Conformal Field Theory, Modularity of $q$-Hypergeometric Functions, and the Bloch Group

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Universit`

a di Pisa

Facolt`

a di Scienze Matematiche Fisiche

e Naturali

Corso di Laurea in Matematica

Anno Accademico 2007-2008

Tesi di Laurea specialistica

Conformal Field Theory,

Modular q-Hypergeometric Functions,

and the Bloch Group

Candidato Daniel Disegni

Relatori Controrelatore

Prof. Don Zagier Prof. Roberto Dvornicich Prof. Stefano Marmi

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Introduction

Modular forms have been among the favourite objects of study of mathematicians for over a century. Still, the first examples of them are to be found back in the work of Euler ([E], 1748), who proved such identities as

∞ X n=0 qn2+n2 (q)n = ∞ Y n=1 1 + qn, (1)

where |q| < 1 and (q)n:= (1 − q) · · · (1 − qn). By setting q = e2πiz with z belonging

to the upper half-plane H, the left-hand side can be written, up to a suitable rational power of q, as η(2z)/η(z), where η(z) = eπiz/12(q)∞is the eta function named after

Richard Dedekind, who proved its modularity in 1877. On the other hand, the left-hand side series above belongs to the class of q-hypergeometric series, which are convergent series of the type F (q) =P∞

n=0an(q) with an(q)/an−1(q) = R(q, qn) for

some fixed rational function of two variables R(x, y).

In general, the problem of understanding which q-hypergeometric series give rise to modular forms or functions is extremely interesting but, at the moment, com-pletely unreachable. Another pair of famous examples of modular q-hypergeometric functions comes from the Rogers-Ramanujan identities:

∞ X n=0 qn2 (q)n = Y n≡±1 mod 5 1 1 − qn (2) ∞ X n=0 qn2+n (q)n = Y n≡±2 mod 5 1 1 − qn (3)

Again, we will see in chapter 1 that up to suitable rational powers of q each of the product expansions can be rewritten as the quotient of a theta series by the Dedekind eta function, thus establishing modularity.

As a more modest problem than the previously stated one, it is quite natural to wonder whether the quadratic forms n2₂+n, n2 and n2+ n appearing in (1), (2) and (3) are the only ones that give rise to modular forms (up to rational powers of q). A more general version of this question can also be asked, involving the r-fold

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q-hypergeometric functions fA,B,C(z) = X n=(n1,...,nr)∈Nr q12n t_An+Bt_n+C (q)n1· · · (q)nr , (4)

for A ∈ Mr×r(Q) a symmetric positive definite matrix, B ∈ Qr a vector, and C ∈ Q

a scalar.

In recent years, a resurgence of interest in q-series has been motivated by their appearance in various contexts in physics (see [A 2] for a review of some of these encounters). In particular, for suitable values of the parameters, the functions fA,B,C(z) are characters of certain quantum field theories with conformal

invari-ance defined on tori, where the argument z is the modular parameter of the torus under consideration (i.e. z = ω2/ω1 if the torus is C/Λ = C/Zω1 ⊕ Zω2). As a

consequence of the physical significance of the set of characters for a given theory, the vector space spanned by them should be invariant under the action of the mod-ular group SL(2, Z) induced by its action on z by M¨obius transformations (which corresponds to a change of basis in the period lattice Λ); the individual characters are expected to be modular functions for some finite index subgroup of SL(2, Z), too. Therefore physicists too were led to the (still very challenging) question of de-termininig for each r ≥ 1 the set of values of A, B and C for which (4) is a modular form.

By estimating the asymptotic expansions for q → 1− of these functions for general values of the parameters and comparing it with the one expected from a modular form, Werner Nahm has been able to formulate a conjectural partial answer to the question just raised, which we now explain. To the matrix A we associate the system of algebraic equations

1 − Qi= r

Y

j=1

QA_jij, (5)

which we will also write in vector notation as 1 − Q = QA, where Q = (Q1, . . . , Qr).

When A is not integral, we should specify what we mean by a solution of (5): we require consistency in the sense that there must be vectors u, v ∈ Cr such that v = Au and Qi = eui = 1 − evi. We then attach to any solution Q of (5) the element

ξQ= [Q1] + · · · + [Qr] ∈ Z[F \ {0, 1}],

where F = Q(ξQ) = Q(Q1, . . . , Qr). It is easily seen that ξQ actually lies in the

Bloch group of F , a very interesting subquotient of Z[F \ {0, 1}] encoding identities for the dilogarithm function Li2(z) = P∞n=1z

n

n2 which we will define and study in

Chapter 2. Then:

Conjecture (Nahm). The following are equivalent for a positive definite symmetric matrix A ∈ Mr×r(Q):

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• there exist B ∈ Qr _{and C ∈ Q such that f}

A,B,C(z) is a modular form;

• for every solution Q of (5), the element ξ_Q is torsion in the Bloch group of Q(ξQ).

The present work can be regarded as an introduction to Nahm’s conjecture, explaining the background and motivation from both mathematics and physics, and describing the evidence which supports it. Chapter 1 is very classical, dealing with the basics in the theory of q-series and modular forms, with a final aim at Rogers-Ramanujan identities and some of their likes. Chapter 2 is, as mentioned, dedicated to the study of (variants of) the dilogarithm function and its functional equations, and their algebraic counterpart which is the Bloch group; while torsion elements of the latter are expected to be related to modularity of the functions (4), hyperbolic three-manifolds have a natural invariant of infinite order in the Bloch group of C, and we will conclude the chapter by briefly surveying the theory. Chapter 3 gives a short introduction to conformal field theory, aimed at explaining how modular forms happen to arise from there, and why they are of the type of interest to us. Finally, the last chapter gives the motivation for Nahm’s conjecture, and describes the state of the art in results and examples. In particular, the rank 1 case is proven (in a somewhat artificial way, so that we should actually say ‘verified’ – even if it is a rather nontrivial verification); we thus give the complete list of all the seven modular forms of type (4) with r = 1 (a result first due to Zagier).

The main references for most of the material covered here are the articles [Za 3] and [Na 2], conveniently published next to each other in the same volume. A bit of originality is claimed only over our proof of the rank 1 case of Nahm’s conjecture, and on the general asymptotic formula in Proposition 4.6 which prepares it.

Acknowledgments

It is a pleasure for me to express my deep gratitude to Prof. Don Zagier, for being as inspiring and effective a supervisor as he is a mathematician; to Prof. Stefano Marmi, for his warm support to this project and many useful suggestions; and to Prof. Roberto Dvornicich, for having accepted to read this thesis, but above all for his major role in transmitting me the sense of beauty of mathematics since even before I started university.

Part of this work has been written during stays at the École Normale Supérieure in Paris and the Max-Planck-Institut für Mathematik in Bonn, supported by the Scuola Normale: I am grateful to all of these institutions.

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Chapter 1

Rogers-Ramanujan Identities

This chapter develops the background on some classical topics, and ends by listing seven modular q-hypergeometric functions (two of which are those of Rogers and Ramanujan). In chapter 4 we will see that these are the only modular forms of type fA,B,C(z) = P_n∈Nq

1

2An2+Bn+C

(q)n , thus establishing the rank 1 case of Nahm’s

conjecture.

1.1 Partitions and q-hypergeometric functions

Throughout this work, unless otherwise explicitly stated, we will set q = e2πiz, where z ranges in the upper half-plane H ∪ {∞}: here ∞ is thought of as lying infinitely high on the imaginary axis, so that e2πi∞ = 0. Therefore |q| < 1, and non-integer powers of q are understood as qα= e2πiα. References for the material in this section are [A 1] and [A 2].

We shall start by using q as a parameter to “deform” classical combinatorial objects (which are usually recovered in the limit q → 1−). The q-deformation of n ∈ N is [n]q := 1−q

n

1−q , and satisfies the basic addition rule [n]q = [k]q+ q

k_{[n − k]} q

for 0 ≤ k ≤ n. The q-factorial is [n]q! :=Qnj=1[j]q. A slightly modified version of it

(the q-Pochhammer symbol) can be defined for any s ∈ C 1 (q)s := ∞ Y j=1 1 − qs+j 1 − qj ,

which is a holomorphic function of s with simple zeroes at all integers ≤ 0, whereas for a positive integer n we have (q)n= (1 − q)n[n]q!; we can also allow s = ∞,

(q)∞= ∞

Y

j=1

(1 − qj).

All the q-objects defined in this section carry combinatorial interpretations in terms of partitions of integers: more precisely, the mth cofficients of their power

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series expansions in the (here formal) variable q equal the number of partitions of m of some particular type.

Definition 1.1. A partition P of the positive integer m into n parts is collection of n non necessarily distinct positive integers aj such thatPn_j=1aj = n. We will write

P = “Pn

j=1aj” = “Pn

0

i=1mibi” with a1 ≤ · · · ≤ an or {a1, . . . , an} = {b1, . . . , bn0}

with b1 < · · · < bn0 and the number b_i occurs m_i times in the partition. Quotation

marks will be dropped when no confusion is possibe. We define two binary partially defined operations on partitions, the concatenation ∪ and the sum ⊕, as follows. If P =Pn1 j=1aj and Q = Pn2 k=1ck with an1 ≤ c1 then P ∪ Q := Pn1 j=1aj+ Pn2 k=1ck. If

with the same notation for P and Q we have n2 ≤ n1, then P ⊕ Q := Pn_j=11 (aj +

cj+n2−n1), where we set ck:= 0 if k ≤ 0.

As a first example, the coefficient of qm in the expansion of 1/(q)n counts the

partitions of m into parts each less than or equal to n (for n a positive integer or n = ∞).

Continuing with our q-analogues, the q-binomial is n k q := [n]q! [k]q![n − k]q! = (q)n (q)k(q)n−k ,

so that the definition makes sense for all k and all n /∈ −N. Then of course for n ∈ N and k ∈ Z, n

k

q vanishes whenever k < 0 or k > n. Our extended definition

will prove convenient in lightening the notation later on: we will use the convention that indefinite summations extend over all integer values of the indices, although often only a finite number of them will contribute. On the contrary, within infinite products the index will always range over a subset of N.

From the addition rule for q-numbers we get the basic recursion for the q-binomial coefficients (with integer n and k)

n k q =n − 1 k q + qn−kn − 1 k q ,

by which we can prove that when 0 ≤ k ≤ n are integers, n k

q is a polynomial in

which the coefficient of qm counts the number of partitions of m into at most k parts, each less than or equal to n − k.

The binomial theorem (1 + x)n=P

k n kx

k _{has at least two analogues.}

Proposition 1.1 (q-binomial theorem). The following formal identity holds for every integer n ≥ 1: n Y j=1 (1 + xqj) =X k xkqk2+k2 n k q .

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Proof. The coefficient of xkqm on each side counts the partitions of m into k unequal parts, each not exceeding n. This is clear for the left-hand side. For the right-hand side, observe that such a partition can be written in the form m = (1 + . . . + k) ⊕ P, where P is a partition of m −k2₂+k into at most k parts each not exceeding n − k.

Proposition 1.2. For every integer n ≥ 1, we have

n Y j=1 1 1 − xqj = X k xkqk2 (1 − xq) · · · (1 − xqk₎ n k q .

Proof. The coefficient of xtqm on the left-hand side counts the partitions of m into t parts, each not exceeding n. Let Q be such a partition, and k ≤ n be the maximal integer such that Q has at least k parts each not smaller than k. Then m can be written as m = P ∪ k · k ⊕ P0, where P is a partition of s into t − k parts each not exceeding k, P0 is a partition of m − k2− s into at most k parts each not exceeding n − k, and P and P0 are uniquely determined by Q. Now it is clear that the coefficient of xtqmin the right-hand side also counts this type of partitions of m. We now move to considering infinite series and products: the following identity of Jacobi is crucial to switch between the two.

Theorem 1.3 (Jacobi’s triple product formula). For all y ∈ C∗ and |q| < 1, X m∈Z ymqm2+m2 = ∞ Y j=1 (1 + yqj)(1 − qj)(1 + y−1qj−1).

Proof (Cauchy). Apply the q-binomial theorem 1.1 with n = 2N , x = yq−N and the change of variables k = M + m, and multiply both sides by x−NqN 2−N2 to

get X m ymqm2+m2 2N N + m q = N Y j=1 (1 + yqj)(1 + y−1qj−1). (1.1)

Now by the combinatorial interpretation we have limN →∞

2N N +m

q= 1/(q)∞, so that

letting N tend to infinity we get exactly what we wanted.

As a simple application, substitute q with q3 and set y = −1/q to get Euler’s “pentagonal number theorem”

(q)∞=

X

m∈Z

(−1)mq3m2−m2 . (1.2)

Another important identity of Euler is obtained by letting n → ∞ in the q-binomial theorem 1.1.

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Theorem 1.4 (Euler). For all |x| ≤ 1 and |q| < 1, ∞ X m=0 xmqm2+m2 (q)m = ∞ Y j=1 (1 + xqj).

We have so far collected some of the basic tools for manipulating and finding product expansions of q-hypergeometric functions (or series).

Definition 1.2. A q-hypergeometric series is a series F (q) =P∞

n=0an(q) such that

for all n ≥ 1 we have

an(q)

an−1(q)

= R(q, qn)

for some rational function R(x, y) with |R(q, 0)| < 1 for all |q| < 1. The usual normalisation is a0= 1.

The function f (z) = F (e2πiz), defined for z ∈ H, is called a q-hypergeometric function.

1.2 Modular forms

Modular forms are complex functions on the upper half-plane H satisfying certain symmetry properties. Their importance in number theory is paramount (let us just mention Wiles’ proof of Fermat’s last theorem), and a number of beautiful treatments of the subject are available (for example [B-G-H-Z], [Miy]). For this reason we will content ourselves with providing the definitions and results that we need, referring the reader to the literature for more details and motivation..

The group Γ(1) := SL(2, Z) acts on H by γz = aγz+bγ

cγz+dγ for γ =

aγ bγ

cγ dγ ∈ Γ(1).

The action extends to a compactification ¯H= H ∪ P1_{(Q), of whose topology we do}

not need the details but just the terminology: a cusp for a subgroup Γ ⊂ Γ(1) is an equivalence class in Γ \ P1(Q) ⊂ Γ(1) \ ¯H(then it is easy to check that Γ(1) only has one cusp). The only analytic functions invariant under the action of Γ(1) are the constants; things can be made more interesting by relaxing the invariance condition in three different directions, i.e. by shifting the natural induced action on functions on H, by allowing for meromorphic functions, and by changing Γ(1) with a finite index subgroup,

For k an integer, define the k-shifted action of GL(2, Q)+ on functions on the upper half-plane by

f |[γ]k(z) = j(γ, z)

−k

f (γ) (1.3)

where j(γ, z) = det(γ)−1/2(cγz + dγ) for γ =

aγ bγ

cγ dγ

∈ GL(2, Q)+_.

Suppose that f is an analytic function on H, invariant under the k-action of a finite index subgroup Γ ⊂ Γ(1). Then invariance under a suitable positive power

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Th ∈ Γ of the translation T = 1 1

0 1 implies that f has a Fourier expansion f (z) =

P

n∈Zanqn/h, where q = e2πiz as usual. We say that f is holomorphic (respectively

meromorphic) at inifinity if an = 0 for all (respectively all but a a finite number

of) n < 0. We say that f is holomorphic (meromorphic) at the cusps if for every cusp s = ρ∞ (ρ ∈ Γ(1)) of Γ, the ρ−1Γρ-invariant function f |[ρ]k is holomorphic

(meromorphic) at infinity.

Definition 1.3. A modular form of integer weight k for a finite index subgroup Γ ⊂ Γ(1) = SL(2, Z) is a holomorphic function on H which is invariant under the k-shifted action (1.3) of Γ, and holomorphic at the cusps in the sense just defined. We denote by Mk(Γ) the space of such functions, and put Mk=S[Γ(1):Γ]<∞Mk(Γ).

A modular function for Γ is a meromorphic function on H which is invariant under the natural action of Γ, and meromorphic at the cusps.

We remark that:

• modular forms have the structure of a graded algebra: Mk(Γ)Mk0(Γ0) ⊂

M_k+k0(Γ ∩ Γ0);

• the quotient of two modular forms of the same weight is a modular function (and viceversa – though this is less obvious);

• it can be shown that Mk= 0 for k < 0;

• the group GL(2, Q)+ _{has an action on modular forms of weight k, given by}

f 7→ f |_[α]_k, which sends Mk(Γ) to Mk(α−1Γα ∩ Γ(1)).

It is well known that SL(2, Z) is generated by T = 1 1 0 1

and S = 0 −1

0 1,

so that to prove modularity for this group it suffices to check periodicity and the behavior under the transformation z 7→ −1/z.

Of special number theoretical interest are the congruence subgroups Γ(N ) ⊂ Γ1(N ) ⊂ Γ0(N ) ⊂ Γ defined for every positive integer N by

Γ0(N ) = a b c d ∈ Γ | c ≡ 0 mod N Γ1(N ) = a b c d ∈ Γ₀(N ) | a ≡ d ≡ 1 mod N Γ(N ) =a b c d ∈ Γ1(N ) | b ≡ 0 mod N .

Modular forms for Γ(N ) are called modular forms of level N. Then if f (z) is a modular form of level N , for all pairs of coprime positive integers p, q, f (p_qz) is a modular form of level N pq.

It is also possible to define modular forms of half-integral weight, but we do not discuss their theory here (see [Sh]): for our purposes, the assertion “f is a modular

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form of half-integral weight k/2” can be taken as equivalent to “f is holomorphic and f2_{is a modular form of integer weight k”. This embeds the algebra of integer weight}

modular forms as the even part of a larger Z-graded algebra. Before constructing examples, we introduce the convenient notation e(x) = e2πix (then q = e(z)), and the convention that all square roots be taken, in what follows, with the argument lying between −π/2 and π/2.

Define a theta series to be a function of the form θ(z) =P

m∈Zε(m)qλm

2

, with λ a rational number and ε an even periodic function. Upon multiplying the argument by a suitable rational number, we can write any such function as a linear combination of the theta series

θN,h(z) =

X

m≡h mod N

qm22N, where N > 0 is even, h ∈ Z/N Z.

We have clearly θN,h(z) = θN,−h(z), and the transformation property under

transla-tion θN,h(z + 1) = e(h

2

2N)θN,h(z) (here we are using the assumption that N is even,

which implies that m mod N determines m2 mod 2N ). We are interested in the behavior under the modular transformation z 7→ −1/z, for which we need Poisson’s summation formula X m∈Z f (m) = X m∈Z ˆ f (m), (1.4)

valid for all functions f : R → C of rapid decay, where ˆf (ξ) = R

Re(xξ)f (x)dx is

the Fourier transform. We recall that the function x 7→ e−πx2 is its own Fourier transform. Applying (1.4) to the function x 7→ e(N₂z(x + h/N )2) yields

θN,h(−1/z) = r −iz N X k∈Z/N Z e(hk N )θN,k(z). (1.5)

Now Euler’s pentagonal number theorem (1.2) can be rewritten as η(z) = θ12,1(z) − θ12,5(z),

where η(z) = q1/24Q∞

n=1(1 − qn) is Dedekind’s eta function. Then in the formula

for η(−1/z) which is obtained by applying (1.5) only terms with odd k contribute, and the other ones add up to give

η(−1/z) =√−izη(z).

This suggests some modularity property of η and the theta series, and indeed one could prove [Sh, 2.1] that for all N and h ∈ Z/N Z the function θN,h(2z) is a

modular form of weight 1/2 and level 4N . For our purposes, the only relevant implication of this will be that θN,h(z)

η(z) is a modular function. Equivalently, the

transformation properties of these functions give an N -dimensional representation ρN : SL(2, Z) → GL(ΘN) (where ΘN is the complex vector space spanned by the

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1.3 The Magnificent Seven

The following is a list of (the) seven modular functions of the form fA,B,C(z) =

P n∈N q12An2+Bn+C (q)n : 1 f2,0,−1/60(z) = θ20,1(z) − θ20,11(z) η(z) ; f_2,1,11/60(z) = θ20,2(z) − θ20,12(z) η(z) ; f_1,0,−1/48(z) = η(z) 2 η(z/2)η(2z); f1,1/2,1/24(z) = η(2z) η(z); f1,−1/2,1/24(z) = 2η(2z) η(z) ; f_{1/2,0,−1/40}(z) = η(z/2)(θ20,1(z/4) − θ20,11(z/4)) η(z)η(z/4) ; f_1/2,1/2,1/40(z) = η(z/2)(θ20,2(z/4) − θ20,12(z/4)) η(z)η(z/4) .

Since this list comes a bit out of the blue, we should say that here the flow of ideas goes in the opposite direction to that of their presentation (though in the same direction as that of the discovery of the results): namely, the numerical asymptotic analyisis from Chapter 4 suggests that these functions and no others should be modular; and it is their actual modularity which allows to trust the numerical results there.

It remains to prove the above identities. Those from the second group are straightforward applications of Euler’s Theorem 1.4. Those from the third group are due to Rogers, and can be proved (with an additional nontrivial argument) from the first two ones (see [A 2, p. 36 or p. 58]). These are the Rogers-Ramanujan identities, and we will treat them in detail. They were first proved by Rogers in a soon forgotten 1894 paper before being rediscovered by Ramanujan in 1915, and are more commonly written as

G(q) := ∞ X n=0 qn2 (q)n = 1 (q)∞ X m∈Z (−1)mq5m2+m2 = Y n≡±1 mod 5 1 1 − qn (1.6) H(q) := ∞ X n=0 qn2+n (q)n = 1 (q)∞ X m∈Z (−1)mq5m2+3m2 = Y n≡±2 mod 5 1 1 − qn, (1.7)

1_{The last two of the identities correct a small mistake in [Za 3, p. 44], where one should read}

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where the equalities at the right are simple consequences of Jacobi’s triple product formula, the issue being proving the ones at the left. We will do this following the “easy” proof by Bressoud [Br], which is inspired by Cauchy’s idea of proof of Jacobi’s triple product formula (Theorem 1.3): we first prove a finite version (with an extra parameter) of the identities, namely the following (with the usual convention of summations on Z).

Lemma 1.5. For all x ∈ C∗ and positive N ∈ N we have 1 (q)2N X m xmq5m2+m2 2N N + m q =X s,t qs2+t2 (q)N −s(q)s−t Qt j=1(1 + xqj)(1 + x −1_qj−1₎ (q)2t .

Proof of identities (1.6) and (1.7) assuming Lemma 1.5. If we set x = −1 in the lemma, then only the t = 0 term contributes on the right-hand side, and we get (1.6) by letting N → ∞ and multiplying both sides by (q)∞. If x = −q, only the

terms with t = 0 and t = 1 contribute on the right, giving in the limit N → ∞ and after multiplication by (q)∞ ∞ X s=0 qs2 (q)s + ∞ X s=1 qs2+1(1 − q2)(1 − q−1) (q)s ;

shifting the second summation index by −1 and merging the two sums gives the left-hand side of (1.7).

We would like to prove Lemma 1.5 by reducing to (1.1). To do this we need to lower the exponent of q on the left-hand side.

Lemma 1.6. For any a ∈ C and positive N ∈ N, we have X m xm_qam2 (q)N −m(q)N +m =X s qs2 (q)N −s X m xm_q(a−1)m2 (q)s−m(q)s+m

Proof. Set n = N − m and x = q2m in Proposition 1.2, then divide both sides by (q)2m to get 1 (q)N +m =X k qk2+2mk (q)k+2m (q)N −m (q)k(q)N −m−k .

This can be used on the left-hand side of the equation to be proved, giving after simplification X m xmqam2 (q)n−m(q)n+m =X m,k q(m+k)2 (q)N −m−k xmq(a−1)m2 (q)k(q)2m+k ,

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Proof of Lemma 1.5. We apply twice (in the second of the equalities below) Lemma 1.6 to the left-hand side of the equation to prove:

1 (q)2N X m xmq5m2+m2 2N N + m q =X m (xq1/2)mq5m2/2 (q)N −m(q)N +m =X s qs2 (q)N −s X t qt2 (q)s−t X m (xq1/2)mqm2/2 (q)t−m(q)t+m .

Then we can conclude by summing the innermost sum using (1.1) with N = t and both sides divided by (q)2t.

We have thus proven that the functions of Rogers and Ramanujan are modular functions for some subgroup of the modular group SL(2, Z). What is even more im-portant for our case is that the vector space R that they span is invariant under the full modular group, i.e. it is an invariant subspace of the representation ρ20 defined

at the end of the preceeding section. Explicitly, as can be seen (for S) by reorga-nizing terms in (1.5) using the symmetry properties of the theta and trigonometric functions, we have ρ20|R(T ) = e(−1/60) e(11/60) , ρ20|R(S) = 2 √ 5 sin(2π 5 ) sin( π 5) sin(π₅) − sin(2π 5 ) ,

in the basis given by q−1/60G(q) and q11/60H(q).

For the functions from the second group (A = 1), T always acts as multiplica-tion by some root of unity on each funcmultiplica-tion, whereas for S we have f1,0,−1/48(Sz) =

f1,0,−1/48(z), and η(2Sz)/η(Sz) = √1₂η(z/2)/η(z); order is restored and a

repre-sentation of SL(2, Z) is found when one writes η(2z)/η(z) = χ0(z) + χ1/2(z), with

χ0(z) − χ1/2(z) = η(z/2)/η(z). (The reason for this somewhat ad hoc solution will

be explained in Section 3.3.) For the functions with A = 1/2 one needs two more functions to get the same result

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Chapter 2

Dilogarithms and Bloch Groups

We introduce the dilogarithm function and some of its variants, and the closely related Bloch group; such objects appear in a variety of contexts in mathematics: we will review their connections to the study of invariants of hyperbolic 3-manifolds.

2.1 Dilogarithms and their functional equations

For every integer m, the mth polylogarithm function [O] is defined by the power series Lim(z) = ∞ X n=1 zn nm,

which converges for |z| < 1. The relation d

dzLim(z) = 1

zLim−1(z) (2.1)

together with the obvious identity Li0(z) = z/(1 − z) shows that the negative

poly-logarithms Lim(z) are rational functions of z with a pole of order −m + 1 at z = 1.

For m = 1 we get the usual logarithm,

Li1(z) = − log(1 − z),

or more precisely the branch of the logarithm which is principal in 1, i.e. Li1(z) =

Rz

0 du

u−1. This explains both the name and the reason why things start to get more

involved (and more interesting). By induction on m using (2.1), we can define Lim,

m ≥ 1 at any z ∈ C \ {1} as soon as we are given a path γ from 0 to z, not passing through 1 or 0 (except at the start):

Liγ_m(z) = Z z

0

Liγ_m−1(t)dt

t (2.2)

where the integral is taken along γ and, for m ≥ 2, Liγ₂(t) denotes consistently the value obtained by restriction of γ to a path from 0 to t. Therefore the analytic

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continuations of the positive polylogarithms exist on any simply connected neigh-borhood ∆ of 0 contained in C \ {1} (just choose a path contained in ∆). Being na¨ıf, we may take ∆ = C \ [1, ∞) and try to define an analytic continuation of Li2

on the whole of C \ {1} by setting, for x > 1, Li2(x) = limz→xLi2(z) where z → x

either from above or from below (this corresponds to rotating the cut slightly be-low or above the real line). This attempt fails since the difference between the two values obtained, calculated by deforming the resulting closed path to the positively oriented circle Cx of radius x centered around the origin, is

I Cx Li1(t) dt t = I Cx Z t 0 du u − 1 dt t = I Cx Z 1 0 tdλ λt − 1 dt t = Z 1 1/x 2πidλ λ = 2πi log x. It follows that while no analytic continuation of Li2 exists on C, the function

DC

0 = Li2(z) − Li1(z) log |z| obtained here is continuous on C \ {1} (remark that

Li1(z) log |z| → 0 as z → 0, and as z → 1 too); its imaginary part turns out to have

many nice features.

Proposition/Definition 2.1. (1) The Bloch-Wigner dilogarithm is the function D : P1(C) → R (well) defined according to (2.2) by

D(z) = =(Liγ₂(z) − Liγ₁(z) log |z|), (2.3) where γ is any path from 0 to z not passing through 0, 1 or ∞ except possibly at the endpoints.

(2) The Bloch-Wigner dilogarithm is a continuous function on P1(C), real ana-lytic on C∗∗= P1(C) \ {0, 1, ∞}, with differential dD(z) = η(z, 1 − z) where for any pair of functions x, y on P1(C),

η(x, y) := log |x|d arg(y) − log |y|d arg(x).

Proof. The form η(x, y) is a well-defined real singular 1-form on P1(C) since d arg(x) = =(dx_x ) is, with singularities where x or y are 0 or ∞. Denote by Dγ(z) the right-hand side of (2.3). A simple calculation establishes the formula dDγ = η(z, 1 − z); in particular η(z, 1 − z) is the differential of the continuous function D0 = =DC0

defined above, and therefore is an exact form. This proves that D(z) = Dγ_{(z) =}

R

γη(z, 1 − z)dz is a real analytic, single-valued function on C

∗∗_{, and continuous at 0.}

The proof that D is well defined and continuous at 1 and ∞ is given in Corollary 2.3.

The relation D(z) = −D(¯z) is clear, so that D vanishes on P1(R). But the Bloch-Wigner function also enjoys many holomorphic functional equations, whose structure we will describe shortly.

The map (x, y) 7→ η(x, y) is clearly antisymmetric in its arguments, and satisfies η(x, yy0) = η(x, y) + η(x, y0).

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Definition 2.1. Let G be an abelian group. The group Λ2G is defined as the quotient group

Λ2G := G ⊗_ZG/R,

where R is the subgroup generated by all elements of the form x ⊗ x for x ∈ G. That is, Λ2G is the set of formal sums with integer coefficients

X

i

ni(xi∧ yi)

with group law given by formal addition and the (redundant) set of rules

x ∧ x = 0, x ∧ y = −(y ∧ x), x ∧ yy0= x ∧ y + x ∧ y0, xx0∧ y = x ∧ y + x0∧ y. By the remark preceeding this definition, we have a well-defined homomorphism which we still call η from Λ2C(t)∗ to the space of real singular 1-forms, and actually since the latter is an R-algebra we can lift η to a map (again denoted by the same letter) on Λ2C(t)∗⊗ Q. Then we have proven the following proposition.

Proposition 2.2. Let Ω1(P1) denote the space of real analytic differential 1-forms on complements of finite subsets of P1(C). Then there is a commutative diagram

Λ2C(t)∗⊗ Q Z[C(t)∗] d ◦ D∗ -∂ -Ω1(P1) η

-where η is the map defined above, ∂ is given by ∂([f ]) := f ∧(1−f ) and then extended by linearity, and for ξ =P ni[fi] ∈ Z[F∗] we define d ◦ D∗(ξ) :=P nid(D ◦ fi).

This is extremely useful for describing functional equations of D.

Corollary 2.3. (1) Let ni and fi be finite collections of integers and non-constant

rational complex functions respectively. If ∂(P ni[fi]) is torsion in Λ2C(t)∗ then X

niD(fi(t)) = constant. (2.4)

(2) The Bloch-Wigner function D is a continuous function on P1(C).

Proof. A priori, since we have not yet given the proof that D is continuous at 1 and infinity, equation (2.4) holds only away from the poles and ‘ones’ of the fi.

Now ξ1= [z] + [1 − z] and ξ2 = [z] + [1/z] give the equations D(z) = −D(1 − z) and

D(z) = −D(1/z) for z 6= 0, 1, ∞ (the algebraic verification that ∂(ξ2) is torsion is

carried out below; for ξ1 this is clear). Since D is continuous at 0, these functional

equations show that it is also continuous at 1 and ∞, with D(1) = D(∞) = −D(0) = 0. Then continuity implies that for any ξ ∈ Ker ∂, the equation (2.4) actually holds on the whole of P1(C).

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We remark that nothing is changed if we replace C(t) by any other field of (reasonably regular) complex functions. We can now list some functional equations for D. We have the six-fold symmetry

D(z) = −D(1 − z) = −D(1 z) = D( 1 1 − z) = D( z − 1 z ) = −D( z z − 1).

Indeed, all of these identities are generated by the first two ones. The very first one is obvious from the criterion just given (and the constant, as in all the other cases, must be set to zero since z 7→ 1 − z has a fixed point), while the second one is proved by

z ∧ (1 − z) + z−1∧ (1 − z−1) = z ∧ (1 − z) − z ∧ (z − 1) + z ∧ z = z ∧ −1 and this is 2-torsion since 2(z ∧ −1) = z ∧ 1 = 2(z ∧ 1) = 0. The relation D(z) = −D(z−1_{) also leads to the extension of D to a continuous function on P}1_{(C), real}

analytic on P1\ {0, 1, ∞}.

In general, to get elements in Ker ∂, and thus functional equations, we might take families of functions (zi)i∈I with I a finite set and 1 − zj ∈ hziii∈I; a simple

way of satisfying this is taking I to be a quotient of Z and 1 − zi = zi−1zi+1. It

is a very easy to check but crucial fact that this implies that zi+5 = zi: we thus

define a 5-cycle to be a ξ =P

i∈Z/5Z[zi] with zi satisfying the above relation; then

∂(ξ) =P

izi∧ (1 − zi) =Pizi∧ zi−1+Pizi∧ zi+1= 0. We may even allow zi = 0

or ∞ provided that for no i we have {zi−1, zi+1} = {0, ∞}. An easy calculation

shows that 5-cycles are parametrised by pairs of functions x = z1 and y = z3, giving

rise to

V (x, y) := [x] + [y] + [ 1 − x

1 − xy] + [1 − xy] + [ 1 − y 1 − xy]. The corresponding relation

D(x) + D(y) + D 1 − x 1 − xy + D (1 − xy) + D 1 − y 1 − xy = 0,

which holds whenever its terms make sense, is known as the five-term relation and is the fundamental functional equation of the dilogarithm. The one-variable identities arise as specialisations of the five-term relation when some zi is 0 or infinity (e.g.

setting z3 = y = 0 yields D(x) + D(1 − x) = 0) but actually much more is known or

conjectured to be true: all of the functional equations of the Bloch-Wigner function should be consequences of the five-term relation. For the case of rational functions of one variable this has been proved by Wojtkowiak: to formulate a precise statement we need to introduce the Bloch group, which we will do in the next section; but the impatient reader can skip to Theorem 2.5.

Functional equations for the classical dilogarithm arise at the same sets of ar-guments which give functional equations of the Bloch-Wigner dilogarithm – but as

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expected they are less clean, involving some constants and 1-logarithms. Explicitly, we have Li2(z) = −Li2( 1 z) − π2 6 − 1 2log 2_{(−z) = Li} 2(1 − z) + π2 6 − log(z) log(1 − z) and the five-term relation1

Li2(x) + Li2(y) + Li2 1 − x 1 − xy + Li2(1 − xy) + Li2 1 − y 1 − xy =π 2

6 − log(x) log(1 − x) − log(y) log(1 − y) + log

1 − x 1 − xy log 1 − y 1 − xy . All proofs are easily done by differentiation and use of the obviuous relation Li2(1) =

π2/6. The latter is one of the eight known special values of the dilogarithm, all of which are immediate consequences of this and the one-variable functional equations:

Li2(0) = 0 Li2(1) = π2 6 Li2(−1) = − π2 12 Li2 1 2 = π 2 12 − 1 2log 2₍₂₎ Li2 3 −√5 2 ! = π 2 15 − log 2 1 + √ 5 2 ! Li2 −1 +√5 2 ! = π 2 10 − log 2 1 + √ 5 2 ! Li2 1 −√5 2 ! = −π 2 15 + 1 2log 2 1 + √ 5 2 ! Li2 −1 −√5 2 ! = −π 2 10 + 1 2log 2 1 + √ 5 2 ! .

There is another natural correction to the classical dilogarithm which gives rise to a well-behaved function, though only defined on P1(R). That is, we can “symmetrise” the form dLi2(z) = − log(1 − z)dz_z to get ω(z) := −1₂log(1 − z)dz_z − 1

2log(z) dz

1−z. A solution to the differential equation dL(x) = ω(x) on P1(R)\{0, 1, ∞}

is given by the Rogers dilogarithm, defined on the interval (0, 1) by L(x) := Li2(x) +

1

2log(x) log(1 − x) − π2

6 for 0 < x < 1 and extended by L(0) = −π2/6, L(1) = 0 and

L(x) = (

−L(1/x) if x > 1,

−π₃2 − L(x/(x − 1)) if x < 0.

Then L is a continuous strictly increasing function on R, real analytic on R \ {0, 1}. At infinity we have limx→−∞L(x) = −π2/3, limx→+∞L(x) = π2/6, so that

1_{Of course, we are presenting the theory in a completely anti-historical way: the five-term relation}

for Li2was discovered (and re-discovered many times) in the nineteenth century, long before Bloch

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L extends to a continuous function ¯L : P1(R) → R/π₂2Z. The same methods which work for D lead to the functional equations

¯ L(x) = ¯L(1) − ¯L(1 − x) = − ¯L(1/x) − ¯L(1), ¯ L(x) + ¯L(y) + ¯L(1 − x 1 − xy) + ¯L(1 − xy) + ¯L( 1 − y 1 − xy) = 0.

We record that our definition of L follows [Ne 2], [D-Z] and [G-Z] rather than the classical one used by [Za 3] and [Na 2], from which it differs by the constant −π2/6. Also for future reference we point out the following (non-projected) version of the five-term relation: L(x) − L(y) + L(y x) − L( 1 − x−1 1 − y−1) + L( 1 − x 1 − y) = 0 for 0 < y < x < 1, (2.5) which is obtained by applying one-variable symmetries to the five-term relation associated to V (x, y/x).

It is quite natural to wonder whether one can combine the real-valued functions D and L (which is defined on the real line, i.e. exactly where D vanishes) into a single, complex-valued function. The answer will be given in Section 2.3.

2.2 The Bloch group

A convenient tool to encode dilogarithm identities is provided by the Bloch group. Definition 2.2. The pre-Bloch group P(F ) of a field F is the quotient of Z[F \ {0, 1}] by all meaningful instances of the relations

[x] + [1 x] = 0, [x] + [1 − x] = 0, [x] + [y] + [ 1 − x 1 − xy] + [1 − xy] + [ 1 − y 1 − xy] = 0.

The symbols [0], [1], [∞] are also allowed for convenience, and intepreted as 0. The Bloch group B(F ) of F is the group defined by the exact sequence

0 −→ B(F ) −→ P(F )−→ F∂ ∗∧ F∗ −→ K2(F ) −→ 0,

where by (Milnor’s) definition K2(F ) := Coker (∂), with ∂([x]) := x ∧ (1 − x).

Equivalently, P(F ) = Z[F ]/h5-cycles, [0], [1], [∞]i. Definitions of the Bloch group vary in the literature according to whether or not one allows degenerate symbols and/or quotients by the one-variable relations when defining P(F ), but they are all equal modulo torsion by results of Dupont and Sah [D-S 1]. In some of the most important cases this is not an issue at all, as follows from the following theorem.

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Proposition 2.4 ([D-S 1, 5.11]). Let F be an algebraically closed field. Then the relations [x] + [1/x] = 0 and [x] + [1 − x] = 0 are consequences of the five-term relation.

By linearity the Bloch-Wigner function extends to a homomorphism D : B(C) → R, and the Rogers dilogarithm extends to a homomorphism L : B(R) → R/¯ π

2

6 Z.

Then Corollary 2.3 says that if P

ini[fi] belongs to the Bloch group of a field of

complex functions, then P

iniD ◦ fi is constant. One may wonder whether the

converse also holds: as anticipated, for the case of rational functions of one variable it is even true that any functional equation of this sort corresponds (modulo constants) to the zero element of the Bloch group, i.e. that any such functional equation is a consequence of the five-term relation.

Theorem 2.5 (Wojtkowiak). (1) If ξ = P

i[xi(t)] ∈ Z[C(t)∗] is associated to a

functional equation of the dilogarithm D(ξ) = 0 , then ξ ≡ constant in the pre-Bloch group P(C(t)).

(2) The map B(C) → B(C(t)) is an isomorphism.

Proof. We refer the reader to [Za 3, II.2] for a simple proof of (1). Part (2) fol-lows from (1) since any ξ ∈ B(C(t)) gives rise to a functional equation by Corollary 2.3, hence it is equivalent to a constant α ∈ C with ∂(α) = 0 in Λ2C(t)∗; special-ization of this relation shows that α ∈ B(C). This proves surjectivity. Injectivity follows from an equally easy specialization argument.

We now describe the beautiful picture surrounding the Bloch groups of number fields. For F a number field with r1 real and 2r2 complex embeddings, B(F ) is

isomorphic modulo torsion to the group Kind

3 (F ), which for us will just be a black

box (which nevertheless motivated Bloch’s original definition of B(F ) – see [Su] and references therein). Now this black box was proven by Borel [Bo] to be canonically isomorphic (again modulo torsion) via the “regulator mapping” to a lattice in Rr2_,

whose covolume is a simple multiple of the zeta value ζF(2). Moreover the following

diagram commutes K₃ind(F )/{torsion} B(F )/{torsion} ∼ -Rr2 reg_F -B(C)r2 D -(σ1_{, . . . , σ}r2₎

-where the σi are representatives of the conjugacy classes of embeddings F ,→ C, and the map that they define makes sense modulo torsion by a theorem of Merkur’ev and Suslin which we will state in a few lines. In particular, ζF(2) is essentially a sum of

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between values of Dedekind zeta functions at positive integers m > 2 and of mth polylogarithms at algebraic arguments is also conjectured to hold ([Za 1], [Za 2]).

Now we investigate the torsion part of B(F ). A torsion element ξ ∈ B(F ) is mapped to zero by the Bloch-Wigner dilogarithm in all real or complex embeddings, and to rational multiples of π2/6 by the Rogers dilogarithm in all real embeddings. Thus we have a way to distinguish elements which are torsion from those which are zero only in the real setting (i.e. we can check that L(ξ)/π₆2 is not an integer). The reason is the theorem of Merkur’ev and Suslin ([M-S]) mentioned earlier, which implies that any torsion element in the Bloch group of a field F becomes 0 in the Bloch group of a finite extension of F obtained by adjoining sufficiently many roots of unity; in particular B( ¯Q) and B(C) are torsion-free. As an example, we have L(1/2) = −π2/12 so that [1/2] is not zero in B(R), but it becomes zero in B(C) since

0 ≡ V (i, i) ≡ [2] + 2([i] + [1 − i

2 ]) ≡ −[2] + 2([i] + [1 − i]) ≡ [1/2].

On the other hand, the relation with the regulator map described above gives a (necessary and) sufficient condition for an algebraic element in the Bloch group to represent zero.

Proposition 2.6. Let ξ ∈ B( ¯Q), then ξ ≡ 0 if and only if D(ξσ) = 0 for every embedding Q(ξ),→ C.σ

It is easy to describe all symbols [x], x ∈ C, which represent torsion in the Bloch groups of their fields of definition: this classification will be used to prove the rank 1 case of Nahm’s conjecture (compare also with the list of special values of the dilogarithm above).

Proposition 2.7. The following are the only nine numbers x ∈ C \ {0, 1} such that [x] is torsion in the Bloch group of Q(x):

x = 1 2, −1, 2, ±√5 ± 1 2 , 3 ±√5 2 .

Proof (sketch). For [x] to belong to the Bloch group we must have x∧(1−x) = 0 in Λ2C∗, which implies log(x) ∧ log(1 − x) = 0 in Λ2(C/2πiQ). The latter is a Q-vector space, therefore p log(x) = q log(1 − x) modulo 2πiQ for some coprime integers p, q, i.e. exponentiating x = tq_{, 1 − x = ζt}p _{for some root of unity ζ and}

t ∈ C∗. Thus x is an algebraic number. Now the torsion condition implies that x is totally real since we must have D(xσ) = 0 for all conjugates of x, and D is non-zero outside the real line (this follows from the interpretation of D(z) as a volume given in Section 2.4); therefore ζ = ±1, and t satisfies tp ± tq _{= 1. By applying one of}

the six-fold symmetries, one can reduce to considering only tp+ tq− 1 = 0, where p > q ≥ 1. When this equation is irreducible, the total reality condition implies that

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it has only real solutions; this condition descends to all derivatives of the polynomial and can be applied to its reciprocal too, imposing the restrictions p − q ≤ 1, q ≤ 2, from which it easy to conclude. Otherwise some additional argument regarding the possible irreducible factors of the polynomial is needed (using [Lj, Thm. 3] – I have not carried out the details in this case). To check that the numbers listed are really (2-)torsion it suffices to apply to them one of the six usual symmetries. For future purposes, we record that if we restrict ourselves to 0 < x < 1 then x corresponds to a solution of 1 − x = xA for A = 1/2, 1, or 2.

The above proof began by showing that the condition x ∧ (1 − x) = 0 is an algebraic condition. In general, it is conjectured that the condition ∂(ξ) = 0 is also algebraic modulo five-term relations, i.e. more precisely:

Conjecture 2.8 (Bloch Rigidity). The map B( ¯Q) → B(C) is an isomorphism. Presently, it is known (by work of Bloch) that D(B(C)) = D(B( ¯Q)). Moreover, Theorem 2.5 can also be interpreted as a rigidity result, since it states that no rationally parametrized families exist in B(C).

2.3 The extended Bloch group and the extended

dilog-arithm

We still need to address the problem of extending the dilogarithm (or something similar to it) to a single-valued complex function. Actually, we are even more inter-ested in finding a complex-valued extension of the Bloch-Wigner dilogarithm defined on a suitable extension of the Bloch group; but let us first analyse the situation in the geometric setting.

What hinders the existence of an extension of the dilogarithm on the doubly punctured Riemann sphere C∗∗= P1\ {0, 1, ∞} is the multi-valuedness of log(z) and log(1 − z), and to circumvent the problem we must lift our function on a suitable cover of C∗∗. Clearly a single-valued extension of Li2 does exist on the universal

cover, but its Galois group is π1(C∗∗) = Z ∗ Z, which makes it quite intricate to

deal with the covering transformations. A good compromise is provided by the universal abelian cover ˆC of C∗∗, on which we will be able to define an extension of the dilogarithm with values in C/4π2Z.

The abelianization of π1(C∗∗) is Z ⊕ Z, with generators corresponding to the

monodromy around 0 and 1, so that ˆC is the Riemann surface for the function z 7→ (log(z), log(1 − z)). A geometric description of ˆC is the following. Let Csplit be

the complex plane split along the lines (1, +∞) and (−∞, 0), i.e. the union of C∗∗\ (1, +∞) ∪ (−∞, 0) and of the doubled lines {x + 0i, x − 0i | x ∈ (1, +∞) ∪ (−∞, 0)}. Then ˆC is the identification space constructed from Csplit× Z × Z, with the glueing

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rules

(x − 0i, r, s) ∼ (x + 0i, r + 1, s) for x ∈ (1, ∞) (x − 0i, r, s) ∼ (x + 0i, r, s + 1) for x ∈ (−∞, 0). We can also give the alternate description

ˆ

C = {ˆz = (u, v) ∈ C2| eu+ ev = 1},

with covering map given by (u, v) 7→ z = eu= 1 − ev; the isomorphism between the two covers is explicitly given by (z; p, q) 7→ (u, v) = (Log (z)+2πip, Log (1−z)+2πiq), where Log (z) and Log (1 − z) are a pair of arbitrary (but fixed) branches of log(z) and log(1 − z).

The covering map ˆC → C∗∗ factors through C \ 2πiZ by (u, v) 7→ v 7→ 1 − ev, and already on the latter space we can define

b

Li2(v) = Li2(1 − ev),

which has derivative bLi0₂(v) = _1−e−v−v; this is a meromorphic function on C with

residues 2πim at the poles 2πim, m ∈ Z, so that bLi2 is a single-valued function on

C \ 2πiZ with values in C/(2πi)2Z.

The variant of the dilogarithm which lends itself to an extension is the Rogers dilogarithm: the reason, as will be clear from what follows, is that it fulfills both the condition of satisfying a “clean” five-term relation at least in some domain, and that of admitting an analyitc continuation on a neighborhood of that domain. Definition 2.3. The extended dilogarirthm is the function ˆL : ˆC → C/4π2Z de-fined by ˆ L(u, v) := Li2(z) + 1 2uv − π2 6 , z = e u _{= 1 − e}v_.

Let us now define the appropriate extension of the Bloch group, following the treament of [G-Z], which slightly improves on the ones of [Ne 2] and [D-Z]. Since we are working over C, by Proposition 2.4 we can describe the Bloch group just in terms of five-term elements, whose labelling we choose in agreement with (2.5) and the cited papers.

Consider the set FT := FT(x, y) := x, y,y x, 1 − x−1 1 − y−1, 1 − x 1 − y | x 6= y, x, y ∈ C∗∗ ⊂ (C∗∗)5,

and the map ρ : (C∗∗)5→ Z[C∗∗_{] given by}

ρ(z0, z1, z2, z3, z4) = 4

X

i=0

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Then B(C) is just the middle homology of the complex FT−→ Z[Cρ ∗∗]−→ Λ∂ 2_C∗,

and the Bloch-Wigner dilogarithm gives a well-defined homomorphism D : B(C) → R. We wish to build a correpsonding “extended” picture.

Let FT+ ⊂ FT be set of 5-tuples FT(x, y) = (z0, z1, z2, z3, z4) with all zi

belong-ing to the upper half-plane H. Then an elementary computation shows that FT+ is exactly the set of all FT(x, y) with y ∈ H and x in the interior of the euclidean triangle with vertices 0, 1, y. Therefore FT+ is connected. By choosing principal branches of log(z) and log(1 − z) we get an inclusion (0, 1) ∪ H ,→ ˆC such that ˆ

L|(0,1) = L mod 4π2. We call cFT +

⊂ ˆ_C5 _{the homeomorphic image of FT}+ _under

this inclusion. Then (2.5) says that the equation ˆ L(x) − ˆL(y) + ˆL(y x) − ˆL( 1 − x−1 1 − y−1) + ˆL( 1 − x 1 − y) = 0

holds on (part of) the boundary of the connected set cFT+; hence by analytic con-tinuation it also holds on the whole of cFT+, and indeed also on the whole connected component cFT of the preimage of FT in ˆC5 which contains cFT

+

. Therefore ˆL is a well-defined function on the extended pre-Bloch group bP(C) := Z[ ˆC]/ ˆρ( cFT), where ˆ

ρ is defined in the obvious way in analogy with ρ. We need only one more ingredient to build the complex from which we will define the extended Bloch group.

Lemma 2.9. The linear map ˆ∂ : Z[ ˆC] → Λ2C defined by∂[(u, v)] := u ∧ v vanishesˆ on ˆρ( cFT).

Proof. Let z = (z0, z1, z2, z3, z4) ∈ FT, and choose a branch Log of the

log-arithm in a simply connected region in C∗∗ containing all the zi and the 1 − zi

for i = 0, . . . , 4. Then ˆρ((Log z0, Log (1 − z0)), . . . , (Log z4, Log (1 − z4))) is clearly

in Ker ˆ∂. This shows on the one hand that ˆρ( cFT+) is in Ker ˆ∂, and on the other hand that for ˆz ∈ ˆC5 in the preimage of FT we have ˆ∂ ◦ ˆρ(ˆz) = w ∧ 2πi for some w = w(ˆz) ∈ C/2πiQ depending analytically on ˆz. Then the desired result follows by analytic continuation.

Definition 2.4. The extended Bloch group bB(C) is the middle homology of the complex c FT−→ Z[ ˆρˆ C] ˆ ∂ −→ Λ2 C, where cFT, ˆρ and ˆ∂ are as defined above.

Then we have proved that there is a well-defined homomorphism ˆ

L : bB(C) −→ C/4π2

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On bB(C) the imaginary part of ˆL is something very well-known: for ˆz = (u, v) we have = ˆL([ˆz]) = =Li2(z) + 1₂=(uv) = =Li2(z) + 1₄(uv − ¯u¯v), whereas D(z) =

=Li2(z)+=(u)<(v) = =Li2(z)+1₄(uv−¯u¯v+u¯v−¯uv). Since the map [(u, v)] 7→ u¯v−¯uv

factors through the wedge product, we get = ˆL([ˆz]) = D(z).

We summarise below what we have just proven and the description of the relation between bB(C) and B(C) provided by [G-Z, 3.5].

Theorem 2.10. There is a commutative diagram with the first row exact

0 - _Q/Z - B(C)b - B(C) - 0 Q/Z id ? 4π2 - _C/4π2_Z ˆ L ? ₌ - _R D ?

In particular this shows that while the “classical” Bloch group is torsion-free, the extended Bloch group has non-trivial torsion which can be detected by the real part of the extended dilogarithm.

As mentioned before, what we have presented here is a refinement of Neumann’s (slightly less) extended Bloch group bB0_{(C), whose construction [Ne 2] is similar to} ours but uses a disconnected cover of C∗∗isomorphic to four copies of ˆC. Accordingly, there is an exact sequence

0 −→ Z/4Z −→ bB(C) −→ bB0_{(C) −→ 0,}

which induces a (slightly less) extended dilogarithm ˆL0 : bB0_{(C) → C/π}2

Z. While b

B(C) is more natural and precise, the group bB0_{(C) has a geometric interpretation,} which we will review in the next section.

2.4 Invariants of hyperbolic 3-manifolds

We briefly survey the beautiful geometric interpretation of the theory developed so far, with the only ambition of conveying the underlying geometrical ideas. For more details we refer to the very enjoyable exposition [Ne 1] and the book [D] and references therein.

The Bloch-Wigner dilogarithm has the same six-fold symmetry as a cross-ratio: that is, the permutation group on three elements S3 has a faithful action on P1(C)

given by sending two distinct transpositions to the the involutions z 7→ 1 − z and z 7→ 1/z, so that D(σ · z) = (−1)σD(z) for all σ ∈ S3; on the other hand the (even)

surjection S4 → S3 given by the action on pairs from 4 elements is such that the

cross-ratio [z0 : z1 : z2 : z3] := (z_(z0₃−z_−z₁3)(z_)(z1₂−z_−z2₀)₎ transforms by [zτ (0) : zτ (1) : zτ (2) :

z_{τ (3)}] = σ · [z0 : z1 : z2 : z3] whenever S4 3 τ 7→ σ ∈ S3. Therefore the function of

4-tuples of elements of P1(C)

e

D(z0, z1, z2, z3) :=

(

D([z0: z1 : z2 : z3]) if the zi are pairwise distinct

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satisfies eD(τ (s)) = (−1)τD(s) for every τ ∈ Se ₄. Hence eD gives a map on the group C3 of oriented nondegenerate 3-chains on P1(C), where we recall that for any

n ∈ N, Cn is the quotient of the free group on (n + 1)-tuples of elements of P1(C)

(n-symplices) modulo the symplices consisting of non-pairwise distinct elements and the relation [zτ (0), . . . , zτ (n)] = (−1)τ[z0, . . . , zn] for τ ∈ Sn+1. We get a complex

C∗ by defining boundary maps ∂n([z0, . . . , zn]) =

Pn

i=0(−1)i[z0, . . . , ˆzi, . . . , zn]. It is

easily checked that the five-term relation for D is equivalent to the relation D ◦ ∂4 =

0, which since H3(P1(C), R) = 0 is in turn equivalent to saying that D factors through ∂3.

We have thus identified the Bloch-Wigner dilogarithm with an additive map on 3-chains which only depends on what they are bounded by. It is not difficult to guess what such a map could be: it is volume, in the following sense. We view P1(C) as the boundary of the compactification of the upper half-space model of the hyperbolic 3-space H3. A hyperbolic polyhedron is the region bounded by a finite number of geodesic planes, and an ideal polyhedron is one all of whose vertices lie on the boundary. Then it was proved by Lobachevsky (see [Mil] for a proof) that eD([z0, z1, z2, z3]) is the hyperbolic volume (which we shall denote by vol) of

the oriented ideal tetrahedron with vertices z0, z1, z2, z3. To visualise the geometric

content of the five-term relation, think of two tetraehedra glued at one face F (these are two terms) and then think of the subdivision of the resulting polyhedron into three tetrahedra which is obtained by drawing the line joining the vetices not lying on F (these are the other three terms).

Before proceeding, we give some context. Thurston’s geometrization conjecture, now widely believed to be proven, roughly asserts that all three-manifolds can be cut into pieces each carrying one of eight geometric structures. Seven of them can be relatively easily classified; the eighth one is hypebolic geometry. In this case the volume is actually a topological invariant (Mostow-Prasad rigidity theorem); however, we can not expect it to be a full invariant since, trivially, the volume of a polyhedron is invariant under “scissors congruence”, i.e. the relation which identifies two polyhedra if they can be cut by geodesic planes into a finite number of isometric polyhedra. However one may still ask how much information about the scissors congruence class of a polyhedron is captured by the volume.

For plane polygons, Euclid defined the area by appealing to scissors congruence (cutting with geodesic lines), and indeed it is an amusing exercise to prove that two polygons of equal area are scissors congruent (and the same is true for hyperbolic and spherical polygons). To define volume he had to appeal to the “exhaustion principle” (i.e. continuity): he did not expect the volume to capture all information about the scissors congruence class of a polyhedron, and neither did Hilbert, who asked as the third of his famous list of problems to find examples of non-scissors congruent tetrahedra with the same volume. The answer was given the very same

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year (1900) by Max Dehn, who defined the scissors congruence invariant δ(P ) :=X

E

`(E) ⊗ θ(E) ∈ R ⊗ R/πQ,

where the sum ranges over all edges of P , and ` and θ denote length and dihedral angle respectively. As an example, a cube always has Dehn invariant 0, whereas a tetrahedron has nonvanishing Dehn invariant. One may wonder whether volume and Dehn invariant suffice to characterize the scissors congrunece class of a polyhedron, and in the euclidean case this was answered affirmatively by Sydler. The same question in the hyperbolic case leads directly to the Bloch group.

To make some precise statements, define the scissors congruence group P(H3) as the quotient of the free group on the congruence classes of hyperbolic polyhedra by the relations [P ] = [Q] + [Q0] whenever Q and Q0 can be glued along faces to form P .2 Then the volume is a well-defined function on P(H3). We want to relate the (pre-)Bloch group to the scissors congruence group, as well as to the group C3defined

above. The latter task is easy: SL(2, C), and indeed also its quotient P SL(2, C), act by isometries on P1(C); if we quotient C3by this action then we just get the free group

on cross-ratios, and if we further quotient out the boundaries of elements of C4we get

exactly the pre-Bloch group P(C). As a digression we record, for future reference, that the same observation yields maps from the third group homology of these groups (considered with the discrete topology) to the pre-Bloch group: explicitly, the maps are given by H3(G, Z) 3 [(g0, g1, g2, g3)] 7→ [g0∞ : g1∞ : g2∞ : g3∞] ∈ P(C).

Moreover Bloch and Wigner proved that these maps give natural exact sequences

0 −→ Q/Z −→ H3(SL(2, C), Z) −→ B(C) −→ 0, (2.6)

and a completely analogous one for P SL(2, C).

The similarity between the exact sequence (2.6) and the one in Theorem 2.10 is not accidental: there are isomorphisms H3(SL(2, C), Z) ∼= bB(C), and H3(P SL(2, C), Z) ∼=

b

B0_{(C). The geometric interpretation of the latter is given at the end of this section.} To see how P(H3) fits in this picture made up of ideal tetrahedra, we remark that:

• a non-ideal tetrahedron can be written as the scissors-congruence difference of two tetrahedra with some vertices at infinity;

• the cross-ratio of a positively oriented ideal 3-symplex (i.e. one for which the orientation induced by the labelling of the vertices coincides with the one inherited from H3) is a number z ∈ H, and reversal of orientation corresponds to complex conjugation of the cross-ratio parameter.

2_{Then actually [P ] = [Q] if and only if they are stably scissors congruent – we will ignore this}

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• any n-symplex s is scissors-congruent to its mirror image (barycentrically sub-divide it by dropping perpendiculars from the circumcenter of each face f to each face of f : then each of the resulting pieces can be labelled by a path of length n (s = fn, . . . , f0) where fi is an i-dimensional face of fj for i < j;

and pairs of pieces whose paths differ only by the element f0 are mirror image

pairs). Therefore scissors congruence does not distinguish orientation. All this should make plausible the following theorem of Dupont and Sah.

Theorem 2.11 ([D-S 1, 4.7]). The scissors congruence group P(H3) is generated by ideal tetrahedra, and the map P(C) → P(H3_{) is the one which annihilates [x] if}

x is real, and if z /∈ R sends [z] to ±[T ], where T is the tetrahedron with cross-ratio parameter z and the sign is chosen according to whether z ∈ H or z ∈ ¯H. The kernel of this map is generated by the mirror-image relations [z] ≡ −[¯z] and [x] ≡ 0 for x ∈ R.

Therefore the pre-Bloch group of C can be thought of as the appropriate recep-tacle for an orientation-sensitive version of the scissors congruence group, of which P(H3_{) is the “imaginary part”. Accordingly, it can be shown [Ne 1, 2.5] that the}

Dehn invariant is (twice) the “imaginary part” of ∂ : P(C) → Λ2C∗.

Let us now come to hyperbolic 3-manifolds, by which we will always mean com-plete, orientable, finite volume ones. Any hyperbolic 3-manifold can be written as M = H3/Γ, where Γ is a discrete group of orientation-preserving hyperbolic isome-tries. By triangulating M (in a weak sense which we will not make precise) into ideal tetrahedra, we can attach to it a class β(M ) ∈ P(C), which can be shown to actually lie in B(C) (plausibility argument: the Dehn invariants of edges meeting at any fixed vertex should add up to zero) and to be independent of the triangulation chosen (a very precise construction has been given in [N-Z]). The invariant β(M ) captures the information about volume. Now a hyperbolic 3-manifold has another important invariant, the Chern-Simons class cs(M ), which lies in R/π2

Z and was shown by Yoshida to be analytically related to volume, in a simple way which will be made more precise later (we will not attempt to define the Chern-Simons invariant precisely: it is a cohomological obstruction to the existence of conformal embed-dings in euclidean space, see [C-S]). To fully capture the Chern-Simons invariant, the class β(M ) is not sufficient; but, as we shall see, it is possible to define a lift

ˆ

β(M ) ∈ bB0_{(C) of β(M ) to solve this problem.}

A simple example can help understand the geometric content of Neumann’s extended Bloch group (see [Ne 2] for details). Kids are taught that the sum of the internal angles of a triangle equals π, but since angles are only defined modulo 2π, it should be more correctly stated that that sum is any odd multiple of π, the (posssible) choice of which is dictated by extrinsic reasons. We will refer to lifts of measures of angles as “log-parameters” for the angles. Then the kids’ story reflects of course the most common choice of log-parameters, but different ones can

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be more natural in some contexts. In particular, in considering the triangulation of a polygon, it is reasonable to require that the log-parameters for the angles meeting at an internal vertex of the triangulation should add up to zero rather than 2π (we call this a “flattening condition”). A similar picture holds for ideal hyperbolic tetrahedra: dihedral angles relative to edges meeting at a fixed vertex add up to π mod 2π, and it can be shown that a choice of log-parameters yields a lift of the cross-ratio parameter of the tetrahedron [z] ∈ P(C) to a class in Neumann’s pre-Bloch group bP0_{(C). When a ξ = β(M ) ∈ B(C) comes from the triangulation of}

a 3-manifold M , we have analogously with the two-dimensional euclidean case a flattening condition for the choice of log-parameters, which uniquely determines an element ˆξ = ˆβ(M ) ∈ bB0_{(C). Then we know that = ˆ}L0( ˆβ(M )) = D( ˆβ(M )) = vol(M ); the keen reader may have already guessed that the upshot is that the real part of ˆL0 gives the Chern-Simons invariant of M . The analytic relation which we alluded to above is thus that these two invariants are the real and imaginary part of a single analytic function: ˆL0( ˆβ(M )) = cs(M ) + ivol(M ) ∈ C/π2Z.

Finally, the promised geometric interpretation of the isomorphism H3(P SL(2, C), Z) ∼=

b

B0_{(C).is the following: a 3-cycle in group homology determines a linear combination} of ideal tetrahedra and flattening conditions for them, in such a way to induce a bijection between our two groups.

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Chapter 3

Modular Forms from Conformal

Field Theory

A conformal field theory (CFT) is a quantum field theory which is invariant under conformal transformations (i.e. transformations which preserve the metric up to a scale). This is a rather peculiar condition for a physical theory, since it forbids the existence of characteristic lengths; however it can be satisfied by systems at critical equilibria, such as are found in condensed matter physics, and in some contexts in string theory. The interest in such theories is nevertheless due not only to their applications, but to their mathematical structure: the symmetry properties of CFTs (especially in two dimensions, the only case condisered here) can sometimes allow for exact solutions. Here we will introduce (two-dimensional) CFT as the underlying physical structure which gives a reason for the modularity of some functions of Nahm’s type (4). Hence the treatment will be rather one-sided, with fundamental features of CFT being completely ignored (a salient example being the “operator product expansion” of fields), and some results which would require much more background being stated without neither proof nor motivation. Nevertheless, CFT is a mathematically rigorous (and deep) theory, and it is possible to give very refined axiomatic treatments of the subject (for example in the language of vertex operator algebra), going however well beyond our scope.

The standard reference for conformal field theory is the excellent book [DF-M-S]. A mathematically oriented treatment of the fundamentals can be found in [Sc] or [Ga] and references therein.

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3.1 Conformal invariance in quantum field theory

The main ingredients of a quantum field theory (QFT) are:

• a manifold M of dimension d + 1 endowed with a Riemannian or pseudo-Riemannian metric, usually corresponding to physical spacetime;

• a complex vector space H endowed with a nondegenerate Hermitian inner product, most often but not necessarily a Hilbert space; physical states are rays in H;

• a distinguished normalised vector |0i ∈ H, called the vacuum, describing a zero or lowest energy state;

• a Z/2Z-graded vector space F = Fb⊕Ff of operator-valued functions (or more

generally distributions) M → End(H), called quantum fields; fields in Fb are

called bosons, fields in Ff fermions. More precisely and more generally, fields

should actually be viewed as sections of some bundle on M with fibre End(H); • a bijective map φ 7→ |φi of F onto a dense subspace of H called field-state identity. In elementary particle theory, fields are built in terms of creation or annihilation operators, so that in this case the interpretation of the field-state identity is that of giving, on the right-hand side of the identity, the state with the particle or excitation mode created by the left-hand side field in the remote past;

• an action functional, depending on some set of fields Φ and their derivatives S =

Z

ddx L(Φ, ∂µΦ),

where the integral is done by choosing a fixed-time “slice” of M , and L is a local functional called the Langragian density.

The physical content of the theory lies in the correlation functions hφ1(x1) · · · φn(xn)i,

which can be defined either in the path integral formalism, involving the action func-tional and a mathematically ill-defined Feynman path integral, or as

hφ₁(x1) · · · φn(xn)i := h0|φ1(x1) · · · φn(xn)|0i.

Their physical significance lies in their being the basic objects needed for calculating probability amplitudes for physical processes (e.g., for the transition from the state created by the left-hand side field to the state created by the right-hand side field). A symmetry of a quantum field theory is specified by the actions of some group (or Lie algebra) of (infinitesimal) transformations of M on the field space F and the state space H, in such a way that the field-state identity is equivariant; since physical