Lecture notes on Chern-Simons-Witten theory

Lecture Notes on THEORY Lecture Notes on THEORY Sen Hu Princeton University ye b World Scientific Singapore .New ...

Author: Sen Hu | E. Witten

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Lecture Notes on

THEORY

Lecture Notes on

THEORY Sen Hu Princeton University

ye b

World Scientific Singapore .New Jersey. London 'HangKong

Published by World Scientific Publishing Co. Ptc.Ltd.

P 0 Box 128, Fmer Road, Singapore 912805 USA ofice: Suite IB, 1060 Main Street, River Edge, NJ 07661 UKofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

LECTURE NOTES ON CHERN-SIMONS-WITTEN THEORY Copyright 0 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book or parts thereof; may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-3908-4 ISBN 981-02-3909-2 (pbk)

This book is printed on acid-free paper. Printed in Singapore by Uto-Print

To my parents To Bo, Maomao and Amy

Preface

More than 15 years ago, Vaughn Jones discovered a n elegant, construction of a new polynomial invariant for knots in three-dimensional space. Jones's discovery has been generalized in many directions and a variety of new definitions have been found. Mathematically rigorous approaches to the Jones polynomial and its generalizations tend to be combinatorial in nature. These definitions often lack manifest three-dimensional symmetry, but this symmetry can sometimes be proved by showing irivariarice urider ari appropriate set of “moves”. There is also an alternative approach to this subject, discovered by “physical” methods a few years after Vaughn Jones’s initial work. In this approach, the basic object of study is a three-dimensional quantum gauge theory in which the action is the Chern-Simons invariant of a connection. Knot invariants and three-manifold invariants can be defined in this theory, at a physical level of rigor. Using the Feynman path integral, the various invariants can be described in a way that manifest the full three-dimensional symmetry. (Indeed, Feynman originally introduced the path integral for a very similar reason - to make manifest the full relativistic symmetry of Quantum Electrodynamics.) On the other hand, by using the relation of the path integral to the Hamiltonian formalism, the combinatorial recipes that are more familiar mathematically can be deduced from this more invariant starting point. But this is a difficult road for mathematicians to pursue. Success depends on putting on a rigorous basis the requisite quantum field theory techniques. This is not likely to be easy. Quantum field theory, in which the quantum concepts are applied t o vii

viii

Prejacc

fields and not just to particles, was in marly ways the greatest and most, difficult achievement in twentieth century physics. It is the basis for most of our present-day understanding of nature. B u l il is a hard siibject, that, has developed in fits and starts and that despile valiant attempts is still largely out of reach mathematically. The gauge theory approach t,o the Jones polynomial and its generalidions places them in this central, yet as of now, mathematically inaccessible part of the physical and mathematical world. What can mathematicians gain by trying anyway, or at least by learning something of what physicists have to say, even if it cannot yet, be fully justified nialherriatically? The combinatorial defiriitions of the knot and three-manifold invariants are beautiful, but they are only one side of the story. Quariturri field thcory exposes a relation of these same invariants to gauge theory on the one hand, to conformal field theory arid stable bundles on R.ierriann surfaces on the other hand, as well as l o olher mathematical theories like Dorialdson theory and t,hc thcory of affine Lie algebras that also have a natural quantum field theory setting. Knowing all sides of the story, or as much as we can learn, is worthy in itself and may well be necessary for understanding applications of the knot and three-manifold invariants. I hope that the present volume by Sen Hu (which is based in part on lectures I gave at Princeton University in the spring of 1989) will help make this subject more accessible to curious mathematicians. Hopefully, the explanations given here will help mathematical readers (who may also want to consult Michael Atiyah’s book The Physics and Geometry of Knots) develop a wider understanding of this subject and its relations t o physics. And perhaps it will impel some to help develop a more complete mathematical exposition of this subject than is now possible. By E. Witten July 22, 2000

Contents

Preface

vii

Chapter 1 Examples of Quantizations 1 . 1 Quantization of R2 . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Classical mechanics . . . . . . . . . . . . . . . . . . . . 1.1.2 Symplectic method . . . . . . . . . . . . . . . . . . . . . 1.1.3 Holomorphic method . . . . . . . . . . . . . . . . . . . . 1.2 Holomorphic representation of symplectic quotients and its quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 An example of circle action . . . . . . . . . . . . . . . . 1.2.2 Moment map of symplectic actions . . . . . . . . . . . . 1.2.3 Some geometric invariant theory . . . . . . . . . . . . . 1.2.4 Grassmanians . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Calabi-Yau/Ginzburg-Landaucorrespondence . . . . . . 1.2.6 Quantization of symplectic quotients . . . . . . . . . . . Chapter 2 Classical Solutions of Gauge Field Theory 2.1 Moduli space of classical solutions of Chern-Simons action . . . 2.1.1 Symplectic reduction of gauge fields over a Riemann surface 2.1.2 Chern-Simons action on a three manifold . . . . . . . . 2.2 Maxwell equations and Yang-Mills equations . . . . . . . . . . 2.2.1 Maxwell equations . . . . . . . . . . . . . . . . . . . . . 2.2.2 Yang-Mills equations . . . . . . . . . . . . . . . . . . . . 2.3 Vector bundle, Chern classes and Chern-Weil theory . . . . . . 2.3.1 Vector bundle and connection . . . . . . . . . . . . . . . ix

7

7 9 11 12 13 14

17 17 17 19 22 22 23 25 25

Contents

X

2.3.2

Curvature. Chern classes and Chern-Weil theory . . . .

26

C h a p t e r 3 Q u a n t i z a t i o n of C h e r n - S i m o n s A c t i o n 27 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Some formal discussions on quantization . . . . . . . . . . . . . 28 3.3 Pre-quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.1 M as a complex variety . . . . . . . . . . . . . . . . . . 31 3.3.2 Quillen’s determinant bundle on M and the Laplacian . 32 3.4 Some Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.1 G = R . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.2 G = S1 = R / 2 x Z . . . . . . . . . . . . . . . . . . . . . 33 3.4.3 T*G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 Compact Lie groups, G = S U ( 2 ) . . . . . . . . . . . . . . . . . 35 3.5.1 Genus one . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5.2 Riemann sphere with punctures . . . . . . . . . . . . . . 36 3.5.3 Higher genus Riemann surface . . . . . . . . . . . . . . 38 3.5.4 Relation with WZW model and conformal field theory . 39 3.6 Independence of complex structures . . . . . . . . . . . . . . . 40 3.7 Borel-Weil-Bott theorem of representation of Lie groups . . . . 44

Chapter 4 C h c r n - S i m o n s - W i t t e n T h e o r y and Thrce Manifold Invariant 4.1 Representation of mapping class group and three manifold invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Kni.,hi k-Zamolodchikov equations and conformal blocks 4.1.2 Braiding and fusing matrices . . . . . . . . . . . . . . . 4.1.3 Projective representation of mapping class group . . . . 4.1.4 Three-dimensional manifold invariants via Heegard decorriposition . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Calculations by topological qiiantum field theory . . . . . . . . 4.2.1 Atiyah’s axioms . . . . . . . . . . . . . . . . . . . . . . 4.2.2 An example: coririecled sum . . . . . . . . . . . . . . . 4.2.3 Jones polynomials . . . . . . . . . . . . . . . . . . . . . 4.2.4 Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Verlinde’s conjecture and its proof . . . . . . . . . . . . 4.3 A brief survey on quantum group method . . . . . . . . . . . . 4.3.1 Algebraic representation of knot . . . . . . . . . . . . . 4.3.2 Hopf algebra arid quantum groups . . . . . . . . . . . .

47

47 48 50 53 57 59 59 GO 60 G1 63 64 64 67

Contents

4.3.3 Chern-Simons theory and quantum groups

xi

.......

68

Chapter 5 Rmiormalized Perturbation Series of Cbern-SimonsWittcn Thcory 71 5.1 Path integral arid morphisni of Hilbert spaces . . . . . . . . . . 71 5.1.1 One-dimensional quantum field theory . . . . . . . . . . 71 5.1.2 Schroedinger operator . . . . . . . . . . . . . . . . . . . 72 5.1.3 Spectrum and determinant . . . . . . . . . . . . . . . . 75 5.2 Asymptotic expansion and Feynman diagrams . . . . . . . . . 77 5.2.1 Asympt.otic expansion of integrals, finitc dimensional case 77 5.2.2 Integration on a sub-variety . . . . . . . . . . . . . . . . 81 5.3 Partition function and topological invariants . . . . . . . . . . . 82 5.3.1 Gauge fixing and Facldeev-Popov ghosts . . . . . . . . . 83 5.3.2 The leading term . . . . . . . . . . . . . . . . . . . . . . 85 5 . 3 . 3 Wilson line arid link invariants . . . . . . . . . . . . . . 88 5.4 A brief introduction on renormalization of Chern-Simons Iheory 89 5.4.1 A regulixation scheme . . . . . . . . . . . . . . . . . . . 90 5.4.2 ‘l‘he Feynman rules . . . . . . . . . . . . . . . . . . . . . 91 Chapter 6 Topological Sigma Model and Localization 95 6.1 Constructing knot invariants from open string theory . . . . . . 95 6.1.1 I n t r o d t d o n . . . . . . . . . . . . . . . . . . . . . . . . 95 6.1.2 A topological sigma model . . . . . . . . . . . . . . . . 96 6.1.3 Localization principle . . . . . . . . . . . . . . . . . . . 97 6.1.4 Largc N expansion of Chern-Sirnons gauge theory . . . 98 6.2 Equivariant cohomology and localization . . . . . . . . . . . . 99 99 6.2.1 Equivariant cohomology . . . . . . . . . . . . . . . . . . 6.2.2 Localization, finite diniensional case . . . . . . . . . . . 100 6.3 Atiyah-Bott’s residue formula and Duistermaat.-Heckman formula101 6.3.1 Complex case, Atiyah-Bott’s residue formula . . . . . . 101 G.3.2 Syrnplectic case, Duister.maat-Heckiiiaii formula . . . . 102 6.4 2D Yang-Mills theory by localization principle . . . . . . . . . . 104 6.4.1 Cohornological Yang-Mills field theory . . . . . . . . . . 104 6.4.2 Relation with physical Yang-Mills theory . . . . . . . . 105 6.4.3 Evaliiation of Yang-Mills thcory . . . . . . . . . . . . . 107 6.5 Combinatorial approach to 2D Yang-Mills lheory . . . . . . . . 110

xii

Contents

Complex Manifold Without Potential Theory by S. S. Chern

113

Geometric Quantization of Chern-Simons Gauge Theory by S. Axelrod, S. D. Pietra and E. Witten

121

On Holomorphic Factorization of WZW and Coset Models

169

Bibliography

I93

Index

197

Afterwards

199

Chapter 1

Examples of Quantizations

Quantization of R2

1.1

1.1.1

Classical mechanics

There are several equivalent formulations of classical mechanics, namely Newtonian, Lagrangian and Hamiltonian formalisms. In Newtonian formalism ,v particles q 1 , ...,q N with masses m l , ..., m N attracting each other are governed by

For N = 2 this is exactly soluble and it was used to explain several laws of Kepler. Lagrange reformulated the above equations into a variational problem: b

8 1 Ldt = 0, where L is a function of

9 1 , ..., qIv,q i ,

The Euler-Lagrange equation

..., qk,

Examples of Quantzratiorls

2

d DL dL . = -l 1. = 1, ,.., N d t aqi 8qi

-(-)

gives thc Ncwton equa.tions (1.1). If we introduce new variablcs pi = furictiorl of q l l writleu as

%

(1.2)

arid lel II = Cipiq1 - L be n

q i y , p l , ...,p~~ thcn Eulcr-Lagrange equations can be re-

A remarkable property is that if

is a transformation induced by

or @ preserves the two form w = C i d p i A dqi, then for the new coordinates & I , ...)Q N , P I ,..., PN, the equations have the same form. In its abstract form classical mechanics can be formulated as follows: Canonical formalism : Let ( M 2 n , w ) be a symplectic manifold. For each H E C o o ( M a nR) , one can associate a Hamiltonian vector field X H on M 2 " , such that

dH(.) =~ ( X H , . ) . The flows generated by X H are called Hamiltonian flows. In the beginning of the 20th century scientists tried to use a model of the solar system to describe atomic structures with some successes. However experiments forced them to abandon the classical picture and adopted the quantum picture.

Classically wc use space-time ( 4 1 , q 2 , 4 3 , 4 4 ) E R4 as the basic variables and each point corresponds to an event. In quantum mechanics one uses the space of probabilities on space time as the basic variables and the observables arc Hermitinn operators on the space of probabilities. This is the most striking idea introduced by Heisenberg. There are several ways of quantization. The Feynman path integral corresponds t o the Lagraiige formalism, Corresponding to Hamiltonian formalism we have geometric quantization. In the following we will describe geometric quantization of (R2n,w ) .

1.1.2

Symplectic method

Let R.2 = T'R' be a symplectic manifold with the symplectic form w = da: A d p with coordinates ( 2 ,p ) . Here is their quantization: take the Hilbert space H ' = L2(R)with the inner product (4,4) = 4(z)4(x')dzdz'. Wc want to construct a map

C"(R2, C ) -+ H e r ( L 2 ( R ) L , 2(R)) Here H e r ( L 2 ( R )L2(R)) , denote the space of Hermitian operators. There are two ways of realizations. ( A ) : 31 = L2(R),consisting of functions of a:, $(x).

p :

*

-+ -2-. . dx

One can verify that [p,z] = -i, here [,I is the Poisson bracket induced from the symplectic form w. (B): X'= L ~ ( R ' consisting ), offunctions o f p , 4 ( p ) .

Examples of Quantizations

4

(A) and (B) are equivalent via Fourier transformation

Let 11s consider quantization of harmonic oscillators. In classical mechanics, harrrionic oscillators is generated by the Hamiltonian

with respect t o the 0 symplectic form w = dx A clp. In quantum mechanics we have the same Hamiltonian. Let

a' is a adjoint of a. Then we have

So B

2 f . It is easy t o verify that [ a , a'] = au* - a'a

= 1,

I H , u] = H a - aH = --a,

[ H ,a'] = Ha* - a*H = a'.

So if $ is an eigen-function of H, then a$! a*$ are also eigen-functions of H , and

Quaratization ~j R2

5

We call a the annilation operator and a+ the creation operator. If x is a ground state , i.e. H X = $x,then

x,a * x ,i a * ) 2 X ,

”’

5,

gives a basis of ’?f with cigenvalues $, $, .... lZ2 . IS a ground state, then a*”x will be For example, x = e-?

where H n ( z ) are Hermite polynomials.

Classically under Lie bracket {, }, symplectic Lie algebra is represented by quadratic forms x 2 p 2 , and ( r p p z ) . This gives representation of the Lie algebra sl(2, R) by ~

+

We can decompose the Hilbert space R into

Examples of Quantizations

6

1.1.3

Holomorphic method

There is a geometric quantization of (R2,w) by a holomorphic method which will be more useful to us. We consider R2 = C . Let

z

1

+ ZZ),r =

= -(p

fi

w

z

1

a

-((p

- iz),

d~ A d p = -id2 A d z .

So iu’ is a form of type ( 1 , l ) .Now consider the space

31 = {halomorphic functions on C } . Let z act on X by multiplication and Z act by product on % to be

&. We define the inner

6

is the conjugate of z. The space 31 can be described in another way, i.e.

then

3 = qc,q,

where I: -+ C is the trivial complex line bundle and a section s(2) of the s a n e as a holomorphic function on C. We define a metric on r ( C ,Lj by

11811;

C is

= e-yS(I)12.

Then (C,II.II) is a Ilermitian line bundle. The metric wc defined can be characterized by the property that it is compatihle with a unitary conncction whose curvature is the ( 1 , l )form w. Here compatibility means

Holomorphic representation of symplectic quotients a n d i t s quantization

7

where D is the covariant derivative. To compute the curvature we pick a holomorphic section s , then

F = iddlog 11s11:. Let us take s = 1, then

11~11;

= e-",

SO

F = iadlog llsll? = i8d(-Zz) = -idZdz = w . Let 31 = I ' p ( C , C).We now construct another quantization as follows: 31 by multiplication, Z acts on 31 as 1 is annihilated by forms a basis of 31. and { ~ ~ . 1 } ~ = 0 , 1 , 2 , Remark: This quantization depends on the choice of a complex structure. It is subtle t o see how it varies as the complex structure varies.

&.

z acts on

1.2

&

Holomorphic representation of symplectic quotients and its quantization

In this chapter we will describe holomorphic representation of symplectic quotients and its quantizations. The novelty is that some symplectic quotients admit holomorphic description via geometric invariant theory and it is very convenient to quantize it in the holomorphic setting. 1.2.1

A n example of circle action

Consider R2" = Cn with the standard symplectic structure w = Cy=ld x i A dyi = i dzi A d z i , or w = -ida, CY = Fidzi. Let us consider a circle action of U(1) on C" by

cr=l

cyZl

zi -+ eaozi, i = 1 , 2 , ..,, n , eie E U(1). This defines a homomorphism: p : U(1)

+ Diff"(R2"),

where DiW (R'") stands for diffeomorphisms of RZnpreserving the symplectic form w .

Examples of Quantizations

8

If we take a derivativr of the above map we would have the moment map: dp : g

+ Sympw(R.2n) ,

where g is the complex Lie algebra C of U(1). SympW(R2n)is the space of symplectic vector fields, i.e. vector fields of' thc form ( - H y , H , ) and H : R'" -+R is n Hamiltonian function. In this case the symplectic vector field is:

It generates a flow dz; dFi _ - Z' - = - z ' dt

"

8 1

dt

and z;& is a first integral. The moment, map is: p ( z ) = C ( Z i Z ; - - 1) i

v i a dp = i v w . The inverse image of p at a regular value gives a symplectic quotient of the U(1) action.

C"//S' = p-l(O)/sl = {(Zl,

...,Zn)IEZi&

= l)/S1 = CP"-l.

i

On the other hand, we may consider

s1= U(1) c GL(1) = c*. And we consider the complexified action: zi + x z i , x E C " . We see that

Holomorphic representation of symplectic quotients

and

its quantization

9

C"//S' = (C")S."C*, where (C")'.' are the set of semi-stable points of the C* action on Cn consisting of non-zero vectors of C n . z is a semi-stable point if and only if C Ziz, is bounded away from zero along the C' orbit C*.z. 1.2.2

M o m e n t m a p of symplectic actions

The construction above applies for much general cases. To proceed further let us consider symplectic action on a complex manifold. Let ( M Z n , w )be a symplectic manifold and let a Lie group G acts on MZnpreserving the symplectic form w . So we have a homomorphism: @ :G

-+

Diff, ( M 2 " ) ,

where Di&(Mzn) is the space of diffeomorphisms of M2" preserving w , i.e. @ ( g ) * ( w ) = w , g E G.

By differentiating @ we have

d@ : g

+ SympW(M2").

Again SympW(MZn) is the space of symplectic vector fields which can be identified with the space of Hamiltonian vector fields, i.e. i, H I with a smooth function H : M2" -+ R. Here i,H is obtained by

W(.,i,H) = dH(.). From this we define a one form: : TM2n

+ g*,

10

Examples of Quantizations

for every E E g. In the case that H 2 ( g ) = 0, or H 1 ( M Z n )= 0, we can always integrate the above one form and get a function:

J

: M2n

-+ g*

such that d J = j .J is usually called the moment map of the action. Here are more examples: 1) Rotational action We have the symplectic manifold M = T*V,V = R3, with the symplectic form w = d a , a = p d q . SO(3) acts on V by rotation. One can calculate the moment functions by

and they are actually the moment functions M I = p 2 ~ 3 - p 3 ~ 2M, 2 = ~ Plq3, M3 = PlqZ - P 2 q 1 . 2) Natural action of V ( n 1) on CP" Here the action is givcn by

3 ~ 1 -

+

p :U(n

+ 1) x CP" + cpn, ( u ,2)

-+

u.2.

One can verify that l h e following furicliori defines the moment map: p : CPn + U ( n + 1)*

a EU(n

+ l),

2*

E C " + I - (0).

3) I,ie group K acts on a complex algebraic manifold X

H o l o m u r p h i c r e p r e s e n t a t i o n of symplecta'c quotients and its q u a n t i z a t i o n

11

Let a Lie group li act on a complex manifold X which is embedded in CP". The action is the induced action of U ( n 1) on C P n . Then the moment map is:

+

p

1.2.3

x + k',

Some geometric invariant theory

The constructions above apply €or much general cases via the Geometric Invariant Theory developed by Hilbert, Grothendick and Mumford. Let X be a complex manifold which acts on by a compact group G. Let ,C + X be an ample holomorphic line bundle, i.e. there exists an integer fi such that X is embedded in C P N by sections in I ' ( X : C N ) ,via

2 3 ('$1 (X)>'"I

SN+I(Z)).

G action lifts t o an action on an arriple holarnorphic line bundle+ When G is compact then G c U ( N + 1) for some N . 'rhen we have G c which acts on CPN as well as on X . From the Geometric Invariant 'l'hcory we hw e

X / / G = x=.=. /Gc.

(1 3)

Here XS.'. arc the set of semi-stable points in X, i.c. thosc point z in X such that invariant polyiiorriials are bounded away from zero on the orbit Gcz. Let u s illustrate it by OIK more example. Consider the circle adion on ( R 2 " , ~whose ) cornplexificatiori is Ihe action C * which acts on ( C n , w ) as

+ ( X q , . , , , xz,, x-%,+1, ..., Xi3z,) The invariant polynomials are zizJ,i 5 s , j > s. r Criis semi-stable if (ZI,..., 2,)

and only if those invariant polynoininls are hounded away froin zero on the orbit of C*.z. Or equivalently C*.z is bounded away from zero, or Ci 2 i z i

Examples of Quantizations

12

is bounded away from zero on the orbit of C * . r . So unstable points z are such that all zi = 0, for i 5 s, or all zj = 0 , j > s. The moment, map of the above action is:

1.2.4

Grassmanians

Given a vector space V over C of dimension N ,the Grassmanian G ( k ,N ; C ) is defined as the space of k-dimensional subspaces in V . It, can be reprcsentated by k linear independent vectors e l , ..., ek C V . Let B c C k N = V x V x ... x V be the space of k linear independent vectors. B is an open dense subset in C k N . G L ( kC , ) acts on B .

G ( k ,N ; C )= B / G L ( k , C).

Let g be a Hermitian metric on V . On V x ... x V = C k N ,we choose a basis

p ,i = 1, ..,, Ic, s = 1, ..., N .

xi,*

Let w = i d4i3A d4iS. Here 4iS = g ; j g S k $ j k . w is invariant under the group action U ( k ) C G L ( k , C ) . In other words, U ( k ) acts on C k N symplectically. The moment map in this case is

p :

ckN-+ u ( k ) * ,( e l , ...,ek) -+ { ( e i , e j ) - dij}.

So we have

p

=0

e l , ..., ek orthonormal.

Here we arrivc at another natural representation of Grassmanians

G ( k ,N ; C ) = pL1(0)/U(k).

Holomorphic r e p r e s e n t d o n of sympkctic quotients arid i t s yuantization

1.2.5

13

Celabi- Yau/Ginzburp Landau correspondence

The following example came from the paper by E. Witten, "Phases of N = 2 theories in two dimensions", Nuclear Physics H 403 (1'393) 150-222. In this paper the N = 2 supersymmetric nonlinear u-model in a special Calabi-Yau target space is reduced to the following problem in algebraic geometry. Let us consider a hypersurface

V = ((~1,~

2 b )l ,

b z ) E C4/alb2 - a2bz = 0).

There is a natural U(1) action on C4 which leaves V invariant,,

~i

-+ h

i , bj

-+ X - l b j , X E U(1).

V / / U ( l ) is a Calabi-Yau threefold because there exists a global holomorphic 3-form on it, i.e.

0 = d a l A daz A dhl A d b z .

The moment map for the U(1) action is U ( n , , b j ) = (1.11~

+

Iu2I2 -

IbiI2 - IhI2 - p ) 2 .

Here T reflects t,he possibility of adding a constant to the Harniltonian. For r > 0, the set of' semi-stablc points is

z+= (V' u VI)/C*,

Vl = ((a,1,a2,0,0) E c". Z+ can be considered as a fiber bundle over { ( a l , a 2 ,0,O) E C ' } , i.e. we have

14

c2+ z+ + CPjl. Zero sections are genus zero holorriorphic curves. For T < 0, the set of semi-stable points is

2- = (V' u V2)/CC,

vz = {(O, 0 , b,,

62)

E

c").

It is also a fiber bundle

cz+ z- + CP:,. We see that there is a transition. T h e singular locus is switched from a-space t o b-space. Such a trarisitiori is called a flop in algebraic geometry, Our point here is that Z+ and Z- came from the same space symplectically. So we expect the quantum theory the same for the two cases. This is the rationale of persistence of quantum theory of space time with respect t o changing of topology in a-models/ string theory. 1.2.6

Quantization of symplectic quotients

We have illustrated examples of quantization in Chapter 1. The constructions there applies for much general cases. 1) Quantization of the cotangent bundle Let N be a manifold and T ' N be its cotangent bundle. On T' h' there is a natural symplectic structure, i.e. w = d a , Q = pidqi, where ( 4 1 , ..., qnj are coordinates for N and (PI: ...,pn)are coordinates €or fibers. The Hilbert space for quantizing T'N is

xi

x=P(Nj. We map coordinates functions to operators on H as follows:

Hulornorphic r e p r e s e n t a t i o n of s y m p l e c t i c q u o t i e n t s a n d i t s q u u n t i z u t i o n

15

2) Quantization of a Kahler rnariifold Let, ( M '", w ) be a Kahler manifold with w the Kahler form so it is naturally a symplectic manifold. We can construct as a holornorphic line bundle C over M 2 " and a Hermitian metric 1 1 1 1 1 on C,such that the curvature of this me& is

Example: On CP" we have the tautologically line bundle Let us consider the metric

11~11:

C + CPn.

= e--zz1s12,

(sI2 is the Euclidean metric on the fiber. Then the curvature

of this metric

is:

In general for a Kahler manifold we can choose

To quantizc the Kahler manifold we take the Hilbert, spacc

R = r ( M 2 n ,c ) , that is the space of holomorphic secttiom of the chosen line bundle. There are natural opcrat,ors such as multiplication by a section or differentiation by a covariant derivat,ive. 3) Quantization of a symplectic quotient, Leh X be a complex manifold which acts on by a compact group G holomorphically. From the Geometric Invariant Theory we have

16

Examples of Quantizations

This suggests we quantize the quotient space by quantizing X and then passing through its quotients. Namely we choose a Hilbert space as:

31 = r ( X / G ,L ) = r ( X " " / G c , L ) = r ( X ' . s . ,.C)Gc. The Hilbert space consists of G c part of the space of holomorphic sections of the bundle L + X'.'. The corresponding operators to coordinate functions are the same as before. Remark: For applications we take X = R2nand G compact Lie groups. One may see [Jeffrey-Kirwan] , [Meinrenken], [Tian-Zhang], for different approaches and recent developments.

Chapter 2

Classical Solutions of Gauge Field Theory

2.1

Moduli space of classical solutions of Chern-Simons action

Our main interest is in gauge field theory. The discovery of a gauge invariant Lagrangian by Yang arid Mills is a breakthrough in gauge field theory. For three-marlifold, Chern-Simons Lagrangian is also invariant under gauge transformatiori up to an integer. It is important to emphasize that the basic variables are gauge equivalent fields. In this section we will describe the moduli space of classical solutions for Chern-Simons action. It is the space of flat connections with some useful additional structures.

2.1.1

Symplectic reduction of gauge fields over a R i e m a n n surface

Readers may also refer to Atiyah and Bott's paper "Philo. Trans. of Royal SOC.London", A308 (1982) 523, for more details. Let C be a Riemann surface, i.e. a topological surface with a choice of complex structure J : T C -+ T C , J 2 = - I d . Let E -+C be a bundle with structure group G. Let A be the space of connections A on the G bundle E . A is a g-valued one form, i.e. A E r ( C , E g g ) , g is the Lie algebra of G. The gauge group = { maps C -+ G} acts on d,the space of connections, via

Classical Solutions of Gauge Field Theory

18

Infinitesimally we have u , a g-valued 0-form on C, and

Du := du $- [ A ,u ] ,A ( € )= A - E D U . There is a natural symplectic structure on A given by

Here 6 is the exterior derivative on A. Du is a symplectic vector field. Remark: 6 acts symplectically on A. Problem: Find the moment map of the 6 action on A. It amounts to solve the equation

bp(u) = i D U W .

Tr((-Du) ASA)

For a one-parameter family of connections A, = A curvature

F, = dA,

+ A, A A,

=F

+ E D B+ O(E’),

D

= ~ W k lTr(uAF)

So bF = D6A. Hence, ~

+ tB, we have the

Moduli s p n c e of classical s o h t i o n s of Chertz-Simon8 a c t i o n

19

+

This means that t h e curvat.ure P = d A A A ,4 is the moment map. By syrnplectic reduction we get the syrriplectic quotient, which sits in the space of gauge equivalent field , p-'(O)/C, i.e. the space of flat connections. For each of such a connection A arid a Loop C in C. we have holonomy Jc'l'rA. This determines a representation p : r l ( C ) + C . It is riot difficult to SCC that the rcprcsentation also determines A up to gniige t,ransformation. Hence we have p - ' ( O ) / G = Horn

( T I(C), G)/G,

where G acts by conjugation. The space of gauge equivalent flat connections inherits t h e natural symplectic structure from d,i.e. w = Tr(G-4A 6 A ) . It can be described in terms of cohomology as follows. The tangent space of p-'(O) is KerD. Here Du = du [A,u] is the covariant derivative. We have then complexes

s,

+

R o ( C ; E @ g )%- R 1 ( C ; E @ g )S 0 2 ( ( C ; E @ g ) The tangent. space of M = p-'(O)/G is

TiM = KerD/ IrnageD = H1(C; E @ g ) . And one can verify easily that

is

il symplectic

a1 and

2.1.2

form on M , i.e. it only depends on the homology classes of

QZ.

Chern-Simons action on a three manifold

Let M 3 be a three manifold, G a Lie group. E + M a principal G-bundle. Let A be a connection on E , i.e. A E R1(M3, E g ) . We consider the

20

Classdcal Solutions of Gauge Field Theory

Chern-Sirnons action

k

Tk(AA dA

+ j2 A A A A A ) ,k E Z’,

taking values in R/2.rrZ. Here G is a Lie group with Lie algebra g on which we endow with a non-degencrate quadratic form, ( a , b ) = ‘l’rab

Again the gauge group is

6 = { maps A4

+ G}

each of them corresponds to a change of trivialization of the bundle E The gauge group 6 acts on the space of connections A via ( g ,A )

+ yAy-’

+M .

+ dgg-’.

One can verify that Chern-Sirrions functional is invariant under the action of gauge transformations up to ail integer which corresponds t o the homotopy class of the gauge transformation. So it is a functional Over the space of gauge equivalent connections valued at R/2nZ. Remark: Pick a four manifold B with aB = M , extend E , A over H , then we have

If we choose a different B’ wit,h a H ‘ = M and consider X = B U (-I?’), here -B’ is a manifold with reversed orientation. Then we have

is an int,eger representing the first Poritryagin class of S . T h e classical solution of Chern-Simons functional are flat connections 011 M’. I t can be seen as follows.

Madirii space of classical sdiitdoian

0.f

Gherri-Sirnons action

Take a one-parameter family of connections A , = A also g- valued I-form.

LC.S (AL) =

k

417

/

T r ( ( A + c B )( d A + t d I I )

+ 32 ( A +

f

$. EH, where

21

B is

H ) A (A t e B ) A ( A + ( B ) )

Sincc B are arbitrary we have E' = 0. Onc knew that flat connect,ions are determined by their holonomy, i.e. by representations T ~ ( M3 ) G. So we have critical points/gauge transformations <==> Homj.rrl ( M ) ,G)/G. Now consider a special 3 manifold M 3 = C x R , where C is a surface. We write connection -4on E -+ M as A = G.+ Aodt, AD is a 0-form. T h e n we have: 1) 6.40 = o <==> d a + a n a = 0, ur a ( t ) is s f a t connection on E X it). D& = 0, this means that u p to gauge transfor2) aL 6a = 0 , ---> mativri, what happens at t f 0 is dctcrmincd by the connection at t = 0. Note that the infinitesimal gauge transformations are:

+

AD+ A0

+

Du Du 6-,

dt

a

du dt

-= -

l3t

+ [Ao,u],

+ a + tDu

So after getting rid of A0 and t we have that the gauge group becomes a gaugc group on the surface and the moduli space becomes

M = p-'(O)/G ,

+ g' , a -+ P' = da + a A a is the moment map. On A, therc is a natural symplectic structure above. where p : A

Classical S o l u t i o n s of Gauge Field T h e o r y

22

2.2 2.2.1

Maxwell equations and Yang-Mills equations Maxwell equations

The following is the well known Maxwell equations:

V B = 0 , v E = 0,

aB

-+

at

V

l3E at

x E = 0,-

-

v x B = 0.

There is a duality to this equation, i.e. it does not change with respect t o the transformation (ElB ) 3 (--B,E ) . This duality leads t o an important recent development of Seiberg-Witten theory. There are also continuous symmetries. It is H. Weyl who made this apparent in terms of gauge symmetry. It is actually a U(1) gauge theory. To achieve this, we write E and B as components of a matrix:

1 0

-El

-E2

-E3 \

F = FP’dxP A d x , is a 2-form. If’ we writc E = (E’dzl + E2dx2 + E 3 d x 3 ) A d t , B = ( B ~ d x ~ + B 2 d x ~ + B 3 d x ~ )thcn A d t Fl = E+*B. IIere, * is the Hodgc operator. * ( d t A d x l ) = d x z A d x 3 , * ( d z iA dt2) = dt A d z 3 , ... It is easy to see that *2 = I d . So we have * F = * E + B . Then it is easy to verify that the Maxwell equations can be expressed as d F = 0, d ( * F ) = 0. Since d F = 0 we have F = dA if the domain is simply connected. A is usually called a vector potential or a gauge. Apparently if we change A to A d A , F does not change. We see that the Maxwell equations can be expressed as a U(1) gauge theory. We consider L a U(1) principle bundle over a domain in R4. Gauge field is U(1) Lie algebra, i.e. R, valued one-form. Gauge transformations are g = e i h l A E R. when the

+

~Wacri,ei[ equations and Yang-Mills e q u a t i o n s

23

bundle is trivial, in genera1 they are sections of the bundle. g acts on A by A' = g A g - ' - i g - l d g s . Interesting quesliuris arise if the bundle L is non-trivial. This motiviatcs P. Ilirac: to search for moriopoles. Monopolcs canrrot exist if the bundle is trivial. If the hiindle is nori-trivial we would have a non-trivial invariant

€or any surfacc C c R4.This i s t h e first Chern class which can be intepreted as flux in physics. 2.2.2

Yung-Mills e q u o t i n n s

Very interesting things happen when wc change the above gaugc group U ( l ) t,o a non-Abelian group, say S U ( 2 ) . Yang and Mills made the change in the mid-fifties to explairi iso-spin in physics, It turns out that thc sctting, which express syrrirrietry vcry well! is most important. Lct E + M bc: a principle bundle with a Lie group G as its fiber. Let A be a connection, i.e. A = C a w a T abe a g valued one form where Ta are generators of the Lie algebra g and w, are one forms. A gauge transformatmiong is a G valued zero form, i.e. a section of the bundIe. g acts on the space of connections by A' = gAg-' gdg-l. Given a connection A ! we consider the covariant derivative D = d A. We have F = D A = d i l A A A is a two-form: it is the curvature of A . A gauge transformation g transforms A into A' whose curvature is F' = gFg-l. We say that F transforms covariantly. Another remarkable property of F is the Bianchi identity DF = 0. Principle bundle arises naturally in geometry and in physics. In both cases one needs to choose local coordinates. Different coordinat,es are related by gauge transformations! and geometric or physical quantities should be gauge invariant or covariant. The following functional on the space of connections, Yang-Mills functional, is gauge invariant,

+

+

+

Here * is the usually Hodge operator induced by a metric on M . Since F transforms covariantly, L Y M is invariant under a gauge transformation.

24

Classical Sulutiuns uJ G a u g e Field Thcory

Thc Euler-Langrange equations of Yang-Mills are:

DF = 0 ,D(*F)= 0. Since D F = 0 is an identity, the non-trivial equations are D ( * F ) = 0. When the dimension of M is four, if a connection A satisfies * F = f F , then it autorrialically salisfies the Yang-Mills equations. Such solutions are called self-dual and antiself-dual instantons. It plays a key role in Donaldson’s theory of differentiable structures of four manifold. Another remarkable property is that F can be used to describe topological properties although it is only a local data. For example, the first Chern class q ( E ) = &TrF E H 2 ( M ,Z) is independent of the representation of connections. This can be seen as follows. Given a connection A , consider its variations A’ = A 6A whose curvature is given by F’ = F DbA. Notice that TrDGA = dTrSA. For any two cycle C, we have

+

+

By the same reasoning plus Bianchi identity, the second Chern class

c?(E)= & ‘ ~ Y ( F A F )is also a topological invariant. It is also the Pontryagin class of E . The second Chern class Q ( E )for an instanton is called the instanton number which is a topological invariant. The Yang-Mills functiorial can be viewed as a functional over the space of gauge equivalent field. Bolt showed that the Yang-Mills functional is a perfect Morse function over the space of gauge equivalent field, and from this he determined cohomologies of the classifying space of the space of gauge transformations [Atiyah-Bott (1982)I. It is interesting to note that the Chern-Simons action C.S. of d M and the second Chern class or Pontryagin class c z ( E ) of M are related by

dC.S. = c@).

Vector bundle, Chern c l m s e s nrid

2.3

2.3.1

Chern- Weil theory

25

Vector bundle, Chern classcs and Chern-Weil theory Vector bundle and connection

Definition: (Vector bundle) Let V be a vector space, M a manifold. We say E + M is a veclor bundle, if E = U,U, x V , wherc M = u,U,, and if lJa n Up # 4 , we identify 11, x V with U p x V by a transition function

=,()Y o , S a p o p ) , % = 90. with (QdJ Here g a p is a cocycle. It satisfies: 1)gaa = 1 1 2)gaP = gargrp, for any point in U, n Up n U,. We have a canonical projection T : E + M , ( 2 ,v) + 2. There are many natural examples of vector bundles such as tagent bundle, cotangent bundle of a manifold and their tensor products. Definition: (Connection) The notion of connection generalizes the concept of directional derivativc. We knew that thc dcrivative -$ acts on the space of functions. For a vector bundle the space of functions is generalized to t h e space of sections

Then a connection can be defined as a linear opcrat,or:

D : r ( E ) -+ r ( E @7 ' * ( M ) ) It satisfies the Lebnitz rule:

D ( f s ) = df @ s -t f D s . In local coordinates, let { e i } be a basis of sections so that every section s can be represcntated by s = Cisie;. Let

26

Classical Solutions o j Gauge Field T h e u r y

where the connection matrix A = ( O t j ) is represented by a rrialrix of one forms. If we choose a different set of local coordinates, they are related by a gauge transformation g : M + C: c IIorn(V, V ) . The connectiori A is then represenled by A' = gAy-l dgg-', where dgg-' is the MaurerCartan form of G. For each connecliori A , we define covariant derivative DA = d+A. It is easy l o see that covariant derivative transforms rovariantly under a gauge transformation: D,t = ,qDA.q-'.

+

2.3.2

Curvatuw, Ch,ern, classes and Chern- W e d t h e o r y

Definition: (Curvature) Curvature of a connection A is defined as R = D i = dA + A A A . It enjoys two important properties: 1) Bianchi identity: Ds1 = 0. 2) For a different set of local coordinates differed by a gauge transformation g , we have fl' = g a g - ' . Definition: (Chern classes and Chern forms) From the Bianchi identity, we can verify that c i ( E ) = &'MI*is a closed form in H 2 i ( M ,R). It is called the i - th, Chern class of E . Chern classes can also be defined by: d e t ( l + &An) = X J + @ ) . If we modify A to A + d A , the curvature is changed to R + D A ~ ABy . this and the Hianchi identity it is easy to verify that, the Chern classes c i ( E ) = &TrRi as cohomology classes are independent of choices of A . This is thc bca.ut,iful Chern-Weil theory.

Chapter 3

Quantization of Chern-Simons Action

3.1

Introduction

From the last chapter we sce that the classical solutions of Yang-Mills over a Rierriann surface or Chcrn-Simons over C x R is the moduli space

Mg = Hom(.rrl(C),G)/G. Here G is any compact semi-simple Lie group with an invariant quadratic form ( , ) = &Tr, k E Zs on its Lie algebra g. There is a natural symplectic form

over the space of connections. So M E inherits this symplectic structure by passing to homology classes of connections. So it is together with w forms a symplectic space. We want to quantize such a space, i.e. to construct a Hilbert space associated to M E or C,

In geometric quantization we first construct a line bundle C --;r M . The Hilbert space will be the space of holomorphic sections, i.e. r ( M ,L’), k E Z+. The construction which we will give at first depends on picking up a complex structure J on C. However one can show that such a construction 27

28

Q u a n t i z a t i o n of Chern-Sirnons Action

is independent of complex structures by constructing a projective flat coiinection on the bundle 31c(G, (, )) -+ Teich, the space of complex structures over the surface. We can then identify different Hilbert spaces by using this fla.t connection. We may take advantage of independent of complex structures. We can decompose C into pants by choosing a set of maximally disjoint, simple closed curves C on C. For each pant decorripositiori we will construct a Hilbert space X by assigning each loop around puncture an irreducible representation of G. Then the Hilbert space R for C can be constructed from (by taking the tensor product,) Hilbert spaces for the collection of pants (conformal blocks). In the following we will realize such a construction for a list of Lie groups. It turns out that it often gives many interesting applications out of those constructions.

1) R 2) S' = R/2rZ This construction leads to classical theta functions on a Jacobi variety. 3) T'G = g' x G, g is the Lie algebra of G regarded as an Abelian group acted on by G. The Lie algebra of T'G is

L = Lie(T*G) Y

gab

+ gG

{ a , b } E L , a , 6 , E g , ( { a ,b } , { a ' , b ' } ) = b ' ( a ) - b(a'). This construction leads to Rcidcrmcistcr torsion or TI invariants. If WE replace T'G by Super T * G ,

it gives Casson invariant [witten- ass on], 4) Compact semi-simple Lie group, c.g. G = S U ( 2 ) . This construction leads to a series of knot invariants including Jones polynomials. 5) Non-compact semi-simple Lie group, e.g. S L ( 2 ,R),S L ( 2 ,C ) with a n invariant quadratic form which are integral and nowdegenerate. 'l'his construction seems to have iriteresting connectioris to Thurston's gcometrization program.

3.2

Some formal discussions on quantization

Let us first consider quantization of the Chern-Simons action informally. It turns out the following informal consideration is very illuminating and it, can be made rigorous mathematically from the work of [Axelrod-DellaPietra-

S Q ~j oCr r n a l discussions o n quantization

29

itt ten]. We convidcr quaritization for a special tiircc manifold M 3 = S x [0, 13. The phase space is now thc space of connections A on C. A is an affine syniplectic space with syrnplectic form w(a,,$) = T k ( a A @ ) .Gauge group acts on A symplcctically. Remark: We also learned in t,hc last chapter that the space of classical solutions M is the space of flat connections. For each flat connection A we have a covariant derivst'ive d~ = d A . Thc tangent space of M is the space of first cohomology H i A ( C ,E @g ) . M is also a symplectic variety with respect to the syrnplectic form w above and one can easily check t,hat w only depend on dA-cohornology classes. This way we push the symplectic form down t o a syrnplectic form on the syrnplectic quotieat. As we have seen in Chapter One it is pretty easy t o quantize an affine symplectic space. W e use holomorphic quantization here. To do t,his we need to have a complex structure on A . There are natural complex structures coming from a choice of complex structure J : TZ:-+ T C , J 2 = - I d , of the underlying surface C. The induced complex structure on A is nothing but t o claim the (l!0) part of the connection A = -4,dz ilzdZ t o be holomorphic. With this complex structure the quantization is quite simple. T h e Hilbert space is then the space uf holomorphic seclions of the trivial line bundle L = A x C. We wish to say a few more words on the bundle C. We define functional derivative or a connection on A to be

sc

+

+

D -

I

I

DA,

6 &A,

--

k

-Ax, 4n

One can check that

So the curvature for this connection is -iw. And formally the first Chern class of the line bundle L is iw. We post the first condition on the big Hilbert space Y! E r ( A ,.CBk) as,

30

Quantization of Chern-Simons Action

D -q DAz

= 0.

Recall that what we want to quantize is the space of gauge equivalent connections d/G. The idea is to quantize affine space of the space of connections and then select the gauge invariant part, i.e.

We also have Gc, the group of complexified gauge group, acts on A. A holomorphic section which is invariant under G is also invariant under Gc. So formally we have

= r h o i (d/Gc C@'). The space A/& is nothing but the space of holomorphic bundles over the surface C. According to a theorem of Narasimhan-Seshadri, the space of stable holomorphic G c bundles is the same as the space of flat G connections on C. This explains why the Hilbert space is rhol(M,L@').This is a well defined Hilbert space even though other Hilbert spaces above are ill defined. To get I ' h o l ( A l L@'")' we need to get the condition to select gauge invariant sections. It is given by

This follows from the fact that F is the moment map for the symplectic action of the gauge group on the space of connections. The above gives a rough idea on how to construct the Hilbert space. We will give rigorous construction of the Hilbert space in the following section. There is an

31

Pre-quorzii+ziition

importan issue of how the Iiilbert space varies as complex structures varies. We shall address $his problem in section 3.5.

3.3

3.3.1

Prc-quantization

M as a c n m p l e z v a r i e t y

'l'he moduli space is

M = IIom(.rrl ( Z ) . G)/G.

+

For each A E M . define its covariant derivative d,t = d -4.Since d i = 0 for a flal connectiori, it induces a cohorriogy on de Rham complexes. Gauge fields are then better expressed in terms of cohornology classes. 'l'he tangent space of M is T M = iYjA(E, E @ g). It's dimension is

dirriM = dim H i (C;E @I g ) = (2y - 2) din1 G. Pick a complex structure J on C, we make ;U a complex variety. If we express connection A on C as -4 = A,dz AZdZ,V = V ( l ~ o+) V('>lI7V(l>O)f= Ci g d z i , V(':')f = Ci then we see t h a t the comis nothing but claiming dzi the holomorphic part and plex structure on d i i the anti-holomorphicpart. J induces a * operator on f2'(C, E @ g ) ,* 2 = I d . Define 1 : T M +, Icy = - * a . I is an integrable complex structure. The symplectic structure w is compatible with the complex structure J , w ( o ! 10) = -JETr(a A * a ) 2 0 and g ( a , p ) = u(cy,IP) is a Kahler metric ( u ( a I, @ )= u((Ia,fl))with the Kahler form w E Q 1 > ' ( M ) . There is also another explicit description of M as a complex manifold by a theorem of Narasimhan and Seshadri [Narasimhan-Seshadri]. Their theorem states that a holomorphic vector bundle E on a compact Riemann surface C is stable if and only if it arises from a unitary flat connection. Here stability means that for each holomorphic sub-bundle U c E,

gd&,

degU rank li

+

degE

<- rank E '

Hence the moduli space is the space of holomorphic bundles over C with respect t o a choosen complex structure. This space has a natural

Quantization of Chern-Simons Action

32

complex structure. The above theorem of Narasimhan and Seshadri has been generalized to Kahler manifolds in high dimensions by DonaldsonUhlenbeck-Yau [Uhlenbeck-Yau] . The above results can be viewed as an infinite dimensional version of geometric invariant theory. It is very interesting that even the existence of Calabi-Yau metrics also admits an interpretation of geometric invariant theory, see [Donaldson]. 3.3.2

Quillen’s determinant bundle o n M and the Laplacian

In geometric quantization, the next step is to construct a line bundle L -+ M . This is given by Quillen’s determinant bundle CA = det dA . Let M = Hom(nl(C,G))/G,A E M be a flat connection on a bundle E over a Riemann surface C,ra nk(E) = r , d e g E = d . The determinant bundle is defined as a bundle C over M . At A , we define the fiber to be

where X(V) is the highest exterior power of V . Let d~ : flo?O(E) + S2O>’(E),fl0,O(E)= kcrdA@Vo, S2°>1(E) = cokerdA@ V,. The Laplaciari can be defined as A = dAd1 didA : VO+ VO.It is natural to define det A = X i = ( Lrt, us consider c ( s ) = Cn then exp(-C’(O)) = A., Quillen also defines a metric ([Quillen])

n

&,

m)2.

+

n,,

He shows that the curvature of the determinant bundle with respect to this metric is equal to the Kiihler form on M .

3.4

3.4.1

Some Lie groups G = R.

The moduli space is

33

SDme Lie groups

M E = Hom(nl(C),R) = H’JC, R) with the standard quadratic form w . It can be quantized by the usual methods as described in chapter 1. 3.4.2

G = S1= R/2xZ

The moduli space is

.,btgl = Hnm(xl(C),S’) = R2”/ZZg= Jac(C). It is the Jacobi variety. Given a complex structure J an C, it induces a complex structure on Jac(C). It is actually given by a basis of holomorphic differentials w 1 , w 2 , ...>ug and Jac(C) = Cg/Z’g is represented by

divided by its periods

where cul! ...,ag;/?I, .,.,figis a basis of HI@,2). To quantize Jac(E:),we first const,ruct a line bundle L -+ Jac(C) with a natural connection A on C whose curvature is --cJ, i . e . , d A = -w Here w is a 2-form on H 1 ( C !R),the tangent space of the moduli space I

Jam!

Note also that w E H’l’(Jac, Z )

-+ H2(Jac,R)

is the first Chern class of C. To quantize Jac(Z), we consider the Hilbert space

Qunntizaliura o j Claern-Sirnoras Action

34

= Space of level r 0 - functions.

In H 1 ( C ,R), pick a set of"a-cycles", ul,..., ag a set of non-intersecting simple closed curves representing H1 ( C ,R) and its dual basis "b-cycles", bl , . . ., b,. We can choose a basis of holomorphic differentials w1 , .. ., w,, such that

s)'

3

- fi. . - 'J.

And 0 function has the following form:

3.4.3

T*G

In this case the moduli space is

M:'

= Horn(?rl(C),?"G)/G,

= T*(Hoin(nl(C), G)/G) = T * ( M g ) . As we discussed earlier, the Hilbert, space is

C o m p a c t Lie groups, G = S U ( 2 )

35

The corresponding topological quantum field theory gives Reiderrieister torsion and q-invarianl. The Super T ' G theory gives Cassori irivariaril [Witten-Casson].

Compact Lie groups, G = S U ( 2 )

3.5

Now we construct geometric quantization of Chern-Simons theory for compact Lie groupsi e.g. S U ( 2 ) . We will describe the Hilbert space for small genus and/or with very few punctures and calculate their dimension. This is very useful l o explain same key fact,s for Jones polynomials.

3.5.1

Genu.9 o n e

Consider a simple closed curve 7 in T2 and aftcr we collapse it we have a sphere with a doublc point. If we forget about t,he double point we have

CPI. Let

t

+ CP'

be a plane biindle, we have q ( f )= 0 So ~=L~+C-~,degL=l.

Now let CP1 = (CP1 - (0)) U (CP1 - {cm})= U 1U U 2 . In take

U1 nU2 we

f=yk ) 0

Ot - k

as the transition map, this defines a bundle on C P 1 . We will construct a linear map 4 : cpl -+ cpz to identify them. Up t o conjugacy we have

So the moduli space is

M:'

= T / W = C'/Zz,

where T c G is a maximal torus and W is the Weyl group. To quantize it, the Hilbert space is

Qunritization of Chern-Simons Action

36

C" (Y'/W).

It is generatted by t,he following basis:

fl

/3

f4

= 1,

= u 2 + 1+ a - 2 ,

= u3 + u2 + u-2

+ u-3,

where fn = {characters of the n dimensional representations of S U ( 2 ) and the first n+1 functions have order of pole 5 n at a}. So we have

'HZ = {the first k+1 characters of S U ( 2 ) ) . In other words, quantization for a given k is equivalent to form a space consisting of meromorphic functions on T / W with poles of order 5 k and it is equivalent t o form the space of representations of S U ( 2 ) of dimension

Fk+1. We have constructed a map such that for a choice of a cycle a and a representation R of S U ( 2 ) of dimension 5 k 1, we have $ ( , , R ) E 3 t ~ .

+

3.5.2

Riemann sphere with punctures

In the study of representation of compact Lie groups, there is an interesting geometric construction by the Borel-Weil-Bott theorem. It is in the same spirit of quantization. They found that the space G / T is naturally a

Gornpacl Lie groups, G = S U ( 2 )

37

syrnplectic space, where T c C; is the maximal torus, with a riaturd syrnylectic structure a . They also found that G / T is isomorphic to G c / B c , the complexifid group module a B o d subgroup, which is sirnilar as we have seen in geometric invariant Iheory. They constructed a natural line bundle whose first Chern class is w . Then m e can form a Hilbert spare, V = Ho(G/T,Lk),the space of holomorphic sections of the line bundle, 'The group C: aclv as right translation and it also acts on the space of sect,ions. This gives a linear representation of G. They showed that all irreducible representation arise from such a construction. This is a very useful fact for us iri the quantization of Chern-Sirnoris theory. Now we consider the case of Riemsnri sphere with several punctures. At each puncture Pi, we associate a repreverilation Ri. To each R+,there is a definite conjugate class ui E G. The moduli space is:

i u ( G , k ; P , : R ,= ) Hom(Ki(C - U P i ) , G)/G

with monodromy around Pi in conjugacy class ui. There is a similar result of Narasiham-Seshedri: h f C , k , C , p , , R ,= { c ] holomorphic G'c bundle un E , plus a reduction of structure group of c to B c at each Pi} Then the Hilbert space is:

The last equality is froin the above Bod-Weil-Hott theorem. Let us consider several examples. We want to determine the dimension O f a ( X , P , , R 1 , , P , , R , ) for small s. a)s=O

There is only one stable bundle c + S2 M(E) is a point consisting of the trivial representation, And dirnI'(pt,C) = 1. So d i r n 3 1 ~= 1. b)s=l

%(E; P, Q ) = RG

Quantization of Chcm-Simons Action

38

So dimX = I , if R is trivial, and 0 otherwise. c)s=2

dimX((C;P I ,R1, P2, R2) = dirn(R1 x R2)(: = 1,

if fiz = dual of RI and 0 otherwise. d)s=3 It is the conformal block N$ = (& e) s = 4

@

Rj @ Rk)G.

dim%(C; Pl,R,,P,,R.z,P3, R.3,P4, R4) = d i m ( R @ RcS,R @ R)G = 2 .

Here R is t,he fundamental representation of S U ( N ) . It can be seen as follows. There is a canonical dccornposition of R @ R = El @ E z ,El is the symrnctrjc part, generated by + ( e i @ e j e j @ e i ) , E2 is the anti-symmetric part, generated by i ( e i 8 e j - e j @ e i ) . So wc have R @ R @ R @ = (El @ E2) @ (El @ E 2 ) . The SU(2) invariant subspace is then generated by El 8 E l and El @ E l , hence it has dimension 2. This is the key t o explain skein relations in knot polynomials.

+

3.5.3

Higher genus R i e m a n n surface

Now we consider the case of higher genus Riemann surface. We want t o quantize M g a t or what amounts t o be the same, to quantize Mchol. Let us first consider the case of genus two surface. We choose 3 simple closed curves on C. After we collapse them we get two copies of three punctured spheres. A bundle E over C is then decomposed into two bundles S and 7 . We think of gluing them together with maps 4i : Sp, + TQ,.For q5i we also have equivalent relation 4i with u4iv, u,w E SL(2, C). Roughly speaking, the Hilbert space is:

= r(G x G x G,L)GLxGR

Compact

Lie

groups,

G = SU(2)

39

Holomorphic function of G c as a representation of GL x GR is

Here Ri runs over all equivalent classes of finite representations of G. And we have

( R@~Rk) 8 Hol Fun(G1 x G2 x G3) = @ R , , R ~ , RC3~ Rj

(Ri@ R j @ R k )

and we want it be GL x GR invariant. The Hilbert space for the case of three punctured sphere Co is 3 1 ~ ,= (Ri 8 Rj @ Rk)G = N i j k . For genus two surface we have

-$..

-

rjk

N Sjk ..

@J Niji;..

+

The sum runs over all first Ic 1 representalioris of S U ( 2 ) . For higher genus surface, we decompose the surface C into three punctured spheres by using a maximal set of simple closed ciirves. For each decomposition of the Riemann siirface, there is a dual graph I’. We assign to each edge c a weight f ( c ) E Pk = (0, 1/2, ..., k/2}. I t obeys three conditions: 1)lf(.1) - f(cz)l 2 f ( C 3 ) 5 f ( C 1 ) -tf ( C z ) , 2)f(c1) f(c2) f ( C 3 ) E z, 3)f(Cl) f(C2) f(c3) I k. Each admissible weight is an clement of basis of the Hilbert space.

+ +

3.5.4

+ +

Relation with WZW model and conformal field theory

We will explain briefly that the quantization of Chern-Sirnons over a disk led to a natural connection with WZW model and conformal field theory. We will see that the moduli space consists of loop group and hence it is naturally connected with rcyrcsentations of affinc Lie algebra.

40

Quariliration of Chern-Simons Action

Now the surface is a disk. The moduli space is the space of flat connections. The gauge group consists of gauge transformations which is an ident,ity on t,he boundary. Since the disk is simply connected, we can find a gniige transformation U such that after applying tjhe gauge transfornlation we have the connection is zero. So we have A' = lJAU-l+dUU-l = 0. Arid t,he connection can be represented as A; = --lJ-'&W, for some U : D + G. The moduli space for the case of disk is then the loop group module the G action by conjugation, i.e. M = L G / G . 'I'here is a canoriical syrriplectic structure coming from its representation as a flat connection, w = JTr(cr A p). One can also construct, a line bundle L over LGIG and a coririection on this bundle whose curvature is w . To quantize, we have the IIilbert space 31 = r h o l ( L G I G ,Lk)).LGacts on LG/G by right translation. IIence this gives a projective representation of the loop group. One might wonder why a version of infinite dimensional Borel-WeilBot,t t,heorem gives many interesting representat,ions of loop groups as well its affine Lie algebras. 'l'he Chern-Simoris acliori reduces to:

It is the chiral Wess-Zumino-Witten (WZW) action. The WZW model is a physical model naturally connected with conformal field theory. We thus suggest a relation between Chern-Simons theory in the bulk with the conformal field theory (CFT) in the boundary. In the case of gauge group G = SL(2,R),SL(2,C ) , it gives AdS/CFT duality, amid the recent proposal in gauge theory/string duality. One should note that the heart of the problem is to prove that the quantization is independent of complex structures. Witten established this for complex gauge groups[~itten-non-compact].

3.6

Independence of complex structures

Since we are aiming at a topological Chern-Simons theory, it is important to see that the quantization of Chern-Simons theory only depends on topological data. In geometric quantization we first choose a complex structure J on C. The space of connections becomes a complex space with ( 1 , O ) part

Independence of complex structures

41

of a connection considered as holomorphic. In the above we construct the Hilbert space by using holomorphic quantization. Kow we shall consider how the Hilbert space varies as compIex structure varies. It is a vector bundle over the space of complex structures. We shall construct a projectively flat connection for this bundle. The space of connections is an infinite dimensional affine space. It is bet,ter to consider quantization of finite dimensional space first. Let ( A ,w ) be a symplectic affine space with w a syrnplectic form. As we have seen in Chapter One we may quantize it by using holomorphic method. To do this we choose a complex structure on A . For example in choosing standard complex structure with standard symplectic form we ma?; write w = Cdpi A dqi = Cdzi A d & . We see this gives a different polarization of the symplectic form. There is an natural line bundle L + A whose first Chern class is w . The Hilbert space is then the space of holomorphic sections, rh01(4, L k ) . It is a vector bundle over the space of complex structures on A. A complex structure is also an almost complex structure J : TM + T M ,J 2 = - I d . Let 65 be a small deformation of J , then we have ( J 6J)2 = - I d . This is J6J 6 d d = 0. Let 6J be a deformation of complex structure J . The following is a connection on the Hilbert s p x c vector bundle.

+

+

[Vi! Vj] = lCwi3: [V;, Vj] = 0, [V;, V,] = 0. It is importmit to note thal: 1) 6% preserves holornorphicity. This can be verified by showing that it. commute with complex derivatives A;. 2) = central. This means that it is a projectively flat connection. Next we consider more interesting case of an afirie symplectic space wilh a Lie group G action which preserves the symplectic form. We then have symplcctic reduction and the symplectic quotient is M = p-'(Oj/G, where p is the moment map. In the c a x of a finite dimensional affine space we can push down objects like symplectic farm, complex structure, natural line bundle, connection etc. from A to M . In other words thosc objects

42

Quentiratdon of Chern-Simona Action

can be constructed equivariantly. And in particular we have the projective flat connection for the HilberL space bundle over complex structures. For the case of infinite dimensional ailinc space of connections we may du similar things except that the big IIilbert spaces we met are often ill defined. We have a very large gauge group acting on the space of connections. And the symplectic quotient is finite dimensional. It would be nice if we could push objects interested to u s down from the infinite dimensional affine space to objects on the symplectic quotiens. .4nd this was done in [AxelrodDella~ietra-Witten]. We first consider the case of U ( l ) gauge group. In this case the space of sections are nothing but 0 functions. We write 0 functions in the form:

It sat isfics :

In other words, 0, are covariant constant sections of a connection over the complex structures on RZn= GL(2n;R)/GL(n; C ) . This implies prcjective flatness of the bundle of the Hilbert space over the space of complex structures. There is another case where we can write down the projective flat connection explicitely. In the case of a punctured sphere, the projective flat connection is given by the Knizhnik-Zamolodchikov equations:

where Q E R1 @ , . . 18R, c I'(M,Lk) is a wave form. All differential operators acting on @ are commutative. In Sec. 4.1.1 we will have more explanat ions. Now we outline our treatment for the general case.

I n d e p e n d e n c e of complex structures

43

Recall in Sec. 3.3 we constructed a natural line bundle L over the symplectic quotient as Quillen's determinant bundle. Lct V be the connection corning from Quillen metric whose curvature is w , i.e. VSVjs - V j V i s = k w i j s , where s is a section. Recall that the Hilbert space is

Formally we have the projective flat connection as

Since we have a large gauge group acting on the space of affine connections we nccd to push down the connection into a connection 011 a line bundle on A/Gc. For a nice derivation of the above connection and a complete proof that the connection is projectivcly flat please see [AxelrodDcllaPictra-Witten]. We shall write down thcir formular presently. Hefore doing that we shall explain how to describe deforrriations of a complex structure in our case. The tangent space of the moduli space is

T,M = H i ( C ;E

@ g)

= T"' @ To)*

An infinitesimal deformation of the complex structure I is

I : T O J+ T110,I I + I

j = 0.

The deformation is given by

G:

3 Tor1 -f, T1'0,

G(cu,cu)= J,Tr(cuA * a ) , T r a 2 E H o ( C ;K 2 ) is a holomorphic quadratic form. In other words, 6 J = Gijwji& @ dzk. Finally the quantum connection is:

xi,j,k

1

6* = 6 + -(ViSJijVj 2t

+ 6 J i j ( V i l o g H ) V j + -21-k *k*+ h J l o g H ) ,

44

Quantization of Chern-Simons Action

where 6 is the derivation of complex structures, H = log A and Witten et a1 proved that S* is a projective flat connection. One may recognize that the first term in the bracket is similar to the one in the finite dimensional case. The rest of the terms in the bracket are due to anomalies. Remark: Given the above projectively flat connection the Hilbert space can be constructed as the space of covariant constant sections of the Hilbert space bundle over the space of complex structures. And mapping class group acts on this Hilbert space and this gives a projective representations of the mapping class group. We shall describe this action in Chapter 4. Remark: There is a nice relation between Chern-Simons-Witten theory and WessZumino-Witten theory because that they share the same form of projective flat connections. In WZW theory the connection has a nice physical origin, that is, it comes from the energy-momemtum tensor. For a nice treatment of WZW model and its relation with Chern-Simons theory please see one of our appendixes.

3.7 Borel-Weil-Bott theorem of representation of Lie groups Let G be a compact corinectrd Lie group. A linear representation of G is a homornorphism 4 : G -+ G L ( V ) with V as a vector space. If there is no invariarit subspare except 0 and V then the representation is called irreducible. For examplc, S U ( N ) acts on the space of homogenous polynomials of N variables of a fixed degree naturally and this gives an irreducible representation of S U ( N ) . For each representation, the trace of 4(g), called character or class function, is invariant under conjugation. It is the fundamental invariant of represent a t ions. For a compact connected Lie group G we have a maximal torus T c G. The conjugacy class of G is the same as T / W , where W = N ( G , T ) / T is the Weyl group, N ( G ,T ) = { g J g T = Tg}. Since T is commut,ative, the representation of T is trivial. It is the direct sum of one-dimensional representations. SO we have 4 1 ~ : T -+ GL(V),q517 = 41 @ ... @ &,q5, are one-dimensional representations. A one-dimensional representation is just a multiplication by a number. If we take the derivative to the representation we would have a linear functional defined on the sub-Lie algebra corresponding to ?‘,dq5i E h* : h -+ R1. We call {d&, ...,d&} c h* the weight system of 4. In the case of adjoint

Borel- Weal-Bott theorem of representation of Lie g m u p s

45

+

representation Ad = dad : C: 4 C:L(g),ad : g h,gh-l,,q, h E C , the wcight system o f A d ( G ) @ C is called the root system, A(G). Let f be a class function on G, i.e. its value only depends on conjugacy classes of GI then we have Weyl's integral formula:

whcrc wc take the Haar measure on G and 1WI the number of Weyl charnbcrs, sign(a)e2"i("(d)iH), t = ExpH, H E h,

Q(ExpH) = UEW

s=

fc

uEA+(C) a , A+ is the set of positive roots. The lQ(l)lz factor is obtained from the consideration of Jacobians. Based on the formula above one can derive the following fundamental Weyl's character formula. Again let 4 : G -+ G L ( V ) be an irreducible representation of G. We then have the highest weight A4 whose multiplicity can be shown to be one. Weyl's character formula is then:

where H E h, c E W = N ( G , T ) / T , d = f CCYEA+(G) a, ( a ( b ) H , ) is the natural pairing. In particular, this implies that the highest weight determines all characters and hence the representation. It is remarkable that all irreducible representations of a compact connected Lie group can be constructed geometrically from the work of BorelWeil-Bott. We outline it as follows. Again let T c G be a maximal torus. It can be shown that G / T is a natural symplectic variety. G / T is also a complex variety whose complex structure comes from G c / B c , B c the Borel subgroup. One can construct a natural line bundle over G/T whose first Chern class agrees with the natural symplectic form. G acts on G / T by translation and hence on the space of cohomology of sections of the line bundle. This induces a linear representation of G. More precisely] we have the following. Let X E h* be an integral valued linear functional.

Q u a n t i z a t i o n of Chern-Sirnons A c t i o n

46

a) If

< X+6,a >= 0, for some a E A, then

H a i k ( G / T ,C x ) = 0 , for any

k.

b) If < X+6, a ># 0, for all a E A . Let q = # { aE At I < X+6, a >< 0). Choose w E W with w(X 6 ) dominant. Put p = w(X 6) - 6:then

+

+

H f l i k ( G / T ,C ) = 0 , k

# q , o r F H k,

= q.

F p is then a finite dimensional irreducible represcntat>ionof G with the highest weight p . For more details, see [Knapp-Vogan].

Chapter 4

Chern-Sirnons-Witten Theory and Three Manifold Invariant

4.1

Representation of mapping class group and three manifold invariant

In quanhirig Chern-Sinions llieury we have constructed a Hilbrrt space vector bundle I'iol(M,C K )+ Teich over the spscc of complcx structures on C. We explained Ilia1 there exists a projcctivc flat conncction of the above vedor bundle The above vector bundle is also a bundle over the moduli space of complex structures, i.e. the quotient space of Teichmuller space module the action of the mapping class group. The mapping class group of C is the group of outer automorphisms of r l ( C ) or which amounts to be the same, the set of different choices of generators of r l ( C ) . It acts on the space of complex structures. With the above projective flat connection over this space, we consider the holonomy of the projective flat connection. This gives a projective representation of the mapping class group. One can use this t o define knot invariant and three manifold invariant. In the case of punctured sphere, the connection is given by the KnizhikZamolodchikov equations. Tsuchiya and Kanie [Tsuchiya-Kanie] evaluated the monodromy of Knizhik-Zamolodchikov equations which gives Jones polynomials. Moore and Seiberg [?] used polynomial equations to define the representation of mapping class groups. Kohno [Kohno] used their representation to define topological invariant of three manifold which gives another realization of Witten's invariant of three manifold. The first rigorous realization of Witten's invariant was given by Reshetikhin-Turaev 47

Chern-Sirnons- Witten Theory and Three Manifold Invariant

48

[Reshetikhin-Turaev] using quantum groups. 4.1.1

Knixhik-Zamolodchikov equations and conformal blocks

As we explained in the last chapter, the Hilbert space for a punctured sphere is simply the tensor products, vj, @ vj, @ ... @ Vj, , of representations. At each puncture Pk we assoiciate one spin jk representation Vj, of sl(2,C) which is an irreducible representation of dimension 2 j 1 , 0 5 jk 5 $, 1 5 k< - n , Ii' is a fixed integer. We know that it is a vector bundle of complex structures over the punctured sphere. In this case it is simply Cn - A , where A = { ( z I , z 2 , ..., zn)}. So we have a trivial vector bundle

+

Vj, 8 vj,

@I ... @

Vj, + Cn - A.

The projective flat connection can be explicitly described in this case. Let { I p } be an orthonomal basis of s l ( 2 , C ) with respect t o the CartanKilling form. Let

is the projection from I$,@ Vj, @ ... @ Vjv, to The Knizhik-Zamolodchikov equations are:

where

l3@

:=O,i=

8 zi

5,.

l , 2,..., n ,

whcre @ is a section of the Hilbert space vector bundle. The fundamental fact is that the left hand first order differential operators are commutative. Hence it gives a projective flat connection. The commutativity follows from the following two facts about Rij,

[nab,n,d]

= 0, u , b , c, d distinct,

Reyrese7itation of mapping class group a n d three manifold invariant

[R,,,

flab

+

49

= 0, a , b , c distinct.

Hence we have an integrable connection

The solutions of Knizhik-Zamolodchikov equations are vectors of the Hilbert space. They are also horizontal sections of the Hilbert space vector bundle. To find a solution is to evaluate the holonorriy of the above connect,ion along a loop. Such solutions are given by iterated path integrals by K. T. Chen [Chen]. We may take advantage of the existence of the flat connection by decomposing the surface into simple pieces. Let ( a 1 ,..., azg-2) be a set of simple closed curves such that C - { a l ,...,azg-2} consisting of pants, i.e. a punctured spheres with three boundary components. We may take the complex structure to be nearly degenerate along the simple closed curves and then the Hilbert space will be the tensor products of those Hilbert spaces of pants which we called conformal blocks, Let C be a pant decomposition. We associate a dual graph y t o it. To each pant we associate a vcrtex and t,o each simple closed curve we associate an edge. Let K > 0 be a positive integer. We construct the Hilbert space X K ( Y ) as follows. It consists of basis

f : edge -+ { 0,1/2, ..., K / 2 } = P K , which satisfies the following conditions:

50

Ghern-Simons- Witten Theory and Three Manifold Invariant

Fig. 4.1 Four punctured sphere

One may think of f ( c ) as the weight of the associating irreducible representation of the gauge group to a simple close curve on the surface. The first two conditions are Clebsch-Gordon conditions. Let j , , j , be weights of representations V,,, Q 2 , we can decompose the tensor product Ql p into irreducible represeutations $j then j , , j , , j must satisfy the first two conditions.

4,

4.1.2

4,

Braiding and fusing matrices

We indicated above that h e IIilbert space for general surfaces can be constructed from conformal blocks. The action of mapping class group also localize to those action on simple pieces. In the rollowing we will analyse those elementary objects. Let us first consider the case of four punctured sphere. Choose a simple closed curve to separate the four punctured sphere into two pieces of three punctured spheres. In graphic representation, Fig. 4.1 illustrates a four punctured sphere and the dual graph of a decomposition of the surface into pants. At each edgc we associate a representation of the gauge group. We have two choices of such separating simple closed curves. For each choice we have a Hilbert space. There is a matrix to connect those two Hilbert spwcs and we call this a fusing matrix, see Fig. 4.2. Algebraically m e have the expression.

Representation of mapping class group a n d three manifold i n v a r i a n t

Fig. 4.2

51

Fusing

Fig. 4.3 Braidirig

We may consider to exchange the two representations j and k of the dual graph. The map between the two Hilbert spaces is called a braiding matrix. See Fig. 4.3. Algebraically it is:

52

Chern-Simons- Witten T h e o r y and Three Manifold Invariant

Fig. 4.4

Yang-Baxter for braiding matriccs

Fig. 1.5 Yang-Baxter for fusing matrices

From the Knizhik-Zamoloddiikov equations, we can prove that braiding matrices and fusing matrices satisfies Yarig-Baxter properties, see Fig. 4.4, Fig. 4.5. Braiding rnatricrs and fusing matriccs are coririecled by:

Representation 01 m a p p i n g class group a n d three rnanijold a'nuarz'lant

V,, Q, 5,+ V , , then

53

we have PC!ai3 = ( - l ) j ' 2 + j : 3 - i C ' ! ' ' 3 . We have Aj = and it arises as ~ o l l o w s . Let w be the integrable I(ni&hik-Zamalodchikov connection. It acts naturally on the space Hom,r(2,C) g...@Vj,> T,,,,). Let d , l . . . i k= 0 be the exceptional divisor corresponding to the inverse image of the subspace of C" defiried by zil = ... = z,,,. Let. CT,jbe a vector in the Hilberk space, then we have the residue of w on the exceptional divisor is:

w,

(KL

R e ~ w , , . . . , ~ = o C y ,= j A(i11

...j

ik)Cy.f,

'B.

k where A ( i i , . - . , & )= Aj(a,l,...,, k ) - C,=lhj(a,p),Aj = A ( i i ,...,i k )

Q 7 , f = I ~ J ~ ~ (C7,j . , , +higher ~ ~ order holomorphic terms) is a solution of Knizhik-Zamalodchikov equations. Next we consider a torus. The Hilbert space is given by Verlinde basis { vg v1/2, . . . I L S ~ , ~ where } , 'ui is an integrable highest weight representation ~

of the affine Lie algebra A i l : ' . Each such representation is determined by an irreducible representation of the underlying Lie algebra. There are two modular transformations:

C

Tvj = exp 2 x G j A j - ~ ) v j1 5 , i,j

24

5 n.

S,T are called switching operators. It gives a representation of the modular group S1,(2,Z) which is the mapping class group of the torus. 4.1.3

Pmjjcctive representation of m a p p i n g class g r o u p

With the construction above of representations of mapping class group of simple pieces we can construct represent atmiorisof rriappirig class group for general surfaces. This was first raa1iz.d by Moore arid Seiberg in [Moore and Seiberg]. Here we follow expositions of Kohno [Kohno]. Let C be a surface of genus g . Let { a i ,Si , &, c i } be a set of simple closed curves.

54

Chern-Simons- Witten T h e o r y and Three M n r ~ i f o l dInvariant

a2

bl

Fig. 4.6

b3

b2

A

surface

of genus g

We use t,hc same notation to denote Ilehn surgery along those curves. It is known that this set of Dehn surgeries generate the full mapping class group. The sct {a;, d,, F;} are disjoint simple closed curves. Thc dual graph for this set of simple closed curves is A. We will constmct a projective representation of the mapping clays group with respect to this dual graph. In this dual graph, {ail&, c i } correspond to edges. The action of mapping class group with respect to X is very easy, they just change the weight of the corresponding edge. For example, we have

TueY(p),f = exp2.rrJrri(A,(,, To describe actions of

l

we

C

-

G)t7(”),J.

have to use switching operators. /31

Pg acts

Rqrcsenlatdon

UJ

mapping class g ~ o u pand three manijdd irauuriurd

55

on the Hilbert space of once punctured sphere. We have ,& = T,, SblTal, where Bl acts on conformal blocks with dual graph I-1 f j b l ) = j l . For 15 k 5 g , we have ~

where b k - 1 , bk are two edges of the conformal blocks r2. Finally we reach the fundamental result of Moore and Seiberg. Let ra be the c y c k group generated by exp 2 n n ( c / 2 4 ) i d c G L ( Z K ( ~ ) ) and c = X Ithen the above construction of & i , &, ,g2,& gives a projective K+2 representation of the mapping class group p~ : M i G L ( r K ( T ) ) / T K , Here is an example of the representation of mapping class group. Let I< = 1,y be the dual graph of the following decompositions: f ( h )= f ( b 2 ) = ...f ( 6 , - 1 ) = O , f ( G k ) = f ( C k ) are admissible weights. So we have the vector space Z1(?) = brgg, dimc V = 2. We have:

56

Chern-Simons- Witten Theory and Three Manifold Invariant

Fig. 4.8

Twice punctured sphere

T = diag(1, i).

We have the representation of

p1 :

M,

-+GL(V@g)/R,.

Representation of mapping class group and three manifold invariant

57

c

Fig. 4.9

A decomposition of surface and its dual graph

PI("2)

wherc

wk,k+l

= w 2 r .-,Pl(a,) = Wg-1,gt

is the operation of w on k-th and (k

+ 1)-th

components of

V@g.

4.1.4

Three-dimensional manifold invar.iants via Heegard decomposition

Let M 3 be a three manifold. It admits a Heegard decomposition:

Chern-Simons- Wztten 'I'heory a n d Three Manifold I n v a r i a n t

58

where C

c.

c M 3 is an embedded surface such that 8 6

= dV2 = C, V, n V2 =

Taking another point of view, we may take a handlebody V l , its second copy V, and a homeorriorpliisrn h : 8Vl + a h . We lhen coristrucl a three manifold M 3 = V1 Uh Vz by identifying poinls in 8V1 wilh points in dVz via h , an element of the mapping class group of the surface SVl . For any triangulation of M 3 , consiclcr its dual arid their regular tabular neighborhood of their 1-skeleton, it gives a Heegard decomposition. For S3, consider the standard sphere S 2 , it gives a Heegard decomposition of the three sphere. From our previous work, we have constructed projective representations of the mapping class group:

P :M

, +GL(ZK(Y))/rK.

Let e7,0 be the vector in Z K ( 7 ) with admissible weight, f : edge(y) PI<,siich that f ( a ) = 0 for any a E edge(7). We have:

~rc(h)e,,o

+

+

P K ( ~ ) o o ~ ~C , oj + n P K ( h ) f , o e , , f .

From our previous work, for two differed decompositions of the surface y(p1),y(pz), we have Z ~ ( y ( p 1 )= ) Z ~ ( y ( p z ) and ) , the isomorphism send o not depend on the vector ey(pl),o t o e y ( p 2 ) , o . So we see that p ~ ( h ) o does

a choice of marking. Thcrc is an important operation on Heegard decomposit,ion. Let M = Vl u h V2 be a Hecgard decomposition of genus g and let S3 = D1 U, Dz be the standard decomposition of S3, where DIand D2 denote solid tori. Ry considering the connected sum of these IIeegard decomposition, we obtain a Heegard decomposition of M # S 3 = M . We deriote by fi : OVl #dD1 + aV2#aD2 the corresponding attaching homeomorphism. This gives a Heegard decomposition of genus g 1. We called this elementary Stabilization. It is known that Hcegard decompositions of a three manifold differ from each other by a numbcr of clcmentary stabilizations. It is easy to show that

+

Crrlctilatiuris by topuluyical quariturri field t h e o r y

59

This follows from

From thc above it is clear that,

$A,

( M 3 ) = SK (0);:PK

(h)oo

is independent of marking and elerncntary stabilization. Hence it is n topological invariant. It is a realization of Witteri's invariant by Kohno. 4.2

4.2.1

Calculations by topological quantum field theory Atiyah's axioms

M. Atiyah formulated axioms of topological quantum field theory for three dimensional manifold as follows: To each surface C c B M 3 , one associate a Hilbert spacc 'HE. To each corbodism a M = C1 u X 2 , one associate a morphism : ' H x , -+%c,. It satisfies the followingaxiom called nnturality. If c?Mf = C1UC2, d M t = Xz U .&, 4 1 : 'He, + X c , , 4 2 : X c , + Xc,, then for M 3 = M ; U c p M:, and ahf" = C1 U X3, we have $ : 'HE, + X c , , and 4 = $ 2 4 1 . This naturality reflects a crucial property of probability. The probability from an initial st,atc to a final s h t e is eqiinl to the sum of product of probability from the initial state to an immediak stjntjeand the probability of the immediate state to the final state. ti'eynrnan's path integral reflects just this property. $J

60

Chern-Simons- Witten Theory a n d Three Manifold I n v a r i a n t

< 411e-tH142 >=

s

2)(pL(d

+(0)€41,0(1)€0Z

H is the Hamiltonian, L is the Lagrangian. A key observation for ChernSimons theory is that the Hamiltonian is zero, so we only need t o consider the inner product in 3 1 ~ We . will introduce some applications assuming that Chern-Simons topological quantum field theory exists. In the next chapter we will explain how Chern-Simons perturbation series makes sense with respect to some regulizations. 4.2.2

A n example: connected sum

Let M I , M2 be two three manifold and M = Ml#M2 their connected sum. Applying the axiom of TQFT(Topologica1 Quantum Field Theory) to this case we have Z ( M ) Z ( S 3 )= Z ( M l ) Z ( M z ) .Or

Z(W - Z(Ml) Z ( M 2 ) ~~

z(s3)

z(s3)

qs3) .

R.cmarks : 1 ) If ml , m2 are two arbitary manifolds without, knots in it, then the above formula implies multiplicity of Reidemeister and Ray-Singer torsion under connected siims. 2) If we consider Chern-Simons theory with the non-compact gauge group S L ( 2 ,R), a leading term in Z ( M ) gives e'vol(M). Hence we have the formula vol(M) = vol(M1) vol(M2).

+

4.2.3

Jone.9 polynomials

Let I, = UC, c 5'" be a link, we define the partition function

Let C = S T l ( N ) and take thc representation to be the defining representation of G. We call Z ( L ) the Jones polynomial. The most remarkable property for Jones polynomials is the skcin relation. We will establish this relation. For a crossirig of the knot we imagine inserting a small sphere around the crossing. The spherc would intersect

Caiculations b y topo/ogical quantum jield theory

Fig. 4.10

61

Skein relation

with the knot at four points. The manifold itself can be representated as M = M L U M R with ~ M R the three ball and M L the rest of the three ball. MR and M L share the same boundary C = S2 - {.z~,z2,t3,z~}.Consider the Feymann path integral on M R , M L , we have vectors II; E ' H R , x E X L respectively. Notice that X L = a;.The partition function is then Z ( L ) = (X>$1. If we replace M R with two other different ways of strings, we would get two other vectors $1, $2. Since dirri'lfx = 2, we have the three vectors $ , $ I , & which satisfy a linear relation: ad+&1 +y$z = 0. If we niultiply the above by x,we then have a Z ( L ) P Z ( L 1 ) yZ(L2) = 0. This is called the skein relation. It is known that this relation is enough to compute knot invariant. Because we can project a knot into a plane and apply this relation, the number of crossings is decreased eventually. The novelty here is that we have achieved a three dimensional interpretation of skein relat,ion without projecting a knot into a plane. Invariants defined this way are intrinsic and natural.

+

4.2.4

+

Surgery

T h e construction of surgery is t h e follow~ing.Consider an embedded torus C in a manifold M ,the manifold can be represented as a union of the solid torus and the test of the manifold with common boundary E. We then consider an element p E S L ( Z , Z ) , and we glue the two manifolds

62

Chern-Simons- Witten Theory a n d Three Manifold Invariant

with boundary C after we apply for p to the torus, then we obtain a new manifold. This construction is called surgery. In Chapter 3, we calculated the dimension of the Hilbert space for a torus. Consider a simple closed curve on Ihe torus. After collapsing the curve we have twice punctured sphere. The Hilhert space is generated by irreducible representations. or level k intcgrable highest weight representations of the loop group. The mapping class group of the torus SL(2,Z) a c t s on the Hilbert space, K = p* : ?P l, We can express it as follows:

where u; is a set of basis of 'H. This is good enough to calculate the change of partition function with respect to surgery. For the loop group, we define the character of an integrable highest weight representation a as:

A

where X is a grading of the representation. Let u s consider S =

Its action on the characters is:

For the group G = S U ( 2 ) ,we have

Here are some interesting applications: 1) Partition function of the inanifold X x

Z(X x Sl)= T k , ,

5''

is

(1) = dim3lx

Calculations by topological q u a n t u m f i e l d theory

63

For exainple, we h a w Z(S2 x S1)= 1. 2) Consider a circle along the S1direction of the manifold S2 x S1and applying surgery S to this, one gets s”. By our surgery formula abovr, we have:

3) Consider three punctured sphere X = S2- ( z 1 , z2, zg}.X x S’can be considered as S2 x S’ with three unknotted Wilson lines. ‘I‘o each line, we associate a representation R;,R j , R k , then we have

4.2.5

Verlinde’s conjecture and its proof

Consider now two unknotted and unlinked knot Rj, Rk in S3. Consider anot,her knot Ri which linked Rj, Rk. We call thisconfiguration L(R;,Rj, &). By applying surgery formula to the similar configurat,ion in S2 x S1,we obtain

m

m

Consider cutting the link at, Ri into two pieces and applying the multiplicative formula, wc have

z(s3;L ( & , Rj, & ) ) So we have

- Ems ? N m j k .

sijSik =-

Let {vi} E

have

k

If we let wi = SO,^

SOi

c,Srwm, then we have

3tTa

be a basis. Then we

64

Ghern-Simons- Witten Theory and Three Manijoid Invariant

lm!f+w

m+G

Fig. 4.11 Algebras t o tangles

zu.2u. f 3

- 6.. 93wj.

So after we apply a unitary representation S t o {will we get a new basis. The Veerlinda algebra is diagonized with respect l o this new basis. 4.3

4.3.1

A brief survey on quantum group method

Algebraic representation of knot

To define knot invariant is to a large extent finding a good algebraic representation of knot. One of the ways is to use tangles. We refer to Sawin's paper for more details [Sawin]. il tangle is the image of a smooth embedding of a union of circles and intervals into the cylinder D x I , where D is Ihc unit disk in C and 1 = [D, I]. TVe want to associate algebras t o tangles. To each point of intersection in the line we associate a vector space V. To two points of intersection of the line we associate tensor product V @ W of the two vector spaces V and W . When we move the line the vector spaces changes so we will associate operators to the vector spaces. By a theorem of hlarkov, a knot can be representated as a braid glued at their two ends. A braid is an element of T I ( G- {q,.,,,Q)). If we want to define knot invariant via braid representation it has to be invariant uIider the following operation.

A brief survey a n quantum group method

Fig. 4.12

65

Braids

Define

i

to be the matrix of V @ V the Yang-Baxtcr relation:

+ V @ V . To respect

the above move, we have

where

i

i

i

A real problem arises fur the above algebraic representations. That is: given an associative algebra B, and two representations p 1 , p2 : B -+ End( V ) ,End( V ’ ) ,is there a natural way to define B terisur product of P I , p 2 , i.e. p : 3! + End(V @ V ’ ) ? A naive definition is p = p1 @ l + l@ p 2 . But it is only a representation as a group, not as an algebra. To cure this problem, we introduce t h e notion of Hopf algebra. Definition (Hopf algebra): A Hopf algebra B is an associative algebra with co-product A : B 3 B @ a, co-unit 6 : I3 3 C and an anti-pole y : B + B. It satisfies the following:

66

Chern-Simons- Witten Theory a n d Three Manifold Invariant

Fig. 4.13 Tangles

(A @ l ) A = (1 @ A ) A ,

(E

@ 1)A = 1 = (1 @ € ) A ,

( i d @y)A = c = (7 @ id)A.

For a Hopf algebra B, and two rcpresentatioris p1, p2 of a, then p = ( p l @ pz)A : B + End(V @ V ‘ ) is a represeritatiori of 0. Here are two examples of Hopf algebra. 1) Let G be a finite group, CC a group algebra. We define A(g) = g @ g , s ( g ) = s - ’ , 4 9 ) = 1. 2) Let U ( g ) be a universal enveloping algebra. We definc A ( z ) = z @J 1 163 2 ,€(X) = 0, y(z) = --2. A Hopf algebra staisfying Yang-Baxter relation is called a quasi-triangular Hopf algebra. We define it as follows. For the tangle in Fig. 4.13, we define uAA’: V @ W 3 W @ V , A ’ ( a )= uAA’(A(.)). Then a quasi-triangular Hopf algebra is a Hopf algebra which also satisfy the following:

+

A’(u) = R A ( a ) R - l ,

A brief survey

67

o n quantum group m e t h o d

It is just another expression of the Yang-Baxter relation. For any quasi-triangular Hopf algebra, one can define knot invariant associated to the algebra by taking the trace of the matrix generated by the braid element on the Hilbert vector space. 4.3.2

Hopf algebra and quantum groups

Let G be any Lie group, g its Lie algebra and U ( g ) be its universal enveloping algebra, i.e. all left invariant differential operators. A quantum group is a deformation of the universal enveloping algebra. Let us illustrate it by an example of SU (2). The universal enveloping algebra of S U ( 2 ) is generated by ( 2 ,y, h } , with relations:

[ h ,I. = 2 2 , [ h ,y] = -2y, ,.[ y] = h. The corresponding Hopf algebra is given by:

A(u) = u @I 1

+ 1 @ a , ~ ( a=) 0, ~ ( a=) --a,

a E {Z,

Y,{A}.

Now we consider deformations of U ( g ) ,denoted as U , ( g ) .The relation is modified as:

(p- ,--ah) [h,]. = 25, [ h ,y] = -2y, [x,y] =

s2

- s-2

,

The corresponding Hopf algebra is defined as: 2 2 A(z) = ~ @ s ’ + s - ~ @ x , A ( y=) y @ ~ ~ + s - ~ @ I y , s = ( a-S) .,s(y) = - S y.

The most inlercsting oncs are those s at, the root of unity, s = e x p ( 5 ) . It gives modular Iiopf algebra. T,et [ k ] = p - 3 - k If { e o , e l , ..., eN-1} gives an N- dimenslorial representat,ion of sl(2, C ) ,then we have N-dimensional representation ofUP(s1(2,C ) )

bY

Chern-Sirnons- Witten Theory arid T h r e e Manifold Invariant

68

where X, Y , A' generated sl(2, C) and is deformed into

Quantum groups can be used to construct quasi-triangular Hopf algebras. One can compute the action of R on V @J W , the composing flip map gives a matrix [Kauffrnan]:

4.3.3

Chcrn-Simons theory and quantum groups

Why docs quantum group exist? Witten showed thal its existence follows from the existence of Chern-Sirnons topological quantum field theory. We give a brief introduction here, for more details please see Witten's paper "Gauge theories, vertex models, and quantum groups", Nuclear Physics

B330 (1990) 285-346. Let us consider the group G = SU(2). In SU(2) Chern-Simons gauge theory, the physical Hilbert space ? I ! ~ , A , A , u + ~of S2 with four charges in representations U , A , A and U 1 is two-dimensional. Proof: We decompose U x A = $Ri, Ri = U - 1,U and (U 1), A 8 (V 1) = @Rj,Rj = u, (U 1) and (U 2) .U, (U 1) of Ri concide with that of Ri. This implies that dim?I!flv,A,A,(U+l) = 2. Since it is two dimensional, any three vectors obey a linear relation. This is the key t o construct an algebra A over C , generated by symbols T- , To and T+ with the following relations. It can be considered as quantum deformation of the S U ( 2 ) Lie algebra.

+

+

+

T-T+

+

+

+ vT+T- + v'T: + v"T0 = 0,

+

A braef survey

T+TL

on

quantum g r o u p m e t h o d

69

+ wT07; + WIT+= 0

The parameter p in the quantum group is identified as q = e x p ( z ) , is the coupling constant of Chcrn-Simons theory. Amon,0' other where things, Witten established that: 1) For every nun-negative integer of half-integer j , the spin-j representation of S U ( 2 ) deform to a representation of the algcbra A. 2) There is a natural tensor product of representation of A , and there is an "R-matrix" relating tciisor products j , Z j , and j z @ j , .

Chapter 5

Renormalized Perturbation Series of Chern-Simons-Witten Theory

5.1

Path integral arid morphism of Hilbert spaces

In our previous work of constructing lopological quantum field theory, we associated a Hilbert space, to each boundary component of a three manifold M 3 . Lel 8Mf = CI U C2, We also need to conslruct, a morphism:

dx, : XEl + X C , . According to Atiyah, such morphisms should satisfy the composition law,

4x = dx,4x, > where dx2 : %c, + YE, is a morphism associated with a manifold M: with boundary C2 U C3 and 4~ : X c , -+X c , is the morphism associated with the manifold M 3 glued from M: and MZ at their common boundary C2. We will construct such morphisms through path integrals.

5.1.1

One-dimensional quantum field theory

In this case we have the following axioms: 1) For every point we associate a Hilbert space:

71

Renormalired Perturbation Series

72

UJ

Chern-Simons- Wittera T h e o r y

2) For every one-dimensional manifold with Riemannian structure we associate a morphism: $hIJ :

H +8.

The morphism satisfies the composition law:

#JI,tl+t2

= $hl,tldI:ta.

So there exists an operator H on %, such that

#I,t

=e

-tH

,

II is called the Hamiltonian. If the one-dimensional manifold is a circle then we have the morphism as;

4 S , t : 31$ --$ %@. In this case 31d = C. From the axioms we have that:

5.1.2

Schroedinger operator

We now consider a special case. Consider the space R2n = T * ( R nwith ) the standard symplectic formw = Cdp'Adx'. The Hilbert space is31 = L2(Rn). We have operator representations xi by multiplication, and p j by - z & .

This can be generalized to any cotangent bundle of a manifold M . The Hamiitonian operator would be -A2 V on M . One-dimensional quantum field theory is essentially the theory of elliptic operators on manifolds. Now we want t o construct morphisms via Feynman path integral. To understand e - t H is to understand its kernel. Given we define the kernel as follows:

+

$(XI,

Path integral and morphism of Hilbert spaces

=

(e-”*)(z”)

1

d Z 1 K ( X t 1x , /;t)*(z‘).

If we use distributive ”wave” functions, e.g. /I

K(Z

12’

>= S ( 3 : - z’),

/

, z , ; t ) = (z”, e - t % ’ ) .

Here are more identities:

I

I1

I+’ >= S(z - 2 ), Iz >= S ( x - X I 1 ) , V(3:)1z1>= V ( X ) d ( X - X I ) = V ( X 1 ) 6 ( Z- z/),

According to Fourier ,

73

then

74

Renormalised Perturbation Series of Chern-Simons- Witten T h e o r y

According to FeyImiaIl, if we take N integral over the space of paths /I

I

Nt(x x ) = {

+ 00,

the integral becomes a.n

maps z : [0, 11 -+R9a1 s.t. z(0) = x’, z(i) = z”}.

We can also consider I h e space:

Path integral and inorphisin of IIdlbcrt spaces

I,

75

I

Ct(z , z ) = { maps ( z , p ) : [0,1] t R.2n, s.t. ~ ( 0 = ) X I , ~ ( t=)2"). /I

I

The integral over C t ( z , x ) gives phase space version:

5.1.3

Spectrum and determinant

Let us consider the space knew its spectrum to be

e

R2 = T*(R')]V = Lz2 2 ' H = 1( p 2

--t*H a

= Fourier trans.

+ x').

We

tS

~

7

,

We warit to calculate this integral. We knew the following formulae of Gaussian integral:

76

&normalized Pecturbaiion Sen'e$ of C'hern-Simons- Witten Theory

We consider operator A on pairs ( p ( l ) ,~ ( i ) ) , 1 -(pz 2

Here, A =

+ x2) - ip-dz = ( p , z ) A ( p ,x ) ~ ' dt

( lx' -I2 ) .

A has two-dimensional invariant subspaces

e v ( p r L x. , l t r . In such a subspace,

So formally, det :i = rIItaEz dct A,,

m= Jm = K%lP+ (7 ) ) 1

1

1

Znn 2

After Ray and Singer, given an operator 0 > 0 and S$n = A,&, formally (' ( s ) = C;Xl log A;, so define C ( s ) = EX';

we

det 8 = e-<'('). Given r E R + , define 0, = T ~ , < + . , s ( s )= E ( T X ~ )= - ~r - 9 < o ( s ) ,then

det(T0)) = exp(-(io(0)j

d

= exp(---(r-"u(s)))ls=0 dS

= do(')det 0 .

If 0 is a finite dimensional matrix on V . then b ( O ) = dim V, detjrO) = rdimdet ( 0 ) . Let 0 be an operator with det 8 = II,( I and consider 8 = ( F ) 2 0 ' then ,

+ (y)'):

d e t d = II,(n2 And in

< function regulation,

T + (-)')). 27r

77

Asymptotic expansion a n d Feynman diagrams

2ir det 0 = ( -)'cO(')

T

If II%

All

converges, then in

det 0'

C function,

And the operator 0" with eigenvalues n2 for n = 1 , 2 , ... has the

C function:

det 0'' = exp(-2C1~iemann (0)). 5.2

Asymptotic expansion and Feynman diagrams

In this section we illustrate calculations of asymptotic expansions for some integrals over a finite dimensional space. We show that it can bc described very well by Feynman diagrams. 5.2.1

Asymptotic expansion of integrals, finite dimensional case

Let X be a finite dimensional manifold equipped with n measure d p . Let f be a smooth function on X. We want to evaluate asymptotic expansion of integral I ( t ) in the following for t + 00,

Renormalized Perturbation Series of Chern-Simons- Witten Theory

78

CsignXi = q(A) = signature(A). Now let us consider more general cases. Let f : X + R be a smooth function with finitely many non-degenerate critical points Pa. At each P, we choose coordinates x;, such that P, has coordinates xr = 0, and

f ( 2 y ) = f(P,)

1 1 + --Cx;xixj + -cxiajk2ixjxk + ... 2! 3!

near P,. And we also wish to pick up coordinates x; so that we have the measure 1

dP = dx( 2...dx a)F

% '

Then for large t we have I ( t ) = X , P ( t ) as asymptotic expansion series in f . Here

I Q ( t )=

dx 1.. .dxn

exp it(f(P,)

with B" a small neighborhood of Pa. To evaluate I " @ ) , let yi = f i x * ,

1 + --CX~xixj + ...) 2!

A s y m p t o t i c expansion and F e y n m a n dingrnrlis

T h e intrgrals we nccd arc (me will omit

And

Hence

Note that

cy

subscript):

79

Renormalized Perturbation Series o j Chcm-Simons- Watlen Theory

80

so

After Feynman this can be conveniently described by Feynman diagrams. In the graph for each line put a factor of (X-')ij. At each vertex put a with N the number of factor of AGk and so on. The numerical factor is symmetries of that graph. In summary we have got the asymptotic expansion series

I(t)=

dpeitf

where C,,, are evaluated by Feynman diagrams.

A s y m p t o t i c expunsion a n d Feyninun diugrums

5.2.2

81

Integration on a sub-variety

Now we consider the case that the set of critical points be a sub-manifold, or a sub-variety in general. In this case G acts on the set of critical points X . Let us assume that G acts on X freely. Then we have d i m X = dim X / G dimG. Or in general let Xi be a component of X and Gi he the stabilizer of a generic x E X i , then d i m x i = dimXi/G d i m G - dimGi. And

+

+

d i m X - dim X,= dirn(X/G) - (dim(X,/G)

~

dirnGi).

Here dim(Xi/G) - dimGi = dim*(Xi/C:) is the formal dimension of X , / G , it can be positive or negative. The contribution of Xi is

We have the formulae for I ( t ) as follows.

dpi is the transverse measure of Xi/G of the orbit space of a component Xi. Let px be a measure on X , we want t o construct a measure ,UX/G on X / G . For this we need to pick up a section s : X/G -+ X . Let W = s ( X / G ) . The tangent bundle to W in X splits,

+

TXW = T ( W ) V(W). On a vector space R a measure is a vector x E ATOP(R). ATOP(T~ ( w ) )2 ATop(T(W)) x AToP(V(W)). The G action gives a map 4 : G -+ V, G = Lie(G). F'IX a measure on G,z E ATop(G), then d ( x ) E ATop(V) = p v . The desired measure pw is defined by px = pv x p w . Let s be defined locally by equations hl = ... = h , = 0 . Then

82

Renormulized f'erturbrrtiora Series of Chern-Sirraona- Willerr Theory

Here iJa is the left, invariant vector Geld on G, 2 E fiber over X / G . 'l'liere is a unique g such that gz E W = s ( X / C ) . Arid

Recall + 6 ( x ) = have

s-",

$ei@t.

I.&

6,be linear functions

on 4'. And we

For the materials above please see Faddeev's book for more details [Faddeev].

5.3

Partition function and topological invariants

Our goal is to get topological invariants for a thrcc manifold M', Lel G' be a compact Lie group, k E Z+. Let A E A be a connection arid A be the space of connections. We knew that the gauge group B acts on the space of connectioris. Wc denote A / 4 as thc space of gaugc equivalent conncctions. The Chern-Sirrions Lagrangian is

Because for two gauge equivalent connections C orlly differs by an integer, e x p ( i k C ) is well defined for gauge cquivalent conncctiorla. We define topological invariants Ihrough the partition h d i o n :

ZG,k( M ) =

L,,

D A exp ik.C.

Partition junction and topological invariants

83

It can also be generalized as follows: given a link L = UiCi, where Ci are knots with Ci n Cj = 4, i # j . For each i pick an irreducible representation Ri of G and we introduce ”holonomy”

W R , ( C ~= ) TrR,Pexp

s,.

A.

We then generalize the partition function to be

It does not depend on metrics on M 3 . It only depends on orientation of M 3 and on orientation of the link L for the path integral is integrated over the space of connections. The problem is how to calculate Z G , ~ ( M )We . first rescale the field A to normalize the quadratic term and the coupling constant is then changed t o $ l c - S . We now consider how to construct perturbation series of the above path intcgral in terms of k - 3 . Another method is to evaluate the integral via surgery explained in the last chapter.

5.3.1

Gauge

fizing and Faddeev-Popoa ghosts

Since the Chern-Simons Lagrangian is gauge invariant the integral is really over gauge equivalent, fields, i.e. over d/G. ‘Yo imitate t,his in gauge theory we need n local section:

To gel a scction we pick a metric on M . The rnctric induces a IIodge operator * : k forms + 3-k forms. Let A = Ao +I?, where A0 is a critical poiril of C, i.e. Ao is a flat connection. We only consider a simple case where we havc an isolated critical point, or it arriounts to be the same, H o = 11’ = 0. Then the scction is defined by requiring that L? obey the differential equation: D* B = 0. Here R = d+& is the covariant derivative. An infinitesimal gaugc transformation (Y transforms B into B - Da. The constraint is then satisfied by solving equation D * L)a = ll * B , or, * D *

Rera orrn alized Perturbot i u n Series of Ch e r n - S i r n a n s - W i t t e n ' I ' h e ~ r y

84

D a = * D * B . This gives a local section

s.

We have HRST transformation

laws:

SC" =

12

f&CbCC.

Here c is a zero form. The first iclcntitsy above expresses thc variation of a conneclion under an infinitesinal gauge transformation. The second equation states that local gauge transformation satisfies Maurer-Cartan equations. The partition function is now:

4 are Lie algebra valued 3-forms. It is the Lagrangian multiplier to the constraint D * B = 0. To pass from the integral over d/G to the integral over A, there is a change of measure. The field c is to rescale the measure of the gauge group by the determinant of the map a -+ * D * D a , d e t ( * D * D) = det D*D, where D* = *D * . It is included in the partition function as follows:

J

Z>cexp(iTrCDiDic) = det(D*D).

The fields C, 4 satisfy SC = Old$ = 0. Here C is a three form, 4 is a zero form, A = D*D is the Laplacian operator. The BRST operator 6 satisfies 6 ' = 0. So S induces a cohomology. The zero order cohomology gives gauge equivalent classes. That is why we say that BRST operator selects gauge invariant objects. In this setting, the new Lagrangian can be written in another form, L' = L - SV. Choose V so that the kinectic energy of C' is non-degenerate. For example, we can choose V as follows.

P a r t i t i o n function a n d topological invariants

85

Then we have

5.3.2

T h e leading term

The leading term is given by:

The functional here is quadratic in H = ( B ,4).We have

+

where L = (*D D*) is a twisted Dirac operator acting on forms. It maps even forms to even forms and odd forms to odd forms and L- is the restriction of L to odd forms, (, ) is the natural inner product

It can be seen as follows. * D B ,D So we have

(H, L-H) =

/

Tr(BA DB)

* 4 are one forms, D * B

is a 3 form.

+ s M T r ( BA * D * 4)+/'r.r(4 * D * B ) .

M

We know that (4,d')= (d', 4).When M is a closed manifold, by applying Stokes theorem, we have

86

Renormalized Perturbation Series of Chern-Simons- Witten Theory

Let us calculate the path integral:

+

/ V H D e D e r x p ( - 2f ( ~ ~ , C - I I ) (c, A c ) ) . We first choose orthonormal eigerifurictioris of lhose two operators. WE have:

We then have

Noticc that:

J'

dEdcexp(uCc) =

J'

d&(

I

+ u ~ c =) U ,

Partition j u n c t i m and topological

inwariants

87

We have det A = I I I z u qz (~L - ) = C2signv3.The determinant of Laplacian operator A can be regularized by using zeta function, det A = exp(-C'(O)), ((s) = Y,u;'. The '7 invariant can be regularized in a similar way, q(L-1 = q ( L - , O ) , q ( L - , s )= CJsignvJ\uJI-s. det(A) is a topological invariant. Its It is shown by Schwarz that dm absolute value is the same as Ray-Singer analytic torsion, and the imaginary part gives the q invariant,

3

Here v ( A a )= lim,,o C,signv, Izi, I-', or formally ? ( A a )= $C,signvi. The q invariant is related to Chern-Simons functional by $ ( q ( i z O )q ( 0 ) ) = * I ( A a ) : where { ( A " ) is the Chern-Sinions action of A" and ~ ( 0 IS) the q invariant of the trivial gauge field A = 0. Q ( G ) is the Casimir for G. For example. we have c z ( S I ' ( 2 ) )= 2 M . Finally, we have the formula €or the leading term as well as the formula €or Abelian Chern-Simons gauge theory,

T*

- 1J-l

is the torsion invariant o f A u .

To define self-linking integrals arid i l u non-bbelian geriaralizations we need a framing of the knot so thal one can consider t.he case z,y, z on the knot to coincide. For a given knot ii', consider a normal vector ficld on K . Imagine moving the knot, slightly in the direction of the vector ficld we then get another knot K ' . Tlic sclf-linking number of Ii' can Lc clefiried as t.he linking numbcr of I< arid I?. Such a definition dcpcnds on a choice of a normal vector fidd which is caILcd a. framing of a knot. It is easy to see that the choice only depends on homotopy classes of the normal vector field. 'rhe changing of framing is mcasurcd by an intcgcr s. It likcs to t w i d K ' around I<)s limes. The yartitioii function changes as 2 -+ cxp(Zxish)Z, where h is t.lie coriformal weight. So wc know IIOW it changes under different framing. In this sense, a choice of framing does not. t,ruuble 11s. For Abclian theory wc have hi = Thc Dehn t w i s t , around a marked point acts as eZKi", .

$.

R e n o r m n l i z e d Perturbntion Series of Chern-Sirnons- Wittera Theory

88

In general, there exists a constant c = c ( C , k ) ,the central charge of Virasoro, such that if framing of M is changed by ‘r units then the partition function transforms as

5.3.3

Wilson line and link invuriunts

Let L be a link with several embedded circles C, in M 3 . For each circle we associate a representation R,. The partition integral is then modified to

Z ( M )=/VAes/MTr(AAdA+

y2 A A A A A ) n T r R , P e x p i

Let A0 be a critical point and let A = A0 the critical point. Then we can expand

+ B be perturbations around

where r corresponds to Green functions of the qunrdratic form by L - . Now suppose that G = I/ (1 1. In this case we have R, is corresponding to n , n E Z. Let A be an one-form. Then we have

For a term of order two, i t corresponds to a wave-line. It then corre= G(z,y), a (1,l) form on M x M . For the flat metric on

sponds to

&

R3, G t J ( x ,y) =

w.

A brief introduction on renormalization of Ghern-Simons theory

-

4Ti

_-k -

89

Link (C1,C,).

In general one can expand holonomy of Wilson line in terms of iterated path integrals from K. T. Chen [Chen]:

PexpkA =1

+ C;=,

S

(C' A(t,)) ...( C'A(t1).

0 < t l < ...< t-51

This was used by D. Bar-Natan to establish a relation of Feynman diagrams in Chern-Simons theory with that of Vassiliev invariants [BarNatan].

5.4

A brief introduction an renormalizationof Chern-Simons theory

Since the seminal work of Witten [Witten] on Chcrn-Simons theory such a theory becomes a model of topological quantum field theory. Witteri's approach is non-perturbative by making a connection with conformal field theory. IKIthis section we continue the work to construct the theory through renorrrialixed perturbation series. Classically the Chern-Simons action enjoys two kinds of symmetries, ~iarrielygauge symmetry and diffeomorphism symmetry. It is remarkable that those two symmetries survive in the quantum level for some regulizntion schemes. This has been explored by sevcral authors [Blasi and Collina], [Cheri-Sernenoff-WII]. 'lhey found that therc are nice cancellations among t,hose divergent terms. The cancellations resemble those in N = 2 supersyrrirrielry theories. The remaining convergent terms are the same as if the coupling constant for the Chern-Simons terms has been shifted by cz(C).

90

5.4.1

Rerrur.rnu[ized P e r l u r h a l i o n Series uJ Chern-Simons- Wittcn Theory

A reguliration scheme

'l'o quantize Chern-Sirrioris Iheory is to evaluate the path inlegral:

= /DAeEIC" The integral is over the space of gauge equivalent fields. The construction of pertiirbatim series depends on a choice of rcgulization scheme. If we use high derivative regulization scheme in t,he Lorcntz gauge, we add following terms to the original Lagrangian,

where we have

Igh

1

d3X(dp?DpC).

The term 1,f is the Lagrarigian multiplier for lhe constraints d*A = 0. The term l g h is the determinant of the Laplace operator. The original path integral is thcn rcplaced by:

2'=

J

'DAPc"?e''o"

Wc can verify that Ittotstill respcct BRST syrrirriehy and diKeomorphism symmetry. The BRST symmetry is:

A brief introduction on renormalization of Chern-Sdmons theory

91

- l 6ca -c b cc , d F = d , & p = O .

2 Assuming we have an infinitesimal diffeomorphisrn rU -+ xu - A", it acts on all the fields above and the diffeomorphisrn symmetry is:

6cU = X"daca

1 + -&x'ea. 3

To verify that Itot respects diffeomorphisrn symmetry it is crucial to observe that d commutes with diffeomorphisrns. 5.4.2

The F e y n m a a r u l e s

The Feynman rules are: I . T h e gluori propagator: 1 G1(z,y) = L_'

2, The ghost propagator: Gzjz,y) =

1

13- 13

+ ULZ'

3. The ghost-glum vertex coming from TcA. 4. The three gluon vertex:

'

92

Renormdired Perturbation Series of Chern-Simon$- Witten Theory

Fig. 5.1 Feynman diagrams

5. The four gluon vertex coming from the Yang-Mills term. Given the Feymann rules above, one can construct Feymariri diagrams. We can organize the diagrams according to the number of loops. Here are some of the one-loop divegent Feynman diagrams and they cancel each other. The solid line corresponds to a gluon propagator and the dashed line corresponds to a ghost propagator. We have three kinds of vertext here, the three gluori vertex, the gluon-ghost-ghost vertex and the four g h o n vertex. We found the following observations crucial. 1. Divergent diagrams cancelled from each other becausc of gluon-ghost symmetry. For any divergent diagram we can always find a loop which contains both gluon and ghost, propagator. If we exchange gluon propagators with ghost propagat,ors we obtain another diagram. Thc resulting diagram is also admissible with a minus sign and it cancclcd the original one. This is true because of the special structure of the Feymann rules. 2. The remaining convcrgent diagrams is the same as if the coupling constant of the Chern-Simons theory shifted by cZ(G). By taking a limit we restore the two symmetries at the quanttum level. T h c only term which respect the two symmetries is the Chern-Simons action itself. And the only

A briej introduction on renorrnalizotion of Chern-Simons theory

93

counter term which survive after taking the limit is the Chern-Simons term. This expalins why we can only have the coupling constant shifted.

Chapter 6

Topological Sigma Model and Localization

6.1 6.1.1

Constructing knot invariants from open string theory

1ntrod.uctzion

A knot is an embedding of a circle into a t,hree-dimmsional space, say K : S1 3 S3. Two knots K l , X a are equivalent, if their complcmcnts in S3 are homeomorphic. So when we talk about a knot, we arc rcally talking about a three manifold, i.e. the complement of a circle in S3. There are two natural ways t o construct knot invariants. One way is what we have been doing based on Chern-Simons-Witten quantum field theory. For any three manifold consider a principle bundle with gauge group G and all gauge fields indexed by the three manifold. We then average over the space of all gauge fields with Chern-Simons Lagrangian as their weight. If such an average make sense it would give topological invariants. In previous chapters we explained how to do this from several points of views. There is another natural construction For a given knot K : S1 + S3, we consider all embeddings GJ : Ii + S3 in the same topological class. We then do an average with respect to all such embeddings. Again if the average make sense then it would give topological invariants because they do not depend on particular embeddings. Witten proposed to do this via open string theory, or topological sigma models. We shall give a brief introduction here. For more details, please see Witten’s paper [Wittenstring]. For the latest development please see Vafa’s recent paper iVafa].

95

Topological Sigma Model a n d L o c a l i z a t i o n

96

6.1.2

A topological sigma model

Witten's idea is to complexify the embedding. He considers a knot as a boundary component of a surface 2= and the target space be T * M 3 . Then he considers all mappings : E + T * M 3 . There is a natural symplectic structure on the target space w = d a , u = Cqidy;. We have a Lagrangian submanifold M 3 C T w M 3 ,i.e. W I M = 0. Choose an almost complex structure J on T * M . Let g = w ( . , J.) > 0 be a metric on T ' M and g is of ( 1 , l ) type, g i j = 0,gG = 0,g;; = gj;; = -iwi;. We then consider all maps Q! : C + '1"M with @(OX) c M . To construct a topological model, we also need t o include fermions. Let : C + X be a givcn map. We then have two kinds of fermions x E r(z,@*(Tx)), 4) E R1(C, @ * ( T X ) )Since . TX = T ( ~ > O ) X C D T ( O ~ ~ ) X , we have $ = $i $'. There are symmetries of those fields due to variations of the maypirig and the derivccl variations on fermions and we shall have a Lagrangian to be invariant under those variations. We have

+

Those are BRST transformations. If we denote A for the collection of fields, we have 614 = -ia{Q, A}, and Q2 = 0. We can choosc Lagrangian to be L = i { Q ,V} t Jx @ * w . For a good theory, we need to choose V such that the kinet,ic part of L is nondegenerate. For example, one can choose V = t d2zgij($iO2@j We then have a supersymmetric Lagrangian to be

+

s,

+ Oz@'$i).

&@*(w) = 27rn i s the degree of the mapping @, It is clearly invariant under Q .

Constructing knot invariants from open s t r i n g theory

97

To quantize the model is to calculate the path integral:

Witten argued that this integral makes sense as an open string theory. It is a topological field theory. The theory does not depend on t : so the perturbation series in t is exact. Usually topological sigma models can be understood by using localization principle. In this case the moduli space is degenerate. However it can be understood by using triangulation of Teichmuller space. Witten also argued that it is equivalent to large N expansion of the Chern-Simons gauge theory.

6.1.3

Localization pr inciple

Let us give a short introduction of localization principle. We will give more detail explanation in the following sections. Let u s first consider the finite dimensional case. Let G be a Lie group and g its Lie algebra, G acts on a symplectic space preserving the symplectic structure. Let Q* (X) be the space of de R.harri complex. Let us consider the space fl*(X)@ Fun(g). Let D = d - iv, where V is the Killing vector field. A form n tZ Q*(X)@ Fun(g) such that D2a= 0 is called an equivariant difkrential form. The induced D cohomology is called equivariant cohomology . Let cy be any equivariant differential form, then we have:

= C, local expressions where X , = {X(l<) = 0)) are components of the fixed point set of the G action, and X is any o m form. It is true because we can take a large t limit and the integral localize to the fixed point set of the G action. We will justify this in Ihe following sections for several interesting cases. However in the case of field theory, w e will be in the infinite dimensional setting. In the cme of open string theory the space X will be the space of fields. The group G will be replaced by the group of symmetries acting on

T o p o l o g i c a l S i g m a Model a n d L o c a l i z a t i o n

98

the fields. The operatcr D will be replaced by the BRST operator Q. We then need to evaluate the integral

The equivariant differential form is now replaced by OH,( P a ,) where H a are submanifolds in A4 and OH,( P a )is the Poincare dual in H * ( M ,R ) . Again the path integral is localized to the fixed point set. The fixed point set is now a vector bundle (given by fermions) over the space of pseudo holomorphic maps (given by bosons). Pseudo holomorphic maps are indexed by the degree of the maps. For large t , the operator Q can be identified with the usual differential operator d. The Q-cohomology is then the same as the usual cohomology H * ( M ,End(E)). The correlation functions are then given by:

where x ( V ) is the Philei- class of tjhe bundle V over M . The above is usually called A model. There is anot,her variant of toyological sigma models called H-model. 6.1.4

Large N expansz'on of Chern-Simons gauge theory

In the perturbative expansion for Chern-Simons gauge theory, the main interaction term is:

Tr(A A A A A ) = AS A A: A A : . Wc then have many Feynman diagrams. t,'Hooft used open Riernarin surface to express those Feynman diagrams. For each line one replaces it by a strip, theri one has a surface with marly holes. Assuming we have h boundary corriponerits and T loops, theri the genus of the surface is g = The free energy or the exparision of Chern-Simons theory is

y.

Equivariant cot~orr~oloyy and localization

99

where k = x. k+N Let, Zq,/, he t,hc pa.rt,ition function of an open string theory on a worldsheet with g handles and h boundaries. Witteri argued that

The claim can be justified by considering triangi~lat~ions of Teichmuller space based on fatgraph method. For references, see [Witten-string]. Most recently Vafa et al. proposed using closed string theory to construct knot invariants, see [ ~ a f a ] .

6.2 6.2.1

Equivariant cohomology and localization Equivariant cohornoloyy

Let X be a manifold, G a Lie group, G acts on X , i.e. we have

p :G

-+ Diff(X).

By taking derivatives, we have

dp : g -+ Vect(X). p induces an action on the de Rham complex R* ( X ) ,by (9,w ) -+ g* ( w ) .

We shall enlarge the space R*(X) into a*(X) @ Fun(g). Here Fun(g) consists of polynomial functions on Lie algebra g . w € R*(X)@ Fun(g) is a differential form on X with coefficients in Fun(g). Again p induces an action on R*(X)@ Fun(g). By taking derivative, we see that infinitesimally, the action is given by w -+ L V W ,V E d p ( g ) . Here LV = div +ivd is the Lie derivative. Those forms w satisfying LVW = 0, V E d p ( g ) are called equivariant differential forms. They are also elements in G-invariant subspace of a * ( X ) 8 F u n ( g ) ,R&(X)= (R*(X)@ Fun(g))G. We define a twisted differential operator D = d iv on R&(X) = ( Q * ( X )8 Fun(g))G. We have D2 = div ivd = L v . So D2 = 0 on R k ( X ) = (R*(X)8 Fun(g))G. Thus D induces a cohomology on the complex R2;;(X), H&(X)= KerD/ImageD is called equivariant cohomology.

+

+

Topological Sigma M o d e l a n d Localization

100

Example (Atiyah and Bott): Let Tnbe a torus which acts on X freely, then H + ( X ) = H ' ( X / T ) . In general we may have fixed points for the action. Let Q be a top equivariant differential form. We will find that the integration of Q usually reduce t o an integral over the set of fixed points. 6.2.2

Localization, finite dimensional case

Witten [Witten-localization] defined equivariant integration as

It is easy to check that JD,O = 0, j3 E n z ( X ) . So we have a map:

1

: H;;(X)

+. c.

If we only integrate the X part, we would have

/

: HE.(X)

-+

H&(pt).

For any one form X E Q> ( X ,)we have

s, s, a=

aetDX.

For e t D X = 1 + t D X , oDX = 0. Picking an orthonormal basis Ta of g, let V ( 4 )= S,d'Va: Va be the vector field representing T,. Then we have

-

1

t2

aexp(tdX - -c,(x(v,))') vol(G)( Z X C ) ~ / ~ 2E

Atiyah-Bott’a residue formula und Duisterrnaat-Heckrnan Jormula

101

A s t -+ 00, outside a neighborhood of X’ = {31X(IJa)= 0, u = 1, ..., s}, the integral hau contribution e--at21 Let X’ = UOEsX,, X, be its connected component. Then we have

2, is determined by local data of IY and the action near X,. We call the above formula the localization principle. 6.3 Atiyah-Bott‘s residue formula and Duistermaat-Heckman formula 6.3.1

Curnplez case, Atiyuh-Bolt’s residue formula

Example (Atiyah and Batt): Consider a U ( l ) action on X . Let Ifbe the vector generated by the action. Let g be a G-invariant Riemannian metric, X = -g(V, .) be a one form. Then

Let i + 0, the above integral will localize at zeros of g(V,V ) . i.e. the zeros of V , At ari isolated point P of V , the Hessian of g(V,V ) is nondegenerate, large t limit is a Gaussian integral. This way, we derive AtiyahBott’s fixed point formula. In the following we derive the original residue formula of Bott for symmetries preserving complex structures. M with Definition (Holomorphic Vector Bundle): A vector bundle E i a complex vector space as their fibers over a complex manifold M is called a holornorphic vector bundle. The complex structure on M induces an almost complex structure J : T c M Q, J 2 = - I d . Let T?”)M = Ker(J i ) , T $ ” ) M = Kcr(J - i ) . Then we have T c M = ‘Tg”’M @ T$’”]M. We called vectors in T g S 1 ) Mtype ( 0 , l ) vectors and vectors in T$’”M type (1,O) vectors. Similar decompositiorls car1 be made for TCM and their tensor products. We then have a notion of type ( p , q ) tensors. Let <, > be a Hermitian structurc on E , i.e. a metric on E whose restriction on each fiber is a quadratic form of type (1,1). Let { s i } be local

+

Topological Sigma Model a n d Localization

102

frames. Let N = { ( s i , s j ) } be a matrix of inner product. We can write d= 8 . 0 ( s ) = (3NN-l is of type ( 1 , O ) is the connection associated with the metric. Q(s) = & ( s ) is its curvature. We have 8 2 = 0. Let E be a complex vector bundle on a complex manifold M . Let c i ( M ) E H Z i ( M 2) , be the i-th Chern class of E . If @ ( c )= @ ( q ..., , cn) is a polynomial in the indeterminates c;, then we have @(,(Ad)) E H * ( M , C ) . We wish to evaluate the integral

a+

We called these characteristic numbers. If wa = C ~ ’ C ~ ~ . . .a1C ~+~ 2u2 , + ... + su, = k , k is callcd the weight of w,. We see that only those inonirnals with k = n contribute to the integral. If A : V -+ V is an endmorphism of a finite dimensional vector space, let del(1 XA) = Ci X i c i ( A ) . We call c,(A) Chern classes for A , let @ ( A )= @{cl(A),..., c n ( A ) ) . Let X be a vector field preserving the complex structure on M . Let L X = ixd dix be its Lie derivative. Restricted to zero’s of X , it induces a map L p ( X )= L x I T M , p E zero(X). Here is Bott’s residue formula which was soon generalized to a much more general case by Atiyah and Bott. Theorem: Let X be a nondegenerate vector field that preserves a complex structure on M . Then for every polynomial @ ( c l ,...,cn) of weight not greater than n , we have

+

+

C p @ ( L ) / c n ( L= ) @(M) Here p ranges over zeros of X and L = Lx I T p ( M ) c, n ( L ) = det L. In the proof, Bott expresses @ ( M )as an exact form away from zeros of X. So the integral is localized to zeros of X which then can be evaluated by local considerations. For a complete elegant proof see: R. Bott, Michigan Journal of Math. 14 (1967) 231-244. 6.3.2

Symplectic case, Duistermaat-Heckman f o r m u l a

Let X be a 2n dimensional compact symplectic manifold with symplectic form w . Suppose that the group U(1) acts symplectically on X , the action

Atiyah-Bott's residue f o r m u l a and Duistermaat-Heckman fo r m u la

103

being generated by a vector field V . The action is said t o be Hamiltonian if there is a function H on X wch that dH = - j v w . We wish to evaluate the following integral:

The Duistermaat-Heckman formula asserts that this integral is given by the semi-classical approximation. If the critical points are isolated points P i , then the formula is

where e ( P i ) is the product of the weights of the circle action in the tangent space at Pi. Proof The G action on X has a moment map p . We pick up an almost complex structure J on X such that w is of type (1,l) and positive, i.e. g ( v , v ) = w ( w , J v ) > 0 , v # 0. Set I = ( p , p ) and X = $ J ( d l ) . Critical points of ,u arc zeros of A . p - ' ( O ) is the set whcre ,u achieve a b s o l u t ~ minimum. Assuming that p - ' ( O ) is a smooth manifold on which G acts freely, then M = p-'(O)/G is a smooth manifold with a natural syrriplectic structurc. Contribution of ,u-l(O) is

Let, Y be an equivnriant neighborhood of p-' (0). We have an equivariant projection T : Y + pil(0)/G. Let -(+, 4 ) / 2 = T*(@),cy = A * ( & ' ) . Then,

By integrating over the fibers of ,u-'(O)

+ p-'(O)/G, we have

Evaluating Gaussian integral gives the iocalization forrriula. For higher critical points of Z, we have

All of them are exponentially small.

2D Yang-Mills theory hy localization principle

6.4

Localization principle also applies to infinite dimensional settings. This principle is extremely useful in topological quantum field theory and in string theory. In the following we introduce Witten's treatment of twodimensional Yang-Mills theory by using localization principle. 6.4.1

Cohomological Yang-Mills field theory

Let C be a surface, E -+ C a principle bundle with fiber G. The gauge group G, consisting of gauge transformations g : C + U , acts on the space of connections A by (g, A ) + g A g - l dgg-'. We will apply localization principle for finite dimensional group action to the currenl case of infinite dimensional group action. T h e tangent space of is the space of one forms with values in the Lie algebra of G'. Infinitesrnally we have:

+

Here

' ~ 6E

TQis an infinitesimal gauge transformation. We can also write

2D Yang-Milla t h e o r y by localization principle

105

Here $i E r ( C , K @ g ) is an anti-commuting one form with values in the adjoint representation of C . It is often called a fermion. d, E r ( C ,g ) is a zero form 011 C with value in the adjoint representation. It is often called a boson. E is an anti-commuting parameter. We can also writc

for every Geld @. Here Q = - D is the analog of twisted differential. We have Q2 = -i6$. So for gauge equivalent field, we have Q2 = 0. Let V be a gauge invariant functional. Let L = -i{Q, V } .The following integral defines a cohomolgical field theory:

The space of connections is an affine space with a natural symplectic form. The gauge group acts on the space of connections which preserves the symplectic form. The moment map of the action is the Yang-Mills functional. So this falls into the case of localization of symplectic action and the path integral is then reduced to an integral over the fixed point, especially the fixed point set of absolute minimals for polynomial expressions. So the above integral localize at fixed points of the group action, and we have

UES

Z, denotes contributions over a component of the fixed point set X u ,it also depends on the normal bundle of the component. An important point is that a properly chosen cohomological theory is equivalent t o the physical Yang-Mills theory. 6.4.2

Relation with physical Yang-Mills theory

The physical Yang-Mills theory is given by

Topologicol Sigma Model o n d Loculiantio~a

106

where L =

& J d2x1 * PA^^

and

*

is the star operator with respect t o a

choseri metric.

If we introduce a scalar field can be written as

C#J with

values in g , t,hen the above iritegral

dp = *(1) is a measure with respect to a chosen metric. Orie can also add ferrriiori Geld 4 E r ( E lR1(E) @ g ) which lies iri the tangent space t,o the space of coririectioris. The path integral is the same as

It is easy to check that the Lagrangian is gauge invariant. The derivation from cohomological Yang-Mills theory t o the physical Yang-Mills theory can be found on pp 32-40 of Witten’s paper [Witten-localization]. In particular, the following precise correspondence between them is established:

< exp(w + c 0 ) P >

Thc left-hand sidc is the cohomological Yang-Mills theory. With the help of localixatiori the original path integral is reduced to the integral over the space of flat connections M . Here w = Tr(i4F +$ A $) is the natural symplectic form on M . Note that T * M = H1(EI7’E@I g). The definition of w only depends on the cohoniological class of i+P+ $$A$.@ = dprl’r+2 is also a well-defined form on M u , 0 are the fundamental BRST invariant observables. There is one more HHS‘I’ observable VC = Tr#$. Here C c C is n simple closed curve. In general, one should evaluate

&

& sc

& s,

+

21) Ynng-Mi115 t h e o r y by localization principle

107

From the physical Yang-Mills theory, after we perform Gaussian integral on $, we eliminate the Vc terrris. This is why we have the above integral which is the same as

S,

exp(w

+ to)

where E = t - 2 Co
Evaluation of Yang-Mills t h e o r y

We now turn to more concrete calculations of quantizations of two dimensional Yang-Mills theory. First we consider the case of cylinder C x [O,T],where C E S1 is a circle. The space of fields is equal to the space of connections Ac on the circle. So the Hilbert space is 31 = C @ ' ( A C )the ~ , space of gauge invariant functions on A c . Since up to gauge transformation all conncctions are classified by their holonomy around thc curve C, we can identify the Hilbert space with 31 = Cw(G)G,i.e. the ring of functions on G which are invariant under the adjoint ackiori, or equivalently functions on the space of conjugacy classes which is the same as functions on the maximal t,oriis modulo the action of the Weyl group. 'I'his is nothing but the characters of irreducible representations of G. We t,ake characters X R to form a basis of X,31= @ R C R . The state associated with C and A is then @ B , ( A )= ,yR,(HolcA),i.e. thc character of the holonomy of A along C with respect to a representation R. In canonical quantization, A"(0)dO and -i& are conjugating variables. So the IIarniltonian is H =

by & n ~ H o l c ( A ) We see that,

$

sc Tr(&)2.

IIere

& acts on 9~

= TrltT"Holc(A).

caTaTa is the quadratic Casiniir of G , 2 ' r ~C , T"T" = c z ( R ) .Thus, we have

108

Tupo[agical Sigma Model arid Localirution

Next, we consider a sphere with three boundaries C1, C2, C,. Kaively W R ~ , R ~ , By R ~ the . consideration of the partition function is Z = Casirnir which is the same for all invariant polynomials one gets that Rl = Rz = R S . Hence the partition function is: Z = x R ~ ~ I I $ l Q ~ ( A I ~ , ) . For a closed surface C of genus g, we cut the surface into 29 - 2 disjoint copies of three punctured spheres along 39 - 3 disjoint simple closed curves. By using the orthogonality relation of group characters which correspond to states R , we have

xR1,R2,R3

Z ( C )= c w y e x p ( R

e2acz (R)

2

1

To compute W R , we first calculate 2 on adisk D , Z ( D ) = 'URexpi-*) Consider:

109

z(s1x

I ) = exp(-

e2uc2(R)

)>

Z(S1 x I ) = Z ( C , ) x Z ( D ) . This implies that V R W R = 1. On D , we fix the holonomy of a connection around 8 0 to take value in U E G. And Z ( U ) = C R V R X R ( U ) On . the other hand, we have Z ( U ) = exp d ( U - 1 ) . By using orthogonality relations for group characters, we find U R = exp(a) dimR. Hence,

Now, consider an example G = S U ( 2 ) . For cadi integer n 2 1, there is a unique representation R, of dimension n. The quadratic Casiniir of representation K, is c a ( ~ , )= we normalize it to c2(R,,) = $. Hence the partition function is:

+.

We will show that Z can be written as a sum over critical points to compare it, wit,h the formula by localization principle. We start, with

By using Poisson summation formula we havc

Topological S i g m a M o d e l a n d L o c a l i z a t i o n

110

89- 1

=

&g-l

e-Q(2g-2) 229-1

(-1

+

c

(2~rn)~ e x p ( - y 1).

mEZ

This agrees with the formula by localization principle from the following consideration. The Eular-Lagrangian equation of Yang-Mills is D f = 0. If f # 0, being covariantly constant, this gives a reduction of the structure group of the connection to a subgroup Ho that commutes with f. Solutions are therefore flat Ho connections twisted by constant curvature line bundles in the U(1) subgroup generated by f. In the case of G = S U ( 2 ) , we get an SU(2) bundle with a covariantly constant splitting as a sum of line bundles. From the classification of line bundles, it follows that the conjugacy class o f f is given by

j=2xm( 0

-i

) 9. This

with m E Z . The value of I at such a critical point is I,,, = shows that it agrees with the formula by localization principlr.

6.5

Combinatorial approach to 2D Yang-Mills theory

Let C be a two dimensional surface, E -+ C be a G bundle with G' a Lie group. Let A be a connection on X,i.e. a valued one form. A s u s ~ a l , F = dA 1/2A A A bc its curvature. F is G valued two form. Pick up a metric on C, it induces a Hodge * operator on forms. Since X is two dirrierisional, f = * F = tF is a 6 valued zero form, here c is the area form. As usual the Yang-Mills Lagrangian is defined as:

+

I ( A ) actually orily depends on A and the measure dp induced by the metric. Let, p = tlp. It is uscful to notice that [ ( A ) is invariant under

1,

c2

+te2,p +lilt.

The quantization of Yang-Mills i n two dimensions is t o compute the parti tion function :

Combinatorial approach t o 2 0 Yang-Mills t h e o r y

111

An allernative Lttgrangiarl is

Their partition function

is the same as two dimensional Yang-Mills because one can integrale same wa.y as in one dimcnsion:

4 the

Lel e + 0, one has a topological field theory Z c ( 0 ) . We want to solve this model. ‘Ihdo so, we triangulate 2’ into plaquettcs ~ i i.e. , polygons with boundary U , , uz,..., U,. bhch w;carries a measure p i , Cp, = p. We have

IIere is a key observation: The conjugacy class of the holonomy is gauge invariant,, the local fador associated with a plaqiiette must be a class function of the holonomy. Class function must be a lincar combination of lhe group characters which forms a basis of class functions. For each plaqucttc and A wc have holorlonly 2.4 = A . Lel N be an irreducible representation of G, wc define ~ ~ ( 2 . 4=) Tr,U. It is Midgal who first made an Ansalz:

sow

And

Topolvyicul Sayma Mvdei and Localiralion

112

zx(PI =

/

n7

du-,nir(ui pi). 1

The crucial propety is that it is invariant under subdivision. The proof is as follows. Let

r = C,dimaXaflilUzFj31/4)et-

(*'I .

be for the plaquette LrlU2W3Ud. We add a line V so it is divided into two small plquettes Tl1U2V and V-'Lr3F4. We have

c

1

The above implies that

J dvr'r''

= r.

This gives invariance under subdivision. As an application, we represmt a close surface C as a disk w i t h boundary ~ l b l n ; l b ; ' c / 2 b 2 n ; ' b z ' . . . , then we have I h e partition function for C as:

Remark: The above partition function can he identified as the symplectic volume.

Remark: A rigorous mathematical justification based on above localization for Yang-Mills is available by works of Liu Kefeng b i u l ] , [Liu2] and

Jeffery and Kirwan [Jeffery-Kirwan].

113 Reprinted from Complez Manijold Without Potential Theory by S . S . Chern, Revised edition, Springer, 1995, pp. 148-154.

7.

Chern-Simons Invariant of Three-dimensional Manifolds In (125). §4 we defined an invariant J

-

I(s) mod 1, f o r a compact

oriented three-dimensional manifold. This has been called a Chern-Simons invariant. It has played an important role i n both mathematics and physics. In f a c t , up t o an additive constant it i s the e t a invariant of the manifold, as introduced by Atiyah, Patodi, and Singer v i a s p e c t r a l theory [l] ; c f . Ref t o $ 7 . It has a l s o been used by W. Thurston in h i s theory of hyperbolic

manifolds. while Robert Meyerhoff har shown t h a t , f o r c e r t a i n hyperbolic manifolds, it takes values which a r e dense oa the u n i t c i r c l e , In mathemat i c a l physics the concept i s found useful i n quantum f i e l d theory, s t a t i s t i c a l mechanics, and the theory of anyons.

We begin by repeating i t s definition. Let H denote t h e manifold, oriented, and l e t P be the bundle of its orthonormal frames, so t h a t we have

where

?r

is the projection, mapping a frame x e l e ~ et o~ i t s o r i g i n x. A

section s : M + P of the bundle s a t i s f i e s the condition

7~

o s

-

identity

and can be viewed a s a f i e l d of frames. It is v e l l known t h a t i n our case such a section always e x i s t s ; we say t h a t M i s parallelizable.

To such a frame f i e l d the Levi-Cavita connection of t h e metric is given by an antisymmetric matrix of one-forms:

and its

curvature by an antisymmetric matrix of two-forms:

114 149 Throughout this section our a m a l l Latin isdicen u i l l run from 1 to 3 . Va haw

(2011

EPij A V j k

dVik

+

@ik*

j We introduce the three-form

and consider the integral

It uill ba proved beloo that for another section t : W

that @(el

-

4

P, @(t>

- Q<s)

mod 1 i s independent of s. This define8 an

is an integer,

so

invariant J(H1

€ a / z ,where J(H1

@(I]

mod 1.

IB [a] we proved the theoremrs: Theorem 7.1. J(!f) i s u conj’omat invariant, 6 . e . i t remains unchanged under a canfarmel transformtion of thc m e t r i c ,

Theorem 7 . 2 . .?(MI

It-

a

critical value a t M if and only if M i s 20-

cuity canjomally f l a t , We w i d to give direct proofs of these theorem in thia section. A) Family of Connections CI. shall uae arbitrary frame f i e l d 8 t o develop the Riemannian

geometry on M. Let feleaeg be a frame, and w’, w a , U s , its dual coframe.

Let the inner product be

Ye introduce gif through tha equations

115 150

and use the g’a to r a h e or lover Indices. as in classical tensor analysis.

Then the connection forms wi1 or

Wij

a r e determined, uniquely, by the

equations

The curvature forms are defined by

By exterior differentiation of (207) we get the Bianchi identity

To avoid confusion notice our convention that the upper index in w i , 0; is the second index. Thus c jW ji gjk Wik (# y i in general). 9

We introduce the cubic form

On our three-manifold M. I is of course closed. But the basic reason f o r its importance and interesting properties is that formally by (2071, (2081, we

have

(210)

which is the first Pontrjagin form. 7. Then

Consider a family of connections on U. depending on a parameter T a l l involve T , and w e have the fundamental formula

wi,j

ni,

116 151

The proof of (211) is straightforvard. It follovs by differentiation of (2091, and using the formulas obtained by differentiation of (2061, (207) vith respect to

7.

It is useful to observe that the last term is a

polarization of the Pontrjagin form. Let P’ be the bundle of all frames of M,

so

that ve have

(212)

vhere i is the inclusion. Then T in (209) can be considered as a form in P‘ and its pull-back i*T is the I given by (202). We will make such identifications when there is no danger of confusion. We

consider @(a) defined by (203). When t is another section.

-

then t(M)

s(M), as a three-dimensional cycle is homologous, modulo

torsion, to an integral multiple of the fiber Px, x E M. But Px is topologically SO(3) and Wij reduces on it to its Maurer-Cartan forms. If j : PX

-

SO(3) + P ie the inclusion,

whose integral over PX is 1. Hence @(s)

mod 1 is independent of s.

We wish to clarify the relation betveen orthonormal frame fields

and arbitrary frame fields. By the Schmidt orthogonalization process P is a retract of P’, under vhich the origin of the frame is fixed. The retraction we denote by r (209)

to be in P’. If

8’

:

:

P’ + P. We consider the form T defined in

M

-+

P’ is a section, then s = r

o

8’

is also

a section and they a r e homotopic through sections. Since dT = 0. w e have (213) Hence J(M)

can be computed through an arbitrary frame field by the left-

hand side of the last equation.

117 152 8 ) Proofs of t h e Theorems

In view of the above remark

we

can, for l o c a l considerations,

use a l o c a l coordinate system ui and t h e r e s u l t i n g natural frame f i e l d

a/&'.

We s h a l l summarize t h e vell-known formulas, which are

(214)

where t h e comma denotes covariant d i f f e r e n t i a t i o n . They imply

and

Ye also introduce the R i c c i curvature and the scalar curvature by

163

These relations imply that the matrix

(221)

i a aymaetric and that the matrix GC. where G = (gij), has trace zero.

Sehauten proved (4, p . 921 that the three-dimensional manifold

H

i8 C O n f O r U J a l l y f l a t ,

if m d only if c

0, i.e. c i j k

0.

By lutegrating (2111, we get (2221

uhere

by (217). We consider a family of metrics g i j ( T 1 aud put

To prove Theorem 7.1

YO

ruppore thie is a conformal family of metrics,

i.r. (226)

'ij

-

Wj.

From (214) ue find aj;

dgi -gihdlml)hk* &h +C(~!.L

(226)

uhere q

- &/&.

+

m

By the 6econd equation of (216) and the equation (217)

ue find h = 0. This proves that

jHT

ia independent of

T,

and hence Theorem

7.1.

To prove Theorem 7 . 2 we consider v i j such that

t h e trace

119 154

v t * 0 . Geometrically t h i s means that ve consider t h e tangent space

of t h e apace of conformal s t r u c t u r e s on M. From (214) ye f i n d (227)

It f o l l o u s that (228)

The term i n t h e middle is zero, because i s symmetric

nij

is antisymmetric and d v i j

i n i, j . To t h e i n t e g r a l of t h e last term ve apply Stokes

theorem t o reduce it to an i n t e p a l i n v o l v i w only t h e

Vij,

and not t h e i r

d e r i v a t i v e s . We uill o m i t the dataila of this computation. The r e s u l t is that the condition

If

the metric is conformally f l a t , ue have C = 0 and hence t h e vanishing

of

t h e above

htEgrEd.

Converrely. at a c r i t i c a l p o i n t of

-

ue muat have Tr(VC)

-

0

f o r dl symmetric v satisfying ~ r ( ~ o - 1 ) O. Hence c is a m u l t i p l r of G-'

-

o r CC I s a multiple of the unit matrix. But GC has trace zero. Hence

i t muat itaalf be zero and ve haye C

0 . Thin proves Theorem 7 . 2 .

121 J. DIFFERENTIAL OEOMETRY 33 (1991) 781-902

GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN Abstract We present a new construction of the quantum Hilbert space of ChernSimons gauge theory using methods which are natural from the threedimensional point of view. To show that the quantum Hilbert space associated to a Riemann surface Z is independent of the choice of complex structure on Z , we construct a natural projectively flat connection on the quantum Hilbert bundle over Teichmiiller space. This connection has been previously constructed in the context of two-dimensional conformal field theory where it is interpreted as the stress energy tensor. Our construction thus gives a (2 + 1)-dimensional derivation of the basic properties of ( 1 + 1 )-dimensional current algebra. To construct the connection we show generally that for affine symplectic quotients the natural projectively flat connection on the quantum Hilbert bundle may be expressed purely in t e r n s of the intrinsic Kahler geometry of the quotient and the Quillen connection on a certain determinant line bundle. The proof of most of the properties of the connection we construct follows surprisingly simply from the index theorem identities for the curvature of the Quillen connection. As an L-xample, we treat the case when Z has genus one explicitly. We also make some preliminary comments concerning the Hilbert space structure.

Introduction Several years ago, in examining the proof of a rather surprising result about von Neumann algebras, V. F. R. Jones [20] was led to the discovery of some unusual representations of the braid group from which invariants of links in S 3 can be constructed. The resulting “Jones polynomial” of links has proved in subsequent work to have quite a few generalizations, and to be related to two-dimensional lattice statistical mechanics and to quantum groups, among other things. Received by the editors November 6, 1989, and. in revised form, January 23, 1990. The first author’s research was supported in part by a National Science Foundation Graduate Fellowship and the Alfred P. Sloan Foundation. The second author’s research was supported in pan by National Science Foundation Grant 86-20266 and National Science Foundation Waterman Grant 88-1 7521.

122

iaa

SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

Tsuchiya and Kanie [38] recognized that the Jones braid representations and their generalizations coincide with certain representations of braid groups and mapping class groups that have quite independent origins in conformal field theory [4] and that have been intensively studied by physicists [ l l ] , [40], [23], [31]. (The representations in question are actually projective representations, for reasons that will be clear later.) The conformal field theory viewpoint leads to a rigorous construction of these representations [34], [39]. Conformal field theory alone, however, does not explain why these particular representations of braid groups and mapping class groups are related to three-dimensional invariants. It was conjectured [2] that some form of three- or four-dimensional gauge theory would be the key to understanding the three-dimensional invariances of the particular braid traces that lead to the Jones polynomial. Recently it has been shown [41] that three-dimensional Chern-Simons gauge theory for a compact gauge group G indeed leads to a natural framework for understanding these phenomena. This involves a nonabelian generalization of old work by A. Schwarz relating analytic torsion to the partition functions of certain quantum field theories with quadratic actions [32], and indeed Schwarz had conjectured [ 331 that the Jones polynomial was related to Chern-Simons gauge theory. Most of the striking insights that come from Chern-Simons gauge theory depend on use of the Feynman path integral. To make the path integral rigorous would appear out of reach at present. Of course, results predicted by the path integral can be checked by, e.g., showing that the claimed three-manifold invariants transform correctly under surgery, a program that has been initiated in [30]. Such combinatorial methods-similar to methods used in the original proofs of topological invariance of the Jones polynomial-give a verification but not a natural explanation of the threedimensional symmetry of the constructions. In this paper, we pursue the more modest goal of putting the Hamiltonian quantization of Chern-Simons gauge theory-which has been discussed heuristically in [42] and in [7]-on a rigorous basis. In this way we will obtain new insights about the representations of braid and mapping class groups that arise in this theory. These representations have been constructed, as we have noted, from other points of view, and most notably from the point of view of conformal field theory. However, threedimensional quantum field theory offers a different perspective, in which the starting point is the fact that affine spaces and their symplectic quotients can be quantized in a natural way. Our goal in this paper is to give a rigorous construction of the representations of mapping class groups that

123 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

789

are associated with the Jones polynomial, from the point of view of the three-dimensional quantum field theory. Canonical quantization. The goal is to associate a Hilbert space to every closed oriented 2-manifold Z by canonical quantization of the ChernSimons theory on Z x R . As a first step, we construct the physical phase space, A‘. It i s the symplectic quotient of the space, s f , of G connections on Z by the group F of bundle automorphisms. It has a symplectic form w which is k times the most fundamental quantizable syrnplectic form wo : here k is any positive integer. A is the finite-dimensional moduli space of flal G connections on Z. We then proceed to quantize .X as canonically as possible. We pick a complex structure J on Z. This naturally induces a complex structure on A‘, making it into a Kihler manifold. We may then construct the Hilbert space &“,(C) by Kahler quantization. If 7 denotes the space of all complex structures on A , we thus have a bundle of Hilbert spaces X ( C ) -+ F.This “quantum bundle” will be denoted 2$. For our quantization to be “canonical” it should be independent of J , at least up to a projective factor. This is shown by finding a natural projectively flat connection on the quantum bun d 1e. The essential relation between Chern-Sirnons gauge theory and conformal field theory is that this projectively flat bundle is the same as the bundle of “conformal blocks” which arises in the conformal field theory of current algebra for the group G at level k . This bundle together with its projectively flat connection is relatively well understood from the point of view of conformal field theory. (In particular, the property of “duality” which describes the behavior of the XJ(2) when Z degenerates to the boundary of moduli space has a clear physical origin in conformal field theory [4], [40j. The property is essential to the computability of the Jones polynomial.) The conformal field theory point of view on the subject has been developed rigorously from the point of view of loop groups by Segal [34], and f’rom an algebra-geometric point of view by Tsuchiya et al. [39]. Also, there is another rigorous approach to the quantization of A‘ due to Hitchin 1181. Finally, in his work on non-abelian theta functions, Fay [8] {using methods more or less close to arguments used in the conformal field theory literature) has described a heat equation obeyed by the determinant of the Dirac operator which is closely related to the construction of the connection and may in fact lead to an independent construction of it. We will be presenting an alternative description of the connection on Z ( 2 )which arises quite naturally from the theory of geometric quantization. In fact, rhis entire paper is the result of combining three simple facts.

124 SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

790

( 1) The desired connection and all of its properties are easily understood for Kahler quantization of a finite-dimensional affine symplectic manifold at . In that case the connection 1-form is a simple second order differential operator on d acting on vectors in the quantum Hilbert space (which are sections of a line bundle over &’ ). (2) By geometric invariant theory we can present a simple abstract argument to “push down” this connection “upstairs” for quantization of g ‘ to a connection “downstairs” for the quantization of A‘. Here, is the symplectic quotient of ~4 by a suitable group of affine symplectic transformations that preserves the complex structure which is used in quantizing A? The one-form d for the connection downstairs is a second-order differential operator on 4. (3) Even in the gauge theory case where the constructions upstairs are not well-defined since at is infinite dimensional, we may present the connection downstairs in a well-defined way. We first work in the finite dimensional case and write out an explicit description of 8 . We then interpret the “downstairs” formulas in the gauge theory case in which the underlying affine space is infinite dimensional though its symplectic quotient is finite dimensional. As is familiar from quantum field theory, interpreting the “downstairs” formulas in the gauge theory context requires regularization of some infinite sums. This can, however, be done satisfactorily. What has just been sketched is a very general strategy. It turns out that we have some ‘‘luck‘‘-the definition of d and the proof of most of its properties can all be written in terms of the Kahler structure of dl and a certain regularized determinant which is independent of the quantization machinery. Since these objects refer only to A? , our final results are independent of geometric invariant theory. One consequence of this independence is that our results apply for an arbitrary prequantization line bundle on A?, and not just for line bundles which arise as pushdowns of prequantum line bundles on a’. The infinite dimensionality of the affine space that we are studying shows up at one key point. Because of what physicists would call an “anomaly”, one requires a rescaling of the connection 1-form from the normalization it would have in finite dimensions. This has its counterpart in conformal field theory as the normalization of the Sugawara construction [ 141, which is the basic construction giving rise to the connection from that point of view. This rescaling does not affect the rest of the calculation

.‘

’For physicists, geometric invariant theory is just the statement that, in this situation, one gets the same result by imposing the constraints corresponding to B invariance before Or after quantization.

125 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

79 I

except to rescale the final answer for the central curvature of the connection. This reproduces a result in conformal field theory. From our point of view, though we can describe what aspects of the geometry of the moduli space lead to the need to rescale the connection, the deeper meaning of this step is somewhat mysterious. Outline. This paper is quite long. Essentially this is because we rederive the connection several times from somewhat different viewpoints and because we describe the special case of genus one in considerable detail. Most readers, depending on their interests, will be able to omit some sections of the paper. For physicists, the main results of interest are mostly in 552 and 5, and amount to a ( 2 + I)-dimensional derivation of the basic properties of ( 1 + 1)-dimensional current algebra, including the values of the central charge and the conformal dimensions. This reverses the logic of previous treatments in which the understanding of the ( 2 + I)-dimensional theory ultimately rested, at crucial points, on borrowing known results in ( 1 + 1)-dimensions. This self-contained (2 + 1)-dimensional approach should make it possible, in future, to understand theories whose ( 1 + 1)dimensional counterparts are not already understood. On the other hand, a mathematically precise statement of the majority of results of this paper is given at the beginning of 54. This discussion is essentially self-contained. In 51, we present a detailed, although elementary, exposition of the basic concepts of Kahler quantization of affine spaces and their symplectic quotients. We define the desired connection abstractly. As an example, in the last subsection we show explicitly how for quantization of the quotient of a vector space by a lattice, the connection is the operator appearing in the heat equation for classical theta functions. The remainder of the paper is devoted to making the results of 51 explicit in such a way that they essentially carry over to the gauge theory case. In 52 we discuss this case in detail and construct the desired connection in a notation that is probably most familiar to physicists. In 53 we present a more precise and geometric formulation of the results of 52 in notation suitable for arbitrary affine symplectic quotients. We derive a formula for the connection that may be written intrinsically on M . This derivation is, of course, only formal for the gauge theory problem. In 54, we state and prove most of the main results. Using an ansatz suggested by the results of 552 and 3 and properties of the intrinsic geometry of & , we find a well-defined connection. The properties of the intrinsic geometry of M which we need follow from the local version of

126 192

SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

the families index theorem and geometric invariant theory. Using these properties, and one further fact, we show that the connection is projectively flat. (Actually, there are several candidates for the “further fact” in question. One argument uses a global result-the absence of holomorphic vector fields on A-while a second argument is based on a local differential geometric identity proved in $7. It should also be possible to make a third proof on lines sketched at the end of $6.) This section is the core section analyzing the properties of the connection and is rigorous since all the required analysis has already been done in the proof of the index theorem. In $5 we shall concentrate on the gauge theory case when C is a torus. We give explicit formulas for our connection and a basis of parallel sections of the quantum Hilbert bundle A?(X), We also show directly that our connection is unitary and has the curvature claimed. The parallel sections are identified with the Weyl-Kac characters for the representations of the loop group of G . This result is natural from the conformal field theory point of view, and was originally discussed from the point of view of quantization of Chern-Simons gauge theory in [7]. In 56 we make some preliminary comments about the unitarity of our connection. In $7 we develop an extensive machinery allowing us to prove in a systematic way the one identity left unproved in 54. Our discussion, however, is incomplete in that we have not checked some details of the analysis of regularization. The appendix contains further formulas relevant to 55. We would like to thank M. Atiyah, V. Della Pietra, C. Fefferman, D. Freed, N. Hitchin, C. Simpson, and G. Washnitzer for helpful discussions.

1. Geometric setup and pushed down connection

In this section we consider the quantization of a finite-dimensional symplectic manifold A which is the symplectic quotient of an affine symplectic manifold af by a suitable subgroup of the affine symplectic group. Quantization of A is carried out by choosing a suitable complex structure J on &#’ which induces one on A . We describe the projectively flat connection whose existence shows that quantization of A is independent of the choice of J . This is an interesting, though fairly trivial, result about geometric quantization. Its real interest comes in the generalization to gauge theory, which will occupy the rest of the paper.

127 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

193

Most of this section is a review of concepts which are well known, although possibly not in precisely this packaging [22], [36], [37], [43].We review this material in some detail in the hope of making the paper accessible. la. Symplectic geometry and Kiihler quantization. To begin with, we consider a symplectic manifold d , that is, a manifold with a closed and nondegenerate two-form w . Nondegeneracy means that if we regard w as a map from w : T d + T * d , then there is an inverse map up’: 7*& .--t T@ . In local coordinates, a’ , if (1.1)

w = wlJda‘

A

da’ ,

and

w- I

rj

0

a

aa‘

i3a’

=w-G3-,

then the matrices wi,and w” are inverses,

Let C ” ( d ) denote the smooth functions on d . Given h E C m ( d ) , we form the vector field 6 = w - ’ ( d h ) called the flow of h . It is a symplectic vector field-that is, the symplectic form w is annihilated by the Lie derivative Ph-since Pvh ( a )= ( i‘h d + d i Vh )w = d ( iVh w ) (since o is closed) and by the definition of 6 one has i,(o) = -dh . Conversely, given a symplectic vector field V , that is a vector field V such that Y V ( w )= 0 , one has a closed one-form (Y = i V ( w ). A function h such that cy = d h is called a Hamiltonian function or moment map for V . If the first Betti number of d is zero, then every symplectic vector field on d can be derived from some Hamiltonian function. The symplectic vector fields on d form a Lie algebra. If two symplectic vector fields V, and Vg can be derived from Hamiltonian functions f and g , then their commutator [ Vr, Vg] can likewise be derived from a Hamiltonian function; in fact

(1.4)

1% %I

= I‘;,,glps’

where [f,glPB denotes the so-called Poisson bracket

Therefore, the symplectic vector fields that can be derived from Hamiltonians form a Lie subalgebra of the totality of symplectic vector fields.

128 794

SCOTT AXELROD, STWE DELIA PIETRA & EDWARD WJTTEN

Essentially by virtue of (1.4) the Poisson bracket obeys the Jacobi identity

so that under the , IPB operation, the smooth functions on d have a Lie algebra structure. It is evident that the center of this Lie algebra consists of functions f such that df = 0 ; in other words, if d is connected, it consists of the constant functions. The smooth functions on &? are also a commutative, associative algebra under ordinary pointwise multiplication, and the two structures are compatible in the sense that These compatible structures [ , IPB and pointwise multiplication give C” (a’ ) a structure of “Poisson-Lie algebra”. According to quantum mechanics textbooks, “quantization” of a symplectic manifold A?’ means constructing -as nearly as possible” a unitary Hilbert space representation of the Poisson-Lie algebra C “ ( d ) , This would mean finding a Hilbert space H and a linear map f + f frQm smooth real-valued functions on a’ to selfadjoint operators on H such that = f -2 and [f=gIpB = i[?, g]. One also requires (or proves from an assumption of faithfulness and irreducibility), that if 1 denotes the constant function on A , then is the identity operator on H . This notion of what quantization should mean is however far too idealized; it is easy to see that such a Hilbert space representation of the Poisson-Lie algebra Cm(d) does not exist. Quantum mechanics textbooks therefore instruct one to construct something that is “as close as possible“ to a representation of C ” ( d ) . This is of course a vague statement. In general, a really satisfactory notion of what “quantization” should mean exists only in certain special classes of examples. The proper study of these examples, on the other hand, leads to much infomation. The examples we will be considering in this paper are affine spaces and their symplectic quotients by subgroups of the affine syrnplectic group obeying certain restrictions. Prequantizalion. If one considers C ” j d ) purely as a Lie algebra, it Hilbert space representation can be constructed via the process of “prequantization”. Actually, for prequantization one requires that 1w represents an inte2? gral cohomology class. This condition ensures the existence of a Hermitian line bundle 2 over a’ with a connection V that is compatible with the Hermitian metric { )x and has curvature -iw . The isomorphism class

(5)

!

129 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

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of 2’(as a unitary line bundle with connection) may not be unique; given one choice of 2 , any other choice is of the form 2‘ = 2 @ S , where S is a flat unitary line bundle, determined by an element of H 1 ( d, U ( 1 ) ) . The problem of prequantization has a solution for every choice of 3 . Let i?? be the group of diffeomorphisms of the total space of the line bundle 9 which preserves all the structure we have introduced (the fibration over d ,the connection, and the Hermitian structure). Let H L 2 ( d 2) , be the “prequantum” Hilbert space of all square integrable sections of 3 . Since i?? acts on 9 , it also acts on H L 2 ( d ,2). An element, D ,of the Lie algebra of i F is just a vector field on A lifted to act on 2 . Acting on H L z ( d , 9) , this corresponds to a first-order differential operator,

D

(1.8)

= V,

+ ih.

Here T is the vector field representing the action of D on the base space and h is a function on d .The conditions that D preserves the connection is that for any vector field u we have

&‘

[V,

+ ih , V,]

= V.+,.

Since the curvature of V is - i o , this is true if and only if T = 5 . One may easily check that the map p,, from C”(S’) to the Lie algebra of 2 ‘7 defined by (1.10)

P,,(h) = $VV*3- h

is an isomorphism of Lie algebras. In addition, the function 1 maps to the unit operator. Prequantization, as just described, is a universal recipe which respects the Lie algebra structure of Cm(S’) at the cost of disregarding the other part of the Poisson-Lie structure, coming from the fact that C ” ( d ) is a commutative associative algebra under multiplication of functions. Quantization, as opposed to prequantization, is a compromise between the two structures, and in contrast to prequantization, there is no universal recipe for what quantization should mean. We now turn to the case of affine spaces, the most imDortant case in which there is a good recipe. Quantization of afine spaces. Let d be a 2n-dimensional affine space, with linear coordinates a’ , i = 1 . . . 2 n and an affine symplectic structure

(1.1 1 )

o = ~ , , d ‘ada I ,

with w,, being an invertible (constant) skew matrix.

130 SCOlT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

796

The Poisson brackets of the linear functions a' are (1.12)

i

[a

j

3

a IPB = CfJ

ij

.

In contrast to prequantization, in which one finds a Lie algebra representation of all of C " ( d ) in a Hilbert space Z , in quantization we content ourselves with finding a Hilbert space representation of the Poisson brackets of the linear functions, that is, a Hilbert space representation of the Lie algebra ^

(1.13)

A

ij

[ a i , a'] = -io

.

Actually, we want Hilbert space representations of the "Heisenberg Lie algebra" ( l. 13) that integrate to representations of the corresponding group. This group, the Heisenberg group, is simply the subgroup of %?that lifts the affine translations. According to a classic theorem by Stone and von Neumann, the irreducible unitary representation of the Heisenberg group is unique up to isomorphism, the isomorphism being unique up to multiplication by an element of V (1) (the group of complex numbers of modulus one). Representing only the Heisenberg group-and not all of C m ( d ) means that quantization can be carried out in a small subspace of the prequantum Hilbert space. Action of the afine symplectic group. Before actually constructing a representation of the Heisenberg group, let us discuss some properties that any such representation must have. The affine symplectic group T'-the group of affine transformations of d that preserve the symplectic structure-acts by outer automorphisms on the Lie algebra (1.13). The pullback of a representation p of (1.13) by an element w of V is another representation p' of (1.13) in the same Hilbert space W . The uniqueness theorem therefore gives a unitary operator U(w): H + H such that p' = V(w)o p . The V(w) are unique up to multiplication by an element of U ( 1 ) , and therefore it is automatically true that for w , w' E H , V(ww') = U ( w ) U ( w ' ) n ( w, w') where n ( w , w') is a U(l)-valued two-cocycle of Y . Thus, the U ( w ) give a representation of a central extension by V (1) of the group W . It can be shown that if one restricts to the linear symplectic group-the subgroup of 5V that fixes a point in &-then (for finite-dimensional affine spaces) the kernel of this central extension can be reduced to 2/22. Now, let us investigate the extent to which a representation p of the Lie algebra (1.13) can be extended to a representation of the PoissonLie algebra C " ( d ) . One immediately sees that this is impossible, since

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one would require both p(a’a’) = p ( a ’ ) p ( a ’ ) and p(a’a’) = p(a’a’) = p ( a ’ ) p ( a ’ ); but (1.14)

p ( a ’ ) p ( a ’ )- p ( a ’ ) p ( a ’ )= -KO1’.

Thus, p cannot be extended to a representation of C ” ( d ) . However, the right-hand side of (1.14), though not zero, is in the center of C “ ( d ) , and this enables us to take one more important step. Defining p(a’a’) = i ( p ( a ’ ) p ( a ’ )+ p ( a ’ ) p ( a ’ ) )>

(1.15) one verifies that

To interpret ( 1.16) observe that the linear and quadratic functions on d form, under Poisson bracket, a Lie algebra which is a central extension of the Lie algebra of the affine symplectic group. In other words, the Hamiltonian functions from which the generators of the affine symplectic group can be derived are simply the linear and quadratic functions on d . Equation (1.16), together with (1.13), means that any representation of the Lie algebra ( 1.13) automatically extends to a projective representation of the Lie algebra of the affine symplectic group W . This is an infinitesimal counterpart of a fact that we have already noted: by virtue of the uniqueness theorems for irreducible representations of ( 1.13), the group “W automatically acts projectively in any such representation. The verification of (1.16) depends on the fact that the ambiguity in the definition of p(a’a’)-the difference between p ( a i ) p ( a ’ ) and p ( a ’ ) p ( a ’ ) -is central. For polynomials in the a‘ of higher than second order, different orderings differ by terms that are no longer central, and it is impossible to extend p to a representation of C “ ( d ) even as a Lie algebra, let alone a Poisson-Lie algebra. It is natural to adopt a symmetric definition (1.17)

I

1

i

l

i

.

.

p ( a ‘ a . . . a “ ) = - ( a lai2. a’”

n!

+ permutations),

but for n > 2 this does not give a homomorphism of Lie algebras. Quantization. There remains now the problem of actually constructing Hilbert space representations of ( 1.13). There are two standard constructions (which are equivalent, of course, in view of the uniqueness theorem). Each construction involves a choice of a “polarization”, that is, a maximal linearly independent commuting subset of the linear functions on d . In the first approach, one takes these functions to be real valued. In the second approach, they are complex valued and linearly independent over C .

132 798

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We will describe the second approach; it is the approach that will actually be useful in what follows. Pick a complex structure J on M , invariant under affine translations, such that o is positive and of type (1 , 1). Then one can find n linear functions z' that are holomorphic in the complex structure J such that (1.18)

c r ~= +idz' A dZ'.

Let 9 be the prequantum line bundle introduced in our discussion of prequantization. We recall that 5 ' is to be a Hermitian line bundle with a connection V whose curvature form is -io. Since the (0, 2) part of o vanishes, the connection V gives 2 a structure as a holomorphic line bundle. In fact, 2 may be identified as the trivial holomorphic line bundle whose holomorphic sections are holomorphic functions tp and with the Hermitian structure ltpI2 = exp(-h) . By,with h = C ,Z'z' . Indeed, the connection V compatible with the holomorphic structure and with this Hermitian structure has curvature 8 8 ( - h ) = Cdt'd'z' = -io. Since H ' ( d , V (1)) = 0 , the prequantum line bundle just constructed is unique up to isomorphism. We now define the quantum Hilbert space X&, in which the Heisenberg group is to be represented, to be the Hilbert space $Z(,W', 2)of holomorphic Lz sections of 3.We recall that, by contrast, the prequantum Hilbert space consists of all L2 sections of 3' without the holomorphicity requirement. The required representation p of the Heisenberg group is the restriction of the prequantum action to the quantum Hilbert space. At the Lie algebra level, the z' act as multiplication operators, (1.19)

i P(Z

) v l = Z'V,

and the Z' act as derivatives with respect to the zi , (1.20)

p ( T i ) t p = -I/.

a

8Z' That this representation is unitary follows from the identity (1.21) which asserts that p ( T i ) is the Hermitian adjoint of p ( z ' ) . (Of course, with the chosen Hermitian structure on 3 , (x , w) = exp(- CiZ i z i ) XV 3

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Irreducibility of this representation of the Heisenberg group can be proved in an elementary fashion. This irreducibility is a hallmark of quantization as opposed to prequantization. Because of the uniqueness theorem for irreducible unitary representations of the Heisenberg group, the Hilbert space XQIJthat we constructed above is, up to the usual projective ambiguity, independent of the choice of J (as long as J obeys the restrictions we imposed: it is invariant under the afine translations, and w is positive and of type ( 1 , 1) ). It automatically admits a projective action of the group of all afine symplectic transformations, including those that do not preserve J . Infinitesimally, the independence of J is equivalent to the existence of a projective action of the Lie algebra of the afine symplectic group. Its existence follows from what we have said before; we have noted in (1.16) that given any representation X I + p ( x ’ ) of the Heisenberg Lie algebra, no matter how constructed, one can represent the Lie algebra of the afine symplectic group by expressions quadratic in the p ( x ’ ) . In the representation that we have constructed of the Heisenberg Lie algebra, since the p ( z ’ ) and ~ ( 3 ’ are)differential operators on d of order 0 and 1, respectively, and the Lie algebra of the afine symplectic group is represented by expressions quadratic in these, this Lie algebra is represented by differential operators of (at most) second order. By the action of this Lie algebra, one sees the underlying symplectic geometry of the affine space d , though an arbitrary choice of a complex structure J of the allowed type has been used in the quantization. Quantization of Kahler manfolds. In this form, one can propose to “quantize” symplectic manifolds more general than afine spaces. Let (d, w ) be a symplectic manifold with a chosen complex structure J such that w is positive and of type ( 1 , 1) (and so defines a Kahler structure on the complex manifold M ). Any prequantum line bundle 2 automatically has a holomorphic structure, since its curvature is of type ( 1 , 1 ) , and the Hilbert space @*(d, 2)can be regarded as a quantization of (a’, w ) . In this generality, however, Kahler quantization depends on the choice of J and does not exhibit the underlying symplectic geometry. What is special about the Kahler quantization of afine spaces is that in that case, through the action of the afine symplectic group, one can see the underlying symplectic geometry even though a complex structure is used in quantization. Most of this paper will in fact be concerned with quantization of special Kahler manifolds that are closely related to affine spaces. So we will now discuss Kahler quantization in detail, considering first some general

134 SCOTT AXELROD, STEVE DELL4 PIETRA %r EDWAIU3 WITTEN

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features and then properties that are special to affine spaces. To begin, we review some basic definitions to make our notation clear. An almost complex structure J on a manifold d is a linear operator from T d to itself with J 2 = - 1 , i.e. a complex structure on T d . On T d @ Q2we can form the projection operators nz = f [ 1 - iJ) and K~ =$( 1 iJ) . The image of K, is the subspace of T d C on which J acts by multiplication by i . It is called TI1'')& or the holomorphic tangent space. Similarly, T ( ' ' ' ) d is the space on which J acts by multiplication by - i , The transpose maps x z and act on T * d @ C . We define T'(',') and Tcio,')as their images. Given local coordinates a ' , we may define ( 1.22) dai = ~ , ( d a ' ) , da i = xSda2 , T h e statement that J is a complex structure means that we may pick our coordinates a' so that - dai actually is the differential of a complex valued function ai and da' is the differential of the complex conjugate a' . We should make contact with more usud notation. The usual complex and real coordinates are i i i z = x +iy f o r i = l , ..., n , (1.23) x' for i = 1 , ..., n , y'-" for i = n + 1 , ..., 212. So we have for i = 1 , ..., n , (1.24) for i = n + I , ..., 2n. We may decompose a 2-form CT as the sum of its (2, 0), (1 , 1) , and (0, 2) components:

+

@J

Il-f

In general, any real tensor can be thought of as a complex tensor with the indices running over i and 7 which correspond to a basis for T d 8 Ctl . We also assume that J is compatible with w in the sense that w ( J v , Jw) = w(v , w) for any v , w E T d . This amounts to the assumption that (1.26)

J'

w =-0J

JJ,",, = - w i , J J k . This is so exactly when w is purely of type ( 1 , 1) .

135 GEOMETRIC QIJANTIZ.4TION OF CHERN-SIMONS GAUGE THEORY

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We may form the map g = w o J from T d to T * d . Equivalently g is the J compatible nondegenerate symmetric bilinear form: ( I.27)

g ( v , u;) = w ( v , l u i ) for

‘u

, ‘UI

E

d.

Finally, we assume that J is chosen so that g is a positive definite metric. In summary, T s t is a complex manifold with a Riemannian metric, g , which is compatible with the complex structure and so that w = - g 0 J is a symplectic form. This is just the definition of a Kahler manifold; w is also called the Kahler form. A connection V on a vector bundle 7 over a Kahler manifold which obeys the integrability condition ( I .28)

0 = [V;,VT]

induces a holomorphic structure on -7,the local holomorphic sections being the sections annihilated by 8;.In particular, since w is of type (1 , I ) , the prequanturn line bundle 2 , which is endowed with a unitary connection obeying (1.29)

P I ,V , l =

-q,,

is always endowed with a holomorphic structure. It is this property that enables one to define the quantum Hilbert space ZQIJas H : 2 ( d , 9). Vuriarion of complex struclure. In general, given a symplectic manifold sf with symplectic structure w , it may be impossible to find a Kahler polarization-that is, a complex structure J for which w has the properties of a Kahler form. If however a Ihihler polarization exists, it is certainly not unique, since it can be conjugated by any symplectic diffeomorphisrn. To properly justify the name “quantization”, which implies a process in which one is seeing the underlying symplectic geometry and not properties that depend on the choice of a Kahler structure, one would ideally like to have a canonical identification of the as J varies. This, however, is certainly too much to hope for. In many important problems, there is a natural choice of Kiihler polarization-for instance, a unique choice compatible with the symmetries of the problem. We will be dealing with situations in which there is not a single natural choice of Kiihler polarization, but a preferred family F. For instance, for d affine we take 7 to consist of translationally invariant complex structures such that w is a Kahler form. In such a case, the spaces ZQ!-, = @>(sf, 2) are the fibers of a Hilbert bundle ZQ over 3 , ZQIS a subbundle of the trivial Hilbert bundle with total space

136 602

SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

A? = HL2( M , 2'x) 7 . We will aim to find a canonical (projective) iden'P tification of the fibers XQl,,as J varies, by finding a natural projectively

flat Hermitian connection 6% on ZQ. The parameter spaces L7 will be simply connected, so such a connection leads by parallel transport to an identification of the fibers of XQ. Let 8' be the subgroup of 8 consisting of those elements whose action on the space of complex structures on d leaves 7 invariant setwise. An element $ E B' maps ZQIJto ZQldJ in an obvious way. Using

-

ZQ

to identify ZQlbJ with ZQIr, we the projectively flat connection 6 get a unitary operator 41 : TQIJ XQl . We consider the association $ + $1 to represent a quantization of the symplectic transformation @ if the 61, are invariant (at least projectively) under parallel transport by . In this case we say that 4 is quantizable. It is evident that the syrnplectic transformations that are quantizable in this sense form a group '27'' ; for any J , 4 + 41 is a projective representation of this group (the representations obtained for different J 's are of course conjugate under parallel transport by dZQ ). Repeating this discussion at the Lie algebra level, we obtain the following definition of quantization of functions h whose flow leaves L7 invariant. (For d affine, h is any quadratic function.) Let 6 J = T V h ( J )E T J S be the infinitesimal change in J induced by h . Le; 6 be the trivial connection on the trivial bundle Zpr . The quantization of h can be written as a sum of first-order differential operators on r(Y , Zpr) , f

Q

(1.30)

ih = ipp,(h) + 6, ,- d q J h + constant. h

The first term is the naive prequantum contribution. The second term represents the fact that the prequantum operator should also be thought of as moving the complex structure. The third term is our use of 6% to return to the original complex structure so that h is just a linear transformation on the fibers of Zpr . To check that (1.30) leaves the subbundle XQ invariant we observe that acting on sections of ZQ

In the first line of (1.31), 6, h

,is the trivial connection acting in the

direction on sections of the trivial bundle

-

qr

J Y . In the second line

'137 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

803

it is the trivial connection on T d 8 C x F 7 . The first equality in { 1.31) follows from the facts that nFv annihilates holornorphic sections and that 8% takes holomorphic sections to holomorphic sections. The second equality follows from (1.9). Equation (1.31) shows that preserves holomorphicity as desired. The requirement that is independent of complex structure is the statement that +

a%h = 0.

( 1.32)

If S q is projectively flat this implies that quantization is a projective representation: (1.33)

Connection for quantizulion of a8ne space. We now turn to the case in which d is an affine syrnplectic manifold and 2 7 consists of translationally invariant complex structures. We take a' to be global affine coordinates. By the uniqueness theorem for irreducible unitary representations of the Heisenberg algebra we know that the projectively flat connection ~ 5 % must exist. It may be defined in several equivalent ways which we shall discuss in turn. 1. We first present a simple explicit formula for ~ 5 % and then show that it corresponds to any of the definitions below. The connection S q is given by 52-( C = d - @ y (1.34.1)

(1.34.2)

8'

=

MQVtVl with

- _

& = - b ( s J w - 1) 4-.

Here 6 J is a one form on Y with values in H o m ( T 9 , Ts') . We call 8"' the connection one-form for d2Q. It is a second-order differential operator on a ' acting on sections of 2. We use the superscript ' u p ' to distinguish it from the connection one-form which we will construct for quantization of the symplectic quotient A . To demonstrate that 6% preserves holomorphicity and is projectively flat, we need the variation with respect to J of the statements that .I2 = - 1 and that J is o-compatible (1.26),that is, (1.35) ( I ,361

0 =JSJ

+ 6 J J = J ' , S J J k t ~ 3 J ' , 3 ' ~= 2idJ',- - 2idJ'r

( O ~ J )=~ (o6.I) ,

Y

+ (wdJ);;

is symmetric.

138 801

S M T T AXELRQD, STEVE DELLA PIETRA & EDWARD WITTEN

Using these identities as we11 as the fact that V has curvature -iw , it is easy to check that S q preserves holomorphicity (so that it does in fact give a connection on the bundle ZQover Y ). To calculate the curvature

Rdzp = (6“$*PI2 we first observe that 4” A bup = 0 because holomorphic derivatives commute. To calculate SdKpwe must remember that the meaning of the indices 1 and 7 change as we change 1 . One way to account for this is to use only indices of type i and explicitly write x, wherever needed. Using the formulas i (1.37.1) dx, = - - 6 J 2

i

6iTF = + - 6 J , 2

(1.37.2) we find (1.38)

-

R f Q = -66”

= -B6JL+Jji - = - i T r ( n z 6 J h S J ) .

This is a two-form on Y whose coefficients are multiplication operators by constant functions, i.e., 8% is projectively flat as desired.

2. The essential feature of the connection that we have just defined is is a second-order differential operator. The reason for that 8”’= 6 this key property is that in quantization of an afine space, the Lie algebra of the afine symplectic group is represented by second-order differential operators. Indeed, a change 6 J of complex structure is induced by the flow of the Hamiltonian function i j (1.39) h = - + ( w J G J ) j j aa . In the discussion leading to ( 1.15), we have already defined the quantization of a quadratic function

h = h,a‘a’

(1.40)

+ h p ‘ + h,

by symmetric ordering, (1.41)

p ( h ) = d = h,+(d’&J

+ &%} + hid’ + h,.

This preserves holomorphicity for any complex structure J and gives a . Accordrepresentation of the quadratic Hamiltonian functions on

&“elJ

ing to (1.30) (dropping the constant), dup= 6 - SRpe is to be simply ih - ip,,(h) , with h in (1.41). This leads to the definition (1.34) of the connection 8%

.‘

2Note that ( I .30),with a constant included, holds true for arbitrary h , and not just those of the form (1.39). By properly including the “melaplectic correction” we can actually find a flat connection and arrange for all unwanted constant factors to vanish.

139 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

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3. One natural candidate, which exists quite generally, for the connection on TQ is the "orthogonally projected" connection for XQconsidered as it subbundle of the trivial Hilbert bundle Rpr. It may be defined by (1.42)

< rypqy' >=< ypw' >

for y , ry' E r ( 7 ,xQ>.

Although we may write this formula down quite generally, there is no general reason that it should yield a projectively flat connection. However, in the case at hand, we may check that this definition agrees with that of points I and 2 above. This is so because 8 ' does not change form after integrating by parts, and @ ''V = 0. This implies that our connection is in fact unitary. 4. Closely related to point 3 is the fact that 6% may also be described as the unique unitary connection compatible with the holomorphic structure on &"n. We discussed the holomorphic structure on ZQpreviously. To describe it explicitly we first note that a complex structure on the space 7 is defined by stating that the forms S J i are of type ( 1 , 0) and that

-

the forms 6JJ, are of type (0, 1) . In other words, the ( 1 , 0) and ( 0 , 1) pieces of d J are (1.43)

6J"'o'

=

n z 8 J z 5 and 6J"") = QJn,.

kt 6'1 ,O) and 6'*. " be the holomorphic and antiholomorphic pieces of the trivial connection on Zpr . Holornorphic sections of %r are those sec-

tions which are annihilated by the 5 operator 6". I' . We define sections of ZQto be holomorphic if they are holomorphic as sections of Z . The Pf integrability condition that we can find a local holomorphic trivialization of ZQis satisfied if we can show that 6"' ') leaves XQ invariant. But this is true since for y a section of ZQ, we have

The statement that 8% as defined in point 1 above is compatible with the holomorphic structure is just the observation that S q " ' ') = 6'" since d only depends on S J " v o ' . lb. Symplectic quotients and pushing down geometric objects. Amne spaces by themselves are comparatively dull. The facts just described get considerably more interest because they have counterparts for symplectic quotients of affine spaces. Our applications will ultimately come by considering finite-dimensional symplectic quotients of infinite-dimensional affine spaces.

140 806

SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

Symplectic quotients. To begin, we discuss symplectic quotients of a general symplectic manifold a? , o . Suppose that a group .Y acts on a? by symplectic diffeomorphisms. We would like to define the natural notion of the “quotient” of a symplectic manifold by a symplectic group action. This requires defining the “moment map”. Let g be the Lie algebra of .Y and T : g -, Vect(a?) be the infinitesimal group action. Since f7 preserves o ,the image of T consists of symplectic vector fields. A comoment map for the f7 action is a .Y invariant map, F , from g to the Hamiltonian functions on a? (where g acts on g by the adjoint action). To express this in component notation, let La be a basis for g and Ta= T(La) . Since T is a representation, we have (1.45)

[Ta

3

Tbl = fabCTc

3

where fab‘ are the structure constants of f7. The comoment map is given by functions Fa whose flow is Ta . Invariance of F under the connected component of f7 is equivalent to the statement that F is a Lie algebra homomorphism, (1.46)

For each A E a? , F,(A) are the components of a vector in the dual space gv . We may view F as a map from a? to gv . Viewed this way it is called a moment map. Since the moment map and (0) c 3 are g invariant, so is F-’(O). The quotient space A = F-’(O)/f7 is called the symplectic or MarsdenWeinstein quotient of d by 27 With mild assumptions, A is a nonsingular manifold near the points corresponding to generic orbits of Z? in F - ’ (0) . We will always restrict ourselves to nonsingular regions of A , although we do not introduce any special notation to indicate this. We have the quotient map:

.’

(1.47)

We may define a symplectic structure, 0 ,on A by (1.48)

&~(6, 6)= ~

w ) for 6,6E TAM.

~ ( 7 1 ,

’This quotient plays a role in elementary physics. If a ’ is the phase space of a physical system, and .F as a group of symmetries, then d is simply the phase space for the effective dynamics after one restricts to the level sets of the conserved momenta and solves the equations that can be integrated trivially duc to group invariance. Alternatively, if the F, are constraints generating gauge transformations of an unphysical phase space, then M is the physical phase space left after solving the constraints and identifying gauge equivalent configurations.

141 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

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By 27-invariance and the fact that (1.49)

w(T,, u ) = o for u E TF-’(o) ,

the definition is independent of our choice of A , v , and w . This is the first example of our theme of “pushing down” geometric objects from &’ to A . The basic principle is that the objects (symplectic form, complex structure, bundles, connections, etc.) that we consider on &’ are 27-invariant so that when restricted to F-l(O) they push down to the corresponding objects on d . Pushing down the prequantum line bundle. In order to push down the prequantum line bundle we must assume we are given a lift of the 27action on d to a 27-action on 9’ which preserves the connection and Hermitian structure, i.e., an action by elements of E . The Lie algebra version of such a lift is just a moment map. We may define the pushdown bundle 2 by stating its sections, ( 1S O )

J

r(A,2) = r(F-l(o),2) ,

The superscript tells us to take the F-invariant subspace. A line bundle on A with (1.50) as its sheaf of sections will exist if 27 acts freely on F-’(O) (or more generally if for all x E F-’(O), the isotropy subgroup of x in 27 acts trivially on the fiber of 9 at x ) . A section w E T ( F - ’ ( O ) ,9) is invariant under the connected component of S precisely if (1.51)

0 = ip(F,)pr = DTo v/ .

The pushdown connection may be defined by (1.52)

V c w = Vvw.

Here u is any vector field on F-’(O) which pushes forward to d on A?. By (1.51) the right-hand side of (1.52) is independent of our choice of o . To show that ( 1.52) is a good definition we must show that the right-hand side is annihilated by VTa. This can be done using (1.29) and (1.49). Similarly, one can check that V has curvature -ia . Pushing down the complex structure. To proceed further, we assume that &’ is an affine space and that F is a Lie subgroup of the affine symplectic group such that (1) there is an invariant metric on the Lie algebra g , and (ii) the action of 27 on d leaves fixed an affine Kahler polarization. We continue to assume that the 27-action on d has been lifted to an action on 2’with a choice of moment map. We let F be the space of Kahler

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SCOlT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

polarizations of M that are invariant under the affine translations and also F-invariant. 7 is nonempty and contractible. Since 27 acts linearly, there is a unique extension of the 9-action to an action of the complexification F, which is holomorphic as a function from F, x d to d . In a closely related context of compact group actions on projective spaces, a basic theorem of Mumford, Sternberg, and Guillemin [ 16; 24, p. 1581 asserts that the symplectic quotient d of at’ by 9 is naturally diffeomorphic to the quotient, in the sense of algebraic geometry, of M by 9,. Since the latter receives a complex structure as a holomorphic quotient, A receives one also. Those results about group actions on projective spaces carry over almost directly to our problem of certain types of group actions on affine spaces, using the fact that subgroups of the affine symplectic group obeying our hypotheses are actually extensions of compact groups by abelian ones. We will not develop this explicitly as actually the properties of the geometry of A? that we need can be seen directly by local considerations near F - ’ ( O ) , without appeal to the “global” results of geometric invariant theory. For instance, let us give a direct description of an almost complex structure 1 obtained on d which coincides with the one given by its identification with M / F c when geometric invariant theory holds. (By the methods of 33a below, this almost complex structure can be shown to be integrable without reference to geometric invariant theory.) Let gc be the complexification of the Lie algebra g . The action of Fc is determined by the action of 2 7 and g, . Since we want it to be holomorphic in 27,, the complex Lie algebra action T, : g, Vect(d ) must be --f

T,(L,) = T, ,

(1.53)

T,(iL,)= JT,.

At every A E F-’ (0) , we have the following inclusion of spaces:

TF-’(O) (1.54)

c

TM

U

T(g)

U

c Tc(gc)

So we have the map (1.55)

TAM

2 [TF-I(O)/T(g)lA

--+

TM/T,(g,).

One can show that this is an isomorphism by simple dimension counting. As a quotient of complex vector spaces, the right-hand side of (1.55) receives a complex structure. Therefore, under the identification (1.55), TAM receives a complex structure. (By F-invariance, the choice of a point A in the orbit above A is immaterial.) The fact that (1.55) is

143 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

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an isomorphism is just the infinitesimal version of the statement that A’ S d / g c . This isomorphism may also be proved using the Hodge theory description that we will present in $3. For instance, surjectivity of ( 1 - 5 5 ) follows since the representative of shortest length of any vector in T d / T c ( g c )actually lies in T F - ’ ( O ) . This argument is just the infinitesimal version of the proof that A E d / F c ,which uses a distance function to choose preferred elements on the Fc orbits. Geometric invariant theory also constructs a holomorphic line bundle L? over sf/q,,such that, if n : d + A’/?is the natural projection, then 9’ = x’(L?’). Moreover, L? has the property that (1.56)

2 y = H * [ d / F C2). ,

HO(d,

This equation is a holomorphic analog of ( 1.50). Under the identification of A with d / q ,the two definitions of 2 agree. Connection for quanlizalion of A . We let
The second to last equality in (1 -57) is due to the fact that, for holomorphic sections, .Yc-invariance is equivalent to F-invariance. The F-action on XQ is just the prequantum action. It is invariant under parallel transport by 13%. Thus we may identify

pQwith a subbundle of

XQ which is preserved under parallel transport by 8% . Therefore S% restricts to the desired projectively flat connection S 4 on the subbundle Of course, the Hermitian structure of PQis the one it inherits as a subbundle of ZQI In finite dimensions, this is a complete description of the projectively flat connection on pQ; there is no need to say more. However, even in finite dimensions, one obtains a better understanding of the projectively flat connection on A$ by describing it as much as possible in terms of the intrinsic geometry of A . Moreover, the main application that we envision is to a gauge theory problem in which S’ and E are infinite dimensional, though the symplectic quotient A is finite dimensional. In

6.

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SCOTT AXELROD, STEVE DELLA PlETRA & EDWARD WITTEN

this situation, the “upstairs” quantum bundle XQ is dificult to define rigorously, though the “downstairs” bundle yQ is certainly well defined. Under these conditions, we cannot simply define a connection on PQby restricting the connection on ZQto the F-invariant subspace. At best, we can construct a “downstairs” connection on by imitating the formulas that one would obtain if ZQ,with a projectively flat connection of the standard form, did exist. Lacking a suitable construction of &“e, one must then check ex postfacto that the connection that is constructed on 2Qhas the correct properties. This is the program that we will pursue in this paper. The desired connection on pQ should have the following key properties. The connection form should be a second order differential operator on .MI since the connection upstairs has this property and since a second-order differential operator, restricted to act on Fc-invariant functions, wiIl push down to a second-order differential operator. The Connection should be projectively flat. And it should be unitary. In the gauge theory problem, we will be able to understand the first two of these properties. In fact, we will see that there is a natural connection on PQthat is given by a second-order differential operator, and that this connection is projectively flat. Most of these properties (except for vanishing of the ( 2 , 0) part of the curvature) can be understood in terms of the local differential geometry of M . Unitarity appears more difficult and perhaps can only be understood by referring back to the underlying infinite-dimensional affine space d . It is even conceivable that a construction of the “upstairs” bundle ZQis possible and would be the best approach along the lines of gauge theory to understanding the unitarity of the induced connection on the gc-invariant subbundle. Ic. Theta functions. As an example of the above ideas, we consider the case in which d is a 2n-dimensional real vector space, with affine symplectic form w and prequantum line bundle 2 , and .F is the discrete group of translations by a lattice A in d whose action on d lifts to an action on 2 . Picking such a lift, and taking the A-invariant sections of over the torus M = & / A . 9, we get a prequanturn bundle To quantize M , we pick an afline complex structure J on d which defines a Kahler polarization; it descends to a complex structure j on A . The existence of the prequantum line bundle 9 , with curvature of type (1 , 1) , means that the complex torus M is actually a polarized abelian variety. The Hilbert space gblJ = H 0 ( M ,2) serves as a quantization of A .

*Q

145 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

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This Hilbert space pQIJ is, in classical terminology, the space of theta functions for the polarized abelian variety (A‘,2) . Classically, one then goes on to consider the behavior as J varies in the Siege1 upper half plane R , which parametrizes the affine Kahler polarizations of d . Thus, the quantum Hilbert spaces PQlJ, as .I vanes, fit together into a holomorphic bundle PQ over R . Since the work of Jacobi, it has been known that it is convenient to fix the theta functions, in their dependence on J , to obey a certain “heat equation”. While it is well known that the theta functions, for fixed J , have a conceptual description as holornorphic sections of the line bundle 9, the conceptual origin of the heat equation which fixes the dependence on J is much less well known. In fact, this heat equation is most naturally understood in terms of the concepts that we have introduced above. The heat equation is simply the projectively flat connection d% on the quantum bundle pQover 12 which expresses the fact that, up to the usual projective ambiguity, the quantization of A is canonically independent of the choice of Kahler polarization j . The usual theta functions, which obey the heat equations, are projectively parallel sections of 2Q. The fact that they obey the heat equation means that the quantum state that they represent is independent of J . Actually, the connection d 4 as we have defined it in equation (1.34) is only projectively flat. The central curvature of this connection can be removed by twisting the bundle 9 by a suitable line bundle over R (with a connection whose curvature is minus that of 8% ). The heat equation as usually formulated differs from d 4 by such a twisting. In the literature on geometric quantization, the twisting to remove the central curvature is called the “metaplectic correction”. We have not incorporated this twisting in this paper because it cannot be naturally carried out in the gauge theory problem of interest. In the rest of this subsection, we shall work out the details of the relation of the heat equation to the connection d 4 . These details are not needed in the rest of the paper and can be omitted without loss. Prequantization of A . A prequantum Herrnitian line bundle on .& with connection with curvature - i o is, up to isomorphism, the quotient of d x C under the identifications ( A , v ) ( A + A, e , ( A ) v ) , where the “multipliers” eA can be taken to be of the form N

(1.58)

e A ( A )= c(A)exp(-$w(A - A , , A)).

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SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WI'ITEN

Here A, is a point in M I A and may be considered as parametrizing the possible flat line bundles over A , and € ( A ) E {& 1) satisfies (1.59)

A section of

€0, +A,)

9 is a function

( 1.60)

w,

=~ ( q G J F 1 ) s :d

+

c

?

q/2n

satisfying

s ( A + A ) = e,(A)s(A).

The connection and metric can be taken to be

Family of complex structures on A . The Siege1 upper half-space d of complex, symmetric n x n matrices 2 with positive imaginary part parametrizes affine Kihler polarizations of d and A . For Z in i-2 we may define the complex structure J, on M as follows. Fix an integral basis A, for A so that in terms of the dual coordinates { x , } on M , w (1.62) -= 6,dx' A d x i f n .

2n

Such a basis always exists (see [15]). (The di are nonzero integers called elementary divisors; they depend on the choice of A .) The complex structure J, is defined by saying that the functions

are holomorphic. In terms of these, w = ni

(1.64)

-

dAi (Im Z);' d A J . ij

The map 2 H J, is a holomorphic map, which may be shown to map onto F. We easily compute (1.65)

( 6 ( l P Jz)L7 0)

= -((d(',') z ) ( I m z ) - ' ) i j .

%

The bundle and the connection 8 % . The quantum Hilbert space pQIJz is the space of holomorphic sections of L? and is thus identified with functions s satisfying (1.60) and (1.66)

0 = V7s(Z, A ) =

147 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

813

As Z varies, such functions correspond to sections of the Hilbert bundle

pQ Y --$

( pulled back to

R ).

It is not hard to write down explicitly the action of the connection ~ 5 % on sections of pQrealized in this way. Recall that ( 1.67)

SaQ = & - M g V . V . 1 J'

Mu

-I

L

u

- : ( ~ J w ) -*

(1.6 1 ) gives the actions of the covariant derivatives V i on s ; and (1.64) -

and (1.65) give expressions for w and S ( l ' o ) J . We need an expression for the action of S on s. For any function s ( Z , A ) , let 6 " s ( Z , A ) denote the exterior derivative in the Z directions when A is considered

independent. It is given by the formula

[

(1.68) d = 6'

+ -(ALi:

0 ((S("o)Z)(I~Z)-')f,6,A1

where the second line takes into account the dependence of the coordinates A' and A' on Z . Substituting this expression and the formulas for V, ,

6

w , and 6 J into (1.67) gives a formula for 6% acting on sections of realized as functions s satisfying (1.60). A more convenient formula, from the point of view of making contact with the traditional expressions for theta functions, however, is obtained by changing the trivialization of 9= d x C so that holomorphic sections are represented by holomorphic functions of Z and A . Such a change in trivialization corresponds to a function g : R x d + U2 satisfying

To obtain the usual theta functions, we take ( 1.70)

g ( 2 , A ) =e xp ($A'(IrnZ)~'(A'-

AL)

If we now write O ( 2 , A ) = s ( Z , A ) g ( Z , A ) , then the conditions that s ( Z , A ) represent a holomorphic section of 2Q are that 8 is holomorphic as a function of 2 and A and has the periodicities (1.71)

e(z,A + 2,) = q;if)e(z, A) e ( Z , A + A r t n ) = c(A,+,)exp(-2niA1

- n i Z , , ) e ( Z ,A ) .

148 814

SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

The classical theta functions satisfy these conditions. The choice of E is called the theta characteristic. Acting on e ( Z , A ) , the holomorphic derivatives become

-

-1 e(z

( A L - A')(AL - A')

,A).

Combining the equations (1.64), (1.65), and (1.72), we find after a short calculation 1 &' i 6 Q = d - -Tr(IrnZ-ld("o)Z), (1.73) Q

4

where

The modified connection d@Q' has zero curvature.

The equation

d%'(' 'o)B(Z,A ) = 0 is the heat equation satisfied by the classical theta functions. It may be shown that the modified connection 8%' is that obtained when account is taken of the "metaplectic correction". Thus the dependence of the classical theta functions on 2 is naturally interpreted from this point of view as the statement that as Z varies, the theta functions e ( Z , .) E PQlz represent the same quantum state.

2. T h e gauge theory problem In this section we will describe the concrete problem that actually motivated the investigation in this paper. It is the problem of quantizing the moduli space A% of flat connections on a two dimensional surface C (of genus g , oriented, connected, compact, and without boundary). This moduli space can be regarded (as shown in [3]) as the symplectic quotient of an underlying infinite dimensional affine space by the action of the gauge group. Our goal is to explain concretely how this viewpoint leads to a projectively flat connection that makes possible quantization of

149 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

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A . In this section, we will aim for simplicity rather than precision and rigor. The precision and rigor will be achieved in later sections (which are independent of this one). Our goal in this section is to explain as directly as possible and without any unnecessary machinery the precise definition of the projectively flat connection that is used for quantization of A , for the benefit of readers who may have use for the formulas. Also, we will explain in a language that should be familiar to physicists how one sees from this (2 + 1)-dimensional point of view a subtlety that is well known in (1 + 1)-dimensions, namely the replacement in many formulas of the "level" k by k + h , with h being the dual Coxeter number of the gauge group. Preliminaries. Let G be a compact Lie group, which for convenience we take to be simple. The simple Lie algebra Lie(G) admits an invariant positive definite Killing form ( , ) , unique up to multiplication by a positive number. If F is the curvature of a universal G-bundle over the classifying space B G , a choice of ( , ) enables us to define an element 1 = ( F , A F ) of H 4 ( B G , R) . We normalize ( , ) so that A/27r is a de Rham representative for a generator of H4(BG, Z)E Z , This basic inner product ( , ) is defined in down to earth terms in the appendix. Let E be a principal G-bundle on the surface C . Let A be a connection on E . Locally, after picking a trivialization of E , A can be expanded A=

(2.1)

C A " . T,,

where T, is a basis of the Lie algebra Lie(G) of G . We can take T, to be an orthonormal basis in the sense that (T, , Tb)= dab. In this basis the Lie algebra Lie(G) may be described explicitly in terms of the structure constants fab' : (2.2)

9

Tbl

= fabcTc*

d

Defining fabc = f a b dcd, invariance of ( , ) implies that fabc is completely antisymmetric. One has b

(2.3) fab'fcd = 2dad where h is the dual Coxeter number of G (as defined in the appendix). Let d be the space of smooth connections on E . d is an affine space; its tangent space T d consists of one forms on C with values in ad(E) . d has a natural symplectic structure, determined by the symplectic form '

7

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SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

The symplectic form that we will actually use in quantization is (2.5)

W=k-wo,

with k a positive integer called the "level". After picking local coordinates on Z,the connection dA can be described by the explicit formula for covariant derivatives

(2.41

DpdJ= (ap+ Ac"To)$

1

with denoting a section of any associated bundle to E , And the curvature form is

F,"" = aPn,"- a,~," -+ h d o ~ p C ~ v d . (2.7) The ''gauge group" B is the group of automorphisms of E as a principal bundle. An element g of the gauge group transforms the connection

4 bY

-

dA g dA * g-' (2.8) F is an infinite-dimensional Lie group whose Lie algebra g consists of the smooth sections of ad(E) . The action of g ou d is described by the map (2.9) c -+ -dAE from T(ad(E)) to T A ( & ) . The action of the group 9 on the space ssf of connections preserves the symplectic structure w . The moment map F : d 4 g" is the map which takes a connection A to its curvature. The curvature F is a twoform with values in ad@) and is identified with an element of gv by the pairing (2.10)

for E ~ r ( a d ( E ) ) . The zeros of the moment map F-' (0) thus consist of flat connections, and the quotient A = F'(O)/F is therefore the moduli space of flat connections on E , up to gauge transformation. If G is connected and simply connected, M is simply the moduli space of flat G bundles over C. If, however, G is not connected and simply connected, there may be several topological types of flat G bundles on C, and A is the moduli space of flat bundles with the topological type of E . The general arguments about symplectic quotients apply in this situation, so that the symplectic structure wo on d descends to a syrnplectic

151 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

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structure on d , which we will call Go. Our normalization convention on ( , ) ensures that t30/271 represents an integral element of H 2 ( L JW) , so that 0,is at leas1 prequantizable. Hulomurphic interpretation. Our goal is to quantize A?' with the symplectic structure w,, and more generally with the symplectic structure li, = k Q D . One of the important ingredients will be a construction of a suitable complex structure on sf, To do so, we pick a complex structure J on E (such that the orientation on Z determined by J is the given one). This choice induces a complex structure .Id on d , as follows. The tangent space TJZ consists of one forms on E with values in ad(E) . Given a complex structure J on I;, we define (2.11)

J,dA

-J d A ,

z=

Relative to this complex structure (2.12) T d= Old 1

6AE T d .

@

T(o'

1) Jf,

where T""'d and T'"''& consist respectively of (0, 1)-forms and ( 1 , 0)-forms on Z with values in ad(E) . (This is opposite to the choice of complex structure on &f which appears frequently in the physics literature in which the holomorphic directions are represented by holomorphic oneforms on C. The above choice, however, is more natural since it is the antiholomorphic one-forms which couple to the 5 operator on C , and it is this operator, which defines a complex structure on the bundle E , which we want to vary holomorphically as a function of the complex structures on Z and d .) It is evident that with the choice (2.12) the symplectic form w on &f is positive and of type (1 , 1) By analogy with the discussion in $1 of group actions on finite-dimensional affine spaces, one might expect that once the complex structure .TN is picked, the action of the gauge group 9 can be analytically continued to an action of the complexified gauge group .Fc (which, in local coordinates, consists of smooth maps of C to G, , the complexification of G ) . It is easy to see that this is so. Once the complex structure J is picked on C, the connection dA can be decomposed as (2.13)

dA = d A + B A ,

where aA and T A are the ( 1 , 0) and ( 0 , 1 ) pieces of the connection, respectively. The Fcaction on connections is then determined by the formula (2.14)

-

a,

+

g . 3A. g - ' .

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SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

It is evident that this action is holomorphic. (2.14) implies the complex conjugate formula (2.15)

with g the complex conjugate of g . Equation (2.14) has the following interpretation. For dimensional reasons, the (0, 2) part of the curvature of any connection on a Riemann surface C vanishes. Therefore, for any connection A , the 3, operator gives a holomorphic structure to the principal G,-bundle E, ( E , is the complexification of E ). The holomorphic structures determined by two such operators T A and 3A,are equivalent if and only gA and BA, are conjugate by a transformation of the kind (2.141, that is, if and only if they are on the same orbit of the action of ' B on d . Therefore, the of equivalence classes of set &/% can be identified with the set dJ''' holornorphic structures on E, . Under a suitable topological restriction, for instance, if G = SO(3) and E is an SO(3) bundle over C with nonzero second Stieffel-Whitney class, 3 acts freely on . In this case, we will simply refer to as MJ; it is the moduli space of holornorphic G, bundles over C , of specified topological type. The subscript in dJis meant to emphasize that these bundles are holomorphic in the complex structure J . In general, reducible connections correspond to singularities in the quotient d / F c;in this case, instead of the naive set theoretic quotient d/.!?,, one should take the quotient in the sense of geometric invariant theory. Doing so, one gets the moduli space JJ of semistable G, bundles on C, of a fixed topological type. For finite-dimensional affine spaces, we know that the symplectic quotient by a compact group can be identified with the ordinary quotient by the complexified group. Does such a result hold for the action of the gauge group on the infinite-dimensional afine space d ? The symplectic quotient of *@' by 27 is the moduli space M of flat connections on E ; the ordinary quotient of .& by gives the moduli space A, of holomorphic structures on E , . In fact, there is an obvious map i : M 4 MJ coming from the fact that any flat structure on E determines a holomorphic structure on Ec . Using Hodge theory, it is easy to see that the map i induces an isomorphism of the tangent spaces of A and M J . Indeed, T A = H ' ( C , ad(E,)). (Here H * @ , ad(E,}) denotes de Rham c o b rnology of E with values in the flat bundle ab(EA).) According to the

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Hodge decomposition, the complexification of H ' ( Z , ad(E,}) is (2.16)

H i ( E , ad{,!?,)) = H " " ) ( Z , ad(E,)) 8 H ( ' " ! ( Z , ad(E,)),

and on the right-hand side we recognize the ( 1 , 0) and (0, 1) parts of the complexified tangent bundle of M, . Actually, it is a fundamema1 theorem of Narasimhan and Seshadri that the map i is an isomorphism between A" and A'. Just as in our discussion of the comparison between syrnplectic quotients and holomorphic quotients of finite-dimensional affine spaces, the symplectic form w becomes a Kahler form on .,&' under this isomorphism. In particular, w is of type ( 1 , 1) , so any prequantum line bundle L? over kf becomes a holornorphic line bundle on LJ (and in fact taking the holomorphic sections of powers of L? leads to an embedding of Mf in projective space). Thus! the Narasimhan-Seshadri theorem gives us a situation similar to the situation for symplectic quotients of affine spaces in finite dimensions. As the complex structure J on C varies, the MJ vary as Kahler manifolds, but as symplectic manifolds they are canonicalky isomorphic to a fixed symplectic variety 1 . Since diffeomorphisms that are isotopic to the identity act trivially on A , isotopic complex structures J and J' give the same Hodge decompositions (2.16), not just "equivalent" ones. Therefore, the complex structure of A'J depends on the complex structure I on 1 only up to isotopy. The moduli space of complex structures on Z up to isotopy is usually called thc Teichmuller space of L ; we will denote it as 7. A point t E LT does not determine a canonical complex structure on Z (it determines one only up to isotopy). But in view of the above, the choice of E does determine a canonical complex structure J, on the moduli space A' of flat connections. The complex structure J , varies holomorphically in t . Therefore, the product 4 x iT, regarded as a bundle over 7 , gets a natural complex structure, with the fibers being isomorphic as syrnplectic manifolds but the complex structure of the fibers varying with I . PrequanlizaEion and the action of the mapping c l a ~ sgroup. There are several rigorous approaches to constructing a prequantum line bundIe .L? over &-that is, a unitary line bundle with a connection of curvature -iO. Since for G a compact. semisimple Lie group, bi(,#?) vanishes and h"(.,&',U (1)) is a finite set, there are finitely many isomorphism classes of such prequantum line bundles. Holornorphically, one can pick a complex structure J on C, and take 9 to be the determinant line bundle Det, of the 8, operator coupled to

154 a20

SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

the associated bundle E(R)determined by some representation R of the gauge group. If this line bundle is endowed with the Ray-Singer-Quillen metric, then the main result of (271 shows that its curvature is -zl(R).6+, , where I(R) is defined by -TrR(TaTb) = 1(R)6,, . Though this construction depends on a choice of complex structure on X , the line bundle Det,, as a unitary line bundle with connection, is independent of the complex structure chosen since the space of complex structures is connected and (according to the remark at the end of the last paragraph) the set of isomorphism classes of prequantum bundles is a finite set. (The isomorphism among the Det, as J varies can also be seen more explicitly by using the Quillen connection to define a parallel transport on the Det, bundle as J varies.) Thus, if k is of the form 2c2(R) for some not necessarily irreducible representation of G , the prequantum bundle can be defined as a determinant line bundle. From the point of view of the present paper, it is more natural to construct the prequantum line bundle by pushing down a trivial prequantum bundle 9 from the underlying infinite-dimensional affine space at' . This can be done rigorously [28].4 Once E is fixed, there is some subgroup rZ,E of the mapping class group of X consisting of diffeomorphisms 4 that fix the topological type of E . rZ has an evident action on 4 coming from the interpretation of the latter as a moduli space of representations of n,(C). The goal of the present paper is to construct an action of rZ,Eon the Hilbert spaces obtained by quantizing A . For this aim, we lift the action of the mapping class group of C on A to an action (or at least a projective action) on the prequantum line bundle 9. Actually, if the prequantum line bundle is unique up to isomorphism, which occurs if G is connected and simply connected in which case H' ( d ,U ( 1)) = 0 , then at least a projective action of the mapping class group is automatic. Even if the prequantum line bundle is not unique up to isomorphism, on a prequantum line bundle constructed as a determinant line bundle one automatically gets a projective action of the mapping class group. (If I#J is a diffeomorphism of 2 , and 2 has been constructed as Det, for some J , then I#J naturally maps det, to det4,, which has a projective identification with det, noted in the last paragraph.) The construction of prequantum line bundles via pushdown is also a natural framework for constructing actions (not just projective actions) of the 4The authors of that paper consider explicitly the case of G = S U ( 2 ) , but it should be straightforward to generalize their constructions.

155 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

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mapping class group. We will now sketch how this construction arises from the three-dimensional point of view. This will be discussed more precisely elsewhere. To start with, we choose an element of H 4 ( B G , Z) . This allows us to define an R/2aZ valued "Chern-Simons" functional of connections on G bundles over three-manifolds with boundary, as discussed in [ 6 ] . The functional S obeys the factorization property that e'S(Mo3E09Ao) = ~ ' ~ ' E (I , A ~I ) . 1 e'S(M2vE2vAz) if the three-manifold Mo with bundle E, and connection A , is obtained by gluing M , , E , , A , to M 2 , E2,A , . The gluing is accomplished by an identification, @ : EJZl + E 2 l Z z ,of the restriction of E, to some boundary component C, of M , with the restriction of E, to some boundary component X, of M 2 . Now fix a bundle E over a surface C . Let A be an element of the space d of connections on E . For i = 1, 2 , let ( M ,, E, , A , ) be obtained by crossing (C,E , A ) with an interval. So for each automorphism @ of E (not necessarily base preserving) we may form M, , E, , A , by gluing. We thus obtain a function p ( @ , A ) = e rS(Mo9Eo,Ao) from Aut(E) x M to U(1). By factorization, p is a lift of the action of Aut(E) on d to the trivial line bundle over d . By restricting to flat connections and factoring out by the normal subgroup Aut'(E) of Aut(E) consisting of automorphism which lift diffeomorphisms of Z which are connected to the identity, we obtain the line bundle L? over A?? with an action of the mapping class group rZ,€= Aut(E)/Aut'(E). Finally, we can introduce the action of the mapping class group rZ,€ on the quantum bundle PQover 7. The fiber h$I, of pQ, over a point t E7, is simply H0(A,, ,p). Obviously, since the mapping class group its action on 7lifts naturally to an action has been seen to act on 2 , on 2Q. Our goal is to construct a natural, projectively flat connection V on the bundle R% --,7 . Naturalness will mean in particular that V is invariant under the action of rZ,E.A projectively flat connection on 7 that is r, & i v a r i a n t determines a projective representation of r z , E . Thus, in this way we will obtain representations of the genus g mapping class groups. These representations are genus g counterparts of the Jones representations of the braid group. The precise connection with Jones's work depends on the following. At least formally, one can generalize the constructions to give representations of the mapping class groups r g q nfor a surface of genus g with n marked points P, , . . , P, . This is done by considering flat connections on C - U,P, with prescribed monodromies around the P, . Jones's

-

a22

SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

representations would then correspond to some of the representations so obtained for I"o,n. For simplicity, we will only consider the case without marked points. Construction of the connection. We will now describe how, formally, one can obtain the desired connection on the bundle #Q-, 7 ,by formally supposing that one has a quantization of the infinite-dimensional affine space &' , and "pushing down" the resulting formulas from d to A . To begin with, we must (formally) quantize d . This is done by using the complex structure Jd on 9 that comes (as described above) from a choice of a complex structure J on E, The prequantum line bundle 2 over a ' is a unitary line bundle with a connection V of curvature -io , To describe this more explicitly, let z be a local complex coordinate on a , and define d/sAZ"(z) and d/dA+"(z) by

for u and H adjoint valued (1, 0) and (0, 1) forms on C and d 2 z = i d z d 7 . (We will sometimes abbreviate J, d 2 z as J, .) Then the conaection V is characterized by

along with

, (In (2.18), d,,(z, w ) d z d C represents the identity operator on T ( K @I ad(E)) ; that is Jxw d7Tdwd,,u, = u, . We also have

lz

d7dz

= i&)

The "upstairs" quantum Hilbert space consists of holomorphic sections Y of the prequantum bundle, that is, sections obeying (2.20)

What we actually wish to study is the object pQIJ = H o ( A J, 9) iniroduced in the last subsection. The latter is perfectly well defined. Formally,

157 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

823

this well-defined object should be the Fc-invariant subspace of the larger space ZQIJ. At present, the Iatter is ill defined. But proceeding formally, we will attempt to interpret +@ I as the invariant subspace of ZQ I J . Supposing for simplicity that Q!is connected, the .Fc-invariant subspace is the same as the subspace invariant under the Lie algebra g, of Fc.The condition for gc-invariance is

q-

(2.21) Here F;i(z) is the curvature of the connection A , which enters because it is the moment map in the action of the gauge group on the space of connections. Also, DF = + A+ is the (0, 1) component of the exterior derivative coupled to A . Green 5 functions. To proceed further, it is convenient to introduce certain useful Green's functions that arise in differential geometry on the smooth surface Z . In what follows, we will be working with flat connections on C-corresponding to zeros of the moment map for the .F-action on .a' For flat connections, the relevant Green's functions can all be expressed in terms of the Green's function of the Laplacian, or equivalently the operator 23 . For simplicity, we shall assume that this operator has no kernel. Let n1 : Z x Z -, Z, for i = I , 2 . be the projections on the first and second factors respectiveIy. Let E, = n:(E),for i = 1 , 2 . Similarly, let K be the canonicaI line bundle of C , regarded as a complex Riemann surface with complex structure J , and let K,= n:(lY) . The Green's function for the operator 58 is a section 4 of ad(E,) @ ad(,!?,)" over Z x C - A ( A being the diagonal), such that

&

(2.22)

D7DZ#=Jz,w)= dUbS7JZ,w).

(Here SYz(z,w ) satisfies J d W d ~ $ ~ v , = , vTz .) It is convenient to =u also introduce

(2.23)

Lzab(Z

1

w,= D l @ i l b ( z w, 3

and (2.24)

-

Lz'b(z

1

w,= D$'b(z,

w).

They are sections, respectively, of K,@ad(E,)@ad(E2)"and R , @ a d ( E , ) g ad(Ez)",over Z x E; - A , and obviously obey (for A a flat connection) (2.25)

D;Lzab(z> w )= -Dizra(,(Z w )= dab&,(Z I

>

w)-

SCOTT AXELROD, STEVE DELLA PIETRA C EDWARD WITTEN

824

On H o ( K @ ad(E)), there is a natural Hermitian structure given by

(2.26) Let A(,,'(z), i = 1 ...(g- I ) Dim(G), be an orthonormal basis for H o ( K Q ad(E)), that is, an orthonormal basis of ad(E)-valued (1, 0) forms obeying

(2.27)

D$(i)z"(z) = 0.

Obviously, their complex conjugates are ad(E)-valued (0, 1)-forms that furnish an orthonormal basis of solutions of the complex conjugate equation

(2.28)

D,l,i);(z) = 0.

Finally, we have

together with the complex conjugate equation. We can now reexpress (2.21) in a form that is convenient for constructing the well-defined expressions on AJ that are formally associated with ill-defined expressions on d . Let

(2.30) And for future use, let

(2.31) Then

The symbol Jz is just an instruction to integrate over the w variable. (2.32) is prove$ by integrating D, by parts and using (2.29). At last, we learn that on a section Y of ZQl, that obeys (2.21), we can write

159 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

825

The point of this formula is that arbitrary derivatives with respect to A,, appearing on the left, are expressed in terms of derivatives .9(i, in finitely many directions that correspond exactly to the tangent directions to A . We also will require formulas for the change in the ,l(;)‘(z) and their complex conjugates x ( i ) o ( z )under a change in the flat connection A . There is some arbitrariness here, since although the sum (2.34) is canonical-it represents the projection operator onto the kernel of gA acting on one-forms-the individual ( 2 ) are certainly not canonically defined. However, expanding around a particular flat connection A , it is convenient to choose the so that, to first order in S A ,

x(i)

(2.35) This choice is natural since A , does not appear in the equation obeyed by the I(;)(in other words this equation depends antiholomorphically on the connection). Varying (2.28) with respect to the connection and requiring that the orthonormality of the ,I( should ;) be preserved then leads to

Construction of the connection. We are now in a position to construct the desired connection on the quantum bundle pQ-, 9. First, we work “upstairs” on AY’ . To define a complex structure on AY’ , we need an actual complex structure on I; , not just one defined up to isotopy. Accordingly, we shall also work over the space of complex structures on I;. We discuss below why, for the final answer, these complex structures need actually only be defined up to isotopy. The projectively flat connection on XQ that governs quantization of AY’ is formally

Given the formulas for the symplectic structure and complex structure of a’ , (2.37) is an almost precise formal transcription of the basic formulaequation ( 1.34)-for quantization of an affine space. However, for finitedimensional affine spaces, one would have t = 1 , as we see in ( 1.34); in the present infinite-dimensional situation, it is essential, as we will see, to permit ourselves the freedom of taking t # 1 .

160 026

SCQrT AXELROV, STEVE D E L U PTETRA & EDWARD WlTTEN

We now wish to restrict 8% to act on ZC-invaiantsections Y of @Q, that is, sections that obey (2.33). On such sections, we can use (2.33) to write

(2.38)

We now want to move G / S A T a ( z ) to the right on {2.41), so thatacting on Fcinvariant sections-we can use (2.33) again. At this point, however, it is convenient lo make the following simplification. Our goal is to obtain a well-defined connection on sections of S? over d x { J } which are holomorphic in A . Since we are working at flat connections, after moving 6/6A,'(z) to the right in (2.38), we are entitled to set F = 0. This causes certain terms to vanish. In moving 6/6A," to the right, we encounter a term

We also pick up a term

where we have blindly used (2.36); the meaning of this formal expression that involves the value of a Green's function on the diagonal will have to be discussed later. With the help of (2.39) and (2.40), one finds that upon moving S / d A t ( ( z ) to the right and setting F = 0 one gets

Now we use (2.33) again. The resulting expression can be simplified by setting F to zero, and using the fact that 9 [ i j F= 0 at F = 0 (by virtue of (2.39) and (2.27)). Also it is convenient to integrate by parts in the w variable in the first line of (2.41), using the delta function to eliminate the

161 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

w integration. After these steps, the connection on

827

pQturns out to be

2 6 Q=S-tb,

(2.42.1)

(2.42.2)

Notice that the combination

(2.43) which is the only expression through which the A ( l ) appear in (2.42), is independent of the choice of an orthonormal basis of the Regularization. The first problem in understanding (2.42) is to make sense of the Green’s functions on the diagonal

(2.44)

d fQc

a

L, d ( Z , z , and

(GbaDwL~‘b(Z.

w))IUI=Z.

These particular Green’s functions on the diagonal havc an interpretation familiar to physicists. Consider a free field theory with an anticommuting spin zero field c and a ( 1 , 0)-form b , both in the adjoint representation, and with the Lagrangian

(2.45)

9=

s,

dzdzDica(z)h;(z).

Let us introduce the current Jzc(z) = 6, h:( z ) c ( z ) and the stress tensor T z z ( z )= b~(z)DZc“(z).Then the Green’s functions appearing in (2.44) are formally

(2.46)

d

f,,L;

d

a JZ,

d

z ) = (Jzc)’,

and

(2.47)

( ~ h a D u , L r n h ( Z9

w))l,l,<-z

= Vzz(Z))’?

where the symbol ( )’ means to take an expectation value with the kernel of the kinetic operator Dz projected out. Interpreted in this way, the desired Green’s functions on the diagonal have been extensively studied

162 828

SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

in the physics literature on “anomalies”. Thus, the crucial properties of these particular Green’s functions are well known to physicists. The interpretation of the Green’s functions on the diagonal that appear in (2.44) as the expectation values of the current J n and the stress tensor T can also be recast in a language that will be recognizable to mathematicians. Introduce a metric gpv on I; which is a Ksihler metric for the complex structure J , Then one has the LapIacian A : r(E,ad(E)) + T(C,ad(E)) , defined by A = -g””D,Dy. Let {2.48)

H = det’(A)

be the regularized determinant of the Laplacian. H is a functional of the connection A (which appears in the covariant derivative Dp ) and the metric g . We can interpret (2.46) and (2.47) as the statements that (2.49)

and (2.50)

Given a regularization of the determinant of the Laplacian-for instance, Pauli-Villars regularization, often used by physicists, or zeta function regularization, usually preferred in the mathematical theory-the right-hand sides of (2.49) and (2.50) are perfectly well defined, and can serve as definitions of the left-hand sides. Since these were the problematic terms in the formula (2.42) for the connection on the quantum bundle, we have now made this formula well defined. It remains to determine whether this connection has the desired properties. Confurma1and diffeomorphisminvariance. In defining the Green’s funti tions on the diagonal, we have had to introduce a metric, not just a complex structure. To ensure diffeomorphism invariance we will choose the metric to depend on the complex structure in a natural way. Before making such a choice, however, we shall explain the simple way in which the connection (2.42) transforms under conformal rescalings of the metric. One knows from the theory of regularized determinants (or the theory of the conformal anomaly in ( t + 1)-dimensions) that under a conformal rescaling g -+ e4g of the metric, with 4 being a real-vahed function on Z , the regularized determinant H transforms as

163 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

829

where S(4, g) is the Liouville action with an appropriate normalization and may be defined by

(2.52) together with the group laws S(0, g) = 1 and S ( 4 1 + ~g2),= S(41J e4Zg). S ( d 2 ,g) . Here, JgR is the scalar curvature density of the metric g . The crucial property of S(4, g ) is that it is independent of the connection A . We conclude that under a conformal rescaling of the metric, the current expectation value defined in (2.49) is invariant. On the other hand, the expectation value (2.50) of the stress tensor transforms as

This means that under a conformal transformation of the metric of C , the connection 6% defined in (2.42) transforms as

(2.54) The second term on the right-hand side of (2.54), being independent of the connection A , is a function on the base in the fibration A x F + 7 . On each fiber, this function is a constant, and this means that the second term on the right in (2.54) is a central term. Up to a projective factor, S;l*b is conformally invariant. The central term in the change of 6% under a conformal transformation has for its (1 + 1)-dimensional counterpart the conformal anomaly in current algebra. To show that our connection lives over Teichmuller space and is invariant under the mapping class group one must check that 8% is invariant under a diffeomorphism of I;, and that the connection form U vanishes for the variation 6Js' = aTwz of the complex structure induced by a vector field w on C . The first assertion is automatic if we always equip C with a metric that is determined in a natural way by the complex structure (for instance, the constant curvature metric of unit area or the Arakelov metric), since except for the choice of metric the rest of our construction is natural and so diffeomorphism invariant. The second point may be verified directly by substituting 6J7' = aTwz into (2.42), integrating by

164 830

SCOTT AXELROD. STEVE DELLA PIETRA & EDWARD WITTEN

parts, and using (2.28). It follows more conceptually from the fact that the connection may be written in a form intrinsic on A (see $4) and from the fact that H and the Kahler structure on A% depend on the complex structure J on C only up to isotopy. Properties to be verified. Let us state precisely what has been achieved so far. Over the moduli space A of flat connections, we fix a Hermitian line bundle 9 of curvature -i& with an action of the mapping class group, The prequantum Hilbert space is r(&,9). The prequantum bundle over Teichmuller space 7 is the trivial bundle = r ( J ,2) x L7. The connection S - t b (with the regularization defined in (2.49), (2.50))is rigorously well defined as a connection on the prequantum bundle. What remains is to show that this connection has the following desired properties. (i) The quantum bundle pQover 7 is the subbundle of %r consisting of holomorphic sections of the prequantum line bundle; that is, &QIJ, =

4r

H0(&, , 9). We would like to show that (with the correct choice of the parameter t ) the connection S - t@ on
+

165 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

831

The anomaly. In (2.30) and (2.31), we have introduced bases 9(i, and g ( m ) of T ( ’ > ’ ) Land T ( ’ ” ) L , respectively. The symplectic structure of .A can be described by the statement that, acting on sections of 2, (2.55) This can be verified directly using (2.18) and the orthonormality of the I ’s (and the fact that a section of 9 over A is the same as a %-invariant section on JZ? ). Now, property (1) above-that the connection 6 - tb preserves holomorphicity-amounts to the statement that, at least when acting on holomorphic sections of 9 ,

(2.56) The analogue of (2.56) would of course be true for symplectic quotients of finite-dimensional affine spaces. For the present problem, (2.56) can be verified directly although tediously. In doing so, one meets many terms that would be present in the finite-dimensional case. There is really only one point at which one meets an “anomaly” that would not be present in the finite-dimensional case. This comes from the term in t[g(m), 81 with the structure

Now, formally the current J,“(z) is defined in a Lagrangian (2.45) that depends holomorphically on the connection A , that is, A , and not A , appears in this Lagrangian. Naively, therefore, one might expect that < J,,(z) >’ or any other quantity computed from this Lagrangian would be independent of A , and therefore annihilated by %(,,,, . However, the quantum field theory defined by the Lagrangian (2.45) is anomalous. As a result of the anomaly in this theory, there is a clash between gauge invariance and the claim that the current is independent of A , . At least for 2 of genus zero, where there are no zero modes to worry about, one can indeed define the quantum current < J,‘(z) >‘ so as to be annihilated by 6 / 6 A Z n ( z ) but , in this case < Jza >’ is not gauge invariant. In the case at hand, we must insist on gauge invariance since otherwise the basic formulas such as the definition of the connection (2.42) do not make sense on the moduli space .A . Indeed, in (2.49) we have regulated the current in a way that preserves gauge invariance. The anomaly is the assertion that the gauge invariant current defined in (2.49)

166 832

SCOTT AXELROD, STEVE DELLA PIETRA & EDWARD WITTEN

cannot be independent of A, ; rather one has

Here h is the dual Coxeter number, defined in (2.3). The . . terms in (2.58), which would be absent for Z of genus zero, arise because in addition to the anomalous term that comes from the short distance anomaly in quantizing the chiral Lagrangian (2.45), there is an additional dependence of (Jz)' on A, that comes because of projecting away the zero modes present in (2.45) in defining ( )'. These I . . terms have analogs for symplectic quotients of finite-dimensional manifolds, and cancel in a somewhat elaborate way against other terms that arise in evaluating (2.56). We want to focus on the implications of the anomalous term. The contribution of the anomalous term to (2.57) and (2.56) is

(2.59)

-

Terms of the same structure come from two other sources. As we see in (2.42), the last term in the connection form d is a second-order differential operator 8,.In computing @(,) @,], one finds with the use of 2.55 a term

(2.60)

-

The last contribution of a similar nature comes from

The ... terms are proportional to 6/dAz and annihilate holomorphic sections of 3. On the other hand,

(2.62) Therefore, on holomorphic sections, after using (2.33), we get

In the absence of the anomalous term (2.59), the two terms (2.63) and (2.60) would cancel precisely if t = 1 . This is why the correct value in

167 GEOMETRIC QUANTIZATION OF CHERN-SIMONS GAUGE THEORY

833

the quantization of a finite-dimensional affine space is f = 1 . However, including the anomalous term, (2.59):the three expressions (2.59), (2.60), and [2.63) sum to zero if and only if t = k / ( k + h ) . The connection on the quantum bundle XQ-+ 7 is thus finally pinned down to be

+

That the connection form is proportional to l / ( k h ) rather than l / k , which one would obtain in quantizing a finite dimensional affine space, is analogous to (and can be considered to explain) similar phenomena in two-dimensional conformal field theory. The rest of the verification of (2.56) is tedious but straightforward. No further anomalies arise; the computation proceeds jusl as it would in the quantization of a finite-dimensional afine space. We forego the details here, since we will give a succinct and rigorous proof of (2.56) in 54.

169 Commun. Math. Phys. 144, 189-212 (1992)

On Holornorphic Factorization of WZW and Coset Models Edward Witten* School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA Received July 19, 1991

Abstract. It is shown how coupling to gauge fields can be used to explain the basic facts concerning holomorphic factorization of the WZW model of two dimensional conformal field theory, which previously have been understood primarily by using conformal field theory Ward identities. We also consider in a similar vein the holomorphic factorization of G / H coset models. We discuss the G / G model as a topological field theory and comment on a conjecture by Spiegelglas. 1. Introduction

The WZW model of two dimensional conformal field thcory [I] is a quantum field theory in which thc basic field is a map g:C+G, 2' being a two dimensional Riemann surface and G being a compact Lie group, which for simplicity we will in this paper take to be simple, connected and simply connected. The basic WZW functional is 1 I ( g ) = 8~ I5 d 2 6 ~ e i j T r ( g - l a i g . g - ' a i g ) ~ i T ( g ) , (1.1) ~

~

where @ is a metric on C, Tr is an invariant form on the Lie algebra of G whose normalization will be specified presently, and r is the Wess-Zumino term 121. The latter has the following description 131 in case X is a Riemann surface without boundary. (For the more general case see [4].) Let B be a three manifold such that BB=C, pick an extension of g over B, which we will also call g, and let 1 r(g)=Bjg*w=- 12a S d~acijkTrg-'aiR.g-'ajg.R-'atg, B

(1 -2)

where m i s the left and right invariant three form on the G manifold defined by 1 12n

' .

w = -Tr(g-' dg A g- ' d g A g - dg) Research supported in part by NSF Grant PHY86-20266

(1.3)

170 190

E. Witten

T(g)is well defied (independent of the choice of B and the extension of g over B) modulo the periods of w.In these formulas, “Tr” is an invariant quadratic form on the Lie algebra of G which we take to be the smallest multiple of the trace in the adjoint representation such that the periods of w are multiples of 2x. (For G = SU(N), “Tr” is simply the trace in the N dimensional representation.) The condition on the periods ensures that the WZW functional I ( g ) is well-defined as a map to C/27cZ, so that e-r(g)is well-defined as a complex valued functional on the space of maps Z-G. The Lagrangian of the WZW model is L(g)= kl(g),where k is a positive integer called the “level,” and the partition function Z of the WZW model is formally defined as a Feynman path integral, Z = J D g e - L = JDge-“. (3.4) 2 depends, of course, on the metric e of C which enters in the definition of I. Conformal invariance of the WZW model means that apart from a relatively elementary factor given by the conformal anomaly, Z depends only on the complex structure determined by p . The WZW model is essentially exactly soluble; the ability to solve it depends on its holomorphic factorization, first investigated by Knizhnik and Zamolodchikov [53. Holomorphic factorization of the WZW model means that locally on the space of complex structures one can write 2 = where the are holomorphic

fs,

functions on the space of complex structures. Globally, as advocated by Friedan and Shenker [ 6 ] , one interprets this formula to mean Z = ( J f j , where f is a holomorphic section of a certain flat vector bundle Y’ over moduli space equipped with a hermitian form ( , }. (The conformal anomaly means that these statements require a somewhat more precise formulation.) The flat bundles that arise in this way have been extensively studied [7,&] and have been seen to have a natural origin in gauge theories [9-12]. To date, the existencc of a holomorphic factorization of the W Z W model has mostly k e n understood using conformal field theory Ward identities, this being the original point of view of Knizhnik and Zamolodchikov. The purpose of the present paper i s to use gauge theories - or more cxactly, coupling ofthe WZW model to gauge fields - to deduce the existence of a holomorphic factorization. Many of the key steps have been previously exploited by Gawedzki and Kupianen 113,141. See also the work of Bernard [l 51 on the heat equation obeyed by characters of afine Lie algebras. Themain novelty which motivated me to write the present paper is the integration over the gauge field and use of the Polyakov-Wiegmann formula to prove that the partition function has a holomorphic factorization; see the derivation of Eq. (2.28). Our treatment will be formal; we will make no claim to analyze the quantum anomalies. Gauged WZW models have been extensively studied [l6-18, 141, mainly with the aim of giving a Lagrangian description of the GKO coset models [19] (whose prehistory goes back to the early days of string theory 1201). After developing our approach to holornorphic:hctorization of the WZW model in Sect. 2, we will apply the same methods to holomorphic factorization of coset models in Sect. 3 (recovering obscrvations of Moore and Scibcrg [Zl] and Gawedzki and Kupianen [I 3, 143, and then we wiH consider the special case of the G/G cosct model, where sharper statements can be made, as this theory is actually a topological field theory. The GIG model has been investigated by Spiegelglas [ 2 2 ] .

171 Holomorphic Factorization of WZW and Coset Models

191

2. Gauge Couplings and Holomorphic Factorization The WZW action functional I ( g ) is invariant under the usual action of G x G (often called G, x G,) on the G manifold. An element (a,b) of G, x G , acts on G by g - + u g b - ’ . ’ Given a Lagrangian with a (global) symmetry, it is usually possible to “gauge” the symmetry, introducing gauge fields and constructing a gauge invariant extension of the original Lagrangian. In particular, gauging the WZW model means generalizing the theory from the case in which g is a map C+G to the case in which g is a section of a bundle X+.Z with fiber G and structure group G , x G , or a subgroup. Letting A be a connection on such a bundle, one aims to find a gauge invariant functional I ( g , A ) (whose variation with respect to g or A is required to be given by a local formula) which reduces, for X the trivial bundle and A=O, to I(g). In the case of the WZW model, such a gauge invariant extension does not exist. There is no problem in gauging the first term in (1.1)- one just replaces derivatives by covariant derivatives. However, the Wess-Zumino term T ( g ) does not have a gauge invariant extension unless one restricts to an “anomaly-free’’ subgroup F of G , x G , (and considers bundles X+Z with structure group F). The condition for a subgroup to be anomaly free can be stated as follows. For any subgroup F of G , x G,, 9, and 8, (the adjoint representations of G , and G,) are F modules. If Tr, and Tr, are the traces in 8, and 9,,then the condition for absence of anomalies is that for any t, t ’ E 9 (the Lie algebra of F ) Tr,tt’-Tr,tt’=O. (2.1) (As will be clear in the appendix, this statement is equivalent to the statement that the class in H3(G,R)represented by (0has an extension in Ha(G,R), where HZf i s the F-equivariant cohomology.) In the appendix, we will review the derivation of (2.1)for the benefit of readcrs not alrcady familiar with such matters and to clarify a few gcomctrical points. Some readers may want to consult the appendix first, but this should not be necessary for readers who are familiar with gauged WZW models or are willing to verify by hand a few easily verified formulas. We will also be interested in gauged WZW actions in cases in which (2.1) is nut obeyed. In such a casc, onc cannot construct a gauge invariant I(g, A ) , but one can find a “best possible” I ( g , A ) , such that thc violation of gauge invariance is a multiple of a standard “anomaly” expression that depends on A but not on g. [A topological explanation of why it is possible to do this will be given in the appendix, where the detailed formula for I ( g , A ) is also explained.] Wc will adopt the following convcntions: z will be a local complex coordinate on Z, d2 z is the measure Idzdrl, and the components of A are defined by A = A,dr + A,dZ. We sometimes use the Levi-Civita antisymmetric tensor density E ~ Jwith E‘’ = - 8’‘ = i. (That is, for one forms u and b, we write J u A h = 5 d2zcijaihj.)Our z b orientation conventions can be most efficiently and usefully explained by saying that the variation of the Wess-Zumino term under 6 g = -gu is

’ If (ihas a non-trivial center Z(G).then Z(G),diagonally embedded in G, x GR,acts trivially on g, so the faithfully acting symmetry group is really (GLx G,)/Z(G). This refinement will not be important until we come to coset models

172 192

E. Witten

For simplicity, we will in this paper consider only the case that G is connected and simply connected, so that a G bundle over Z, is trivial. 2.1. The Holomorphic Wave-Function

To begin with, we consider the case that F = G,. In this case, (2.1j is not obeyed, so a gauge invariant functional I(g,Aj does not exist. However, we take 1

2.1r

I(g, A) =I(g)+

1 d2z Tr A,g-' 8,g - - j d2z Tr &A,,

4n

(2.3)

E

which is as close as there is to a gauge invariant functional, in the following sense. Under an infinitesimal gauge transformation, sg= -gu ,

6Ai=-D,u= -d,u-[Ai,u],

(2.4)

one has Sl(g,A)=

1

- JdZzTru(d,A,-&4,)= 4% J

i

-

I TrudA,

4n:1

(2.5)

an expression which depends on A but not on g or the complex structure of Z. Equation (2.3) is the unique extension of I ( g ) with this property. We now formally define a functional of A by Y ( A ) = jDge-kmA)

=jDgexp(-kI(g)--

k 1d2zTrA,g-'3,g+- k jd2zTrAIA,

2n x

4x z

Note that wedonut treat Aasaquantumvariable; thatis,wedonotintegrateover A. This would not be sensible as I(g,A) is not gauge invariant. Now, !P obeys two key equations. First,

and second

6

(D.raT

k + &&A,-

"

-Fzz

2n

P=O.

Equation (2.7) is proved simply by acting with the left-hand side on the integral representation of Y', and differentiating undcr the integral sign. Equation (2.8)is a consequence of the standard anomaly Eq. (2.5). By differentiating under the integral sign, one finds that the left-hand side uf (2.8) eyuats

- k.- Dg e - krlg.A)[D,(g- 'D,g) + F,,] , 2x

(2.9)

where we have introduced covariant derivatives Dg =dg -gA. The quantity in brackets in (2.9) i s the equation of motion of the g field - the variation of I ( g , A) under 6g= -gu. Therefore, upon integrating over g, (2.9) vanishes, by idtegration by parts in g space.

173 193

Holomorphic Factorization of WZW and Coset Models

To elucidate these equations, it is useful to first introduce the operators

D - d

k

& I , 4a A,

DA,

D d k +-AA,. DA, SA, 4n Note that for z, w 6 C,

1-

[-LD

DA,(z)’ DA,(w)

=k 6(z, w).

2n:

(2.10)

(2.11)

In terms of these operators, (2.7) is simply D

-Y=O, DAZ

(2.12)

and (2.8) becomes (2.13) which in view of (2.12) can be written in a way that does not refer to the complex (2.14) These equations are closely related to the basic formulas that appear in canonical quantization of 2 + 1 dimensional Chern-Simons gauge theory, as explained, for instance, in Sect. 2 of [11] or in [lo]. Let d be the space of connections on (the trivial F bundle over) C. d has a symplectic structure that can be defined purely in differential topology, without choosing a conformal structure on C. It can be defined by the formula (2.15) where a, and a, are any two adjoint-valued one forms representing tangent vectors to d.“Prequantization” of d (in the sense of Kostant [23] and Souriau [24]) means constructing a unitary complex line bundle 9 with a connection whose curvature is -iw. Equation (2.10) can be regarded as a formula defining a connection on the trivial complex line bundle 9= d x C over d (which we take with the standard unitary structure). This connection according to (2.11) has curvature -ikw. The factor of k means that 9, with this connection, can be identified as Y e t ,with Y the basic prequantum line bundle. Hence (2.10) and (2.11) actually describe prequantization of d,with the symplectic structure kw. The notion of prequantization obviously does not depend on a choice of polarization or complex structure, and indeed, though (2.10) and (2.11) are written in terms of a local complex coordinate on C, they are actually entirely independent of the conformal structure of C. Y ( A )is best regarded not as a “function” on d but as a section of the prequantum line bundle Z e k . The complex structure enters when one wishes to quantize d.A choice of complex structure on C determines a complex structure on d - in which the A , are holomorphic and the A , are antihdomorphic. This complex structure is a

174 E. Witten

i 94

“polarization” which permits quantization: the quantum Hilbert space consists of holomorphic sections of 5fBk.Equation (2.12) means simply that Y ( A )is such a holomorphic section. Now let us discuss the meaning of (2.13). Let P be the group of gau_ge transformations, that is, the group ofmaps of C to F. Acting on functions on d ,F is generated by the operators 6 Di (2.16) 6Ai’ To find a I? action on sections of Z a k ,one must “lift” the vector fields in (2.16) appropriately. This can be done in a standard fashion (for instance, see Sect. 2 of [l 11); the I? action on sections of 2 is generated by the operators ik .. D. D - 8°F.. ~

IDAi

4.n

”’

(2.17)

(The second term is the contribution of the classical “rnomcnt map.”) We can thus see the meaning of (2.13) or (2.14) - !P is gauge invariant, as a section of LfBk. The two conditions we have found that Y is holomorphic and gauge invariant mean together that Y can be regarded as a physical state of 2 i1 dimensional Chern-Simons gauge theory (with gauge group F). (See 19-11] for more background.) This in fact can be rcgarded as the essential relation betwcen thc WZW modcl and Chcrn-Simons theory. Wc will now recall a fcw furthcr facts about the Chern-Simons theory. (The facts summarized in the next three paragraphs are not all strictly needed for reading the present paper, but they help put the discussion in context.) The I? action on sections of Y B does k not depend on the conformal structure of C, but something new happens once such a conformal structure is picked. A connection A on (the trivial F bundle over) a complex Riemann surface C determines an operator which defines a complex structure on the bundle. Gauge transformations act by aA-faAAf- for f:C-F, but as this formula makes_sense for f : C+F, ( F , is the complexification of F), one actually gets an action of F , (the group of maps of C to F,) 0%d.A E invariant section of LZBk which is al%o holomorphic is automatically F , invariant. Let V be the space of holomorphic, F , invariant sections of LFBk.Vis the space of physical states in Chern-Simons gauge theory, at level k. From what we have said above, Y is a vector in K A P invariant section of 2 @k is the same as a section of an appropriate_pushover the quotient space d / F , . The down line bundle, which we will also call 2@k, quotient d / F , , with the quotient taken in an appropriate sense, is the moduli space of stable holomorphic F , bundles over C, or (by a theorem of Narasimhan and Seshadri) the moduli space A‘ of flat F connections on C, up to gauge transformation. This is a compact complex manifold, and in particular, the vector space which can be identified as H o ( A ,LZmk), is finite dimensional. So far, when we have made statements that depend on the complex structure of C, we have considered C with a f i x e d complex structure. Permitting the complex structure of C to vary, we get not a single vector space V but a family of vector spaces parametrized by the space Y of complex structures on C, or in short, a vector bundle Y over Y . The bundle Y-Y has a natural projectively flat connection (which is essential for the topological invariance of Chern-Simons theory); the holomorphic structure is obvious, and the anti-holomorphic structure, which we will recall at an appropriate point, is less obvious. ~

~

aA

’,

175 Holomorphic Factorization of WZW and Coset Models

195

2.2. The Norm of the Wuve Function By now we have defined, for every complex Riemann surface Z, a vector space I/ consisting of holomorphic, gauge invariant sections of the line bundle PBk over the space ,dof connections. A natural Hermitian structure on Y is given formally

(Formally, D A is the measure on .ddetermined by its symplectic structure, and it is natural to divide by the volume of Fz (? because of the gauge invariance of ‘Y1Y 2 . ) In genus one, this Hermitian structure can be worked out explicitly (that is, reduced to an explicit description of an inner product on the finite dimensional vector space V ) ,by actually computing the integral over the infinite dimensional F , orbits [14, 251. In genus > 1, such an explicit evaluation is not known. We want to compute the norm of the vector Y introduced in the last subsection, with respect to this Hermitian structure. To this aim, we first want an integral expression for Y. This could be obtained by just complex conjugating the definition (2.6) of Y , but instead, we prefer to introduce a conjugate WZW model describing a map h:Z+G. This time, we introduce a gauge field B gauging the subgroup G, of G, x G,. This is again an anomalous subgroup, so a gauge invariant action I(h, B) extending the WZW action I(h) docs not exist. The best that one can do, analogously to (2.3), is

I‘(h, B ) = l ( h ) -

1 j d Z zTrB,i?,h. h 2n z

-

’

1 --

5 dZz TrB,B,.

47L z

(2.19)

Under the infinitesimal transformation (2.20)

6 B , = -D,u,

Gh=uh,

one has N ( h , B )=

J d2z Tr u(i?,B,- a,B,).

I -

412 r

(2.21)

As in (2.6), we now define

x(B)= J D ~ E - ~ ” ( * * ~ ) 4x r

C o m p a a ( 2 . 6 ) and (2.22),it is evident that in fact x is the complex conjugate of P, r ( A )= W A ) . We now come to the key step in the present paper. We use these integral representations to compute IY12: 1

J DA m Y ( A )

va@j d

‘y‘2=

-

(

k J DgDhDA exp - k I ( g ) k l ( h ) J d2z Tr A,g- d,g vol (G) 27L H k k (2.23) +I d2z TrA,ii,h h - ’ + 2x 5r d 2 z TrAJ, 27L J ~

-

~

176 196

E. Witten

Notice that the integrand is invariant under gauge transformations of the form

Gh=uh,

Sg=-gU,

SA,=-D,u.

(2.24)

This follows from the cancellation between (2.5) and (2.21). We can perform the integral over A, using the fact that the exponent in (2.23) is quadratic in A and the operator appearing in the quadratic term is a multiple of the Gaussian integration over A gives

At this point we may use a formula of Polyakov and Wiegman [26]:

I(gh)=I(g)+I(h)-

1 Jd'zTrg-'a,ga,h.h-'. 2n I

~

(2.26)

The proof of this formula follows from the following: (i) the formula is obviously valid if h = l ; (ii) the left- and right-hand sides are both invariant under h-wh, g+gw-t, for arbitrary w : C + G . To demonstrate (ii), it suffices to check infinitesimal invariance under 6g = -gu, 6h = uh. This can easily be verified directly. Actually, a more conceptual proof of (ii) follows from our above calculation. We know that (2.23)is invariant under (2.24),and integrating out A, an operation that will preserve this symmetry, one deduces that the exponent on the right-hand side of (2.25) has the desired symmetry. Therefore, replacing the double integral over g and h by a single integral over [ L g h , and canceling the factor of vol(8) in the process, we get

I Y 1'

= J Dfe-k'(/).

(2.27)

The right-hand side of (2.27) is by definition thc partition function Z(C) of the WZW modcl (with symmetry group G and lcvel k) so we have learned Z(C)=1'YI2,

(2.28)

which, though still in need of further elucidation, is the statement of holomorphic factorization of the WZW model. 2.3. Varying the Complex Structure of' Z

So far, we have considered the surface 2' with a ,fixed complex structure. At this level, Z(Z) is a number; Y is a vector in a fixed vector space !I Equation (2.28) is a relation between them. In this form, the relation is not very remarkable. It gains interest when one permits the complex structure of .E to vary. We will work over thc space Y of all conformal classcs of metrics on 2. Evcry conformal mctric Q determincs a complcx structurc. For any given Q. wc can dcfinc a vector space V, consisting of holomorphic and gauge invariant sections of the

* We can assume a regularization in which the determinant of a multiple of the identity is one. With an arbitrary rcgulariration, such a determinant is a factor of the form erx''), whcrc c is a universal constant, indrpcndcnt of Z, and x(Z)is the Euler characteristicof 2. Such a bctor can be rcmovcd by adding to the WZW action a multiple nf J I/pR, where R is the scalar curvature of a 1

metric p that is compatible with thc complcx structurc of Z

Holomorphic Factorization of WZW and Coset Models

191

prequantum line bundle 9 @ over k d.The V, vary as fibers of a vector bundle V over 9’.A section of Y is a function Y ( A ;e) of connections and conformal metrics which, in its dependence on A for fixed @, obeys (2.12) and (2.14). The space 9’ is a complex manifold, whose exterior derivative has the standard decomposition d = 8 8.We will write 8’.O ) and 6‘’. ’) respectively for the 8 and a operators of 9’.One can write these explicitly in the form

+

(2.29)

The bundle V has a (projectively) flat structure, which is defined by giving compatibly a holomorphic structure and an antiholomorphic structure. The holomorphic structure is the “obvious” one. Y ( A ;e) is said to be holomorphic, in its dependence on e, if it is annihilated by

p.1 ) = p .

I)

(2.30)

For the antiholomorphic structure, we cannot simply use the operator SC1,O), since this does not commute with the operator on thc Icft-hand side of(2.12).Rather, as cxplained in [Il, lo], Y ( A ;e) is antiholomorphic if it is annihilatcd by (2.31) It is now just a matter of differentiating under the integral sign to show that Y ( A ;e) as defined in (2.6) is annihilated by V(l-o). This has essentially been done in [13]. We have

(

S“.o)Y = J D g e - k r ( A * g)

k

J d2z&,,e2” Tr(g-’D,g)’), 87c1

~

(2.32)

where Dig = aig -gAi. Similarly,

(2.33) so that

Combining the pieces, we get

p.0’y = 0

(2.35)

as was claimed. Now, let e,, a = 1, ...,dim V be a basis of orthonormal, covariantly constant sections of Y (over some open set in moduli space). Y can be expanded in this basis as

w . 4 ; e)=

c a

e1.m

(2.36)

178 198

E. Witten

with some expansion coeficients f , . Equation (2.35) means simply that the fk(e) are anti-holomorphic functions on Y in the usual sense. Consequently, (2.28) amounts to an expression (2.37) for the WZW partition function as a finite sum of norms ofholomorphicfunctions. The stress tensor of the WZW model is usually defined as

6 k K,= 2 I(& A ) = - - Tr(g-'D,g)' 4n

(2.38)

PZZ

The current is

s

k

Jz=-1(g,A)=-gg-'Dzg SAz 2n

(2.39)

The fact that

K , = - (n,kj. Tr J t ,

(2.40)

which obviously was the main point in the derivation of (2.35), is known as the (classical form of the) Sugawara-Sommerfield construction. It is well known that when J , is defined as a quantum operator, T r J i must be defined with some point splitting or other regularization; this has the effect of replacing k by k + h (h being the dual Coxeter numbcr of G). Scc [13, Eq. (49)] for some discussion of this in the present context. Obviously, our discussion has been purely formal, and we have made no attempt to pruve that the key statements, such as the statement (2.28) of holomorphic factorization, survive the quantum anomalics. A proper treatment would have to address theconformal anomalies that affect both 2 and 'P and show that the left- and right-hand sides of (2.28) have the same conformal anomaly and are equal. Finally, the gauge invariant functional

(2.42)

that appeared in the exponent in (2.23) deserves some comment. Let G' be the compact, connected, and simply connected group G'= G x G . The pairfg,11) : C+G x G can be rcgarded as a map of C to G'. The G' WZW actiun i s just I(R, h) =I(g) + I(h). Let F be the subgroup of G). x G Rconsisting of clcmcnts of the form ((1, u), (C', 1)). In other words, k' acts by ( g , h ) + ( g a - ' , u h ) . rhen F is an anomaly free subgroup of Cl x C i [in the sense that (2.1)is obeyed]. Therefore a gauge invariant action I ( g , h , A ) , reducing to I ( g , h ) at A = & exists. It is precisely (2.41). Our computation of holomorphic factorization amounted to demonstrating that if ZJZ) is the partition function of the WZW model with target G, and ZGr!F(X) is the partition function ofa gauged WZW modcl with targct C' and gauge group F. then

Z,(Z) =z,,m

I

(2.42)

179 Holomorphic Factorization of WZW and Coset Models

199

Holomorphic factorization has its origin, from this point of view, in the fact that when one computes the action (2.41) of the gauged G I F model, it turns out to be the sum of a functional of g and a functional of h. Since exponentiating the action (to get the integrand of the path integral) turns sums into products, this leads to the ability to factorize Z,.,,,(Z) in the fashion that we have described.

3. Holomorphic Factorization of Coset Models

So far we have considered gauged WZW models only as a technical tool in order to understand ordinary WZW models. The gauged WZW models are, however, interesting models of conformal field theory in their own right. For every anomalyfree subgroup F of G , x G, (that is, every subgroup obeying the condition in (2.1)), one has a corresponding gauge invariant generalization of the WZW action, which, upon quantization, leads to a conformal field theory model. The models that arise this way are equivalent to coset models, as has been shown by several authors cited in the introduction. The most standard examples of anomaly-free subgroups of G, x G , are the following. Let G,,, be the diagonal subgroup of G,, x G, (acting by g+aga-', a E C). Let H be any subgroup of Cadj.Such an H is always anomaly free. Let B be an H-valued connection. Since H is an anomaly-free group, a gaugeinvariant extension I(g,B) of the WZW action I ( g ) exists. Explicitly, it is

We want to understand thc holomorphic factorization of the corresponding coset modcl partition function

This model (with the diagonal cmbcdding of H in G,. x C , ) is sometimes called "the" G / H model, and corresponds to the diagonal modular invariant, as will be clear. In this paper we will only consider these standard anomaly-free subgroups, but the generalization of the considerations to other cases should be apparent.

3.1. Holomorphic Wove Function

As in holomorphic Iactorization of the original WZW model, we now consider a subgroup F of G, x G , which is not anomaly free.3 In fact, we takc F=H,, x G,, where H,. is the subgroup of C,, coming from the cmbcdding of H in G . An F connection is a pair (B, A),wherc B and A are H and G connections, rcspectively. A The arguincnt could a1w he expressed In terms of il certain anomaly free subgroup of GLx C; where G'=C x G. This formulation would proceed in paratlel with the last paragraph of Sect. 2

180 E. Witten

200

gauge invariant action I(g, A, B ) extending the WZW action does not exist, since the subgroupF ofG, x G, is not anomaly free. Analogous to (2.3), there is instead a best possible action, uniquely determined by requiring that the violation of gauge invariance is independent of g and of the conformal structure of Z.This action is

I I(g, A , B ) = l ( g ) + - J d'z TrA,g-'d,g27L E

+ 27l1 zj d'z -

TrB,gA,g-'-

1 . 1 dZz trB,d,g.g-' 2n E 1 - 1d2z Tr(A,A,+B,BJ.

-

4n E

(3.3)

Under Gg=ag-gu, 6 A i = - D i u 9 6Bi=-Dia (3.4) (here u and v are zero forms valued, respectively, in the Lie algebras of G and H), we have 1 S l ( g , A, B ) = - j d2z Tru(d,A,- i?=-4,-CZB, 471 2

--i

+ Q,)

j Tru(dA-dB).

(3.5)

471 1

Before proceeding, let us make a few comments on the global structure. If G has a nontrivial center Z(G),then Z(G),diagonally embedded in G , x G,, acts trivially in the WZW model (since g=ago for u t Z(G)).The symmetry group that acts faithfully in the WZW model is hence really ( G , x G,)/Z(G).Similarly, F = H, x G, does not act faithfully; the group that acts faithfully is F ' = ( H , x GR)/Z,whcre Z = H n Z ( G ) . To make thc most precise statements in what follows, it is best to think of the pair ( A , B ) as a gauge field with structure group F',The group of maps of 1 to F will be called p.The complexilication of F' will be called F& and the group of maps of Z , to F& will be callcd The groJpF of maps of? to H, G, and their cornplexifications H,: and G , will bc called H,G, A,, and G,. Now, as in Sect. 2, we introduce the holomarphic wave function

X(A,8)= J D g e - k " d J . B ' .

(3.6)

x obeys certain conditions anatogous tn those studied in Sect. 2. To exhibit these, we let d be the space of F-valued connections on I,W the apace of H-valued connections,and O = d x g . We want to considere asasymplectic manifold, with thc symplectic structure given by the formula o(a,,h,;a,,b,)=-

1

1 JTra,~a,-- (Trb,~b,

271 P

2n i.

(3.7)

(Here the a,and bj are respectively one forms with values in thc Lie algebras of G or H . The pairs (u,, b,) and(a2,6,)define tangent vectors to V .The "Tr"in the second expression on the right of (3.7) is the quadratic form on the H Lie algebra that is induced from the embedding of H in G.)The minus sign before the second term in (3.7) is characteristic of coset models. Prequantization means construcling a line bundle Y over V with a unitary connection of curvaturc --iw.

1st Hoiomorphic Factorization of WZW and &set Models

201

Rather as in Sect. 2, the line bundle over V that is relevant is the trivial line bundle endowed with a connection described by the following formulas: D

S k 6A, 4a A'' D - 6 k - - -A , , DA, 6A, 4x D - S k _ _ - + -&, DB, SB, 471 -

DA,

+

~

D DB,

-

6 63,

(3.8)

k 4x Bz .

Computing the curvature of this connection, we see that the trivial line bundle endowed with this connection is isomorphic to 9 0 Q k ,which is how we will refer to it henceforth. The action of the gauge group (that is, the group ofmaps of Z to G x H) on W lifts to an action on POQk. The lift is described at the Lie algebra level by the obvious generalization of (2.17); the G action is generated by the operators

D

D; __ DA,

ik .. 4a

- -&"F,j(A),

(3.9)

and the H action is generated by (3.10)

Here F ( A ) and Fin) are the curvatures of A and 3, re~pcctively.~ The analogs of (2.12) and (2.14) are easy to find. x obeys first of all (3.11) T h i s bas the following interpretation. Pick on V a complex structure that comes from the standard complex structure on d and the opposite complex structure on a.(Thus, A, and B, are holnmorphc, and A , and B, are antiholomorphic.) The (n,2) part of the curvature of thc connection (3.8) vanishes, so Fekhas a natural structure of holomorphic line bundle over g.Equation (3.11) mcaiis that x is a holomorphic section of this line bundle. x also obeys the analog of (2.14), namely

As in the discussion of (2.14), this equation means that x is gauge invariant in th_c appropriate scnsc: it is invariant under the natural lift of the action of the group F of gaugc transformations 10 an action on sections of Y@jk. If H (or G) is not connected and simply connected, describing a lift o f thc gaugc group to act on Yak requircs morc than the lift of the Lie algebra described by these l ~ r m u l a sThe . lull story is naturally described using the gauge theory approach to prequantszation of the space ul conncctions 127-291 and will not bz explained here, Lhough the exisccncc nf a natural lift is essential later when we consider the role of thc ccnter or G

182 E. Witten

202

3.2. The Space of Conformal Blocks

Let W be the space of holomorphic sections of TBk which are invariant - such as x. We will devote this subsection to a detailed characterization of W Wis a finite dimensional vector space which can be give_nthe following concrete descripti0n.h which is F invariant is automatically also Fc holomorphic section_of 2’@’k invariant. Let &‘=V/FCAccording to the Narasimhan-Seshadri theorem, 9is the moduli space of flat F’-valued connections on Z, up to gauge transformation. W gets a complex structure from its interpretation as the quotient of the complex manifold V by the complex group Fc. The holomorphic line bundle SBk over %‘ pushes d_own to a holomorphic line bundle, which we will call by the same name, over 8. Fk invarisnt sections of 6p@’over V are pullbacks of sections of pBk over 9, so W = H o ( 9 , T * k ) .This is the space identified in [21,14] as the space of conformal blocks of the coset model. W is finite dimensional, since 9 is compact. In fact, if 2 is trivial, then W = A x A”,where A=d/& and A*=S/&. As is apparent from (3.111, the complex is the structure on A is the standard one, and the complex structure on opposite one. We will refer to A”with the opposite complex structure as 2.If 9cLl is the standard prequantum line bundle over A and 2,*,,, is the standard prequantum line bundle over -f (and we denote their pullbacks to x &” by the same symbols), then 5YBk= 2’:f@5?&-k’. The minus sign, of course, comes from has curvature of type (1,l): it is the minus sign in the second term in (3.7). [As Y(21 naturally holomorphic both in the standard complex structure on A- and the opposite one.] Consequently, if Z is trivial, W = H O ( 9 , 8 @ k ) = H * [ * R Ix 2 - , 9 $ ; @ 9 & - k ~ ) =

P(&, Y$;)@P(R, Lf;)(-kI),

(3.13)

The space of conformal blocks of the WZW model with target group G, studied in Sect. 2, was (3.14) v, =H * ( A , qy) Likewise, the space of conformal blocks nfthe WZW model with target group H is

.

vrr H%v, q:;). =;

(3.15)

Here we take JV with its standard complex structure, and a positive tensor power of Y(2,. Upon reversing the complex structure on A‘ and 9’‘2), we see that, if hF is the dual vector space to V,, then

v,. =HO(2, Y Consequently, if 2 is trivial,

- L’) .

*(

w7=vc@qp~

(3.16) (3.17)

Now, we want to find the appropriate statement that holds when Z is not trivial. First of all, the natural projection of F 4 F ‘ induces a natural map i : PAP”. i is not quite an ernbcdding; the kernel consists of constant gauge transformations 62 elements of the center of E. i is also not quite surjective; the quotient Z ’ = p i i ( F ) consists of “twists” by elements of Z in going around closed one-cycles in Z (described explicitly at the end of this subsection), so in fact 2;‘ = Hom(N,(Z, Z). Thus we have an exact sequence

O-ti[P)+P

-

Z’-0.

a,

(3.1 8)

183 Holomorphic Factorization of WZW and Coset Models

203

Similarly, after complexification (which does not affect finite_grou_psand so leaves Z and Z' unmodified), we have a natural projection i: F,+F& and an exact sequence 0-r i(Fc) Pc+Z'+O (3.19) with the same Z. We can take the quotient of Q by Fc by first dividing by i(pc) and then dividing by Z'. As W'/i(Fc)= .Ix 3, we get a natural action of Z' on .X x 3, and 9=(Ax R ) / Z 8 *

(3.20)

From this it Pollows that, if X z ' denotes the 2' invariant part of a vector space X , then

w =(VG@V,.)"'

.

(3.21)

The Z action on A ' x . 3 that enters here is easy to describe explicitly. is the moduli space of According to the Narasimhan-Seshadri theorem, .X x representations of the fundamental group of C in G x H.For C of gcnus r, such a representation is given explicitly by holonomies (gi,hl), ..., (g2*,h2J about 2r generating cycles (modulo conjugation, and subject to a well-known relation). 2' acts by ( g l , hl), ..., (gZr,hzr)+(zlgl,zlhl),,.., (ZzrgZr.ZZrhZr), with zlr ..., z Z rbeing arbitrary elements of 2. 3.3. Holomorphic Factorization The vector space W has a natural Hermitian structure formally given by

(It is convenient to divide by vol(G). vol(8), and not by vol(P), which differs from this by a factor of #Z, the number of elements in Z.) We want to show that the partition function of the GIH coset model is ZG,H(4

= lXIZ '

(3.23)

The reasoning required is very similar to that in Sect. 2, so we will be brief. One first introduces a conjugate WZW model, with gauge group G, x H,. The action, for h : C +G, and A and B gauge fields of G, and H,, is

+ 2n1 J dZzTr B,h- '8,h

1

j d2z Tr A,&h h- ' 2x2 1 1 +J d2z Tr A,hB,h-' - - J dZzTr(A,A, + B,B,) . (3.24) 2n I 4n 2 We thus get an integral expression for x(A,B): x(A,B )= j" Dh e-kr'(h,A,B). (3.25) I'(h,A , B) = I(h)

-

2

Combining (3.6) and (3.25), we get

-

~

184 E. Witten

204

Writing out the exponent on the right-hand sideof(3.26) explicitly, one sees that it is quadratic in A. The integral over A is Gaussian, therefore. After doing this integral one finds that, using the Polyakov-Wiegman formula, the integral over g and h collapses to an integral over f = gh. The remaining functional integral is precisely the definition (3.2) of the partition function Z,,,(Z) of the G/II model, completing the formal proofof(3.23).These steps proceed precisely in parallel with the corresponding points in the derivation of (2.28), and will not be elaborated further. It remains to consider what happens when the complex structure of 1 varies, Again, the parallel with Sect. 2 is so close that we can be brief. When the mrnplex structure of ,Z varies, W varies, as the fiber of a vector bundtc W Over the space Y of complex structures on E. w' has a projectively flat connection, gwcn by formulas analogous to those of Sect. 2. The holomorphic structure of f 'is defined by saying that a section z(A, B ; pj is holomorphic if it is annihilated by the (0,lj part of the

The antiholomorphic structure is defined by the (1,O) part of the connection (3.28)

me

justification of these formulas is that V commutes with the operators on the left hand side of (3.1 1). Alternatively, one can deduce these rormulas systematically by working out the Bogoliubov transformation that compensates for a change in polarization of 0. The fact that B , appears in (3.27) and A , in (3.28) of course reflects the ubiquitous reversal of sign and change ofcomplex structure of the coset model.] Precisely as in Sect. 2, by difkrentiating the definition of x under the integral sign, one shows that x is antiholomorphic,

v"

'O'X(A,B;@)= 0 .

(3.29)

(3.23) and (3.29) make up what i s usually called holomorphic factorization of the GIH model. 4. The GIG Model

In this section, we will consider the special case of the G/H coset model for H = C. This casc is particularly simple, being a topological field theory, and as a result sharper statements can be made. The understanding of these statements also illuminates the "ordinary" models, even the original WZW model, as we will see. The action of the G/G model is the familiar G/H action,

(4.1j

specialized to the casc H = G . Thus, B is now a gauge field valued in the Lie algebra of G . ( E is of course gauging the adjoint subgroup of G, x GR,so the covariant

185 Holomorphic Factorization of WZW and Coset Models

205

+

derivative of g will be Dig= aig [Be g].) The main novelty of the case G = H is that this model is a topological field theory, in the sense that (for instance) the partition function is independent of the metric of C. We will first prove this directly, and then reformulate the argument in the language of holomorphic factorization. For the direct proof, we note that under an infinitesimal change in the metric of C,the action of the gauged WZW model changes according to the following formula: H(g, B) =

J 6e,,$'

871 r

Tr(g- 'D,g)'+

j dZz6e,,~zL Tr(D,g. g- ')' 871 I

.

(4.3)

The variation of the partition function is hence

+ J d z z 6 ~ , , ~ 'Tr(D,g-g-')"), "

(4.4)

and we must show that this vanishes. To do this, we will show that the integrand in (4.4) is a total derivative in function space. In fact, since the variation of the action in a change of the connection B is

Assuming that one can integrate by part in function space, the left-hand side of(4.6) vanishes, and this means that the first term on the right-hand side of (4.4) can be discarded. The second term on the right-hand side of (4.4) similarly vanishes since

4.1. Factorization

The attentive reader will note that the key fact in the last paragraph was the Sugaward-Sommerfield construction (2.40), which also played a key role in the analysis ofholomorphic factorization of general WZW and coset models. In fact, it is illuminating to recast the above argument in the language of holomorphic factorization.

186 206

E. Witten

Precisely as in the general discussion of coset models, we consider the gaugmg of HI, x G, (which now is G, x G,). The closest to a gauge invariant action is (3.171, which wc repeat for convenience:

1 +1dzz Tr B,gA,g2n E

-

~

4x x

d2z Tr(A,A,

+ &,B,f.

(4.8)

The novelty, compared to the case of arbitrary H , is that ROW there is a kind of symmetry betwcen thc Gr. and ci, gauge fields B and A : (4.8) is invariant under reversing the complex structure (or equivalently, the orientation) of E,exchanging g with g - l , and exchanging A and B. [Alternatively, if one exchanges g with g-' and exchanges A and B, while leaving the orientation of Z fixed, then (4.8) is complex conjugated.] We want to 5ee the consequences of this. Just as in the general case of arbitrary ff,onc introduces the holomorphic wave function X(A,B) = 5 Dg p - w A , R ) . (4.9)

The general arguments specialized to this case show that the norm of

Ix12= z

x is

C ! m

(4.10)

=0 .

(4.11)

and that x is anti-holomorphic, p1,0'*

The novelty is the symmetry between A and B, which reverses the complex structures, and so makes it apparent that must also be holomorphic, V(0,1'*

= 0.

(4.12)

Equations (4.11) and (4.12) can both be proved by using the general definition (3.27) and (3.28) of the connection and differentiating under the integral sign, as in the proof of (2.35). Equations (4.11) and (4.12) together mean that x is covariantly constant, and hence 1x1' is a constant. From the factorization Iaw(4.10) we thusdeduce again that Z,&) is independent of the metric. To probe more deeply, we now recall the genera! description in Sect. 3 of the vector bundle W in which holomorphic factorization of the coset model takes place. We had (4.13) x E WXJG*lZ' > with v, = H O ( A , 9"k) , (4.14) VH= HO(./V, Y O k ) . Setting H = G and interpreting V,@ V,, as Hom( V,, VG),we have

x E (Hom(vG,v,))z'.

(4.1 5)

Now, Hom( V,, VG)contains a canonical (and 2-invariant) element, the identity map 1 ; it is natural to ask whether x = 3 .

187 Holomorphic Factorization of WZW and Coset Models

207

The “symmetry” between A and B H s it obvious that x is hermitian (in the natural norm on VG).Indeed x(A, B ) = x f B ,A )because (4.8) is complex conjugated if one exchanges A and B while mapping g+g- ’. With our methods it is also easy to prove that p=x. (4.16) This amounts to the statement that (4.17)

This is proved by replacing each of the three copies of x that appear in (4.17 ) with the integral representation (4.9); performing the Gaussian integral over B, and using the Polyakov-Wiegmann formula Gust as in the original proof of holomorphic factorization in Sect. 2.2). Equation (4.16) means that x is the orthogonal projection operator onto a subbundle Y ’of Y“ (whose inclusion in Y is compatible with the projectively flat connection on Y lsince y, is covariantly constant). It will be evident presently that holomorphic factorization can be carried out in Y‘. One expects that x = 1 and V ’ = f ^but , the methods of this paper do not seem to suffice for proving this. The fact that x is covariantly constant means that if e,(A; e) is an orthonormal basis ofcovariantly constant sections of Y ’ ,then x(A, B ; e)= 1 Qiiei(A;p)epB;e), i,j

with some constants Qi,? The fact that x 2 = x (and x = 1 when restricted to V’, by definition of V ’ )means that Qi.j = 6,. So (4.18)

We can thus compute the norm of x to get ZG,G(C)= 1x1’ = dim (9’”‘).

(4.19)

One expects that Y = V ,but in any case, if this is not true, it is V ’that should be called the space of conformal blocks in the WZW model. (This will be even more apparent in the next subsection.) So we have established that the partition function of the GIG model is the number of conformal blocks of the WZW model, a result that has been conjectured by Spiegelglas [22], with considerable evidence. 4.2. Relation to the WZW Model and “Ordinary” Coser Models Now we will see what we can learn about the original WZW model, and general coset models, by applying our knowledge of the G / G model. The reason that one can learn something interesting is that, upon returning to the definition (4.9) of x, and noting that I(g,O,O) is the original action of the WZW model, we see that the partition function of the WZW model is ZG(C)= x(0,O;e) .

(4.20)

In view of (4.18), we get therefore (4.21)

m

E. Witten

This formula expresses the partition function of the WZW model in terms of quantities that naturally arise in quantizing the moduli space d of G-valued connections, namely the orthonormal parallel sections ei(A;Q). As a check, let us verify that (4.21) is compatible with the earlier description of Z,(c) as the norm squared of a holomorphic section of T : Z,(Z) = 1 YIZ.

(4.22)

Recalling the definition (2.6)of Y , we see that Y ( A ;e)= ,y(A,0; e),so from (4.18) we get

dimc’

-

‘W;el = 1 44 deAO; el.

(4.23)

i= 1

As the ei are orthonormal, insertion of this in (4.22) gives back (4.21). In a similar fashion, one can also obtain a formula for the partition function of the G/H model. Recalling the definitions (4.14)of V, and V,, we see that there is a : which takcs a section of Y@“ ovcr , and I restricts it to natural map r G / HV,+V,, A”. (As AV and N arc the moduli spaccs of holomorphic GE and Ha:bundles, respectively, there is a natural inclusion of .Ar in A.)Taking complex conjugates, there is also a natural map r&: V:-+”. These maps do not respect the unittary struclures. For every H , wc have holomorphic factorization zG;H(z)=

I%G/dA,

B;@)I2

1

(4.24)

where ,yGIH is the functional defined in (3.6). Inspecting the dehition, we see that

(4.25)

Alternatively,

with fj an orthonormal basis of parallel sections of YT&. Formulas of this type were suggested in [13, 141. If one takes H to be the trivial group (with only the identity element), then (4.26) reduces, as it should, to (4.21j. Appendix

The purpose of this appendix is to clarify the geometric meaning of the classical gauged WZW actions on which this paper is based. Some readers may wish to consult this appendix before reading the body of the paper (see also [4,29, 301). The problem can be clarified by formulating it in more generality than we actually need. We consider an arbitrary connected manifold M with a closed threeform w whose periods are multiples of 2n, so that w is related to a class in H 3 ( M ,Q. We let ,Z be an oriented two dimensional surface without boundary. To simplify the considerations that follow, we assume that r r , ( M ) = x 2 ( M j = 0 , so that a

189 209

Holomorphic Factorization of WZW and Coset Models

continuous map X :C+M is automatically nullhomotopic. (The main novelty that arises if one relaxes this condition is that one must use integral cohomology instead of just working with differential forms.) We suppose given the action of a compact Lie group F on M and we suppose that w is F invariant. To simplify the story, we suppose that F is simple and simply-connected. (Again, if these conditions are relaxed, the main novelty that arises is that one must use integral F-equivariant cohomology, rather than the de Rham model that will appear presently.) We describe the Lie algebra of F with generators T, and relations

CT,,T,I=E*T,. Let T ( X )= J x * w , B

where B is any three manifold with aB = L,and an arbitrary extension of X over B has been chosen. r has values in RJ2nZ. We wish to construct a gauge invariant generalization of r. The action of I: on M is generated by vector fields 5.Introducing a gauge field A = C A"7,, with structure group F, we want to find a generalization T ( X , A ) of r a

that is invariant under for

E'

6X=tUV,, dA"= -DC", an infinitesimal gauge transformation. The variation of

(A.3)

r is

6r = 2J Eax*(iye(w)).

(A.4)

(iy is the operation of contracting with a vector field K) Additional terms that can be added to (A.2) to cancel this exist only if there are one-forms laon M such that and moreover such that

+ i",(U = 0.

(A.6) If such 1, exist, then, by averaging suitably over the compact group F , one can suppose that they transform in the adjoint representation of F. In this case, the desired gauge invariant generalization of T is i".(&)

1

T ( X ,A ) = r ( X )- J A" A X*(l,) - - J A" A Ab . X*(iv,l,). r 22

(A.7)

Equations (AS) and (A.6) have a geometrical meaning, in terms of the so-called F-equivariant cohomology of M , denoted H;(M). A de Rham model for this equivariant cohomology, explained in [31, 321, can be described as follows. Let Q*(M)be the de Rham complex of M , and let S * ( 9 ) be a symmetric algebra on the Lie algebra 9 of F , with generators @ considered to be of degree two. Let W* =(Q*(M)@S*(st))F(with denoting the F invariant part). In W*, introduce the differential D=d+

C@iy,. (1

210

E. Witten

Ifw is a closed form on M, an element c3 E W* is called an equivariant extension of w if DG=O and = o =a.The meaning of (AS) and (A.6)is simply that they are the conditions for w to have an equivariant extension. In fact,

is an equivariant extension of w if and only if the An obey (A.5) and (A.6) and transform in the adjoint representation of F . Now let us specialize to the case ofactual interest in this paper in which M is the group manifold of a simple, compact, connected, and simply-connected Lie group G, and 1 12K

w = --Tr(g-'dg)'.

(A.10)

Mureover, F is a connecled subgroup of G , x G,. The embedding oPF in G,> and G R is determined by an embedding of Lie algebras which we can writc as T,+IT,,L,

(A.l I j

T.,R l '

The vector fields V, are described by the formula

& =IEn(Th.Lg-gT,,,)-

(A.12)

One has i,~= dL,

(A.13)

with

These An transform in the adjoint representation of F. The non-uniqueness in the is A,,+%,+dwa, where the w, are zero forms in the adjoint choice of the i, representation of F. Equation (A.14)is the unique universal formula that works for any F. One now computes that 1 ivaO-A+ kb(&)= g

Tr(To,LTb,L-

T,,R'G,R).

(A.15)

[Note that the possible w, do not contribute since i,~(dw,)=f,',w, is antisymmetric in a and b.] Thus the equivariant extension W of w and the corresponding gauge invariant extension T(g,A ) of T exist precisely if F is such that the right-hand side of (A.15)vanishes. This is the criterion that was stated in (2.1).The gauge invariant extension of I-, when F is such that (A.15) vanishes, is explicitly

Tk,A ) = m -

1

E A"

A

1

T r (T,,

g-

+ T,.R g - 'd d .

A" A Ab Tr(T,,.g- l&,Lg- q,Rg-lT,&). (A.16) 8, Even when (A.15)does not vanish, (A.16) is the closest that there is to a gauge invariant extension ofr(g,A), in the sense that the variation of (A.16)under a gauge transformation depends only on A and is independent of g. This fact, which played an important role in the body of the paper, reflects the fact that the 1, obeying (AS) --

191 Holomorphic Factorization of WZW and Coset Models

211

exist for any F ; only the validity of (A.6) depends on F. This means that although an extension W of w obeying DW = 0 may not exist, w always has an extension such that DW E S*(F). (A.17) (And such an 6 is unique if one wishes a formula that works universally for any F.) This relation precisely ensures that the violation of gauge invariance depends on A and not g. Geometrically, the reason that (A.17) has a solution is as follows. The equivariant cohomology of G is the cohomology of the homotopy quotient G//F = G x ,EF. If one computes the cohomology of G / / F from the spectral sequence of the fibration G//F+BF, one sees (since w is a three dimensional class, and the nontrivial cohomology of BF begins in dimension four) that the only obstruction to existence of an equivariant extension W of w comes from H4(BF). In fact, the invariant quadratic form on the F Lie algebra that appears on the right-hand side of (A.15) represents the obstruction class in H4(BF), via the Chern-Weil homomorphism. The cohomology of BF is isomorphic to S*(F), so the obstruction is an element of S*(F). The gauge invariant generalization of the WZW Lagrangian is (A.18) with * the Hodge star operator, d , the gauge-covariant extension of the exterior derivative, and r(g,A ) given in (A.16). The first term depends on the conformal structure of Z, and the second has a topological origin that we have attempted to elucidate in this appendix. The properties of the WZW model depend on a peculiar interplay between the two terms, some aspects of which we have seen in this paper. All the particular formulas for gauged WZW Lagrangians given in this paper are various specializations of (A.18). References 1. Witten, E.: Non-Abelian bosonization in two dimensions. Commun. Math. Phys. 92, 455 (1984) 2. Wess, J., Zumino, B.: Consequences ofanomalous ward identities. Phys. Lett. 37B,95(1971) 3. Witten, E.:Global aspects of current algebra. Nucl. Phys. B223, 422 (1983) 4. Felder, G., Gawedzki, K., Kupianen, A.: Spectra of Wess-Zurnino-Witten models with arbitrary simple groups. Commun.Math. Phys. 117, 127 (1988) GawedAi, K.: Topological actions in two-dimensional quantum field theories. In: Nonperturbative quantum field theory. 't Hooft, G. et al. (eds.). London: Plenum Press 1988 5. Knizhnik, V.G., Zamolodchikov, A.B.: Current algebra and Wess-Zumino model in two dimensions. Nucl. Phys. B247, 83 (1984) 6. Frieda< D., Shenker, S.: The analytic geometry of two-dimensional conformal field theory. Nucl. Phys. B281,509-545 (19x7) 7. Verlinde, E.:Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B300, 351 8. Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett. B212, 360 (1988); Classical and quantum conformal field theory. Nucl. Phys. B 9. Witten, E.:Quantum ficld theory and the Jones polynomial. Commun. Math. Phys. 121,351

(1989) 10. Elitzur, S., Moore, G., Schwimmer, A,, Seiberg, N.: Remarks on the canonical quantization of the Chern-Simons-Witten theory. IAS preprint HEP-89/20

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11. Axelrod, S., DellaPietra, S., Witten, E.: Geometric quantization of Chern-Simons gauge theory. J, Diff. Georn. 33, 787 (1991) 12. Hitchin, N.: Flat connections and geometric quantization. Commun. Math. Phys. 131, 347 (1 990) 13. Gawedzki, K.: Constructive conformal field theory. In: Functional integration, geometry, and strings. Hava, Z., Sobczyk, J. (eds.). Boston, Basel: BirkhPuser 1989 14. Gawedzki, K., Kupianen, A.: G/H conformal field theory from gauged WZW model. Phys. Lett. 215B. 119 (1988); Coset construction from functional integrals. Nucl. Phys. B 320 (W), 649 (1989) 15. Bernard, D.:On the Wess-Zumino-Witten models on the torus. Nucl. Phys. B303.77 (1988); On the Wess-Zumino-Witten models on Riemann surfaces. Nucl. Phys. A309, 145 (1988) 16. Guadagnini, E., Martellini, M., Minchev, M.: Phys. Lett. 191B, 69 (1987) 17. Bardacki, K., Rabinovici, E., Saring, B.: Nucl. Phys. B299, 157 (1988) Altschulcr, A., Bardacki, K., Rabinovici, E.: Commun. Math. Phys. 118, 241 (1988) 18. Karabali, D., Park, 9.-H., Schnitzer, H.J., Yang, Z.: Phys. Lett. 216B, 307 (1989) Schnitzer, H.J.: Nucl. Phys. 8324, 412 (1989) Karabali, D., Schnitzer, H.J.: Nucl. Phys. B329, 649 (1990) 19. Goddard, P., Kent, A,, Olive, D.: Phys. Lett. B152, 88 (1985) 20. Bardacki, K., Halpern, M.B.: Phys. Rev. D3, 2493 (1971) Halpern, M.B.: Phys. Rev. D4, 2398 (1971) 21. Moore, G., Seiberg, N.: Taming the conformal zoo. Phys. Lett. B 22. Spiegelglas, M.:Lecture at IAS (October, 1990),Setting Fusion Rules in Topological LandauGinzburg. Technion preprint; Spiegelglas, M., Yankielowicz, s.: G/G Topological Field Theory by Cosetting. Fusion Rules As Amplitudes in G/G Theories. Preprints (to appear) 23. Kostant, B.: Orbits, symplectic structures, and representation theory, Proc. of the U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965);Quantization and Unitary Representations. Lecture Notes in Math., vol. 1170, p. 87. Berlin, Hridelberg, New York: Springer 1970; Line Bundles and the Prcquantized Schrodinger Equation. COIL Group Theoretical Methods in Physics (Marseille, 1972) p. 81 24. Souriau, J.: Quantification gcometrique. Commun. Math. Phys. 1, 374 (1966). Structures des systemes dynamiques. Paris: Dunod 1970 25. Gawedzki, K.: Quadrature of conformal field theories. Nucl. Phys. 328, 733 (1989) 26. Polyakov, A.M., Wiegman, P.B.: Theory ofnon-ahclian Goldstonc bosons in two dimensions. Phys. Lett. B 131, 121 (1983) 27. Rarnadas, T.R., Singer, LM., Weitsman, J.: Some comments on Chern-Simons gauge theory. MIT prcprint 28. Freed, D.: Preprint (to appear) 29. Axelrod, S.: Ph. D. thesis, Princeton University (1991), Chapter four 30. Hull, C.M., Spence, B.: The geometry of the gauged sigma model with Wess-Zumino Term. Queen Mary and Wcstfield Collegc prcprint Q M W 90/04 31. Atiyah, M.F., Bott, R.: The moment map and equivariant cohornology. Topology 23,1(1984) 32. Mathai, V., Quillen, D.: Supcrconnections, Thom classes, and cquivariant diffcrcntial forms. Topology 25, 85 (1986)

Communicated by A. Jaffe

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+-

Index

adjoint representation, 99 affine Lie algebra, 37 asymptotic expansion, $3 Atiyah's axioms, 55 Atiyah-Bott's residue formular, 95

Dirac operator, 81 Duistermaat-Heckman formular, 95 elliptic operators, 68 equivariant cohomology, 91 equivariant differential form, 91 Euler-Lagrange equation, 1, 24

Borel-Weil-Bott theorem, 34, 38, 41 braiding matrix, 4 i BRST operator, 80 BRST symmetry, 86 RRST transformations, 90

Fiddeev-Popo\~ghost, 79 Feynman diagrams, 76 Feynman path integral, 68 Feynman rules, 87 fiber bundle, 13 flop, 14 framing, 83 fusing matrix, 46

canonical fornidisrn, 2 circle, action, 7 Calabi-Yau, 13 characters, 34, 42, 58 Chern-Simons action, 7 , 27 Cherri class, 24, 2 5 , 31 Chern form, 26 Chern-Wed lhcory, 26 class function, 41 Clebst.~i-(;ortlori conditions, 46 complex structure, 17, 29 conformal block, 28, 35: 44 conforrrtal field theory, 37 C~IIIIOC tion, covariant derivative, 19, 23, 2 5 , 29, 79 critical points, 21

gauge ecpivalcrit field, 19, 20 gauge group, 17, 23 gauge field,17 gauge transformation, 17, 20, 21, 23 Gaussian integral, 71 georrietrir invariant theory, 11, 15, 31 C inzburg- Landau, 13 Grassmanian, 1 2 hariiioriic oscillators, 4 Heegard decompositinr:, 53 Hermitian metric, 15 highest weight, 41, 42

determinant.tiirridlc, 30 197

198

Index

holonomy, 19, 21: 78, 84 holornorphic vector bundle, 29, 95 holomorphic differentials, 31 holomorphic method, 6 Hodge operator, 22, 23, 24 Hopf algebra, 61 integrabte highest weight representation, 58 Jacobi variety, 31 Kahler form, 29, 30 Kahler manifoId, 15, 29 Kahler metric, 29 Knizhik-Zamalodchikov equations, 38, 43, 48 localization! 89 loop group, 37, 58 mapping class group, 43, 46, 49: 54 Maxwell equation, 22 modular transformations, 49 moduli space, 17. 21, 27, 28, 29. 30, 31

moment map, 8, 18, 2 1 monodromy, 3.5, 43 monopole, 33 morphism, 67, 68 partition function, 78 Pontryagin classes, 20, 24 principle bundle, 22 projective flat connection, 38, 43 quantum group, 60, 64 QuiUen metric, 40 Ray-Singer analytic torsion, 82 regulization scheme, 85 renormalization, 85 self-dual, 34

semi-stable, 9 skein relation, 36, 56 spectrum, 71 stable bundle, 35

structure group, 17 supersymmetric Lagrangian. 90 symplectic action, 9 symplectic method, 3 symplectic quotient, 7 , 15: 19 symplectic reduction, 17, 19 symplectk variety, 42 Teichmuller space, 43, 91 topological sigma model, 90 universal enveloping algebra, 63 vector bundle: 25 Verlinde basis, 49 Weyl’s character formular, 41 Weyl group, 33! 41 W Z W model, 38 ’fang-Baxter, 48, 62 Yang-kIjh equation, 7, 22, 23, 24, 27

Afterwards

This monograph arises from E. Witten’s lectures on topological quantum field theory in the spring of 1989 at Fine Hall of Princeton University. At that time E. Witten unified several important mathematical works in tcrms of quantum field thcory, most notably, Donaldson polynomial, Gromov/ Floer homology and Jones polynomials. In his lectures Witten explained his three-dimensional intrinsic construction of Jones polynomials via Chcm-Simons gauge theory. His construction leads to many beautiful applications such as skein relation, surgery formula and a proof of Verlinde’s formula. The heart of the construction is to quantize Chern-Simons action. He used geometric quantization technique. Recall that a quantum state is a probability over the classical space. In this case the classical moduli space consists of unitary Aat connections. The Hilbert space is then the space of holornorphic sections of the natural determinant bundle over the moduli space, or non-Abelian theta functions. To have a topological theory one hais to show that the quantization is independent of other choices, in this case the choices of complex structures. It is shown that there exists a projective flat connection for the Hilbert space bundle over the space of complex structures [Axelrod-DellaPietra-Witten], [Hitchin]. Knizhik-ZamoIodchikov equations are the explicit expression of the projective flat connection for the case of punctured sphere. In Atiyah’s axioms of topological quantum field theory, one also needs to construct rnorphisms between Hilbert spaces corresponding to cobordisms of manifolds. Witten illustrated that such morphisms are given by the Feynman path integral with Chern-Simons action. He explained relevant background such as Feynman-Kac formula and Feynman diagrams 199

200

Aft emardds

and made it more accessible to mathematicians. Therc are some interesting works by theoretical phycisists to try t o make sense of the perturbation series. It is found that symmetries play a most important role here. After all, Chern-Sinions action is the only Lagrangian to enjoy both gauge symmetry and diffeomorphism symmetry. Divergent Feymann diagrams cancelled from each other because of the two symmetries. We have added some materials to fill details left out and t o update somc: new developments. In Chapter 4 wc explained the approach batled on representations of mapping class groups [Moore and Seiberg] , [Kohno]. Due t o time constraint and limitation of my knowledge we omitted many important approaches, e.g. quantum group method [Rcshetikhin-Turaev], [Kauffman], Vassiliev invariant [Birman and Lin] , [Bar-Natan], [Kontsevich] , perturbation series [Axelrod-Singer], among others. We hope interested readers can find some materials from those references. We also found it very amusing to define knot invariant via string theory or topological sigma models. In Chapter 6 we gave a very brief introduction of this method based on Witten’s paper[Witten-strind. There are very interesting conjectures by Vafa [Vafa] about constructing knot invariant from closed string theory recently. We explained localization principle which is usually used to establish mathematical bases for topological sigma models. Localization principle has become a powerful tool in dealing with many problcmv in topological quantum field theory and in string theory. It is a pleasure for me t o study the notes I took from Witten’s lectures. It is not surprising that it takes quite long to absorb and understand some of the materials. I found it very rewarding to study them. I wish to thank E. Wittcn for answcring some of the naive questions from me and for providing a nice Forward. I wish tn khank S. S. Chern for his genuine interest and encouragement. I wish to thank Dr. K. K . Phua who encouraged m e t o write up thcse notes. I wish to thank B. Abikoff, S . Axelrod, Chen Wei, P. Deligne, Jing &in, Lin Xiassong: Liu Kefeng, H. Verlinde, Paul Yang and Zhang Weiping for many helpful discussions. I wish to thank Lu Jitan and Kiln Tan for their professional editorial work. And last but not least I thank my wife, Hou Bo, for her support. Without all of t h e interest and encouragcmcnt above I would not be able t o finish this work. By Sen Hu November 11, 2000