Topological Representations of the Quantum Group Uq (Sl2(C )) in Two

Topological Representations of the

Quantum Group U sl C in Two

Dimensional Conformal Field Theory

Als Habilitationsschrift dem Fachbereich Physik der

Westfalischen WilhelmsUniversitat M unster vorgelegt von

Christian Wieczerkowski

M unster

Institut f ur Theoretische Physik I

Westfalische WilhelmsUniversitatM unster

WilhelmKlemmStr

M unster

Topological Representations of the

Quantum Group U sl C in Two

Dimensional Conformal Field Theory

Als Habilitationsschrift dem Fachbereich Physik der

Westfalischen WilhelmsUniversitat M unster vorgelegt von

Christian Wieczerkowski

M unster

Institut f ur Theoretische Physik I

Westfalische WilhelmsUniversitatM unster

WilhelmKlemmStr

M unster

Acknowledgement

This work is based on the articles listed b elow which have b een written together with

Giovanni Felder and Mauro Crivelli I am grateful for a very pleasant collab oration I

would like to thank in particular Giovanni Felder for his supp ort and encouragement

in the technical part of this study

Sources

G Felder and C Wieczerkowski Topological Representations of the Quantum

Group U sl C Commun Math Phys

G Felder and C Wieczerkowski Fock Space Representations of A and Topo

logical Representations of the Quantum Group U sl C New Symmetry Prin

ciples in Quantum Field Theory eds J Frohlich G t Ho oft A Jae G Mack

P K Mitter and R Stora Plenum Press New York

M Crivelli G Felder and C Wieczerkowski Generalized Hyp ergeometric Func

tions on the Torus and the Adjoint Action of U sl C Commun Math Phys

M Crivelli G Felder and C Wieczerkowski Topological Representations of

U sl C on the Torus and the Mapping Class Group Lett in Math Phys

G Felder and C Wieczerkowski The KnizhnikZamolo dchikov Equation on the

Torus Vancouver Centre de Recherches Mathematiques CRM Pro ceedings and

Lecture Notes Vol

G Felder and C Wieczerkowski Conformal Blo cks on Elliptic Curves and the

KnizhnikZamolo dchikovBernard Equation Commun Math Phys

Contents

Introduction

Prologue

Scale Invariance

EnergyMomentum Tensor

Minimal Mo dels

U Current Algebra

Conformal Blo cks

Integral Representations

Lo cal Systems

Topological Representations of the Quantum Group U sl C

Introduction

Lo cal Systems on Conguration Spaces

Colored Braid Group oids

RMatrix Representations

Lo cal Systems

The Topological Action of U sl C

The SU case

Families of Lo ops

Op erators

Relations

Intersection Pairing

Reection and Duality

Intersection Pairing

Tensor Pro ducts and Copro duct

The mo dule A w w

r s

Tensor Pro ducts and Intersection Pairing

The Lo cal System

Mono dromy Action of the Braid Group oid and Universal RMatrix

Lo cally nite homology

App endix

Fock Space Representations of A and Topological Representations

of U sl C

Fock Space Representations of A

Intertwiners

Chiral Primary Fields

Generalized Hyp ergeometric Functions on the Torus and the Adjoint

Representation of U sl C

Introduction

Generalized Hyp ergeometric Functions on the Torus

Lo cal Systems over Conguration Spaces on the Torus

Braid Group on the Torus

Coloured Braid Group oid and Lo cal System

Torus with punctures

Topological Representations of U sl C

Lo cal system from multivalued forms

Families of Lo ops and Op erators

The Torus with One Puncture

The Torus with Many Punctures

On the adjoint representation

Conjecture on lo cally nite homology

Prop erties of W j

Prop erties of z j

Topological Representations of U sl C on the Torus and the Map

ping Class Group

Introduction

Conguration spaces and lo cal systems on the torus

Representation of X

Conguration spaces and braid groups on X

Lo cal systems over X

r ed

sl C Topological representations of U

Families of nonintersecting lo ops with values in the lo cal system

r ed

sl C Topological action of U

Action of the Mapping Class Group

Generators of M X

Compatibility of lo cal systems

L Action of M on A X X

r r r

r ed

sl C Action of M X on U

Action of S

Identication of the S and the T transformation

r ed

Universal elements of U sl C

q K

Identication of U T

r ed

Trace on U sl C

q K

S transformation

Identication of S

The KnizhnikZamolo dchikovBernard Equation on the Torus

Introduction

Lie algebra valued meromorphic functions

Meromorphic vector elds

Conformal blo cks

KnizhnikZamolo dchikovBernard equation

Weyl invariance

Conformal blo cks on elliptic curves and the KnizhnikZamolo dchikov

Bernard equations

Introduction

Conformal blo cks on elliptic curves

Elliptic conguration spaces

Meromorphic Lie algebras

Tensor pro duct of ane KacMo o dy algebra mo dules

Vector elds

Conformal blo cks

Flat connections theta functions

The at connection

Theta functions

Ane Weyl group

Mo dular transformations

Mono dromy pro jective representations of SL Z

The vanishing condition

Pro of of Theorems

Examples

The KnizhnikZamolo dchikovBernard equations and generalized clas

sical YangBaxter equation

The KZB equations

The classical YangBaxter equation

App endix

Lie algebras of meromorphic functions

Connections on ltered sheaves

Summary

Introduction

Conformal invariance has a long history in physics Maxwells theory of electro dynam

ics and also Einsteins general relativity are examples of conformal invariant classical

eld theories Cunningham Cu and Bateman Ba were the rst to observe that

Maxwells equations are covariant not only under the Lorentz group but also under the

larger conformal group Conformal transformations are co ordinate transformations

under which the metric tensor g x is multiplied by a spacetime dep endent scalar

factor x They include in particular Poincare transformations homogeneous and

inhomogeneous Lorentz transformations where the conformal factor is one They also

include dilatations where the conformal factor is a constant dierent from one and

so called sp ecial conformal transformations As a brief introduction to conformal in

variance in classical physics we mention the article FRW by Fulton Rohrlich and

Witten In two dimensions conformal invariance is a particularly p owerful concept

If one considers two real co ordinates x y as one complex number z x iy then

conformal transformations consist of substitutions z f z where f is a holomorphic

function It is not surprising that holomorphic mappings therefore play an imp ortant

role in the electro dynamics of two dimensional systems

The standard mo del of particle physics is conformal invariant as a classical eld

theory if all particle masses are put to zero In quantum eld theory an early hop e was

that conformal invariance might b e an approximate symmetry at high energies The

reason b ehind this hop e was that conformal invariance should take place at energies

where the particle masses b ecome negligible It was argued that theories should not

have an intrinsic scale at high energies and therefore b e scale invariant Unfortunately

renormalization tends to break scale invariance The renormalization of a quantum

eld theory amounts to the subtractions of singularities due to vacuum uctuations on

all length scales Their subtraction requires the introduction of a renormalization scale

which breaks the conformal invariance At low energies scale invariance is furthermore

broken explicitely in the standard mo del due to nonvanishing particle masses Scale

transformations were subsequently rened to the renormalization group an imp ortant

structure underlying quantum eld theory An introduction to this sub ject can b e

found in Co by Coleman

The construction of conformal invariant solutions to quantum eld equations has

remained a promising enterprise The idea is to use the constraints imp osed by con

formal invariance to obtain truly nonp erturbative information ab out quantized eld

theories In conjunction with the idea of op erator pro duct expansions this program is

known in particle physics as conformal b o otstrap It still awaits its fruitful completion

in higher dimensions than two A status of the work preceding the present b o ost in

two dimensions together with references to the original articles can b e found in the

rep ort DMPPT by Dobrev Mack Petkova Petrova and Todorov

In two dimensions the conformal b o otstrap is a story of astounding success The

seminal work of Belavin Polyakov and Zamolo dchikov BPZ started an avalanche

of activity which is now an entire branch of theoretical physics The wealth of results

obtained in its development has had deep impact on various branches of physics and

mathematics and we may well exp ect more surprising results to come

The motivation in physics to study two dimensional conformal invariant eld the

ories is at least threefold One motivation is string theory String theory as lectured

for instance by Green Schwarz and Witten GSW is a candidate for a quantum

theory of all interactions including gravity Conformal invariance comes in string the

ory as reparametrization invariance of world sheets Two dimensional conformal eld

theories are the classical solutions of string theory A particular conformal eld theory

determines the string vacuum and enco des information ab out the number of space

time dimensions and the gauge group of its low energy limit Not every conformal eld

theory however is an acceptable string vacuum One has constraints b oth from the re

quirement of internal consistency for example the vanishing of the conformal anomaly

and mo dular invariance and also from the requirement that it should repro duce the

four dimensional standard mo del as its low energy limit The investigations of these

constraints is an ambitious ongoing program

The second motivation is the study of two dimensional statistical systems at the

critical p oint Ca Two dimensional systems come in statistical physics and in solid

state physics in the form two dimensional lms in three dimensions layered systems

in three dimensions which are eectively two dimensional and as surface eects of

three dimensional bulk systems Examples of which are the quantum Hall eect and

layered mo dels of high T sup erconductivity From a theoretical p oint of view two

dimensional systems are highly valuable as mo dels of critical b ehavior b ecause of their

solvability They also serve as a lab oratory to test approximations with exact results

In the statistical physics of critical systems one addresses the following interrelated

questions What is the classication of universality classes of critical b ehavior What

are the criteria to decide to which universality class a given system b elongs What

are the values of critical exp onents and universal amplitudes What are the nite

size eects when a critical system is put into a nite volume What is the eect of

p erturbations which drive a system away from criticality Conformal invariance is a

p owerful to ol in their investigation The strategy is to identify universality classes with

conformal eld theories The critical exp onents of a universality class are enco ded

in the sp ectrum of anomalous dimensions of the scaling op erators in the conformal

eld theory Conformal invariance puts restrictive constraints on their p ossible values

Finite size eects are studied as the resp onse of a conformal eld theories to a variation

of the size and shap e of the Euclidean spacetime region Last not least the eect of

p erturbations is studied by conformal p erturbation theory A basic concept in this

strategy is to view the critical system from the p oint of a Euclidean eld theory

Critical b ehavior is a collective phenomenon due to uctuations of a statistical

system on all length scales An illustrative example in statistical physics is the two

dimensional Ising mo del Its critical prop erties can b e investigated by means of the

renormalization group BGZ WK The link to conformal eld theory is the

concept of scale invariance In Wilsons renormalization group WK it comes ab out

as follows Applying the renormalization group in the infrared direction one lo oks

at a system on larger and larger scales Technically one averages out short distance

uctuations and studies the resulting eective theory of long distance uctuations

After having integrated out all uctuations b etween two length scales say L and

L L L one compares the result with that of a pure scale transformation by

L L Supp ose rst that your system is noncritical in the sense that uctuations

on a particular scale are dominating Such a mo del is solved in a single step by

averaging out precisely these uctuations The analysis of critical systems require an

innite number of steps when in each step a nite p ortion of uctuations is averaged

The idea is then to study the ow of eective theories always rescaling to a unit

scale where the eective distributions can b e compared Universality classes come

out as domains of attraction Two dierent systems in the same universality class

tend to converge to the same eective distribution Critical systems come out as

renormalization group xed p oints Applying the scale analysis to a critical system one

ows into a xed p oint which characterizes the universality class The two dimensional

and also the three dimensional Ising mo del has such a nontrivial infrared xed

p oint This xed p oint enco des the critical prop erties of the Ising mo del for instance

its critical exp onents Their investigation is a truly nonp erturbative problem The

to olb ox of the renormalization group for mo dels in any number of dimensions contains

the expansion the N expansion and numerical techniques All of them make

approximations which are dicult to control In two dimensions conformal invariance

provides a metho d which is truly nonp erturbative and exact Critical prop erties are

therefore advantageously studied in terms of conformal eld theory For instance

the critical exp onents can b e inferred from the sp ectrum of anomalous dimensions

Conversely if the critical indices are known from other calculations they can b e used

to identify the conformal eld theory Other prominent examples b esides the Ising

mo del are the tricritical Ising mo del and the three states Potts mo del Do They

b elong to a series of conformal eld theories called minimal mo dels BPZ whose

symmetry prop erties form a sub ject of this thesis

A critical system is scale invariant in the sense that lo oking up on it with a p o orer

magnifying glass repro duces the same picture up to a trivial scale transformation

Belavin Polyakov and Zamolo dchikov BPZ argue that scale invariance implies

invariance under sp ecial conformal transformations A system which is Euclidean in

variant to b egin with is consequently invariant under the full conformal group The

argument relies on a eld theoretic description of the critical system For the Ising

mo del this can b e done quite explicitely see for instance ID For a general lattice

eld theory it requires to p erform a scaling limit of the lattice correlation functions

Roughly sp eaking the continuum eld theory is contained in their long distance b e

havior The argument concludes that the scaling limit of a critical system is a massless

Euclidean eld theory It encloses a traceless stressenergy tensor which generates

the spacetime symmetries Provided that the eld theory satises general axioms it

follows that tracelessness implies conformal invariance In two dimensions the technical

conclusion is that the stressenergy tensor is a Lie eld and that its charges satisfy

the commutation relations of two copies of the Virasoro algebra This fact is known as

L uscherMack theorem LM and is the starting p oint of BPZ

Two dimensional conformal eld theories exhibit a wealth of interesting structures

A basic structure is the factorization into holomorphic and antiholomorphic chiral

eld theories The observation is that if one changes from real Euclidean co ordinates

x and x to the complex ones z x ix andz x ix two dimensional conformal

eld theory on the plane presents itself in terms of quantities which dep end either on

z or on z Such quantities are called chiral In this thesis we will restrict our attention

to the holomorphic chiral sector A more involved structure which is a main issue

in this thesis is the multivaluedness of chiral conformal correlators as functions of

the insertion p oints In mathematical terms chiral conformal correlators have a non

trivial mono dromy In two dimensions the exchange of two insertion p oints in a chiral

correlator dep ends on the path along which they are exchanged In dimensions higher

than two any two such paths can b e continuously deformed into one another In two

dimensions this is not the case As a consequence the mono dromy of chiral correlators

is not governed by the p ermutation group but rather by the braid group In a nutshell

two dimensional eld theories allow for more than the BoseFermi alternative We

mention that braid group statistics is a fascinating topic in itself whose consequences

are far from completely explored An illuminating reference to braid group statistics

are the lecture notes Fr by Frohlich

The conformal eld theories which we will lo ok up on b elow therefore furnish in

deed mono dromy representations of braid groups The complete representation theory

of braid groups is a dicult mathematical problem We will restrict our attention to

those representation which arise in the conformal eld theory of the minimal mo dels due

to BPZ and the WessZuminoWitten mo dels KZ based on SU They turn

out to b e so called Rmatrix representations The particular representation of the braid

group which is realized in a given conformal eld theory is an extremely imp ortant

structural datum The investigations of this thesis were initiated by the observation

that the Rmatrix representations from minimal mo dels and WessZuminoWitten

mo dels based on SU coincide with Rmatrix representations of the quantum group

U sl C Abstractly sp eaking one observed that conformal eld theory comes to

gether with data from the representation category of quantum groups A natural

question to ask is whether this coincidence can b e traced back to a quantum group

symmetry or at least to a quantum group action on auxiliary quantities entering the

construction of chiral correlators This question will receive a p ositive answer in form

of top ological representations of quantum groups connected to integral representations

of chiral correlators

Quantum groups emerged from the solution of two dimensional integrable mo dels

by means of the algebraic Bethe ansatz Their foundation as a mathematical ob ject

was layed by Drinfeld D Quantum groups have undergone an intense insp ection

since then For our purp oses the notion of U sl C as a quantum deformation of

the universal envelopping algebra of sl will suce The deformation is such that the

resulting ob ject remains a Hopf algebra In practice one deformes b oth the relations of

the Lie algebra and for instance the copro duct The copro duct is necessary in order

to preserve the notion of a tensor pro duct of two representations The result in the case

of U sl C is a Hopf algebra which is coasso ciative but nonco commutative The

nonco commutativity comes in form of a universal Rmatrix which satises a Yang

Baxter equation The YangBaxter equation again is the link to the algebraic Bethe

ansatz B Quantum groups therefore provide a to ol to solve integrable mo dels based

on the algebraic Bethe ansatz The representation theory of U sl C turns out to

degenerate in the case when the complex deformation parameter q b ecomes a ro ot of

one In conformal eld theory precisely this degenerate case turns out to b e realized

An account of the representation theory of U sl C in the degenerate case has b een

given by Frohlich and Kerler FK We mention that the decomp osition of its adjoint

representation in this case has b een accomplished only very recently by Ostrik O

That such a structure plays a role in quantum theory is an exciting surprise Since

the advent of quantum mechanics it is taught that symmetries take place in the form of

representations of symmetry groups In the presense of a symmetry quantum states can

b e organized into multiplets which form representations of the symmetry group and

the symmetry implies selection rules for transitions A question which was neglected

b efore the app earance of quantum groups is whether one can also quantize the notion of

symmetry The most general notion of a symmetry compatible with the framework of

quantum theory known to day is that of a weak quasiHopf algebra The quantum group

U sl C falls into this category An explanation of the app earance of quantum group

data in conformal eld theory is to implement the quantum group as a global symmetry

algebra The idea there is to decomp ose the Hilb ert space of states into sup erselection

sectors carrying quantum group quantum numbers and to construct eld op erators

making transitions b etween sup erselection sectors sub ject to braid relations This

picture has b een worked out by Mack and Schomerus MS A source of troubles are

unwanted and unphysical nondecomp osable nite dimensional representations

In this thesis we pro ceed in a dierent direction The quantum group data braiding

and fusion rules comes enco ded in chiral conformal correlators They form the building

blo cks of a conformal eld theory The most p owerful metho d for their investigation

are free eld representations We will address the question why quantum group data

app ears in free eld representations In free eld representations one works with an

enlarged theory The quantum group turns out to act on this enlarged theory More

over this action can b e describ ed explicitely the representations can b e explicitely

constructed identied and analyzed The result is a top ological representation

The third motivation to study two dimensional conformal eld theory and its sym

metries is its mathematical b eauty which has attracted a lot of workers to the eld

Two dimensional conformal eld theory is to a considerable extent the representation

theory of innite dimensional Lie algebras the Virasoro algebra and current algebras

As a consequence it yields b eautiful representation theoretic interpretations of critical

exp onents in two dimensions When formulated on higher top ologies intricate connec

tions with complex geometry for instance mo dular prop erties of higher genus surfaces

app ear The main motivation to study higher genus surfaces comes from string theory

But the lowest higher top ology the torus plays also an imp ortant role in statistical

physics Ab out half this thesis is devoted to the mo dications that come ab out in the

theory of top ological representations in the transition from genus zero to genus one

Last not least top ological representations are a theory of a large class of sp ecial func

tions hypergeometric functions and generalizations thereof As a mathematical sub ject

they have b een established by the work of Schechtman and Varchenko SV and by

Felder and the author FW The sub ject is still in motion In particular top ological

representations on higher genus surfaces are still to b e prop erly understo o d

Prologue

Two dimensional conformal eld theory has its origin in the statistical physics of two

dimensional systems at a critical p oint It has furthermore attracted a lot of interest

b ecause of its applications to string theory and also b ecause of its mathematical b eauty

We mention the seminal pap er BPZ the collection ISZ of reprints and the

lectures Gi Ca ID as a rst guide to the literature

Scale Invariance

The scaling limit of a critical mo del denes a two dimensional Euclidean quantum eld

theory This eld theory is not only invariant under the group of Euclidean motions

but also under dilatations One can argue BPZ Ca that it is then invariant

under the conformal group An ob jective of conformal eld is the classication of all

two dimensional critical phenomena The intense work in conformal eld theory since

BPZ has changed the ob jective to a considerably broader nature The connection

of two dimensional conformal eld theory with quantum groups is a structural question

which has b een p osed in the course of this developement

EnergyMomentum Tensor

We assume spacetime to b e two dimensional Euclidean space Complex co ordinates

are introduced by z x ix and z x ix It is common strategy to consider z

and z as indep endent variables and to set them complex conjugate in the very end of

the reasoning The innitesimal conformal transformations are given by z z z

The corresp onding generators on functions are l z They span an innite

n z

dimensional Lie algebra with relations

l l n ml

n m nm

In eld theory coordinate transformations are generated by the charges constructed

from an energymomentum tensor In conformal eld theory the energymomentum

tensor is traceless and decomp oses in a holomorphic z z comp onent T z and an anti

holomorphic zzcomp onent T z It has an expansion

T z L z

where the co ecients satisfy the relations of the Virasoro algebra

L L n mL n n L C

n m nm nm n

Thus in conformal eld theory we are not dealing with the Witt algebra but rather

its central extension This conformal anomaly was derived in LM and is the

starting p oint of BPZ Analogous equations hold for the antiholomorphic comp o

nent of the energymomentum tensor Eq plays the role of an innite symmetry

algebra in conformal eld theory

Minimal Mo dels

In a given conformal eld theory the central charge C takes a denite value c and

is the most imp ortant single datum In the Ising mo del c Sp ecial values of c

have the imp ortant prop erty that they yield a nite set of irreducible highest weight

representations of Mo dels with this prop erty are called rational First examples

are the minimal mo dels of BPZ They have the prop erty of b eing solvable in a

strong sense Integral representations are known for their correlation functions The

integrals have b een computed explicitely in many cases Let us briey discuss the setup

of minimal mo dels of BPZ following F The Hilb ert space of a minimal mo del

is a direct sum

0 0

H H H

n n n n

n n

of irreducible highest weight representations for two copies V ir V ir of The

generators are L and L acting as L and L resp ectively The central charge

n n n n

p p

where p and p are two p ositive integers without common divisor They lab el the series

of minimal mo dels The highest weight mo dule H is constructed as a quotient of a

n n

Verma mo dule V h c with highest weight

n n

n p n p p p

n n

p p

by its maximal prop er submo dule LV h c The representations are lab elled by

n n

pairs n n of integers sub ject to n p n p and pn p n A result of

FQS shows that a mo del is unitary if and only if p p For p p we

obtain the series of unitary minimal mo dels with central charge

where p The Ising mo del is the rst of which with p Let us also

have a brief lo ok at the eld content of minimal mo dels It is convenient to introduce

the abbreviations

N n n M m m L l l

for double indices With each representation is asso ciated a conformal primary eld

L L L

C z z z z

N M N M N M

M L

In the renormalization group terminology it is a scaling eld Here C are C valued

N M

structure constants The eld z is a holomorphic chiral primary elds It maps

N M

H to H is zero on the other H and is a conformal eld of weight h

M L K N

L k L k

L z z z k h z

n N

N M N M

We remark that denes an action of V ir on a space of op erators generated from the

primary eld The op erators obtained from the primary eld by acting with pro ducts

of L n are called descendants The family of eld op erators forms a conformal

multiplet BPZ In principle all matrix elements of chiral primary elds can b e

computed up to normalization constants by means of The normalization constants

can b e xed by

v v

L M

N M

where v denotes the highest weight vector in H It follows that the theory is

M M

completely determined by the structure constants Their evaluation is the sub ject of

the conformal b o otstrap BPZ Correlation functions of primary elds factorize

into

z z i F z z F z z z z h

k k k N N

The index stands for a sequence of intermediate representations M M The

constants are pro ducts of structure constants The factors in are correlation

functions

z F z z v z v

k M k M

N M N M

k k

Here M M denotes the vacuum representation The correlation functions

are called chiral conformal blo cks They are the main ob jects of this investigation

In principle they can b e computed by doing the sum over intermediate representa

tions Again there exists a more p owerful to ol the metho d of integral representations

In a nut shell the program is to deduce integral representations for the solutions of

dierential equations for the conformal blo cks which follow from The result is

generally a multiple contour integral Boundary data is then enco ded in the choice of

integration contours At this p oint the quantum group enters the scene The quan

tum group organizes which contours yield the physical conformal blo cks By lo cality

is required to b e single valued This is not the case for the conformal blo cks

They are generally multivalued functions of the insertion p oints Up on ana

lytic continuation they dene a representation of the braid group on the conguration

space of insertion p oints This representation is one of the interesting features of two

dimensional conformal eld theory More on this will b e said in the bulk of this thesis

U Current Algebra

Integral representations for the conformal blo cks of minimal mo dels are eciently de

duced from free eld representations See DF F and references therein Since

they form the p oint of departure of this thesis we give a brief derivation for the case

of minimal mo dels We do not follow the standard free eld computations but rather

choose an alternative geometrical construction We use an adaptation of the approach

in F for WessZuminoWitten mo dels to the case of U current algebra We

nd it amusing by itself It also underlies the more involved constructions of the last

chapters

We b egin with an introduction of prerequisites simultaneously xing the notation

Let g C b e the complexied Lie algebra of U The asso ciated lo op algebra

is Lg C t the space of complex valued formal Laurent series in t The lo op

algebra is again a Lie algebra and is ab elian in this case It has a central extension

Lg Lg C k This central extension is a nonab elian Lie algebra with bracket

f k g k resf g k

where f and g are formal Laurent series resf is the co ecient of t and are

complex numbers and k denotes the central element The derivative with resp ect to t

is adjoined as an additional element L through

L f k f

In terms of the generators a t n Z the bracket b ecomes

a a n k

n m nm

Analogous to one asso ciates with a U current j z a z We

z Z

remark that Lg can b e decomp osed into n g n where n t C t Fur

thermore Lg C L is integer graded by the assignments deg t n deg k

deg L

The next brick in this construction are highest weight mo dules V where is a

real number called the charge of the mo dule V is dened as b eing generated from

a cyclic vector v with dening prop erties

a v n

a v v

and k v v The mo dule is spanned by nite linear combinations of vectors

a a a v with n n n The space V comes equipp ed

n n n k

with the structure of a highest weight mo dule over the Virasoro algebra through the

following basic construction One denes

L a a n a

n m nm n

where is another real parameter The colons mean normal ordering They order the

annihilators a n to the right For instance

L a a

m m

On every element of V of nite weight only nitely many terms in the sum con

tribute Therefore is well dened on V It can b e shown that satisfy the

relations with central charge

This construction is therefore in particular suited for the case of minimal mo dels v

is also Virasoro highest weight vector It satises

L v n

L v h v

with highest weight

We remark that this representations of the Virasoro algebra is not irreducible in the case

of minimal mo dels The irreducible representation can b e obtained through a b eautiful

cohomological construction We refer to F for this so called BRSTcohomology

Conformal Blo cks

Conformal blo cks were introduces as correlation functions of chiral conformal elds

They dep end on the p ositions of the eld op erators which we take to b e noncoinciding

To describ e this situation we introduce a conguration space

n n

fz z g C C n

i j

of n noncoinciding p oints in the complex plane Chiral conformal elds come as con

formal multiplets This is formalized by asso ciating with each p oint z a highest weight

mo dule V The highest weight vector v corresp onds to a chiral primary eld

j j

at z Other vectors corresp ond to descendant elds This corresp ondence is slightly

inaccurate b ecause we are using Fock spaces at this moment instead of irreducible

Viraso mo dules We should b etter sp eak of free eld or U conformal blo cks

The dening prop erty of these conformal blo cks is a U Ward identity which we

explain next Let Mz z b e the space of meromorphic functions holomorphic

on C n fz z g They can b e thought of as g valued functions and form an ab elian

Lie algebra Let f b e such a meromorphic function At each z we can p erform a

Laurant expansion and interpret the result as an element of Lg But Lg Lg although

the inclusion is not a Lie algebra homomorphism Lg acts on V Therefore we

have linear maps

Mz z EndV f f z t

j n j j j

Doing this for all p oints z z we obtain a linear map

n n

Mz z EndV V

n n

This map has the remarkable prop erty of b eing a Lie algebra homomorphism The

reason is that the residues coming from

f g res f g

sum up to zero thanks to the residue theorem For this purp ose we b etter formulate

the theory on the Riemann sphere C P rather than on the plane C In other words we

have an action of the space of g valued meromorphic functions on the tensor pro duct

of highest weight mo dules

Conformal blo cks are distinguished by b eing invariant under this action A confor

mal blo ck at xed p ositions z z is a linear form

F V V C

such that for all f Mz z and all elements u V V

n n

hF f ui

The bracket h i means the evaluation of a linear form on a vector The space of these

invariant linear forms is denoted by

E z z Hom V V C

n n

Mz z

This is equivalent to a U Ward identity A case of particular interest is the value

of an invariant linear form on the pro duct of all highest weight vectors

v v v

This case corresp onds to a pro duct of chiral primary elds An interesting question is

wether the value of a conformal blo ck on general vectors is already xed by its value on

the pro duct of highest weight vectors This is indeed the case For f z z z

n it follows that

j i

a z z a f v v

i j

where a is meant to act on the j th factor of v But the Ward identity then implies

that

D E

F a v z z hF v i

i i j

i j

Iterating this pro cedure we can evaluate the conformal blo ck on any vector in the

pro duct mo dule

The next question to b e asked is how a conformal blo cks b ehave under variation of

the p ositions z z The answer to this question relies on a remarkable dierential

equation to which we turn our attention now The idea is to view conformal blo cks as

invariant sections of a vector bundle Let U C b e an op en subset Consider then

the innite rank trivial vector bundle

U Hom V V C

C n

Let E U denote the space of sections F U Hom V V C of this

C n

vector bundle such that F is holomorphic in z z and such that F z z

n n

is an element of E z z for all z z U Elements of E U are called

n n

holomorphic conformal blo cks The trivial vector bundle comes together with a

at connection

F F L F r dz r r

z z i z

i i i

dened on holomorphic sections This connection is called FriedanShenker connection

A straight forward computation proves that

r F f r F f F f

z z z

i i i

As a consequence leaves invariant the space of sections E U To b e precise an

invariant section is mapp ed to an invariant dierential form Conformal blo cks are

distinguished by the prop erty of b eing horizontal sections That is a conformal blo ck

is an element of F E U which satises

In the case of general current algebras where one starts this construction from a general

simple complex Lie algebra is the celebrated KnizhnikZamolo dchikov dierential

equation KZ A more explicit form of is

D E

hF z z ui F z z L u

z n n

We have already seen that it is sucient to compute the pairing with the pro duct of

highest weight vectors In this case we have that

i i i i

L v a a v a v

From we then immediately deduce the U KniznikZamolo dchikov equation

i j

hF z z v i hF z z v i

z n n

z z

i j

j i

It tells the b ehavior of conformal blo cks up on variation of the conguration z z

It is not dicult to solve in the sp ecial case that

Recall that is the U charge of V Eq therefore requires the total charge

i i

to b e zero Notice that charge neutrality is required by invariance of a conformal blo ck

under the constant function The result is

i j

z z hF z z v i

i j n

choosing an overall normalization constant equal to one Eq is a well known

expression from many b o dy theory of U charged particles The p oint with charge

neutrality is that b ehaves at innity like

j 6

z z

as a function of z for xed z z and is therefore regular at innity In the non

neutral case is still a solution to but is singular at innity Thus it would

have to b e interpreted as a conformal blo ck with another eld lo cated at innity

Integral Representations

With this formalism we are ready to derive integral representations The case of total

charge neutrality has b een solved explicitely The key idea is to reduce the general

nonneutral case to the neutral one at the exp ense of additional so called screening

charges Screening charges are dened by h That is they corresp ond

to highest weight mo dules of conformal weight one The solution to this quadratic

equation denes two charges

Integral representations can b e derived if the excess charge can b e comp ensated for by

screening charges With this we mean that we have charge neutrality in an extended

system

r r

The total number of screening charges will b e denoted by r r r Let w w

and w w b e the p ositions of the p ositive and negative screening charges resp ec

tively To simplify the notation we put

w w z z z w w w

r r

and dene furthermore

a v v v

v a v v

j i

Then we dene an extended conformal blo ck which dep ends on the screening charge

p ositions w by

D E E D

F z w u F z w u v v

The right hand side is a charge neutral conformal blo ck and is explicitely known u is

an element of V V As it stands do es not satisfy invariance under

Mz z due to the presence of sreening charges The idea is now to integrate

out the sreening charges in a way to obtain an invariant expression Eq is by

construction invariant under the extended symmetry

f f f

z w

in a selfexplanatory notation The problematic piece is f Notice that we seek

invariance for functions f which are regular on the p ositions of the screening charges

The trick is b est explained by computing

f a v ff w a f w a g a v

ff w L f w g v

where f is assumed to b e regular at w At this p oint it is required that the charge b e

a screening charge ie the conformal weight b e one As a consequence

Eo n D E D

f w F z w u v v F z w f u v v

j j

n D Eo

f w F z w u v v

j j

The strategy is then to integrate the screening charges over contours

such that is zero for all functions f Mz This can indeed b e accomplished

and is a main theme of this thesis Naively one would exp ect that any collection of

closed curves encircling the singularities would do the job This is not the case b ecause

the integrand is generally multivalued and one has to account for phase factors In

the more complicated case of the torus one even has to account for a general case of

certain matrices Once this has b een achieved one has an integral representation

D E

h F z ui F z w u dw

with all desired prop erties The integral is then in particular Mz invariant

and satises the atness condition The integrand follows directly from the neutral

expression It is a single valued function times

Y Y Y

i i i j

z w z w z z

i i i j

j j

ij ij ij

Y Y Y

w w w w w w

i j i j i j

ij ij ij

Notice that The single valued function comes from evaluating by

means of This evaluation is a form of Wicks theorem and is left to the reader

Notice that we do not take the screening charges to b e highest weight vectors

Lo cal Systems

Our investigation of top ological representations of quantum groups b egins with this

integral representation for minimal mo dels The function is multivalued on the

n r

conguration space C of charges and screening charges We dene it by analytic

continuation from a reference p oint Analytical continuation asso ciates with each lo op

in the conguration space b eginning and ending at the reference p oint a phase factor

This is a one dimensional lo cal system With the lo cal systems from minimal mo dels on

C P we asso ciate top ologigal representations of the quantum group U sl C They

serve in the rst place to explain what kind of integration contours yield physical con

formal blo cks They also answer structural questions ab out the conformal blo cks One

such structural question is the b ehavior of conformal blo cks up on analytic continuation

of the insertion p oints For minimal mo dels on C P one nds an Rrepresentation of

the braid group where the Rmatrix b elongs to U sl C The top ological repre

sentation of U sl C gives an explanation of this fact Thus conformal blo cks come

enco ded with quantum group data The quantum group data can b e reconstructed

to a large extent from the lo cal system alone In particular another class of mo dels

the WessZuminoWitten mo dels built on SU p ossess the same lo cal system and

therefore the same quantum group structure This may b e viewed as a kind of quan

tum group universality Quantum group symmetry was used on the side of integrable

lattice mo dels in PS to identify scaling limits The connection b etween conformal

eld theory and quantum groups is therefore also a contribution to the classication

program The constructions of this thesis are built up on integral representations of the

form and generalizations thereof A surprisingly rich structure emerges in partic

ular when one considers conformal eld theories on a toroidal space time Connections

with quantum groups innite dimensional Lie algebras and complex geometry ap

p ear They form a whole menu of interesting problems among which the top ological

representations of quantum groups

Except for the last two chapters which rely on the theory of current algebras the

basic input from conformal eld theory is the lo cal system Each chapter is preceded

by a short introduction to the particular sub ject under investigation Chapter one to

four form one logical unit and chapter ve and six another one They can b e read

indep endently Chapter ve is a subcase of chapter six We have included it in order

to make the essence readable to a broader audience

Topological Representations of the Quantum Group

U sl C

We dene a top ological action of the quantum group U sl C on a space of homology

cycles with twisted co ecients on the conguration space of the punctured disc This

action commutes with the mono dromy action of the braid group oid which is given by

the Rmatrix of U sl C

Introduction

In the free eld representation of conformal eld theory based on SU one is led to

consider integrals of the form DF ZF

G w w dz dz f z z w w

C s r r s

Y Y Y

n n n

j i j

z z z w w w

i j i j i j

ij ij ij

In this formula n n are p ositive integers f is a single valued meromorphic func

tion symmetric under p ermutations of the z variables with p oles on the hyperplanes

fz w g The parameter is equal to k for the WessZuminoWitten WZW

i j

mo del based on SU at level k and is equal to p p for minimal mo dels with central

charge c p p pp

For each integration cycle C in the r th homology group with co ecients in the lo cal

system given by the mono dromy of the dierential form in G is a many valued

analytic function on the conguration space C C fw w C jw

s i

w i j g To compute its transformation under analytic continuation along paths

exchanging the punctures w one needs to know the mono dromy action of the braid

group oid on homology Examples of this computation by contour deformation have

b een worked out by several authors among others GN DF TK FFK La

in dierent languages It generalizes the computation of Gauss for the hypergeometric

function It has b ecome clear that the mono dromy is describ ed by the Rmatrix more

precisely by the j symbols of the quantum group U sl C The top ological p oint

of view we adopt here is closest to PS

In this chapter we prop ose an explanation of this fact It consists of two parts

First one considers a space of relative cycles on which U sl C acts The action is

describ ed purely in top ological terms and commutes with the mono dromy action of the

braid group The absolute cycles are then given by the highest weight vectors in the

space of relative cycles We have then schematically the following dictionary b etween

top ological and algebraic entities

Relative cycles Elements of the tensor

pro duct of Verma mo dules

i n

Absolute cycles Highest weight vectors in

i n

Intersection pairing Covariant bilinear form

Mono dromy action of the Rmatrix representation of

braid group oid on relative the of braid group oid on

cycles V

i n

Moreover the quotient of the space of absolute cycles by the cycles in the null

space of the intersection pairing is closed under braiding and is given by the fusion rule

sub quotient More precise denitions and corresp ondences are explained in the bulk

of this chapter

Our approach is rather elementary and based on the concept of families of loops

rather than on the in some sense more natural homology groups directly We exp ect

that our construction extends to lo cally nite homology but this would require a

somewhat more sophisticated machinery

We p oint out that part of our results can b e understo o d as a top ological version

of results known in the literature on free eld representation of conformal eld theory

PS GP BMP and in particular GS The results in BMP suggest

that our construction extends to groups of higher rank Here we present the purely

top ological results in this sub ject which can b e read without knowledge in conformal

eld theory See FSa F G for applications of these concepts to conformal eld

theory While this work was completed we received some interesting preprints SV

where related results were obtained

This chapter is organized as follows in section we introduce the concept of braid

group oid representations and lo cal systems in a rather general context In section

we sp ecialize to SU and explain the action of U sl C on relative cycles Section

contains the discussion on intersection pairing In section we show that the

representation of U sl C on relative cycles is isomorphic to the tensor pro duct of

Verma mo dules one for each puncture In section we compute the mono dromy

action of the braid group oid on relative cycles The app endix contains a summary of

results on U sl C

Lo cal Systems on Conguration Spaces

Colored Braid Group oids

Let X b e a connected twodimensional manifold p ossibly with b oundary k a p ositive

integer the number of colors and n n nonnegative integers the numbers of

strands with given color Set n n Dene the conguration spaces

C X C X X n fz z g S S

n n n ij i j n n

k k

where the symmetric group S acts by p ermutations on the rst n variables S

n n

on the subsequent n variables and so on It is understo o d that the factors S with

n should b e omitted in An element of C X can also b e thought as a

i n n

sequence Z Z of pairwise disjoint subsets of X with cardinalities jZ j n

k i i

Fix a base p oint x of C X and let O b e the orbit of x under the symmetric

n n x

group S Thus O can b e identied with the right coset space

n x

S O S S

n x n n

The colored braid group oid B X x is the space of paths in X starting and

n n

ending in O up to homotopies preserving endp oints viewed as a subgroup oid of the

fundamental group oid of C X The group oid G B X x is indexed by

n n n n

k k

O and has comp onents lab eled by the endp oints

G G

The multiplication law G G G is the comp osition of paths Since X is

connected braid group oids corresp onding to dierent choices of base p oints are iso

morphic Any such isomorphism can b e describ ed as the comp osition with a homotopy

class of paths connecting the base p oints If k G is a group the braid group on n

X x can b e describ ed in terms of the braid strands on X The group oid G B

n n

group B X x Let h B X x S b e the canonical pro jection homomorphism

n n n

Then for O there is a onetoone map

fg B X x hg g G

such that g g g g

For X C call x C X an admissible base point if x is the image of a p oint

n n

in C with

Rez Rez

1 1

Acting as z z z z

1 n

(1) (n)

Supp ose now that X C For any two admissible base p oints there is a unique homo

topy class of paths in the space of admissible base p oints connecting them Therefore

the corresp onding colored braid group oids can b e uniquely identied and we can omit

the dep endence on x in the notation with the agreement that G B C is dened

n n

using any admissible base p oint

An element in O can b e describ ed by a color map

f ng f k g

such that j ij n The corresp ondence b etween and is the following Let

x S Then

X X

n n i i i

j j

Let i n b e the standard generator of B C that exchanges the ith

i n

strand with the i st one and let h denote the corresp onding transp osition

i i

Then the system

G i n O

i x

i i i

is a system of generators of G

Let R The inclusion C fRez ag C C induces an isomorphism

n n

i i

B fRez ag B C The same holds for the subset fRez ag Let

n n

i i

n n n i k The inclusion

i i

0 00

C fRez ag C fRez ag C C

n n

i i

Z Z Z Z Z Z Z Z

k k k k

induces an injective homomorphism of group oids

0 0 0

B C B C B C

n n n

i i

More precisely we have a map O O O dened by restriction to the orbits

O O of admissible base p oints and maps in an obvious notation

G G G

00 00 0 0

with and compatible with the comp osition law Intuitively

this homomorphism is simply the juxtap osition of colored braids

RMatrix Representations

A representation of a group oid G G with index set I on a family of complex

vector spaces V is an index preserving homomorphism from G to the group oid

Hom V V of invertible linear maps of the vector spaces V In other words

a representation of G is a family of maps

G Hom V V

such that g g g g To simplify the notation we will often omit the

lab el thinking of as the restriction of a map dened on G

Denition Let U k b e vector spaces and for each pair let R b e

an invertible element of End U U An Rmatrix representation of the group oid

B C is a representation on the family of vector spaces lab eled by O

V U U

such that on generators

PR P u v v u

i i

denotes PR acting on the ith and j th factor in the tensor pro duct where PR

Prop osition I Let k be a xed positive integer A family of vector spaces

U k and a family R of invertible elements of End U U denes

a representation of B C for al l n n if and only if the YangBaxter

n n k

i k

equation

R R R R R R

holds on U U U II Let be the homomorphism

0 0 00 00

B C B C B C n n n

n n i

n n n n

k i i

k k

0 00

and set Then for al l O O

n n

i i

00 00 0 0

g g g g

where and

Example Let U C k and identify V C C with C Let

q b e any non zero complex numbers Then q denes an Rmatrix

i i

representation of B C

n n

Example Let A b e a quantum universal enveloping algebra D with universal

Rmatrix R A A and let b e nite dimensional representations of A on spaces

U Then R R denes an Rmatrix representation of B C

n n

Lo cal Systems

Let M x b e a top ological space with base p oint and M its universal covering space

with right action of M x M is the space of homotopy classes of paths in M

originating at x For any representation M x GLV on a vector space

V one denes a lo cal system L as the vector bundle M V over M with the

identication m v m v M x and pro jection m v m the

covering pro jection on the rst argument Thus a lo cal system is the same as a at

vector bundle with holonomy and sp ecied trivialization of the b er over the base

p oint

This construction has the following slight generalization Let O b e a nite subset of

M and G the subgroup oid of the fundamental group oid of M consisting of homotopy

classes of paths whose endp oints are in O For O let M b e the universal covering

space of the space with base p oint M The group oid G acts on the disjoint union

t M on the right by comp osition of paths given by maps M G M Let b e

a representation of G G on a family of vector spaces V These data

O O

dene a lo cal system as the vector bundle

M V L t

with identication m v m v G Such a lo cal system is

the same as a at vector bundle over M together with a family of vector spaces V and

isomorphisms of the b ers over O with V such that parallel transp ort op erators

are given by Lo cal horizontal sections are continuous sections which lo cally can b e

written as m m v with constant v and m covering m

Let M M b e top ological spaces and O M O M b e nite subsets A

homomorphism of lo cal systems L over M to L over M is a map L L mapping

b ers to b ers linearly and sending lo cal horizontal sections to lo cal horizontal sections

Lemma Let f be a map from M to M such that f O O and let f

HomV V be linear maps indexed by O such that the diagram

V V

y y

V V

f f

is commutative for al l O G Then f lifts uniquely to a homomorphism

L L of the local systems associated to also denoted by f which reduces to

f on the ber V over O

Let b e a representation of B C and let L b e the corresp onding lo cal system

n n

Here is an explicit description of L in terms of transition functions Fix an admissible

base p oint x and dene the cells C C C as follows let x S and

dene

n o

C z z C C jR e z Rez

n i

The cells C are pairwise disjoint their union is dense in C C and each cell

contains precisely one p oint in O for any choice of admissible base p oint x

Let C b e the closure of the cell C For y C C let b e any path going

n n n

i i i

Dene the lo cally constant from to y in C and continuing from y to in C

transition function g y Then L is the at vector bundle over C C

L t C V

with identication

v V v V y v y v y C C

if and only if v g y v

Then the parallel transp ort Let b e any path whose endp oints lie in C

op erator along is an op erator in HomV V in the trivialization Therefore we

have an extension of the denition of to all homotopy classes of paths with endp oints

in C

Let now b e the representations asso ciated with a family of Rmatrices as

in Prop osition and L the corresp onding lo cal systems on C C Let a

n n

i i

the restriction of L R C fRez ag C fRez ag Denote by L

n n

i i

L to C C C C resp ectively Let L L b e the at bundle over

0 0 0

n n n

i i i

n n

i i

0 0 0

C dened by taking the tensor pro ducts of the b ers C C C

n n

i i

0 00

Prop osition The maps C C C C C C lifts to a homomor

n n

i i

phism

L L L

0 0 0

n n

i i

sending local horizontal sections to local horizontal sections The lift is xed by setting

the homomorphisms of Lemma equal to the canonical homomorphisms V V

0 0 0

L Proof Let C C and C C Then L C C C C

0 00 0 00

00 0

n n

n n n n n n

i i

i i i i

i i

is the vector bundle

0 00

V V C 0 C t

O O

The map maps C C to C and the transition functions are given by

n n

i i

tensor pro ducts of transition functions The claim follows then from Prop osition

and Lemma

If n and n then C C C and the b er of L over

any p oint is canonically identied with U We have the following sp ecial case of the

preceding Prop osition

Prop osition Let a R f k g and z z be complex numbers with

Rez a Rez Then the maps

C C C C

n n n n n

k k

C C C C

n n n n n

k k

given by Z Z Z Z fz g Z lift to homomorphisms

k k

U L L

n n n

n n

L U L

n n n

n n

These homomorphisms preserve horizontal sections and are isomorphisms on each ber

The Topological Action of U sl C

The SU case

Let us sp ecialize the general discussion to the case of interest to us Let D b e the unit

disc fjz j g and w w b e s distinct p oints in its interior Dene X w w

s r s

to b e the b er over w w of the bration C D C D In other

s r

words X w w is the space of subsets of D n fw w g with r elements

r r s

Let n n b e p ositive integers and q C n fg The family of onedimensional

Rmatrices

R q R R q j s

j j

denes a representation of B C and a lo cal system over C C and also on

r r

C D by restriction Let L w w b e the restriction of this lo cal system to

r r s

X w w We will often omit the w dep endence in the notation and write X L

r s r r

when no confusion arises

In the following construction it is useful to choose also two p oints on the b oundary

fZ X jZ P g By of D For deniteness choose P P Denote X

Prop osition the inclusions

Z Z fP g X X n X

r r

lift to homomorphisms L j L j

r r

X nX X

Families of Lo ops

In the following we x s distinct p oints w w in the interior of the unit disc and

denote by X the set D n fw w g

Denition A nonintersecting family of lo ops in X based at the p oint P is a nite

sequence X of curves in X such that

P t P for t

j j j

If t s and t s then t s and j k

j k

For all j the homotopy class of is nontrivial

A nonintersecting family of lo ops can also b e represented as a map from the r cub e

to X It is the restriction of a continuous map dened on the op en r cub e

with op en r faces

f g Q

dening an inclusion Q X of a closed subset of X

r r r

Denition A homotopy of nonintersecting families of lo ops is dened to b e a homotopy

h X such that for all s h s is a nonintersecting family of

lo ops Two families are said to b e homotopic if there is a homotopy h such that

h and h

Consider the space A A w w of nite linear combinations

r r r

where are homotopy classes of families of lo ops and are horizon

tal sections of the pullback bundle L over the contractible space Q mo dulo the

r r

equivalence relations

f f for any orientation preserving or reversing isometry

f of the cub e

with homotopy If for some i is homotopic to the comp osition

i i

X and s s

r i r

are all nonintersecting families of lo ops then

r r r

i i

where are dened by restriction of

It is understo o d that horizontal sections over homotopic families of lo ops are canon

ically identied by parallel transp ort so that the expressions make sense

Let b e suciently small that the closed discs of radius centered at w are disjoint

b e the spaces obtained from X and contained in the interior of the unit disc Let X

r r

by removing p oints fz z g such that jz w j Elements of A X X

r i j r r

represent relative lo cally nite cycles in H X X L with co ecients in the lo cal

r r r

system L Thus we have a linear map

A w w H X X L

r r s r

r r r

On the other hand we can also view a family as a map from to X by the

formula

t t f t t g

r r r

and a section of L is mapp ed under to a section of L and we also have

r r

a linear map

X L A w w H

r r r s

Op erators

A and then compute their commutation We dene a set of op erators acting on

relations

Let b e the path

X t e

r r

Let i b e the inclusion t t t t Dene a

r r

linear op erator F A A that adds a loop

r r

where is the section over Q such that i on This denition

makes sense since we can assume that the representative do es not intersect

except at the endp oints

r r

Introduce the face maps

t t t t t t e

r i i r

t t t t t t e

r i i r

and the linear op erator that kil ls a loop

e e E

i r r

ir ir

denotes omission The third op erator is the diagonal op erator K dened on A as

n r

K q

The relation b etween E and the b oundary op erator is explained by the

Prop osition The diagram

A H X X L

r r

r r r

y y

H A X L

r r

is commutative

Relations

Theorem The operators E F K obey the relations

K E q EK K F q FK EF FE K K

We have a representation of the quantum group U sl C on A w w

q r r s

Proof The rst two relations follow from the denition The third relation is b est

checked in an explicit trivialization We can assume that f g is in

some cell C Denote by the horizontal section of L which takes the value over

Let in the trivialization over C b e the paths the p oint

t t

i r

n o

to the cell containing the p oint P going from the cell C We

have the explicit expressions

r r

i r r r r

i i

n o

Denoting by the paths t t we nd representa

n r

tion factors q q and We then compute

r r r

EF E

r r

i r r r

i i

r r r

n r n r

i i

q q

The pro of is complete

From Prop osition and Theorem follows

Corollary Singular vectors in A w w ie vectors in Ker E represent

r s

L X absolute cycles in H

r r

Intersection Pairing

Reection and Duality

Let L b e the lo cal system dual to L ie the at line bundle with holonomies

which is the representation obtained from by replacing q by its inverse

and let b e the reection sending x iy to x iy The reection maps orbits of

admissible base p oints to orbits of admissible base p oints and preserves holonomies

and lifts therefore to an involutive homomorphism of lo cal systems

L w w L w w

r s s

The lift is sp ecied by setting the maps of Lemma equal to the identity

Denote by A w w the space of linear combinations with ho

motopy classes of nonintersecting families of lo ops based at P and horizontal

w w mo dulo the ab ove equivalence relations The reection sections of L

induces an isomorphism

w w A w w A

s r s

which denes an action of U sl C on A

Intersection Pairing

In this subsection we assume that all families of curves are smo oth maps on

Let b e a family of curves based at P and b e a family of curves based

at P Supp ose that and intersect transversally in a nite number of p oints lying

in the interior of X Thus the set of t t such that t t is nite contained in

r r

and the tangent map D D is nonsingular at any such t t The

intersection index t t at t t is then dened to b e if the tangent map preserves

the orientation and otherwise The orientation of T X C is conventionally

t r

dened via the identication

x iy x iy x x y y

r r r r

r r

of C with R

Denition The intersection pairing is the complex bilinear form

A w w A w w C

r r r r r

r r and such that which is zero on A A

t t ht t i

0 0 0

tt t t

on A A h i denotes duality of b ers

It is p ossible to give a more explicit formula for Let t b e the tangent vector

at t to a smo oth curve

Prop osition Suppose that and intersect transver

sal ly Let T be the set of t t such that t t and

ij ij

sign Im t t be the intersection index of and at t t Let for S

i r

i j j

n o

r r

T t t j t t T j r

j j j

Then

X Y X

sign ht t i

j j

j S

tt T

Proof The condition t t is equivalent to t t for all j and some

j j

p ermutation Thus

ft t jt t g T

for by the second prop erty of nonintersecting families of and T T

curves For t t T t t is the sign of the determinant of the matrix

t Re t Re

j ij j j ij

t Im t Im

j ij j j ij

which is easily put in blo ck form by p ermuting rows and column and the result follows

Theorem Fix w w let be the intersection pairing corresponding to

w w and let be the intersection pairing corresponding to w w I

s s

For al l a A w w b A w w

r s r s

a b b a

II Let T denote transposition with respect to Then

T T T

E F F E K K

ie is a covariant bilinear form

Proof Set a and b Lo oking at the denition of intersection

pairing we see that since preserves the pairing b etween b ers it is sucient to prove

that the intersection index t t is the same on b oth sides of the equation Let

t t b e an intersection p oint of with Identify the tangent space at a p oint

of the unit r cub e in a canonical way with R and the tangent space at a p oint in X

with R as ab ove Then the intersection index o ccurring on the left hand side is the

r r r

sign of the determinant of the r r matrix D D R R R The matrix

is the diagonal matrix with entries We have

detD D det D D

r r

detD D

The sign of the last determinant is the intersection index o ccurring on the right hand

side K K follows immediately from the denition Next we show that

F E The third relation follows then from Let b e a family of

lo ops based at P and one based at P Let us as in the pro of of

Theorem denote by the section of L w w which takes the value over

r r

the p oint in the trivialization over which is assumed to

b e in a cell Supp ose that intersects in a p oint which is in some cell C and let

t t denote the value of the parameters at the intersection p oint Denote by the

path s s ts s t s and by Then we have the path s

ij i

the explicit expression

X X Y

sign

j j j j

j j

tt T

Let b e the path t exp it We have to compute

r r

at It can b e assumed by p ossibly applying a homotopy that intersects each

is close to zero with p ositive of exactly two p oints namely when the parameter t

j j

intersection index and when t is close to one with negative index see Fig In

Therefore the corresp onding paths b oth cases the parameter t of is close to

r j

asso ciated to these intersections can b e replaced by the trivial path and by the paths

dened by

t t

j r

We are in p osition to complete the calculation

X X

r i r

sign

X Y

j j j j

j j r i r i

tt T

Figure The p oints of intersection of

with

Tensor Pro ducts and Copro duct

In this section we give explicitly the structure of A w w as a U sl C

r s q

mo dule

The mo dule A w w

r s

The spaces A w w constitute a complex vector bundle over C D This

r s

bundle comes with a at GaussManin connection induced by the connection of L

The holonomy of this connection will b e computed in the next section Let us notice

that the spaces A w w are isomorphic although not canonically isomorphic

r s

and we can x w w as we like For deniteness we choose w w to b e

s s

admissible so that Rew Rew To describ e A w w as a linear

s r s

space we choose a basis as follows Fix a nonintersecting family of lo ops

so that winds around w as shown in Fig a Introduce the shorthand notation

i i

r r

to denote a homotopy class of nonintersecting families of lo ops constructed as follows

j j

Let i s j r b e slight homotopic deformations of such that

i i

j r

lies inside and such that is a nonintersecting

s s

b e the asso ciated homotopy class Dene a horizontal family of lo ops Let

section denoted by over this family to b e the section which takes the value with

resp ect to the trivialization over a p oint with co ordinates ob eying

Rew Rew Rez Rez

Rez Rez Rew

r r

If r r run over all nonnegative integers with total sum r the families of lo ops

form a basis of A w w

r r

Figure The lo ops used to dene a basis of A w w and A w w

r s s

Theorem The U sl C module A w w is isomorphic to the tensor

q r s

product of Verma modules

V V

n n

with action of U sl C given by the sfold coproduct If Rew Rew

q s

an isomorphism is explicitly given by

r r r r

s s

v F F v

n n

F by denition of F For higher r we Proof For s we have

have to show that the action of the generators on the basis is indeed given by the

copro duct For the diagonal generators K K this follows from the denition For

the other raising and lowering op erators the pro of follows from contour deformation

To compute the action of F we must deform the added lo op to the comp osition of

lo ops homotopic to

r r r r

s s

The co ecient is up to a sign the transition function we pick up by going from

is trivialized to the p oint where the the p oint where the section over

r r

j i j

section over is trivialized The sign is and comes

n r

j j

j i

from reordering the lo ops Thus q and we get the result

F K K F

r r

of Similarly by computing the contribution prop ortional to

s i

E we see that we get the same terms as in the computation of E

n r

j j

j i

that we pick up by going from the vicinity of w except for the factor q

to P and we obtain the result

K K E E

This concludes the pro of

Remark We see that the tensor pro duct also has a top ological interpretation let

S S b e the upp er and lower halves of the unit circle We can think of D nfw w g

with w w i j and w int D as the result of glueing s punctured discs D n fg

i j i

in such a way that S of the ith disc is identied with S of the i st disc This

construction gives an identication of A w w with A A so that

r s

r r r r

s s

is identied with

The mo dule A w w b eing isomorphic to A w w also has the

s r s

structure of a tensor pro duct of Verma mo dules In order to achieve compatibility

b etween tensor pro duct structure and bilinear form one has to choose the isomorphism

in a sp ecial way Let b e the nonintersecting family depicted in Fig b and as ab ove

r r

dene and a horizontal section taking the value with resp ect to

the trivialization over a p oint with

Rew Rew Rez Rez Rez Rez

r r r

Let furthermore b e the automorphism of U sl C dened on generators by

H H E K E F FK

The following dual version of Theorem is proven exactly as Theorem

w w is isomorphic to the tensor Theorem The U sl C module A

s q

product of Verma modules

V V

n n

with action of U sl C given by the twisted coproduct If Rew

Rew an isomorphism is explicitly given by

r r r r

s s

v F F v

n n

Tensor Pro ducts and Intersection Pairing

The isomorphisms describ ed in the preceding Theorems dene intersection pairing as

a bilinear form on V V

n n

Theorem The intersection pairing coincides with the product of the Shapovalov

bilinear forms on V In particular it is symmetric and degenerate It reduces to a

nondegenerate symmetric bilinear form on the fusion rule subquotient

Proof For s the highest weight vector v of V has v v and one

n n n n

T T T

has E F F E K K which are the characterizing prop erties of the

Shap ovalov bilinear form on the Verma mo dule V The choice of identication of

A A with the pro duct of Verma mo dules is chosen in such a way that the weight

t t ht t i of each intersection p oint factorizes into s factors equal to the

i i

weights of the corresp onding intersections in

i i

The Lo cal System

The result of Theorem can b e cast into the formalism of lo cal systems To b e more

precise introduce the dep endence of the lab els n n explicitly in the notation

A A w w jn n

r r s s

Fix a base p oint w w such that Rew Rew The spaces

dene a at vector bundle over C D with GaussManin connection induced by

V w is identied with V the connection on L The b er over w

n n r

by the explicit isomorphism of Theorem Similarly for any p ermutation S

using the trivial V with V w we can identify the b er over w

n n

identication

jn n w w jn n A w A w

s r r

Mono dromy Action of the Braid Group oid and Universal

RMatrix

D with D fz In the following we will consider the conguration space C

n n

C j jz j g n r and n n

Let p C D C D b e the pro jection given by omitting the rst r

entries of z z w w p denes a b er bundle over C D with b ers

r s

p w w X w w In particular X w w D n fw w g is

s r s s s

the punctured unit disc In the following we will restrict our attention to fw w g

int D

Fix a base p oint x w w C D with Rew Rew We

s s

construct a nonab elian representation of the colored braid group oid B D x

G Note that O S x and G i f s g G G is generated by

x s S

C D is a smo oth parametrized curve with and S Here

i i

x which implements a counter clo ckwise exchange of w and x and

Let the representation space V asso ciated with S b e A w w

s r

It admits the explicit decription as linear span

j j

A w w C

s r

The sum is over j j f p g such that j j r Thus we

s s

r s

N rs

C with N r s The have an identication A w w

simplest nontrivial case is s with N r s r Let b e represented by the

deformation homomorphism V V asso ciated with Introduce the

i i

n n

q number notation n q q and for k n

n n n k

q q q

k k

q q q

Prop osition I

j j

j j k j k j

i i

i i s

k k n j k n j k

i i

q q

n j k

i q

j j

j j k j k j

i i

i i s

k k n j n j k

i i

q q

n j k

i i q

Proof Without loss of generality we can restrict the pro of to the case s i

and id The lo ops used in this pro of are represented in Fig The matrix

representation of is computed by consecutive deformations and sub divisions of the

individual lo ops in

j j j j n j n j j n j j

q q

Sub divide the last lo op in a and a lo op to obtain

j j

j n j j j j

Figure The lo ops app earing in the pro of of Prop osition The p oints marked

with a cross are the p oints used to dene the section

Iterate this sub division until there is no lo op left over The result is

j j

X X

i j k j k k k n

q q

i i j

The sum over ordered k tuples is now p erformed with Gauss formula

i k j

q q

i i j

Then sub divide the lo ops in and lo ops This decomp osition yields

j k j k

j k

i h

j k

j k ln j k l j k l j k l

Insert and into and reorder the double sum to obtain

j j

k k j n j k n j k

i h

l lj n k j k j k

Then p erform the second sum with the q binomial formula

k k

Y X

k k k k n j l lj n k

n j l q q

l l

Insert and into to nd The matrix representation of is

computed following the same lines

The imp ortant consequence of Prop osition is that the deformation homomor

phism V V written as an op erator has a universal form which resembles

i i

the universal Rmatrix of the quantum group algebra U sl C Let p N fg b e

the smallest p ositive integer such that q

Theorem Denote by X the operator X acting on the

ith factor of V V Take n n p Then

n n s

q q

k k H H k k

i i

q q E F

i i i

and

q q

k k k k k H H

i i

F E q q

i i i

Proof Recall the denition of the op erators E F H and of They act as

i i i i

j n j

i q i q

j j j j j

s s

s i i s

q q

j j j j j

s s

s i i s

j j j j

s s

n j H

i s i s

and

j j j j

s s

s i s

Compare and with and to conclude and

The content of Theorem exceeds pure nomenclature since the op erators E F

i i

and H have a top ological interpretation They satisfy the commutation relations

H E E H F F E F H

i j j ji i j j ji i j j q ji

We have identied A w w with the tensor pro duct V V of

r s n n

U sl C Verma mo dules the identication b eing

j j j j

s s

F v F v

s n n

Moreover we have identied E HomA w w A w w

i r r

s s

with the element

E E

of U sl C Here E stands in the ith entry Similarly we have pro ceeded with F

q i

and E is identied with the ith transp osition We have proved that this identica

i i

tion is a quantum group algebra homomorphism and a mo dule isomorphism

The observation of this section is now that

with R U sl C the universal Rmatrix in an obvious normalization and R

its inverse acting on the ith and i st entry

It follows that denes an Rmatrix representation of B X The YangBaxter

equations for the top ological Rmatrix follows from prop erties of the universal R

matrix

Having constructed an N r sdimensional Rmatrix representation of B D

we also have a rank N r s lo cal system L D over C D Let C D b e

the intersection of the cell C with C D

r N rs

L D t C D C

with the equivalence relation over C D C D given by multiplication with

the matrices and resp ectively The b er space p w w is

in the basis A w w The parallel transp ort matrix asso ciated with

is the universal Rmatrix in the representation n n

i i

This representation of the braid group oid is not irreducible in general In particular

it has as invariant subspaces the null space of the bilinear form which denes a

subbundle of our at bundle invariant under parallel transp ortThese subspaces are

describ ed explicitly in the App endix

Lo cally nite homology

We have worked on the spaces A rather than on homology groups directly We now

formulate some conjectures on the relation to homology and the structure of the corre

sp onding lo cally nite homology groups These conjectures follow from the assumption

that our quantum group action extends to an action on homology and by applying

the computations of FSa F which are not completely rigorous to the situation

studied here As usual we assume that s distinct p oints w w in the interior of

the unit circle s p ositive integers n n and a complex number q are

given If q is a ro ot of unity we furthermore assume that n p where p is the

smallest p ositive integer such that q For small enough the lo cally compact

spaces X X are dened as in and we have a lo cal system L over X

r r r

Conjecture If q is not a root of unity the map

A w w H X X L

r r s r

r r r

is an isomorphism of vector spaces

If q is a ro ot of unity let U sl C b e Lusztigs version of U sl C Lu with

p p L

b e the Verma mo dule over the quantum generators H E F E p F p Let V

group U sl C with vacuum vector v so that H v n v and E v

n n n n

E pv There is a canonical Hopf algebra homomorphism U sl C

n q

L L

U sl C so that V is also an U sl C mo dule For any H diagonalizable

q n

U sl C mo dule M denote by M the eigenspace of H to the eigenvalue n

q n

Conjecture If q is not a root of unity there are isomorphisms

H X X L V V

r n n n sr

r r r i

H X L Ker E jV V

r n n n sr

r r

If q is a root of unity there are isomorphisms

lf L L

H X X L V V

r n sr

r r r n n i

lf L L

H X L Ker E jV V

r n sr

r r n n i

App endix

We summarize some known facts ab out U sl C following essentially D RT

Fix a nonzero complex number q Let U sl C b e the algebra with unit over C

with generators E F H and relations

H H

H E E H F F E F q q

H H

We often denote K q Of course q is not welldened in the algebra but its action

on mo dules where H takes integer values is A more precise denition is the following

let U sl C b e the complex algebra with unit with generators E F H K K and

relations

H E E H F F E F K K

K K K K K H HK

U sl C is a Zgraded algebra with the assignment deg E degF

deg H deg K Let G b e the category of Zgraded left U sl C mo dules

q q

M M such that

nZ n

For all M there exists an N such that E

HM nM and K M q M

n n n n

The degree of homogeneous element of a mo dule in G is called weight Following

common usage we refer to ob jects in G as Zgraded U sl C mo dules

q q

Let n b e an integer and q C n fg The Verma module V is the quotient of

U sl C by the left ideal generated by E K q and H n with left action

of U sl C The mo dule V is in G and is generated by a highest weight vector v

q n q n

image of of weight n A basis of V is given by the vectors F v j

n n

and one has the explicit formulae

j n j

j j j j

F v HF v n j F v EF v

n n n n

q q

The notation we use here for q numbers are

j j j l

j j

j j q q

l l

j j j

If q is a ro ot of unity we dene a number p as the smallest p ositive integer such that

ip p

q e

for some integer p If q is not a ro ot of unity we set p

Prop osition I If q is not a root of unity V is irreducible for n It

contains a proper submodule SV generated by the singular vector F v if n The

n n

quotient V S V is an irreducible ndimensional representation

n n

ip p

II If q e is a root of unity then V contains a proper submodule SV generated

n n

by the singular vector F v where n p and n nmo d p The quotient V S V

n n n

is irreducible of dimension n

The Shapovalov form on V is the symmetric bilinear form V V C

n n n

uniquely characterized by

v v and

n n

E F H H V

The null space of is SV

The action of U sl C on tensor pro ducts of mo dules in G is dened by the coas

q q

so ciative copro duct U sl C U sl C U sl C dened on generators

q q q

H H H K K K

E E K E F F K F

The action on tensor pro ducts with s factors is given by U sl C U sl C

q q

s s

U sl C with The universal Rmatrix of

U sl C is the formal series

q q

k k H H k k

q R E F q

A singular vector is a vector annihilated by E

This series is well dened on any tensor pro duct mo dule in G since only nitely many

terms are nonvanishing when R acts on a vector Also singular denominators cancel

Let denote the pro duct of Shap ovalov forms V V V

n n n

V C

Prop osition Let R R be the R matrix acting on

the ith and i st factor in V V and P the transposition

n n ii s

Then

i i s

R P R P

ii ii ii ii

for al l V V

n n

Let W V V b e the space of singular vectors of weight n in V

n n n n

V The family of vector spaces W V V S carries an Rmatrix

n n s

n n

representation of the colored braid group oid B As a consequence of Prop osition

we have

Prop osition Let F V V be the quotient of W V V by

n n n n n n

s s

V the nul l space N of restricted to W V The representation of

n n n

B on fW V V g reduces to a wel lde ned representation on

n S

n n

fF V V g

n S

n n

V The sub quotient F V is called the fusion rule subquotient of V

n n n n

V with weight n It can b e characterized more explicitly

Prop osition Let p n n n Then

dimF V V N

n n n

n n

if jn n j n minn n p n n

otherwise

of weight n which is V there is a singular vector in V Thus if N

n n

not in the null space of Corresp ondingly we have a homomorphism

V V V C

n n n

n n

Supp ose in the following that p n n n Introduce the path space

n as the space of complex linear combinations of sequences m m P

s s

m n m

s i

i s N of integers in p such that N N

n n n m n m

s i i

Prop osition The homomorphism

P W V V

n n n

n n

m n m

m m C v C C

s n

n n n m n m

composed with the canonical projection W F gives an isomorphism

n n

P F V V

n n n

n n

The pro ofs of the last two prop ositions can b e extracted from FK noticing that

since the vectors of the form with some SV are in the null space of

s j n

we can replace everywhere V by the irreducible quotient V S V

n n n

and Topolog Fock Space Representations of A

ical Representations of U sl C

We apply the top ological representations of U sl C to the Fock space representa

tions of the untwisted ane KacMo o dy algebra A We show how singular vectors

in quantum group Verma mo dules determine Fock space representations of BRST op

erators primary elds and conformal blo cks

Fock Space Representations of A

Let us recall the basic facts ab out Fock space representations of A The algebra A

a f g and n Z and a central element K with relations is generated by J

J J J J J K

n m nm n m

n K K J J J J

n nm m n

supplemented by a derivation d satisfying

a a

d J nJ d K

n n

Following Wa BF FF we then require a b osonic system together with

n Z with a free b osonic eld Let b e the algebra generated by and

relations

nm n m n

m m

Furthermore we require a Fock space F with vacuum vector v such that for n

v v

Let b e the algebra generated by a n Z with relations

a a n

n m nm

generated from a vacuum and F the Fock space of charge J where

satisfying for n vector v

a v v

n n

Jk Jk

Then dene and F F F This tensor pro duct of Fock spaces

comes equipp ed with the structure of a highest weight mo dule over A at level k

through the following construction due to Wa Let us introduce eld op erators

X X

n n

z z z z

n n

a z j z

and z such that j z i z In terms of these we can comp ose currents

J z z id

J z z z id id j z

J z z z k z id z j z

Then it can b e shown that the co ecients of the developments

n a a

z J J z

satisfy the relations The free eld stress energy tensor

id j z j z T z z z id

agrees with the Sugawara form Expanding as usual

T z L z

F also b ecomes a highest weight mo dule over the Virasoro algebra Vir with central

charge c k k The generators of A have the explicit form

J id

n n

J id a

m n

n nm

X X

id a J k n

m nm n m n

n nml

The generators with n generate an sl subalgebra F is in particular a mo dule

over sl The vacuum is also an sl highest weight vector with

J v J v J v

Jk Jk Jk

As an sl mo dule F thus contains a highest weight representation with spin J gen

erated from the vacuum This representation is irreducible since J The

basic ob jects in the following constructions are vertex op erators which we take to b e

dened as

V z

I k

n n

X X

I I z z I I

exp exp a a exp exp b lnz a

n n

Here a b and we have an op erator from F to F To b e precise

n n

Jk J I k

V z expi z denes a bilinear form on F F restricted dual

J Kk Jk

Finally let

I a

V z V z z

I k I k

splitting o the factor which dep ends on the zero mo de of j z

Intertwiners

As a rst application of the top ological representations we construct op erators map

ping one Fock space to another which intertwine the representations of A and con

sequently Vir As a basic ingredience introduce

V z z V z

the screening op erator V z maps F to F Its op erator pro duct expansion

Jk J k

with the stressenergy tensor is given by

T z V w V w V w O

z w z w w

where O sums up regular terms It shows that V has conformal weight The

op erator pro duct expansions of V with the currents have the form

J z V w O

n o

J z V w id V w

w z w

J z V w O

They follow from basic op erator pro duct expansions of the free eld diverging terms

cancelling each other neatly Due to the total derivative V z do es not intertwine

the action of A on F with that on F To pro duce intertwiners we have to

Jk J k

integrate pro ducts of screening op erators This gives

Q V z V z dz dz

R R

V z V z z z dz dz

R i j R

ij R

Y Y

z V z V z z z dz dz

R i j R

i ij R

a map from F to F

Jk J Rk

Taking matrix elements these integrals give rise to top ological representations of

U sl C at q exp p ossibly a ro ot of Let X D n fg n fz

q R ij R i

z gS b e the conguration space of R indistinguishable screening charges in the disc

j R

D fjz j g punctured at Fix a p oint p on the b oundary of D eg p

Let L b e the lo cal system determined by the multi valued form on X obtained by

R R

evaluating the integrand on Fock vacua Note that dierent values of J yield dierent

Let A b e the space of linear combinations of families of nonintersecting

lo ops in X based at p together with sections of L mo dulo equivalence relations

R R

reecting the p ossibility to homotopically deform or reparametrize Let E A

A F A A and K A A b e the top ological op era

R R R R R

tors introduced ab ove They satisfy the U sl C relations E is the combinatorial

b oundary op erator comp osed with a map which identies R chains in X with

R chains in X We have shown that

K E q E K K F q F K E F K K

endowing A with the structure of a mo dule over the quantum group algebra

p p

U sl C In the application we have in mind q In this case E and

K This is proved by an explicit computation using a basis to describ e A w

as a space Families of lo ops which contain p homotopic deformations of a single lo op

represent null homologous cycles To prove this one retracts the p lo ops to p p oints

on a single lo op path ordering the arguments The prefactor from the path ordering

vanishes This can b e taken into account by adding another relation in the denition of

the space A This relation puts families of lo ops which contain p homotopic lo ops

equivalent to the null family As a consequence then also F Using a generalized

version of Poincare duality this could p ossibly b e understo o d in terms of the kernel

of the top ological intersection pairing For q a ro ot of unity the conclusion is that we

r ed

sl C the algebra nd a representation of the reduced quantum group algebra U

obtained from U sl C dividing by the ideal generated by the central elements E

p p

F and K Let us only consider this case in the following For a detailed account

r ed

on U sl C and its representation theory we recommend FK

r ed

sl C Verma A is isomorphic to the U As a quantum group mo dule

q R

mo dule V n n J generated from a singular vector v n such that

E v n K v n q v n

A V n b e the isomorphism For R p we dene weight Let

R nR

spaces V n C F v n K acts on V n by multiplication with q

R R

Restricting our attention to the most interesting case let k p b e a

p ositive integer and let n Z The BRST construction of BF FF then pro duces

unitary integrable irreducible highest weight representations of A with n p

n J on the homology Here we work directly with the representations on the

Fock spaces

Theorem I Consider Q as a map from V n to the space of linear operators

Hom F F Then

C Jk J Rk

Q ker E V n V n Hom F F

R R Jk J R

maps to the space of A intertwiners II Let n J and n n mod n

p For n

kerE Cv n

For n p

kerE Cv n CF v n

III For n p the only nonvanishing intertwiner besides the identity is Q

with R n and C F v n unique up to a normalization constant

Thus the structure of singular vectors in U sl C Verma mo dules in our case a

ro ot of provides all data needed to decide wether intertwiners exist and furthermore

gives explicit formulas for them

Corollary I For n J let F F For n p the nonvanishing

n Jk

intertwiners form an innite sequence

pn n

Q Q

F F F

pn n n

II This sequence is a complex

The homology of this complex is isomorphic to the irreducible A highest weight

mo dule of weight J and level k Further intertwiners can b e constructed out of cy

cles with np path ordered arguments n f g They corresp ond to p owers

of generators F p in Lusztigs version of U sl C They can b e used to con

struct complexes whose homology gives nonintegrable A mo dules BMP Here

n n

n q q and n n

Chiral Primary Fields

The second application of top ological representations concerns the construction of chi

ral primary elds w F U F U is a representation of A at level

Jk I Kk I

to b e dened b elow Consider op erators

z z V z

I I k

mapping F to F The op erator pro duct expansion with T z

Jk J I k

I I

T z w w O w

I I I

z w z w

shows that z has conformal weight h I I The op erator pro duct

I I

expansion with the current J z has the form

a I a

J z w w D J O

I k I

z w

I a

D J is a spin I representation matrix of sl For u U the representation space

of the representation D dene

u w w u

I I

The op erator pro duct expansion implies that

I a n a

u w D J u w w J

With these preparations in mind let us consider the op erators

w V z V z dz dz w

I R R I

from F to F with K J I R Using Wicks theorem

Jk Kk

minfI Rg

X X Y

C I z w

I f Rg

jI j

Y Y

I R

w z z w

ifRgnI

we normal order the integrand C I is a combinatorial factor counting the

number of contractions The result is

Z minfI Rg

X X

C I w dz dz

I R

I f Rg

jI j

I R I a

w z V w V z V z w

i I k k k R

R R

Y Y Y Y

I a

w z z z z w z

i i j i

i i

ij R

ifRgnI

The chain C is taken to b e an element of A w The lo cal system L w involved

R R

in the denition of A w is determined by the mono dromy of the multi valued R

form obtained by evaluating the integrand on Fock vacua As a U sl C mo dule

A w has b een shown to b e isomorphic to the tensor pro duct of the space

Verma mo dules V n V m Let V n V m b e the weight space on which K

nmR

acts by multiplication with q

Lemma Consider w as a map from V n V m to the space of bilinear

operators Hom F U F with K J I R Here U is the sl representa

C Jk I Kk I

tion space For C ker E V n V m V n V m we nd the

R R

commutation relation

I a a n

D J u w J u w w

I I

n C

For C a chain which represents an absolute cycle in homology we obtain a chiral

primary eld Its construction go es as follows Expanding

R R

nh h h

K I J

w u u w

I n I

C C

it follows that

I a a

D J u u J

I n I mn

Let

h h h

K I J

U U w C w w

I I

a I a n

b e the A mo dule with J acting as D J w d as w w and K as zero

Then

nh h h

K I J

u u w

I n

denes a map from F U to F To b e precise for u U

Jk I Kk I

h h h

K J I

u w Hom F F w C w w

I C Jk Kk

is a formal p ower series in w whose co ecients are linear op erators from the Fock space

F to the Fock space F For u U we obtain a well dened op erator through

Jk Kk I

R R

u u w w dw

I I

C C

Here C is moved with w through the GaussManin connection on C n fg That is we

asso ciate op erators to Laurent p olynomials This map is an A homomorphism

Theorem Consider as a map from V n V m to the space of bilinear

operators Hom F U F with K J I R Then

C Jk I Kk

ker E V n V m V n V m Hom F U F

Jk I Kk

R R

maps to the space of A intertviners

The general decomp osition of the tensor pro duct of Verma mo dules V n V m

which contains among other things the structure of singular vectors has b een carried

out in FK For the sake of brevity it cannot b e repro duced here Let us only

mention the following partial results

Prop osition Let n J m I n m p and R n m

Then

ker E V n V m V n V m

R R

Rl l

C x F v n F v m

with x l R determined by x and the recursion relation

nRl

R l n R l x q l n l x

q q l q q l

x x

where x q q

u with K The intertwiner corresp onding to this cycle is denoted by

J I R It is unique up to a normalization constant which can b e xed as follows

using the three p oint function

K J I

h h h K K

K I J

e w v i w N hv

Jk Kk

JI JI

involving a classical sl ClebschGordan co ecient and the fusion co ecient N

The three p oint function is conveniently normalized such that the fusion co ecient

takes the values or u is the chiral primary eld in the Fock space

representation

Prop osition Let n J m I and l K be such that

n m l p Dene a triple to be admissible if jI J j K I J

I J K Z and I J K p Then the chiral primary eld has the fol lowing

K K

admissible not admissible N II For properties I For

JI JI

The quantum group data enco ded in the fusion rules of w is here inherited

from the representation V n V m The most imp ortant prop erty of w is BRST

invariance It is a necessary condition to pro ceed through the BRST construction Let

us mention that BRST invariance also has a cohomological meaning on the quantum

group level

Conformal blo cks are vacuum exp ectation values of pro ducts of chiral primary elds

A pro duct

J J

u u u

s s

J I J I J I

s s

denes an element of

Hom U F U F

I J k I J k

Using such op erators can b e expressed in terms of free elds Pro ducts of ex

pressions of the form are well dened as absolutely convergent matrix pro d

uct and give rise to elements of when integrated as in provided the

time ordering condition is satised This condition states that U z U z

p p

j for all i j Here U stands is well dened whenever jz j jz U z U z

i i pq p

j q

for or a vertex op erator This condition gives the cycle C of integration for

used to integrate the factors the expression as a pro duct of nested cycles C

J I

i i

The resulting cycle is identied with a singular vector of V m V m with

m J I J in the top ological representation space Algebraically the ex

i i

J J

i i

with an element is understo o d as follows Identify C pression of C in terms of C

J I J I

i i i i

Hom V n V n V m kerj

U sl i i i V n V m

i i

where m J and R I J J Then

i i i i i i

C C C C

J I J I J I

s s

Hom V n V m V m V m

s s

U sl

kerE j

V m

These quantum group intertwiners and path spaces have b een investigated in FK

Generalized Hyp ergeometric Functions on the Torus

and the Adjoint Representation of U sl C

We study the homology groups with co ecient in lo cal systems arising in the free

eld representation of minimal mo dels of conformal eld theory on an elliptic curve

with punctures We dene an action of the quantum enveloping algebra U sl C

on a space of relative cycles extending the results obtained previously for the sphere

Absolute cycles are identied with singular vectors In the case of one puncture we nd

that the resulting top ological representation is essentially the adjoint representation

Introduction

The studies La FW SV indicate that there exists a dictionary b etween homol

ogy of certain conguration spaces with co ecients in lo cal systems and representation

theory of quantum enveloping algebras D The examples of lo cal systems providing

such connections come from integral representation of conformal blo cks of conformal

eld theory BPZ DF GN ZF M CF FGK The idea is that in

some sense the charges generating half of the quantum group symmetry in the free

eld representation in conformal eld theory are given by integrals over screening op

erators PS BMP GP GS In the previous chapter FW we have shown

the existence of an action of U sl C on certain relative lo cally nite homology

groups on conguration spaces on the sphere In this case the lo cal system is given by

the integrand of the free eld representation of conformal blo cks of the SU WZW

mo dels or minimal mo dels

In this chapter we consider the situation of the torus for which one knows explicit

integral representations F BF FG We restrict our attention to the case of

minimal mo dels which is the simplest The main dierence is that the lo cal system

is not given by a line bundle as in the case of the sphere but rather a vector bundle

From the p oint of view of free elds this follows from the fact that the space of free

eld conformal blo cks on the torus is higher dimensional

We nd again an action of the quantum enveloping algebra of sl C on relative

cycles in such a way that absolute cycles are identied with singular vectors as in the

case of the sphere The resulting representation is a tensor pro duct of Verma mo dules

with U sl C with the adjoint action We hop e that this work will lead to a clearer

understanding of the role of quantum groups for higher genus Riemann surfaces

Generalized Hyp ergeometric Functions on the Torus

In the free eld representation of minimal mo dels with central charge c p

p pp on the torus C Z Z one is led to consider integrals of the form

G z z j

C s

p p

X Y

i j

W j E z z j dz dz

i j s sr r

ij sr r

where

i s

n n

s i s r

s r i s r r

mp m p

m m

p p

parametrize the exp onents b elonging to integrated screening variables

and

exp W W jp p W j

p p

sr r

W z p p

i i

z z j

i j

E z z j i

i j

with

i n i

e e

iz n i n

e z j

iz n i n

e z j

the Dedekind eta function and the Jacobi theta functions The values of the exp onents

are constrained to satisfy

s s

X X

r xp n n r xp

i i

for some integer x reecting charge conservation In the following we restrict our

attention to the case when r and n n admitting screening

charges only That is we assume that

n p

i s

p p

s i s r

p p

with the neutrality condition

When studying integrals of this form one is entering the following kind of problems

To b egin with assume that

n n n n n n

for some k and n n f g such that n n s

n n k k

and n r Let C Z Zb e the torus with mo dular parameter Then we have

a vector of multivalued forms with comp onents

z z jz z j

s s sr

i j

W j E z z j dz dz

i j s sr

ij sr

on the conguration space

S X n fz z gS

n n ij i j n

Since W j W j we can restrict to the range p p Other

p p

prop erties of W j are summarized in App endix A

We will often use the identication m m m p mp of Zpp Z with

Z where is the lattice generated by p p and p

Let n n n and

S n fz z gS X

n ij i j n n

on the rst n variables is a bration with b ers The pro jection p X X

n n

X z z p z z

r s s

These are conguration spaces of r indistinguishable particles on the punctured torus

to obtain a vector of multivalued n fz z g Fix and z z X

s s n

r forms on The p ositions z z which are presently kept xed should not

b e confused with the p ositions of the screening charges z z Let us suppress

s sr

the dep endence on the former and the mo dular parameter in our notation The r forms

are multivalued on X single valued on the universal covering space X with

r r

base p oint and dene a p pdimensional representation of X through

p p

with x x the right action of the fundamental group on the universal covering

space The representation matrices can b e computed by analytic continuation Let c

p p b e singular r chains in the universal covering space The equivalence

c is compatible with the pairing relation c

p p

In other words we can view c as a singular r chain with co ecients in the space of lo cal

horizontal sections of the vector bundle of rank p p

p p

L X C x v x v

r r

We thus need to examine the singular homology group H X L with co ecients in

r r

the lo cal system asso ciated to the representation of X As a supp ort condition

we will require that the chains are lo cally nite lf DM ES BM p ossibly

innite linear combinations of simplices on X fw w X j jw z j g

r r i j

Elements of H X L pro duce when paired with a generalized hypergeometric

r r

function on the torus provided the integral is convergent

Lo cal Systems over Conguration Spaces on the Torus

Braid Group on the Torus

Let T S S n f g and dene

C T T n fx x gS

n ij i j n

the conguration space of n indistinguishable particles on the torus Here S denotes

the symmetric group acting as x x x x Let

B T C T

n n

b e the braid group on the torus A convenient choice of base p oint is

n n

N N N N

for some N n For i n dene elements B T as represented by

i i n

the paths C T

i i i i

t t t t

i i

N N N N

moving the particle in p osition i along an A and B cycle resp ectively For i

n dene elements B T as represented by

i n

x t x t

i i i

x t i i t

i i t x t

t cos t sin t

implementing a counter clo ckwise exchange of the particle in p osition i with that in

p osition i It is convenient also to introduce the abbreviations

n n n

in terms of which

n i i i

i n n i i

The group B T is generated by and i n A detailed investigation

n i

of B T can b e found in Bi S Cr

Coloured Braid Group oid and Lo cal System

Let n n n f g jnj n n and dene

k k

jnj

S C T T n fx x gS

n n ij i j n

the conguration space of particles with colours f k g identically coloured parti

cles b eing indistinguishable Let C T b e the base p oint The orbit of un

S der the action of S can b e identied with the right coset space I S S

n jnj n n n

An element I can in turn b e describ ed by a colour map f jnjg

T b e the space of paths starting in and f k g For I let B

jnj

ending in up to homotopies preserving the end p oints Dene

T B B T

jnj

the coloured braid group oid on the torus Multiplication is comp osition of paths In

idid

T C T The coloured braid group oid B T can b e particular B

n n

jnj

describ ed in terms of the braid group B T Let B T S b e the canonical

n n

jnj

homomorphism There exist onetoone maps

fg B T j g g B T

jnj

having the prop erty

g g g g

Using this we can write down generators for B T The generators are

i jnj

is the cyclic p ermutation i i mo d jnj and is the ith transp osition A

representation of B T on a family of nite dimensional vector spaces V indexed

by I is a family of maps

B T Hom V V

Hom V V the group oid of invertible linear mappings from B T to

b etween the vector spaces V satisfying the representation prop erty

g g g g

The dimension of a representation is d dimV A ddimensional representa

tion of B T denes a at rank d vector bundle over C T with distinguished

n n

trivializations over the p oints I

The representation restricted to C T gives a at vector bundle C T

n n

V It comes with an identication of the b er over with V The identication of

id id

the b er over with V is uniquely given by the condition that the parallel transp ort

along any path from to is

To do explicit calculations it is convenient to introduce lo cal trivializations of L

Dene cells lab eled by elements of I

g x x C T fx x C T j x

n n

jnj

i jnj

The union of the closures of these cells is C T and every cell contains precisely one of

the p oints in the S orbit of Since cells are contractible we have an identication

jnj

of the restriction of L to C T with the trivial at bundle C T V This

r n n

trivialization will b e used often b elow

Torus with punctures

T on Let n n n s jn j and r n The pro jection p C T C

k k n n

the rst s variables is a bration with bres

C T n fx x g p x x

r s s

Note that C T n fx x g is the conguration space of r indistinguishable particles

r s

on T n fx x g the punctured torus Cho ose a base p oint x x C T

s s nid

In the language of categories B T is the set of morphisms of a category whose ob jects are

elements of I A representation is a functor to the category of nite dimensional vector spaces

and let x x b e the base p oint of C T n fx x g Then B T

s sr r s n r

C T n fx x g is a subgroup oid of B T and we have a homomorphism

r s n

B T B T The at vector bundle corresp onding to the pull back of a

n r n

representation is just the restriction to C T n fx x g of the at vector bundle

r s

over C T asso ciated to

Topological Representations of U sl C

Lo cal system from multivalued forms

The multipliers of the multivalued r forms up on analytic continuation dene a par

ticular representation of the coloured braid group oid This representation denes in

turn a lo cal system over the conguration space The singular homology groups we

investigate have co ecients in this lo cal system

Recall the basic data which we start from n n n f g jnj

n k Then s r n r and k Q k such that

k k

idi Remember that f s r g f k g denotes the colour map

asso ciated with I Given these data we consider

X Y

i j

z z j f z j E z z p p

sr i i i j

i ij sr

p p Fix the mo dular parameter Then f f is a

p p

multivalued analytic function on the conguration space C with values in C

Let T x x x x to obtain a dieomorphism C T C

n n

Dene f f Fix a base p oint x x in Y C T such that

sr n n

x x For I let f b e the singlevalued function

p p

on the universal covering space Y with values in C dened as the analytic

T T T id

continuation of f from the base p oint where it takes the value f f

An element g B T induces a map Y Y through x xg

g n n g

The p oint xg is represented by a path from to comp osed with a path from

to px p Y Y b eing the covering pro jection Then

n n

p p

T T

g M f f

p p

denes a p pdimensional representation of B T on V C An explicit

calculation by analytic continuation yields

s r

M exp i

p p

o n

exp i s r M

p p sr

exp f i i i g M

If s r is a screening charge it follows that

M exp i

exp i M

These matrices deserve an abbreviation since they will o ccur frequently b elow Let

q exp ip p and

q A q B

p and i with the convention q exp ip If i n ip p

it follows that

n i

M q

If i i this representation matrix takes the form

q M

completing the description of the representation of B T asso ciated with the mul

tipliers of f

T and C T n fx x g Let n n n jn j s p C T C

r s k n n

p x x In the following x x will b e xed as ab ove Pulling back the

s s

ab ove representation of B T with the homomorphism B T B T we

n n r n

obtain a representation of B T We take the tensor pro duct of this representation

n r

with the pullback of the totally antisymmetric representation of S by the canonical

homomorphism B T S The result is the representation denoted by

n r r

asso ciated with the multivalued r forms This representation induces a lo cal

system at vector bundle L x x C T n fx x g

r s r s

the op en disc of radius centered at x i s Let b e a small number D

the conguration space C Y of r indistinguishable D Denote by Y Y T n

r i

are subsets Z Y of cardinality r Fix p oints y p oints on Y Thus elements of Y

y D such that y x y and dene Y fZ Y jy Z g The bijections

Z Z fy g Y n Y Y

r r r

lift to isomorphisms L jY n Y L jY The lift is of course not unique To

r r

x it it is sucient to dene the isomorphism from the b er of the base p oint to the

b er of its image We dene it to b e the identity map in the distinguished trivialization

introduced ab ove

Families of Lo ops and Op erators

We generalize the previous construction on the sphere to the torus To make this

chapter selfcontained we briey recall the notion of families of lo ops and top ological

op erators with quantum group relations adapted to the torus Let x x

C T Y C Y and y Y b e as ab ove The p osition of the punctures

n id r

will b e kept xed in the following

A nonintersecting family of lo ops based at y is a family of smo oth

homotopically nontrivial embedded lo ops starting and ending at y with no mutual

intersections except at the endp oints Homotopies of families of lo ops are dened Non

intersecting families of lo ops can b e represented by embeddings of the op en r cub e

with op en r faces Q into X

r r

Let A b e the space of linear combinations of homotopy classes

of nonintersecting families of lo ops with co ecients in the space

of horizontal sections of L over Q Horizontal sections corresp onding to homo

r r

topic families of lo ops are canonically identied by parallel transp ort so the denition

represent lo cally nite relative homology classes in makes sense The elements of A

by a subspace which maps to zero in L We consider a quotient of A Y Y H

r r r r

where the equivalence relation is given by A homology Let A

r r

f f for any orientation preserving or reversing isometry

of the cub e Q

when

i r r r

i i

in such a way that if the homotopy ever is homotopic to the comp osition

i i

is denoted by h s s h s is a nonintersecting family

of lo ops for all s The sections are the restrictions of

whenever at least p lo ops in the family are in the same class

in Y y

The third identication is p eculiar to the case when q is a ro ot of unity if n lo ops

say in a nonintersecting family are all homotopic to a lo op

n r

then the corresp onding lo cally nite homology class is prop ortional to the class of a

relative cycle parametrized by t t t t t Q as

n n r r

t t t t t t

r n n r

The prop ortionality factor is

and vanishes if n p We dene now op erators E F and K acting on the space

A and compute their commutation relations The op erator E is a close relative

r r

of the b oundary op erator Dene

e e E

r i r

ir ir

r r

with e the standard face maps denotes the omission of

i i

Intuitively the ith particle is moved to y and then taken out The op erator F adds a

lo op along the b oundary of the hole around the rst puncture

r r C

with Y t x t Here and is the

C C r C

horizontal section of L with i where i is the inclusion t t

This denition makes sense since we can assume that do not intersect

r C

except at the endp oints The op erator K is simply dened as

n r

K j q

n r Note that b oth the Charge neutrality requires that n r

construction of E and F make use of the isomorphisms relating lo cal systems

over dierent conguration spaces

Theorem The operators E F and K satisfy the relations

F K E F K K F q E K K E q K

In other words the operators E F K and K dene a representation of U sl C

A on

r r

Proof The pro of is a rep etition of that of Theorem The rst and the second

relation are immediate consequences of the denition of E F and K The third

relation is b est proved using an explicit trivialization Without loss of generality we

can assume that x x C T for some I Denote

s r n n

by v the section with the value v in the trivialization over C T We can further

assume that x x y C T for some I y in

s r n i n

t p osition i Let b e the paths t x x

i r s

Using

E v

i r

i i i i

it follows that

E F v E v

r r C

0 0

i r

i i

i r C

i i i i

0 0 0 0

r r

F E v

n r n r

v q q

proving the third relation

The Torus with One Puncture

We have proved ab ove that A comes equipp ed with the structure of a U sl C

r r

mo dule A legitimate question to address is what kind of mo dule this is In this section

we will consider the torus with a single puncture and nd the adjoint representation

of U sl C

as a space we choose a basis as follows Let and b e lo ops To describ e A

A B

in Y based at y such that winds around an Acycle and winds around a B

A B

cycle For j j f g such that j j r let b e homotopic

A B A B

B B

j i

deformations of and homotopic deformations of such that lies

B A

A A B

i i i j

on the left of and ab ove and is a nonintersecting family

B A A B A

j j

A B

This family of lo ops is drawn in Fig We choose of lo ops also denoted by

A B

representatives in such a way that x C T We dene v to

r nid

j j

A B

which takes the value v in the trivialization b e the horizontal section over

A B

p p

over C T Let e p p b e the standard basis of C Then

nid

p p j j

A B

A C e

j j r

r A B

A B

In this basis the op erators E F and K are represented by the following matrices

Lemma Applying E F and K to the elements of the basis of A we obtain

B q

j j j j

j j j

B A B A

A B B

e q B q E e

B A B A

q q

A q

j j

j j j j

A B

A B A B

e q A q

A B

q q

j j j j

j j j

A B A B

A B A

e q B q B A F e

A B A B

j j

B A

e q B A A

B A

j j j j

j j

A B A B

A B

e q K e

A B A B

j j

A B

Figure The family of lo ops

A B

j j

B A

with the convention that if j or j

B A

Proof The action of E is explicitly computed from

j j j j

A B A B

e E e

id id

A B i i A B

i i

j j

A B

j j

A B

id id

A B i i

i i

The paths have representation matrices

r r i

q B q i j

r n r r i

q Aq j i j j

B A B

To compute the action of F we have to deform the added lo op to the comp osition of

two and two lo ops The result is

A B

j j j j

B A B A

F e e

B A B A

j j j

B A B

B B A A B

j j

B A

B B A A A

j j

B A

B B A A B

A transition function is picked up when the cell on which the bundle is trivialized

changes The paths have braid group oid representation matrices

q B A

r n r

q AB A

The last representation matrix can b e simplyed using AB q BA Finally the lo ops

are reordered with the help of an isometry of the unit cub e The action of K follows

immediately from its denition

The matrices acting on the sections are

Ae q e B e q e

Using these formulae we conclude that the representation of U sl C on A can

b e split into a direct sum of isomorphic representations

n n

Dene for n n q q n n n

q q q q q q q

Lemma I A decomposes into a direct sum of subspaces

j j p p

A B

0 0

n p A C e

n pnp

n j j

A B

invariant under the action of U sl C II The action of E F and K on A takes

the explicit form

j j

A B

0 0

E e

n pnp

A B

o n

B q

j j

j j j

A B

B A B

0 0 0 0

q e q e

n pnp n pnp

A B

q q

j j j n pp n

0 0

A q A B q

j j

n pp n

A B

0 0

e q

n pnp

A B

q q

j j

A B

0 0

F e

n pnp

A B

0 0

j j

j j n pp n

A B

0 0

q j n pp n e

B q n pnp

A B

o n

0 0 0 0

j j

n pp n j n pp n

A B

0 0 0 0

q e q e

n pnp

A B

j j j j

j j

A B A B

A B

0 0 0 0

q e K e

n pnp n pnp

A B A B

0 0

n n

Denote A Let U sl C b e the Hopf algebra with generators E A A

q r

F K with the relations of Theorem and copro duct E E K E

F F K F K K K see Section Let I b e the

p p p

sl C ideal generated by the central elements E F and K and U

sl C by the adjoint action We dene the U sl C I U sl C acts on U

q p q

idemp otents

n m m

T q K n Z

p p

As b efore we set q exp ip These idemp otents have the following prop erties

0 0

K T q T T T T T n n Z

n n np nq

nn pp

0 0 0 0 0 0 0 0

T n m p n Z T T

nn pp nm nn pp mn pp

0 0

For each n Z the elements T n p build a basis of the

nn pp

subalgebra generated by K

A U sl C as follows Denition For n p dene linear maps

j j

B A

0 0 0

n pnp n

B A

0 0

B q

j pj j j j n pp n

A B B B A

0 0

q F E T

n pp n

q q

we are ready to state the main result of this Having introduced the maps

section

Theorem I For X U sl C and n p the diagram

0 0

n n

A A

0 0

n n

y y

sl C sl C U U

q q

sl C with to the module U is a homomorphism from A commutes That is

0 0

is twoto is an isomorphism If p is even the adjoint action II If p is odd

n n

p n

X g one with image the submodule fX U sl C jK X

Proof I is checked by an explicit calculation using Lemma and Lemma To

j l

prove I I we notice that F E T with j l n p and build a basis

of U sl C For p o dd we see from Denition using the third eq of that

is bijective If p is even the image is the subspace spanned by the basis vectors

with

Let us work out the interplay b etween top ological and algebraic ob jects a little

further We have introduced F A A as the op erator which adds a lo op

r r

and identies the the section as describ ed ab ove using the p oint y With any lo op

Y based at y such that y we can asso ciate an op erator L

such that i generalizing A A

r r

F L Two sp ecial cases are F L and F L See Fig for a

C L A R D

graphical representation

and F F F Theorem I For n p F maps A to A

L R LR

II The diagrams

0 0

n n

A A

r r

0 0

n n

y y

U U sl C sl C

q q

X FXK

Figure The lo ops

A D

and

0 0

n n

A A

r r

0 0

n n

y y

U sl C U sl C

q q

X XFK

commute

Proof I is homotopic to the comp osition Using the equivalence

C A

relations imp osed on A it follows that

r C r A r D

the sections b eing identied as ab ove I I The counterpart of F on U sl C follows

from

j j j j

A B A B

0 0 0 0

A e e

n pnp n pnp

A B A B

The action of F on U sl C is computed with using the explicit form of

R n

F F F and ad X FXK XFK

L R F

The Torus with Many Punctures

Combining the ab ove results with previous work on top ological representations of

U sl C on the disc the representation on A can b e identied The result is a

r o r

tensor pro duct of U sl C Verma mo dules one for every additional puncture with

the algebra itself The latter is understo o d as the representation space for the adjoint

action

Figure The family of lo ops of eq

The starting p oint is again an explicit description of A as a space in terms of a

basis Fix a nonintersecting family of lo ops

j j j

B A

j j j

B A

B B A A

s s

It is understo o d that i s and k j are homotopic deformations of

i i

k k

such that lies inside See Fig Let this family b e parametrized such that

i i

x x T C

id n

Dene a horizontal section over this family denoted by v giving it the value v in

T Then the distinguished trivialization over C

id n

p p

M M

j j j

A B

C e A

A B s r

j j r

Generalizing the case with one puncture we dene the following map

V n V n A Denition For n p dene maps

U sl C as follows

j j j

A B

0 0

n pnp

A B s

j j

A B

0 0

F v n F v n e

s n pnp

A B

denotes the map of Denition for the torus with one puncture V h is where

the U sl C Verma mo dule generated from a singular vector v h with E v h

and K v h q v h

Theorem I For n p the maps A V n V n

U sl C are onetoone and onto if p is odd and twotoone if p is even II For

X U sl C let

X X X

i i

n s

Denote B V n U sl C Then the diagram

j q

0 0

n n

A A

s s

0 0

y y

n n

0 0

n n

B B

s s

X X adX

i i i

commutes That is the topological action of U sl C on A is given by the coprod

uct on the tensor product of Verma modules with the algebra itself

Proof We will give an explicit pro of for s The generalization to s is obvious

and will b e omitted Since

j j j

A B

0 0

E e

n pnp

A B

j n j

q q

j j j

A B

0 0

n pnp

A B

q q

B q

j j j

n j j j j

A B

X A B B

0 0

e q q B q

n pnp

A B

q q

A q

j j j

j j j j

B A

A B A B

0 0

e q A q

n pnp

B A

q q

it follows that

j j j

A B

0 0

E e

n pnp

A B n

j j j

B A

0 0

e E K E

n pnp

B A n

Where we identify the copro duct E E K E To compute the action of

F the added lo op has to b e homotopically deformed and split into a comp osition of

lo ops and By a deformation pro cedure it is shown that

B A

j j j

B A

0 0

F e

n pnp

B A

j j j

j j

A B

0 0

q e

n pnp

A B

j j j

A B

A A B

0 0

e q B A q B

n pnp

A B

j j j

A B

0 0

A q B A e

n pnp

A B

As a consequence

j j j

B A

0 0

F e

n pnp

B A n

j j j

B A

0 0

e F K F

n pnp

B A

where F F K F Finally

j j j

A B

0 0

K e

n pnp

A B

j j j

n j j j

A B

0 0

q q e

n pnp

A B

so that

j j j

A B

0 0

K e

n pnp

A B

j j j

A B

0 0

K K e

n pnp

A n B

proves the assertion since K K K The right action of K is a consequence

of the charge neutrality condition

Thus we have proved that the top ological action of U sl C on the torus with

many punctures algebraically repro duces the copro duct

On the adjoint representation

Let q expi p p where p and p are relative prime integers with p U sl C

is dened as the unital algebra over C generated by E F and K sub ject to the

relations

K K K E q EK

K F q FK E F K K

sl C U sl C I obtained In the following we will consider the quotient U

q p

p p p

by dividing by the Ideal I generated by the central elements K E and F

From U sl C it inherits the copro duct

K K K

E E K E F F K F

and the antipo de

S E K E S F FK S K K

j l n

Theorem The monomials F E K j l p n p form a

sl C PBWbasis of U

Using the notation X X X the adjoint representation of U sl C

i i

sl C is given by acting on U

ad Y X YS X

i i

In particular the action of the generators E F and K is

ad X EX K XK E

ad X FXK XFK

ad X K XK

In order to identify the U sl C mo dules A n p with U sl C we

j l

consider for each n the basis given by F E T j l n p In

n p

addition to the prop erties the idemp otents T satisfy

ET T E FT T F

n n n n

which together with

n n

q K q K

n n

F E F n

q q

n n

q K q K

n n

E F E n

q q

and allow us to compute explicitly the action of the generators The result is

j l

Lemma Let n p The action of E F and K on the basis F E T

n p

j l n p is given explicitly by

lj j l j l

0 0 0 0 0 0

T q T ad F E T F E

nn pp nn pp nn pp

j j n l n pp

q q

j l

0 0

F E T

nn pp

q q

0 0 0 0

nn pp nn pp j l j l

0 0 0 0 0 0

T q T q ad F E T F E

nn pp nn pp nn pp

l l n n pp

0 0

q q

nn pp j l

0 0

q F E T

nn pp

q q

j l lj j l

0 0 0 0

F E T q F E T ad

nn pp nn pp

Conjecture on lo cally nite homology

We conclude by stating a conjecture on the lo cally nite middledimensional homology

groups with co ecients in the lo cal systems L Let as b efore q b e a ro ot of unity

and p b e the smallest p ositive integer such that q Dene quantum binomial

co ecients as

lim q q

n m m

q q

Let A b e the asso ciative Zgraded algebra with unit generated by K K of degree

zero E of degree and F of degree n n with relations

K EK q E K F K q F

n n

EF F E F q K q K n

n n n

n m

F F F

n m nm

K K K K F

This is half of Lusztigs construction of the quantum group at ro ot of unity It is

obtained by formally setting F F n for q generic and taking the limit when

n q q

q go es to a ro ot of unity

There is a homomorphism U sl C A given by E E K K

and F F Thus U sl C acts on A via the adjoint action U sl C A A

q q

The U sl C mo dule A is Z x where x x aS x x a x

q j

j j j j

graded for the grading of U sl C dened by degE degF degK

N p

Let A b e the quotient of A by the ideal generated by the central element E

N p

N The algebra U sl C acts on A since E commutes with the action

q N

on A Multiplication by E denes embeddings of U sl C mo dules

A A

N N

deg x deg x These maps are of degree zero for the shifted degree on A given by

N p Dene the graded U sl C mo dule A to b e the direct limit of the mo dules

A with the shifted degree A basis of A is given by the classes of

N pl

F E T A j l N n f p g

j n N

In this expression N is any number such that N p l The degree of is

the subspace of homogeneous elements of degree d j l Denote by A

An alternative description of the U sl C mo dule A was essentially suggested

to us by D Kazhdan Let Z b e the subalgebra of the center of U sl C generated

p p

by E Then A A C t with adjoint action of U sl C where E acts on A

Z q

by multiplication and on C t as ddt The isomorphism relating the two denitions is

clx A x t N

For simplicity we state our conjecture in the case of p o dd

Conjecture Suppose that p is odd I The action of U sl C on families of

L and there is a degree zero isomorphism X X loops extends to an action on H

r r

lf r

of graded U sl C modules H X X L A II There is a degree

q r r

r r n

r r lf

A KerE A X zero isomorphism of graded vector spaces H

n r

This conjecture is parallel to the one formulated in FW for the case of the sphere

To prove it one should understand b etter lo cally nite homology In the remaining of

this section we describ e these isomorphisms Let as in X b e

A B

A and B lo ops on the oneholed torus X based at a p oint y on the b oundary of the

hole Consider the lo cally compact cells in X

C f t t t t X j

lr B B l A l A r r

t t t t g

l l r

where denotes closure in X We orient C using the standard orientation of the

r lr

parameter space R t and choose as section over it the section taking the value e

over a p oint in one of the cells dened in where a trivialization is xed The

L represented by C with this section will b e denoted by C X X class in H

r lr lr r

r r

L is a direct sum of submo dules lab eled by X X H The U sl C mo dule

r r q

r r r

n p spanned by C with n p np n p l

Each of these submo dules is isomorphic to A The isomorphism is

pN l

C F E T

lr r l

for some ro ot of unity dep ending on the choice of trivialization The second isomor

phism is obtained through the identication of KerE with Ker

The space of cycles obtained here is bigger than the space of cycles relevant for

conformal eld theory The cycles for conformal eld theory should b e computed using

the cohomological metho ds of FSa FS but the details remain to b e understo o d

By construction there is a pro jective action of the mapping class group PSL Z on

relative homology which commutes with the action of the quantum group It will b e

describ ed b elow

Prop erties of W j

Dene

iW p p

W j W jp p

0 0 0

iW np p ip p np p

sr s

X X

z n z p W p

i i i

is i

A straightforward computation yields

ilp

W l p j e W j

ilW l p p

W j W l p j e

and

i p p

W j e W j

p p

0 0

i p p i W p p

p 0

W j e W j e

p p

To verify the last identity note that

p p

i p p

e W j j

p p p p

p p

i p p

W j e j

p p p p p p

and

z j i e z j

Prop erties of z j

Following Jacobi one denes

iz n i n

e z j

In terms of an innite pro duct

i n i n i

e cos z e z j e sin z

It satises the following identities

iz i

z j z j z j e z j z j z j

p p

i z j z j i e z j z j

It has simple zero es on the lattice Z Z and no others Consider the fractional p ower

z j Q n Z a multi valued function on C n Z Z Up on analytic continuation

along straight paths from z to z and z resp ectively it has the prop erty that

i z

z j e z j

i z iz i

e z j z j

with

z n z m

A B

z fx y C jm x m n y n g

as follows from an explicit calculation Note that z is constant on every translated

fundamental domain

Topological Representations of U sl C on the

Torus and the Mapping Class Group

We compute the mapping class group action on cycles on the conguration space of

the torus with one puncture with co ecients in a lo cal system arising in conformal

eld theory This action commutes with the top ological action of the quantum group

U sl C and is given in vertex form

Introduction

We consider top ological representations of U sl C app earing in free eld represen

tations of conformal eld theories on the torus based on SU Topological representa

tions of quantum groups on the complex plane were introduced in FW SV La

The torus has b een investigated in CFW

Fock space traces of pro ducts of vertex op erators yield multivalued holomorphic

dierential forms on conguration spaces over the torus Quantum groups D Lu

FK RT enter through their action on a certain space A of linear forms on the

space of holomorphic multivalued dierential forms with given mono dromy These

forms are given by integration on pro ducts of lo ops Singular vectors with resp ect with

this action give cycles and dene thus linear forms on cohomology We consider the

torus with one puncture together with the lo cal system given in CFW asso ciated

to the mono dromy of the dierential forms We restrict our attention to the quantum

group U sl C at q a pth ro ot of unity The top ological action of U sl C has

q q

b een identied in CFW with the adjoint representation in the sense that the space

A is isomorphic to U sl C as U sl C mo dule with the adjoint action The main

q q

input from conformal eld theory BPZ DF ZF F BF SV G GK

FSa FSb is the form of the lo cal system

An imp ortant feature of conformal eld theories on the torus is mo dular invariance

FSb A natural question to p ose is the meaning of mo dular transformations on

the side of top ological representations The rst observation is that the lo cal system

coming from conformal eld theory is compatible with mo dular transformations in a

sense to b e dened b elow As a consequence the mo dular group acts on the space A

The second observation is that we can explicitly compute the action of the mo dular

Figure The lo ops and

group on A by contour deformation metho ds Using the identication of A with the

quantum group algebra we obtain the action of the mo dular group on the latter

Since the action of the mo dular group commutes with the top ological action of

U sl C it commutes on the algebraic side with the adjoint action The result is a

vertex form of the generators T and S of the mo dular group These generators are

expressed in terms of the universal Rmatrix of an enlarged version of U sl C the

K generated algebra and the Haar measure on U sl C A representation of the

mapping class group in SOS form arises in the study of threemanifold invariants of

Reshetikhin and Turaev RT

We obtain as a byproduct the quantum group interpretation of mo dular invariance

Namely the action of the mo dular group leaves invariant the subspace of singular

vectors in the adjoint representation

Similar formulas have b een discovered indep endently in the context of braided

groups of monoidal categories by Lyubashenko and Ma jid LM

Conguration spaces and lo cal systems on the torus

Let X b e a torus with one puncture

Representation of X

Let D w fz C jjz w j r g the op en disc We represent X D n

R R

w the disc with two holes the b oundaries of which we identify Eg D

i R

i i

we take w R and w R dene w R e w R e and identify

z z Let p R serve as a base p oint X p is generated by elements

and as represented by the lo ops in X based at p shown in Fig For later purp ose

we introduce the abbreviations and

Conguration spaces and braid groups on X

For r we dene conguration spaces

r r

X X n fz z X jz z g S

r ij r r i j r

S is the symmetric group acting from the right Let x x b e a base

r r r

p oint in X X is the braid group with r strings on X We call a base p oint

r r r

admissible if R x x R R Braid groups dened with resp ect to

dierent admissible base p oints are canonically isomorphic We will always assume

to b e an admissible base p oint such that fx x g X X is generated

r r r

by elements i r and Intuitively interchanges x with x

i i i i

counterclockwise while and move x along the resp ective lo ops other comp onents

of b eing kept xed An abundance of relations hold among these generators We

will not present them here

Lo cal systems over X

Let p b e an o dd p ositive integer Put q e and dene p by p matrices A and B

with entries

A q B

mn mn mn mnpl

They satisfy AB q BA Let V C The assignments

q i r A B

r i r r

dene a pdimensional representation X GLV It is the mono dromy

r r r

representation asso ciated with multivalued dierential forms on X mentioned ab ove

is the direct sum of two equivalent pdimensional irreducible representations

b e the subspace of X consisting of congurations which contain p among Let X

to b e the bijection which X their comp onents We then dene X n X

r r

inserts p The family of representations r is compatible in the following sense

b e the isomorphism induced by X Let X n X

r r r r r

r r

then

r r r r

With we asso ciate the lo cal system L X X V a at vector

r r r r

r r

bundle over X with distinguished trivialization over the holonomy asso ciated with

r r

elements of X b eing Due to the compatibility can b e lifted to L

r r r r r r

L X L X We dene L x v x v which is checked

r r r r r

X nX X

to b e well dened

r ed

Topological representations of U sl C

We summarize briey the constructions leading to top ological representations of the

r ed

quantum group U sl C adjusting the notations to the present setup

Families of nonintersecting lo ops with values in the lo cal system

f g and b e lo ops X Let Q

r r

starting and ending at p nonintersecting except at p Dene Q X

r r r

j k

b e the corresp onding embedding Denote by Q X j k r a family of

r r

nonintersecting lo ops obtained by homotopic deformation of j lo ops and k lo ops

given by

j k j j r

t t t t t t

r j j r

It represents a lo cally nite r chain in X with b oundary in X

We lift it to take values in X We sp ecify the lift by choosing an admissible

r r

p oint on its image connecting this p oint to the base p oint by an admissible path

j k

Then the equivalence class v v V denes a family of nonintersecting lo ops

in X with values in L X The space of families of nonintersecting lo ops in X with

values in L X is denoted by A X X L or shorter by A Its precise deni

r r r r r

tion contains equivalence relations reecting the p ossibility of homotopic deformation

j k

reparametrization and splitting of lo ops see FW The elements e

j k minfr p g such that j k r and n p constitute a basis Here

e A family which contains p homotopic lo ops is put equivalent to zero

n m nm

Therefore we restrict ourselfs to r p

r ed

sl C Topological action of U

The basic ingredience of top ological representations are op erators

E A A F A A K A A

r r r r r r

dened by

j k j k

E e L e

n r r n

j k j k j k

F e q e

n n

j k r j k

K e q e

n n

They are shown to satisfy the relations of U sl C

K E q EK K F q FK E F K K

and also the additional relations

p p p

E F K

r ed

sl C Thus dening U A comes equipp ed with the structure of a mo dule

r ed

over U sl C We identify this representation as the adjoint representation Let

r ed

A U sl C b e the map

j k k pj

e N j k nF T E

n n

j j n j j j k j k j k nj n

N j k n q q q

r ed

An explicit computation proves that is a homomorphism of U sl C mo dules

Moreover it is onetoone and onto Here

nm m

T q K

r ed

The actions of U sl C on itself by left multiplication and by right multiplication

twisted with the antipo de also have top ological counterparts The op erator which

implements left multiplication by F is called F The op erator which corresp onds to

right multiplication by F is denoted by F On the top ological side they are given

j k nj k j k

F e q e

L n n

j k j k j k

F e q e

R n n

Intuitively F adds a lo op while F and F add and lo ops resp ectively The

L R

interpretation of A as a bimo dule will not b e worked out here

The imp ortant formula of this section to keep in mind is

Action of the Mapping Class Group

Let DiX b e the group of dieomorphisms which leave X fz C j jz j Rg

invariant Let Di X b e the subgroup of Di X consisting of dieomorphisms ho

motopic to the identity The mapping class group of X is dened as

M X DiX Di X

A reference on mapping class groups is Bi

Generators of M X

M X is generated by Dehn twists On the torus we have two kinds of Dehn

twists T and T T is dened as follows Consider the annulus fz X j r

jz w j r g with small We dene a map of this annulus to itself by

ijz j

T w z w e z with a smo oth function interpolating b etween r

and r We say that T is the Dehn twist asso ciated with the lo op

t w r e t The Dehn twist T is asso ciated with the lo op

t w r t w r t See gure The orientations we use are shown by

arrows T and T leave the base p oint invariant Recall that is a conguration

r r

on X Thus we have a map from M X to Aut X the automorphisms

r r

of X r

r r

Compatibility of lo cal systems

We dene a representation X GL V to b e compatible with M X

r r r

if T for all T M X This means that for every T there exists a matrix

D T GL V such that

T D T D T

If a representation is compatible with M X then we have an action of M X

on L given by

L T L L x v T x D T v

r r r

which is well dened due to equation

The family of lo cal systems X GL V r is indeed compatible

r r r

with M X For the generators T and T the compatibility is proved by dening

p p

X X

ll l ll l

D T q A D T q B

l l

and by noting that the action of T has the form T and T while

that of T has the form T and T Note also that D T and D T

have matrix elements

l m

D T q q

mn mn

X X

D T q

mn mnpk l

k Z l

Action of M on A X X L

r r r

The mapping class group M acts on A as follows T M acts by

L T v T D T v

r r r

j k

We compute the action of T and T on the basis elements e of the linear

space A X X L

r r r

j k

Action of T Let us dene Let r j k The action of T on e

is seen to have the form

j k j k

L T e D T e

r n n

The rst problem is to decomp ose again in terms of the basis of A X X L

r r r

The decomp osition is p erformed with the help of

j l k

j j j l j l j lk

j j j l j l j lk j

j l k lk j l k

e e

n j lk n

j l k lk j l k

e q e

n n

k l lk

We use q B and absorb the factor by an isometry of

j k l

Q The ordering of lo ops according to deformation and the assignement of

j lk

the comp onents of t t Q to the individual lo ops indicated by

j k l j lk

sup erscripts should b e clear from the notation used in Iterate to obtain

minfjpk g

j s k s j k

c j k e e

s ns n

with co ecients

Y X

k j i sj k s

q q c j k

i i j s

using Gaus formula

i ssj

q q

i i j s

The q binomial co ecient is dened as

s j s

q q

Putting and together it follows that

j k

L T e

r n

minfjpk g

sj k s j s k s n

q e dq

with

d q

Thus gives the matrix elements of L T in terms of the basis with elements

j k

Action of T Let r j k The action of T has the form

j k j k

L T e D T e

r n n

which we again will express in terms of the basis of A X X L Using it

r r r

follows that

j k s s j k

b j k e e

s ns n

smaxfj k pg

with co ecients

Y X

k i sk s

q q b j k

l i i k s

Note that b j k c k and yield

s s

j k

L T e

r n

X X

j k s s llsk s

e q

nls

smaxfj k pg

L completing the calculation of the action of T on A X X

r r r

r ed

sl C Action of M X on U

r ed r ed

We have identied A X X L as a U sl C mo dule as U sl C with the

r r r

r q q

adjoint action The action of M X on A X X L commutes with the top o

r r r

r ed

sl C In this section we identify the action of M X on logical action of U

r ed

L By construction it commutes sl C dened by its action on A X X U

r r r

r q

with the adjoint action

r ed

sl C b e the restriction of the map L U Action of T Let A X X

r r r r

q r

dened ab ove Then

minfjpk g

n o

sj k s j k n

q L T e dq

r n r

k s pj s

N j s k s n s F T E

r ed r ed

sl C by sl C U using We dene U T U

q q

U T L T

r r r

Thus what is left to compute U T is the ratio of normalization constants

N j s k s n s j s

s sj k sn s

N j k n j

We conclude that

minfjpk g

s sssn k pj n

q U T F T E dq

k s pj s

F T E

r ed

sl C giving the action of U T on U

Action of T Let r j k Using we obtain

n o

X X

j k llsk s

q L T e

r n r

smaxfj k pg q

s pj k s

N j k s s n l s F T E

nls

Insert

N j k s s n l s

N j k n

k s

j k s

k sk sk sj nsls

k s

to conclude that

k pj

U T F T E

k s

X X

k sk sk sj nslsl

k s

smaxfj k pg

j k s k

s pj k s

k s F T E

q nls

j s

q q

completing the calculation of U T

Action of S

We dene S T T T The action of S on X p is given by S

and S Recall that That is S maps to and to

conjugated by This transformation is known as the S transformation We

put

D S D T D T D T

D S q

D S p erforms a discrete Fourier transformation on V X GL V is

r r r

compatible with S the equivalence b eing given by D S We thus have an action

L S on A X X L It has the form

r r r r

j k j k

L S e D S e

r n n

r ed

r j k As b efore we deduce from an action U S on U sl C However

we do not expand in terms of the basis of A X X L as we did in the case

r r r

of L T and L T although this could b e done Instead we compute directly the

r r

image of under Using

j k j j k j k

e q F e

n R n

it follows that

j k k j k

n r

j j k k k j nj k pk j

q N j k n T E F

j pl

By reexpressing in terms of the basis F T E and applying the

k r

L could b e obtained We expansion of in terms of the basis of A X X

r r r

conclude that

n o

pq k

j k k j k

L S e

r n r

d j

j j k k k nj j n pk j

q N j k n K E F

using

nlj n j n

q T K

The nal result is

k pj

U S F T E

k pq

k j k j j k k k nj j n pk j

q K E F

d j

r ed

U S is the algebraic version of the S transformation It is a mapping of U sl C

to itself onetoone and onto which commutes with the adjoint action

Identication of the S and the T transformation

We identify the op erations U T and U S in terms of the quasitriangular structure

r ed

of U sl C Let us rst adjust the normalization of D T and D S as follows

N d N

D T D T D S D S

d pq

r ed

With this change of normalization U T and U S act on U sl C by

k pj

U T F T E

minfjpk g

k s pj s s sssn n

F T E q N q

and

k pj

U S F T E

k j k j j k k k nj j n pk j

N q K E F

r ed

sl C Universal elements of U

q K

r ed

sl C It is known to b e a ribb on Let us consider the K generated version of U

Hopf algebra The universal Rmatrix is

p p

X X

n nn n n nm n m

A A

q E F q K K R

p n

n mn

The asso ciated central element V is

p p

X X

n nnnmmm n n

V q F H E

n m

nm m

q K H

Identication of U T

Let N q Then

r ed r ed

U sl C U sl C

q q

U T

y y

r ed r ed

U sl C U sl C

q K q K

commutes That is U T is identied with the multiplication by the inverse of the

central element V of This is shown by

k pj

V F T E

p p

X X

s sssll k s pj s

q F H T E

ls ns

with

H T H H H

ls ns ls ns nps

H H

ln ns lnp nps

comparing the result with

r ed

Trace on U sl C

q K

r ed

Let U sl C C b e the linear map such that

q K

p p

E F H n p

j k

E F H else

r ed r ed

sl C we have sl C For X E F K and Y U is a trace on U

q K q K

XY YX

S transformation

r ed

Let R sl C the universal Rmatrix Dene a linear map S U

i i

q K

r ed

U sl C by

q K

S X K X

j l l j

with the trace of A short computation reveals that

pr ps

S T E F

r s

r r sssr n r nr s

q F K E

r s

q q

r ed r ed

We thus obtain a map S U sl C U sl C by restriction

q q

Identication of S

Put the normalization constant in to b e

q N

The transformation is identied as

U S X S X

with S the transformation To verify we compute the inverse transforma

tion of It is seen to b e

pk j k k j

U S T E F

pN k

j j k k j k nk j k nk pj

q F K E

Setting k r and j p s and comparing with the result

follows

The KnizhnikZamolo dchikovBernard Equation

on the Torus

In this chapter we give a description of conformal blo cks of the WessZuminoWitten

mo del based on sl C in the sense of TUY in terms of solutions of the Knizhnik

Zamolo dchikovBernard equation We discuss the role of a doubly ane version of the

the Weyl group

Introduction

Two dimensional conformal eld theory asso ciates to a punctured Riemann surface

a complex vector space of conformal blo cks It also tells how the vector spaces are

identied when the punctures and the complex structure are varied For the case of

WessZuminoWitten WZW mo dels this construction can b e done in mathematically

well dened terms TUY In the case of the sphere the description can b e made

very explicit in terms of the KnizhnikZamolo dchikov dierential equation The aim

of this chapter is to show that such a description in terms of dierential equations is

also p ossible in the genus one case The KnizhnikZamolo dchikov equation is replaced

by a genus one generalization due to Bernard Bea Beb

The construction is similar to the case of the sphere see eg F but requires a

twist The reason for this is the following conformal blo cks are dened as linear forms

on a certain innite dimensional space ob eying some invariance condition Thanks to

this invariance in the case of the sphere a conformal blo ck is uniquely determined by its

restriction on a nite dimensional subspace of primary states isomorphic to a tensor

pro duct of nite dimensional representations of a nite dimensional Lie algebra On

higher genus surfaces this is not the case The solution to this problem is to introduce

additional parameters and consider parametric families of conformal blo cks that are

invariant under a Lie algebra dep ending on the parameters Spaces of conformal blo cks

corresp onding to dierent values of the parameters are then identied thanks to a at

connection The p oint is that horizontal conformal blo cks are determined as functions

of these parameters by their values on primary states

In this chapter we consider the WZW mo del based on the Lie algebra sl C on

the torus We construct conformal blo cks for the torus with given family of punctures

and mo dular parameter as certain multilinear forms invariant under a Lie algebra of

twisted Lie algebra valued meromorphic functions Under variations of the punctures

they satisfy the KnizhnikZamolo dchikovBernard KZB equation We derive the KZB

equation from the condition that conformal blo cks b e at with resp ect to the Friedan

Shenker FS connection We then introduce a doubly ane version of the Weyl group

and formulate an invariance condition on conformal blo cks as solutions to the KZB

equation

Some op en ends of this construction will b e left untouched including the construc

tion of integral representations SV Ch the connection with quantum groups

FW SV CFW mono dromy prop erties TK and the role of integrability

TUY

Two dimensional conformal eld theory has its origin in the work of BPZ Its

formulation in terms of complex geometry can b e found in FS The basic references

to the WZW mo del are W KZ The notion of conformal blo cks on a Riemann

surfaces was formulated in mathematical terms in TUY and by Beilinson and

Feigin The KZB equation on the torus was rst derived in Bea It has also

app eared recently in FG

Lie algebra valued meromorphic functions

The innite symmetry of the WZW mo del on the torus can b e formulated in terms of

a Lie algebra of twisted meromorphic Lie algebra valued functions

Let g sl C with Cartan generators E F and H and invariant bilinear form

X Y trXY normalized such that H H

In our construction we will use the following kinds of conguration spaces Fix

H f C j Im g and let L Z Z C We then dene

n n

fz z C j z z mo d L g

n i j

and

n n n

C C n

The space C is a covering of the nth conguration space over the torus

C L Let us also introduce

n n

D fz z C C j L g

and

n n n

C C n D

The Lie algebra is now dened as follows For z z let g z z

n n

b e the Lie algebra of meromorphic functions X C g holomorphic on C nL z z

such that

X z X z X z exp i ad X z

Here Lz z z L denotes the union of the punctures and their trans

n j j

lates by the lattice L The Lie algebra consists of meromorphic functions on the

plane p erio dic along the Ap erio d and twisted along the Bp erio d with p ossible p oles

on Lz z and no others

Let Lg g C t and Lg Lg C the central extension of Lg asso ciated to

the twococycle

X f Y g k X Y res f tg t

with k a p ositive integer With a highest weight g mo dule V one can asso ciate a highest

weight Lg mo dule V The g action is rst extended to b g C t C by letting

g tC t act by zero and C by multiplication Then V is given by U Lg V

U b

and V as the g submo dule We will always view g as the Lie subalgebra g t of Lg

V of V This construction as well as prop erties of V can b e found in K

As an input we require a family of nitedimensional irreducible highest weight

g mo dules V V We can think of the mo dule V as b eing attached to the p oint

n j

z The next step in our construction is an action of the Lie algebra g z z on

j n

Let X X z t b e the Laurent expansion V the tensor pro duct V

j j

of X at z viewed as a formal p ower series in t As an element of Lg it is included in

Lg But Lg acts on V Therefore we can dene

k k

X X X

A computation reveals that in End V V we have the equation

X z Y z res X Y X Y k

z z

But X Y is doubly p erio dic so that the sum of residues vanishes We therefore

V obtain an action of g z z on V

Meromorphic vector elds

Analogous to the action of Lie algebra valued meromorphic functions there is an action

of doubly p erio dic meromorphic vector elds on V V

Let us rst consider any highest weight g mo dule V Sugawaras construction yields

a pro jective representation of VectS C t on V Let T T and T form an

a b

orthogonal basis of g normalized such that T T For example we may put

E F T E F and T H Then one denes for n Z T

X X

a a

T T L

nm m

a m

a a m

with T T t As elements of End V the L ob ey the Virasoro relations

C n

L L n mL nn

n m nm

The pro jective representation is now given by with central charge c

X X

t L

n t n n

n n

VectS also acts by derivations on Lg The explicit formula is

t X t t X t

It extends to an action on Lg by letting vector elds act by zero on the center

For z z C let Vectz z b e the Lie algebra of meromorphic

n n

vector elds on C holomorphic on C n Lz z such that z z and

z z

For Vectz z let z t denote the Laurent expansion of

n j j t

at z viewed as an element of VectS Through Sugawaras construction it is mapp ed

to an element of EndV We then dene as ab ove

It turns out that in End V V we have the equation

z z res

z z

But the sum of residues is again zero so that we obtain an action of the Lie algebra

Vectz z on V V Moreover it can b e shown that

X X

for Vectz z and X g z z That is this action intertwines the

n n

natural action of vector elds on functions

Conformal blo cks

For z z we dene the space E z z of conformal blo cks as

n n

the vector space of linear forms G V V C such that for X g z z

and v V V G satises hG X v i That is G is required to b e invariant

under the action of g z z

We will also want to vary the parameters z z The b ehavior under vari

ations of these parameters is determined by a further condition This condition tells

that conformal blo cks are at with resp ect to the FS connection

n n

In the following we let z z take values in an op en subset U

We denote by P the set

n n

P fz z z U C j z Lz z g

n n

Then we dene g U to b e the Lie algebra of meromorphic functions X U

n n n

C g holomorphic on U C n P such that for z z U

X z z g z z It generalizes our previous denition

n n

To introduce the FS connection we require the following auxiliary function Let

n n n

U C C b e a meromorphic function holomorphic on U C n P such

that for z z U z z z z satises z z and

n n

z z i The example which we will have in mind is

z z

where z z is a Jacobi theta function see GR But the following

considerations will b e indep endent of the particular choice of

We dene rst order dierential op erators on g U by the formulas

X D X X H X X D

z z

j j

We leave it as an exercise to show that they map g U to itself

Let us now introduce rst order dierential op erators on the space of holomorphic

V the space of multilinear formas on V V functions G U V

n n

as follows

G GL r G G G H G r

z z

j j

Here H is given by and by

The FS connection is the connection

dz rz d r r

j j

n n

on the innite rank trivial vector bundle U V V For G U

V V and X g U it ob eys

rG X rG X G DX

where D denotes the expression

dD dz D D

j z

This prop erty ensures that the connection restricts prop erly to the subbundle of con

formal blo cks The restriction of r to conformal blo cks is at

n n

Let E U b e the space of holomorphic functions G U V V such

that for z z U Gz z is an element of E z z Due

n n n

n n

to the FS connection leaves E U invariant Elements of E U are called

holomorphic conformal blo cks

The horizontality condition is that G E U is required to satisfy the equations

r G r G

for xed mo dular parameter Eq completes the construction of holomorphic

conformal blo cks for the purp ose of these notes It is also p ossible to vary consistently

the parameter but we will not need this here

KnizhnikZamolo dchikovBernard equation

In this section we will consider holomorphic conformal blo cks G restricted to the sub

space V V of V V The value of a conformal blo ck on general vectors

in V V can b e computed as the image of certain dierential op erators in

with co ecients dep ending on z z acting on G evaluated on certain vectors

in V V

The condition can b e written in a more explicit form when the conformal

For v V V V blo cks are restricted to the subspace V V of V

n n

G reads the horizontality condition r

hG v i hG k H H E F F E v i

Recall that V is annihilated by E F and H with n

j n n n

To pro ceed we require another auxiliary function h U C C It is the

meromorphic function h z hz z z given by the ratio

of Jacobi theta functions

The functions E h z z and F h z z are elements of g U As

j j

functions of z they have simple p oles at z z For w V V we then compute

j n

hG E h z z w i

hG fE E h z z E gw i

k j

and

hG F h z z w i

hG fF F h z z F gw i

k j

But due to g U invariance b oth right hand sides are zero As a result we can express

j j

hG E w i and hG F w i in terms of G evaluated on certain vectors in V V

For w V V the second horizontality condition r G takes the form

z z

k j

j k

hG w i hG fH H gw i

z z

k j

From it we obtain an expression for hG H w i

Both horizontality conditions together in this explicit form yield the dierential

equation

k j

k G GH G

k j

with

k j k j

k j

h z z E F h z z F E

k j k j

z z

k j

H H

z z

k j

This equation is called the KZB equation The tensor is a unitary solution of

a genus one generalization of the classical YangBaxter equation The YangBaxter

equation is a consistency condition reecting the fact that the KZB equation comes

from an integrable connection

The calculation of this section shows one part of the following theorem

n n

Theorem Let U be any suciently smal l neighborhood of a point in Let

n n

E U be the space of holomorphic functions on U with values in the linear forms

on V V invariant under the action of the Cartan subalgebra C H Then the

natural map

n n

E U E U

restricts to an isomorphism between the space of conformal blocks obeying the horizon

tality condition and the space of local solutions in E U of the KZB equation

Weyl invariance

In this section we introduce the notion of solutions invariant under a doubly ane

version of the Weyl group of sl C

For H let W b e the group of transformations of C given by

where fg and L That is W is a semidirect pro duct of Z

with Z Z

Let us x z z C For W the meromorphic Lie algebras

g z z and g z z are isomorphic We have explicit formulas for

n n

isomorphisms

g z z g z z

n n

for the generators and of W They are given by

i i

X z exp ad exp ad X z

H E F

X z X z X z exp iad X z

We have seen that g z z acts on V V This action has b een denoted

by There exists an action of W on V V such that for W

X X

For the generators of W this action can b e written explicitly in the form

i i

H exp E F exp

id exp iz H H

V That is the action os W is intertwined by isomorphisms of V

For G E z z it follows that G E z z At this p oint

n n

we want to vary the parameters again For the generators of W we dene functions

g z z exp k

g z z g z z

n n

Let b e a generator of W We then dene transformed conformal blo cks by

G z z g z z Gz z

n n n

It can b e shown that G is a holomorphic conformal blo ck provided that G is holo

morphic conformal blo ck G satises the KZB equation if G do es A solution is then

called invariant under W if it ob eys

G z z Gz z

n n

The condition of invariance under the doubly ane Weyl group and regularity of G as

a function of C selects a nite dimensional space of solutions to the KZB equation

If the level is a p ositive integer and the highest weights are integrable see K

the space of conformal blo cks contains according to TUY for but the pro of

in TUY extends to the case of general a nite dimensional subspace of conformal

blo cks that are dened on the tensor pro duct of irreducible quotients of Verma mo dules

The horizontal sections in this subspace are conjecturally contained in the ab ove space

of solutions see FG where this is proven in a slightly dierent setting Moreover

in this case one can write down integral representations of solutions

Conformal blo cks on elliptic curves and the Knizhnik

Zamolo dchikovBernard equations

We give an explicit description of the vector bundle of WZW conformal blo cks on ellip

tic curves with marked p oints as subbundle of a vector bundle of Weyl group invariant

vector valued theta functions on a Cartan subalgebra We give a partly conjectural

characterization of this subbundle in terms of certain vanishing conditions on ane

hyperplanes In some cases explicit calculation are p ossible and conrm the conjec

ture The FriedanShenker at connection is calculated and it is shown that horizontal

sections are solutions of Bernards generalization of the KnizhnikZamolo dchikov equa

tion

Introduction

The aim of this chapter is to give a description of conformal blo cks of the Wess

ZuminoWitten mo del on genus one curves as explicit as on the Riemann sphere

Let us recall the wellknown situation on the sphere One xes a simple nite

dimensional complex Lie algebra g with invariant bilinear form normalized so that

the longest ro ots have length squared and a p ositive integer k called level One then

considers the corresp onding ane KacMo o dy Lie algebra the one dimensional central

extension of the lo op algebra g C t asso ciated to the co cycle cX f Y g

X Y res dfg Its irreducible highest weight integrable representations of level value

of central generator k are in one to one corresp ondence with a certain nite set I of

nite dimensional irreducible representations of g These representations extend by

the Sugawara construction to representations of the ane algebra to which an element

L is adjoined such that L X f X f Then to each ntuple of distinct

p oints z z on the complex plane and of representations V V in I one

n n k

asso ciates the space of conformal blo cks E z z It is the space of linear forms

of the corresp onding level k representations of the ane V on the tensor pro duct

algebra which are annihilated by the Lie algebra Lz z of g valued meromorphic

functions with p oles in fz z g and regular at innity The latter algebra acts on

by viewing Lz z as a Lie subalgebra of the direct sum of n copies of the V

lo op algebra via Laurent expansion at the p oles The central extension do es not cause

problems as the corresp onding co cycle vanishes on Lz z in virtue of the residue

theorem

It turns out that the spaces E z z are nite dimensional and are the b ers of

a holomorphic vector bundle over the conguration space C diagonals carrying the

i i

at connection d dz L L acts on the right of a linear form given in terms of

the Sugawara construction We use the notation X Id X Id to denote

the action of on the ith factor of a tensor pro duct

This part of the construction generalizes to surfaces of arbitrary genus see TUY

What is new is that one has to also sp ecify lo cal co ordinates around the p oints z to

give a meaning to the Laurent expansion and that the connection is in general only

pro jectively at ie the curvature is a multiple of the identity

To give a more explicit description of the vector bundle of conformal blo cks on the

sphere and in particular to compute the holonomy of the connection one observes that

the map E z z V given by restriction to V V is injective Thus we

n i i i

can view E as a subbundle of a trivial vector bundle of nite rank This subbundle

can b e describ ed by an explicit algebraic condition FSV After this identication

the connection can b e given in explicit terms and the equation for horizontal sections

reduces to the famous KnizhnikZamolo dchikov equations

i j

X X

T T

a a

k h z z z z

z n n

z z

i j

j j i

In this equation h is the dual Coxeter number of g and T a dimg

is any orthonormal basis of g We view here the dual spaces V as contragradient

representations Let us now consider the situation on genus one curves which we

view as C Z Z for in the upp er half plane Let us denote by E z z the

space of conformal blo cks Again by TUY this is the nite dimensional b er of a

holomorphic vector bundle with at connection on the elliptic conguration space C

of n tuples z z with Im and z z mo d Z Z if i j

n i j

The trouble is that the restriction to V is no longer injective the reason b eing

i i

that there are no meromorphic functions on elliptic curves with one simple p ole only

The way out is the following construction which brings the mo duli space of at G

bundles into the game Consider the Lie algebras Lz z parametrized by

in a Cartan subalgebra h of Zp eriodic meromorphic functions X C g with p oles

at z z mo dulo Z Z such that X t exp i ad X t

and we can dene a space of twisted conformal blo cks These algebras act on V

E z as space of invariant linear forms see The original space of conformal

blo cks is recovered by setting

It turns out that E z is again the b er over z of a holomorphic vector

n n

bundle E over C h with at connection whose restriction to C fg is E Thus

we can by parallel transp ort in the direction of h identify the space of sections E U

of E over an op en set U C with the space of sections of E which are horizontal

in the direction of h

hor

E U E U h

The p oint is now that the restriction map

hor

E U h V O U h

h i i

to V is injective Prop osition Comp osing these two maps we may view the vector

bundle of conformal blo cks as a subbundle of an explicitly given vector bundle on C

of nite rank Indeed we show that the image is contained in the space of functions

on U h which have denite transformation prop erties of theta function type under

translations of by Q Q where Q denotes the coro ot lattice Moreover the theta

functions in the image are invariant under a natural action of the Weyl group and ob ey

a certain vanishing condition as the argument approaches ane ro ot hyperplanes We

conjecture that these conditions characterize completely the image This conjecture is

conrmed in some cases including a sp ecial case which arises EK in the theory of

quantum integrable many b o dy problems see we describ e explicitly the space of

conformal blo cks in the case of sl n where the representation is any symmetric

p ower of the dening N dimensional representation

The characterization of conformal blo cks in terms of invariant theta functions ob ey

ing vanishing conditions was rst given in the sl case by Falceto and Gawedzki

FG who dene conformal blo cks as ChernSimons states in geometric quantiza

tion

After the identication of conformal blo cks as subbundle of the invariant theta

function bundle we describ e the connection in explicit terms Theorem and

get a generalization of the KnizhnikZamolo dchikov equations These equations were

essentially written by Bernard Bea Beb in a in a slightly dierent context and

were recently reconsidered from a more geometrical p oint of view in FG They have

the form see Sect

X X

j jl

h z z

z j l

ll j

X X

i H z z

j l

for some tensors H g g given in terms of Jacobi theta functions Here is

related to by multiplication by the WeylKac denominator Thus the right way to

lo ok at these equations is to view u as a section of a subbundle of the vector bundle

over the elliptic conguration space of n tuples z z whose b er is a nite

dimensional space of invariant theta functions

In this chapter we do not discuss an alternative approach to conformal blo cks on

elliptic curves which is in terms of traces of pro ducts of vertex op erators Bernard

Bea showed that such traces ob ey his dierential equations Using this formulation

integral representation of solutions were given in the sl case in BF To complete the

picture one should show that those solutions are indeed theta functions with vanishing

condition

Let us also p oint out the pap er EFK that shows that the same space of invariant

theta functions with vanishing condition can b e identied with a space of equivariant

functions on the corresp onding KacMo o dy group

Conformal blo cks on elliptic curves

Elliptic conguration spaces

Let H f C j Im g b e the upp er half plane and for H denote by L

the lattice Z Z C Let n b e a p ositive integer We dene the elliptic conguration

space to b e the subset of C H consisting of p oints z z so that z z

n i j

mo d L if i j

The space of p oints z C with xed is a covering of the conguration space

of n ordered p oints on the elliptic curve C L

Meromorphic Lie algebras

Let g b e a complex simple Lie algebra with dual Coxeter number h and k b e a

p ositive integer Fix a Cartan subalgebra h of g and let g h g b e the

corresp onding ro ot space decomp osition The invariant bilinear form is normalized in

such a way that for long ro ots see H We choose an orthonormal

basis h of h The symmetric invariant tensor C g g dual to admits then

a decomp osition C C with C h h h h and C g g

We dene a family of Lie algebras of meromorphic functions with values in g

parametrized by C h

Denition For z z z C and h let Lz b e the Lie

algebra of meromorphic functions t X t on the complex plane with values in g

such that

X t X t X t exp i ad X t

n n

z L More generally for any op en set U C h and whose p oles b elong to

let L U b e the Lie algebra of meromorphic functions t z X t z on

C U with values in g whose p oles are on the hyperplanes t z r s i n

r s Z and such that for all z U the function t X t z b elongs to

Lz Similarly dene LU for an op en subset U of C to b e the Lie algebra of

meromorphic functions t z X t z on C U with values in g whose p oles are

on the same hyperplanes and such that for all z U the function t X t z

b elongs to Lz

An explicit description of these Lie algebras is given in App endix An imp ortant

prop erty is that they have a ltration by lo cally free nitely generated sheaves Let

O U b e the algebra of holomorphic functions on an op en set U C h and for

U b e the O U submo dule of L U consisting of any nonnegative integer j let L

functions whose p oles have order at most j Similarly we dene L U for op en sets

n j

U are sheaves of O mo dules U C The assignments U L U U L

is a locally free locally nitely generated sheaf of O modules Prop osition L

U In other words every point in C h has a neighborhood U such that L

n n

C O U as an O U module for some n Moreover for each x C h every

U for some j and U x The same results X Lx extends to a function in L

hold for L

The pro of is contained in App endix see Corollary

Tensor pro duct of ane KacMo o dy algebra mo dules

Let Lg g C t b e the lo op algebra of g Fix a p ositive integer k N Let

Lg Lg C K b e the central extension of Lg asso ciated with the co cycle

cX f Y g X Y resf g dt

where the residue of a formal Laurent series is given by res a t dt a Thus

the Lie bracket in Lg has the form

X f K Y g K X Y f g cX Y K

With every irreducible highest weight g mo dule V is asso ciated an irreducible highest

weight Lg mo dule V of level k Its construction go es as follows The action of

g is rst extended to the Lie subalgebra b g C t C K of Lg by letting

g tC t act by zero and the central element K by k Then a generalized Verma

mo dule V U Lg V is induced It is freely generated by any basis of V as

U b

a g t C t mo dule The p olynomial subalgebra Lg g C t t C of Lg

V the generalized is Zgraded with degX t j Since V U Lg

U b Lg

Verma mo dule is naturally graded By denition the irreducible mo dule V is the

quotient of the generalized Verma mo dule by its maximal prop er graded submo dule

We will consider integrable mo dules V of xed level k If we x a

set of simple ro ots and denote by the corresp onding highest ro ot V

is integrable of level k if the irreducible g mo dule V has highest weight in the the

subset

I f P j i l k g

k i

of the weight lattice P Let v b e the highest weight vector of V and e a generator of

g Then the maximal prop er submo dule is generated by e t v

The grading extends to V by setting degv v deg v With our

n i

convention all homogeneous vectors have nonnegative degree

Fix n highest weight g mo dules V j n such that the corresp onding Lg

are integrable of level k and let H and z z complex numbers mo dules V

as an Lg mo dule which is attached with z z mo d L if i j We think of V

i j

to the p oint z

n n

In the following we will use the abbreviations V V V and V

V V

We now construct an action of Lz on V For X Lz let X

X z t g C t b e the Laurent expansion of X at z viewed as a formal Laurent

j j

series in t Then

X X X

denes a Lie algebra embedding of Lz into Lg Lg As a vector space

Lg Lg is embedded in Lg Lg The embedding is of course not a Lie

algebra homomorphism Since Lg Lg acts on V we obtain a map from

Lz to End V This map will also b e denoted by Thanks to the residue

theorem it turns out to b e a Lie algebra homomorphism

Prop osition For X Y Lz

X Y X Y

Proof In End V we have the equation

res X t Y t dt X Y X Y k

But X t Y t is doubly p erio dic by adinvariance of so that the sum of

residues vanishes

Vector elds

The Lie algebra VectS C t of formal vector elds on the circle acts by deriva

d t

d d

tions on Lg Let us denote this action simply by t X t t X t It

dt dt

extends to an action on Lg by letting vector elds act trivially on the center The

Sugawara construction yields a pro jective representation of VectS on V for any

nite dimensional g mo dule V The Sugawara op erators L EndV are dened

by choosing any basis fB B g of g with dual basis fB B g of g so that

B B and setting

b ab

X X

a nm m

L B t B t n

n a

k h

L L L

These op erators are indep endent of the choice of basis and ob ey the commutations

relations of the Virasoro algebra L L n mL n n with central

n m nm

charge c k dimg k h Then

X X

L EndV t

n n n

n n

denes a pro jective representation of VectS on V with the intertwining prop erty

d d

X t t X t for any X t Lg Note that all innite sums are actually t

dt dt

nite when acting on any vector in V

Conformal blo cks

For a Lie algebra mo dule V we denote by V the dual vector space with natural right

action of the Lie algebra The notation h v i will b e used to denote the evaluation of

a linear form on a vector v

Denition The space of twisted conformal blocks asso ciated to data g k V V as

ab ove is the space E z of linear functionals on V annihilated by Lz z

h n

If then E z E z is called the space of conformal blocks at z

Let us vary the parameters let U b e an op en subset of C h Then the space of

holomorphic functions U V ie of functions whose evaluation h ui on

any xed vector u V is holomorphic on U is a right L U mo dule

Denition The space E U of holomorphic twisted conformal blocks on U C h

is the space of V valued holomorphic functions so that for all op en subsets U of

U the restriction of to U is annihilated by L U We also dene the space E U

of holomorphic conformal blocks on U C by replacing L by L

With this denition the assignments U E U U E U are sheaves of O

mo dules

n n

Lemma Let U be an open subset of C h resp of C Then E U

resp E U if and only if is holomorphic on U and x E x resp E x for

al l x U

Proof It is obvious that if is holomorphic and if x E x for all x U then

E U Let E U and x U To show that x E x we have

h h h

to show that every element X of Lx is the restriction of an element of L U for

some neighborho o d U of x But this follows from Prop The same applies in the

untwisted case

Flat connections theta functions

The at connection

For each op en subset U of C h we have dened a Lie algebra L U acting on

n n

V U the space of holomorphic functions on U with values in V It is con

venient to extend this denition Let G b e the simply connected complex Lie group

whose Lie algebra is g and for z g C G let Lz g b e the Lie algebra of

meromorphic g valued functions X t on the complex plane whose p oles mo dulo L

b elong to fz z g and with multipliers

X t X t X t Adg X t

If U is an op en subset of C G dene L U to b e the Lie algebra of meromorphic

functions on U C z g t whose p oles are on the hyperplanes t z n m

n m Z and restricting to functions in Lz g for xed z g U As ab ove

we introduce the space E z g of Lz g invariant linear forms on V and the

sheaf U E U of L U invariant holomorphic V valued functions

G G

Let z t b e a meromorphic function on C C whose p oles b elong to the

hyperplanes t z n m and such that as function of t C

z t z t z t z t i

Although the construction do es not dep end on which we chose we will always set

z t t z

t log tj

ij ij t

e tj

for deniteness

Let A z g t b e a meromorphic function on C G C dep ending linearly

on Y g whose p oles as a function of t b elong to fz z g and such that

A z g t A z g t

Y Y

A z g t Adg A z g t Y

Y Y

j i j

A we may take A to b e If A e

j Z

itz

e Adg

Y A z g t

itz

Adg e

Note that for xed z t and Y A extends to a regular function of g G

Denote by the derivative in the direction of the left invariant vector eld on G

asso ciated to Y g f g lim the partial f g exp Y and by

Y t z

derivatives with resp ect to the co ordinates z t of C C

The prop erties imply the

G The dierential operators Prop osition Let U be an open subset of C

D X x t X x t

z z

i i

x t X x t D X x t X x t

D X x t X x t A x t X x t x z g U C

Y Y Y

map L U to itself

Therefore we have a connection D L U U L U dened by D

G G

P P

for any basis of left invariant vector elds on dz D d D D

j z a

G with dual basis

We pro ceed to dene a connection on E Consider rst the following dierential

n n

op erators on the space V U of V valued holomorphic functions on the op en set

U C G

r v x v x L v x

z z

i i

r v x v x x v x

r v x v x A xv x x U

Y Y Y

In this formula the denition of the op erator taking the Laurent expansion at the

p oints z see is extended to general meromorphic g valued functions and vector

elds considered as a function of the variable t C For a meromorphic vector eld

d d d

t on the complex plane we set with z t C t

i i i

dt dt dt

n n

d r Let r V U U V U denote the connection dz r

j z

Prop osition The connections D r obey the compatibility condition

rX v DX v X rv X L U v V

Proof This is veried by explicit calculation

This has the following consequence Dene r on holomorphic functions on U

ie such that h x v i is holomorphic on U for all with values in the dual V

v V by the formula hr x v xi dh x v xi h x rv xi

Corollary The connection r preserves twisted holomorphic conformal blocks ie

it maps E U to U E U

G G

Prop osition The connection r on E U is at

Proof For X Y g the curvature F X Y r r r is given by the

X Y

expression

F X Y A A A A A

X Y Y X X Y X Y

Note that the co cycle

A A dt

X Y

g t vanishes indeed the integrand I t is Zp erio dic and ob eys I t I t

for some Zp erio dic function g t and the integration cycle can b e decomp osed into

a sum of contours b ounding some fundamental domains The contributions of the four

g tdt edges cancel by p erio dicity except for a term

Thus we can write F as

F X Y F X Y

F X Y A A A A A

X Y Y X X Y

X Y

Now as a simple calculation shows F X Y viewed as a function of t C with values

is g is Zp erio dic and ob eys F X Y t Adg F X Y t as a consequence of

Thus F X Y is in the image of L U and vanishes on invariant linear forms

A similar reasoning applies to the commutators r r r r X g These

z X X

commutators are also in the image of L U and thus vanish on invariant forms We

are left with the commutator r r which vanishes except p ossibly for i The

pro of that it vanishes also for i will b e given later on see

The group G acts on V since the co cycle vanishes on g Lg Denote this

action simply G V h v hv

Prop osition For al l h G X AdhX is a Lie algebra isomorphism from

Lz g to Lz hg h Thus the map X X with X z g AdhX z h g h

h h

is an isomorphism from L U to L U for any open U C G where U

G G

fz hg h jz g U g Moreover for any X L U AdhX h X h

and thus z g z hg h h denes an isomorphism E U E U

h h G G

This isomorphism maps horizontal sections to horizontal sections

Proof The rst statement follows immediately from the denitions The fact that

AdhX h X h is also clear once one notices that the co cycle dening the

central extension vanishes if one of the arguments is a constant Lie algebra element

and r and we have r r X g The Finally h commutes with r

X h h z

AdhX

latter identity follows from the equality see

AdhA z g t A z hg h t

X AdhX

Thus preserves horizontality

The existence of a connection implies as in TUY that the sheaf U E U is

the sheaf of holomorphic sections of a holomorphic vector bundle whose b er over x

is E x This follows once one notices that E U is actually a subsheaf of a lo cally

G G

free nitely generated sheaf carrying a connection whose restriction to E is r Details

on this p oint are in App endix

To make connection with the previous sections consider the pull back of E by

the map exp i from h to G It is the vector bundle E on C h Let us

introduce co ordinates on h with resp ect to some orthonormal basis h Then the

pullback connection on E is given by and in the direction of

r h z

Moreover we can use the connection to identify by parallel translation the space of

conformal blo cks E U with the space of twisted conformal blo cks in E U G or

in E U h such that r X g or r resp ectively Here we use

h X

the fact that G and h are simply connected

The p oint of this construction is given by the following result Let V U b e the

space of holomorphic functions on an op en set U with values in the nite dimensional

n n n

space V We also set for any op en subset U of C G or C h resp ectively

hor

E U f E U jr X g g

G G X

hor

E U f E U jr X h g

h h X

Prop osition The compositions

hor n

E U E U G V U G

G G

hor n

E U E U h V U h

h h

where the rst map sends a holomorphic conformal block to the unique twisted holo

morphic conformal block horizontal in the G resp h direction which coincides with

on U fg resp U fg and the second map is the restriction to V are injective

Proof The rst map is an isomorphism to the space of twisted holomorphic conformal

blo cks horizontal in the G resp h direction The fact that the second map is injective

follows from the fact that using the invariance and the equation r resp

r one can express h v i for any v V linearly in terms of the restriction

of to V

We may and will thus view the sheaf E of sections of the vector bundle on C of

conformal blo cks as a subsheaf of V U h The next steps are a characterization

of this subsheaf and a formula for the connection after this identication

Theta functions

Let Q fq h j exp iq Gg b e the coro ot lattice of g

Denition Let z C and V V b e nite dimensional g mo dules and k

a nonnegative integer The space z of theta functions of level k is the space of

holomorphic functions f h V such that

i f h

ii One has the following transformation prop erties unter the lattice Q Q h

f q f

z q f q f exp ik q q ik q i

The space of such theta functions is nite dimensional as can b e easily seen by Fourier

series theory Denote by W b e the Weyl group of g generated by reection with

resp ect to ro ot hyperplanes It is known that this group is isomorphic to N H H

N H G b eing the normalizer of H exph For w W let w N H b e any

representative of the class of w in N H H The Weyl group acts on the space of theta

functions Indeed if f z then w f f w w also in z the

k k

coro ot lattice and the invariant bilinear form are preserved by the Weyl group and is

indep endent of the choice of representative w by i Let z denote the space

of W invariant theta functions

Theorem Let g A l D l E E E F or G Then the image

l l

of is contained in the space of holomorphic functions V U h such that for

n W

al l z C z belongs to z and such that for al l roots X g

and nonnegative integers p

p p

z X O

In the remaining cases we have

Theorem Let g A B or C l Then the image of is contained in

l l h

n n

the space of holomorphic functions V U h such that for al l z C

z belongs to z and such that for al l r s f g X g and

nonnegative integers p

p p j

X O r s z z exp i c

j rs

as r s with c c r

r r

The pro of of these theorems will b e completed in We conjecture that the space

of functions describ ed in Theorems actually coincides with the image of

This conjecture is veried in a simple class of examples in b elow

The fact that the formulation of the result is simpler for certain Lie algebras is due

to the following prop erty shared by the Lie algebras of Theorem for each ro ot

and integer m there exist an element q in the coro ot lattice with q m For the

other simple Lie algebras this is true only if m is assumed to b e even More on this in

Ane Weyl group

Since H acts trivially on E the Weyl group acts on the right on the values of E

h h

w acts as w and this is indep endent of the choice of representative

hor

Prop osition Let E U h Then for al l q Q

z q z

For al l w W

z w z w

Proof The value of at z q is obtained from z by parallel transp ort

along some path from to q Recall that is the pull back of a section of

hor

E U G to U h The image of the path in G is closed and contractible G is

simply connected which proves the rst claim

From Prop and the fact that w Adw we see that if is horizontal

then also is horizontal But these horizontal sections coincide at and thus

everywhere

Mo dular transformations

a b

The group SL Z acts as follows on C h if A SL Z

c d

z a b

A z

c d c d c d

Lemma Introduce the linear functions t it for h For al l x

C h A SL Z the map X xX with

a b

xX t exp ad c tX c dt A

c d

is a Lie algebra isomorphism from Lx to LAx

Proof This follows directly from the denitions

n n n

We have dened an action Lx V V for all x C h Let us denote

it as X v X v see to emphasize the xdep endence

Lemma Dene a linear isomorphism v xv from V viewed as Lx

module to V viewed as LAxmodule if x z

ick c

x exp exp x xc d

A A A

c d c d

This map has the intertwining property

x X xX x

A x Ax A A

for al l X Lx

Note that the choice of co ecient is irrelevant for the validity of the Lemma How

ever it is imp ortant for compatibility with the connection see b elow

We should also add a remark ab out the p ower of c d The exp onent L

is diagonalizable with nite dimensional eigenspaces However the eigenvalues are

fractional in general and the p ower is dened for a choice of branch of the logarithm

for each A SL Z This is made more systematic in the next subsection

Proof This is again straightforward The only subtlety is that a priori there could

b e a contribution from the central extension in the computation of the intertwining

prop erty However the central term app earing in this computation is prop ortional to

the sum of the residues of the comp onent of X along which is doubly p erio dic By

the residue theorem this sum vanishes

We can thus dene linear maps by x Ax x Lemma

A A

is an isomorphism from E U to E A U implies then that

h h

Lemma Let A be the pul l back on oneforms of the map x Ax dened on

some open U C h and r E U U E U be the connection dened

h h

in We have

r A r

A A

This fact can b e derived from a straightforward but unfortunately lengthy calculation

The main identity one uses is

t a b

c dt ict

c d c d

maps horizontal sections to horizontal sections Moreover Lemma ensures that

since A c d do es not have comp onents in z or direction maps

sections which are horizontal in the h direction to sections with the same prop erty

hor hor

E U h E A U h

h h

Let us apply this in the sp ecial case A

Prop osition Let V U be in the image of Then for al l q Q

z q z q z exp iq k iq q k i

hor

Proof We have E A U h Thus

z z

A A

for all coro ots q by Lemma and z U Explicitly

z z

z q q z

A A

Inserting the formula for we obtain

j j

L L iq k iq q k

j j

z q z e exp

x q

n j

with x z Now we use the fact that on V acts as iz q

x q j j

and that L acts as a multiple of the identity to conclude the pro of

Mono dromy pro jective representations of SL Z

In this subsection we assume that n set z and show that a central extension

of SL Z acts on the space of horizontal sections of the bundle of conformal blo cks

The fact that we have a central extension comes from the necessity to choose a

In fact L is diagonaliz branch of the logarithm to dene the expression c d

able with nite dimensional eigenspaces and any two eigenvalues dier by an integer

Moreover L acts by a nonnegative rational multiple CasV k h of the identity

on V V for any integrable V of level k Let L j id r s N We introduce

V V

a central extension

ZsZ SL Z

of SL Z by the cyclic group of order s The group consists of pairs A where

A SL Z has matrix elements a b c d and is a holomorphic function on the upp er

half plane such that c d The pro duct is A B AB A

Then this group acts on V valued functions on H h as ab ove but keeping track of

the choice of branch

A A

This action preserves the connection The inversion here is to correct for the wrong

order up to ambiguity in the choice of branch Thus we conclude that

A B BA

acts on the space of global horizontal sections on H h of E This mono dromy

s h

representation restricts to the character m exp imr s of ZsZ

In the case of V trivial representation this mono dromy representation is just the

representation of SL Z on characters of ane Lie algebras see K It would b e

interesting to calculate this mono dromy representation explicitly for general V Some

progress in the sl case was made in CFW where a connection with the adjoint

representation of the corresp onding quantum group was established

The vanishing condition

Let G b e a simple complex Lie group with Lie algebra g h a Cartan subalgebra g

h g a Cartan decomp osition and H exp h Supp ose that G EndV

is a nite dimensional representation of G Thus V is also a g mo dule For K G

or H let I K V b e the space of holomorphic functions on K with values in V such

that g h K ug hg g uh The Weyl group W acts on I H V let w b e a

representative of w W in N H Then w f h w f w h is well dened for

f I H V since H acts trivially on the image of functions in I H V We denote

by I H V the space of Weylinvariant functions in I H V

Lemma The restriction map I G V I H V is injective Its image is the

W W

space I H V of functions u in I H V such that for al l positive roots X g

p N and m Z

p p

X uexp i O m

as m

Proof The b ehavior of functions in I G V under conjugation by N H implies Weyl

invariance

Let X g and h Then

Adexp i X e X

If u I G V uexpX exp i is a holomorphic Q p erio dic function of h

thus a holomorphic function on H On the other hand by

X X

uexpX exp i u exp exp i exp

i i

e e

exp e X uexp i

i p p

e X uexp i

for some M We see that the latter expression is holomorphic on the ane hyperplanes

m if and only if for all p X u vanishes there to order at least p

To conclude the pro of we use some facts ab out conjugacy classes in algebraic

groups see eg S Chapter Let for each ro ot and integer m H H

b e the set of elements of the form exp i such that m The conjugacy

classes containing elements in H H H form the dense op en subset G of

ss m ss

regular semisimple elements in G Its complement contains the set H consisting of

conjugacy classes of elements of the form expX exp i where lies on precisely

one of the distinct H This elements are regular as they are regular in the identity

comp onent of the stabilizer of exp i see S which is the direct pro duct

of a torus of dimension rank times the SL subgroup asso ciated with By the

ab ove reasoning a Weyl invariant function on H extends uniquely to an equivariant

holomorphic function on G and the vanishing conditions imply that it extends to a

holomorphic functions on G H The complement of G H consists of higher

ss ss

co dimension classes whose closure intersects H and of classes whose closure do not

intersect H H Counting dimensions shows that this complement is of co dimension

at least two so by Hartogs theorem our vanishing conditions are sucient to have an

extension to all of G

By Weyl invariance we may replace the set of p ositive ro ots in the formulation

of the lemma to a subset of ro ots consisting of one ro ot for each Weyl group orbit

Also we may restrict the values of m by Q p erio dicity of uexp i Indeed if the

vanishing condition holds at m it also holds at m q for all q Q

We thus have the following result The action of the ane Weyl group W

W Q on Z is dened by

w q m w m q

W W

Lemma The subspace I H V I H V is characterized by the vanishing

condition for m in any fundamental domain for the action of W on Z

From Bou we see that in the cases A l D l E E E F G a

l l

fundamental domain is f F g where runs over a fundamental domain F

consisting of one or two elements of W If g A B C then we have to add

l l

where is a long ro ot

As a corollary we obtain a more precise characterization of the image of i Let us

identify functions on H with Q p erio dic functions on h via the map exp i

and view V as a representation of G by hg u v i hu g v i

Corollary The image of E U by is contained in the space of functions

n n n W

V U h such that for al l z C z belongs to I H V

hor hor

E A U h implying further vanishing Moreover if E U h then

h h

a b

On V L acts by a scalar The conditions let x z and A

c d

to V is restriction of

j j

c d

x Axc d xe

It follows that for all p

j p p

z X O m Ax exp

c d

if m Changing variables this implies that

j p p

z X O ma c z exp i

a c

Since any pair a c of relatively prime integers app ear in the rst column of some

SLZ matrix we obtain the result

Corollary The image of E U by is contained in the space of functions

n n

V U h such that for al l z C r s p Z p r s

j p p

z exp i z X O r s

r s

as r s

Pro of of Theorems

Theorems follow from Prop ositions and Corollaries

together with the fact that twisted conformal blo cks are annihilated by h L U

Examples

Here we give an explicit description of the space of conformal blo cks in some sp ecial

cases The discussion parallels the constructions in FG where ChernSimons states

in the case of sl are studied First of all consider the case of one p oint z with the

trivial representation Then the vanishing condition is vacuous and we are left to

classify scalar Weyl invariant theta functions of level k This space coincides with the

space spanned by characters of irreducible highest weight Lg mo dules in accordance

with the Verlinde formula

Next we consider the case of one p oint z with a symmetric tensor p ower of the

dening representation C of sl

If N the problem is reduced to describing the space of Weyl invariant theta

functions of level k with the prop erty that

p p

u O e

for all p and ro ot vectors e g Actually it is sucient to consider one

ro ot since the Weyl group acts transitively on the set of ro ots of sl

j N

The symmetric p ower S C has a nonzero weight space if and only if j is a multiple

of N Let us set j l N and denote by the elements of the standard basis of C

lN N N l

Then the weight zero subspace of S C C S is onedimensional and is

l l

The following considerations apply also to spanned by the class of v

the case l if we agree that S C is the trivial representation

The Weyl group of sl is the symmetric group S and is generated by adjacent

N N

transp ositions s j N If we identify the Weyl group with N H H

then a representative in N H of s is given by s if r j j s

j j r r j j j

s It follows that S acts on the weight zero space by the l th p ower of the

j j j N

alternating representation w v w v

l l

The next remark is that e v but e v We thus see that

O as approaches the hyperplane If is a Weylinvariant theta

function it then also vanishes to order l on all hyperplanes n m n m Z

Therefore the quotient of by the l th p ower of the WeylKac denominator

is an entire function as has simple zero es on those hyperplanes Moreover is a

scalar theta function of level N the dual Coxeter number of sl and w

We conclude that the space of conformal blo cks at xed is contained in the space

of functions of the form

u v

where u is an entire Q p erio dic scalar function on h such that uw u for all

w S and

k N l

u q q u

q exp iq iq q q Q

We have assumed here that N In the case where the vanishing condition must

b e satised also at other p oints on h one can pro ceed in the same way noticing that

the Weyl denominator vanishes there to o

A basis of Q p erio dic functions with multipliers is easily given using Fourier

series The basis elements are lab eled by P k N l Q where the weight

lattice P is dual to Q if k N there are no nonzero conformal blo cks The Weyl

group acts as w

Therefore the dimension of our space is the number of orbits of the Weyl group in

P k N l Q This number is wellknown a fundamental domain in P for the action

of the semidirect pro duct of the Weyl group by the group of translations by k N l Q

is the set of weights in the dilated Weyl alcove I see

k N l

More explicitly if are simple ro ots fundamental weights with

i i i j ij

and n then I if and only if the integers n satisfy the inequalities

i i i k N l i

n k N l n i N

i i

The number of N tuples of integers with these prop erties is calculated to b e

k N l

This is the formula for the dimension of the space of Weylinvariant theta functions

of level k extending to holomorphic functions on SL with values in the l N th

symmetric p ower of the dening representation of sl We now show that this coincides

with the Verlinde formula V which according to TUY Fa give the dimension

of the space of conformal blo cks

Let I b e the set of integrable highest weights of level k It consists of dominant

integral weights with k The dimension of the space of conformal blo cks with

one p oint to which an irreducible representation of highest weight I is attached

is given by the formula

d N

in terms of the structure constants N of Verlindes fusion ring A convenient formula

for these constants in terms of the classical fusion co ecients m the multiplicity

of a in the decomp osition of the tensor pro duct of b with c was given in GW and

K Exercise

Let W W b e the group of ane transformations of h generated by the Weyl

group W and the reection s at the hyperplane f h j k h g is the

highest ro ot and h the dual Coxeter number Let b e half the sum of the p ositive

ro ots of g and dene another action of W on h by w w Let

W f g b e the homomorphism taking reections to

Then for all a b c I

w a a

m N

bc bc

w W

are given in terms of mo dular trans Actually in Verlindes formula the co ecients N

formation prop erties of characters They are uniquely determined by the equation

S S S

db dc da

S S S

d d d

where according to K

S a

exp i

S k h

Here is the character of the representation of G with highest weight a

Let us check that the two formulas agree this is essentially the solution to Exercise

of K Let w W and q k h Q and supp ose that b oth a and w a q

are dominant integral weights Then it is easy to see from the Weyl character formula

see H that if k h P

exp i w exp i

w aq a

There is a unique element in each ane Weyl group orbit in the shifted Weyl alcove

I Using these facts and the formula for the multiplicities in the decomp osition of

we deduce tensor pro ducts n

a b c

Let us apply this to our example Identify h with C C Then integral

weights are classes a a a of ntuples of integers dened mo dulo Z

The Weyl group S acts in the obvious way and a weight is dominant if a a

N j j

The ane reection s is

s a a a k N a a a k N

N N N

r N

and N N Let c r b e the highest weight of S C Then

the decomp osition rules of tensor pro ducts say that m if a b l j N

j j j

for some integers l such that l a a j N and l r Otherwise

j j j j j j

m As a dominant weight a b elongs to I if and only if

a a k

We need two prop erties see GW of the co ecients N valid for any a b c I

a a a

i N m for all a b c I and ii N N where a a

k N

bc bc bc

k a a a We will also use iii Each orbit of W acting via on h

N N

contains at most one p oint in I

Let us now x c N l and do the classical calculation rst

i a a l for al l j Lemma Let c N l Then m

j j

f N g

is nonzero if and only if there exist nonnegative integers Proof The co ecient m

l l summing up to N l such that l a a if j and a l a l

N j j j N N

a It follows that l l for all j and this solution ob eys the inequality i a a l

j j j

for all j

Lemma Let c N l with N l k and suppose a I Then N

if a a k l

Proof In this case a k a a and since k a a l m

N N

by prop erties i ii by Lemma Therefore N

Lemma Let c N l with N l k Then N if and only if

a a l j N and a a k l

j j N

Proof We need to prove only the if part We do this by showing that only the

rst term in the sum is nonzero Let us supp ose that a ob eys the hypothesis of

a w a

the Lemma and that m m with w and derive a contradiction Since

ac ac

b w a is dominant and is not in I by iii we have b b k Let us choose

k N

the representative in a with a and identify a a with the row lengths of

N N

a Young diagram Then b is obtained by adding N l b oxes to this Young diagram in

such a way that a b a Then w with w b a is the unique element

i i i

mapping b to I This element is constructed as follows i Add for all j N j b oxes

to the j th row of b this adds ii Draw a vertical line at distance k N from the

end of the N th row to the right of it the only b oxes to the right of this line are in the

rst row and their number is at most N l k iii Take these b oxes and add them to

the N th row ie act by s see p ermute the rows to get a Young diagram ie

act by an element of W iv Subtract N j b oxes from the j th row j n

We obtain in this way a diagram which has N l b oxes more than the original diagram

with row lengths a and whose rst row has b k k b oxes The two diagrams

i N

are equivalent meaning that the latter is obtained from the former by adding the same

number of b oxes to each row This number is at least k a which by hypothesis

is strictly larger than l We need thus more than N l b oxes and this is a contradiction

The dimension of the space of conformal blo cks can b e now computed note that

a a l ie a a l N j maps bijectively the set of weights ob eying the

j j

conditions of Lemma onto I whose cardinality coincides with the dimension

k lN

of the space of invariant theta functions with vanishing condition

We conclude that the space of invariant theta functions satisfying our vanishing

condition coincides with the space of conformal blo cks in accordance with our conjec

ture

The KnizhnikZamolo dchikovBernard equations and gen

eralized classical YangBaxter equation

The KZB equations

The KnizhnikZamolo dchikovBernard KZB equations rst written in Beb for a

holomorphic conformal blo ck E U are the horizontality conditions r where

is identied with its image by To write these equations explicitly let us compute

the expression of the connection r on E U viewed as a subsheaf of V U h via

It is convenient to introduce functions w C expressed in terms of the

function

t log tj

w tj j

w j tj

See App endix for details on these functions We use the notation

k h

and the abbreviation X for X t We also identify g as a Lie subalgebra of Lg

X X g Let C e e see Then we can write L as

X X X

e e h h L

n n n n

Now let U C and E U which we identify via with a function on U with

n n

values in V We then have for xed u V

X X

hr ui h ui h h h e e ui

Recall that vectors in V are annihilated by X with X g n We now use the

invariance of under the action of L The functions t e t z are elements

of Lz They have simple p oles at t z with residue e As a consequence of

the invariance of we have

j k

h e ui h e ui z z h e ui

k j

k k j

V We can use this identity to compute the value of on vectors e u for all u

The atness condition r translates to

h ui ui z z h h h h ui

k j

k k j

h h that To compute further we need the commutation relation e e

follows from e e h e e h We therefore obtain the formula

ui h ui h h ui hr

k j

h z z ui

k j

where is essentially the WeylKac denominator for any choice of

p ositive ro ots

Y Y Y

dim g

n rank g n i i i

q q q e e e

q e and with the abbreviation

t tC tC

P P

ij j i

We also use the standard notation to denote Y if X X Y This

s s

notation will b e used b elow also in the case i j The indep endent factors in do

not play a role here but will provide some simplications later In deriving we

have used that by the classical pro duct formula for Jacobi theta functions is up

to a indep endent factor the pro duct Before continuing we can use

the formula to complete the pro of of Prop

End of the proof of Prop What is left to prove is that r r on E U But from

by the ab ove formula for r it follows that r vanishes Indeed we have h

z j z j

j j

h invariance and the other terms cancel by antisymmetry As r preserves conformal

blo cks we have r r and the claim follows from the fact that r commutes

j z

with r with j

A more involved but similar calculation gives a formula for r also essentially due to

Bernard which will b e given here without full derivation

One of the ingredients is Macdonalds or denominator identity see K

X X

_ _ _

i h q h q ih q w

e w e

w W

q Q

implying one form of Fegans heat kernel identity

Here is half the sum of all p ositive ro ots of g W is the Weyl group and w

is the sign of w W The complex dimension of g enters the game through the

Freudenthalde Vries strange formula h dim g

Let us summarize the results We switch to the more familiar left action notation

by setting hX v i h X v i if X is in a Lie algebra and is in the dual space to

g mo dule We also need the following sp ecial functions of t C expressed in terms of

t t and Weierstrass elliptic function with p erio ds

t t I t

J t t t w t

w t w w

These functions are regular at t Introduce the tensor

J tC Ht I tC

Theorem The image by of a horizontal section of E U obeys the KZB

equations

X X

j jl

h z z

z j l

ll j

X X

i H z z

j l

where z z and H are the tensors respectively

Remark For n these equations reduce to thus is a V valued function

of and only and

X X

CasV e e i

where z z z O z and CasV is the value of the quadratic Casimir ele

ment C in the representation V This equation was considered recently by Etingof

and Kirillov EK who noticed that if g sl and V is the symmetric tensor pro d

lN N

uct S C e e l l Id on the one dimensional weight zero space of V and

the equation reduces to the heat equation asso ciated to the elliptic CalogeroMoser

SutherlandOlshanetskyPerelomov integrable N b o dy system

X X

i l l N N l l

i j

See also for a description of the space of conformal blo cks in this case

The classical YangBaxter equation

The tensor z z g g ob eys the unitarity condition

Let us remark that the fact that the connection is at is then equivalent to the identity

X X X

h h h

in g g g This identity may b e thought of as the genus one generalization of the

classical YangBaxter equation It admits an interesting quantization F

App endix

Lie algebras of meromorphic functions

We have the following explicit description of Lz z Let w C b e

n w

meromorphic Zp erio dic functions on the complex plane whose p oles are simple and

b elong to L and such that

t t i

t e t

w w

t t

Such functions exist for w C L and are unique if we require that t t

They can b e expressed in terms of the Jacobi theta function

t log tj

t w j j

tj w j

n i n it

e tj

Here a prime denotes a derivative with resp ect to the rst argument

Prop osition For fg h and z C the meromorphic

functions of t dened as limits at the removable singularities Z

X e t z

X t z t z l n

X t z j l n

X g or h if are wel l dened provided j Im j Im and belong to

Lz z If runs over fg and X runs over a basis of g h if

then these functions form a basis of Lz

Proof It is easy to check that these functions b elong to Lz Let L z

b e given by the functions in Lz whose p ole orders do not exceed j By the

RiemannRo ch theorem

dj dimL z dimg j n

if j Indeed L z is the space of holomorphic sections of the tensor pro duct

of a at vector bundle on the elliptic curve by the line bundle asso ciated to j D where

D is the p ositive divisor z

The functions given here are linear indep endent as can b e easily checked by lo oking

at their p oles and have the prop erty that for j the rst dj functions b elong to

L z

To obtain a basis outside the strip j Im j Im we can transp ort our basis

using the following isomorphisms

Prop osition Let z C and q q P Then the map sending X

Lz to the function

t exp it ad q X t

is a Lie algebra isomorphism from Lz to Lz q q

j j

n n n j

U to U L For any op en subset U of C C h or C G dene L U L

G h

b e the space of functions in LU L U L U resp ectively whose p ole orders do

h G

not exceed j

Corollary The sheaves L L are locally free nitely generated for al l j

U for Moreover for each x C h every X L x extends to a function in L

some j and U x

The pro of in the case of L is obtained by setting simply

We wish to extend this result to L Let us rst notice that the function t is

G w

actually a meromorphic function of e Thus if g exp i the functions in Prop

can b e written as f Adg t z X where the meromorphic function f is regular

as a function of the rst argument in the range corresp onding to j Im j Im

Therefore we may extend the denition of the basis to give a basis of L z g for g in

some neighborho o d of g exp iu with Adu unip otent commuting with Adg

It is clear that the multipliers are correct if g is on some Cartan subalgebra but such

g s form a dense set in G The p ole structure do es not change if the neighborho o d

is suciently small In this way by choosing prop erly the Cartan subalgebra we nd

lo cal bases of L in the neighborho o d of all p oints in G whose semisimple parts are of

the form exp i with in some Cartan subalgebra and j Im j Im for all

Prop osition Let z C and q q P Then the map sending X

L z g to the function

t exp it ad q X t

is a Lie algebra isomorphism from L z g to L z exp iq q g

G G

With the Jordan decomp osition theorem we get a lo cal basis around all p oints of G

and we obtain

is locally free nitely generated for al l j More Prop osition The sheaf L

U for some over for each x C G every X Lx extends to a function in L

j and U x

Connections on ltered sheaves

Let S b e a complex manifold and denote by O the sheaf of germs of holomorphic

sections on S A sheaf of Lie algebras over S is a sheaf of O mo dules L with Lie

bracket L L L a homomorphism of sheaves of O mo dules ob eying antisymmetry

and Jacobi axioms A sheaf of Lie algebras L over S is said to b e lo cally free if it is

lo cally free as O mo dule ie if every x S has a neighborho o d U such that as

O U mo dule LU W O U for some complex vector space W In this case LU

e is freely generated over O U by a basis e e with Lie brackets e e f

k i j

with nitely many nonzero summands for some holomorphic functions f on U

We will consider the case in which the sheaf L of Lie algebra is ltered by lo cally

free sheaves of O mo dules of nite type In other words L admits a ltration

j j j

L L L L L

j n

O inclusions induced from inclusions with L lo cally isomorphic to some C

j l j l n n

j j

and such that L L L In particular L is lo cally free C C

ie every p oint of S has a neighborho o d such that the statement holds for the restriction of the

sheaf to this neighborho o d

A sheaf of Lmo dules is a sheaf V of O mo dules with an action L V V which is

assumed to b e a homomorphism of O mo dules The image sheaf of this homomorphism

is denoted by LV In the ltered situation it is assumed further that V is ltered by

lo cally free nitely generated O mo dules

j j j

V V V V V

j l j l

and that the action is compatible with the ltration ie L V V In par

ticular V is lo cally free and we can dene a dual sheaf V lo cally as V U

Hom V U O U If V U is of the form V O U for some vector space V

O U

then V U is the space of functions u on U with values in the dual V such that hu w i

is holomorphic for all w V The dual sheaf V has a natural structure of a sheaf of

right Lmo dules and we have a natural pairing h i V V O

We can dene the asso ciated graded ob jects

j j

Gr L L L

j j

Gr V V V

with the understanding that V L

Then Gr L is a graded sheaf of Lie algebras acting on the graded sheaf Gr V of

O mo dules and homogeneous comp onents are lo cally free and nitely generated

The sheaf of coinvariants is V LV and the sheaf of invariant forms E is lo cally

given by

U E U f V U j X X LU g

In the ltered situation LV is ltered with LV L V and we have

r sj r s

induced homomorphisms

j j j

V LV V LV V LV V LV limV LV

j j

Lo cally V LV U is the quotient V U by the submo dule of linear combinations

r s

of elements of the form X v X L v V with r s j

A connection r on a sheaf of O mo dules V is a C linear map V V where

is the sheaf of holomorphic dierential forms on S such that for all op en sets

U S

rf v f rv df v

for any f O U v V U The notation r is used to denote the covariant derivative

in the direction of a lo cal holomorphic vector eld if rv v r v v

i i i i i

A connection D on a sheaf of Lie algebras is furthermore assumed to have covariant

derivatives b eing derivations for all lo cal vector elds

D X Y D X Y X D Y X Y LU

and a connection r on a sheaf of Lmo dules with connection D is assumed to b e

compatible with the action ie

r X v D X v X r v X LU v V U

Such a connection induces a unique connection also called r on V such that for all

op en U S u V U v V U

dhu v i hru v i hu rv i

Let r b e a connection on a sheaf V of O mo dules If V is ltered by free nitely

generated O mo dules V we say that r is of nite depth if there exists an integer d

j j d

such that rV V The smallest nonnegative such integer will b e called

depth of the connection

Theorem Let L be a sheaf of Lie algebras and V a sheaf of Lmodules over a

complex manifold S Suppose that L and V have a ltration by locally free nitely gener

ated O modules and compatible connections D and r of nite depth If Gr V Gr L Gr V

has only nitely many non zero homogeneous summands then the sheaf of invariant

forms E is locally free and nitely generated

Proof Let z S and U b e a neighborho o d of z such that the restriction of V to U

is free Thus there exist vector spaces V V such that

j j

V U V O U V U V O U

The assumption that Gr V Gr L Gr V has vanishing comp onents of degree N means

that if j N and v V U we have a decomp osition not necessarily unique

v v X v

j N

for some v V and X LU By iterating this we see that we can take v V

The rst consequence of this is that the restriction map E U V U is injective

for all suciently large l

The second consequence is that we can replace the connection by a connection which

preserves V U for some large l and coincides with the given one on the image of

invariant forms The construction go es as follows

Let us choose a basis e e of V with the prop erty that for all j a basis of

j j

V is obtained by taking the rst dimV elements of this sequence View V as the

subspace of constant functions in V U and choose a decomp osition for all e

X e e e

i i

with e V U Dene a new connection r by

re re

This formula uniquely determines a connection r on the restriction of E to U The

dual connection on V U also denoted r is dened as usual by hr e i dh e i

i i

h r e i By construction this dual connection coincides with r on invariant forms

N d

and if d denotes the depth of the connection r it maps V U to itself

If we introduce lo cal co ordinates t t around z with z at the origin we see

that we have to solve the following problem given a subsheaf E of a nitely generated

free sheaf F on an op en neighborho o d U of the origin in C with connection r on

F preserving E show that there exists an op en set U U containing z such that

E U is a free O U mo dule Write F as F O for a vector space F We may

assume that U is a ball centered at the origin

Lemma Let be the vector eld t on C and r be a connection on a free

i t

nitely generated sheaf of O modules F F O on a ball U centered at the origin of

C For each F there is a unique F U such that and r

The pro of is more or less standard the F valued holomorphic function on U is a

solution of the system of linear dierential equations

n n

X X

t t A tt t

i i i

i i

for some holomorphic matrixvalued functions A with initial condition It

is convenient to rewrite this equation in the form

xt B x txt B x t t A xt

i i

In this form we can apply the standard existence and uniqueness theorem the unique

solution with initial condition is given by the absolutely convergent Dyson series

B x t B x t dx dx t

m m

The domain of integration is the simplex x x It is clear from

m m

this formula that is holomorphic on U This concludes the pro of of the lemma

Let E b e the subspace of F consisting of all values at of sections of E U where

U runs over all op en balls contained in U and centered at the origin Let e e

b e a basis of F such that the rst s e build a basis of E The homomorphism of

O U mo dules

E O U F U h h

is injective since vanishes if and only if vanishes We claim that the image of is

precisely E U if U is small enough Let E U Then we can write as

a te t t

j j

for some holomorphic functions a t By assumption a if j s Since

j j

re we have

n r

X X

t a te t r t

i t j j

i j

But r preserves E and therefore a t if j s It follows that a t

i t j j

a if j s We have shown that E U is contained in the image of the

homomorphism Now let for j s t b e sections of E U such that

e Such sections exist by denition of E for some neighborho o d U Then

j j

the construction ab ove gives

a te t t

j l l j

The holomorphic matrixvalued function a t is the unit matrix at t and is thus

invertible for t U if the ball U is small enough We conclude that e E U

which completes the pro of

Let us see how this applies to our situation following TUY For us S is either

n n n

of C C h C G and L is the corresp onding sheaf of Lie algebras which we

denoted L L L resp ectively The mo dule V is the free graded O mo dule V O

h G

The key observation is that GrL consists of the degree j part of g C t O

for all suciently large j Moreover GrV V canonically since V is graded and

the action of elements of suciently high degrees in GrL on GrV comes from the

action of g C t on the factors V

The fact that Gr V Gr L Gr V has only nitely many non trivial homogeneous

comp onents follows then from the fact that V t g C t is nite dimensional

for all p ositive integers N which is proved in TUY using Gabb ers theorem

Summary

In this thesis we considered two classic animals in the rational conformal zo o the

minimal mo dels BPZ and the WessZuminoWitten mo dels W KZ TK

We restricted our attention to correlation functions in the analytic chiral sector the

conformal blo cks In b oth cases integral representations for the conformal blo cks are

known from DF ZF CF SV F BF They are given by multiple con

tour integrals which generalize Gauss integral representation for the hypergeometric

function The integrals are many valued functions on conguration spaces Their

mono dromy yield representations of braid groups An amazing observation is that the

mono dromy of these integrals computed by contour deformation GN DF TK

FFK La is given by quantum group data FW GS PS AGS Thus

one has an explicit connection to another eld of two dimensional symmetry the the

ory of quantum groups D Ji Lu FK FRT and integrable lattice mo dels

PS Pa Ka

Chapter two was devoted to this connection We introduced a top ological quantum

group action on an enlarged space of integration contours which lab el conformal blo cks

The physical conformal blo cks were identied with singular vectors in the quantum

group mo dule We identied the quantum group representation with a pro duct of

Verma mo dules This top ological quantum group representation was our main sub ject

The analysis was done in entirely in terms of integral representations for conformal

blo cks It did not make use of the prop erties of eld op erators In chapter six and

chapter seven we furthermore used a scheme which dened conformal blo cks without

direct reference to eld op erators Recall also chapter three for the op erator origin of

integral representations

This p oint of view leaves aside some imp ortant issues like braid group statistics

Fr sup erselection sectors DHRand their quantum symmetry MS FRS

FK To make contact with these sub jects one should reconstruct the conformal

eld theory FFK Another imp ortant issue also left aside is BRSTinvariance

Topological representations should also have a meaning in terms of the BRST complex

F BMP We hop e to return to this question in future work

Conformal mo dels exhibit very interesting structures when they are formulated on

higher top ologies FS In chapter four we investigated the top ological representation

on a toroidal spacetime There it was more intricate than on the sphere We were led to

integrate multicomponent many valued dierential forms to obtain conformal blo cks

Again a fundamental question was the mono dromy of the outcome The quantum group

provided an ecient and b eautiful answer The mono dromy was again given by an R

matrix but this time in the adjoint representation CFW rather than in a Verma

mo dule We mention that Rmatrix representations are a wide sub ject themselves

with applications to link invariants KR invariants of three manifolds RT state

sum invariants TV mono dromy representations Ko of braid groups and braid

group statistics Fr

Chapter ve addressed the interplay b etween the top ological representation and

mo dular transformations of spacetime It turned out that the dierential forms from

conformal eld theory gave rise to top ological representations which worked nicely

together with the mo dular group The outcome was a representation of the mo dular

group on the quantum group representation It was given in terms of universal quantum

group elements

Chapter six and chapter seven were devoted to a mathematical construction of

conformal blo cks on the torus using metho ds of TUY This approach was purely

representation theoretic and did not require the construction of eld op erators Dif

fering from the sphere we were led to consider twisted conformal blo cks on the torus

Bea We derived the KnizhnikZamolo dchikovBernard equation in this twisted

framework and discussed the role of the mo dular group As a byproduct we arrived at

a genus one generalization of the classical YangBaxter equation We remark that the

analogous construction on higher genus surfaces Beb in this framework is still an

op en and imp ortant problem Progress in this direction was recently made in I

All this is one glimpse of the outstanding coherency and unity of two dimen

sional conformal eld theory and its branches in the physics and mathematics of

integrable systems and lattice mo dels Pa PS Ka quantum groups D

Ji FRT Lu innite dimensional Lie algebras K low dimensional top ol

ogy RT TV string theory Po FMS DFMS GW GSW Riemann

surfaces FS Beb FG braid groups and braid statistics Fr MS Chern

Simons theory W GK and many more areas

Thanks

I thank Giovanni Felder and Mauro Crivelli for sharing their enthusiasm and ideas

on top ological representations with me Global thanks to J urg Frohlich Krzystof

Gawedski Thomas Kerler Gerhard Mack Philipp e Ro che Volker Schomerus and

Alexander Varchenko for discussions on quantum groups Thanks to J urg Frohlich for

giving me the opp ortunity to work at the ETH Z urich in a most stimulating atmosphere

Thanks also to Jacques Magnen and Vincent Rivasseau for their kind hospitality at the

Ecole Polytechnique Thanks to Gernot M unsterManfred Stingl and all members of

the Institut f urTheoretische Physik I for their interest in my work and many discussions

ab out quantum eld theory Thank you Karin for your love and supp ort and a lot of

roundtrips to Z urich and Paris

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