n∑

i=1

εveikei +N∑

j=1

εvejpej) (5.45)

where for each internal edge ei ∈ Eint(Γ), the associated momentum is denoted by kei,similarly pej is used for each external edge ej ∈ Eext(Γ). Denote the total external

momentum Pv =∑N

j=1 εvejpej at each vertex. In the above delta function, n =♯Eint(Γ), N = ♯Eext(Γ) and V = ♯V (Γ) for simplicity.

Then the transition amplitude IΓ is given by Feynman rules

IΓ = C(Γ)

∫ n∏

i=1

dDkei(2π)D

(i

k2ei −m2 + iǫ)

V∏

v=1

δv(kei, pej) (5.46)

109

where C(Γ) is some constant relevant to Γ, we can just drop such constant in compu-tation since they can be easily restored later. As a convention, iǫ is always absorbedinto m2 in the Feynman propagator.

By introducing the Schwinger parameters te, the Feynman propagator becomes

i

k2e −m2=

∫ ∞

0

dteeite(k2e−m2) (5.47)

Meanwhile, write the delta function in the integral form by Fourier transform

δv(kei, pej) =

∫dDxve

ixv·(∑n

i=1 εveikei+Pv) (5.48)

Put these together, then integrate over the internal momentum variables kei and thevertex variables xv, one can rewrite the amplitude as

IΓ =∫ ∏n

i=1

dDkei(2π)D

∫∞0dteie

itei (k2ei−m2)∏V

v=1

∫dDxve

ixv·(∑n

i=1 εveikei+Pv)

=∫ ∏n

i=1 dteie−iteim

2 ∏v d

Dxvei∑V

v=1 xv·Pv∫ dDkei

(2π)Deiteik

2ei+i

∑Vv=1 εveixv·kei

=∫ ∏n

i=1 dteie−iteim

2 ∏v d

Dxvei∑V

v=1 xv·Pv 1(−4πitei )

D/2 e− i

4tei(∑V

v=1 εveixv)2

=∫ ∏n

i=1 dteie−itei

m2

(−4πitei )D/2

∫ ∏v d

Dxve− i

4tei(∑V

v=1 εveixv)2+i∑

v Pv·xv

= δ(∑

v Pv)∫ ∏n

i=1 dteie−iteim

2

(−4πitei )D/2

∫ ∏v d

Dxve− 1

2(∑

v,v′i

2teiεveiεv′ei

xvxv′)+i∑

v Pv·xv

= δ(∑

v Pv)∫ ∏n

i=1 dteie−iteim

2

(−2itei )D/2 det(DΓ(t))1/2

e−∑

v,v′ [D−1Γ (t)]v,v′PvPv′

(5.49)where the (V − 1)× (V − 1) matrix DΓ(t) is defined as

[DΓ(t)]v,v′ =

n∑

i=1

i

2teiεveiεv′ei (5.50)

Each element [DΓ(t)]v,v′ can be viewed as a collection of edges connecting distinctvertices v and v′ and for such edge e ∈ Eint(Γ) one associates the inverse of Schwingerparameter 1/te. Ignoring possible negative signs, the determinant det(DΓ(t)) givesall possible paths connecting all the vertices. In other words, one can define the firstKirchhoff-Symanzik polynomial as

UΓ(t) = det(DΓ)n∏

i=1

tei =∑

T

∏

e/∈Tte (5.51)

where T ranges over spanning trees of Γ.The second Kirchhoff-Symanzik polynomial VΓ(t, P ) involves the external momen-

tum Pv and elements of the inverse matrix [D−1Γ (t)]v,v′ . We don’t give the details here,

the interested reader can consult the textbook [63]. The second graph polynomialdoes not appear in our Feynman graph evaluation of 3d gravity with matter, thereason will be explained later.

110

In sum, the amplitude can be expressed in the parametric form

IΓ(t, P ) = δ(∑

v

Pv)

∫ n∏

i=1

e−iteim2

dteie−VΓ(t,P )/UΓ(t)

U1/2Γ (t)

(5.52)

5.6 Kirchhoff polynomial with κ-correction

In this section, we will derive the parametric representation of the spin foamamplitude (5.42) for a spherical graph Γ. Notice that external momenta can be gaugedaway [41], so we only integrate over internal momenta. This gauge fixing process killsgraph polynomials involving external momentum and we are mainly interested in thegeneralized first Kirchhoff polynomial.

As before, we introduce the Schwinger parameters te for each edge e ∈ E(Γ)and the new Feynman propagators are written as

i

|Pe|2 − sin2 θeκ2

=

∫ ∞

0

dteeite(P 2

e − sin2 θeκ2

) (5.53)

Note that taking the no gravity limit κ → 0 would recover the classical Feynmanpropagator and reduce the κ-deformed star product back to the usual product offields, so it is reasonable to expect that the new Kirchhoff graph polynomial is theusual Kirchhoff polynomial plus some κ-related correction in quantum gravity.

Plug the integral form of Feynman propagators into the spin foam amplitude(5.42) and use the new addition in the momentum space,

I(Γ, θ) =∫ ∏

v d3Xv

∫ ∏e d

3Pe

∫∞0dtee

ite(P 2e − sin2 θe

κ2)ei

∑

v Xv·(⊕e∈∂vεvePe)

=∫ ∏

e dtee−ite

sin2 θeκ2

∫ ∏v d

3Xv

∫d3Pee

iteP 2e +i

∑

v Xv·(⊕e∈∂vεvePe)(5.54)

Since the trivalent graph Γ is embedded in the triangulation ∆, there are exactlythree incoming or outgoing edges at each vertex.

⊕e∈∂vεvePe = εveiPei ⊕ εvejPej ⊕ εvekPek = εveiPei + εvejPej + εvekPvek

−κεveiPei × εvejPej − κεveiPei × εvekPek − κεvejPej × εvekPek

(5.55)

The associative addition of 3-momentum has no cyclic symmetry, so we have to fixthe ordering of Pi, Pj , Pk as a convention. As usual, we assume an ordering on theedges ei, i = 1, 2, · · ·n ≡ ♯E(Γ), then the convention is that i < j < k is alwayssatisfied at each vertex.

Then the last integration on the momentum space is

I(t, X) =∫ ∏

e d3Pee

ite∑

e Pe·Pe+i∑

e

∑

v εveXv ·Pe−iκ∑

e6=e′∑

v εveεve′Xv ·Pe×Pe′

=∫ ∏

e d3Pee

ite∑

e Pαe ·Peα−iκ

∑

e<e′∑

v εveεve′εαβγXαv Pβ

e P γ

e′+i

∑

e

∑

v εveXαv ·Peα

(5.56)

In the matrix form,

I(t, X) =

∫d~Pe−

12~P tA~P+ ~J·~P (5.57)

111

where ~P = (P 1e1 , P

2e1, P

3e1, · · · , P 1

en, P2en, P

3en)

t. If we define

Te =

−2ite 0 00 −2ite 00 0 −2ite

, (5.58)

Mee′ = 2iκ

∑

v εveεve′εα11Xαv

∑v εveεve′εα12X

αv

∑v εveεve′εα13X

αv∑

v εveεve′εα21Xαv

∑v εveεve′εα22X

αv

∑v εveεve′εα23X

αv∑

v εveεve′εα31Xαv

∑v εveεve′εα32X

αv

∑v εveεve′εα33X

αv

(5.59)

Mee′ is a skew-symmetric matrix, then

A =

Te1 Me1e2 Me1e3 · · · Me1en

0 Te2 Me2e3 · · · Me2en...

......

. . ....

0 0 0 · · · Ten

3n×3n

(5.60)

A is an upper triangular matrix resulted from our convention on the momenta aroundeach vertex.

J = (i∑

v

εve1 ~Xv, · · · , i∑

v

εven ~Xv) (5.61)

We take the integration over the usual R3 instead of B3κ, because the error term

is obviously less than κ3. To see this, recall that in dimension one, the Gauss errorfunction is defined as

erf(x) =2√π

∫ x

0

e−t2dt.

and the complementary error function is defined as

erfc(x) = 1− erf(x) = 2√π

∫ ∞

x

e−t2 dt

For large x, the asymptotic expansion of the complementary error function is givenby

erfc(x) ∼ e−x2

x√π

∞∑

n=0

(−1)n (2n− 1)!!

(2x2)n,

where (2n − 1)!! is the double factorial. We can generalize this formula to higherdimensions and up to a constant related to π, the error term in three dimension isway less than κ3 which will be dropped immediately in a linear approximation.

Hence by the Gaussian integral in three dimension, the integral over the momen-tum ~P is approximated by

I(t, X) =1

(detA)1/2e

12JA−1Jt

=1∏n

i=1(−2itei)3/2e

12JA−1Jt

(5.62)

112

Since A is an upper triangular matrix, its inverse can be computed explicitly. Theinverse matrix is another upper triangular matrix, its diagonal elements are justT−1ei

and the sub-diagonal elements are −T−1eiMeiei+1

T−1ei+1

. We write the rest of the

elements outside as a higher order correction matrix O(κ2)B. We will only use linearκ-approximation and drop higher order terms in the later calculation.

T−1e =

i/2te 0 00 i/2te 00 0 i/2te

(5.63)

A−1 =

T−1e1 −T−1

e1 Me1e2T−1e2 0 · · · 0

0 T−1e2

−T−1e2Me2e3T

−1e3

· · · 00 0 T−1

e3· · · 0

......

.... . .

...0 0 · · · T−1

e3 −T−1en−1

Men−1enT−1en

0 0 0 · · · T−1en

+O(κ2)B

(5.64)Now the spin foam amplitude looks like

I(Γ, θ) =∫ ∏

e

dtee−ite

sin2 θeκ2

(−2ite)3/2∫ ∏

v

d3Xve12JA−1Jt

(5.65)

we will go further to take care of the integration over the vertices and define a newgraph polynomial.

Let us look at 12JA−1J t closer

12JA−1J t = −1

2

∑vv′

~Xv(∑n

i=1 εveiT−1eiεv′ei)

~X tv′+

12

∑vv′

~Xv(∑n−1

i=1 εveiT−1eiMeiei+1

T−1ei+1

εv′ei+1) ~X t

v′

= −12

∑vv′(

∑ni=1 εvei

i2tei

εv′ei)~Xv · ~Xv′+

2κ∑

vv′v′′(∑n−1

i=1 εveii

2teiεv′′eiεv′′ei+1

i2tei+1

εv′ei+1) det( ~Xv, ~Xv′ , ~Xv′′)

(5.66)The first part again tells us that if e is some edge connecting two end points

then we just associate the parameter 1/te correspondingly. The usual first Kirchhoffpolynomial then can be recovered if we consider all possible spanning trees in thegraph Γ.

The second part includes a determinant det( ~Xv, ~Xv′ , ~Xv′′) = εα,β,γXαvX

βv′X

γv′′ , we

ignore the possible signs εα,β,γ as usual. We view the determinant as a means to couplethree vertices together. For two adjacent edges connecting these three vertices, weassign a term κ

tei tei+1. It can be thought of as a “quantum tunnelling” between two

vertices connected indirectly.

113

If we introduce a new notation ei ≺ ej, which means that edges ei and ej areadjacent in some spanning tree T of Γ and i < j, then our definition of the generalizedfirst Kirchhoff polynomial is given as follows.

Definition 34. The spin foam graph polynomial including a κ-linear quantum gravitycorrection can be defined as

UΓ(t) =∑

T ⊂Γ

(∏

e∈Tte + κ

∏

ei≺ej

teitej) (5.67)

where T ranges over those spanning trees of Γ. Here we use parameters te insteadof 1/te because we have to respect the fact that in any spin foam model graphs arealready assumed to be dual graphs. It consists of two parts, the classical part andthe correction part.

Lemma 11. The graph polynomial UΓ(t) is independent of the ordering the edges.

Proof. We only have to take care of the correction term, and there are two subcasesdue to the triangulation.

Fix a spanning tree T , those outermost vertices do not contribute to the κ-correction of the polynomial UT (t). Among other vertices, if the valence of the vertexis 2, it gives one pair of adjacent edges, then teitej will be produced as a correctionterm in UT (t); and if the valence of the vertex is 3, then it gives three pairs of adjacentedges, which produce a term teitej · tejtek · teitek = t2eit

2ejt2ek in the κ-correction part of

UT (t).So whatever the ordering of the edges is, the polynomial can be read off directly

from combinatorial information of a graph.

Let us determine the degree of UT (t) by introducing a function on vertices:

d(v) =

0 if val(v) = 12 if val(v) = 26 if val(v) = 3

(5.68)

We then have

deg(UT (t)) =∑

v∈Td(v) and deg(UΓ(t)) = maxT deg(UT (t)). (5.69)

5.7 Hamiltonian action at vertices

Two dynamical processes are of special interest for us in the spin foam model.When a Hamiltonian acts on a specific vertex, it could split into a small triangle thenexpand through spacetime; or in the opposite direction, some small triangle could

114

collapse into a vertex resulted from some action. In both cases, the graph remainstrivalent but the triangulation changes.

Let us look at the change of the polynomial UΓ(t) in both cases. First supposewe have one vertex v and three edges a, b, c around v in Γ, after excitation, v splitsinto a triangle with edges α, β, γ. There are two possible subcases depending on thevalence of the vertex in spanning trees T ⊂ Γ.

If the valence of the vertex is 3 in some spanning tree T , that is, assume edgesa, b, c are still in T and the original polynomial of T is

U0 = abcP (t) + κab · bc · caQ(t) = abcP (t) + κa2b2c2Q(t) (5.70)

where we just use a, b, c instead of ta, tb, tc for simplicity. The polynomials P (t), Q(t)collect the contributions from other parts of the graph.

To get a spanning tree T based on T ∪ αβγ, we just break the loop αβγ and wehave three possible cases T = T ∪ αβ, T = T ∪ αγ and T = T ∪ βγ. Then the newpolynomial is

U1 = abcP (t)(αβ + αγ + βγ) + κQ(t)(aα · aβ · αc · αβ · βb+aα · αγ · αc · γc · γb+ aβ · βγ · βb · γc · γb)

= abcP (t)(αβ + αγ + βγ) + κabcQ(t)(aα3β3 + cα3γ3 + bβ3γ3)= abcP (t)(αβ + αγ + βγ) + κabc(αβγ)3Q(t)( a

γ3 +bα3 +

cβ3 )

= abcP (t)ν + κabcQ(t)(a(ω/γ)3 + b(ω/α)3 + c(ω/β)3)

(5.71)

where we denote ν ≡ αβ + αγ + βγ and ω ≡ αβγ.In the inverse process, if the triangle αβγ collapses into a vertex v, it is clear that

we immediately get U0 from U1 by the mappings:

ν 7→ 1; a(ω/γ)3 + b(ω/α)3 + c(ω/β)3 7→ abc (5.72)

Secondly, if the valence of the vertex is 2 in some other spanning tree T ′, assumethat passing from Γ to T ′, edge c is deleted and a, b are still in T ′. The polynomialrelated to T ′ is

U′0 = abP ′(t) + κabQ′(t) (5.73)

Again suppose the vertex v explodes into a triangle αβγ, and the new T ′ derivedfrom T ′ is just T ′ = T ∪ αβ, T ′ = T ∪ αγ and T ′ = T ∪ βγ. It is easy to see thepolynomial related to T ′ is

U′1 = abP ′(t)ν + κQ′(t)(aα · aβ · αβ · αb+ aα · αb · γb · αγ + aβ · βγ · γb)

= abP ′(t)ν + κabQ′(t)(aα3β2 + bα3γ2 + β2γ2)= abP ′(t)ν + κabQ′(t)(aα(ω/γ)2 + bα(ω/β)2 + (ω/α)2)

(5.74)

Conversely, if the triangle αβγ shrinks to a vertex v and v has valence 2 in somespanning tree, we should apply

ν 7→ 1; aα(ω/γ)2 + bα(ω/β)2 + (ω/α)2 7→ 1 (5.75)

115

and easily get U′0 from U′

1.We can then derive the general relation between UΓ,UΓ\e and UΓ/e.

UΓ = UΓ\e + e · UΓ/e (5.76)

where the action by e means for a classical term, we just multiply it by e; for acorrection term, we multiply it by

∏e∩ek 6=∅ eek.

We can also expand the polynomial with respect to e ≡ tn:

UΓ = UΓ|tn=0 +∂UΓ

∂tn|tn=0tn +

∂2UΓ

∂t2n|tn=0

t2n2!+ ∂3UΓ

∂t3n|tn=0

t3n3!+ ∂4UΓ

∂t4n

t4n4!.

= P0 + P1tn + P2t2n + P3t

3n + P4t

4n

(5.77)

where Pi(t1, · · · , tn−1), i = 0, · · ·4 are the i-th derivative evaluated at tn = 0. Thepossible biggest power of tn is 4 when e connects two vertices of valence 3 in somespanning tree.

Notice that this is not a genuine deletion-contraction relation like the one satisfiedby the original Kirchhoff polynomial in the classical case, since here in (5.76), in theterm e · UΓ/e the action of e is not just multiplication by the corresponding variable,but it involves also a multiplication by the more complicated term

∏e∩ek 6=∅ eek. In

particular, this means that one cannot directly apply to this case the techniques de-veloped in [2], [3], to study the graph hypersurfaces based on the deletion-contractionproperty. In the next section, we use a more direct method, based on counting pointsover finite fields, to investigate the properties of the graph hypersurface of the tetra-hedron graph.

5.8 Graph hypersurfaces and motives

The graph hypersurfaces for the classical Kirchhoff polynomial have been exten-sively studied from the point of view of motives and periods (see for instance [89]for an overview). In general, one can investigate the motivic properties of a varietydefined over Z by either computing its class in the Grothendieck ring of varietiesK(VarZ) or by considering the reductions modulo primes and computing the num-ber of points #X(Fq) over finite fields. We consider explicitly the case of the graphhypersurface of the tetrahedron graph, with the quantum gravity correction.

We are interested in possibly singular hypersurfaces from generalized Feynmangraph evaluations. It is known [12] that graph hypersurfaces are general enough togenerate the Grothendieck ring of varieties, but many interesting graph hypersurfacesstill turn out to be mixed Tate. Although an individual graph is not mixed Tate, thesum over graphs could be mixed Tate [17].

We recall the definition of polynomially countable based on counting Fq-rationalpoints of XΓ, where the affine hypersurface is defined as

XΓ = t ∈ An | UΓ(t) = 0 (5.78)

with n = ♯E(Γ) for any graph Γ.

116

Figure 5.1: Tetrahedron with vertices and edges labeled

The simplest triangulation of a sphere is given by the tetrahedron as in the figure,which is a fundamental object in 2+1 loop quantum gravity.

Definition 35. Consider the function FΓ : q 7→ ♯XΓ(Fq) defined on the set of primepowers q = pn. We say that the graph hypersurface XΓ is polynomially countable ifFΓ is a polynomial in Z[q].

Actually, this definition suits for any scheme X of finite type over Z, but we onlyfocus on graph hypersurfaces in this work.

Thus with this definition, the result of [12] tells us that FΓ is not polynomiallycountable for almost all graphs, and [17] shows the sum of all FΓ for connected graphswith n edges is polynomially countable for any n ≥ 3.

The relation between the two concepts, mixed Tate and polynomially countable,is a little subtle: for mixed Tate motives [X ], the numbers ♯X(Fp) are polynomialin Z[p] for almost all primes p. The “for almost all p” excludes the cases that, forinstance, the variety may have bad reduction at finitely many primes and that canalter the behavior of ♯X(Fp). Conversely, assuming the Tate conjecture holds andknowing ♯X(Fp) are polynomial in Z[p] for almost all p would imply the motive ismixed Tate.

As the simplest example in loop quantum gravity, we think of the tetrahedron asa triangulation of the 2-sphere. The hypersurface is tested over different finite fieldsto count the rational points, it turns out that the hypersurface of the tetrahedron isgeneral and complicated, which is not polynomially countable.

Example 45. Let us consider the triangulation ∆ of the 2-sphere as our graph Γ ≡ ∆in the figure. Obviously, the tetrahedron has 16 spanning trees: abc, abd, abf, ace, acf,ade, adf, aef, bcd, bce, bde, bdf, bef, cde, cdf, cef .

As we can see, the spanning trees can be divided into two types: T1 with one vertexof valence 3 like abc and T2 with two vertices of valence 2 like abd. Accordingly, there

117

are two types of polynomials UT1 and UT2, then UΓ =∑

T1UT1 +

∑T2UT2. It is easy

to get the graph polynomial for an individual spanning tree, for instance

Uabc = abc + κa2b2c2; Uabd = abd + κa2bd (5.79)

Now the total graph polynomial looks like:

UΓ = abc + κa2b2c2 + ade+ κa2d2e2 + bef + κb2e2f 2 + cdf + κc2d2f 2

+abd + κa2bd+ abf + κab2f + ace+ κa2ce+ acf + κac2f+adf + κad2f + aef + κae2f + bcd + κbc2d+ bce+ κb2ce+bde + κbde2 + bdf + κbdf 2 + cde+ κcd2e + cef + κcef 2

(5.80)

If we delete some edge, say e, then the spanning trees are reduced to abc, abd, abf ,acf, adf, bcd, bdf, cdf . In terms of the graph polynomial, we have UΓ\e = UΓ|e=0, thatis

UΓ\e = abc+ κa2b2c2 + cdf + κc2d2f 2 + abd + κa2bd + abf + κab2f+acf + κac2f + adf + κad2f + bcd+ κbc2d+ bdf + κbdf 2 (5.81)

On the other hand, if we contract the same edge e in Γ, then Y , Z are identifiedas one vertex and the spanning trees are ac, ad, af, bc, bd, bf, cd, cf compared to theoriginal ace, ade, aef, bce, bde, bef, cde, cef . In other words, we set e = 1 in theclassical part and set those adjacent pairs eek = 1 in the correction part to get thepolynomial UΓ/e .

UΓ/e = ac + κac+ ad+ κad+ af + κaf + bc+ κbc+bd+ κbd+ bf + κbf + cd+ κcd+ cf + κcf

(5.82)

We expand the graph polynomial as UΓ = G+ eF + e2E, where G = UΓ\e and

F = ac + ad+ af + bc+ bd + bf + cd+ cf + κa2c + κb2c+ κcd2 + κcf 2 (5.83)

E = κaf + κbd + κa2d2 + κb2f 2 (5.84)

As mentioned before, if an algebraic variety X is mixed Tate as a motive, then itsclass [X ] in the Grothendieck ring of varieties is a polynomial function with integercoefficients of the Lefschetz motive L = [A1].

Number of points over finite fields, as an additive invariant, factors through theGrothendieck ring

Nq : K(VarFq)→ Z; with [X ] 7→ ♯X(Fq) (5.85)

Suppose [X ] =∑

i aiLi ∈ K(VarFq), then (for almost all primes p) the number of

points ♯X(Fq) will be equal to∑

i aiqi since A1 has q points over Fq.

When it is hard to compute the Grothendieck class of the graph hypersurface[XΓ] ∈ K(VarFq), one can instead check whether XΓ is polynomially countable, i.e.

118

verify FΓ for different prime powers q to see if ♯XΓ(Fq) fits a polynomial∑

i aiqi for

some integer coefficients ai.The case of the hypersurface of the tetrahedron shows that such κ-correction

terms introduces significant complexity. Without the correction terms, [X ]classic =L5 + L3 − L2 shows that the graph hypersurface of the tetrahedron is a mixed Tatemotive.

However, if we take the correction terms into account, we obtain a very differentresult.

Proposition 5. The hypersurface of the tetrahedron defined by UΓ with the parameterκ = 1 is not polynomially countable.

Proof. By computer calculation, for the fields Fp with p = 2, 3, 4, 5, 7, 11, we find thatthe best polynomial fit for the counting of points #XΓ(Fp), by a polynomial withrational coefficient, is given by

#XΓ(Fp) = −37951160480

p5 +116827

576p4 −24982339

12096p3

+5588389

576p2 −631697377

30240p +

1188935

72.

(5.86)

It is proved in Proposition 6.1 of [97] that if the counting function q 7→ #X(Fq)of a variety is given by a polynomial function with rational coefficients, then thepolynomial must in fact have integer coefficients. Thus, the fact that the best poly-nomial fit for the data #XΓ(Fp) for the first few prime powers p = 2, 3, 4, 5, 7, 11 is apolynomial with rational non-integer coefficients implies that the counting functionq 7→ #XΓ(Fq) for the tetrahedron graph must in fact be non-polynomial.

From the relation between mixed Tate motives and polynomially countability, thisproposition suggests that the hypersurface of the tetrahedron may also not be mixedTate. It would be interesting to see if the Feynman amplitudes of the spin foam modelcould then be related to explicit periods that are not mixed Tate periods (multiplezeta values).

5.9 Conclusions and discussion

A new effective noncommutative field theory was proposed by Freidel and Livine[40] in the study of 3d loop quantum gravity. The perturbative expansion of thisnoncommutative field theory gives the spin foam amplitude for any trivalent Feynmangraph embedded in a triangulation of the spacetime manifold. In other words, we canstudy the Ponzano-Regge model coupled with matter by computing Feynman diagramevaluations as in quantum field theory.

In order to look closer at the geometric aspect of Feynman graphs, one may con-sider parametric representations of Feynman integrals. After introducing Schwinger

119

parameters, we derive a generalized Kirchhoff polynomial based on the noncommuta-tive star product. This Kirchhoff polynomial consists of two parts, the classical partagrees with the usual first Kirchhoff-Symanzik polynomial, and the κ-correction parttakes care of the quantum gravity effect in the κ-linear approximation.

There are two simple Hamiltonian actions on the nodes, one of which splits a nodeand creates a new triangle in quantum history, the other action does the oppositewhose effect is to collapse a triangle into a node. As a result, these Hamiltonianactions change the corresponding Kirchhoff polynomials and a deletion-contractionrelation is derived.

Feynman diagram integrals provide surprising connections with multiple zeta val-ues, which are periods of mixed Tate motives. Hence it makes perfect sense to testthe motivic complexity of graph hypersurfaces derived from the Kirchhoff polynomial.In our case, we use computer calculations to count the points of the correspondinghypersurface variety over finite fields. As a simplest example, the hypersurface of thetetrahedron is not polynomially countable, which implies it may not be mixed Tatemotive.

Notice that the star product in 3d quantum gravity is different from the Moyalproduct, and the parametric representation of the Moyal noncommutative field theoryis much more complicated. It would be interesting to study the analytic and geometricproperties of generalized Feynman diagram Moyal expansions.

We only consider spherical graphs in our work, for more general graphs there isa nontrivial braiding which make it a noncommutative braiding field theory. In thisdirection, there are some similar properties of generalize Feynman graphs betweenthe κ-deformation and the Moyal field theories. It would be interesting to identifythose graphs and introduce some Moyal product methods into the study of spin foammodels in loop quantum gravity.

An important question in Feynman motives is to determine the condition, underwhich a Feynman graph gives a multiple zeta value, or geometrically, a Feynman graphhypersurface gives a mixed Tate motive. It would be great to make any progress inthis direction.

120

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BIOGRAPHICAL SKETCH

Dan Li was born on September 22, 1981 in Shanghai, China. He received his bachelor’sdegree from Shanghai Donghua University in Applied Mathematics. He also obtainedhis master’s degrees from Shanghai Fudan University in Pure Mathematics. In the Fallof 2007, he started his doctoral study at Florida State University under the supervisionof Dr. Matilde Marcolli. Dan has presented a talk at the 2012 AMS Joint MathematicsMeetings held in Boston. He has also attended the 8th and 9th Spring Instituteon Noncommutative Geometry and Operator Algebras at Vanderbilt in Nashville.In addition, he has attended the Trimester Program on Geometry and Physics atHausdorff Research Institute for Mathematics in Bonn. Dan’s mathematical interestsrange from mathematical physics, noncommutative geometry to algebraic geometry.

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