Surveys in Differential Geometry, Vol. 5: Differential Geometry Inspired by String Theory

Volume V

Surveys in Differential Geometry: Differential Geometry inspired by String Theory

A supplement to the Journal of Differential Geometry Edited by S.-T. Yau

'

International Press

JOURNAL OF DIFFERENTIAL GEOMETRY C.C. HSIUNG Lehigh University Bethlehem, PA 18015

Editors-in-Chief S.-T. YAU Harvard University Cambridge, MA 02138 Editors

JEFF CHEEGER New York University New York, NY 10012

H. BLAINE LAWSON State University of New York Stony Brook, NY 11794

SIMON K. DONALDSON Stanford University Stanford, CA 94305

RICHARD M. SCHOEN Stanford University Stanford, CA 94305

MICHAEL H. FREEDMAN University of California

La Jolla, CA 92093

Associate Editors SHIGEFUMI MORI Faculty of Sciences Nagoya University

Nagoya 464, Japan NIGEL HITCHIN Mathematics Institute University of Warwick Coventry CV4 AL, England

ALAN WEINSTEIN University of California Berkeley, CA 94720

Surveys in Differential Geometry, Differential Geometry inspired by String Theory, S.-T. Yau, Editor-in-Chief. ISBN 1-57146-070-5 International Press Incorporated. Boston PG Box 43502 Somerville, MA 02143 All rights are reserved. No part of this work can be reproduced in any form, electronic or mechanical, recording, or by any information storage and data retrieval system, without the specific authorization from the publisher. Reproduction for claassroom or personal use will, in most cases, be granted without charge. Copyright ® 1999 International Press Printed in the United States of America The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Surveys in Differential Geometry, vol. 5

Introduction In this volume of Surveys in Differential Geometry, we collect papers on the part of geometry that is related to the modern development of string theory. Recall that in the first volume of Surveys in Differential Geometry, we have the paper by Witten on strings with gravity which was applied to calculate chern numbers for certain line bundles over moduli space of curves. The geometric ideas initiated by consideration of string theory have been tremendous successful, and we believe that it is time to collect articles that can survey the subject in a reasonable way. The papers in this volume only represent some part of the interaction between string theory and geometry that is important and rigorous. We have the paper by Bismuit and Labourie on the Verlinde formula, which is an accumulation of ideas from physicists and mathematicians. We have the paper of Aspinwall on K3 surface and string duality, which is a simplest nontrivial space to test some very nontrivial duality theory in string theory. We have the paper by Bryan and Leung on counting curves on surfaces, which is also inspired by works on branes in string theory. We have the paper of Kefeng Liu on how to apply ideas of localization through heat kernel to various geometric model, which in particular inspired the previous mentioned work of Bismuit and Labourie. Finally, we have collected a few papers related to mirror symmetry among Calabi-Yau manifolds. While there have been tremendous activities in this subject in the past ten years, the final rigorous treatment of the formula by Candelas and others on toric varities is only accomplished in the paper of Lian-Liu-Yau, which we reproduce here. Li and Tian's paper surveys both quantum cohomology and the vitual cycles which are part of the whole theory. In 1996, Strominger-Yau-Zaslow come up with a geometric construction of mirror manifolds, which needs a lot more rigorous mathematical

treatment. Mark Gross's paper represents a good survey in the topological and some geometrical side of this.

We hope these papers will provide a good starting point for geometers who may want to get into the subject of the interaction of geometric and string theory.

Table of Contents K3 Surfaces and String Duality Paul S. Aspinwall ..........................................................1

Symplectic Geometry and the Verlinde Formulas Jean-Michel Bismut and Frangois Labourie ............................... 97

Counting curves on irrational surfaces Jim Bryan and Naichung Conan Leung .................................. 313

Special Lagrangian Fibrations II: Geometry. A Survey of Techniques in the Study of Special Lagrangian Fibrations Mark Gross ............................................................. 341

Mirror Principle I Bong H. Liars, Kefeng Liu, and S.-T. Yau ................................ 405

Mirror Principle II Bong H. Lian, Kefeng Liu, and S.-T. Yau . ............................... 455

Differential Equations from Mirror Symmetry Bong H. Lian and S: T. Yau ............................................. 510

Heat Kernels, Symplectic Geometry, Moduli Spaces and Finite Groups Kefeng Liu .............................................................. 527

A Brief Tour of GW Invariants Gang Tian and Jun Li ...................................................543

Surveys in Differential Geometry Vol. 1: Lectures given in 1990, edited by S.-T. Yau and H. Blaine Lawson Vol. 2: Lectures given in 1993, edited by C. C. Hsiung and S.-T. Yau Vol. 3: Lectures given in 1996, edited by C.C. Hsiung and S.-T. Yau Vol. 4: Integrable Systems, edited by Chuu Lian Terng and Karen Uhlenbeck

Vol. 5: Differential Geometry inspired by String Theory, edited by S.-T. Yau

Surveys in Differential Geometry, vol. 5

K3 Surfaces and String Duality Paul S. Aspinwall Dept. of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855

ABSTRACT The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface.

The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review "old string theory" on K3 surfaces in terms of conformal field theory. The type HA string, the type IIB string, the Es x Es heterotic string, and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface itself.

01999 international Press

PAUL S. ASPINWALL

2

CONTENTS

1.

Introduction

2. Classical Geometry 3. The World-Sheet Perspective 4. Type II String Theory 5. Four-Dimensional Theories 6. The Heterotic String Acknowledgements References

4 23 33 4.1

67 89 89

1. Introduction The notion of "duality" has led to something of a revolution in string theory in the past year or two. Two theories are considered dual to each other if they ultimately describe exactly the same physics. In order for this to be a useful property, of course, it is best if the two theories appear, at first sight, to be completely unrelated. Different notions of duality abound depending on how the two theories differ. The canonical example is that of "S-duality" where the coupling of one theory is inversely related to that of the other so that one theory would be weakly coupled when the other is strongly coupled. "T-duality" can be defined by similarly considering length scales rather than coupling constants. Thus a theory at sniall distances can be "T-dual" to another theory at large distances. In the quest for understanding duality many examples of dual pairs have been postulated. The general scenario is that one takes a string theory (or perhaps Alltheory) and compactifies it on some space and then finds a dual partner in the lhrm of another (or perhaps the same) string theory compactified on some other space. In this form duality has become a subject dominated by geometrical ideas since most of the work involved lies in analyzing the spaces on which the string theory is compactified. One of the spaces which has become almost omnipresent in the study of string duality is that of the K3 surface. We will introduce the K3 surface in section 2 but let as make a few comments here. Mathematicians have been studying the geometry of the K3 surface as a real surface or a complex surface for over one hundred years

In these lectures we will always be working with complex numbers and so "surface" will mean a space of complex dimension two, or real dimension four. A curve will be complex dimension one, etc. Physicists' interest in the K3 surface (for an early paper see, for example, [2]) was not sparked until Yau's proof [3] of Calabi's conjecture in 1977. Since then the K3 surface has become a commonly-used "toy model" for compactifications (see, for example, [4]) as it provides the second simplest example of a Ricci-flat compact manifold after the torus. The study of duality is best started with toy models and so the K3 surface and the torus are bound to appear often. Another reason for the appearance of the K3 surface, as we shall see in these lectures, is that the mathematics of the heterotic string appears to be intrinsically bound to the geometry of the K3 surface. Thus, whenever the heterotic string appears on one side of a pair of dual theories, the K3 surface is likely to make an appearance in the analysis. [1].

K3 SURFACES AND STRING DUALITY

3

The original purpose of these lectures was to give a fairly complete account of the way K3 surfaces appear in the subject of string duality. For reasons outlined above, however, this is almost tantamount to covering the entire subject of string duality. In order to make the task manageable therefore we will have to omit some current areas of active research. Let us first then discuss what will not be covered in these lectures. Note that each of these subjects are covered excellently by the other lecture series anyway.

Firstly we are going to largely ignore M-theory. M-theory may well turn out to be an excellent model for understanding string theory or perhaps even replacing string theory. It also provides a simple interpretation for many of the effects we will discussing.

Secondly we are going to ignore open strings and D-branes. There is no doubt that D-branes offer a very good intuitive approach to many of the phenomena we are going to study. It may well also be that D-branes are absolutely necessarily for a complete understanding of the foundations of string theory. Having said that M-theory and D-branes are very important we will now do our best to not mention them. One reason for this is to able to finish giving these lectures on time but another, perhaps more important reason is to avoid introducing unnecessary assumptions. We want to take a kind of "Occam's Razor" approach and only introduce constructions as necessary. To many people's tastes our arguments will become fairly cumbersome, especially when dealing with the heterotic string, and it will certainly be true that a simpler picture could be formulated using Mtheory or D-branes in some instances. What is important however is that we produce a self-consistent framework in which one may analyze the questions we wish to pose in these lectures. Thirdly we are going to try to avoid explicit references to solitons. Since nonperturbative physics is absolutely central to most of the later portions of these lectures one may view our attitude as perverse. Indeed, one really cannot claim to understand much of the physics in these lectures without considering the soliton effects. What we will be focusing on, however, is the structure of moduli spaces and we will be able to get away with ignoring solitons in this context quite effectively. The only time that solitons will be of interest is when they become massless. Our main goal is to understand the type II string and the heterotic string compactified on a K3 surface and what such models are dual to. Of central importance will be the notion of the moduli space of a given theory. In section 2 we will introduce the K3 surface itself and describe its geometry. The facts we require from both differential geometry and algebraic geometry are introduced. In section 3 we will review the "old" approach to a string theory on a K3 surface in terms of the world-sheet conformal field theory. In section 4 we begin our discussion of full string theory on a K3 surface in terms of the type IIA and type IIB string. The start of this section includes some basic facts about target-space supergravity which are then exploited. The heterotic string is studied in section 6 but before that we need to take a long detour into the study of string theories compactified down to four dimensions. This detour comprises section 5 and builds the techniques required for section 6. The heterotic string on a K3 surface is a very rich and complicated subject. The

PAUL S. ASPINWALL

4

analysis is far from complete and section 6 is technically more difficult than the preceding sections.

Note that blocks of text beginning with a "M' are rather technical and may be omitted on first reading.

2. Classical Geometry In the mid 19th century the Karakorum range of mountains in Northern Kashmir acted as a natural protection to India, then under British rule, front the Chinese and Russians to the north. Accordingly, in 1856, Captain T. G. Montgomerie was sent out to make some attempt to map the region. From a distance of 128 miles he measured the peaks across the horizon and gave them the names Ki, K2, K3, ... ,

where the "K" stood simply for "Karakorum" [5]. While it later transpired that most of these mountain peaks already had names known to the Survey of India, the second peak retained the name Montgomerie had assigned it.

It was not until almost a century later in 1954 that K2, the world's second highest peak, was climbed by Achille Compagnoni and Lino Lacedelli in an Italian expedition. This event led shortly afterwards to the naming of an object of a quite different character. The first occurrence of the name "K3" referring to an algebraic variety occurs in print in (6]. It is explained in (7] that the naming is after Kummer, Kahler and Kodaira and is inspired by K2. It was Kummer who did much of the earliest work to explore the geometry of the space in question.'

2.1. Definition. So what exactly is a K3 surface? Let us first introduce a few definitions. For a general guide to some of the basic principles used in these lectures we refer the reader to chapter 0 of [8]. First we define the Hodge numbers of a space X as the dimensions of the Dolbeault cohomology groups hn,9(X) = dim(HP,v(X))

(1)

Next consider the canonical class, K, which we may be taken to be defined as

K = -ci(Tx),

(2)

where Tx is the holomorphic tangent bundle of X. A K3 surface, S, is defined as a compact complex Kiihler manifold of complex dimension two, i.e., a surface, such that hr,o(S) = 0 (3)

K = 0.

Note that we will sometimes relax the requirement that S be a manifold. The remarkable fact that makes K3 surfaces so special is the following THEOREM 1. Any two KS surfaces are difeomorphic to each other. 'This may explain the erroneous notion in some of the physics literature that the naming is Ka after "Kummer's third surface" (whatever his first two surfaces may have been). To confuse the issue slightly there is a special kind of K3 surface known as a "Kummer surface" which is introduced in section 2.6.

KS SURFACES AND STRING DUALITY

5

Thus, if we can find one example of a K3 surface we may deduce all of the topological invariants. The simplest realization is to find a simple example as a complex surface embedded in a complex projective space, i.e., as an algebraic variety. The obvious way to do this is to consider the hypersurface defined by the equation

f = xo + xi + x2 + x3 = 0

(4)

in the projective space p3 with homogeneous coordinates [x0, x1, X2, x3]. It follows from the Lefschetz hyperplane theorem [8] that h1"0 of such a hyper-

surface will be zero. Next we need to find if we can determine n such that K = 0. Associated to the canonical class is the canonical line bundle. This is simply the holomorphic line bundle, L, such that c1 (L) = K. From our definition of K it follows that the canonical line bundle for a manifold of dimension d can be regarded as the dth exterior power of the holomorphic cotangent bundle. Thus a section of the canonical line bundle can be regarded as a holomorphic d-form. The fact that K = 0 for a K3 surface tells us that the canonical line bundle is trivial and thus has a holomorphic section which is nowhere zero. Consider two such sections, s1 and $2. The ratio s1 /s2 is therefore a holomorphic function defined globally over the compact K3 surface. From basic complex analysis it follows that s1/s2 is a constant. We see that the K3 surface admits a globally defined, nowhere vanishing, holomorphic 2-form, (1, which is unique up to a constant. It also follows that h2'0(S) = 1. Let us try to build fl for our hypersurface of degree n in P3. First define affine coordinates in the patch x0 56 0:

yl =

xl 2°,

1/2 =

12 Zo,

Y3 =

x3 2°

(5)

An obvious symmetric choice for 0 is then

_

dy1 A dye

8f /Oys

_

dye A dy3

_

8//81/1

dy3 A dy1

8f /81/2

(6)

This is clearly nonzero and holomorphic in our patch x0 0 0. We can now consider another patch such as x1 54 0. A straight forward but rather tedious calculation then shows that 1 will only extend into a holomorphic nonzero 2-form over this next patch if n = 4. Our first example of a K3 surface is called the quartic surface, given by a hypersurface of degree 4 in P3. We could have arrived at this same conclusion in a somewhat more abstract way by using the adjunction formula. Consider the tangent bundle of S, which we denote Ts, together with the normal bundle Ns for the embedding S C P3. One can then see that TS

NS = Tr5Is,

(7)

where TpsJs is the restriction of the tangent bundle of the embedding p3 to the hypersurface S. Introducing the formal sum of Chern classes of a bundle E (see, for example, [9])

c(E)=1+c1(E)+c2(E)+...,

(8)

c(Tps1s) = c(Ts) A c(Ns).

(9)

we see that

PAUL S. ASPINWALL

6

Treating a wedge product as usual multiplication from now on, it is known that [8]

c(Tph) = (1 + x)k1"1,

(10)

where x is the fundamental generator of H2(Pt,7L). Since H2 is dual to H2(k-1) which is dual to H2(k-1), we may also regard x as the homology class of a plane Pk-1 C fl embedded in the obvious way by setting one of the homogeneous coordinates to zero.2 Stated in this way one can see that cl (Ns) is given by nx. Thus we have that x) 4

c(Ts) = 2-+ 1+nx

(11)

= 1+(4-n)x+ (6 -4n+rt22)x2. Again we see that n = 4 is required to obtain K = 0. Now we also have c2 which enables us to work out the Euler characteristic, X(S), of a K3 surface: X (S) =

Js

C2(TS)

= f is A

cs(Ts)

Pa

(12)

= fy 4x.6x2 P = 24,

where is is the 2-form which is the dual of the dual of the hypersurface S in the sense explained above. One may also show using the Lefschetz hyperplane theorem

that rrl(S) = 0.

(13)

We now have enough information to compute all the Hodge numbers, 10'4. Since

7r1(S) = 0, we have that the first Betti number b1(S) = dim H' (S) = h' ° + h°'' must be zero. The Euler characteristic then fixes b2(S) which then determines h1'' since we already know h2'0 = 1 from above. The result is ho,o

h°'1 h2'° h1'1 h°'2 h2'1 h1'2

1

h1"0

hz,2

0

=

1

0

20 0

1.

(14)

0 1

2Throughout these lectures we will often use the same notation for the 2-form and the associated 2(k - 1)-cycle.

K3 SURFACES AND STRING DUALITY

7

2.2. Holonomy. Before continuing our discussion of K3 surfaces, we will take a detour and discuss the subject of holonomy which will be of considerable use at many points in these lectures. Holonomy is a natural concept in the differential geometry of a manifold, M,

with a vector bundle, R : E -> M, with a connection. Consider taking a point, p E M, and a vector in the fibre, e1 E 7r I (p). Now, following a closed path, r, starting and ending at p, parallel transport this vector according to the connection. When you are done, you will have another vector e2 E r-1 (p). Write e2 = gr(el),

where gr(e1) is an element of the structure group of the bundle. The (global) holonomy group of E -a M is defined as the group generated by all such gr(el) for all closed paths F. The holonomy, SAM, of a Riemannian manifold, M, is defined as the holonomy of the tangent bundle equipped with the Levi-Civita connection from the metric. The holonomy of a Riemannian manifold of real dimension d is contained in O(d). If it is orientable this becomes S0(d). The study of which other holonomy groups are possible is a very interesting question and will be of some importance to us. We refer the reader to (10, 11] for a full discussion of the results and derivations. We require the following: 1..S)M C U(2) if and only if M is a Kahler manifold. 2. SjM C SU(d) if and only if M is a Ricci-flat Klihler manifold. 3. ,f)M C Sp(4) if and only if M is a hyperkbhler manifold. 4. S)M C Sp(d). Sp(1) if and only if M is a quaternionic Kfhler manifold .3 5. A "symmetric space" of the form G/H where G and H are Lie groups has holonomy H. Actually in each case the specific representation of the group in which the fibre transforms is also fixed. A celebrated theorem due to Berger, with contributions from Simons (11], then states that the only other possibilities not yet mentioned are $M 9G2 where the fibre transforms as a 7 or SJM °-° Spin(7) where the fibre transforms as an 8. There is a fairly clear relationship between the holonomy groups and the invariant forms of the natural metric on M in the first cases. For the most general case we have that the form d dx' ®dx'

(15)

on Rd admits O(d) as the group of invariances. In the complex K5hler case we consider the Hermitian form d/2

d2' ®dz'

(16)

{.1

on Cd/2 which admits U (sl) as the invariance group.

3The "." denotes that we take the direct product except that the Z2 centers of each group are

identified.

8

PAUL S. ASPINWALL

For the next case we consider the quaternionic numbers, Ill. In this case the natural form d/4

dS' 0 dr'

(17)

on lHld/4 is preserved by Sp(.). Note that writing quaternions as 2 x 2 matrices in the usual way gives an embedding Sp(4) C SU(a). Thus, a hyperkiihler manifold is always a R.icci-fiat Kahler manifold. In fact, one is free to choose one of a family of complex structures. Let us denote a quaternion by q = a + bI + cJ + cK, where

a, b, c, d E R, I2 = J2 = K2 = -1 and IJ = K, JI = -K, etc. Given a hyperkahler structure we may choose a complex structure given by q, where q2 = -1. This implies a = 0 and b2 + c2 + d2 = 1.

(18)

Thus for a given hyperkiililer structure we have a whole S2 of possible complex structures. We will see that the K3 surface is hyperkiihler when equipped with a Ricci-fiat metric.

Because of the fact that quaternionic numbers are not commutative, we also have the notion of a quaternionic Kahler manifold in addition to that of the byperkahler manifold. The space IHP admits an action of Sp(n). Sp(1) by multiplication on the right by n x n quaternionic matrices in Sp(n) and by a quaternion of unit norm on the left. This also leads to the notion of a manifold with a kind of quaternionic structure - this time the "quaternionic Ki filer manifold". The main difference between this and the hyperkahler manifold is that the extra Sp(l) can act on the S2 of complex structures between patches And so destroy any global complex

structure. All that remains is an S2 bundle of almost complex structures which need have no global section. Thus a generic quaternionic Kiihler manifold will not admit a complex structure. When this bundle is trivial the situation reduces to the hyperkahler case. As an example, the space 1Wn is quaternionic Kiihler. Note that

the case n = 1 is somewhat redundant as this reduces to Sp(1). Sp(1) = SO(4), which gives a generic orientable Riemannian manifold.

2.3. Moduli space of complex structures. We now want to construct the moduli space of all K3 surfaces. In order to determine the moduli space it is very important to specify exactly what data defines a particular K3 surface. By considering various data we will construct several different moduli spaces throughout

these lectures. To begin with we want to consider the K3 surface purely as an object in algebraic geometry and, as such, we will find the moduli space of complex structures for K3 surfaces. To "measure" the complex structure we need some relatively simple quantity which depends on the complex structure. This will be provided by "periods" which are simply integrals of the holomorphic 2-form, Cl, over integral 2-cycles within S. To analyze periods we first then require an understanding of the integral 2-cycles H2(S, Z).

KS SURFACES AND STRING DUALITY

9

Since b2(S) = 22 from the previous section, we see that H2 (S, Z) is isomorphic

to Z22 as a group." In addition to this group structure we may specify an inner product between any two elements, ai E H2(S,Z), given by al -a2 = #(ai r) a2),

(19)

where the notation on the right refers to the oriented intersection number, which is a natural operation on homology cycles [9]. This abelian group structure with an inner product gives H2 (S, Z) the structure of a lattice. The signature of this lattice can be determined from the index theorem for the signature complex [9]

T=J 3(c2l - 2C2)=-3X(S)= -16.

(20)

Thus the 22-dimensional lattice has signature (3,19). Poincard duality tells us that given a basis {et} of 2-cycles for H2(S,Z), for each e; we may find an ey such that ei.ej* = 5ij,

(21)

where the set {ej*} also forms a basis for H2 (S, Z). Thus H2 (S, Z) is a self-dual (or unimodular) lattice. Note that this also means that the lattice of integral cohomology, H2(S, Z), is isomorphic to the lattice of integral homology, H2(S, Z). The next fact we require is that the lattice H2 (S, Z) is even. That is, e.e E 2Z,

b'e E H2(S,Z).

(22)

This is a basic topology fact for any spin manifold, i.e., for cj(TX) = 0 (mod 2). We will not attempt a proof of this as this is rather difficult (see, for example, Wu's formula in [13]). The classification of even self-dual lattices is extremely restrictive. We will use

the notation I'to refer to an even self-dual lattice of signature (m, n). It is known that m and n must satisfy

m - n = 0 (mod 8)

(23)

and that if m > 0 and n > 0 then r,.,, is unique up to isometries [14, 15]. An isometry is an automorphism of the lattice which preserves the inner product.

4Actually we need the result that H2 (S, Z) is torsion-free to make this statement to avoid any finite

subgroups appearing. This follows from irl (S) = 0 and the various relations between homotopy and torsion in homology and cohomology [12].

PAUL S. ASPINWALL

10

In our case, one may chose a basis such that the inner product on the basis elements forms the matrix

-Es

-Es U U

U/ where -E8 denotes the 8 x 8 matrix given by minus the Cartan matrix of the Lie algebra Es and U represents the "hyperbolic plane"

U'f (?

/

.

(25)

Now we may consider periods Wi =

f

' ci.

e

(26)

We wish to phrase these periods in terms of the lattice r3,19 we have just discussed. First we will fix a specific embedding of a basis, lei), of 2-cycles into the Lattice r3,19. That is, we make a specific choice of which periods we will determine. Such a choice is called a "marking" and a K3 surface, together with such it marking, is called a "marked K3 surface". There is the natural embedding r3,19 ee H2(S,Z) C H2(S,R) = j,3, t9 (27)

We may now divide f t E H2 (S, ( as

fl = x + iv,

(28)

where z, V E H2 (S, Ilk. Now a (p, q)-form integrated over S is only nonzero if p = q = 2 [8] and so

Js

fl A fl = (z + iy).(x + iy) = (x.x - y.y) + 2ix.?l

=0. Thus, x.x = y.y and x/.y = 0. We also have

Js fz n l = (x +iy).(x - iy) = (x.x + y.y) (30)

=

f 110112 > 0.

K3 SURFACES AND STRING DUALITY

11

The vectors x and y must be linearly independent over H2(S, R) and so span a 2-plane in H2 (S, R) which we will also give the name 1. The relations (29) and (30) determine that this 2-plane must be space-like, i.e., any vector, v, within it satisfies v.v > 0.

Note that the 2-plane is equipped with a natural orientation but that under complex conjugation one induces (x, y) -+ (x, -y) and this orientation is reversed. We therefore have the following picture. The choice of a complex structure on a K3 surface determines a vector space R3,19 which contains an even self-dual lattice r3,19 and an oriented 2-plane Cl. If we change the complex structure on the K3 surface we expect the periods to change and so the plane fl will rotate with respect to the lattice r3,19. We almost have enough technology now to build our moduli space of complex structures on a K3 surface. Before we can give the result however we need to worry about special things that can happen within the moduli space. A K3 surface which gives a 2-plane, Cl, which very nearly contains a light-like direction, will have periods

which are only just acceptable and so this K3 surface will be near the boundary of our moduli space. As we approach the boundary we expect the K3 surfaces to degenerate in some way. Aside from this obvious behaviour we need to worry that some points away from this natural boundary may also correspond to K3 surfaces which have degenerated in some way. It turns out that there are such points in the moduli space and these will be of particular interest to us in these lectures. They will correspond to orbifolds, as we will explain in detail in section 2.6. For now, however, we need to include the orbifolds in our moduli space to be able to state the form of the moduli space. The last result we require is the following THEOREM 2 (Torelli). The moduli space of complex structures on a marked KS surface (including orbifold points) is given by the space of possible periods. For an account of the origin of this theorem we refer to [16]. Thus, the moduli

space of complex structures on a marked K3 surface is given by the space of all possible oriented 2-planes in IR3,19 with respect to a fixed lattice rs,19.

Consider this space of oriented 2-planes in R. Such a space is called a Grassmannian, which we denote Gr+(a, k3,19), and we have 0+(3,19) ( 31) (0(2) x 0(1,19)) + This may be deduced as follows. In order to build the Grassmannian of 2-planes in 83,19, first consider all rotations, 0(3,19), of k3,19. Of these, we do not care about internal rotations within the 2-plane, 0(2), or rotations normal to it, 0(1, 19). For oriented 2-planes we consider only the index 2 subgroup which preserves orientation on the space-like directions. We use the "+" superscripts to denote this. This Grassmannian builds the moduli space of marked K3 surfaces. We now want to remove the effects of the marking. There are diffeomorphisms of the K3 surface, which we want to regard as trivial as far as our moduli space is concerned, but which have a nontrivial action on the lattice H2(S, Z). Clearly any diffeomorphism induces an isometry of H2(S, Z), preserving the inner product. We denote the full group of such isometries as 0(r3,,9) 5 Our moduli space of marked K3 surfaces can Gr+(fJ, ks,19) a,

*

5Sometimes the less precise notation 0(3,19; Z) is used.

PAUL S. ASPINWALL

12

be viewed as a kind of Teichmuller space, and the image of the diffeoulorphisms in O(13,19) can be viewed as the modular group. The moduli space is the quotient of the Teichmuller space by the modular group.

What is this modular group? It was shown in [17, 181 that any element of 0+(r3,19) can be induced from a diffeomorphism of the K3 surface. It was shown further in [19] that any element of 0(r3,19) which is not in 0+(173,19) cannot he induced by a diffeomorphism. Thus our modular group is precisely 0+(r: ,9). Treating (31) as a right coset we will act on the left for the action of the modular group. The result is that the moduli space of complex structures on a K3 surface (including orbifold points) is

.W. °-` O+(x3,19)\O+(3,19)/(0(2) x 0(1,19))+.

(32)

When dealing with A it is important to realize that O+(f. ,9) has all ergodic action on the Teichmiilier space and thus.///, is actually not Hausdorff. Such unpleasant behaviour is sometimes seen in string theory in fairly pathological circumstances [20] but it seems reasonable to expect that under reasonable conditions we should see a fairly well-behaved moduli space. As we shall see, the moduli space

.4f' does not appear to make any natural appearance in string theory and the related moduli spaces which do appear will actually be Hausdorff.

2.4. Einstein metrics. The first modification we will consider is that of considering the moduli space of Einstein metrics on a K3 surface. We will always assume that the metric is Kahler. An Einstein metric is a (pseudo-)Riemannian metric on a (pseudo-)Rieniallnian manifold whose Ricci curvature is proportional to the metric. Actually, for a K3 surface, this condition implies that the metric is Ricci-flat [21]. We may thus use the terms "Einstein" and "Ricci-flat" interchangeably in our discussion of K3 surfaces. The Hodge star will play an essential role in the discussion of the desired moduli space. Recall that [9, 10] a A */3 = (a, (3)w9,

(33)

where a and ,B are p-forms, wg is the volume form and (a, 0) is given by (34) (a, fl) = p1 J E g0291112a ny,... .. ,... dxldx2 ... , i,iaji12 in local coordinates. In particular, if a is self-dual, in the sense a = *a, then

a.a > 0 in the notation of section 2.3. Similarly an anti-self-dual 2-form will obey a.a < 0. On our K3 surface S we may decompose

H2(S,R) =Yo+E

(35)

where .3t° represents the cohomology of the space of (anti-)self-dual 2-forms. We

then see that dim.3G°+ = 3,

dim.3e- = 19,

(36)

from section 2.3.

The curvature acts naturally on the bundle of (anti)-self-dual 2-forms. By standard methods (see, for example, [10]) one may show that the curvature of the bundle of self-dual 2-forms is actually zero when the manifold in question is a K3

K3 SURFACES AND STRING DUALITY

13

surface. This is one way of seeing directly the action of the SU(2) holonomy of section 2.2. Since a K3 is simply-connected, this shows that the bundle 3l°+ is trivial and thus has 3 linearly independent sections. Consider a local orthonormal frame of the cotangent bundle {eI, e2, e3, e4 }. We may write the three sections of .310+ as 81 = el A e2 +e3 A e4 82 = el A e3 + e4 A e2

(37)

as = el A e4 + e2 A e3.

Clearly an SO(4) rotation of the cotangent directions produces an SO(3) rotation of 3t°+. That is, a rotation within .3t°+ is induced by a reparametrization of the + is fixed. underlying K3 surface. One should note that the orientation of Let us denote by E the space 3t°+ viewed as a subspace of H2(S, R). Putting dzl = el + ie2 and dz2 = e3 + ie4 we obtain a Kahler form equal to sl, and the holomorphic 2-form dzl A dz2 is given by 82 + is3. This shows that E is spanned by the 2-plane S1 of section 2.3 together with the direction in H2 (S, R) given by the Kahler form. This fits in very nicely with Yau's theorem [3] which states that for any manifold M with K = 0, and a fixed complex structure, given a cohomology class of the Ki£hler form, there exists a unique Ricci-flat metric. Thus, we fix the complex structure by specifying the 2-plane, 0, and then choose a Kahler form, J, by specifying another direction in H2(S, R). Clearly this third direction is space-like, since Vol(S) = r J A J > 0, s

(38)

and it is perpendicular to 11 as the Kahler form is of type (1, 1). Thus E, spanned by 11 and J, is space-like. The beauty of Yau's theorem is that we need never concern ourselves with the explicit form of the Einstein metric on the K3 surface. Once we have fixed fl and J, we know that a unique metric exists. Traditionalists may find it rather unsatisfactory that we do not write the metric down - indeed no explicit metric on a K3 surface has ever been determined to date - but one of the lessons we appear to have learnt from the analysis of Calabi-Yau manifolds in string theory is that knowledge of the metric is relatively unimportant. As far as our moduli space is concerned, one aspect of the above analysis which is important is that rotations within the 3-plane, E, may affect what we consider to be the Kahler form and complex structure but they do not affect the underlying Riemannian metric. We see that a K3 surface viewed as a Riemannian manifold may admit a whole family of complex structures. Actually this family is parametrized by the sphere, S2, of ways in which E is divided into 0 and J. This property comes from the fact that a K3 surface admits a hyperkahler structure. This is obvious from section 2.2 as a K3 is Ricci-flat and Kibler and thus has holonomy SU(2), and SU(2) Sp(1). The sphere (18) of possible complex structures is exactly the S2 degree of freedom of rotating within the 3-plane E we found above.

14

PAUL S. ASPINWAI.I,

Some care should betaken to show that the maps involved are su jective [22, 23] but we end up with the following (24]

THEOREM 3. The moduli space of Einstein rnetr ics, for it KS sul fare (including orbifold points) is given by the Grussmannian of arwented 3-planes within the space 1113"19 modulo the effects of diffeomorphisraas carting on the lattice H2(S,Z). In other words, we have a relation similaru to (32): olKE °-a 0+(r3,19)\0+(3,19)/(50(3) x 0(19)) x R+,

(39)

where the 118+ factor denotes the volume of the K3 surface given by (3S). This is actually isomorphic to the space

.,WE °-a 0(r3,19)\0(3,19)/(0(3) x 0(19)) x I1t.a.,

(40)

since the extra generator required, -1 E 0(3,19), to elevate 0{'(3, 19) to 0(3,19), is in the center and so taking the left-right coset makes no difference. Note that (40) is actually a Hausdorff space as the discrete group 0(I';a,Ig) has a properly discontinuous action [25]. Thus we see that the global description of this space of Einstein metrics on a K3 surface is much better behaved than the moduli space of complex structures discussed earlier.

2.5. Algebraic K3 Surfaces. In section 2.4 we enlarged the uloduli space of complex structures of section 2.3 and we found a space with nice properties. In this section we are going to find another nice moduli space by going ill the opposite direction. That is, we will restrict the moduli space of complex structures by putting constraints on the K3 surface. We are going to consider algebraic K3 surfaces, i.e., K3 surfaces that may be embedded by algebraic (holonaorphic) equations in homogeneous coordinates into PN for some N. The analysis is done in terms of algebraic curves, that is, R.ienuuul surfaces which have been holomorphically embedded into our K3 surface S. Clearly such a curve, C, may be regarded as a homology cycle and thus an eleuueui. of.H_(S,2 ). By the "dual of the dual" construction of section 2.1 we may also regard it as all element C E H2(S,Z). The fact that C is holomorphically embedded may also be used to show that C E H1A(S) [8]. We then have C E Pic(S), where we define

Pic(S) =HZ(S,Z)flH"(S),

(41)

which is called the "Picard group", or "Picard lattice", of S. We also define the "Picard number", p(S), as the rank of the Picard lattice. Any element, of the Picard group, e E Pic(S), corresponds to a line bundle, L, such that cI (L) = c: [8]. Thus the Picard group may be regarded as the group of line bundles on S. where the group composition is the Whitney product. As the complex structure of S is varied, the Picard group changes. This is because an element of Ha (S, Z) that was regarded as having type purely (1. 1) may pick up parts of type (2, 0) or (0,2) as we vary the complex structure. A completely generic K3 surface will have completely trivial Picard group, i.e., p = 0.

The fact that S contains a curve C is therefore a restriction oil the complex structure of S. An algebraic K3 surface similarly has its deformations restricted as 6Note that the orientation problem makes this look more unlike (32) than it needs to! We encourage the reader to not concern themselves with these orientation issues, at least on first reading.

K3 SURFACES AND STRING DUALITY

15

the embedding in PN will imply the existence of one or more curves. As an example let us return to the case where S is a quartic surface

f =xo+xi+x;+x3=0,

(42)

in P3. A hyperplane P'' C P3 will cut f = 0 along a curve and so shows the existence of an algebraic curve C. Taking various hyperplanes will produce various curves but they are all homologous as 2-cycles and thus define a unique element of the Picard lattice. Thus the Picard number of this quartic surface is at least one. Actually for the "Fermat" form off given in (42) it turns out7 that p = 20 (which is clearly the maximum value we may have since dimH""I(S) = 20). A quartic surface need not be in the special form (42). We may consider a more general 1

aijkl 4-414-131

.!

(43)

for arbitrary aijki E C. Generically we expect no elements of the Picard lattice other than those generated by C and so p(S) = 1. Now consider the moduli space of complex structures on a quartic surface. Since C is of type (1, 1), the 2-plane Si of section 2.3 must remain orthogonal to this direction.

We may determine C.C by taking two hyperplane sections and finding the number of points of intersection. The intersection of two hyperplanes in P3 is clearly PI and so the intersection C fl C is given by a quartic in PI, which is 4 points. Thus C.C = 4 and C spans a space-like direction. Our moduli space will be similar to that of (32) except that we may remove the direction generated by C from consideration. Thus our 2-plane is now embedded in H2,19 and the discrete group is generated by the lattice AC = rs,h9 fl Cl. Note that we do not denote AC as r2,,9 as it is not even-self-dual. The moduli space in question is then Quartic °-` O(AC)\ 0(2,19)/(0(2) X 0(19)). (44) This is Hausdorff. Note also that it is a space of complex dimension 19 and that a simple analysis of (43) shows that f = 0 has 19 deformations of aijkl modulo reparametrizations of the embedding P3. Thus embedding this K3 surface in P3 has brought about a better-behaved moduli space of complex structures but we have "lost" one deformation as (32) has complex dimension 20. One may consider a more elaborate embedding such as a hypersurface in p2 X PI given by an equation of bidegree (3, 2), i.e.,

f=

,o+I+a2-s

'laoata.,lrobi xo X1 'T2 'Yo yi'.

(45)

no+elva

This is an algebraic K3 since p2 X PI itself may be embedded in Ps (see, for example,

[8]). Taking a hyperplane PI x PI C P2 X PI one cuts out a curve C,. Taking the hyperplane p2 x {p} for some p E PI cuts out C2. By the same method we used for the quartic above we find the intersection matrix r2 31 (46) 3 0J 71 thank M. Gross for explaining this to me.

PAUL S. ASPINWALL

16

Thus, denoting Ac,c2 by the sublattice of 1'3,19 orthogonal to C, and C2 we have

4' 0(Ac,c2)\ 0(2,18)/(0(2) x 0(18)).

(47)

This moduli space has dimension 18 and this algebraic K3 surface has p(S) = 2 generically. In general it is easy to see that the dimension of the nroduli space plus the generic Picard number will equal 20. Note that the Picard lattice will have signature (l,p - 1). This follows as it is orthogonal to S1 inside l3,i9 and thus has at most one space-like direction but the natural Kiihler form inherited from the ambient PN is itself in the Picard lattice and so there must be at least one space-like direction. Thus the moduli space of complex structures on an algebraic K3 surface will always be of the form

AAlg

0(A)\ 0(2,20 - p)/(0(2) x 0(20 - p)),

(48)

where A is the sublattice of H2(S,Z) orthogonal to the Picard lattice. This lattice is often referred to as the transcendental lattice of S. Note that this lattice is rarely self-dual.

2.6. Orbifolds and blow-ups. When discussing the moduli spaces above we have had to be careful to note that we may be including K3 surfaces which are not manifolds but, rather, orbifolds. The term "orbifold" was introduced many years ago by W. Thurston after their first appearance in the mathematics literature in [26] (where they were referred to as "V-manifolds"). The general idea is to slightly enlarge the concept of a manifold to objects which contain singularities produced by quotients. They have subsequently played a celebrated rule in string theory after they made their entry into the subject in [27]. It is probably worthwhile noting that the definition of an orbifold is slightly different in mathematics and physics. We will adopt the mathematics definition which, for our purposes, we take to be defined as follows:

An orbifold is a space which admits an open covering, such that each patch is diffeomorphic to alt' /Gi. The Gi's are discrete groups (which may be trivial) which can be taken to fix the origin of IL". The physics definition however is more global and defines an orhifold to be a space of the form M/G where M is a manifold and G is a discrete group. A physicist's orbifold is a special case of the orbifolds we consider here. Which definition is applicable to string theory is arguable. Most of the vast amount of analysis that has been done over the last 10 years on orbifolds has relied on the global form MIG. Having said that, one of the appeals of orbifolds is that string theory is (generically) well-behaved on such an object, and this behaviour only appears to require the local condition. The definition of an orbifold can be extended to the notion of a complex orhifold where each patch is biholomorphic to C"/Gi and the induced transition functions are holomorphic. Since we may define a metric on C"/Gi by the natural inherited metric on C", we have a notion of a metric on an orbifold. In fact, it is not hard to extend the notion of a Kiihler-Einstein metric to include orbifolds [28]. Similarly, a complex orbifold may be embeddable in PN and can thus be viewed as an algebraic variety. In this case the notion of canonical class is still valid. Thus, there is nothing to stop the definition of the K3 surface being extended to include orbifolds. As we

K3 SURFACES AND STRING DUALITY

17

will see in this section, such K3 surfaces he naturally at the boundary of the moduli space of K3 manifolds.

An example of such a K3 orbifold is the following, which is often the first K3 surface that string theorists encounter. Take the 4-torus defined as a complex manifold of dimension two by dividing the complex plane C2 with affine coordinates (zl, z2) by the group Z4 generated by Zk H. Zk + 1,

Zk H zk + i,

k = 1, 2.

(49)

Then consider the Z2 group of isometries generated by (zi, z2) -* (-z1, -z2). It is not hard to see that this Z2 generator fixes 16 points: (0,0), (0, (0, Thus we have an orbifold, which we will (y, 0), ... , (2 + 2 , 1 + (0, J + 2i),

2),

2i),

denote So. Since the Z2-action respects the complex structure and leaves the Kahler form invariant we expect So to be a complex Kahler orbifold. Also, a moment's thought

shows that any of the non-contractable loops of the 4-torus may be shrunk to a point after the Z2-identification is made. Thus rq (So) = 0. Also, the holomorphic 2-form dzl A dz2 is invariant. We thus expect K = 0 for So. All said, the orbifold So has every right to be called a K3 surface. We now want to see what the relation of this orbifold So might be to the general class of K3 manifolds. To do this, we are going to modify So to make it smooth. This process is known as "blowing-up". This procedure is completely local and so

we may restrict attention to a patch within So. Clearly the patch of interest is C2/Z2. The space C2 /Z2 can be written algebraically by embedding it in C3 as follows. Let (xo, xl, x2) denote the coordinates of C3 and consider the hypersurface A given by (50) f = xoxi - xa = 0. A hypersurface is smooth if and only if 8f/8xo = ... = 8f/8x2 = f = 0 has no

solution. Thus f = 0 is smooth everywhere except at the origin where it is singular. We can parameterize f = 0 by putting xo = C2, xi = 172, and x2 = tn. Clearly then (¢, 77) and (-g, -r7) denote the same point. This is the only identification and so f = 0 in C3 really is the orbifold C2 /Z2 we require. Consider now the following subspace of C3 x P2: {(xO,xi,x2),[80,81,82] E C3 X 1P2; x{8j = xj8i, Vi, j}.

(51)

This space may be viewed in two ways - either by trying to project it onto the C° or the 1F2. Fixing a point in lF2 determines a line (i.e., Cl) in C3. Thus, (51) determines a line bundle on 1F2. One may determine ci = -H for this bundle, where H is the hyperplane class. We may thus denote this bundle Opz(-1). Alternatively one may fix a point in C3. If this is not the point (0, 0, 0), this determines a point in p2. At (0,0,0) however we have the entire 1P2. Thus Op2 (-1) can be identified pointwise with C3 except that the origin in Cs has been replaced by 1P2. The space

(51) is thus referred to as a blow-up of Cs at the origin. The fact the Opa(-1) and CB are generically isomorphic as complex spaces in this way away from some subset means that these spaces are birationally equivalent [8]. A space X blown-up at a point will be denoted k and the birational map between them will usually be

18

PAUL S. ASPIN WALL

written

7:.k -> X.

(52)

That is, it represents the blow-down of X. The ]?2 which has grown out of the origin is called the exceptional divisor. Now let us consider what happens to the hypersurface A given by (50) in C3

as we blow up the origin. We will consider the proper transform, A C X. If X is blown-up at the point p E X then A is defined as the closure of the point set rr'1(A \ p) in X. Consider following a path in A towards the origin. In the blow-up, the point we land on in the exceptional P2 in the blow-up depends on the angle at which we approached the origin. Clearly the line given by (xot, x1t, x2t), t E C, xoxl -xa = 0, will land on the point [so, 81, 82] E P2 where again soil - s2 = 0. Thus the point set that provides the closure away from the origin is a quadric soil - sz = 0 in p2. It is easy to show that this curve has X = 2 and is thus a sphere, or P1.

K3 SURFACES AND STRING DUALITY

19

We have thus shown that when the origin is blown-up for A C C3, the proper transform of A replaces the old origin, i.e., the singularity, by a ?1. Within the context of blowing up A, this P1 is viewed as the exceptional divisor and we denote it E. What is more, this resulting space, A, is now smooth. We show this process in figure 1. Carefully looking at the coordinate patches in A around E, we can work out the

normal bundle for E C A. The result is that this line bundle is equal to OpI (-2). We will refer to such a rational (i.e., genus zero) curve in a complex surface as a "(-2)-curve". Let us move our attention for a second to the general subject of complex surfaces with K = 0 and consider algebraic curves within them. Consider a curve C of genus g. The self-intersection of a curve may be found by deforming the curve to another one, homologically equivalent, and counting the numbers of points of intersection (with orientation giving signs). In other words, we count the number of points which remain fixed under the deformation. Suppose we may deform and keep the curve algebraic. Then an infinitesimal such deformation may be considered as a section of the normal bundle of C and the self-intersection is the number of zeros, i.e., the

value of c, of the normal bundle integrated over C. Thus c1(N) gives the selfintersection, where N is the normal bundle. Note that two algebraic curves which intersect transversely always have positive intersection since the complex structure fixes the orientation. Thus this can only be carried out when c, (N) > 0. We may extend the concept however when c1 (-N) < 0 to the idea of negative self-intersection. In this case we see that C cannot be deformed to a nearby algebraic curve. The adjunction formula tells us the sum c1 (N) + c1(T ), where T is the tangent bundle of C, must give the first Chern lass of the embedding surface restricted to C. Thus, if K = 0, we have C.C = 2(g - 1).

(53)

That is, any rational curve in a K3 surface must be a (-2)-curve. Note that (53) provides a proof of our assertion in section 2.3 that the self-intersection of a cycle is

always an even number - at least in the case that the cycle is a smooth algebraic curve. Actually, if we blow-up all 16 fixed points of our original orbifold So we obtain a smooth K3 surface. To see this we need only show that the blow-up we have done

does not affect K = 0. One can show that this is indeed the case so long as the exceptional divisor satisfies (53), i.e., it is a (-2)-curve [8]. A smooth K3 surface obtained as the blow-up of T4/Z2 at all 16 fixed points is called a Kummer surface. Clearly the Picard number of a Kummer surface is at least 16. A Kummer surface need not be algebraic, just as the original T4 need not be algebraic. Now we have enough information to find how the orbifolds, such as So, fit into

the moduli space of Einstein metrics on a K3 surface. If we blow down a (-2)curve in a K3 surface we obtain, locally, a 0/Z2 quotient singularity. The above description appeared somewhat discontinuous but we may consider doing such a process gradually as follows. Denote the (-2)-curve as E. The size of E is given by the integral of the Kiihler form over E, that is, J.E. Keeping the complex structure of the K3 surface fixed we may maintain E in the Picard lattice but we may move

20

PAUL S. ASPINWALL

J so that it becomes orthogonal to E. Thus E has shrunk down to a point -- we have done the blow-down.

We have shown that any rational curve in a K3 surface is an element of the Picard lattice with C.C = -2. Actually the converse is true [16]. That is, given an element of the Picard lattice, e, such that e.e = -2, then either e or -e gives the class of a rational curve in the K3 surface. This will help us prove the following. Let us define the roots of r3,19 as {a E r3,19; a.a = -2). THEOREM 4. A point in the moduli space of Einstein metrics on a K3 surface corresponds to an orbifold if and only if the 3-plane, E, is orthogonal to a root of r3,19.

If we take a root which is perpendicular to E, then it must be perpendicular to fZ and thus in the Picard lattice. It follows that this root (or minus the root, which is also perpendicular to E) gives a rational curve in the K3. Then, since J is also perpendicular to this root, the rational curve has zero size and the K3 surface must be singular. Note also that any higher genus curve cannot be shrunk down in this way as it would be a space-like or light-like direction in the Picard lattice and could thus not be orthogonal to E. What remains to be shown is that the resulting singular K3 surface is always an orbifold. For a given point in the moduli space of Einstein metrics consider the set of roots orthogonal to E. Suppose this set can be divided into two mutually orthogonal sets. These would correspond to two sets of curves which did not intersect and thus would be blown down to two (or more) separate isolated points. Since the orbifold condition is local we may confine our attention to the case when this doesn't happen. The term "root" is borrowed from Lie group theory and we may analyze

our situation in the corresponding way. We may choose the "simple roots" in our set which will span the root lattice in the usual way (see, for example, [29)). Now consider the intersection matrix of the simple roots. It must have -2's down the diagonal and be negative definite (as it is orthogonal to E). This is entirely analogous to the classification of simply-laced Lie algebras and immediately tells us that there is an A-D-E classification of such events. We have already considered the Al case above. This was the case of a single isolated (-2)-curve which shrinks down to a point giving locally a C2/Z2 quotient

singularity. To proceed further let us try another example. The next simplest situation is that of a C2 /Z3 quotient singularity given by ([;, 17) H (4, w2p), where w is a cube root of unity. As before we may rewrite this as a subspace of 0 as (54) f = xoxl - xa = 0. The argument is very similar to the C2/Z2 case. The difference is that now as

we follow a line into the blown-up singularity and consider the path (sot, xlt, x2t) within xoxl - xa, the x2 term becomes irrelevant and the closure of the point set

becomes sosl = 0 within p2. This consists of two rational curves (so = 0 and sl = 0) intersecting transversely (at [so, s1, s2] = [0, 0,1]). Thus, when we blow-up C2/Z3, we obtain as an exceptional divisor two (-2)-curves crossing at one point. Clearly this is the A2 case.

Now consider the general case of a cyclic quotient C2/Z.. This is given by f = xoxl - xa = 0. At first sight the discussion above for the case n = 3 would appear to be exactly the same for any value of n > 3 but actually we need to be

K3 SURFACES AND STRING DUALITY

21

careful that after the blow-up we really have completely resolved the singularity. Consider the coordinate patch in Op7(-1) written as (51) where 82 # 0. We may use yo = so/s2, y1 = 31/32 as good affine coordinates on the base p2 and y2 = x2 as a good coordinate in the fibre. Since xo = Yoy2 and x1 = Y1y2, our hypersurface becomes

y2(yoyl - y2 -2) = 0.

(55)

Now y2 = 0 is the equation for the exceptional divisor P2 C P3 in our patch. We are interested in the proper transform of our surface and thus we do not want to include the full p2 in our solution - just the intersection with our surface. Thus we throw this solution away and are left with

yoyl - y2-2 = 0.

(56)

If n = 2 or 3 this is smooth. If n > 3 however we have a singularity at yo = y1 = y2 = 0. This point is at so = s1 = 0 which is precisely where the two P"s produced by the blow-up intersect. One may check that the other patches contain no singularities. What we have shown then is that starting with the space C2 /Z,,, n > 2, the blow-up replaces the singularity at the origin by two P1is which intersect at a point but, in the case n > 3, this point of intersection is locally of the form C2 To resolve the space completely, the procedure is clear. We simply repeat the process until we are done. Note that when we blow up the point of intersection of two P1is intersecting transversely, the fact that the P1's approach the point of intersection at a different "angle" means that after the blow-up they pass through different points of the exceptional divisor and thus become disjoint. We show the process of blowing-up a CI/Zs singularity in figure 2. In this process we produce a chain of 5 P's, E1, ... , E5, when completely resolving the singularity. We show the PI's as lines.

interClearly we see that resolving the C1 /Z singularity produces the section matrix for the (-2)-curves. Thus we have deduced the form of the A-series. We now ponder the D- and E-series. Consider the general form of the quotient C2/G. We are interested in the cases which occur locally in a K3 surface in which K = 0. This requires that G leaves the holomorphic 2-form dz1 A dz2 invariant. This implies that G must be a discrete subgroup of SU(2). One may also obtain this result by noting that the holonomy of the orbifold near the quotient singularity can be viewed as being isomorphic to G. The subgroups of SU(2) are best understood from the well-known exact sequence 1 -+ Z2 --> SU(2) -a SO(3) -a 1.

(57)

Thus any subgroup of SU(2) can be projected into a subgroup of SO(3) and considered as a symmetry of a 3-dimensional solid. The cyclic groups Z,, may be thought of as, for example, the symmetries of cones over regular polygons. The other possibilities are the dihedral groups which are the symmetries of a prism over a regular

polygon, and the symmetries of the tetrahedron, the octahedron (or cube), and the icosahedron (or dodecahedron). Each of these latter groups are nonabelian and correspond to a subgroup of SU(2) with twice as many elements as the subgroup of SO(3). They are thus called the binary dihedral, binary tetrahedral, binary octahedral, and binary icosahedral groups respectively. In each case the quotient C2 /G

PAUL S. ASPTNWALL

22

n=6 n= 4

FIGURE 2. Blowing up C2/Zs.

can be embedded as a hypersurface in C. This work was completed by Du Val (see [30] and references therein), after whom the singularities are sometimes named. The case of an icosahedron was done by Felix Klein last century [31] and so they are also often referred to as "Kleinian singularities". Once we have a hypersurface in Cs we may blow-up as before until we have a

smooth manifold. The intersection matrices of the resulting (-2)-curves can then be shown to be (minus) the Cartan matrix of D,,, Ee, E7, or E8 respectively. This process is laborious and is best approached using slightly more technology than we have introduced here. We refer the reader to [16] or [32] for more details. The results are summarized in table 1(see [33] for some of the details). We have thus shown that any degeneration of a K3 surface that may be achieved by blowing down (-2)-curves leads to an orbifold singularity. One might also mention that in the case of the A blow-ups, explicit metrics are known which are asymptotically flat [34, 35, 36]. Unfortunately, since the blow-up inside a K3 surface is not actually flat asymptotically, such metrics represent only an approximation to the situation we desire. As we said earlier however, lack of an explicit metric will not represent much of a problem. This miraculous correspondence between the A-D-E classification of discrete subgroups of SU(2) and Dynkin diagrams for simply-laced simple Lie groups must

count as one of the most curious interrelations in mathematics. We refer to [37] or [38] for the flavour of this subject. We will see later in section 4.3 that string theory will provide another striking connection. We have considered the resolution process from the point of view of blowing-up by changing the Kahler form. In terms of the moduli space of Einstein metrics on a K3 surface this is viewed as a rotation of the 3-plane E so that there are no longer

K3 SURFACES AND STRING DUALITY Group

Generators

Cyclic

001

(

Binary

Dih ed r

l

oahedral

si

1

x2+Y2=+_"+,=0

e

Ea,

/ 1

\\1

0

'

j'J , /

e=e

-+I 1 °

0

Resolution

x2 +Y2 + o" =0

,7

D4,

Octtahedral

Hypersurface

e-.

o 0

_, (3

0

Tettral edral

J.

0

0

13

23

17

rle

D"+2

x' + 1,3 + °

EB

x2+y3+y=9 = 0

E7

x2}Y3+s =0

Be

TABLE 1. A-D-E Quotient Singularities.

any roots in the orthogonal complement. Since E is spanned by the Kahler form and fi, which measures the complex structure, we may equally view this process in terms of changing the complex structure, rather than the Kahler form. In this language, the quotient singularity is deformed rather than blown-up. The process of resolving is now seen, not as giving a non-zero size to a shrunken rational curve, but rather changing the complex structure so that the rational curve no longer exists. This deformation process is actually very easy to understand in terms of the singularity as a hypersurface in C. Consider the Au_1 singularity in table 1. A deformation of this to

x2 +y 2 + z" + a,_2z"-2 +

all = 0

(58)

will produce a smooth hypersurface for generic values of the ai's. (Note that the an_lz"-1 term can be transformed away by a reparametrization.) It is worth emphasizing that in general, when considering any algebraic variety, blow-ups and deformations are quite different things. We will discuss later how the difference between blowing up a singularity and deforming it away can lead to topology changing processes in complex dimension three, for example. It is the peculiar way in which the complex structure moduli and Kahler form get mixed up in the moduli space of Einstein metrics on a K3 surface that makes them amount to much the same thing in this context. The relationship between blowing up and deformations will be deepened shortly when we discuss mirror symmetry.

3. The World-Sheet Perspective In this section we are going to embark on an analysis of string theory on K3 surfaces from what might be considered a rather old-fashioned point of view. That is, we are going to look at physics on the world-sheet. One point of view that was common more than a couple of years ago was that string theory could solve difficult problems by "pulling back" physics in the target space, which has a large number of dimensions and is hence difficult, to the world-sheet, which is two-dimensional and hence simple. Thus an understanding of two-dimensional physics on the world-sheet would suffice for understanding the universe. More recently it has been realized that the world-sheet approach is probably inadequate as it misses aspects of the string theory which are nonperturbative in

PAUL S. ASPINWALL

24

the string coupling expansion. Thus, attention has switched somewhat away from the world-sheet and back to the target space. One cannot forget the world-sheet however and, as we will see later in section 5, in some examples it would appear that the target space point of view appears on an equal footing with the world-sheet point of view. We must therefore first extract from the world-sheet as much ru we can.

3.1. The Nonlinear Sigma Model. "Old" string theory is defined as a twodimensional theory given by maps from a Riemaun surface, E, into a target manifold X:

x:E-*X.

(59)

In the conformal gauge, the action is given by (see, for example., [39] for the basic ideas and [40] for conventions on normalizations)

r r (gij - Bij) DA 52i daz - 21r f DR (22) d2z + ... , S = 8aa' l

i

s

(60)

where we have ignored any terms which contain fermions. The terms are identified as follows. g,, is a Riemannian metric on X and I3,j are the components of a real 2-form, B, on X. tk is the "dilaton" and is a real number (which might depend on x) and R(2) is the scalar curvature of E. This two-dimensional theory is known as the "non-linear a-model".

In order to obtain a valid string theory, we require that the resulting twodimensional theory is conformally invariant with a specific value of the "central charge". (See [41] for basic notions in conformal field theory.) Conformal invariance puts constraints on the various parameters above [42, 43]. In general the result is in terms of a perturbation theory in the quantity a'/R2, where R is some characteristic "radius" (coming from the metric) of X, assuming X to be compact. The simplest way of demanding conformal invariance to leading order is to set the dilaton, ', to be a constant, and let B be closed and g;j be Ricci-flat. There are other solutions, such as the one proposed in [44] and these do play a role in string duality as solitons (see, for example, [45]). It is probably safe to say however that the solution we will analyze, with the constant dilaton, is by far the best understood. In many ways one may regard this conformal invariance calculation to be the string "derivation" of general relativity. To leading order in a'/R.'2 we obtain Einstein's field equation and then perturbation theory "corrects" this to higher orders. Nonperturbative effects, i.e., "world-sheet instantons", should also modify notions

in general relativity. Anyway, we see that in this simple case, a vacuum solution for string theory is the same as that for general relativity - namely a Ricci-fiat manifold.

There is a simple and beautiful relationship between supersymmetry in this non-linear a-model and the Kiihler structure of the target space manifold X. We have neglected to include any fermions in the action (60) but they are of the form z/i' and transform, as far as the target space is concerned, as sections of the cotangent bundle. A supersymmetry will be roughly of the form

6,x' = F1 ,W.

(61)

The object of interest here is 1j'. With one supersymmetry (N = 1) on the worldsheet one may simply reparameterize to make it equal to M. When the N > 1

K3 SURFACES AND STRING DUALITY

25

however we have more structure. It was shown [46) that for N = 2 the second 14 acts as an almost complex structure and gives X the structure of a Kahler manifold. In the case N = 4, we have 3 almost complex structures, as in section 2.4, and this leads to a hyperkabler manifold. The converse also applies. Note that this relationship between world-sheet supersymmetry and the complex differential geometry of the target space required no reference to conformal invariance. In the case that we have conformal invariance one may also divide the analysis into holomorphic and anti-holomorphic parts and study them separately. In this case we have separate supersymmetries in the left-moving (holomorphic) and right-moving (anti-holomorphic) sectors. The case of interest to us, of course, is when X is a smooth K3 manifold. From what we have said, this will lead to an N = (4,4) superconformal field theory, at least to leading order in a'/R2. Actually for N = 4 non-linear v-models, the perturbation theory becomes much simpler and it can be shown [47] that there axe no further corrections to the Ricci-flat metric after the leading term. Additionally, one may present arguments that this is even true nonperturbatively [48]. Thus our Ricci-flat metric on the K3 surface is an exact solution. One must contrast this to the N = 2 case where there are both perturbative corrections and nonperturbative effects in general.

3.2. The Teichmuller space. The goal of this section is to find the moduli space of conformaily invariant non-linear o-models with K3 target space. This may be considered as an intermediate stage to that of the last section, where we considered classical geometry, and that of the following sections where we consider supposedly fully-fledged string theory. This will prove to be a very important step however. Firstly note that there are three sets of parameters in (60) which may be varied to span the moduli space required. In each case we need to know which deformations will be effective in the sense that they really change the underlying conformal field theory. Here we will have to make some assumptions since a complete analysis of these conformal field theories has yet to be completed.

First consider the metric gij. We have seen that this must be Ricci-fiat to obtain conformal invariance. We will assume that any generic deformation of this Ricci-flat metric to another inequivalent Ricci-flat metric will lead to an inequivalent conformal field theory. Since the dimension of the moduli space of Einstein metrics on a K3 surface given in (40) is 58, we see that the metric accounts for 58 parameters.

Next we have the 2-form, B. This appears in the action in the form

4E)

B.

(62)

Thus, since the image of the world-sheet in X under the map x is a closed 2-cycle, any exact part of B is irrelevant. All we see of B is its cohomology class. As b2 of a K3 surface is 22, this suggests we have 22 parameters from the B-field. Lastly we have the dilaton, 4'. This plays a very peculiar role in our conformal field theory. Since 4' is a constant over X, by assumption, we may pull it outside the integral leaving a contribution of 24'(g - 1) to the action, where g is the genus of E. In this section we really only care about the conformal field theory for a fixed E and so this quantity remains constant. To be more complete we should sum over

26

PAUL S. ASP[NWALL

the genera of E. Taking the limit of a' -i oo, the world-sheet image in the target space will degenerate to a Feynman diagram and then g will count the number of loops. Thus we have an effective target space coupling of (63) A = e4. Anyway, since we want to ignore this summation of E for the time being we will ignore the dilaton. See, for example, [49] for a further discussion of world-sheet properties of the dilaton from a string field theory point of view. All said then we have a moduli space of 58 + 22 = 80 real dimeusionLs. To proceed further we need to know some aspects about the holonomy of the moduli space. The local form of the moduli space was first presented in [50] but we follow more closely here the method of [51]. Let us return to the moduli space of Einstein metrics on a K3 surface. Part of the holonomy algebra of the symmetric space factor in (40) is S0(3) = su(2). This rotation in R3 comes from the choice of complex structures given a quaternionic structure as discussed in section 2.2. Thus, this part of the holonomy can be understood as arising from the symmetry produced by the 82 of complex structures. This su(2) symmetry must therefore be present in the non-linear a-model. In the case where we have a conformal field theory however, we may divide the analysis into separate left- and right-moving parts. Thus each sector must have an independent su(2) symmetry. Indeed, it is known from conformal field theory that an N = 4 superconformal field theory contains an affine Su(2) algebra and so an N = (4,4) superconformal field theory has an su(2) ® su(2) 5 SO (4) symmetry. This symmetry acts on the tangent directions to a point in the moduli space (i.e., the "marginal operators") and so will be a subgroup of the holonomy. Thus the 50(3) appearing in the holonomy algebra of the moduli space of Einstein metrics is promoted to 50(4) for our moduli space of conformal field theories. Now we are almost done. We need to find a space whose holonomy contains SO(4) as a factor and has dimension 80. One could suggest spaces such as A x B, where A is a Riemannian manifold of dimension 4 and B is a Riemannian manifold of dimension 76. Such a factorization is incompatible with what we know about the conformal field theory, however. Analyzing the marginal operators in terms of superfields shows that each one is acted upon non-trivially by at least part of the SO(4). Given the work of Berger and Simons therefore leaves us with only one possibility.'

THEOREM 5. Given the assumptions about the effectiveness of deformations on the underlying conformal field theory, any smooth neighbourhood of the moduli space of conformally invariant non-linear a-models with a K3 target space is isomorphic

to an open subset of

_

0(4,20) 0(4) x O(20)

(64)

We now want to know about the global form of the moduli space. Here we are forced to make assumptions about how reasonable our conformal field theories $Actually we should rule out the compact symmetric space possibility. This is done by our completeness assumption, as we know we may make a K3 surface arbitrarily large.

27

K3 SURFACES AND STRING DUALITY

can be. We will assume that from theorem 6 it follows that the moduli space of conformal field theories is given by .4, G, \ g,, where G, is some discrete group. That is, .°l, is the Teichmnller space. All we are doing here is assuming that our moduli space is "complete" in the sense that there are no pathological limit points in it possibly bounding some bizarre new region. While this assumption seems extremely reasonable I am not aware of any rigorous proof that this is the case.

3.3. The geometric symmetries. All that remains then is the determination of the modular group, G,. To begin with we should relate (64) to the Teichmuller spaces we are familiar with from section 2. This will allow us the incorporate the modular groups we have already encountered. This is a review of the work that first appeared in [52, 53]. The space .%, is the Grassmannian of space-like 4-planes in R4'20. We saw that the moduli space of Einstein metrics on a K3 surface is given by the Grassmannian of space-like 3-planes in ]3,19 and this must be a subspace of .l, since the Einstein metric appears in the action (60). This gives us a clear way to proceed. Let us introduce the even self-dual lattice r4,20 C 81,20. It would be nice if we could show that this played the same role as x3,19 played in the moduli space of Einstein metrics. That is, we would like to show G, a, 0(r4,20). We will see this is indeed true. First we want a natural way of choosing r3,19 C r4,20. To do this fix a primitive element w E r4,20 such that w.w = 0. Now consider the space, w1 C ]41.20, of all vectors x such that x.w = 0. Clearly w is itself contained in this space. Now project onto the codimension one subspace w1/w by modding out by the w direction. We now embed w1/w back into 84,20 such that w1

n r4,20

' r3,1,.

(65)

W

It is important to be aware of the fact that all statements about r3,19 are dictated by a choice of w. The embedding w1/w C R4'20 can be regarded as a choice of a second lattice vector, w', such that w' is orthogonal to w1/w, w`.w' = 0 and w'.w = 1. As we shall see, the choice of w" is not as significant as the choice of w. Now perform the same operation on the space-like 4-plane. We will denote this plane II C R4'20. First define E' = II n w1 and then project this 3-plane into the space w1/w and embed back into R1,2e to give E. This E may now be identified with that of section 2.4 to give the Einstein metric on the K3 surface of some fixed volume.

Fixing E we may look at how we may vary H to fill out the other deformations.

Let II be given by the span of E' and B', where B' is a vector, orthogonal to E', Note normalized by B'.w = 1. We may project B' into w1/w to give B E 183,19.

that R3"19 is the space H2 (X, R) and so B is a 2-form as desired. Lastly we require the volume of the K3 surface. Let us decompose B' as

B' =aw+w`+B,

(66)

We claim that a is the volume of the K3 surface. To see this we need to analyze the explicit form of the moduli space further.

28

PAUL S. ASPINWAL1.

We have effectively decomposed the Teichmiiller space of conformal field theca ries as 0(3,19) :, 0(4,20) xR (67) x -` O(4) x 0(20) 0(3) x 0(19)

where the three factors on the right are identified as the Teicluniiiler spaces for the metric on the K3 surface (given by E), the B-field, and the volume respectively. Each of the spaces in the equation (67) has a natural niet ric, given by the invariant metric for the group action in the case of the symmetric splices. This can be shown to the Zauullodchikov coincide with the natural metric from conformal field theory metric - given the holonomy arguments above. The isomorphism (67) will respect this metric if "warping" factors are introduced as explained in [54). It, is these warping factors which determine the identification of the volume of the K3 surface as a in (66). This will be explained further in [53]. The part of Go we can understand directly is the part which fixes no but acts on W -L/w. This will affect the metric on the K3 surface of fixed volume and the B-field. We know that the modular group coming from global diffec uorpllisnls of the K3 surface is 0+(r3,19), which should be viewed as 0-1'(H2(.V,Z)). The discrete symmetries for the B-field meanwhile can be written as 13 = B + c, where

e E H2(X,Z). To see this note that shifting B by a integer element will shift the action (60) by a multiple of 27ri and hence will not effect the path integral. Thanks to our normalization of B', a shift of B by an element of r3,t9 amounts to a rotation of II which is equivalent to an element of O(r4,20). Note that this can also be interpreted as a redefinition of w'. That is, the freedom of choice of w' is irrelevant once we take into account the B-field shifts. We may also consider taking the complex conjugate of the nor-linear er-model action. This has the effect of reversing the sign of B while providing the extra element required to elevate 0+(x3,19) to 0(r3,19). The result is that the subgroup of Go that we see directly front the non-linear or-model action is a subgroup of 0(]'4,20) consisting of rotations and translations of r3,19 C x4,20. This may be viewed as the space group of r3,19i or equivalently, the semi-direct product Go D O(r3,10) V x3,19.

This is as much as we can determine from

w.L /w.

3.4. Mirror symmetry. To proceed any further in our analysis of G. we need to know about elements which do not correspond to a manifest symmetry of the non-linear a-model action (60). This knowledge will be provided by mirror symmetry. Mirror symmetry is a much-studied phenomenon in Calabi-Yau threefolds (see, for example, [55] for a review). The basic idea in the subject of threefolds is that the notion of deformation of complex structure is exchanged with deformation of complexified Kiihler form. The character of mirror symmetry in K3 surfaces is somewhat different since, as we have seen, the notion of what constitutes a deformation of complex structure and what constitutes a deformation of the Kahler form can be somewhat ambiguous. Also, we have yet to mention the possibility of complexifying the Ki hler form, as that too is a somewhat ambiguous process.

K3 SURFACES AND STRING DUALITY

29

Indeed mirror symmetry itself is somewhat meaningless when viewed in terms of the intrinsic geometry of a K3 surface and does not begin to make much sense until viewed in terms of algebraic K3 surfaces. The results for algebraic K3 surfaces were first explored by Martinec [56], whose analysis actually predates the discovery (and naming) of mirror symmetry in the Calabi-Yau threefold context. See also [57, 58] for a discussion of some of the issues we cover below. Recall that the Picard lattice of section 2.5 was defined as the lattice of integral 2-cycles in H2 (X, Z) which were of type (1,1). The transcendental lattice, A, was defined as the orthogonal complement of the Picard lattice in H2(X,Z). The signature of A is (2,20 - p), where p is the Picard number of X. It is clear that r4,2o c I'3,19 a U, where U is the hyperbolic plane of (25). We will extend the Picard lattice to the "quantum Picard lattice", T, by defining

T = Pic(X) a U,

(69)

as the orthogonal complement of A within F4,20. Given a K3 surface with an Einstein metric specified by E and a given algebraic structure, we know that the complex structure 2-plane, fl, is given by En (A ®z R).9 The KInler form direction, J is then given by the orthogonal complement of Sl C E. Accordingly, J lies in Pic(X) ®z R. We want to extend this to the non-linear picture. Keep Sl defined as above, or equivalently,

S2=ii n(A®z R),

(70)

and introduce a new space-like 2-plane, U, as the orthogonal complement of Cl C 11. Clearly we have

u=IIn(T®z R).

(71)

Our notion of a mirror map will be to exchange

p : (A,fl) ++ (T,U)

(72)

Note that as U encodes the Kiihler form and the value of the B-field we have the notion of exchange of complex structure data with that of Kfhler form + B-field as befits a mirror map. The moduli space of non-linear v-models on an algebraic K3 surface will be the subspace of (64) which respects the division of 11 into Cl and U. This is given by o,aig °-`

0(2,20 - p)

0 ( 2 ) x 0 ( 20 - p)

x

0(2, p)

0 (2 ) x O (p)'

(73)

where the first factor is the moduli space of complex structures and the second factor is moduli space of the Kiihler form + B-field. Given an algebraic K3 surface X with quantum Picard lattice T(X) we have

a mirror K3 surface, Y, with quantum Picard lattice T(Y) such that T(Y) is the orthogonal complement of T(X) C r4,2o. Translating this back into classical ideas, Pic(Y) e U will be the orthogonal complement of Pic(X) C r3,19. If X is such that the orthogonal complement of Pic(X) C rs,19 has no U sublattice then X has no classical mirror. 9The notation A ®z R denotes the real vector space generated by the generators of A.

PAUL S. ASPINWALL

30

Such mirror pairs of K3 surfaces were first noticed some time ago by Arnold (see for example, [59]). The non-linear o-model moduli space gives a nice setting for this pairing to appear. Now we also have the mirror construction of Greene and Plesser [60] which takes an algebraic variety X as a hypersurface in a weighted projective space and produces a "mirror" Y as an quotient of X by some discrete group such that X and Y as target spaces produce completely identical conformal field theories. An interesting case is when X is the hypersurface

f =xo+x3+x. +z 2 =0 in P{21,Ia,a,1}

(74)

X is a K3 surface which happens to be an orbifold due to the

quotient singularities in the weighted projective space. Blowing this up provides a K3 surface with Picard lattice isomorphic to the even self-dual lattice r,,;,. Thus in this case A T % r2,10. What is interesting is that in this case, the Green-Plesser construction says that the group by which X should be divided to obtain the mirror is trivial. This mirror map does not act trivially on the marginal operators however but one may show actually exchanges the factors in (73). This is little delicate and we refer the reader to [53] for details. Anyway, what this shows is that the G reenePlesser mirror map, which is an honest symmetry in the sense that it produces an identical conformal field theory, really is given by the exchange, It, in (72). This implicitly provides us with another element of G0., namely an element of O(r4,20) which exchanges two orthogonal x2,10 sublattices. It is now possible to show that this new element, together with the subgroup given in (68), is enough to generate 0(14,20). Thus G, at least contains 0(r4,20). To prove the required result that G, L O(r4,20) we use the result of [25] which states that any further generator will destroy the Hausdorff nature of the moduli space contrary to our assumption. We thus have

THEOREM 6. Given theorem 5 and the assumption that the nuxluli space is Hausdorff, we have that the moduli space of conformally invariant non-linear amodels on a KS surface is 0(ra,20)\ 0(4, 20)/(0(4) x 0(20)) (75) Lastly we should discuss the meaning of r4,2°. The lattice r3,19 was associated

to the lattice H2(X, Z). Extend the notion of our inner product (19) to that of any p-cycle (or equivalently !-form) by saying that the product of two cycles is zero unless their dimensions add up to 4. Now one can see that the lattice of total cohomology, H"(X, Z), is isomorphic to ra,2o. This gives a tempting interpretation. We will see later that this really is the right one.

This tells us how to view w. A point in the moduli space A, determines a conformal field theory uniquely but we are required to make a choice of to before we can determine the geometry of the K3 surface in terms of metrics and B-fields. This amounts to deciding which direction in -4,20 will be H°(X,Z). This choice is arbitrary and different choices will lead to potentially very different looking K3 surfaces which give the same conformal field theory. This may be viewed as a kind of T-duality.

K3 SURFACES AND STRING DUALITY

31

3.5. Conformal field theory on a torus. Although our main interest in these lectures are K3 surfaces, it turns out that the idea of compactification of strings on a torus will be intimately related. We are thus required to be familiar with the situation when the a-model has a torus as a target space. The problem of the moduli space in this case was solved in [61, 62]. This subject is also covered by H. Ooguri's lectures in this volume and we refer the reader there for further information (see also [63]). Here we will quickly review the result in a language appropriate for the context in which we wish to use it. To allow for the heterotic string case, we are going to allow for the possibility that the left-moving sector of the a-model may live on a different target space to

that of the right-moving sector. We thus consider the left sector to have a torus of nL real dimensions as its target space and let the right-moving sector live on a nR-torus. Of course, for the simple picture of a string propagating on a torus, we require nL = nR. The moduli space of conformal field theories is then given by

O(rnL,nR)\ o(nL, nR)/(O(nL) x O(n)?))

(76)

The even.self-dual condition, required for world-sheet modular invariance [61], on

the lattice rn.,nR enforces nL - nR E 8Z. Note that whether we impose any supersymmetry requirements or not makes little difference. The only Ricci-flat metric on a torus is the flat one, which guarantees conformal invariance to all orders. The trivial holonomy then means that a torus may be regarded as Kihler, hyperkahler or whatever so long as the dimensions are right. Let us assume without loss of generality that nL < nR.10 Our aim here is to interpret the moduli space (76) in terms of the Grassmannian of space-like nLplanes in R"L_"R. Let us use U to denote a space-like nL-plane. Any vector in RnL,nR may be written as a sum of two vectors, PL + pR, where

pLEH pa E

II

(77)

and ITl is the orthogonal complement of R. Thus pL.pL > 0 and PR.PR < 0. The winding and momenta modes of the string on the torus are now given by points in the lattice rnL,nR according to Narain's construction [61]. The left and right conformal weights of such states are then given by zpL.pL and -apR.pR, thus allowing the mass to be determined in the usual way. It is easy to see how the moduli space (76) arises from this point of view. The position of 11 determines pL and pR for each mode. The modular group O(rfL,fR) merely rearranges the winding and momenta modes.

Let us try to make contact with the a-model description of the string. We choose a null (i.e., light-like) nL-plane, W, in rnL,fR ®z R °-° R11-111 which is "aligned" along the lattice rnL,ff. By this we mean that W is spanned by a subset of the generators of rfL,fR. We then define W` as the null nL-plane dual to W, 10 This means that the heterotic string will be chosen to be a superstring in the left-moving sector and a bosonic string in the right-moving sector. Unfortunately this differs from the usual convention. This is imposed on us, however, when we consider duality to a type I1A string and the natural orientation of a KS surface.

PAUL S. ASPINWALL

32

and the time-like space V g R"R'fL so that rfL,n,, ®a lg S' W EH W, (+) 4'.

(78)

This decomposition is also aligned with the lattice so that and rnL,nR fl v generate rnL,nn, We may write 11 as given by

n 1 ,11, t,,,,,,,,, MI",

11= {x + O(x) + A(x); V:r. E ll'},

(79)

where

0:W--;W' (80)

A:W -aV Viewing 0 as

0:WxW-*1I,

(81)

we may divide,0 into symmetric, G, and anti-symmetric parts, B, such that 4, _ B + G. Physically W represents, under the metric G, the target space in which the string lives. To be precise, the target space nL-torus is W rfL,nR n

(82)

W

W` is then the dual space in which the momenta live. Clearly B is the B-field. Lastly V represents the gauge group generated by the extra right-moving directions. To be precise, the group is given by

iR n i . (83) rnL, We will discuss in section 4.3 how this group can be enhanced to a nonabelian group U(1)nR-nL L

for particular H's. The quantity A then represents u(1)-valued 1-forums. We will follow convention and refer to these degrees of freedom within A as "Wilson Lines".

One may write G, B and A in terms of matrices to make contact with the expressions in [62, 63]. This is a somewhat cumbersome process but the reader should check it if they are unsure of this construction. Here we illustrate the procedure in the simplest case, namely that of a circle as target space. Here we have nL = nR = 1 and rx,1 °-` U as in (25). An element of 0(1, 1) preserving the form U may be written in one of the forms

t 0

for

-t l ( -l

0

t-1 0

l (

_t

l Ot

(84)

where t is real and positive. We divide this space by O(1) x O(1) from the right and by O(rl,I) from the left. Both these groups are isomorphic to Z2 X Z2 and are given by (84) with t = 1. The result is that the moduli space is given by the real positive line mod Z2 represented by

t C 0

where t is real and positive.

t0'

/

(t01

(85)

K3 SURFACES AND STRING DUALITY

33

0

FIGURE 3. The moduli space picture for a string on a circle. Let us chose a basis, {e, e*}, for r1,1 so that e.e = e*.e* = 0 and e.e* = 1. Now let the space-like direction, 11, be given, as in figure 3, by II = {xe + xe* tan B; x E ]g},

(86)

where t = tan 0. (Note that 11 and 111 may not look particularly orthogonal in figure 3 but don't forget we are using the hyperbolic metric given by U!) Denoting a state in I'i,l by ne + me*, for m, n E Z, one can show PL.PL = a n2 tan O + nm +

In 2 tan In2 0

(87)

m2

PR-PR =-an2tan0+nm- 2tanB' Since G is given by tan 8, the radius of the circle is proportional to tan 0. identification in (85) then gives the familiar "R t-> 1/R" T-duality relation.

The

4. Type II String Theory Now we have the required knowledge from classical geometry and "old" string theory on the world-sheet to tackle full string theory on a K3 surface. We begin with the most supersymmetry to make life simple. This is the type NIA or 11B superstring which has 2 supersymmetries when viewed as a ten-dimensional field theory. The heterotic string on a K3 surface constitutes a much more difficult problem and we won't be ready to tackle it until section 6.

PAUL S. ASPINWALL

34

d

N

11 10

1

Rep Ill -

2

ia'6

9 8

2 2

IlIr6

7

2

6 5

4

IEI`I

110

4

4 8

3

16

C

TABLE 2. Maximum numbers of supersyrumetries.

4.1. Target space supergravity and compactification. We begin by switching our attention from the quantum field theory that lives in the world-sheet to the effective quantum field theory that lives in the target space in the limit that a'/R2 -> 0. Of particular interest will be theories with N supersyrnrnetries in d dimensions (one of which is time).

A spinor is an irreducible representation of the algebra S0(l,d - 1) and has dimension 2[41-1, where the bracket means round down to the nearest integer. This is not the only information about the representation we require however. A spinor may be real (R), complex ('p, or quaternionic'r (1111) depending on d. The rule is

if d = 1, 2,3 (mod 8) O if d = 0 (mod 4) I ll if d = 5, 6, 7 (mod 8). A complex representation has twice as many degrees of freedom as a real representation. A quaternionic representation has the same number of degrees of freedom as a complex representation (as, even though one may define the representation naturally over the quaternions, one must subject it to constraints [291). In terms of the language often used in the physics literature: in even numbers of dimensions we use Weyl spinors; real spinors are "Majorana" and quaternionic spirrors are "sytnplectic Majorana" r2 We may now list the maximal number of supersynunetries in each dimension subject to the constraint that no particle with spin > 2 appears. It was determined

in [64] that N = 1 in d = 11 was maximal in this regard. To get the other values we simply maintain the number of degrees of freedom of the spinors. For example, in reducing to ten dimensions we go from 32 real degrees of freedom per spinor to 16 real degrees of freedom per spinor. We thus need two spinors in ten dimensions. The result is shown in table 2. When d E 4Z + 2, the left and right spinors in the field theory are independent and the supersymmetries may be separated. As we did in section 3.1, we will denote a theory with NL left supersymmetries and NR right supersymmetries as having 11Sometimes the terminology "pseudo-real" is used. 32Note that two symplectic Majorana spinors make up one quaternionic spinor.

K3 SURFACES AND STRING DUALITY

35

N = (NL, NR) supersymmetry. For purposes of counting total supersymmetries,

N=NL+NRNow we want to see what happens when we compactify such a theory down to a lower number of dimensions. That is, we replace the space 1l 1,do -1 by Hl,d, -1 X X,

for some compact manifold X of dimension do - d1. To see what happens to the supersymmetries we need to consider how a spinor of 5o(l,do - 1) decomposes

under the maximal subalgebra s0(l,do - 1) D so(1,d1 - 1) a so(do - d1). The holonomy of X will act upon representations of so(do - d1). Any representation

of so(1,d1 - 1) eso(do - d1) which is invariant under this action will lead to representations of 50(1, d1- 1) in our new compactified target space. This tells us how to count supersymmetries. As the first example, consider toroidal compactification. In this case the torus is flat and so the holonomy is trivial. Thus every representation survives. Compactifying N = 1 supergravity in eleven dimensions down to d dimensions results

in reproducing the maximal supersymmetries in table 2. Note that for d = 10 the supersymmetry is N = (1,1) and for d = 6 the supersymmetry is N = (2, 2). One simple way of deducing this immediately is from the general result [65] that compactification of eleven dimensional supergravity on any manifold results in a non-chiral theory.13

Naturally our next example will be compactification on a K3 surface. In this case the holonomy is SU(2). Let us consider the case of an N = 1 theory in ten dimensions compactified down to six dimensions on a smooth K3 surface. The required decomposition is

50(1, 9) D 50(1, 5) a 30(4) el 50(1, 5) a 5u(2) a su(2).

(88)

The spinor decomposes accordingly as 16 -1 (4, 2,1) a (4,1, 2). We may take the last su(2) in (88) as the holonomy. Thus the (4,2, 1) part is preserved. It may look like we have 2 spinors in 6 dimensions at this point but remember that our spinor in ten dimensions was real and spinors in six dimensions are quaternionic. Thus the degrees of freedom give only a single spinor in six dimensions. That is, N = 1 supergravity in ten dimensions compactified on a smooth K3 surface will give an N = 1 theory in six dimensions. The general rule is that compactification on a smooth K3 surface will preserve half as many supersymmetries as compactifying on a torus. In the case that the number the supersymmetries or the number of dimensions is large, the form of the moduli space of possible supergravity theories becomes quite constrained. Holonomy is again the agent responsible for this. Let us write the notion of extended supersymmetry very roughly in the form

,Q'} =611P, (89) where Q are the supersymmetry generators and P is translation (we may ignore any central charge for purposes of our argument). Now such a relationship must clearly be preserved as we go around a loop in the moduli space. However, the supersymmetries may transform among themselves as we do this. This gives us a restriction on the holonomy of the bundle of supersymmetries over the moduli {

"Fortunately recent progress in M-theory appears to tell us that compactification on spaces which are not manifolds can circumvent this statement (see, for example, (66]).

36

PAUL S. ASPINWALL

space. Comparing (89) to the analysis of invariant forms in section 2.2 tells us immediately what this restriction is. In the case that the spinor is of type Qt, C, or R, the holonomy algebra will be (or contain) SO(N), u(N), or sp(IV) respectively. Note that when we have chiral spinors in 4Z + 2 dimensions we may factor the holonomy into separate left and right parts since these sectors will not mix. Now the tangent directions in the moduli space are given by massless scalar fields which lie in supermultiplets. These multiplets have a definite transformation property under the holonomy group above. For a thorough account of this process we refer to [67]. This relates the holonomy of the bundle of supersyrmetries to the holonomy of the tangent bundle of the moduli space. This knowledge can tell us a great deal about the form of the moduli space. As an example consider N = 4 supergravity in five dimensions from table 2. We see immediately that the holonomy algebra of the moduli space is sp(4). All analysis of the representation theory of the supergravity nlultiplet shows that the

massless scalars transforms in a 42 of 5p(4). The only possibility from Berger and Simons result, and Cartan's classification of noncompact symmetric spaces, is that the moduli space is locally Es(e) / Sp(4)., where the tilde subscript denotes a quotient by the central Z2. The moduli spaces for all the entries in table 2 are given in 1681-

4.2. The IIA string. The type IIA superstring in ten dimensions yields, in the low-energy limit, a theory of ten-dimensional supergravity with N = (1, 1) supersymmetry. If we compactify this theory on a K3 surface then each of these supersymmetries gives a supersymmetry in six dimensions and so the result is all N = (1,1) theory in six dimensions. The local holonomy algebra from above must therefore be sp(1) cy sp(l) su(2) ® 5u(2) 50(4). There are two types of supermultiplet in six dimensions which contain moduli fields: 1. The supergravity multiplet contains the dilaton which is a real scalar. 2. Matter multiplets each contain 4 real massless scalars which transform as a

4 of so(4). Thus the moduli space must factorize (at least locally) into a product of a real line for the dilaton times the space parametrized by the moduli coming from the matter multiplets. Thus, given the assumption concerning completeness, the Teichmuller space is of the form 0 (4, m)

x Il$,

(90)

0(4) x 0(m) where m is the number of matter multiplets. As well as the metric, B-field and dilaton, the type II string also contains "Ramond-Ramond" states (see, for example, [39)). For the type IIA string these may be regarded as a 1-form and a 3-form.14 A p-form field can produce massless fields upon compactification by integrating it over nontrivial p-cycles in the compact manifold. As a K3 surface has no odd cycles, no moduli come from the R-R sector. Thus, comparing (90) to (75) we see that m = 20. 14In a sense all odd forms may be Included [69].

K3 SURFACES AND STRING DUALITY

37

The moduli space of conformal field theories may be considered as living at the

boundary of the moduli space of string theories in the limit that 4) -+ -oo, i.e., weak string-coupling. Thus 0(I.4,30) acts on this boundary. Given the factorization of the holonomy between the dilaton and the matter fields (as they transform in different representations of So(4)) the action of O(r4,2o) also acts away from the boundary in a trivial way on the dilaton. It is believed that moduli space is given simply by

a

(91)

That is, there are no identifications which incorporate the dilaton. Certainly any duality which mixed the factors would not respect the holonomy. Thus the only possibility remaining would be an action of the form 4) - -4 which would be a strong-weak coupling duality (acting trivially on all other moduli). Instead of such an S-duality, a far more curious type of duality was suggested in (70, 71] as we now discuss.

The form of the moduli space of toroidal conformal field theories (76) in section 3.5 bares an uncanny resemblance to that of the moduli space of K3 conformal field theories, A,. Indeed they are identical if nL = 4 and nR = 20. This is precisely the moduli space of conformal field theories associated to a heterotic string compactified

on a 4-torus. Recall that the heterotic string consists of a superstring in the leftmoving sector and a bosonic string in the right-moving sector. As such there are 16 extra dimensions in the right-moving sector which are compactified on a 16-torus and contribute towards the gauge group. In ten dimensions the heterotic string is an N = (1, 0) theory and thus yields an N = (1,1) theory in six dimensions when compactified on a 4-torus. The suggestion, which at first appears outrageous, is that the type IIA string compactified on a K3 surface is the same thing, physically, as the heterotic string compactified on a 4-torus. Although the moduli space of conformal field theories is identical for these models, the world-sheet formulation is so different that any

notion that the two conformal field theories could be shown to be equivalent is doomed from the start. The claim however is not that the conformal field theories are equivalent, but that the full string theories are equivalent. The heterotic string has a dilaton, just like the type HA string, and so the moduli space of heterotic string theories is also . 0, x f The only way we can map these two moduli spaces into each other without identifying the conformal field theories is thus to map 4) from one theory to -4i from the other. This then, is the alternative to S-duality for the type IIA string. PROPOSITION 1. The type IIA string compactifced on a K3 surface is equivalent to the heterotic string compactified on a 4-tomes. The moduli spaces are mapped to each other in the obvious way except that the strongly-coupled type IIA string maps to the weakly-coupled heterotic string and vica versa.

It would be nice at this point if we could prove proposition 1. Unfortunately it appears that string theory is simply not sufficiently defined to allow this. We should check first that this proposition is consistent with what we do know about string theory. The fact that the moduli spaces of the type HA string on a K3 surface and the heterotic string on a 4-torus are identical is a good start. Next we may check that

38

PAUL S. ASPINWALL

the effective six-dimensional field theory given by each is the same. This analysis

was done in [71]. The result is affirmative but there is some new information. To achieve complete agreement the flat six-dimensional spaces given by the two string theories are not identical but instead the metrics are scaled with respect to each other. That is, let go denote the six-dimensional space-time metric and P the dilaton, then '§Het = -'bf1A (92)

96,1iet = e34i'.' 96,IIA

Other checks may be performed too. Since the strings are related by a strongweak coupling relations, the fundamental particles in one theory should map to solitons of the other theory. This has been analyzed to a great extent but we refer to J. Harvey's lectures for an account of this. To date nothing has been discovered in known string theory to disprove proposition 1. Assuming there are no such inconsistencies one may wish to boldly assert that proposition 1 is true by definition. That is, whatever string theory may turn out to be, we will demand that it satisfies proposition 1. This is the point of view we will take from now on.

4.3. Enhanced gauge symmetries. One of the interesting questions we are now fully equipped to address is that of the gauge symmetry group of the sixdimensional theory resulting from either a compactification of a type IIA string on a K3 surface, or a heterotic string on a 4-torus. Firstly we may consider this question from the point of view of conformal field theory. The type IIA string produces gauge fields from the R-R sector. The 1-form gives a U(1). The 3-form may be compactified down to a 1-form over 2-cycles in b2(S) = 22 ways. Lastly, writing the 3-form as C3, we have a dual field C3 given by dC3 = *dC3,

(93)

where C$ is a one form and also gives a U(1). Thus, all told, we have a gauge group of U(1)24. Note that it is not possible to obtain a nonabelian gauge theory, as far as the conformal field theory is concerned, because the R-R fields are so reluctant to couple to any other fields (see, for example, [72]). Now consider the heterotic string picture. As explained in section 3.5, generically we obtain a U(1)'8 contribution to the gauge group from the extra 16 rightmoving degrees of freedom for the heterotic string. We also obtain 4 "Kaluza-Klein" U(1) factors from the metric from the 4 isometries of the torus. Lastly, the B-field

is a 2-form and so contributes bl(T4) = 4 more U(1)'s. All told we have U(1)24 again as befits proposition 1. Things become more interesting however when we realize that the heterotic string can exhibit larger gauge groups at particular points in the moduli space. What happens is that some winding/momentum modes of r4,20 may happen to give physical massless vectors for special values of the moduli. Such states will be charged with respect to the generic U(1)24 and so a nonabelian group results. The heterotic string is subject to a GSO projection to yield a supersymmetric field theory and this effectively prevents any new left-moving physical states from becoming massless. This may be thought of as a similar statement to the assertion

K3 SURFACES AND STRING DUALITY

39

that the type HA string could never yield extra massless vectors. The right-moving sector of the heterotic string is not subject to such constraints however, and we may use our knowledge from section 3.5 to determine exactly when this gauge group enhancement occurs. The vertex operator for one of our new massless states will be as follows. For

the left-moving part we want simply 8X" to give a vector index. For the rightmoving part we require another operator with conformal weight one. This results in a requirement that we desire a state with pL = 0 and pR.pR = -2. Thus we require an element a E F4,zo which is orthogonal to 11 and is such that a.a = -2. The charge of such a state, a, with respect to the U(1)24 gauge group is simply given by the coordinates of a. This follows from the conformal field theory of free

bosons and we refer to (41) for details. Comparing this to the standard way of building Lie algebras, we see that a looks like a root of a Lie algebra, whose Cartan subalgebra is U(1)e24 (except that the Killing form is negative definite rather than positive definite).

In the simplest case, there will be only a single pair ±a which satisfy the required condition. This then will add two generators to the gauge group charged with respect to one of the 24 U(1)'s. Thus the gauge group will be enhanced to SU(2) X U(1)23. Note that this enhancement to an SU(2) gauge group can also be seen in the simple case of a string on a circle as was pictured in figured 3. If 0 = 45° then the lattice element marked a in the figure (together with -a) will generate SU(2).

The general rule then should be that the set of all vectors given by

.d_ {a E r4,2o fl n'; a.a = -2},

(94)

will form the roots of the nonabelian part of the gauge group. Note that the rank of the gauge group always remains 24. Note also that the roots all have the same length, i.e., the gauge group is always simply-laced and falls into the A-D-E classification. This allows us to build large gauge groups. One thing one might do, for example,

is to split r4,20 °-` r4,4 ® rg ® r3, where rs is the Cartan matrix of Es (with a negative-definite signature). We may then consider the case where n C r4,4 ®z R. This would mean that any element in rs ®rs is orthogonal to 11 and thus the gauge group is at least Es x Es. Looking at (79) we see that this is equivalent to putting

A = 0. Thus we reproduce the simple result that the Es x Es heterotic string compactified on a torus has gauge group containing Es x Es if no Wilson lines are switched on. Proposition 1 now tells us something interesting. Despite the fact that the conformal field theory approach to the type IIA string insisted that it could never have any gauge group other than U(1)24, the dual picture in the heterotic string dictates otherwise. There must be some points in the moduli space of a type IL4 string compactified on a K3 surface where the conformal field theory misses part of the story and we really do get an enhanced gauge group. Since we know exactly where the enhanced gauge groups appear in the moduli space of the heterotic string and we know exactly how to map this to the moduli space of K3 surfaces, we should be able to see exactly when the conformal field theory goes awry.

40

PAUL S. ASPINWALL

We will determine just which K3 surfaces give rise to this behaviour for the type IL4 string. To do this we are required to choose w as in section 3.3 so that, we can find a geometric description. Let us first assume that we may choose u, so that w1 (95) aE , `daE.sa7 w

Let w* be the same vector as was introduced in equation (66). Any vector in II can Thus, the stateuu:nt be written as a sum x + bw* + cu,, where b, c E !!I and a: E that a is orthogonal to II implies that a is orthogonal to E. Now we use the results of section 2.6 which tell us that this implies that the. K3 surface is an orbifold. To be precise, the set sat corresponds to the root diagram of the A-D-E singularity given in table 1. What we have just shown is a remarkable fact (acid was first shown by Witten in [71]).

PROPOSITION 2. If we have a K3 surface with an orbifold singularity then a type IIA string compactified on this surface can exhibit a nonabelian gauge group such that the A-D-E classification of orbifold singularities coincides perfectly with the A-D-E classification of simply-laced Lie groups.

This proposition rests on proposition 1 and the assumptions that went into building the moduli spaces. As an example, we see that an SU(n) gauge group corresponds to a singularity locally of the form O" /Z,,. Note that the orbifold singularity is not a sufficient condition for an enhanced gauge symmetry. For II to be orthogonal to a we also require that. B', and hence B, is also perpendicular to a. One way of stating this is to say that the component of B along the direction dual to a is zero. Note that the volume of the Iii surface does not matter in this context.

It is probably worth emphasizing here that this statement about the B-field can be important [73]. Orbifolds are well-known as "good" target spaces for string theory in that they lead to finite conformal field theories. It might first appear then that we are saying that the conformal field theory picture is breaking down at a point in the moduli space where an enhanced gauge group appears but when the conformal field theory appears to be perfectly reasonable. This is not actually the case. The enhanced gauge symmetry appears when B = 0 along the relevant direction. Conformal field theory orbifolds however tend to give the value B = 2. Thus, the point in the moduli space corresponding to the happy conformal field theory orbifold and the point where the enhanced gauge group appears are not the same. There is good reason to believe the conformal field theory at the point where the enhanced gauge group appears is not well-behaved [74]. What happens if we relax our condition on w given in (95)? Now the situation is not so clear. E need not be perpendicular to any a and so the K3 surface may be smooth. If this is the case, the volume of the K3 surface cannot be arbitrarily large. This is because in the large radius limit, II is roughly the span of E and w,

but a is not perpendicular to w. In fact, the volume of the K3 surface must be of order one in units of (a')2. Thus we see, assuming the B-field is tuned to the right value, that an enhanced gauge group arises when the K3 surface has orbifold singularities and is any size, or if the K3 surface is very small and is given just the right (possibly smooth) shape.

K3 SURFACES AND STRING DUALITY

41

Note that to get a very large gauge group we probably require the K3 surface be singular and it to be very small. This is necessary, for example, if we want a gauge group Es x Es x SU(2)4 X U(1)4. Note that this latter group is "maximal" in the sense that the nonabelian part is of rank 20, which is the maximal rank sublattice that can be orthogonal to 11. We have proven the appearance of nonabelian gauge groups by assuming proposition 1. Instead one might like to attempt some direct justification. This is probably best seen by using Strominger's notion of "wrapping p-branes" [75]. The general idea is that the R-R solitons are associated with cycles in the target space and the mass of these states is given by the area (or volume) of these cycles. Thus, as the K3 surface acquires a quotient singularity an S2 shrinks down and thus a soliton becomes massless. We will not pursue the details of this construction as solitons he somewhat outside our intended focus of these lectures. We also refer the reader to [76, 77] for further discussion.

4.4. The IIB string. The type JIB superstring in ten dimensions yields, in the low-energy limit, a theory of ten-dimensional supergravity with N = (2, 0) supersymmetry. If we compactify this theory on a K3 surface then each of these supersymmetries gives a supersymmetry in six dimensions and so the result is an N = (2, 0) theory in six dimensions. The local holonomy algebra from above must therefore be 5P (2) el so (5). There is only one type of supermultiplet in six dimensions which contains massless scalars

and that is the matter supermultiplet. Each such multiplet contains five scalars transforming as a 5 of so(5). Thus, given the assumption concerning completeness, the Teichmiiller space is of the form [78] (96) 0(5,M) 0(5) x O(m)' where m is the number of matter multiplets. In addition to the metric, B-field and dilaton, moduli may also arise from the R-R fields. The type IIB string in ten dimensions has a 0-form, a 2-form and a

self-dual 4-form. The 0-form gives bo (S) = 1 modulus. The 2-form gives b2 (S) = 22 moduli. The 4-form gives b4 (S) = 1 modulus. One might also try to take the dual of the 4-form to give another modulus. This would be over-counting the degrees of freedom however, as the 4-form is self-dual. Thus, the number of moduli are given by Metric 58 B-field 22 Dilaton 1 1 0-form 2-form 22 4-form 1 Total 105 To get the dimension of the Teichmiilller space correct we require m = 21.

To find the discrete group acting on the Teichmiiller space we go through a procedure remarkably similar to that of section 3.4 where we found the moduli space of conformal field theories. Firstly we may consider the Teichmiiller space as the Grassmannian of space-like 5-planes in K5,21. Denote the 5-plane by e. Then

PAUL S. ASPINWALL

42

we choose a primitive null element, u, of r5,21. Now we have u1/u = 104,20. We may define II' = 8 fl u1. Define the vector R' by demanding that II' and R' span

8, R' be orthogonal to 11' and u.R' = 1. Then project II' and R into u1/u to obtain the space-like 4-plane, R, and the vector, R, respectively. Clearly we now interpret If in terms of the underlying conformal field theory on the K3 surface. R represents the degrees of freedom coming from the R-R sector. The six-dimensional dilaton may be deduced from R' - R just as the volume of the K3 surface was deduced from B' - B in section 3.4. It is important to note that this construction provides considerable evidence for our identification of 11(4,2° with H*(S,Z) in section 3.4. This is because R, which represents the R-R degrees of freedom, is a vector in 1114,20 - the same space in which 11 lives. We also saw how these moduli arise from H° ® H2 93 H4 -- H*. (which, as the reader will have Generating the discrete modular group, guessed, will turn out to be O(ra,21)) is slightly different to the way we built O(r4,20) in section 3.4, but we show here that we can reduce it to the same problem. First note that we have O(r4,20) C G,, . That is, any symmetry of the conformal field theory will be a symmetry of the string theory (just as any symmetry of the classical geometry is a symmetry of the conformal field theory). The next ingredient we use will be that of S-duality of the type IIB string in ten dimensions [70, 71]. This asserts that there is an SL(2, Z) symmetry acting on the ten-dimensional dilaton and the axion (i.e., the R-R 0-form). This group is generated by a strong-weak coupling interchange of the form'lo,un - -' io,un, and a translation of the axion by one. While one might assert this S-duality statement as a distinct conjecture, it is certainly intimately related to other duality statements. One simple way of "deriving" it is to'see that it is almost an inevitable consequence of M-theory [79, 80] (see also J. Schwarz's lectures). We will also see in section 5.1 that it follows from proposition 1 and mirror symmetry. To embed SL(2, Z) into r5,21 we note that SL(2, Z) is the group of automorphisms of Z2, which means that it is the group of isometries of a null lattice of

rank two. We may split r5,21 '-` r3,i9 e r2,2 and then let the SL(2, Z) act on a null 2-plane in the r2,2 part. If we identify r3,19 with H2 (S, Z) then we see that the group of classical symmetries of the K3 surface (i.e., 0(x3,19)) commutes with S-duality. This is exactly what we desire from the effective target space theory. Now the shift of the axion is a shift in the H°(S,Z) direction. Since O(r4,20) acts transitively on primitive null vectors, we immediately see that any shift by

an element of x4,20 is a symmetry of the string theory. This is an analogue of the "integral B-field shift". That is, shifting any R-R modulus by an element of H*(S,Z) is a symmetry of the string theory. The other generator of SL(2, Z) may be taken as one which exchanges the two null vectors generating the null 2-plane in r2,2. That is, we swap the axion direction, H° (S, Z), with u, a null vector outside R4,20. This is the exact analogue of mirror symmetry (which exchanged an element of H2(S) with H°(S)). We have reduced the problem to one completely analogous to finding the modular group for conformal field theories on a K3 surface. Thus we deduce that we can generate all of O(x5,2i) (as asserted first in [71]). Assuming the moduli space is Hausdorff we have

K3 SURFACES AND STRING DUALITY

43

H*(S)

FIGURE 4. The S-dualities of the type IIB string. PROPOSITION 3. The moduli space of type IIB string theories compactified on a KS surface is

_cm = 0(rb,21)\0(5,21)/(0(5) x 0(21)).

(97)

This proposition depends on the S-duality conjecture for the type IIB string in ten dimensions, theorem 6, and the completeness and Hausdorff constraints. We should mention that there is a strong-weak coupling duality in the resulting six-dimensional theory. Let u* be a null vector in r5,21 dual to u such that u and u* span the r1,1 sublattice orthogonal to H*(S,Z) r4,20. One of the elements in O(r5,21) exchanges u and u* and has the effect of reversing the sign of the sixdimensional dilaton. Note that this is not at all the same as the element of O(r5,21) which changed the sign of the ten-dimensional dilaton as the latter exchanged u with H°(S, Z) as shown in figure 4. In general an S-duality in a given number of dimensions will not give rise to an S-duality in a lower number of dimensions upon compactification. In the type JIB string however, we see S-duality in both six and ten dimensions.

The behaviour of the type IIA and type IIB string can be contrasted. The strongly-coupled type IIA string on a K3 surface is dual to a different string theory (the heterotic string) which is weakly-coupled. The type JIB strongly-coupled string on a K3 surface is dual to a weakly-coupled version of itself. Finally in this section let us mention some strange properties of the type IIB string on a K3 surface. We know that when the type HA string is compactified on an orbifold with B=O then an enhanced gauge symmetry appears. This indicated some divergence within the underlying conformal field theory. It should be true therefore that a type IIB string compactified on the same space must have some interesting nonperturbative physics since it is associated with the same divergent conformal

44

PAUL S. ASPINWALL

field theory. It was explained in [74] that these new theories are associated with massless string-like solitons which appear at these points in the moduli space.

5. Four-Dimensional Theories Now let us explore what happens when we compactify string theories down to four dimensions. This process need not involve a K3 surface in general but we will find that in all the easy cases K3 surfaces will be present in abundance!

5.1. N = 4 theories. Supersymmetries are not chiral in four dimensions and so, in contrast to the six-dimensional case, there is only one kind of N = 4 theory.

The local holonomy algebra must be u(4) =2 u(1) ® Su(4) rs u(1) s 50(6). There are two types of N = 4 supermultiplet in four dimensions which contain moduli fields:

1. The supergravity multiplet contains the dilaton-axion field which is a complex object under the u(1) holonomy. 2. Matter multiplets each contain 6 real massless scalars which transform as a 6 of 50(6). Thus the moduli space must factorize into a product of a complex plane, for the dilaton-axion, times the space parametrized by the moduli coming from the matter multiplets. Thus, given the assumption concerning completeness, the Teichmiiller space may be written in the form SL(2) 0(6,m) x 0(6) x O(m) U(1) '

(98)

where m is the number of matter multiplets. A very simple way to arrive at this theory is to compactify a heterotic string on a 6-torus. The first factor of (98) is then clearly the moduli space of the conformal field theories on the torus with 6 left-moving dimensions and 22 right-moving dimensions - that is, m = 22. The SL(2)/ U(1) term then comes from the dilaton-axion system: the dilaton being the string dilaton as usual and the axion from dualizing the B-field to obtain a scalar. The type IIA string may be compactified on K3 x T2 to obtain an N=4 theory too. The conformal field theory on a K3 has 80 real deformations and for T2 it has 4 deformations. The 1-form R-R field gives br (K3 x T2) = 2 moduli and the 1-form R-R field gives b3(K3 x T2) = 44 moduli. The 3-form may also be compactified down to a 2-form in bi (K3 x T2) = 2 ways and then dualized to give 2 more scalars. Adding the dilaton-axion we have 80 + 4 + 2 + 44 + 2 + 2 = 134. This implies m = 22 again. Actually proposition 1 tells us that this must be the same theory as the heterotic string on a 6-torus. Let us examine the moduli space of conformal field theories on a 2-torus. As far as the Teichmiiller space is concerned we have 0(2,2) 0(2) x 0(2)

-

SL(2) U(1)

X

SL(2) U(1)

(99)

up to Z2 identifications. One of the SL(2)/U(1) factors may be regarded as the complex structure of the torus and the other SL(2)/U(l) factor represents the Kiihler form and B-field on the torus. We refer to [63] for a review of this. What

KS SURFACES AND STRING DUALITY

45

does the second factor in (98) represent? There are many ways of approaching this problem. Here we use a trick following [81] that will come in use later on. Begin with a six-dimensional field theory given by the heterotic string compactified on a 4-torus. The effective field theory in six dimensions will be roughly of the form

S = f d6X gee 24'a.xet (R + ...)

(100)

Now compactify over a 2-torus of area AHd, (as measured by the heterotic string).

S

f d4X

rd4X gge "----t(R+...)>

(101)

= fd4X 94Anw(R+...), where AIM is the area of the 2-torus when we consider building the same theory by compactifying the type IIA string on K3 x T2. We have made use of (92) to derive this. Looking at the second line of (101) we see that the area of the T2 is actually playing the role of the coupling constant. Thus, in going from the heterotic string description of the situation to the type IIA description of the same situation, the dilaton of the heterotic string has been replaced by the area of the T2 of the type

HA string. Thus, the SL(2)/ U(1) factor in (98) must represent the Knhler form and B-field of the 2-torus in the type IIA picture. We know from conformal field theory that SL(2, Z) acts on the SL(2)/U(l) part of the Teichmuller space giving the Kiihler form and B-field of the 2-torus. Combining this knowledge with what we found from the heterotic string, we see that 0(r6,22) x SL(2, Z) acts as the modular group for our N = 4 theory. PROPOSITION 4. The type IIA string compactified on K3 x T2 is equivalent to the heterotic string compactified on a 6-torus and they form the moduli space

.AN=4 = (O(r6,22)\0(6,22)/(0(6) x 0(22))) x (SL(2, Z)\ SL(2)/ U(1)). (102)

This rests on the same assumptions as proposition 3.15 The SL(2,Z) factor of the modular group acts as an S-duality in the effective four-dimensional theory. Thus we have derived, from proposition 1 the existence of Montonen-Olive S-duality [83, 84, 85] for N = 4 theories in four dimensions. (Again we are going to neglect to discuss solitons - see [82] for such analysis.) Lastly we may consider the type IIB string compactified on K3 x T2. Again there are a multitude of ways of arriving at the desired result. One of the easiest ways is to take following proposition from [86, 87]: 11Depending on one's tastes, in the case of the moduli space of the beterotic string on a 6-torus one may wish to consider this statement as more fundamental than proposition 1 as it may be analyzed directly in terms of solitons [82].

46

PAUL S. ASPINWALL

PROPOSITION 5. The type RA superstring and type JIB superstring compactifled down to nine dimensions on a circle are equivalent except that the radii of the circles are inversely related.

One also needs to shift the dilaton of one theory relative to the other to achieve the same target space effective field theory for the two string theories. Thus, since the T2 of K3 x T2 contains a circle, the type IIB theory compact-

ified on K3 x T2 can also be bundled into proposition 4. Note that an R ti 1/R transformation on one of the circles in the T2 is a mirror map in the sense that the notions of deformation of complex structure and complexified Kibler form are interchanged. Thus, the SL(2, Z)\ SL(2)/ U(1) factor in the moduli space in (102) represents the complex structure moduli space of the 2-torus in the case of the type JIB string. Thus the SL(2, Z)\ SL(2)/ U(1) factor in the moduli space in (102) can play 3 roles [88, 89]: 1. The dilaton-axion variable in the case of the heterotic string. 2. The area and B-field of the 2-torus in the case of the type IIA string. 3. The complex structure of the 2-torus in the case of the type IIB string.

Comparing the heterotic string to the type IIA string in this setup may be regarded as fairly profound. The dilaton of the heterotic string, i.e., the coupling of the space-time field theory, is mapped to an area in the type IIA theory, i.e., the coupling of the world-sheet field theory. Thus, in a sense we are mapping the space-time field theory associated to one string theory to the world-sheet field theory associated to another. One might take this as evidence that neither the target space point of view nor the world-sheet point of view of string theory may be regarded as more fundamental than the other since they may be exchanged. The SL(2, Z) S-duality of the type IIB string in ten dimensions is now sitting in the group 0(r6,22) In fact, the above analysis can be used to "prove" the existence of this S-duality group. This can be viewed as an analogue of the deduction of this same S-duality group from M-theory as was done in [79, 80].

Note that we can play the same game as in section 4.3 to find the enhanced gauge groups. In this case we have a space-like 6-plane in r6,22 ®z J8 and we look for roots perpendicular to this plane. The gauge group is always of rank 28.

5.2. More N = 4 theories. Does the moduli space (102) represent all possible N = 4 theories in four dimensions? It seems unlikely as one expects to be able to build theories with a gauge group of rank < 28. Consider compactification of the type II string. All we demand to obtain the desired theory in four dimensions is that the manifold on which we compactify have SU(2) holonomy. The only complex surface with SU(2) holonomy is a K3 surface. We have more possibilities in complex dimension three however. Thus we expect that the moduli spaces we discussed in section 4 to give the complete story for N = 2 theories in six dimensions but we are not done yet for N = 4 theories in four dimensions.

Note that this is a similar statement to the one that Seiberg gave using anomalies [50]. When using a conformal field theory to compactify a ten-dimensional theory to six dimensions one may consider the case of a type IIB string compactified to a chiral N = (2, 0) six dimensional theory and analyze the anomalies. The Hodge numbers of a K3 surface are found to be necessary for a consistent theory.

K3 SURFACES AND STRING DUALITY

47

We should add that one can find further theories in six dimensions if one is willing to drop the requirement that the compactification has some conformal field theory description (and switching to something like M-theory instead). See [90] for an example. Let us consider how to build a complex threefold with holonomy SU(2). First we note the existence of a covariantly-constant holomorphic 2-form and thus h2'0 = 1. The Dolbeault index [9] may then be used to establish h1"0 = 1. Thus our manifold cannot be simply-connected. Now we may use the Cheeger-Gromoll theorem [91] which tells us that the universal cover of the manifold is isometric to M x R' for some compact simply-connected manifold, M. It is clear that we require that the universal cover to be K3 x 112. In other words, any complex threefold with holonomy SU(2) is isometric to KS x T2 or some quotient thereof. To build more N = 4 theories we will consider compactifying type II strings on a quotient of K3 x T2. This quotient must of course preserve the global SU(2) holonomy and thus any element of the quotienting group must preserve the holomorphic 2-form on the K3 surface. Any such action has fixed points on the K3 surface. Thus, to avoid getting a quotient singularity, any such action on the K3 surface must be accompanied by a translation on the r. That is, the quotienting group must have translations in 112 as a faithful representation. An immediate consequence is that the quotienting group must be abelian. The classification of such groups, G, has been done by Nikulin [92] and we list

the results in table 3. M is the rank of the maximal sublattice of H2(K3,Z) that transforms nontrivially under G. This action of G on re,22 °-` r4,20 ® I'2,2 is now determined. Firstly, the action on K3 is a geometric symmetry and so must preserve w and w'. The K3 part of the action is then determined by the action on H2(K3,Z) ?d r3,19 C r4,20 For the

explicit form of the action on the lattice H2(K3,Z) we refer the reader to [93]. Lastly we need the action on r2,2. This encodes the action of G on T2 which we require to be a translation. We also want this action to be geometric and therefore left-right symmetric. This forces the shift to be a null direction. This is sufficient to determine the shift up to isomorphism. Now that we know the action of G on rs,22 we may copy the description of the quotienting procedure over into the heterotic string picture. The result is that we are now describing an asymmetric orbifold of a heterotic string on 2's. Such objects were first analyzed in [94]. An asymmetric orbifold is an string-theory orbifold in which the left-movers and right-movers of the conformal field theory are not treated identically. Because of this the geometric description of the quotienting process in terms of target space geometry is obscure. Also the chiral nature of quotienting can produce anomalies. One manifestation of this can be lack of modular invariance of the resulting conformal field theory. Let us consider an asymmetric orbifold of a toroidal theory built on a lattice A. Consider an element of the quotienting group, g E G, and represent it as a rotation

Z2 Z2xZ2

G

M

1

8

12

Z2 X Z4

Z2XZg

Z3

16

18

12

TABLE 3.

Zg X Zg Z4 Z4 X Z4 Z5 Zg Z7 Z8 16

14

18

ikulin's K3 quotienting groups.

16

16

18

18

PAUL S. ASPINWALL

48

of the lattice followed by a shift, b E A ®z R. Let the eigenvalues of the rotation be of the form exp(2iri.r1). A necessary condition for modular invariance is that [94, 95] IgI

(Eri1 - r5) +12.6- ) E Z;

tlgEG,

(103)

3

where 191 is the order of g. One may check the groups in table 3 and show that this condition is indeed satisfied in every case. Did we have the right to expect that (103) should be satisfied for all the groups

in table 3? To this author the result seems a little mysterious. Indeed, it is the case that if one considers quotients which destroy the N = 4 supersymmetry then one need not be so lucky [96, 97]. For now though, since we are concerned with N = 4 theories at this point, we may content ourselves with the knowledge that the anomalies appear to be looking after themselves and press on. Now let us determine the moduli space. Firstly any deformation of the original

theory of the type II string on K3 x T2 or the heterotic string on TO which is invariant under G will be a deformation of the resulting quotient. Secondly we need to worry in string theory that we may introduce some "twisted marginal operators"

- that is, massless modes associated with fixed points. Since there are no fixed points of G (at least in the type II picture) we may ignore the latter possibility. To obtain the first type of deformation we may simply restrict attention to the invariant sublattice AG c r6,22 under (the rotation part of) G. Note that all of the space-like directions of r6,22 are not rotated by G and so the resulting invariant sublattice, AG, will have signature (6, m), where m. = 22 - M from table 3. What we have done is to build a moduli space of the form

O(AG)\O(6,m)/(O(6) x 0(m)),

(104)

of an N = 4 theory in four dimensions which we viewed either as a type IIA string on a freely-acting quotient of K3 x T2 or an asymmetric orbifold of a heterotic string on TO.

Are there any further possibilities beyond those listed in table 3? We restricted ourselves to classical symmetries of the K3 surface. There should certainly be more symmetries from the stringy geometry of the K3 surface when it is Planck-sized. One may also look at M-theory to provide more possibilities. We refer the reader to [93, 98] for further discussion. Although (104) looks suspiciously like a Narain moduli space for a heterotic string on a torus, it is important to notice that AG is, in general, not self-dual. Any attempt to describe this theory in a straight forward way as a toroidal compactification is doomed as it would imply that the theory is not modular invariant.

On a similar point we have to be careful when considering which enhanced gauge groups can appear. One can view the moduli space as a Grassmannian of space-like 6-planes in Ills,'" but it is no longer the case that only elements, v E AG,

perpendicular to this 6-plane with v.v = -2 will give massless vector fields. The simplest way to approach the question of enhanced gauge groups is as follows (see (99, 100] for the original analysis in terms of heterotic strings). Consider the original theory before we divide by G. In this case, we know what the roots of the enhanced gauge group are, given the space-like 6-plane. As G has a nontrivial

49

KS SURFACES AND STRING DUALITY

Bn

Cn

F4

C

G2

FIGURE 5. Quotients of simply-laced groups.

O m=22

M=10 m=14

FIGURE 6. The moduli space of N = 4 theories.

action on the lattice r6,22, it may also act on the roots of the gauge group. Since our desired theory is the invariant part of the original theory under the quotient by G, the resulting gauge group will be the invariant part of the original gauge group under the action of the discrete group G. The problem we have therefore is as follows. Given a simply-laced gauge group

and an action of a discrete group, G, on the roots of this gauge group, find the subgroup of the gauge group which is invariant under this action. This will be the enhanced gauge group of the desired quotient theory. Fortunately this is a wellknown problem in Lie group theory (see, for example, exercise 22.24 in [29]). The outer automorphism of the group, given by an action on the roots can be written as a symmetry of the Dynkin diagram in the obvious way.16 The results are shown in figure 5 and show that non-simply-laced Lie groups can result. In particular one may show that any Lie group (of sufficiently small rank) can appear as an enhanced gauge symmetry. Lastly let us note an important point about the N = 4 moduli spaces. This is that they are disconnected from each other as shown in figure 6. If components of the moduli space with different values of m were to touch each other then, at such "Except for the case of Su(2n + 1) in which case the outer automorphism yields so (2n + 1) as the invariant subalgebra.

50

PAUL S. ASPINWALL

a point of contact, we would have a theory with very special properties. As one approached such a theory from within the interior of one of the regions, extra states would become massless to furnish the deformations into the other region. This does not happen according to the conformal field description of either the heterotic string

or the type II strings. Thus it would appear unreasonable to expect it to occur in the full string theory. This is to be contrasted with the behaviour of N = 2 theories in four dimensions, as we discuss in section 5.7.

5.3. Generalities for N = 2 theories. Now that we have understood the main features of the moduli space of N = 4 theories in four dimensions we are ready to embark on a study of the much richer field of N = 2 theories. Much of the recent interest in duality was sparked by Seiberg and Witten's work on N = 2 Yang-Mills field theory [101, 102]. Here we are hoping to analyze full string theory in the same context. Thus we expect the subject to be at least as rich as Seiberg-

Witten theory. In the short period that N = 2 theories have been studied in the context of string duality, the subject is already vast and it will be difficult to do justice to it here. As in the rest of these lectures, we will attempt to confine our attention to matters related to the moduli space of theories. How can we obtain an N = 2 theory in four dimensions from string theory? Two answers immediately appear given the usual holonomy argument. Firstly one may take a heterotic string theory in ten dimensions and compactify it on a complex threefold with SU(2) holonomy. We have already discussed such manifolds in section 5.2 and found that they are of the form K3 x T2, or some free quotient thereof. Secondly one might take a type II string and compactify it on a complex threefold of SU(3) holonomy, i.e., a Calabi-Yau manifold. Given the story for N = 4 theories above it is tempting to conjecture that there may be dual pairs of such theories. That is, we wish that a heterotic string when compactified in a specific way on K3 x T2 be physically equivalent to a type II string compactified on a Calabi-Yau threefold. This story began with the papers of [103, 96] and, as we shall see, the full picture is still to be uncovered. There is one immediately apparent curiosity which is associated with such a

conjecture. This is that there are a very large number of topological classes of Calabi-Yau threefolds. The exact number is not known since a classification remains

elusive. Indeed, one cannot rule out the possibility that the number is infinite. Contrasted to this are the few manifolds of K3 x T2 and its quotients. At first it might appear that only a tiny fraction of the type II compactifications can have heterotic partners. This argument is flawed, however, as the heterotic string requires

more data to specify its class than just the topology of the space on which it is compactified. The heterotic string in ten dimensions has a gauge group which is either Es x Es or Spin(32)/Z2. Let us consider the Be x Es string for purposes of discussion. This "primordial" gauge group must be compactified in addition to the extra dimensions.

The generally accepted way to do this is to take a vector bundle E - X over the compactification manifold, X, with a structure group contained in Es x A. The embedding of this structure group into the heterotic string's gauge group then gives a recipe for compactifying the heterotic string on X including the gauge degrees of freedom.

K3 SURFACES AND STRING DUALITY

51

This is exactly what we were doing in section 3.5. In the case of a heterotic string we considered a rank 16 principal U(1)18-bundle over a torus. The structure group was embedded as the Cartan subgroup of Es x E8 and the connection on the bundle was specified by the parameters of the matrix A. The equations of motion demand that A be a constant and so the bundle is flat. The Narain moduli space then gives the full moduli space of such flat vector bundles on a torus. In the case of compactification over a more general manifold, one way of solving the equations of motion [104] is to demand that the vector bundle be holomorphic and that the curvature satisfies g'3Fi3 = 0.

(105)

One also requires c1 (E) E H2 (X, 2Z) and that (106) c2(E) = c2(Tx), for anomaly cancelation. Clearly the torus fits into this picture if we replace the principal U(1)16-bundle by the associated sum of holomorphic line bundles. The analysis of this bundle for the case of a heterotic string on a K3 surface is going to be much harder than the toroidal case because now the bundle cannot be flat as it must satisfy c2(E) = 207 We can see hope then that the large number of choices of possible Calabi-Yau manifolds for compactification of the type II string might be matched by the large number of choices of suitable bundles over K3 x T2 for the heterotic string compactification. It will take a fairly long argument before we are able to give an explicit example of such a pair so we will discuss the situation in general first. Let us start in our

usual way by thinking about the holonomy of the moduli space. For an N = 2 theory in four dimensions the holonomy algebra is u(2) u(1) ® sp(1). There are two types of N = 2 supermultiplet which contain massless scalars: 1. The vector multiplets each contain 2 real fields which form a complex object under the u(1) holonomy. 2. The hypermultiplets each contain 4 real massless scalars which transform as a quaternionic object under the sp(1) holonomy. Thus, at least away from points where the manifold structure may break down, we expect the moduli space to be in the form of a product

AN=2 = .fV xH,

(107)

where dlv is a Kahler manifold spanned by moduli in vector supermultiplets and .4H is a quaternionic Kiihler manifold spanned by moduli in hypermultiplets. It can be shown [105] that .I ' is not hyperkahler. The effective four-dimensional theory always contains a dilaton-a3don system. This will govern the string-coupling. As it plays such an important role we will look first at whether the dilaton-axion lives in a hypermultiplet or a vector multiplet. One way to do this is simply to count the dimensions of the moduli space and use the fact that .41v has an even number of real dimensions and IH has a multiple of four dimensions. For a more direct way of justifying which kind of supermultiplet the dilaton-axion lives in see [97]. "As is common, we assume integration over the base K3 surface in this notation.

PAUL S. ASPINWALL

52

Consider first the type IIA superstring on a Calabi-Yau manifold X. First consider the moduli space of underlying conformal field theories (see, for example, [40] for a full discussion). We have hl"l(X) complex dimensions of moduli space

coming from the deformations of KWer form and B-field and we have h2,'(X) complex dimensions coming from deformations of complex structure. The R-R sector moduli come from a 3-form giving b3(X) = 2(h2,'(X)+1) real deformations. Finally we have 2 real deformations given by the dilaton-axion. Given that h''(X) and h2"1(X) can be even or odd, the only way to arrange these deformations in a way consistent with the dimensionality of the moduli space is to arrange: There are h''(X) vector supermultiplets. There are h2"1(X) + 1 hypermultiplets, one of which contains the dilatonaxion. Next consider the type JIB superstring compactified on a Calabi-Yau manifold, Y. The moduli space of conformal field theories is as for the type IIA string with hl,' (Y) complex deformations of complexified Kahler form and h''' complex deformations of complex structure. The R-R moduli consist of one from the 0-form,

b2(Y) = h" (Y) from the 2-form, b4 (Y) = h'' (Y) from the self-dual 4-form and one more from dualizing the 2-form. Lastly we have two more moduli from the dilaton-axion. Now we are forced to arrange as follows. There are h2,1 (y) vector supermultiplets.

There are hl,'(Y) + 1 hypermultiplets, one of which contains the dilatonaxion. Note that the type HA picture and the type IIB picture are related by an exchange

h" (X) +* h2" (y) (108) h2"1(X) ++ h1"1 (Y).

If we have a pair of Calabi-Yau varieties, X and Y, such that type IIA string theory compactified on X is equivalent to type JIB string theory on Y then we may use this as definition of the statement that "X and Y are a mirror pair". See [106] for further discussion of this point. Lastly we consider the heterotic string compactified on a product of a K3 surface and a 2-torus. First let us assume that there are no nasty obstructions in the moduli space and we can count the number of deformations of the K3 surface, the bundle over the K3 surface, the 2-torus, and the bundle over the 2-torus and simply sum the result. There are 80 deformations of the K3 surface as far as conformal field theory is concerned and the resulting moduli space is a quaternionic Kiihler manifold. It was shown by Mukai [107] that the moduli space of holomorphic vector bundles over the K3 will be hyperki hler. Thus it appears very reasonable to expect that the complete moduli space coming from the K3 surface will be a quaternionic KiWer manifold. Certainly its dimension is a multiple of four assuming unobstructedness. The moduli space coming from the torus will be locally of the form 0(2, m)/(0(2) x 0(m)) and has an even number of dimensions. Thus we assemble the supermultiplets as follows The vector multiplets come from the 2-torus, together with its bundle, and the dilaton-axion.

K3 SURFACES AND STRING DUALITY

53

The hypermultiplets are associated to the K3 surface, together with its bundle.

Note that we are assuming we can give simple geometric interpretations to all the moduli. We will see later that there may be other vector or hypermultiplet moduli we have not included in the above lists.

5.4. KS fibrations. Suppose we are able to find a pair of theories, one a type II string on a Calabi-Yau manifold and the other a heterotic string theory compactified over some bundle on K3 x T2. The first thing one would do would be to line up the moduli spaces of the two theories so that the parameters of one theory could be understood in terms of the other. We begin by analyzing what would happen in the case of the vector multiplet moduli space. The is the first step required before we can actually propose a dual pair of such theories. We now follow an argument first presented in [108]. For definiteness we choose a type IIA string rather than a type IIB. The reason for this is that we will ultimately be able to tie our analysis to proposition 1, which was also phrased in terms of the type IIA string. The holonomy argument leads to a factorization of the moduli space, which in turn has another consequence given the fact that all the matter fields are related to the moduli by supersymmetry. This is that the couplings between fields in the vector multiplets can only depend on the moduli from the vector multiplets and the couplings between fields in the hypermultiplets can only be affected by the moduli from the hypermultiplets. This may also be deduced directly from the supergravity Lagrangian [109]. Let us consider just the moduli space, .acv, coming from the scalars in the vector multiplets. We know this is a complex Kahler manifold. Further analysis of the supergravity Lagrangian puts more constraints on the geometry of the moduli space [110]. A manifold satisfying these extra conditions is called "special Kahler".18 The main importance of special Kahler geometry is the fact that all the information we

require about the theory is encoded in a single holomorphic function if on the moduli space. If we use specific complex coordinates, the "special coordinates", on the moduli space, the metric is of the form

007

K=-log (2(9+A -(q4 -e) \8-9h5 =

8,//

(109)

OK 8gi8g5

When viewed from the point of view of the heterotic string, we expect the dilaton-axion to be contained in this moduli space. Let us suppose for the time being that all the other moduli can be understood from the world-sheet perspective of the heterotic string. This should mean that we have a moduli space of conformal field theories spanning all but one of the complex directions in the moduli space with the extra dimension being given by the dilaton-axion system. In the limit that the string coupling becomes very small, i.e., the dilaton approaches -oo, we expect 1sThe quaternioaic Kiihler manifold parametrized by the hypermultiplets is also subject to extra constraints [111].

54

PAUL S. ASPINWALL

that the moduli space as described by the conformal field theory becomes exact. Thus, in the limit of small dilaton, the moduli space should factorize into a product of the moduli space of conformal field theories, and the extra bit spanned by the dilaton-axion. In [112] precisely this problem was analyzed. It was discovered that the only way a special Kahler manifold could factorize was if it became locally a product of the form

0(2,m)

x SL(2)

(110)

0(2) x O(m) U(1) ' This of course is excellent news. The first term in (110) looks suspiciously like the Narain moduli space for a 2-torus and the second term looks like a dilaton-axion. This is exactly what we wanted. Note that the form (110) is only expected in the

limit that the dilaton approaches -oo. Away from this limit we expect the two factors to begin to interfere with each other. Now we want to carry this information over to the type IIA compactification on the Calabi-Yau manifold, X. All of the vector multiplet moduli are expected to be associated to deformations of the Kahler form and B-field on X. Let us fix

some notation. In contrast to the K3 surface, degrees of freedom of the Kahler form and the B-field for the Calabi Yau threefold, X, can be nicely paired-up. Introduce a basis of divisors, or 4-cycles, {Dk}, spanning H4(X,Z), where k = 0,... , hl,' (X) - 1. Dual to the dual of these divisors we have a basis of 2-forms, {ek}, generating Ha(X,Z).19 Expand out the Kahler form and B-field as h1-'-1

B + iJ = E (Bk + iJk)ek,

(111)

k=0

for real numbers BA;, Jk. We will take eo to correspond to the generator associated to the direction in moduli space given by the heterotic dilaton-axion. Our information for the heterotic side is in terms of the local form of the moduli space. Thanks to special Kahler geometry we can translate this into information concerning couplings between certain fields. The fields we are interested in are the superpartners of the moduli of the vector superfields - i.e., the gauginos, the vector bosons and the moduli themselves. One may consider couplings in the effective action of the form of "Yukawa couplings", i.e., jk = (ai1ij0k), or other terms equivalent by supersymmetry. It was shown in [113] that, to leading order in the non-linear a-model, the coupling between three fields is given by

nijk = #(D; n D j n Dk), (112) where the D's are the divisors in X associated to the fields. The reader is also referred to [114] for an account of this 20 It is also known from special Kahler geometry that kijk =

8,

(113)

84aOgj8gk

We now have some approximate knowledge about both the heterotic string and the type IIA string. In the case of the heterotic string we know that, in the small 19Fbr simplicity let us assume all cohomology is torsion-free.

2ONote that [114] explicitly refers to a heterotic string compactified on a Calabi-Yau manifold whereas we are considering a type HA string. Most of the calculations are unaffected however.

KS SURFACES AND STRING DUALITY

55

dilaton limit, the moduli space factorizes in the form (110) and in the case of the type IIA string, we know that, in the small a'/R2 limit, the couplings are of the form (112). We can make a useful statement about a heterotic-type II dual pair if both of these approximations happen to be simultaneously true. Note that, for the heterotic string, the dilaton lies in a vector multiplet and that, in the type IIA string, the size parameters lie in vector multiplets. Thus we wish to assert that the moduli spaces of the theories are aligned in the right way so that as the dilaton in the heterotic string approaches -oo, some size in the Calabi-Yau space on which the type IIA string is compactified is becoming very large. To picture this presumed aligning of the moduli spaces it is best to picture exactly what makes corrections to the approximations we are considering. In the case of the heterotic string, corrections arise from instantons in the Seiberg-Witten theory [101]. The action of such an instanton becomes very large, and hence the contribution to any physical quantity becomes very small, when the dilaton becomes close to -oo. In the case of the non-linear o-model, corrections come from worldsheet instantons [115]. A world-sheet instanton takes the form of a holomorphic map from the world-sheet into the target space. We will be interested only in treelevel effects and so as far as we are considered world-sheet instantons are rational curves. The action for such an instanton is simply the area of the curve. Thus the effect of an instanton gets weaker as the Kahler form is varied so as to make the rational curve bigger. The picture one should have mind therefore is that as the heterotic dilaton is decreased down to -oo, some rational curve (or some set or family of rational curves) is getting bigger. The important thing is that no curve should shrink down during this process. Let us assume we are now at the edge of our moduli space where, in the type IIA interpretation, all of the rational curves are very large compared to a'. Thus, by assumption, the heterotic string's dilaton is close to -oo. We can take the moduli space given by (110) and deduce the form of 9. We may then translate this into a statement about'ctjk and thus about the topology of X from (112). The result is that [112]

#(DonDonDo)=0 #(Do n Do n D;) = 0,

i = 1, ... , h1" - 1

(114)

#(DonD{nDj)=,,j, where n;j is a matrix of nonzero determinant and signature Suppose that X is such that there is some smooth complex surface embedded in X whose class, as a divisor, is Do. From (114) we see that Do n Do = 0 and so the normal bundle for this surface is trivial. It then follows from the adjunction

formula, and the fact that X is a Calabi-Yau space, that the tangent bundle for this surface has trivial cl. Thus, the surface representing Do must either be a K3 surface or an algebraic 4-torus (also known as an "abelian surface"). The fact that the normal bundle is trivial also suggests that the K3 or abelian surface can be "moved" parallel to itself to sweep out the entire space X. That is, X is a fibration where the generic fibre is either a K3 surface or an abelian surface.

as

PAUL S. ASPINWALL

We can make the above more rigorous by appealing to a theorem by Oguiso [116] which states that THEOREM 7. Let X be a minimal Calabi-Yau threefold. Let D be a nef divisor

on X. If the numerical D-dimension of D equals one then there is a fibmtion : X -> W, where W is F1 and the generic fibre is either a K3 surface or an abelian surface.

The numerical D-dimension of a divisor is the maximal number of times it may be intersected with itself to produce something nontrivial. We see above that Do has D-dhnension equal to one. The statement that D is "nef" is the assertion that for any algebraic curve, C C X, we have that

#(D f1 C) > 0, dC C X. (115) We can come very dose to proving that Do is nef following our assumption about the way that moduli spaces are aligned. The special coordinates on a our moduli space of type IIA string theories may be written in the form

qk = e2"(.').

(116)

We are in an area of moduli space where all qk
II qk (D,,nc)

(117)

k=ii

Clearly, if C is not nef then negative powers of qk appear and the instanton sum will fail to converge, in contradiction to expectation. W The above argument that C is nef is not actually complete. When one computes the instanton sum, one also has to compute the coefficient in front of the monomial of the form (117). This can be done using the methods discussed in [117, 1181. In the simple case of an isolated curve, the coefficient is simply one (although extra contributions arise from multiple covers). C may not always be isolated however. One may be able to deform C into a whole family of rational curves. In this case the coefficient might be zero. One cannot then rule out that #(D n C) < 0 as the field theory is simply unaffected by C.

An example of a case where rational curves don't count in this way is that of N = 4 theories. Compactifying the type IIA string on K3 x T2, there are no instanton corrections but the K3 surface may contain rational curves. Each rational curve will clearly he in the form of a family C x T2 inside K3 x T2. An algebraic surface of the form IP' x T2 is known as an "elliptic scroll" 2' Since the calculation in [117, 118] is essentially local we may generalize this result to any N = 2 Calabi-Yau compactification. That is we claim that rational curves

in an elliptic scroll do not contribute to the instanton sum. Note that rational curves in a K3 surface are unstable in the sense that a generic deformation of complex structure of the K3 surface will kill them. The work of Wilson [119) showed that essentially the same is true for elliptic scrolls in Calabi-Yau manifolds. Thus, if we choose a generic complex structure on the Calabi-Yau manifold then there will be no curves lying in an elliptic scroll. If we assume that the only curves contributing zero to the instanton sum lie in an elliptic scroll then we may complete our proof that Do is nef by choosing a generic complex structure. We do not know if this assumption is valid but it seems reasonable.

Note that the fibration ! : X -+ W is allowed to "degenerate" at a finite number of points in W. Indeed, if this were not the case then X could not be a Calabi-Yau variety. Note that the degenerate fibre need not be some degenerate 21An "elliptic curve" is the algebraic geometer's name for an algebraic curve of genus 1.

K3 SURFACES AND STRING DUALITY

57

limit of a K3 surface. It could be a perfectly smooth manifold (which is neither a K3 or abelian surface). Finally we would like to know if the generic fibre is a K3 surface or an abelian surface. We can determine this by finding c2, and hence the Euler characteristic, of the fibre. This may be determined by using the holomorphic anomaly of [120]. The result is that the Euler characteristic of the generic fibre is 24 and so the fibration is of the K3 type. We refer the reader to [108] for details of this calculation. We thus arrive at the following: PROPOSITION 6. Given a dual pair of theories, one of which is a heterotic string compactified on K8 x T2 (or some free quotient thereof) and the other is a type HA string compactified on a Calabi-Yau manifold X, then if there is a region of moduli space in which both the respective perturbation theories converge, then X must be of the form of a KS-fibration over V.

This proposition depends upon our statement about zero contributions from rational curves only arising from elliptic scrolls. The fact that the base of the fibration is a P'I may be deduced from the fact that HI of the base injects22 into the total cohomology of X and that X has b, = 0.

Not only do we know that X is a K3 fibration, but we also know that the divisor Do is the generic K3 fibre. This tells us immediately which deformation of X corresponds to a deformation of the heterotic dilaton - it is the component of the KAbler form that affects the area of the curve dual to Do. Let us first assume that the K3-fibration of X has a global section. This means that as well as the fibration map:

§:X->W,

(118)

we also have a holomorphic embedding

ry:W-8X.

(119)

The image of this embedding is a rational curve in X. Clearly this curve is dual to Do. Thus the value of the dilaton in the heterotic string is given by the area of this rational curve. The instanton sums converge as the dilaton approaches -oo and as this rational curve gets very large. Thus the weakly-coupled limit of the heterotic string is given by the limit of X in which the base P°I swells up to infinite size. Note that if the K3 fibration does not have a global section, but rather a multivalued section, we may play a similar game. In this case the multi-section will not define an embedding of the base FI into X but rather an embedding of a multiple

cover of X. Thus, the algebraic curve within X whose area gives the heterotic dilaton will be of genus greater than zero.

One should note that what we have discussed here for the N = 2 dual pairs is actually very similar to the N = 4 case discussed above. In that case we had a type HA string compactified on K3 x T2 or a quotient thereof. This space may be viewed as a smooth K3-fibration over T2 (that is, an elliptic curve). This fibration is trivial in the case that the target space is K3 X T2 but becomes nontrivial when we take a free quotient. We saw in section 5.1 that the dilaton for the heterotic string is given by the area of the T2 - i.e., the size of the base of the fibration. 22This follows from the Leray spectral sequence for a fibration.

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PAUL S. ASPINWALL

The general picture we see is the following. If we have a heterotic-type IIA dual pair in four dimensions, we may expect that the type IIA string is compactified on

a K3-fibration over a 2-torus in the case of N = 4 supersymmetry, or, a K3-fibration over a Pl in the case of N = 2 supersymmetry.

In either case, the area of the base of the fibration gives the value of the heterotic dilaton.

The link to proposition 1 is clear in the N = 4 case. For the N = 2 case we should note that the K3 surface on which the heterotic string is compactified might be viewable as an elliptic fibration itself - that is, a fibration over PI with generic fibre given by a T2. Thus the K3 x T2 (or quotient thereof) upon which the heterotic string is compactified may be viewed as an "abelian fibration" over PI - i.e., a fibration over Pl with generic fibre given by a V. When viewed in this way the relationship between the N = 2 theories in four dimensions given by the heterotic string and the type IIA string may be viewed as a fibre-wise application of the duality of proposition 1. Such "fibre-wise duality", which was first suggested in [97], is a potentially very powerful tool. It has been extended to fibre-wise mirror symmetry in [121] and has recently been applied to the problem of mirror symmetry itself in [122]. Both of these developments deserve to be covered here in some detail since they both have

direct relevance to K3 surfaces, but we do not have time to do so. We refer the reader to [123, 124] for more details of the latter in the context of K3 surfaces. Let us note that the general appearance of K3-fibrations in the area of heterotictype II duality was first noted in [125] where some naturalness arguments based on the work of [126] were presented. It is interesting to note that in this context it was really the type JIB string that being studied rather than the type IIA. This raises the question as to whether the mirror of a K3 fibration is another K3 fibration, which again raises the possibility of some fibre-wise duality argument.

5.5. More enhanced gauge symmetries. So far we have identified one of the moduli lying in a vector multiplet. This is the dilaton-axion in the heterotic string, or the component of the complexified Kiihler form associated to the base of the K3-fibration on which the type IIA string is compactified. Now we wish to turn our attention to some of the other vector multiplet moduli.

Let us first look at things from the type IIA perspective. We know that our Calabi-Yau space, X, is a K3-fibration and we wish to analyze elements of H2(X), or equivalently, H4(X). In general, given a fibration X -+ W, the cohomology of X may be determined from the cohomology of the base together with the cohomology of the fibre. The mechanism by which this happens is called a "spectral sequence". We do not wish to get involved with the technicalities of spectral sequences here and refer the reader to [127] for the general idea. The result is that the contributions to H4(X) are as follows (see figure 7): 1. The generic fibre, Do, will generate an element of H4(X). 2. Take an algebraic 2-cycle in the fibre, i.e., an element of H2(Do) and use it to "sweep out" a 4-cycle in X by transporting it around the base, W. Note that this 2-cycle needs to be monodromy invariant for this to make sense. Thus the 2-cycle might be an irreducible curve in Do which is monodromy

59

K3 SURFACES AND STRING DUALITY Bad Fibre

W FIGURE 7. X as a K3 fibration over W.

invariant or it may be the sum of two curves which are interchanged under monodromy, etc.

3. When we have a bad fibre which is a reducible divisor in X, we may vary the volumes of the components of this bad fibre independently. Thus such fibres will contribute extra pieces to H4(X). The second class above is clearly generated by elements of the Picard group of the generic fibre. A monodromy-invariant element of the group C, will sweep out a divisor Di, where Ci = Do n Di. Thus

#(Do n Di n D,)x = #((Do n Di) n (Do n D;))Do (120)

= #(C1 n C,,)D,,

where the subscript denotes the space within which we are considering the intersection theory. This agrees very nicely with our earlier result of (114). We see that nib is simply the natural inner product of the monodromy-invariant part of the Picard lattice of the generic fibre. As we mentioned in section 2.5, the Picard lattice has

signature (+, -, -, ... , -). When we take the monodromy-invariant part we retain the positive eigenvalue as the Kiibler form restricted to the generic K3 fibre must be monodromy invariant and the generic fibre has positive volume. Thus iii has the correct signature. Now consider the third class. We denote such an element by Ba. Since this class is supported away from the generic fibre, we have Don B. = 0. Now comparing to (114) we are in trouble. The moduli coming from such vector supermultiplets cannot live in the space (110) we expected from the heterotic string. What can have gone wrong? The only place our argument was flawed was in the perturbative analysis of the heterotic string. It turns out that the classes B. cannot be understood perturbatively from the perspective of the heterotic string. We will encounter such

objects later in section 6.2 but in this section we will restrict our attention to perturbative questions.

60

PAUL S. ASPINWALL

We now know exactly how to interpret the Teichmiiller space SL(2) 0(2,m) 0(2) x 0(m) x U(1) '

(121)

both in terms of the heterotic string compactified on K3 x T2 and its supposed type HA dual compactified on X. The second term comes from the heterotic dilatonaxion and the complexified Kahler form associated to the class Do E H4(X). The first term is considered to be associated to the Narain moduli space of the heterotic string on T2, or, from what we have just said, the complexified Kahler form on the (monodromy-invariant part of) the K3 fibre. We computed the stringy moduli space of the Kahler-form and B-field on an algebraic K3 surface earlier and obtained the second term of (73). Fortunately it turns out to have just the right form! Note that we have reduced our problem once again to a comparison of a heterotic string on a torus (this time a 2-torus) with a type IIA string on a K3 surface (this time an algebraic fibre in X). We should be able to use the same old arguments we used in sections 4.3 and 5.2 to obtain the enhanced gauge group. Let us first assume there is no monodromy acting on the Picard group of the K3 fibre for X. We expect the slice of the moduli space coming from varying the complexified Kahler form on the generic K3 fibre to be of the form

0(T)\0(2,p)/(0(2) x 0(P)),

(122)

where T is the quantum Picard lattice introduced in (69). Identifying this with the Narain moduli space of Tz that we expect to see in the weak-coupling limit of the heterotic string, we see that the Narain lattice for the T2 is isomorphic to T.23 That is, we have a gauge bundle of rank p - 2 compactified over T. To obtain the enhanced gauge groups we take (122) to be the Grassmannian of space-like 2-planes in R2,P and look for points where the 2-plane becomes orthogonal to roots in the lattice T. The roots will give the root diagram of the enhanced gauge group in the usual way. Let us clarify all this general discussion with an example taken from [103]. Let us take Xo to be the hypersurface zi + za + x312 + x424 + x314 = 0,

(123)

in the weighted projective space p4( 12,8,2,1,1)1 The weighted projective space contains various quotient singularities. We will blow these up and take the proper transform of Xo to be X. There is a Z2-quotient singularity along the locus [z1, z2, z3, 0, 0]. We may blow this up, replacing each point in the locus by PI. The projection of X onto to this P1 will be the K3-fibration. That is, roughly speaking, we view [z4, z5] as the homogeneous coordinates of the base W F1. To find the fibre fix a point in the base by fixing z4/z8. This projects 1 12,8,2,,,,) onto the subspace P{12,8,2,I} Now 1112,8,2,I} may be viewed as a Zrquotient of P{s,4,1,1} by taking 04 K -z4. Such 23Note that there is no reason to expect that T should be self-dual. In many proposed dual pairs (e.g., some of those of [1031) it is not self-dual. On the heterotic side this says that the heterotic string on T2 is not modular Invariant. We shall assume that modular invariance is satisfied once the K3 factor is taken into account too. In general one might worry that strange effects such as those encountered in [96] might cause problems with this. We will assume here that modular invariance looks after itself.

K3 SURFACES AND STRING DUALITY

61

codimension quotients are equivalent to reparametrizations in complex geometry and so the fibre may be taken to be the hypersurface zl + za + z32 + z42 = 0,

(124)

in the weighted projective space 1F{e,4,1,1} This is a K3 surface as expected. Note that we still have a Z2-quotient singularity in the fibre along [z1, z2, 0, 0]. This intersects the generic K3 fibre at a single point. Thus for a smooth X we blow-up again to introduce a single (-2)-curve into each generic fibre. Now let us work out the Picard lattice of the generic fibre. A generic hyperplane

in 1{6,4,1,1} may be written as az3 + bz4 = 0 for some a, b E C. Any two such hyperplanes will intersect at the point [z1, z2, 0, 0] but this is exactly where we are blowing up. Thus the hyperplane doesn't intersect itself at all but will intersect the (-2)-curve (i.e., the exceptional divisor) once. The Picard lattice, for a generic value of complex structure of X, will be generated by the hyperplane and the single (-2)-curve and has intersection matrix

\ 0 -2 /

.

(125)

A simple change of basis shows that this is r1,1 L U. Note that neither the hyperplane nor the exceptional (-2)-curve are effected by monodromy around W and so we needn't worry about monodromy invariance in this case. The bad fibres occur when we fix a point on the base W such that z224 4 +z24 5 = 0. Thus, at 24 points, we have fibres

zi + z2 +x62 = 0,

(126)

in P{6,4,1,1} Although singular, this equation does not factorize and so the bad fibres are irreducible. Thus there are no contributions to h" from bad fibres. Thus, h" = 1 + 2 = 3 since we have the generic fibre itself together with p = 2. Now T = U ® U °-' r2,2 and the moduli space of vector multiplets coming from the quantum Picard lattice will be

o(r2,2)\0(2,2)/(0(2) x 0(2)).

(127)

This is exactly the moduli space for a string on T2. Thus if the type IIA string compactified on X is dual to a heterotic string on K3 x T2 then the T2 part of the latter has none of the gauge group from the ten-dimensional string wound around it. All must be wound around the K3 factor. The lattice r2,2 contains root diagrams for Su(2) ® su(2) and su(3) and so we should be able to obtain these gauge symmetries for suitable choices of vector moduli. Clearly the (-2)-curve in the generic K3 fibre may be shrunk down to a point to obtain SU(2). To obtain more gauge symmetry one must shrink the K3 fibre itself down to a size of order (a')2 as discussed in section 4.3. There is an important point we should note in general about enhanced gauge groups. It is known that in N = 2 theories quantum effects may break the gauge group down to a Cartan subgroup of the classical gauge group. Exactly how this happens depends on whether any extra hypermultiplets are becoming massive when the point of classical enhanced gauge symmetry occurs. We don't want to discuss hypermultiplets yet but, at least in simple cases, there are usually a small number,

62

PAUL S. ASPINWALL

if any, of such massless particles and the theory will be "asymptotically free". In this case the gauge group will be broken. Thus we only really expect the nonabelian gauge group to appear in the heterotic string for the case that the coupling tends to zero. In the type IIA picture this corresponds to the base PI blowing up to infinite size. This means that we are "decompactifying" the type IIA picture so that it is compactified, not on X, but on the generic fibre - a K3 surface. In this respect we really are saying little more than we already said in section 4.3. We can obtain enhanced gauge symmetries when we have a theory in six dimensions from a type IIA string corllpactified on a certain K3 surface. This is approximately true in four dimensions so long as the base PI is very large. For the heterotic string, the gauge group is broken in the quantum theory by Yang-Mills instantons. For the type IIA string we note that world-sheet instantons wrapping around the base PI presumably play an analogous role. This is another example of one string's target space field theory being another string's world-sheet field theory as we saw in section 5.1. It would be interesting to see if these instautons can be explicitly mapped to each other. To determine the gauge group we should say what happens when there is mou-

odromy in the Picard lattice of the generic K3 fibre as we move about W. There really is no difference between this and the N = 4 analogue we discussed in section 5.2. The monodromy of the Picard lattice should be translated into an action on the heterotic string on T2 and divided out. Thus we expect an asymmetric orbifold of T2 for the heterotic dual. When finding the enhanced gauge group we should take the monodromy-invariant subdiagram of the root diagram. This may lead to non-simply-laced gauge groups again. One should also worry about the global form of the gauge group. That is, one may have the simply connected form of the group or one may have to mod out by

part of the center, e.g., the gauge group might be SU(2) or SO(3). All we have said above is only really enough to determine the algebra of the gauge symmetry. We will evade this issue where possible in these lectures by only specifying gauge algebras rather than gauge groups. There will be times later on in these lectures when we have to confront this problem, however. Let us mention here that compactifying the type JIB string, rather than the type IIA string, on a K3 fibration, can lead to a very direct link between the geometry of the Calabi-Yau threefold and Seiberg-Witten theory as explored in [128, 129J. Therefore an understanding of mirror symmetry within K3 fibrations may shed considerable light on some of the details of string duality. Finally let us note that further analysis may be done on the moduli space of vector multiplets to check that string duality is working as expected. We refer the reader to [130, 131] for examples and especially to [132] where further direct links to geometry were established.

5.6. Heterotic-heterotic duality. We have seen how a type IL4 string compactified on a Calabi-Yau space with a K3 fibration may have as a dual partner a heterotic string compactified on K3 x T2. An obvious question springs immediately to mind. What happens if X can admit more than one K3 fibration? One might expect it may be dual to more than one heterotic string. This implies that we can find pairs of heterotic strings that are dual to each other. We will follow this line of

K3 SURFACES AND STRING DUALITY

63

FIGURE 8. The Hirzebruch surface F,,.

logic to analyze a case introduced in [133]. This geometric approach was discussed originally in [134, 135). We are going to consider the example we introduced in the last section based on the hypersurface in P4(,2,,,2,1,1) given by (123). In the last section we projected onto the last 2 coordinates of the '{I2,E,2,1,1} to obtain a K3 fibration over P1. Now let

us project into the last 3 coordinates. Our base space will now be II°{The fibre will be an algebraic 2-torus, that is, an elliptic curve. Thus X may be considered as an elliptic fibration as well as a K3 fibration. The space P{is singular. Writing the homogeneous coordinates as [xo, xI, x2), there is a Z2 quotient singularity at the point [1, 0, 0). This may be blown up, introducing a (-2)-curve. This exceptional curve provided the base of the K3 fibration in the last section. We may also use it to write the blow-up of F{2,1,1} as a fibration. From the same argument as we used in the last section, it is straight forward to see that the fibre will be P'. Thus our blown-up Pt 2,1,1} is a fibration with base space given by ?t and with fibre given by P1. Note that the fibre never degenerates. Such complex surfaces are called "Hirzebruch surfaces". These objects will turn out to be important for the analysis of the heterotic string on a K3 surface so we discuss the geometry here in some detail. A F1 bundle over Fl may be regarded as the compactification of a complex line bundle over I?' by adding a point to each fibre "at infinity". Such line bundles are

classified by an integer - the first Chern class of the bundle integrated over the base 1P1. We use the notation F,,, to denote the Hirzebruch surface built from the line bundle with cl = -n. Assume first that n > 0. Denote the base rational curve by Co. The line bundle with cl = -n represents the normal bundle to Co and so the self-intersection of Co equals -n. Thus Co is isolated (assuming n > 0). Denote

the fibre by f. Clearly #(f n f) = 0 and #(f n Co) = 1. We may introduce a class C, = Co + n f . This intersects f once and Co not at all. C, is a section of the bundle, and hence a rational curve, away from the isolated section Co. The self

intersection of C, is +n. Note that C, can be deformed. Thus we view F as a Fl bundle over P1 with one isolated section, of self-intersection -n, and a family of sections, disjoint from the isolated section of self-intersection +n. This picture of a Hirzebruch surface, with two sections in the class CI shown, is drawn in figure 8. If n = 0 this picture degenerates and we simply have Fo Fr x Fr

84

PAUL S. ASPINWALL

Note that if n < 0 we may exchange the roles of Co and C1 and recover the surface F_n. Thus we may assume n > 0. For the case we are concerned with in this section Co is the exceptional divisor with self-intersection -2 and so we have F2 as the base of X as an elliptic fibration. It turns out however that F2 is somewhat "unstable" in this context. Note that one may embed P{2,1,1} into P3 as follows. Denote the homogeneous coordinates by [xo, XI, x2] and [yo, y1, y2, y3] respectively. By putting Yo = xo v1 = X2 (128)

y2=X2 ys = XIx2,

we embed p2as the hypersurface Viva - ys = 0. (129) This is singular, as expected, but we may deform this hypersurface to a generic quadric. It is well-known that a generic quadric in P3 is isomorphic to PI x PI ?, F0 (see, for example, [8]). Thus P{2,1,1} may be blown-up to give F2 but deformed to give F0. A generic point in the moduli space of X is actually an elliptic fibration over Fo rather than F2.24 We have arrived at the result that X is an elliptic fibration over F0. This may be viewed as a two-stage fibration. X may be projected onto a P1 to produce a K3

fibration. The fibre of this map may be projected onto the other PI to write the K3 as an elliptic fibration. Since the base space of the elliptic fibration is P1 x PI, if one of the P's in the base may be viewed as the base for X as a K3 fibration, then so may the other PI. Thus X can be written as a K3 fibration in two different ways. This suggests that there should be two heterotic strings dual to the type IIA theory on X and hence dual to each other. What is the relationship between these two heterotic strings? One of the heterotic strings will be compactified on S1 x 1' and the other on S2 x T22, where Si and S2 are K3 surfaces. We examined the moduli space coming from the vector multiplets in section 5.5 and saw that the 3 moduli described the dilaton-axion and the Narain moduli of the T2 (without any Wilson lines switched on). The dilaton is given by the area of the base FI in X. Let us determine the area of the heterotic T2 in terms of the moduli of the K3 fibre within X. The generic K3 fibre within X has Picard number 2 and has a moduli space of Ki hler form and B-field given by (127). Let I'2,2 be generated by the null vectors w and v together with their duals w' and v'. For simplicity we will avoid switching on any B-fields. Thus we consider a point in the moduli space (127) to be given by a space-like 2-plane, U, spanned by w' + aw and v` + 13v, for a, 0 > 0. Following our construction to determine the Kiihler form, we have that U f1 wl is spanned 24Th see this note that when we blow-up the fibration of X over ll'a {2,1,1} to a fibration over F2 we introduce an elliptic scroll just as in section 5.4. The results of [1191 tell us that the (-2)-curve will therefore vanish for a generic complex structure.

K3 SURFACES AND STRING DUALITY

65

by v" +,3v which is contained in wl/w. Thus, as promised, B is zero. From our analysis in section 3.3 we see that the Kahler form determines the volume to be

J.J = a.

(130)

We also have that the direction of J is given by v" + Qv. Thus the Kahler form is

J= 2#

+

2

v.

(131)

We have seen that the generic K3 fibre is itself an elliptic fibration over Hat. Knowing the intersection numbers of the curves within this fibration together with the positivity of the Kahler class determines the class of the base P1 to be v" - v and that of the elliptic fibre to be v. Thus, the area of the p1 within the K3 is given by

J.C = 2 -

2a.

(132)

Now let us reinterpret Zl in terms of the moduli of the T2. From section 3.5 we obtain a map from W into W' given by25

0=l

0

\

I

.

(133)

This is symmetric and thus gives the metric. Therefore, the area of the T2 is given by We may now obtain an interesting result confirming the analysis of [133] by

det(o) =.

going to the limit where a and Q are taken to be very large. In this case, our heterotic string is compactified on K3 x T2 where the 2-torus is very large. In the type HA string, the generic K3 fibre contains a rational curve which becomes very large. From the equation (132), the area of this rational curve is proportional to the heterotic string's 2-torus in this limit. This limit as a, /3 - oo may be viewed as a decompactification of the model to a six-dimensional theory given by the heterotic string compactified on a K3 surface. Let us consider the coupling of this six-dimensional theory, As. This is given by the four-dimensional coupling prior to decompactification and the area of the T2. The former is given by the size of the base Pt of the K3 fibration, which we refer to as P1. The latter is given by the area of the base of the K3 fibre itself written as an elliptic fibration, which we will refer to as P. We have that AS = A. Area(T2) ..,

Area(P21)

A rea( 1P1)'

( 134)

The other heterotic string is obtained by exchanging the roles of IPl and F21. Thus we have PROPOSITION 7. Let X be the Calabi-Yau manifold given by a resolution of The two heterotic string theories dual to the type IIA string theory on X decompactify to two heterotic string theories compactified on KS surfaces. From (134) these two six-dimensional theories are the degree 24 hypersurface in Ho{12,a,2,i,i}

'Note that W is spanned by w' and v'; and W is spanned by w and v. There is no simple choice of conventions which would have circumvented this notational irritation!

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PAUL S. ASPINWALL

S-dual in the sense that the coupling of one theory is inversely proportional to the coupling of the other theory.

The main assumption underlying this proposal is that these heterotic string theories actually exist. We will get closer to identifying these theories in section 6.3.

We may vary the complex structure of X and this should correspond to deformations of the K3 together with its vector bundle on the heterotic side. Note that the K3 of one of the heterotic strings will, in general, have quite different moduli than the K3 of the other heterotic string since exchanging the roles of IPl and IP need not be a geometrical symmetry of X -- only a topological synmmetay. Before we can explicitly give the map between these K3 surfaces and their vector bundles we need to map out the moduli space of the moduli from the hypenmrltiplets. This remains to be done. There are many examples of Calabi-Yau manifolds which admit more than one K3 fibration. This will lead to many examples of heterotic-heterotic duality. In most cases however the result will be a rather tortuous mapping between four-dimensional theories and will not be as simple as the above example.

5.7. Extremal transitions and phase transitions. We will take our first tentative steps into the moduli space of hypermultiplets in this section. This will deal with the simplest aspects of the bundle over K3 - namely when this bundle, or part of this bundle, becomes trivial. Our heterotic string theory is compactified on a bundle over K3 x T2 which we view as the product of a bundle over K3 and a bundle over T'=. Generically one would expect the deformation space of this bundle structure to be smooth. Where this can break down however is when part of the bundle becomes trivial. To deform away from such a bundle we may wrap the trivial part. around either the K3 surface or the T2. Thus we obtain branches in the nioduli space. A deformation in the K3 part will be a hypermultiplet modulus while a deformation in the T2 part will be a vector modulus. Our picture therefore for a transition across this branch will consist of moving in the moduli space of hypermultiplets until suddenly a vector scalar becomes massless and can be used as a marginal operator to move off into a new branch of the moduli space, at which point some of the moduli in the hypermultiplets in the original theory may acquire mass. When we compactify the Es x Es or Spin(32)/Z2 heterotic string on a vector bundle, the original gauge group is broken by the holonomy of the bundle. Thus, when we move to a transition point where part of the vector bundle becomes trivial we may well expect the holonomy group to shrink and thus the observed gauge group is enhanced. Since we have only N = 2 supersymmetry this observed gauge group enhancement may get killed by quantum effects but should be present in the zero-coupling limit. The picture is the geometric version of the "Higgs" transitions explored in, for

example, [102]. We wish to see how this transition appears from the type IIA picture. We have already seen what the type IIA picture is for moving within the moduli space of vector scalars to a point at which an enhanced gauge symmetry appears.

This is where we vary the Kahler form on the generic K3 fibre to shrink down

K3 SURFACES AND STRING DUALITY

67

some rational curves (or go to Planck scale effects). Thus, every K3 fibre becomes singular. That is, we have a curve of singularities in X. Now given such a singular Calabi-Yau manifold, it may be possible to deform this space by a deformation of complex structure to obtain a new smooth Calabi-Yau manifold. This would correspond to a deformation of each singular K3 fibre to obtain a smooth K3 fibre. This process will decrease the Picard number of the generic fibre (as we have lost the class of rational curves we shrank down) and will change the topology of the underlying Calabi-Yau threefold. In the type IIA language then, this Higgs transition consists of deforming the Kiihler form on X to obtain a singular space and then smoothing by a deformation to another smooth manifold. Such a topology-changing process is called an "extremal transition". One example of an extremal transition is the "conifold" transition of [136]. A conifold transition consists of shrinking down isolated rational curves and then deforming away the singularities. These were explored in the context of full string

theory in [75, 137] (see also B. Greene's lectures). In our case however we are not shrinking down isolated curves but whole curves of curves and so we are not discussing a conifold transition. There has been speculation [138, 137] that the moduli space of all Calabi-Yau threefolds is connected because of extremal transitions (based on an older, much weaker, statement by Reid [139]). See [140, 141] for recent results in this direction. Certainly no counter example is yet known to this hypothesis. The heterotic picture

of these extremal transitions is simple to understand in the case of singularities developing within the generic fibre. Such specific extremal transitions are certainly not sufficient to connect the moduli space and we will require an understanding

of nonperturbative heterotic string theory to complete the picture. One obvious example to worry about is when the Calabi-Yau threefold on the type HA side goes through an extremal transition from something that is a KS fibration to something

that is not. It is not difficult to see that such a transition must involve shrinking down the base, W, of the fibration and thus going to a strongly-coupled heterotic string. It is not surprising therefore that we cannot understand such a transition perturbatively in the heterotic picture. Given a dual pair of a type IIA string compactified on a Calabi-Yau manifold and a heterotic string compactified on K3 x T2 we may generate more dual pairs by following each through these phase transitions we do understand perturbatively. Such "chains" of dual pairs were first identified in [142] and many examples have

been given in [143]. In order to understand where these chains come from, and indeed the original Kachru-Vafa examples of dual pairs in [103], we need to confront

the issue of compactifying the heterotic string on a K3 surface, a subject we have done our best to avoid up to this point!

6. The Heterotic String This section will be concerned with the heterotic string compactified on a K3 surface. In particular we would like to find a string theory dual to this. An obvious answer is another heterotic string compactified on another K3 surface, as we saw in section 5.6. We will endeavor to find a type II dual. It turns out the we will not be able to find a type II dual directly but will have to go via the construction of

as

PAUL S. ASPINWALL

section 5. This process (in its various manifestations) is often called "F-theory". A great deal of the following analysis is based on work by Morrison and Vafa [144, 135, 145].

6.1. N = 1 theories. The heterotic string compactified on a K3 surface gives a theory of N = 1 supergravity in six dimensions. From section 4.1 the holonomy algebra of the moduli space coming from supersymmetry will be sp(1). There are two types of supermultiplet in six dimensions which contain moduli fields: 1. The hypermultiplets each contain 4 real massless scalars which transform as quaternionic objects under the sp(1) holonomy. 2. The "tensor multiplets" each contain one real massless scalar. Thus, at least away from points where the manifold structure may break down, we expect the moduli space to be in the form of a product WN=1 =

T X 'H,

(135)

(possibly divided by a discrete group) where 11T is a generic Riemannian manifold

spanned by moduli in tensor supermultiplets and A "H is a quaternionic Kahler manifold spanned by moduli in hypermultiplets. For a review of some of the aspects of these theories we refer to [146]. The six-dimensional dilaton lives in a tensor multiplet. An interesting feature of these theories is that it appears to be impossible to write down an action for these theories

unless there is exactly one tensor multiplet 26 Thus moduli in tensor multiplets other than the dilaton should be regarded as fairly peculiar objects. It is useful to compare N = 1 theories in six dimensions with the resulting N = 2 theory in four dimensions obtained by compactifying the six-dimensional theory on a 2-torus. To explain the moduli which appear in four dimensions we need to consider another supermultiplet of the N = 1 theory in six dimensions. This is the vector multiplet which contains a real vector degree of freedom but no scalars. Each supermultiplet in six dimensions produces moduli in four dimensions as follows:

1. The 4 real scalars of a hypermultiplet in six dimensions simply produce the 4 real scalars of a hypermultiplet in four dimensions. 2. The 1 real scalar of a tensor multiplet together with the anti-self-dual twoform compactified on T2 produce the two real scalars of a vector multiplet in four dimensions. 3. The vector field of a vector multiplet compactified on the two 1-cycles of T2 of produces the two real scalars of a vector multiplet in four dimensions. That is, hypermultiplets in six dimensions map to hypermultiplets in four dimension but both tensor multiplets and vector multiplets in six dimensions map to vector multiplets in four dimensions. We should emphasize that quantum field theories in four and six dimensions are quite different. In particular, conventional arguments imply that six-dimensional quantum field theories should always be infra-red free and therefore rather boring. 26This is because the gravity supermultiplet (of which there is always exactly one) contains a self-dual two-form and the tensor multiplets contain anti-self-dual two-forms. It is problematic to write down Lorentz-invariant actions for theories with net (anti-)self-dual degrees of freedom [147].

KS SURFACES AND STRING DUALITY

69

This notion has been revised recently in light of many of the results coming from string duality [146, 148] where it is now believed that nontrivial theories can occur in six dimensions as a result of "tensionless string-like solitons" appearing. Such theories potentially appear when one goes through extremal transitions between tensor multiplet moduli and hypermultiplet moduli. Despite the strange properties of these exotic six-dimensional field theories we will be able to avoid having to explain them here. This is because all of our discussion will really happen in four dimensions - the six-dimensional picture is only considered as a large T2 limit. It is important to realize however that some of the things we will say, based on four-dimensional physics, may well be rather subtle in six dimensions. An example of this will be when certain enhanced gauge symmetries are said to appear when the hypermultiplet moduli are tuned to a certain value. If massless tensor moduli also happen to appear at the same time (which will happen for Es as we shall see) then any conventional description of the resulting six-dimensional field theory is troublesome. Declaring what the massless spectrum of such a theory is not an entirely well-defined question and one should move the theory slightly by perturbing either a massless hypermultiplet modulus or a tensor modulus before asking such a question.

6.2. Elliptic fibrations. Our method of approach will be to consider a type IIA string compactified on a Calabi-Yau threefold, X, dual to a heterotic string compactified on K3 x T2 and then take the volume of the 2-torus to infinity thus decompactifying our theory to a heterotic string on a K3 surface. Actually, decompactification is a rather delicate process and perhaps we should be more pragmatic and say that we will consider a heterotic string on K3 x T2 and try to systematically ignore all aspects of the type IIA string on X coming from the T2 part of the heterotic string. Much of the analysis we require we have already covered in section 5.6. There, for a specific example, we did precisely the decompactification we require. We need to consider how general we can make this process. Clearly we should insist that our moduli space of vector multiplets in the four-dimensional theory corresponding to the heterotic string on K3 x T2 contain the space

O(r2,2)\O(2,2)/(O(2) x 0(2)).

(136)

That is, we have the moduli space of the string on T2. If this were not the case, we could not claim that we had really compactified on a true product of K3 x T2. From the analysis of section 5.4 we expect X to be a K3 fibration. Given the appearance of r2,2 in (136) we expect from section 5.5 that the Picard lattice of the generic fibre of X must contain the lattice r1,1. A K3 surface with a Picard lattice containing r1,1 is an elliptic fibration. To

see this, let v and v` be a basis for r1,1. The class v - v' is a primitive element of self-intersection -2 and thus (see, for example, [16]) either this class, or minus this class, corresponds to a rational curve within the K3. Now either v or v` is a nef curve of zero self-intersection. This will be the class of the elliptic fibre. Note that we have a rational curve in the K3 surface intersecting each fibre once - our

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PAUL S. ASPINWALL

elliptic fibration has a global section. What's more, it is easy to see that this section is unique as there is no other (-2)-curve.27 We thus claim that X is a K3 fibration and that each K3 fibre can be written as an elliptic fibration with a unique section. Together these give [149] PROPOSITION 8. If a heterotic string compactified on 10 x T2 is dual to a type

IIA string on X and there are no obstructions in the moduli space to taking the size of T2 to infinity and thereby ignoring the effects associated to it, then X is an elliptic fibration over a complex surface with a section. This proposition is subject to the same conditions as proposition 6 and to the caveat in the footnote above. Let us denote this elliptic fibration asp : X -> 0. 0 itself is a g1 fibration over lP1, 8 -+ W. The simplest possibility is that 0 is the Hirzebruch surface F,,. This need not be the case however - there may be some bad fibres over some points in W. The subject of elliptic fibrations has been studied intensively by algebraic geometers, thanks largely to Kodaira [150]. This can be contrasted to the subject of K3 fibrations, about which relatively little is known.

Let us fix our notation. Let W, the base of X as a K3 fibration, be l1 with homogeneous coordinates [to,t1]. We will also use the affine coordinate t = tl/to. 8 is a P1 fibration over W. Over a generic point in W, the 1P1 fibre will have homogeneous coordinates [so,si] and affine coordinate s = si/so. Now we want to write down the fibre of X as an elliptic fibration over 0. Any elliptic curve may be written as a cubic hypersu face in P2. Writing this in affine coordinates we may put the fibration in "Weierstrass form" for a generic point (s, t) E 0: (137) y2 = xs + a(s,t)x + b(s, t), where x and y are affine coordinates in a patch of P2 and a and b are arbitrary polynomials. If 0 itself has bad fibres - that is, it is not F - then more coordinate

patches need to be introduced to give a global description of the elliptic fibration. Note that not any elliptic fibration can be written in Weierstrass form. Homogenizing the coordinates of P2 putting x = xt /xo and y = x2/xo, we see that [0, 0,1] always lies in (137), which gives the fibration a global section. Thus only elliptic fibrations with a section can be written in this form. Luckily this is the case we are interested in. The j-invariant of the elliptic fibre (137) is 4a3/6, where 6 is the discriminant of (137) given by 6 = 4a3 + 27b2.

(138)

If 6 = 0 then the elliptic fibre is singular. The hypersurface, or divisor, 6(w, z) = 0 in 0 thus gives the locus of bad fibres. It is commonly called the discriminant locus. "Actually we have cheated here. We have assumed that the bundle over K3 may be chosen so that it breaks the Be x Es or Spin(32)/Z2 gauge group. Then the Naraiu moduli space for the T2 really is given by (136). There can be times however, as we shall see soon, when this is not possible and then the Narain moduli space for the 2-torus becomes larger. This allows for more than one section of the elliptic fibration. We claim this is not important for the examples we discuss in these lectures, however, as we may begin in a case where the gauge group is broken by the K3 bundle and then proceed via extremal transitions to the case we desire, preserving the section of the elliptic fibration if X.

K3 SURFACES AND STRING DUALITY

71

I0

(n+5 lines)

I*

n

(n fines)

in

l

II

-<

II*

III

III *

IV --y-

IV

*

FIGURE 9. Classification of elliptic fibres.

There are only so many things that can happen at a generic bad fibre of an elliptic K3 surface and these have been classified (see, for example, [16]). Let us take

a small disc D c C, embedded in ®, with coordinate z which cuts a generic point on the discriminant locus transversely. Let z = 0 be the location of the discriminant locus. Let us analyze the restriction of the fibration of X to the part which is fibred over D, We are thus considering an open set of a fibration of a complex surface. Assuming that the total space of the fibration is smooth and there are no (-1)curves, which will certainly be true if we are talking about a Calabi-Yau manifold, the possibilities for what happens to the fibre when z = 0 is shown in figure 9. The case Io is when there is no zero of b and the elliptic fibre is smooth. In all other cases the lines and curves in figure 9 represent rational curves. Case I1 is a rational curve

with a "double point", i.e., it looks locally like y2 = x3 at one point and case II is a rational curve with a "cusp", i.e., it looks locally like y2 = x3 at one point. All the other cases consist of multiple rational curves. All of the singular fibres should be homologous to the smooth fibre. To achieve this, some of the rational curves in the bad fibre must be counted more than once to obtain the correct homology. The multiplicity of the curves are shown as the small numbers in figure 9. If omitted, multiplicity one is assumed.28

"There is also a possibility that I fibres can appear with multiplicity greater than one. We ignore this as the canonical class of such a fibration cannot be trivial.

PAUL S. ASPINWALL

72

L

K N

>0 >0

Fibre

0

to

T'

0 1

>0

IN

-4N-t

>1

2

1

>2

3

II III

At

2

4

IV

.42

6

I*

IV* III* II*

D4 DN_2 Eo E7

0

>2

>2 >3 2

3

>3

>7 IN-o

4

3

>5

>4

5

8 9 10

Es

TABLE 4. Weierstrass classification of fibres.

The possibilities listed in figure 9 can also be classified according to the Weierstrass form. On our disc, D, the polynomials, a and b, in (137) will be polynomials in z. Let us define the non-negative integers (L, K, N) by a(z) = z' ao(z)

b(z) = zKbo(z)

(139)

b(z) = zNa0(Z),

where ao(z), bo(z), and do(z) are all nonzero at z = 0. The triple (L, K, N) then determines which fibre we have according to table 4. See [321 for an explanation of this. To be precise, the Weierstrass form of the fibration in x, y and z will produce a surface singularity which, when blown-up, will have the fibres in figure 9. This is closely linked to the results in table 1. It is important to note that this classification procedure only applies to a smooth

point on the discriminant locus. Only in this case can we characterize the bad fibre in terms of the family of elliptic curves over our small disc, D. When the discriminant is singular the nature of the bad fibre need not be expressible in terms of the geometry of a complex surface - it will be higher-dimensional in character. In general, for Calabi-Yau threefolds, we should expect to encounter some singular fibres not listed above. Such exotic fibres are important in string theory but we will try to avoid such examples here as it makes the analysis somewhat harder. For our elliptic fibration, p : X -+ e, a knowledge of the explicit Weierstrass form is enough to calculate the canonical class, K,x. This may be done as follows. In homogeneous coordinates, the Weierstrass form is xOx2 = xi + axox, + bxo,

(140)

giving a cubic curve in P2. Now since the elliptic fibration is not trivial, this p2 will vary nontrivially as we move over e. We may describe such a p2 as the projectivization of a sum of three line bundles over e. We are free to declare that xo is a section of a trivial line bundle. We may then find a line bundle, 2°, such and x2 is a section of in order to be compatible with that xl is a section of (140). It also follows that a is a section of 24 and b is a section of 26.

KS SURFACES AND STRING DUALITY

73

Now let us consider the normal bundle of the section, a, given by [xo, xI, x21 = [0, 0, 1], embedded in X. We may use affine coordinates bi = xI /X2 and C2 = xo/x2 whose origin gives the section a. Note that l;I is a section of 2-I and 6 is a section

of 2-s. Near a, (140) becomes 6 = {l .

(141)

Thus, £I is a good coordinate to describe the fibre of the normal bundle of a in X. This implies that this normal bundle is given by 2-I. We may now use the adjunction formula for a C X to give

Kx1, = K, +-V.

(142)

Actually, this is the only contribution towards the canonical class of X. That is,

Kx = P` (Ke + 2).

(143)

For the case we will be interested in, we want Kx = 0 and so .2= -Ke. We will give examples of such constructions later. Note that if there is a divisor within 8 over which a vanishes to order > 4 and b vanishes to order > 6, we may redefine 2to "absorb" this divisor and lower the

degrees of a and b accordingly. This is why no such fibres appear in table 4. As this will change Kx, such occurrences cannot happen in a Calabi-Yau variety. Let us consider a K3 surface written as an elliptic fibration with a section. The Picard number of the K3 is at least two - we have the section and a generic fibre as algebraic curves. If we have any of the fibres I,,, for n > 2, or 4, III, IV, II", III', or IV' we will also have a contribution to the Picard group from reducible fibres. Each of these fibres contains rational curves in the form of a root lattice of a simply-laced group. Let us denote this lattice V. The possibilities are listed in table 4. Thus, shrinking down these rational curves will induce the corresponding gauge group for a type HA string. We know that for a Calabi-Yau manifold compactified on a K3 fibration, the moduli coming from varying the Kiihler form on the K3 fibre map to the T2 part of the heterotic string compactified on K3 x T2. In particular, the act of blowing-up rational curves in the K3 to resolve singularities, and hence break potential gauge groups, is identified with switching on Wilson lines on T2. Thus, to ignore Wilson lines, these rational curves must all be blown down and held at zero area. That is, any of the fibres I,,, for n > 2, or In, III, IV, II", III`, or IV" appearing in the elliptic fibration will produce an enhanced gauge symmetry in the theory. From section 5.6 and, in particular (131), the size of an elliptic fibre within this K3 will be fixed to some constant a120 as a,,6 -+ oo to make the T2 infinite area. Thus this size is "frozen out" as a degree of freedom. To ignore the 2-torus degrees of freedom for the type IIA string compactified on X we should take the K3 fibre within X, consider it as an elliptic fibration with an elliptic fibre of frozen area and blow down any rational curves which may take the Picard number of the K3 fibre beyond 2. In summary, any degrees of freedom coming from sizes within the elliptic fibre structure are ignored.

Consider the base, 8, as a Pi bundle over W. Suppose we have bad fibres in this case. These must correspond to reducible fibres. Now, when we build X as a K3 fibration over W, these reducible fibres in 8 will build reducible fibres in X. This is exactly case 3 listed at the beginning of section 5.5. That is, varying

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PAUL S. ASPINWALL

the size of irreducible parts of these reducible fibres will give moduli in vector supermultiplets which cannot be understood perturbatively on the heterotic side. The important point to note here is that these degrees of freedom will not go away when we unwind any Wilson lines around T2 and take its area to infinity. Thus, these degrees of freedom must be associated to the six-dimensional theory of the heterotic string compactified on a K3. From section 6.1 these moduli must therefore come from tensor multiplets in the six-dimensional theory. It is the peculiar nature of tensor moduli which prevented us from having a perturbative understanding of these moduli in section 5.5. We saw in section 5.6 how the size of 14' and the size of the fibre of 0 were used to produce the area of the heterotic string's T2 and the size of its dilator. The area of the T2 is lost as a degree of freedom in our six-dimensional theory. We see then that the number of tensor multiplets in our theory will be the Picard number of 43 minus one. In the case that A is F,,, that is, there are no bad fibres, the number of tensor multiplets will be one and this single multiplet contains the heterotic, dilaton as a modulus. Questions concerning hypermultiplets between the four-dimensional theory and the six-dimensional theory are unchanged. In particular, we retain the relationship from section 5.3 that the number of hypennultiplets is given by h-,' (X) + 1. Now we have counted massless tensor multiplets and hyypermultiplets, let us count massless vector multiplets. We know that any vector multiplet in the four-

dimensional theory must have its origin in either a vector multiplet or a tensor multiplet in six dimensions. Thus we can count the number of six-dimensional vectors by subtracting hu"l(O) -1 from the number of four-dimensional vectors. ® Many of these vectors can be seen directly in terms of the enhanced nouabelinu gauge symmetry but there is an additional contribution. In section 5.5 we saw how H=(S), where .1 is a K3 fibration, could be built from elements from the base, from the generic fibre and from the bad fibres in a fairly obvious way. Now we want to consider if the Caine thing is true for an elliptic fibration. Analyzing the spectral sequence one can see that H2($, :) contributes, for the base, to give the tensor multiplets and Ho(e, R2p.Z) contributes for the fibres. This latter piece accounts for the enhanced nonahelian symmetry discussed above. The object

of interest will be the term Ht(e,Rlp.rr). In the case of K3 fibration, this is trivial since Hl(K3) = 0. This ceases to be true for an elliptic fibration however. 'T'his contribution may be associated to the group of sections of the bundle as seen in (1511. Later on, in section 6.5, we will consider a case where this is a finite group, but here we note that if this group is infinite, then its rank will contribute to the dimension of H2(X). We am then that if the elliptic fibration has an infinite number of sections, there will be massless vector fields beyond those accounted for from the nonabelian gauge symmetry from

bad fibres. These may contribute extra u(1) terms to the gauge symmetry (1451 and, conceivably, more nonabelian parts. As this part of the gauge group is rather difficult to analyze we will restrict ourselves to examples in these lectures where there is no such contribution.

6.3. Small instantons. Our goal in this section will be to find the map from at least part of the moduli space of hypermultiplets for a type IIA string on a Calabi-Yau manifold to the moduli space of hypermultiplets of the heterotic string

compactified on a K3xT2. As we have seen, the T2 factor is irrelevant for the hypermultiplet moduli space and we are free to consider the latter as a heterotic string on a K3 surface so long as the Calabi-Yau is an elliptic fibration with a section. We will be able to go some way to determining "which" heterotic string a given elliptic threefold is dual to.

KS SURFACES AND STRING DUALITY

76

The best policy when finding a map between two moduli spaces is to start with a particularly special point in the moduli space of one theory with hopefully unique properties which allow it to be identified with a correspondingly special point in the other theory's moduli space. This special point will usually be very symmetric in some sense. The trick, invented in [135] and inspired by the work of (152], is to look for very large gauge symmetries resulting from collapsed instantons in the heterotic string.

Recall that the K3 surface, S, on which the heterotic string is compactified comes equipped with a bundle, E -3 S, with c2 (E) = 24. Fixing S, the bundle E will have moduli. A useful trick when visualizing the moduli space of bundles is to try to flatten out as much of the connection on the bundle as possible. Of course, the fact that c2(E) = 24 makes it impossible to completely flatten out E but we can concentrate the parts of the bundle with significant curvature into small regions isolated from each other. In this picture, an approximate point of view of at least part of the moduli space of the bundle can be viewed as 24 "instantons" localized in small regions over S each of which contributes one to c2(E). As well as its position, each instanton will have a degree of freedom associated to its size - that is, the characteristic length away from the centre of the instanton where the curvature becomes small. At least from a classical point of view, there is nothing to stop one shrinking this length scale down to zero.29 Such a process naturally takes one to the boundary of the moduli space of instantons. Let us consider the Es x Es heterotic string. The observed gauge group of a heterotic string theory compactified on S will be the part of the original Es x Es which is not killed by the holonomy of E. It is the "centralizer" of the embedding of the holonomy in Es x Es - i.e., all the elements of Es x Es which commute with the holonomy. What is the holonomy around a collapsed instanton? In general the global holonomy is generated by contractable loops due to the curvature of the bundle, and from non-contractable loops in the base. The curvature is zero everywhere when the instanton has become a point. Also, if S is smooth we may look at a 3-sphere surrounding the instanton to look for holonomy effects. Since 7rs(S3) = 0 we cannot generate non-contractable loops. Thus, the holonomy of a point-like instanton is trivial.30 One possibility is to shrink all 24 instantons down to zero size. When we do this, E will have no holonomy whatsoever and the resulting heterotic string theory will retain its full Es x Es gauge symmetry. As this is such a big gauge group it is a good place to start analyzing our duality map. We thus want to find a Calabi-Yau space on which we may compactify the type IIA string to give an Es x Es gauge symmetry. That is, we want an elliptic fibration over F such that when viewed as a K3 fibration, the generic K3 fibre has two Es singularities. Let us discuss what this implies about the discriminant locus within

o2 F.

First, let us be a little sloppy with notation and not distinguish between curves and their divisor class (roughly speaking, homology class) in 0. Thus we use Co to 21Although there is no reason to suppose that one may do this independently for all 24 instantons. 3ONote that this need not be the case if S has an orbifold singularity and the instanton sits at this point. Then we can only surround the instanton by a lens space which is not simply connected. One should note that we have conveniently ignored those loops which happen to go through the point where the point-like instanton sits.

PAUL S. ASPINWALL

76

denote the class of base, i.e., the (-n)-curve within O, and f to denote the class of the generic ]t°1 fibre. Let us determine Ke in terms of these classes. Consider the adjunction formula for a curve C E O. Integrating this over C we obtain its Euler characteristic

X(C) = -C.(C+Ke).

(144)

Knowing that Co and f are spheres is enough to determine

KF" _ -2Co - (2 + n) f.

(145)

Let us use the letters A, B, and A to denote the divisors associated to the equations a = 0, b = 0, and 6 = 0 respectively. From (143), the classes of these divisors will be

A=8Co+(8+4n)f B=12Co+(12+6n)f

(146)

A=24Co+(24+12n)f, in order that X be Calabi-Yau. The locus of Es singularities in X will map to curves in O. As these E8's are independent, we want to make these curves disjoint sections of O, that is, one curve will be in the class Co (the isolated zero section of the Hirzebruch surface as a IF' bundle over ]tot) and the other in the class C. = Co + of (a section in the nonisolated class which we view as a section "at infinity" of the Hirzebruch surface). From table 4 we see that we want II" fibres over these curves. These two curves of II* fibres will account for a large portion of the A, B, and A divisors. Let us write A', B', and A' for the remaining parts of the divisors not contained in the curves Co and C. From table 4 we have

A'=A-4(Co)-4(Co+nf)=8f B'=B-5(Co)-5(Co+nf)=2Co+(12+n)f

(147)

tY=4, -10(Co)-10(Co+nf)=4Co+(24+2n)f. This means that what is left of the discriminant, A', will collide with the curve Co a total number of Co.O' = 2(12-n) times and with the curve C a total number of 2(12 + n) times. These 48 points of intersection are not independent. The reason that A' collides with the curves of II" fibres is precisely because B' also collides with these curves. Each time B' hits these curves, the degree of the discriminant will rise by 2 and hence A' hits them twice in the same place. To see exactly what shape this intersection is one may explicitly write out the equations. The result is that A' crosses itself transversely at these points as well as hitting Co or C,,. We

see then that, generically, A' collides with Co at 12 - n points and with C. at 12 + n points. Within A', away from these collisions, we expect the discriminant to behave reasonably nicely.31 The result is shown in the upper part of figure 10. We know how to deal with all of the points on the discriminant from Kodaira's list in figure 9 except for the 24 points of collision of A' with the two lines of II" fibres. Here we have to work harder to obtain a smooth model for X. a1Although it will have cusps.

K3 SURFACES AND STRING DUALITY

77

12+n

I

12- n

12+n S0 0

FIGURE 10. 24 small instantons in the Es x Es string.

Let us focus on one of these points where either Co or C. hits 0' twice within 0. Blow up this point of intersection, it : 0 --> 0. This will introduce a new rational

curve, E, in the blown-up surface 0 such that Kg = 7r"Ke + E. Pulling back .P onto © will show that a now vanishes to degree 4 on E and b vanishes to degree 6

on E. Introduce 2" = 7r'2- E and write a and b in terms of .2' instead of Y. Now a and b will not vanish at all on a generic point on E. Note that the effect on Kx of blowing up the base, 0, and then subtracting E from .2'nicely cancels out and so X is still Calabi-Yau. Thus, to obtain a smooth X we need to blow up all 24 points of collision. This process is shown in the lower part of figure 10. The dotted lines represent the 24 new P"s introduced into the base. Note that they are not part of the discriminant. From our interpretation of moduli in section 6.2 we see that in terms of the underlying six-dimensional field theory we have 24 new moduli from tensor multiplets as soon as we try to enhance the gauge group to Es x Es in this way. Remember that in the language of the heterotic string this large gauge group was meant to be the result of shrinking down 24 instantons to zero size. What we have therefore shown here is that shrinking down 24 instantons to zero size results in the appearance of 24 new tensor moduli.

78

PAUL S. ASPINWALL

Given that the appearance of tensor multiplets should he a nonperturbative phenomenon in the heterotic string, it would seem unreasonable to expect them to appear when the target space and vector bundle is smooth. At least in the case that the underlying K3 surface is large, it is then clear that each tensor modulus can be tied to each shrunken instanton in this picture. That is, one point-like instanton will result in one tensor modulus appearing. Suppose we try to give size to some of the instanton. This should correspond to a deformation of the complex structure of the Calabi-Yau threefold. We know that this should result in the disappearance of (at least) one of the tensor multiplets and thus (at least) one of the blow-ups in O. Thus the deformation has to disturb one of the curves of R" and thus lower the size of the effective gauge group. What is important to notice is that 12 + n of the small instantons are thus embedded in

the Es factor associated to C. and 12 - n of the instantons are embedded in the Es associated to Co. To put this another way, when we smooth everything out to obtain a theory with no extra tensor moduli we expect to have an (Es x Es)-vector bundle which is a sum of two Es-bundles, one of which has c2 = 12 + n and the other with c2 = 12 - n. This pretty well specifies exactly which heterotic string our type IIA string is dual to. For a G-bundle on a K3 surface, where G is semi-simple and simplyconnected, the topological class of this bundle is specified by a map H,1 (K3) -* 7r3(G) (for a clear explanation of this see [153]). This maybe viewed as given by the second

Chern class of each sub-bundle associated to each factor of G. In our case we fix the total second Chern class and so the only freedom remaining is specified by how c2 of the bundle is split between the factors of G. Thus n determines the class of our Es x Es bundle. We have arrived at the following: PROPOSITION 9. Let E be a sum, El ® E.2, of two Es-bundles on a smooth

KS surface such that c2(EI) = 12 + n and c2(E2) = 12 - n. Then a heterotic string compactifted on this bundle on KS is dual to (a limit of) a type IIA string compactiled on a Calabi-Yau threefold which is an elliptic fibration with a section over the Hirsebruch surface F,,. Our main assumption here is that the type IIA string on the Calabi-Yau manifold really is dual to a heterotic string. If it is, then we have certainly identified the correct one subject to the provisos of proposition 6. This proposition first appeared in [135]. Although virtually all of the mathematics above has been copied from that paper our presentation has been slightly different. Rather than take the limit of decompactifying a T2 in a heterotic string on K3 x T2, the line of attack in [135] was effectively to decompactify the dual type IIA string (or its mirror partner, the type IIB string) to a twelve-dimensional theory compactified on a Calabi-Yau manifold. It is not clear whether this twelvedimensional "F-theory" exists in the usual sense of ten-dimensional string theory or eleven-dimensional M-theory or whether it serves simply as useful mnemonic for the above analysis. Another point of view of F-theory is to think of it as the type IIB string compactified down to six dimensions on 0 (see, for example, [154] for

some nice results along these lines). The fact that e is not a Calabi-Yau space is corrected for by placing fixed D-branes within it. Again this is essentially equivalent to the above. The key ingredient to associate to the term "F-theory" is the elliptic

fibration structure. Whether one wishes to think of this in terms of a mysterious

KS SURFACES AND STRING DUALITY

79

twelve-dimensional theory or a type IIA string or a type IIB string is up to the

reader. The above proposition establishing the link between how the 24 of the second Chern class is divided between the two Es's, and over which Hirzebruch surface X is elliptically fibred, is in agreement with all the relevant conjectured dual pairs of [103] and [142] for example. The case of n = 12 was established in [132]. The appearance of tensor moduli for small instantons was first noted in [146].

Since we are not allowing ourselves to appeal to M-theory or D-branes in these lectures we will not reproduce the argument here but just note that all necessary information appears to be contained in the type IIA approach we use here.

6.4. Aspects of the E8 x E8 string. Let us now follow the analysis of [135, 145] and continue to explore the duality between the type IIA string on X and the Es x E8 heterotic string. In the previous section we looked at the case of fixing hypermultiplet moduli in order to break none of the Es x Es gauge group. We should ask the opposite question of what the gauge group is at a generic point in the hypermultiplet moduli space. We shall do this as follows. If any of the E8 x E8 gauge group remains unbroken we expect either of our curves Co or C,. to contain part of the discriminant locus, A. Let us focus on Co. Split off the part of the discriminant not contained in Co by putting

A=NCo+a,

(148)

where N > 0 and A' does not contain Co. Since the only way that the intersection number of two algebraic curves in an algebraic surface can be negative is if one of the curves contains the other, we have A'.Co > 0.

(149)

Following (146) we have, for n > 0, N>12-24.

(150)

n Similarly we may analyze the divisors A and B to obtain the respective orders, L and K, to which a and b vanish on Co. This gives

L>4-8n (151)

K>6_

12

n

We may now use table 4 to determine the fibre over a generic point in Co. Repeating this procedure for the other "primordial" Es along C. shows that no singular fibres are required there for n > 0. We see that in the case n > 2, we will have singular fibres over Co generating a curve of singularities within X. Thus we expect an enhanced gauge group. Loosely

speaking, the gauge group can be read from the last column of table 4. The only thing we have to worry about is the monodromy of section 5.2 - it may be that there is monodromy on the singular fibres as we move about Co. If A'.Co = 0 then there can be no monodromy since the fibre is the same over every point of Co. This occurs for n = 2,3,4,6,8,12. When n = 7,9, 10, 11, the fibre admits no symmetry

m

PAUL S. ASPINWALL

n L K N Fibre <2 0 0 0 10 4

IV

Su(3)

6

Io

50(8)

3 3 3

4 4

W

3

5 5

8 8 9 9 10

2

4 5

2

7 8

>9 4

5

Ho Es

2 3

3

6

0

Mon.

Z2

F4

IV*

E6

111*

ET

III* II*

E7

E6 50(8) G2

su(3) su(2) 5u(2)

Es

TABLE 5. Generic gauge symmetries, G.

and thus there cannot be any monodromy. Therefore, the only time we have to worry about monodromy is for the IV* fibre in the case n = 5. There is indeed monodromy in this case [149]. (See [155] for an account of this in terms of "Tate's

algorithm" or [156] for an alternative approach.) Thus, whereas one associates E6 with a type IV* fibre, this becomes F4 from figure 5 when n = 5. The gauge algebras for generic moduli are summarized in table 5. This agrees nicely with the heterotic picture. For a given value of c2 of a bundle, find the largest possible structure group, H, of a vector bundle. Then the desired gauge group, G, will be the centralizer of H within E8 x Es. One "rough and ready" approach to this question is as follows. Consider an H-bundle E with fibre in an irreducible representation, R, of the structure group. The Dolbeault index theorem on the K3 surface then gives [9]

dimH°(E) - dimH'(E) +dimH2(E) =

r

Jis

td(T) Ach(E)

= 2rank(R) -1(R)c2(E),

(152)

where 1(R) is the index of R using the conventions of [157]. If E really is a strict H-bundle with fibre R, it should have no nonzero global sections, otherwise the structure group would be a strict subgroup of H. Thus H°(E) is trivial. Similarly, by Serre duality, we expect the same for H2(E). Thus, the right hand side of (152) must be non-positive. That is, 2rank(R) c2(E)

1(R)

'

(153)

for any irreducible representation, R. The bundle with c2 = 12 + n may have the full Es as structure group so one Es of the E8 x Es will be broken generically for any n > 0. Table 5 then shows how the other Es is broken down to G by a bundle with structure group Ho and c2 = 12 - n. For example, the 3 of Su(3) has 1(3) = 1 and so c2 of a generic .5u(3)-bundle with no global sections is at least 6. This is why it appears on the row for n = 6 in the table. Note that there need not exist a bundle that saturates the bound in (153) and so we cannot reproduce all of the rows in table 5. For example, in the case n = 3 one sees that (153) does not rule out a bundle with the full Es structure group but we see that only an E6-bundle is expected. See [133] for a discussion of this. It is

K3 SURFACES AND STRING DUALITY

81

interesting to note that proposition 9 implies that su(3) must appear as the gauge symmetry in the case n = 3 and so the Es-bundle must not exist. If a smooth Es-bundle on a K3 surface with c2 = 9 is discovered it will violate proposition 9. The cases n = 9,10,11 are somewhat peculiar since we appear to be suggesting that we have a bundle with trivial structure group and yet c2 > 0. Clearly this is not possible classically. The fact that classical reasoning is breaking down somewhat can be seen from the fact that the reasoning of section 6.3 applies to this case and we have 12 - n tensor multiplets. That is, we have 12 - n point-like instantons which cannot be given nonzero size.

The fact that n < 12 can be understood from both the type HA side and the heterotic side. In the case of our elliptic fibration over F,,, if n > 12 then (L, K, N) as determined from (150) and (151) is at least (4, 6,12). As discussed in section 6.2, this means that we may redefine 2to absorb Co to reduce the fibre to something in the list in figure 9. This kills Kx = 0 however and so we do not have a Calabi-Yau space. On the heterotic side, c2 < 0 would be a clear violation of (153). One might worry about "point-like anti-instantons" but such objects would break supersymmetry and as such do not solve the equations of motion. In section 5.7 we discussed extremal transitions between Calabi-Yau manifolds which, in the heterotic language, corresponded to unwrapping part of the gauge bundle around the K3 surface and rewrapping it around the T2. Such transitions are not of much interest to us in this section as we are concerned only with the K3 part of the story. There are other possible extremal transitions which will effect us though. One kind which is of interest are ones which will take us from an elliptic fibration over P. to another elliptic fibration over In the heterotic string this will correspond to a transition from splitting the Es x Es bundle into two bundles

with c2 equal to 12 + n and 12 - n, to a splitting of 12 + n - i and 12 - n +

1

respectively. In this way we may "join up" all the theories considered so far into one connected moduli space. This phenomenon was first observed in [146] but we will again follow the argument as presented in [135, 145]. As explained in section 6.3, when the Es x Es heterotic string has a point-like instanton, we expect the dual Calabi-Yau space for the type 11A string to admit a blow-up in the base, ®, of the elliptic fibration. When all 24 instantons are pointlike we saw this as a collision between a curve of H* fibres and other parts of the discriminant locus as shown in figure 10. Let us concentrate on what happens when one of these points is blown up. As in section 6.3, we use A' to denote the part of discriminant left over after we subtract the contribution from the two curves of 11* fibres. In the first diagram in figure 11 we show locally how A' loops around a collision between it and a line, C,,, of 11* fibres together with the class, f, that passes though this point. Now when we do the blow-up by switching on a scalar in a tensor multiplet we go to the middle diagram. The exceptional divisor is a line of self-intersection -1. The line that was in the class f also becomes a (-1)-line. The middle diagram of figure 11 is obviously symmetric and we may blow-down the latter (-1)-curve to push the loop onto the bottom line of 11* fibres as shown in the last diagram. The effect of this is to change Co ,into a line of self-intersection n - 1 and Co into a line of self-intersection -n + 1. Thus we have turned F into F,,-1. It is also clear that we have moved the small instanton from one of the Es's into the other

PAUL S. ASPINWALL

82

(- n)

(- II)

Fn

(- I,+I) FI

- I

FIGURE 11. The transition n.. + a - 1.

Es. Thus we may connect up all our theories which are elliptic fibratiorrs over F,,. Note that the use of tensor moduli means that we do not expect a perturbative interpretation of this process in the heterotic string language.

6.5. The Spin(32)/Z2 heterotic string. Now that we call coutvnt ourselves with the knowledge that we know how to build a type IIA dual to the generic E8 x E8 heterotic string on a K3 surface we turn our attention to the Spin(32)/Z2 heterotic string. Whereas the topology of the E8 x E8 bundle required specifying how the 24 instantons were divided between the two E8's, the Spin(32)/Z2 heterotic string is quite different. In this case the gauge group is not simply-connected and the topology of the bundle is not simply specified by c2.

It will be important to recall some fundamentals of the construction of the Spin(32)/Z2 heterotic string from [158]. The 16 extra right-movers of the het:erotic string are compactified on an even self-dual lattice of definite signature. There are two such lattices, which we denote r8 a r8 and rr6. The former is two copies of the root lattice of E8 x Es with which we have become well-au:quainte.d in these talks. The second lattice is the "Barnes-Wall" lattice [15]. This may be constructed by supplementing the root lattice of 50(32) by the weights of one of its spinors. Such spinor weights are never of length squared 2 are so do not give rassless states. Thus,

as far as massless states are concerned, the lattice is the root lattice of SO(32) and the string states fill out the adjoint representation. Massive representations may fill out spinor representations for one of the spinors and but we never have representation in the vector representation or the other spinor representation. The gauge group can be viewed as a Z2 quotient of Spin(32) which does not admit vector representations. Hence 7rr of our gauge group is Z2. When we try to build Spin(32)/Z2-bundles the situation is very similar to real vector bundles over M where the fact that irr(SO(d)) c Z2 leads to the notion of the second "StiefelWhitney" class, w2, of a bundle as an element of H2(M, Z2). Here we have a similar object characterizing the topology of the bundle which we denote w2. If w2 34 0 then bundles with fibre in the vector representation are obstructed just as spinor representations are obstructed for non-spin bundles with w2 # 0. See [159] for a detailed account of this. Thus, rather than being classified by how 24 is split between second Chern class, which was the case for the E8 x E8 string, the topological class of an Spin(32)/Z2

K3 SURFACES AND STRING DUALITY

83

heterotic string compactified over a K3 surface is characterized by w2 E H2(S, Z2). Whereas elements 62 are in one-to-one correspondence with the homotopy classes of Spin(32)/Z2-bundles on a fixed (marked) K3 surface, in our moduli space we are also allowed to vary the moduli of the K3 surface. Thus two elements of H2(S,Z2)

should be considered to be equivalent if they can be mapped to each other by a diffeomorphism of the K3 surface. That is, w2 E

r3,19/2P3,19 o+(r3,19)

(154)

It was shown in [159] that there are only three possibilities: 1. w2 = 0, 2. 102 96 0 and 4112.4112 = 0 (mod 4),

3. 62 0 0 and 4u2.4ii2 = 2 (mod 4). We will focus on the case t 2 = 0 in this section. This will allow us to use the same arguments as in the last section about shrinking instantons down to retrieve the entire primordial gauge group. If w2 were not zero, the topology of the bundle would obstruct the existence of arbitrary point-like instantons at smooth points in the K3 surface. See [159] for an account of this.

We have seen already that if a perturbatively understood heterotic string is dual to a limit of a type HA string on Calabi-Yau threefold, X, then X must be an elliptic fibration with a section over F,,, where 0 < n < 12. Thus, if our duality picture is going to continue working for any Spin(32)/Z2 heterotic string then we must have already encountered it in disguise as the Es x Es heterotic string for a particular n. The statement that the Spin(32)/Z2 heterotic string compactified on a K3 surface is the same thing as an Es x Es heterotic string compactified on a K3 surface should not come as a surprise as the same thing has been known to be true for toroidal compactifications for some time [61, 160]. The identification of the dilaton as the size of the base of X as a K3 fibration has nothing to do with whether we deal with the E8 x Es or the Spin(32)/Z2 string and so the duality between these theories cannot effect the string coupling. Thus there must be some T-duality statement that connects these two theories. This may be highly nontrivial however, as it may mix up the notion of what constitutes the base of the bundle over K3 and what constitutes the fibre. A construction of such a T-duality at a special point in moduli space was given in [159]. Now let us return to the issue concerning the extra states in the r16 lattice not

contained in the root lattice of 50(32). What is the dual analogue of these extra massive states coming from the spinor of 50(32)? Let us think in terms of the type IIA string compactified on a K3 surface versus the Spin(32)/Z2 heterotic string compactified on a 4-torus as in section 4.3. We may switch off all the Wilson lines of the heterotic string compactification by rotating the space-like 4-plane, 11, so that r4,20f1II1 o r16. Now view this as the "fibre" of the type HA string compactified on X versus the heterotic string compactified on K3 x T2. This restores the full 50 (32) gauge symmetry and so can be thought of as shrinking down all 24 instantons. As discussed in section 5.5 we may now vary the T2 and the Wilson lines by varying

the 2-plane, U, in T oz lit = lR24, where T is the quantum Picard lattice of the generic fibre of X as a K3 fibration. Thus we see that T Lw r2,2 ® r16. In other

84

PAUL S. ASPINWALL

words, the Picard lattice of the generic KS fibre of X is rl,, Ha r,s

r1,17 and is therefore self-dual. Let the limit of a type IIA string compactified on Xo be dual to the Spin(32)/Za

heterotic string compactified on a K3 surface (times a 2-torus of large area) with W2 = 0 and all the instantons shrunk down and let X be the blow-up of X0. This theory will have an so(32) gauge symmetry. We know the following: 1. X is a K3 fibration and an elliptic fibration with a section over a Hirzebruch surface. 2. Xo contains a curve of singularities of type Die,. 3. The generic K3 fibre of X has a self-dual Picard lattice (of rank 18). Let us construct X. Table 4 tells is that the base, 6, of X as an elliptic fibration will contain a curve of Ii2 fibres. We may put this curve along Co, the isolated section of F,,. A generic fibre of X as a K3 fibration will be a K3 surface built as an elliptic fibration with a I12 fibre (and, generically, 6 Ir fibres). Let us denote this K3 surface as S. What is Pic(St)? Let a denote the section of St as an elliptic fibration "at infinity" guaranteed by the Weierstrass form. Let R be the sublattice of Pic(St) generated by the irreducible curves within the fibres not intersecting a. Let ' be the set of sections. One may show [161]

disc(R) = I4'I2 disc(Pic(Ss)),

(155)

where disc denotes the "discriminant" of a lattice, i.e., the determinant of the inner product on the generators.

In our case, R is generated by a set of rational curves forming the Dynkin diagram for D16. Thus, disc(R) is the determinant of the Cartan matrix of D16 which is 4. In order for Pic(St) to be unimodular we see that we require exactly two sections. Writing an elliptic fibration with two sections in Weierstrass form is easy. We are guaranteed one section at infinity. Put the other section along (y = 0, x = p(s, t)). Thus the general Weierstrass form with two sections is Y2 = (x - p(s,t))(x2 + As, Ox + q(s,t)),

(156)

where a(s, t) = q(s, t) - p(s,t)2

(157) b(s, t) = -p(s, t)q(s, t).

From (146) the divisors P and Q, given by the zeros of p and q, are in the class

4Co + (4 + 2n) f and 8Co + (8 + 4n) f respectively. The discriminant is S = 4a3 + 27b2 (158)

= (q + 2p2)2(4q - p2).

The fact that the discriminant factorizes will have some profound consequences. In terms of divisor classes let us write A = 2M1 + M2, where M1 is the divisor given by q + 2p2 and M2 corresponds to 4q - p2. We know that 2M1 +M2 contains 18Co from the I12 fibres. Using the fact that a and b vanish to an order no greater than 2 and 3 respectively along Co and the

K3 SURFACES AND STRING DUALITY

85

fact that M1 and M2 must contain a nonnegative number of Co and f, one may show that Mi contains 8Co and M2 contains 2Co as their contributions towards the I12 fibres. What remains is

Mi=(8+4n)f (159)

M2=6Co+(8+4n)f. Putting M2'.Co > 0 fixes n < 4. There are no other constraints. We have not yet achieved our goal in determining what n is for the Spin(32)/Z2 heterotic string. All we know is that n < 4. Note however that along the 8 + 4n zeros of Ml we have a double zero of A. Thus, generically we have 8 + 4n parallel lines along the f direction of 12 fibres. Thus means the gauge algebra acquires an extra 5u(2)8+4n factor. The necessary appearance of an extra gauge symmetry can ultimately be traced to the way the determinant factorized. We will denote this as sp(1)8+4n to fit in with later analysis. It is very striking how different this behaviour is from the Es x Es analysis of the last section. In the latter case we acquired 24 massless tensors when the instantons were shrunk down to zero size, whereas for the Spin(32)/Z2 heterotic string we acquire new massless gauge fields. The natural thing to do would be to identify each sp(1) with a small instanton. To do this fixes n = 4 as one can have 24 small instantons, at least in the case that the underlying K3 surface is smooth and tit = 0. Thus we arrive at the following PROPOSITION 10. The Spin(32)/Z2 heterotic string compactifled smoothly on a

K3 surface with W2 = 0 is dual to (a limit of) the type IIA string compactified on an elliptic fibration over F4. This proposition depends on the assumptions governing proposition 6 and the somewhat schematic way we associated each of the sp (1) factors to a small instanton

to shown=4. Another way [135] to argue n = 4 is that the generic point in moduli space of n = 4 theories has gauge symmetry 50(8) and then compare to the analysis in section 6.4. We can try to break as much of 5o(32) as possible by finding the largest subalgebra of so(32) consistent with (153). Since 02 = 0 we may consider vector representations in which case 50(24) is the largest such algebra with c2 = 24 and this breaks 50(32) down to 50(8). A glance at table 5 then confirms that n = 4.

Actually, the analysis of the Spin(32)/Z2 heterotic string was first done by Witten [152], where the appearance of the sp(1) factors from small instantons was argued directly. Unfortunately this involves methods beyond the scope of these talks. It is worth contrasting Witten's method with the above. It is a remarkable achievement of string duality that the same answer results. We may consider a specialization of the moduli to bring together some of the components of Mi along f. Suppose we bring k of the 24 lines together. This will produce an Ilk line of fibres. As usual we have to worry about monodromy to get the full gauge group. In this case the transverse collision between the Ilk line and the I12 line produces monodromy acting on the Ilk fibres to give an sp(k) gauge symmetry.32 This corresponds to k of the 24 point-like instantons coalescing [152]. 82Actually, since the f curves are topologically spheres, there had better be more than one point within them around which there is nontrivial monodromy. There are three other points within each f curve where there is monodromy provided by non-transverse collisions of f with what is

PAUL S. ASPINWALL

86

I2

2

12

I`

1

1

1

6

2

I1

I,

CO

CO

l i

f f f f

f

f

FIGURE 12. Three 50(32) point-like instantons coalescing to form 5)7(3). We show an example of this in figure 12. As an extreme case, all 24 instantons may be pushed to the same point. This results in a gauge algebra of so (32) 0) SP (24).

As we said above, it is interesting to contrast the behaviour of the point-like E8 instantons with the point-like 50(32) instantons. The former produced massless tensor multiplets whereas the latter produce extra massless vector multiplets to enhance the gauge group. Of course, since both theories are meant, to live in the

same moduli space for n = 4, it should be possible to continuously deform one type of point-like instanton into another. This may well require a deformation of the base K3 surface as well as the bundle. The T-duality analysis of [159] should provide a good starting point for such a description.

6.6. Discussion of the heterotic string. We have analyzed both the Es x Es and Spin(32)/Z2 heterotic string compactifred on a K3 surface in terms of a dual type IIA string compactified on an elliptically fibred Calabi-Yau threefold. In both cases we discovered curious nonperturbative behaviour associated to point-like instantons. It is worth noting that in the case of the Spin(32)/Z2 string we appear to have acquired a free lunch concerning the analysis of the gauge group. To see this note the following. In old perturbative string theory a type IIA string compactified on a Calabi-Yau manifold never has a nonabelian gauge group. Similarly a heterotic

string may have a subgroup of the original Es x E8 or Spin(32)/Z..i as its gauge group together with perhaps a little more from massless strings. This latter can only have rank up to a certain value limited by the central charge of the corresponding conformal field theory (see, for example, [108] for a discussion of this). In the case of point-like instantons we have claimed above to have found gauge algebras as large

as 50(32) ® 57(24). Such a gauge group must be nonperturbative from both the type IIA and heterotic point of view. When discussing duality one would normally claim that its power lies in its ability to relate nonperturbative aspects of one theory to perturbative aspects of another. This then allows the nonperturbative quantities to be calculated. In the above we appear to have discovered things about a subject that was nonperturbative in both theories at the same time! How did we do this? The answer is that we first left of the discriminant. We show this on one of the f curves in figure 12. We omit this on the other f curves to make the figure more readable.

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related perturbative nonabelian gauge groups in the heterotic string to curves of orbifold singularities in the type IIA picture. To do this the curves of singularities were always formed by each K3 fibre of X, as a K3 fibration, acquiring a singular point. There is no reason why a curve of singularities need be in this form - it may be completely contained in a bad fibre and not seen by the generic fibre at all. Why should the type HA string intrinsically care about the fibration structure? Assuming it does not, this latter kind of curve should have just as much right to produce an enhanced gauge group as those we could understand perturbatively from the heterotic string. This is exactly the type of curve that gives the 5p(24) gauge symmetry which is now nonperturbative in terms of the heterotic string. The example of heterotic-heterotic duality studied in section 5.6 gives a clear picture of how nonperturbative effects are viewed in this way. Begin with a perturbative gauge group in the heterotic string. This corresponds to a curve of singularities in the type IIA's K3 fibration passing through each generic fibre. Heterotic-heterotic duality corresponds to exchanging the two PI's in the base of the Calabi-Yau viewed as an elliptic fibration. This gives another K3 fibration but now the curve of singularities lies totally within a bad fibre. Thus it has become a nonperturbative gauge group for the dual heterotic string.

An issue which is very interesting but we do not have time to discuss here concerns the appearance of extra massless hypermultiplets at special points in the moduli space. This question appears to be rather straight forward in the case of hypermultiplets in the adjoint representation, as shown in [162]. The more difficult question of other representations has been analyzed in [96, 145, 149, 155, 163]. It is essential to do this analysis to complete the picture of possible phase transitions in terms of Higgs transitions in the heterotic string. As usual, in discussions about duality, it also hints at previously unsuspected relationships in algebraic geometry. Another very important issue we have not mentioned so far concerns anomalies. The heterotic string compactified on a K3 surface produces a chiral theory with potential anomalies. This puts constraints on the numbers of allowed massless supermultiplets (see [164] for a discussion of this). One constraint may be reduced to the condition [165, 166, 167] 2?3 - 29nT - nH + ny = 0,

(160)

If and nv count the number of massless tensors, hypermultiplets, and vectors respectively. The reason we have been able to ignore this seemingly important constraint is that, assuming one does the geometry of elliptic fibrations correctly, it always appears to be obeyed. At present this appears to be another where nT,

string miracle! lM As an example consider the following. Compactify the Es x Es heterotic string, with c2 split as 12 + n and 12 - n between the two Es factors, on a K3 surface so that the unbroken gauge

symmetry is precisely Es. This means that the corresponding elliptic threefold, X, has a curve of II` fibres which we may assume lies along Co in the Hirzebruch surface P,,. We will

calculate nT, ng, and nv. What is left of the divisors A, B, and A after subtracting the contribution from the curve of II' fibres is given by

A' =4Co+(8+4n)f B' = 7Co + (12 + 6n)f

A' = 14Co+(24+12n)f.

(161)

88

PAUL S. ASPINWALL

B' collides with Co generically 12 - n times. Each of these points corresponds to a point-like instanton required to produce the Ea gauge group. Each such point must be blown up within the base to produce a Calabi-Yau threefold. Thus we have 12 - n nassless tensor rnultiplets in addition to the one from the six-dimensional dilatou. Since the gauge group is Es, we have 248 massless vectors furnishing the adjoint representation. To count the massless hypermultiplets we require h2"1(X). It is relatively simple to compute h1,1 (X) (assuming there is a finite number of sections) as this is given by 2, from

the Hirzebruch surface, plus 12 - n from the blow-ups within the base, plus 1 from the generic fibre, plus 8 from II* fibres. That is, hl-'(X) = 23 - it. Now h2,' may be determined

from the Enter characteristic of X, X(X) = 2(h3"1 - h2"1). To find this, recall that the Euler characteristic of a smooth bundle is given by the product of the Euler characteristic of the base multiplied by the Enter characteristic of the fibre. Thanks to the nice way Euler characteristics behave under surgery, we may thus apply this rule to each part of fibration separately. The Baler characteristic of any fibre is given by N in table .1. Over most of the base, the fibre is a smooth elliptic curve which has Euler characteristic zero. Thus only the degenerate fibres contribute to our calculation. We have a curve of It* fibres over Co which contributes 10 x 2. The rest of the contribution conies from A'. Let its determine the geometry of 0'. Firstly we know that, prior to blowing up the base, A' has 12 - n double points as seen in the upper part of figure 10. Secondly, whenever A' and B' collide, which happens at A'.B' = 24n+ 104 points, A' will have a cusp (assuming everything is generic). Natively, the Euler characteristic of 0' can be given by the adjun

-0'.(A' + K) _ -596 - 130n.

(162)

However, each cusp will increase this by value by 2 and each double point by 1. In addition, when we do the blow-up of the base, the double points will be resolved and an additional 1 must be added for each double point. Thus X(0') = -596 - 130n + 2(24n + 104) + 2(12 - n) (163)

= -364 - 84n.

Over most of the points in A' the fibre will be type h but there will be type II fibres over each cusp. Thus the Euler characteristic of X is given by X(X) = 10.2 + 1.(-364 - 84n - (24n + 104)) + 2.(24n + 104) (164)

= -240 - 60n, and so h2,1(X) = 143 + 29n. To obtain the number of hypermultiplets we add one to this figure from the four-dimensional dilation in the type IIA string. In summary, we have

rrr=13-n ny =144 + 29n

(165)

ny = 248.

It can be seen that this satisfies (160). See [164, 145] for more examples.

In section 6.5 we analyzed the Spin(32)/Z2 heterotic string compactified on a K3 surface with w2 = 0. What about the case w2 $ 0? An example of this was studied in [159] based on the construction of Gimon and Polchinski [168]. This uses open string models which we have not discussed here. In this case, tut is dual to

E Ci, where {Ct} are sixteen disjoint (-2)-curves. The fact that such a class z in H2(S, Z) can be seen in (16]. Such a tut is conjectured to be dual to the case lies of the Es x Es heterotic string with n = 0. It would be interesting to analyze this from the elliptic fibration point if view presented here. One indication that things will work is that the algebra Ep(8) appears naturally in the nonperturbative group in the n = 0 case from the analysis following (159). This agrees with the model

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of Gimon and Polchinski. The problem is that we have the full so(32) present as a gauge symmetry which does not appear in Gimon and Polchinski's model. One can also show that massless tensor multiplets appear from the type IIA approach. These issues are resolved in [156]. An interesting issue we had no time to pursue is that concerning what happens as we take the coupling of the heterotic string to be strong. All of the above analysis was done for a weakly-coupled heterotic string. It has been noted that one can expect some kind of phase transition to occur as the coupling reaches a particular value [133]. This can be analyzed in the context of elliptic fibrations [145].

As concluding remarks to the discussion concerning the heterotic string on a K3 surface we should note that this analysis is far from complete. In the case of the type HA and type IIB string we were able to give a fairly complete picture of the entire moduli space. For the heterotic string we have been able to study a few components of the moduli space and a few points at extremal transitions. A first requirement to study the full moduli space will be a classification of elliptic fibrations. A more

immediate shortcoming in our discussion is that we do not yet have an explicit map between the moduli of the elliptically fibered Calabi-Yau manifold and the K3 surface and bundle on which the heterotic string is compactified. This must be understood before we can really answer questions such as what happens when point-like instantons collide with singularities of the K3 surface.

Acknowledgements I would like to thank my collaborators M. Gross and D. Morrison for their contribution to much of the work discussed above. I would also like to thank them for answering a huge number of questions concerning algebraic geometry many of which I should have looked up in a book first. I thank J. Louis, with whom I have also collaborated on some of the work covered above. It is a pleasure to thank 0. Aharony, S. Kachru, E. Silverstein, and H. Tye for useful conversations and of course B. Greene and K. T. Mahanthappa for organizing TASI 96. The author was supported by an NSF grant while at Cornell University, where part of these lecture notes were written, and is supported in part by DOE grant DE-FG02-96ER40959 at Rutgers University.

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Surveys in Differential Geometry, vol. 6

Symplectic Geometry and the Verlinde formulas Jean-Michel Bismut and FranCois Labourie ABSTRACT. The purpose of this paper is to give a proof of the Verlinde formu-

las by applying the Riemann-Roch-Kawasaki theorem to the moduli space of flat G-bundles on a Riemann surface E with marked points, when G is a connected simply connected compact Lie group G. Conditions are given for the moduli space to be an orbifold, and the strata are described as moduli spaces for semisimple centralizers in G. The contribution of the strata are evaluated using the formulas of Witten for the symplectic volume, methods of symplectic geometry, including formulas of Witten-Jeffrey-Kirwan, and residue formulas. Our paper extends prior work by Szenes on SU(3) and Jeffrey-Kirwan for SU(n) to general groups G.

CONTENTS

Introduction 98 1. Simple Lie groups and their centralizers 103 2. Fourier analysis on the centralizers of semisimple Lie groups 126 3. Symplectic manifolds and moment maps 153 4. The affine space of connections 170 194 5. The moduli space of flat bundles on a Riemann surface 6. The Riemann-Roch-Kawasaki formula on the moduli space of flat bundles 248 7. Residues and the Verlinde formula 277 8. The Verlinde formulas 302 308

References

Supported by Institut Universitaire de France (I.U.F.).

Supported by Institut Universitaire de France (I.U.F.). 01999 International Press

07

m

JEAN-MICHEL BISMUT AND FRANQ01S LABOUIVE

Introduction The Verlinde formula [62], [3] computes the dimension of spaces of holomorphic

sections of canonical line bundles over the moduli space A4 of seruistahle GC_ bundles on a Riemann surface E with marked points, with C, a connected and simply connected compact Lie group. For a given "level" p, the Verlinde formula is a sum over the finite collection of weights parametrizing the representations of the central extension of the loop group LG at level p. This formula, discovered by Verlinde in the context of quantum field theory, has received a number of rigorous proofs first for G = SU(2) by Thaddeus [57], Bertram and Szenes [7] and Szenes (52] (see also Donaldson [17] and Jeffrey-Weitsman [31] for related questions), and for more general groups by Tsuchiya-Ueno-Yarnada [59], Beauville-Laszlo [4] (for G = SU(n)), Faltings [20], Kumar, Narasimhan, Ramanathan [38] for general groups G. A common feature of many of these proofs is that establishing the fusion rules for the Verlinde numbers is an essential step in the proof, a second step being the description of the fusion algebra. The purpose of this paper is to give a proof of the Verlinde formula for connected

simply connected compact Lie groups, by methods of symplectic geometry. This program has been already carried out by Szenes [53] for SU(3), and Jeffrey-Kirwan [30] in the case of SU(n). The main point of the paper is to extend their approach a to general groups. More precisely, we will obtain the Verlinde formula by an application of Riemann-Roch. The theorem of Narasimhan-Sheshadri [46] asserts that 1 can be identified with the set of representations of al(E) with values in G, with given conjugacy classes of holonomies at the marked points. For generic choices of holouornies, this last space is a symplectic orbifold (M/G, w), which carries an orbifold Herrnitian line bundle with connection (AP, V-0), such that cr(A', V") = W. In particular

the orbifold M/G is complex. This orbifold carries a canonical Dirac operator Dp,+, unique up to homotopy. Its index Ind(Dp,+) is the Euler characteristic of Y. We will compute the index Ind(Dp,+) using the formula of Riemann-Roch-Kawasaki [32, 33]. We show that, for p large, it is given by the Verlinde formula. The fact that p has to be large may be related to the fact that a priori, higher cohomology may well not vanish for small p. In fact, the Verlinde formula computes dirn H°(M, A"), while Ind(Dp,+) is the corresponding Enter characteristic (we will come back to

this point at the end of the introduction in connection with results by Teleman [55, 56]). For non generic holonomies, we also show that small perturbations of the holonomies still produce an orbifold moduli space. For suitable perturbations, we show that for any p, the index of the corresponding Dirac operator Dp,+ is still given by the Verlinde formula. The typical case where such a perturbation is needed is when E does not have marked points. Our proof contains various interrelated steps. A first step is the description of the strata of the orbifold moduli space M/G. These strata are in fact moduli spaces for the semisimple centralizers in G. Up to oonjugacy, there is only a finite family of such centralizers. In general,

they are non simply connected. The strata split into a union of substrata indexed by the fundamental group of the centralizers. The description of the geometry of M/G involves results contained in Sections 1, 4, 5, 6. Observe

that if G = SU(n), there are no non trivial semisimole centralizers, which

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99

explains the smoothness of the moduli space M/G (this case which was already considered by Jeffrey-Kirwan [30]). A second step consists in reproving Witten's formula [63] for the symplectic volume of the moduli spaces. Another step is the detailed construction of the orbifold line bundle A". Also we have to compute the action of the finite stabilizers of elements of M on AP. This is done in Sections 4 and 5. In Section 5, we show that the formalism of the moment map can be applied to each stratum of the moduli space. In Section 6, we use a formula of Witten [64] , Jeffrey-Kirwan [28] (see Vergne [61]) and Liu [39, 40] to express the

contribution of each stratum as the action of a differential operator on a locally polynomial function on a maximal torus T. This locally polynomial function is just the symplectic volume of a deformation of the moduli space MIG. Our treatment of these formulas is very close in spirit to Liu [39, 40]. Witten's formula [63] for the deformed symplectic volume of each stratum is a Fourier series on T. In order to calculate the contribution of each stratum explicitly, it is of critical importance to express Witten's formula using residues techniques, which will make obvious the fact that the given Fourier series is indeed a local polynomial on T. These residue techniques are developed in Section 2. In Section 7, we give a residue formula for the index of the Dirac operator on M/G, by putting together the contribution of all strata. Another step is to express the Verlinde sums as residues. This step, which is carried out of Section 7, has many formal similarities with what is done in Section 2 for the Fourier series on T. In Section 8, a comparison of the results of Sections 6 and 7 leads us to our main result.

We now review our techniques in more detail.

1. The residue techniques Residue techniques play an important role in the whole paper, in order to convert the Witten Fourier series [63] for the symplectic volumes into expressions which make them local polynomials in an "obvious" way, so that differential operators can be applied to these polynomials. Similar residue techniques are also applied to the Verlinde sums.

Szenes [53, 54] initiated the use of residue techniques to treat Verlinde formulas. In [53], Szenes applied such residue techniques to the case of SU(3) and obtained the corresponding Verlinde formulas. In [54], Szenes developed a cohomological approach in terms of arrangement of hyperplanes to treat the Witten sums for any group G.

In [30], Jeffrey and Kirwan gave a proof of the Verlinde formulas for SU(n) using residue techniques from a point of view which is very different from ours. In fact they use non abelian localization formulas [64], [28] applied to an extended moduli space. In the spirit of the formula of Duistermaat-Fleckman [19], they use residue techniques to evaluate the Fourier transform of the contribution of the fixed points. In particular they reobtain Witten's formulas [63] for the symplectic volume of the moduli space of G = SU(n).

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The strategy used in the present paper goes in some sense in an opposite direction. First, as in Liu [39, 40], our computations are local, and only use the action of differential operators on the symplectic volumes of the uoduli spaces. Then, we express the symplectic volumes as residues in order to evaluate explicitly the action of certain differential operators on the syrplectic volumes . In the present paper, multidimensional residues are used in a rather "naive" way. The Witten sums are expressed as sums over a lattice identified to Z. We compute the given sums by summing in succession in the variables k1, ... , k, E Z, and by applying standard residue techniques to these one dimensional suns. Handling the recursion requires the development of a trivial, but heavy linear algebra. We believe that Szenes's techniques [54] can put put to fruitful use to give a more conceptual approach to this part of our work.

2. A combinatorial description of the moduli spaces Let 01,... 0Os be s adjoint orbits in G. Put X = G29 x ]1; Oj. ¢:X-+ G be given by 9

(0.1)

Let

K

Sb(u1,vi,... ,u9,ve,w1,... ,w,,) _ 11 ['uz,v;] H wj. i=1

j=1

Put M = ¢'1(1). Under a genericity assumption on the Oj,1 < j < s , the condition (A) of Definition 5.17, which requires that s > 1, in Theorem 5.18, we show that M is a smooth manifold on which G acts locally freely. The moduli space is the orbifold M/G. To prove Witten's formula [63], we show in Theorem 5.45 that the image by of the Haar measure on X has a density with respect to the Haar measure on G, which is essentially given by the symplectic volume of the quotient fibres of 0, which are themselves moduli spaces with an extra marked point. This approach was initiated by Liu [39, 40], who showed in particular that the differential of 0 can be expressed in terms of the combinatorial complexes which compute the cohomology of the flat adjoint vector bundle E. Liu then obtains the intersection numbers of the moduli spaces by applying certain differential operators to the symplectic volumes. Inspired by Witten [63, 64], Liu gave a special role to the heat kernel on the group G to establish Witten's formula, while in our approach, we do not use any heat kernel. Needless to say, the heat kernel on G remains crucial in understanding connections with 2-dimensional Yang-Mills theory, and also with Witten's non Abelian localization [64]. 3.Moment maps and the quantization conjecture In Section 5.10, under genericity assumptions, we show that a G-invariant neighborhood X of M in X can be equipped with a symplectic form, that G acts on X with a moment map, and that the standard symplectic structure on the quotients on the fibres of ¢ included in X come from the symplectic structure on X. We can then use directly the formulas of Witten [64] and Jeffrey-Kirwan [28] to express the integrals of certain characteristic classes in terms of differential operators acting on the symplectic volume of the fibres. When the genericity assumptions are not verified, we replace the given moduli space by a generic perturbation, which still carries a Dirac operator to which the

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Riemann-Roch-Kawasaki theorem [32, 33] can be applied. We then show that the above index results still hold. We will now put our results in perspective from the point of view of geometric quantization, especially in connection with the Guillemin-Sternberg conjecture [23]. By Atiyah-Bott [2, Section 9], we know that when there is one marked point with central holonomy, M/G is a symplectic reduction of the affine space A of Gconnections with respect to the action of the gauge group EG, which acts on A with

a moment map p, which is the curvature, so that M/G = l-1(0)/EG. Similarly the line bundle A' is itself the reduction of a universal line bundle LP on A. This theory can be extended to the case with marked points (this we do in part in Sections 4 and 5). If A was instead a compact manifold, and EG a compact connected Lie group, the Guillemin-Sternberg conjecture [23] asserts that the Riemann-Roch number of (M/G, AP) is equal to the multiplicity of the trivial representation in the action of EG on the cohomology of LP. The Guillemin-Sternberg conjecture has been proved in various stages, the most general result being given by Meinrenken [41],[42] (for a more analytic proof, see Tian and Zhang [58]). In Meinrenken's formalism, when the considered group does not act locally freely on 1A-'(0) , one replaces 0 by any regular value of p close to 0 , and one still gets a corresponding version of Guillemin-Sternberg's conjecture. Chang [16] initiated the study of Verlinde formulas in the context of geometric quantization. A new twist was introduced to the story of the proof of the Verlinde formula in work by Meinrenken and Woodward [43], [44], Alekseev, Malkin, Meinrenken [1], and later work by these authors. In [43, 44], Meinrenken and Woodward gave a symplectic proof of the fusion rules for the Verlinde numbers. In [1], the authors develop a theory of group actions with moment maps taking their values in the given Lie group G. This theory is in fact a theory of the standard moment map for an action of a central extension of the loop group LG. The space X is the prototype of such a manifold, the moment map being just 0. These authors then develop a localization formula in equivariant cohomology, which is an analogue of the formula of Duistermaat-Heckman [19], Berline-Vergne [5]. By using the theory of symplectic cuts and the previous results by Meinrenken [41, 42] on the Guillemin-Sternberg conjecture, they announce a proof of the Verlinde formulas. In some way, our paper represents a direct attempt to prove Verlinde formulas directly, by a method which resembles the proof given by Jeffrey-Kirwan [29] of the Guillemin-Sternberg conjecture. The proof of [29] consists in extracting the Riemann-Roch number of p'1(0)/EG from the Lefschetz formulas. In [55, 56], Teleman gave a proof of the vanishing of the higher cohomology groups of \P for a small perturbation of the moduli space M/G. As a consequence, one should always have Ind(DP,+) = dimHO(.M, AP). As explained before, we only prove such an equality for large p, and otherwise, we have to perturb the moduli space by a perturbation which can be `large' for small p, so that there could be a wall-crossing discrepancy between Ind(Dp,+) and dimHO(M). Inspection of the proof shows that this discrepancy also vanishes for g large enough, but Teleman's results indicate that the perturbation described above should never be needed to get the above equality.

102

JEAN-MICHEL SISMUT AND FRANQOIS LABOURIE

Our paper is organized as follows. In Section 1, we establish basic simple facts on compact simply connected simple Lie groups and their semisimple centralizers. In Section 2, we develop our basic residue techniques in several variables. In particular, we express certain Fourier series on T, which are local polynomials, as residues. In Section 3, we reestablish well-known results on symplectic actions with moment maps, and we give a proof of the formula of Witten and Jeffrey-Kirwan. In Section 4, we construct the canonical line bundle L on the moduli space of Gconnections on the Riemann surface E with fixed holonomy at the given marked points. We show that a suitable central extension of the gauge group EG acts on L, and we compute the action of certain stabilizers on L and on a related line bundle ay. In Section 5, we describe the moduli space MIG. We show that the formalism of the moment map can be applied to the action of G on M. We apply the formula of Witten [64) and Jeffrey-Kirwan [28] to the moduli spaces associated to semisimple centralizers. Also we give a formula for c1(TM/G). In Section 6, we apply the theorem of R.iemann-Roch-Kawasaki to the orbifold M/G, and we give a residue formula for the index Ind(D,,+). In Section 7, we give a residue formula for the Verlinde sums. Finally in Section 8, we compare Ind(DP,+) and the Verlinde formula, and give a number of conditions under which they coincide. The results contained in this paper were announced in [12].

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1. Simple Lie groups and their centralizers Let G be a connected and simply connected compact simple Lie group. The purpose of this Section is to give the basic facts which will be needed in the description of the strata of the moduli space of flat G-bundles on a Riemann surface E. This involves in particular a complete description of the semisimple centralizers in G. This Section is organized as follows. In Section 1.1, we recall elementary properties of roots and coroots. In Section 1.2, we construct the basic scalar product on the Lie algebra g of G. In Section 1.3, we compute the volume of a maximal torus T. In Section 1.4, we relate the quadratic form attached to a representation to the basic quadratic form. In Section 1.5, we introduce the dual Coxeter number. In Section 1.6, we construct an embedding of the center Z(G) in the Weyl group W. In Section 1.7, we give simple properties of the element p/c E T, in particular in its relations with the center Z(G). In Section 1.8, we review elementary properties of the characters of G. In Sections 1.9-1.12, we describe the semisimple centralizers, and give some of their properties. In Section 1.13, we consider the intersection of an adjoint orbit with such a centralizer. Finally in Section 1.14, we recall various properties of the stabilizers of elements of g, and we construct the symplectic structure on the coadjoint orbits, and the corresponding line bundles.

I.I. Roots and coroots. Let G be a connected simply connected simple compact Lie group of rank r. Let g be its Lie algebra, let g' be its dual. Let T be a maximal torus in G, let t be its Lie algebra, let t' be its dual. Let W be the corresponding Weyl group. We will denote the composition law multiplicatively in G, but often additively

in T. Let r c t be the lattice of integral elements in t, i.e. (1.1)

l'={tEt,exp(t)=1inT}.

Let A = I" C t' be the lattice of weights in t', so that if h E r, )1 E A, (1.2)

(A, h) E Z .

Let R = {a} C A be the root system of G. Then R is a finite family of elements

of A, which span t'. Let W C A be the lattice generated by R, let W r be the lattice dual to W. Let CR = {ha} C t be the family of coroots attached to R. Let GTR C t be the C t' be the corresponding dual lattice. Since G lattice generated by CR, let is simply connected, by [15, Theorem V.7.1J, (1.3)

t = CR,

A=G t

.

Let Z(G) be the center of G. By [15, Proposition V.7.16], (1.4)

R /MR = Z(G)

.

Let R,, Re and CR,, CRe be the short, long roots and coroots. Note that R, corresponds to Cite and Re to CR,. Recall that G is said to be simply laced if all the roots (or coroots) have the same length. In this case, all the roots will be

JEAN-MICHEL BISMUT AND FRANgOIS LABOURIE

104

considered as long, and the coroots as short, so that

R. = 0, CRR=0.

(1.5)

In the sequel, when G is simply laced, any statement concerning R. or CRt should be disregarded. OR, be the lattices generated by Rt, B,, Let Re, R CR,. It follows from the classification of Lie groups that

R, = R . Note that when G is simply laced, the second equality in (1.6) is empty.

1.2. The basic scalar product on the Lie algebra. II

DEFINITION 1.1. Let (, ) be the G-invariant scalar product on g such that if II is the corresponding norm, if ha E CR.,

(1.7)

Ilhali2 = 2.

If G is not simply laced, there is one m E N (equal to 2 or 3) such that if ha E OR,, (1.8)

IIha1I2 = 2m.

By[15],iff,f'ECR, E Z.

(1.9)

Using (, ), we may and we will identify t and t". By [15, V, eq. (2.14)], under this identification, if a E R, if ha E CR corresponds to a, then (1.10)

a = IIh2II2 ha .

By (1.10) , we find that

CRcRnT.

(1.11) Also, by (1.6), (1.8), (1.10), (1.12)

CR = R1,

CRl = mR. By (1.12), (1.13)

mRCRRCR, mCRCZ°R.eCCR.

From (1.12),(1.13), (1.14)

m

m

m

Also (1.15)

r=CRCRnR cCR =A.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

105

Observe that the Weyl group W preserves all the objects which have been constructed above. Also note that

T=t/7R.

(1.16)

Moreover T' = t/R is a maximal torus in the adjoint group G' = G/Z(G). 1.3. The volume of T. Let Vol(T) be the volume of T for the metric (, ). PROPOSITION 1.2. The following identity holds

Vol(T)2 = ICR*/CRI.

(1.17)

PROOF. Consider the exact sequence of lattices

0->CC ->CR ->CR /CR->0,

(1.18)

which induces the exact sequence (1.19)

0 -> CR /GAR -+ t/GCR -> t/CR* -> 0

From (1.19), we obtain (1.20)

Vol(t/Z R) = IcW /URIVol(t/CR) .

Now t/CR and t*/CR = t/CR are dual tori. Therefore (1.21)

Volt/GR Vol(t/CR) = 1.

By (1.20), (1.21), we obtain (1.17). The proof of our Proposition is completed. PROPOSITION 1.3. The following identity holds (1.22)

Vol(T)2 = IZ(G)I IR/RtI.

In particular, if G is simply laced, (1.23)

Vol(T)2 = IZ(G)I.

PROOF. Clearly (1.24)

ICR /CRI = If* iii IR/UR-I

Also by (1.4), (1.25)

(CR*/R)* = R /ZR = Z(G) .

From (1.12), (1.17 ), (1.24), (1.25), we get (1.22) . The identity (1.23) follows.

1.4. The quadratic form attached to a representation. Assume temporarily that G is a compact connected semisimple Lie group, which is not necessarily simply connected. Otherwise, we use the notation of Sections 1.1-1.3. Let a : G -+ Aut(V) be a finite dimensional representation of G. DEFINITION 1.4. If A, B E g, put (1.26)

(A, B)° = 4 DV [a(A)a(B)] .

Then (, ) is an ad-invariant symmetric bilinear form on g. Let x E R i-* [x] E [0, 1[ be the periodic function of period 1 such that for x E [0,1 [, [x] = X.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

106

PROPOSITION 1.5. If U E R, h E r, then (u, h)° E Z .

(1.27)

PROOF. Let M C A be the set of T-weights in the representation a. Then (1.28)

(u, h)° = - E (A, u) (A, h). ACM

Also for A e M, (A, h) E Z. Therefore, mod(Z), (1.29)

(u, h)° = -

[(a, u)] (a, h) ACM

=-F (E A,h). *Q0,11

Also the image of u in T = t/r lies in Z(G).Therefore the representation a splits into a sum of representations on which u acts like eliA8, 0 < a < 1. The corresponding T-weights are given by {A E M, [(a, u)] = s}. Since G is semisimple

E x=0

(1.30)

((aM ,a)Jaa

From (1.29), (1.30), we get (1.27) . The proof of our Proposition is completed.

1.5. The dual Coxeter number. Again we assume that G is a connected and simply connected simple compact Lie group. Also we use the assumptions and notation of Sections 1.1-1.3. Let K C t be a Weyl chamber. Let R+ be the corresponding system of positive roots, so that

R = R+ U R_.

(1.31)

Then (1.32)

K= ft E t, for any a E R+, (a, t) >01.

Let P C t be the alcove in K whose closure contains 0. Then (1.33)

P={tEt, for anyaER+, 0<(a,t)<1}.

Since G is simple, the adjoint representation of G' on g is irreducible. The corresponding T-weights are given by {0} U R. Let as e R+ f1 Rt be the corresponding highest root. Then (1.34)

P = It E K, (ao, t) < 11 .

DEFINITION 1.6. Let p E t' be given by (1.35)

P=! L

a.

aER+

By [15, Proposition V.4.12], if a E R+ is a simple root, (p, ha) = 1. In particular p E Get", and p E K. DEFINITION 1.7. Let c E N be the dual Coxeter number, given by (1.36)

c = (p,hao) + 1.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

107

Let -r : G' - Aut(g) be the adjoint representation. Then by [47, p 285], [22, eq. (1.6.45)],

-2c(,).

(1.37)

PROPOSITION 1.8. If t E t, (1.38)

ct =

(a,t)a.

aER+ In particular

cW C R.

(1.39)

PROOF. Since {0} U R are the T-weights of r, if t, 9 E t, (1.40)

(t,t')'=-2 E (a,t)(a,t'). *ER+

Using (1.37), (1.40), we get (1.38). From (1.38), (1.39) follows.

0

Recall that we have identified t and t' by (, ). Then ao = ha0. In particular, for any a E R+, (1.41)

0 < (a, p/c) < 1,

i.e. p/c E P. DEFINITION 1.9. For t E t, put (1.42)

o(t) = 11 (eix(a,t) - e 'x(O,t)). sER+

Equivalently, (1.43)

o(t) = e2ix(p,t) 17 (1 - e 2ix(a,t)) aER+

By (1.43), we find that o(t) descends to a function defined on T = t/tel. Recall that W is the Weyl group of G . If w E W, set (1.44)

e. = det(wIt).

Then by [15, Theorem VI.1.7], if w E Wt E t, (1.45)

o(wt) = eWo(t).

Put (1.46)

I=IR+I.

Now we recall a result stated in [3, Lemma 9.17]. PROPOSITION 1.10. The following identity holds (1.47)

(i1o(ep/0))2 = ICR /,Z

1.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

108

1.6. An embedding of the center in the Weyl group. By [15, Theorem

V.4.1],wKf1K00ifandonly ifw=1. Take q E Z. Then UwEW w(qP n CR) is a disjoint union of finite sets. PROPOSITION 1.11. T h e set U wEw w(qP n CR) embeds naturally as a subset of CR /qVR. More precisely, (1.48)

U w(qP n CW) = {A E CR /qCR, a2(A/q) 0 01. wEW

Also

cP n CR = (PI.

(1.49)

PROOF. Let W,,ff = W x CR be the affine Weyl group. By [15, Proposition V.7.10], Wff acts freely and effectively on the set of alcoves in t. Thus we get n K. By [15, Note V.4.14], (1.48). If A E cP n GAR , then A E ?'

A-pECR =CR nK.

(1.50)

Also, by (1.36),

(p,h,0)=c-1.

(1.51)

Since coo = h.., by (1.34), since ,\ E eP,

(A,hao)
(1.52)

By (1.50)-(1.52), we obtain (1.53)

0 < (A - p,ha,) < 1.

Since (A - p, h.(,) E N, from (1.53), we obtain A = p.

(1.54)

The proof of our Proposition is completed.

13

PROPOSITION 1.12. Let f be a W-invariant function on W/qCR. Then (1.55)

E f(A) =1WI ae8k'/gL7F

f(A) \EgP6CR

PROOF. This is a trivial consequence of Proposition 1.11.

PROPOSITION 1.13. The set gPnZ°R embeds naturally into {A E CR /qR, a2(A/q) # 0}.

PROOF. By (1.33), (1.42), if A E qP, then a2 (,\/q) # 0. Let J1, A' E qP n GR, and assume there is p E R such that (1.56)

A - A' = qp.

By (1.33), for a E R.,., (1.57)

-q<(a,A-X)
and so (1.58)

-1 < (a, p) < 1.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

109

Since pER,for aER+,(a,µ)EZ. By (1.58), we get

p=0.

(1.59)

The proof of our Proposition is completed. PROPOSITION 1.14. If A E GAR`, a2 (A/q) # 0, there is w E W, A' E qP fl UR such that wA - A' E g7 R*.

PROOF. Take A E U R--. Then by [15, Proposition V.7.10], there exists w E W,

h E P, f E UN, such that

q = wh + f .

(1.60)

Clearly of E q rR. Also by (1.45), a2(A/q)

(1.61)

= a2(h).

So if a2(A/q) 34 0, then a2(h) 0 0, so that h E P. The proof of our Proposition is completed.

Recall that Z(G) = RICK. Also W acts trivially on Z(G) C G. Equivalently

if fER,wEW wf - f E V-R.

(1.62)

If g E G, let Z(g) C G be the centralizer of g, let 3(g) be its Lie algebra. By [14, Corollaire 5.3.1J , since G is simply connected, Z(g) is a connected Lie subgroup of G.

An element t E T is said to be regular (resp. very regular) if 3(t) = t (reap. Z(t) = T). By the above, t E T is regular if and only if t is very regular. Let Tres C T be the set of regular elements in T. By [15, Proposition V.7.10] , P embeds into Tree. More precisely,

Treg = U wP

(1.63)

WC -W

and the union in (1.63) is disjoint. Let u E Z(G). Then u + P C T is another alcove in Tre$. Therefore there is a well-defined wu E W such that

u + P = wuP .

(1.64)

PROPOSITION 1.15. The map u E Z(G) i-+ w E W embeds Z(G) as a commutative subgroup of W. In particular IZ(G)I divides JWI.

PROOF. If u E Z(G), wu = 1, then u + P =Pin T. Therefore there is v E R mapping into u E R/7R such that v + P = P in T. By proceeding as in (1.57), (1.58), we find that v = 0, i.e. u = 1. If u,u' E Z(G), then (1.65)

u+u'+P=u'+wuP=wu(u'+P)=wuwu,Pin T.

From (1.65) , we get (1.66)

wu+u, = wuwu,

The proof of our Theorem is completed.

JEAN-MICHEL BISMUT AND FRANCOIS LABOURIE

110

By Proposition 1.13 , we can view qP n Z R* as a subset of GAR*/qW , which itself is stable by W. In particular if A E qP n CR , w E W, wA will be viewed as an element of W/qW. So the equality wA = A says here that wA - A E qR*.

PROPOSITION 1.16. If A E qP n GR C r/qR , w E W, then wA E qP n fl*lqR if and only if there is u E Z(G) such that w = wu.

GAR C

PROOF. Take A E qP n ?M, u E R representing an element of Z(G) _ R *17R--. Since A/q E P, by (1.64) , there is p E P such that

wua/q - p - u E CR.

(1.67)

Therefore

wuA - gp E qR .

(1.68)

Then A' = qp E qP n CR*, and wua - A' E qR . Conversely, if A, A' E qP n M, u E W are such that

wA-A'=qu,

(1.69) then

wA/q=X/q+u.

(1.70)

From (1.70) , since A/q, A'/q E P, we get w = wu. The proof of our Proposition is completed.

Put

h-

(1.71)

IWI IZ(G)1

Let w1, ... , w° E W be distinct representatives in W of the classes of W/Z(G). n

t

THEOREM 1.17. The set Uw'(gP fl ' ) is a disjoint union, which embeds 1

into

More precisely n

(1.72)

U wi(gP n (in ) _ {A E CR /qR , o'z(A/q) 0 0). 1

PROOF. Our Theorem follows from Propositions 1.14 and 1.16.

THEOREM 1.18. Let f a W-invariant function on CR /qR . Then (1.73)

E f(A) = IZ GI)1 E f(A)

aEZ'7F"/a7r'

AEgPn

'U/v)fo

PROOF. Our identity follows from Theorem 1.17. REMARK 1.19. Our Theorem is also a consequence of Proposition 1.12. In fact,

if f is a W-invariant function on CR*/gR, (1.74)

E f (A) = I z7R

eZW*/qVW

2(a/V)oo

> f(,\) I

r,eZ''1I*/e1' .2(a/9)$O

Now R/GR= Z(G), and so using Proposition 1.12, we finally obtain (1.73).

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

111

1.7. The element plc. PROPOSITION 1.20. Ifs E t, t E P, then (1.75)

E(a,s)[(a,t)] = 2(ct - P, s).

aER

PROOF. By (1.35),(1.38), (a, s) [(a, t)]

(1.76) aER

=

E ((a,s)(a,t) - (a,s)(1 - (a,t)))

aER+ 1: (a,s)(a,t) - (2p,s)

aER 2(ct - p, 8)

The proof of our Proposition is completed.

Let t E Tres. Then t determines a Weyl chamber K and an alcove P of the above type. In the sequel, we consider t as an element of P C T. The element p is still given by (1.35 ). Of course p depends implicitly on t E Tres. THEOREM 1.21. The following identity holds

ct - P= 1

(1.77)

E[(a,t)]a.

aER

In particular the map t E Tres H ct-p E t descends to a map Ttes = T.,/Z(G) -3 t. PROOF. By Proposition 1.20 , we get (1.77) . Also if u E Z(G), if t E Treg is replaced by t + u E Tag, [(a, t)] is unchanged. By (1.77) , we find that t E Treg H ct - p E t descends to a map from T,:g into t. The proof of our Theorem is completed.

Observe that since p E CR and 2p E R, then u E Z(G) = R /CR y exp(2iir(p, u)) = ±1 is a character of Z(G). Recall that K is a Weyl chamber, and that Z(G) has been embedded in W, by an embedding which depends explicitly on K.

THEOREM 1.22. If to E W, then wp/c - p/c E ZR if and only if to = 1. Also

if w E W , then wp/c - p/c = u E V if and only if w = w,,, so that w E Z(G). The map to E Z(G) H wp/c- p/c E R /GVR = Z(G) is a group isomorphism. Also if u E Z(G), exp(2iir(p, u)) = E,,,,.

(1.78)

PROOF. By (1.41), p/c E P. Using [15, Proposition V.7.10] , we find that if wp/c - p/c E ZR, then w = 1. By Theorem 1.21, if u E Z(G) = R*/ZR,

p/c + u = w (p/c) in t/Z°R.

(1.79)

By (1.79) , we find that w (plc) - plc E R". Conversely let w E W be such that

u=w(p/c)-p/cER'.

(1.80)

By (1.79), (1.80), we get (1.81)

w (p/c) = w(p/c) in T = t/CR,

JEAN-MICHEL BISMUT AND FRANcOIS LABOUR.IE

112

so that w = wu. Clearly if u E Z(G), by (1.45), (1.82)

a(wue°I `) = e.. a(eP/`)

Also by (1.79) and by the above, o'(wue°/o) = v(ePI`+u) = e2in(t,,,.)o(c°1,)

(1.83)

Since v(eo/c) 54 0, from (1.82), (1.83), we get (1.78) . The proof of our Theorem is completed.

Let P be the closure of the alcove P. By (1.33), (1.84)

P={tEt,foranyaER+, 0<(a,t)<1}.

PROPOSITION 1.23. For h E Z.R, then Fn (P + h) # 0 if and only if h = 0.

PROOF. Recall that Wall = W x CR. By [15, Lemma V.4.3], if f E P, v E

Waff, then of E P if and only if of = I. In particular, if f E P, It E CR, then f + h E P if and only if h = 0. The proof of our Proposition is completed. PROPOSITION 1.24. The center Z(G) = W/ CR embeds as a finite subset of P. PROOF. If U E Z(G) = R /V-R, there is an alcove Q containing 0 such that -it is represented in t by an element v E Q. By [15, Theorem V.4.1], there is w E W such

that wQ = P. Also since u E R /V-R, then we - v E CR. Therefore wv E P still represents u. So any element u E Z(G) has a representative in P. By Proposition 1.23, this representative is unique. The proof of our Proposition is completed. REMARK 1.25. If u E Z(G), we still denote by u the corresponding representative in P. Then if w E W, wu E wP is the unique representative of u in wP. Of course, if wu E P, the above implies that we = u. However this also follows from (15, Theorem V.4.1].

Let wo E W be the unique element of the Weyl group such that w K = -K. THEOREM 1.26. Let u E P be the unique representative in P of an element of Z(G). Then if t E F, if [t + u] E wuP represents t + u E T, then (1.85)

[t + u] = t + wuu.

Also (1.86)

wuu = won.

In particular (1.87)

wuP = wuu + P.

Moreover (1.88)

u = p/c - wup/c in t.

In particular, if t E P, (1.89)

wup - c[t + u] = p - ct in t.

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113

PROOF. Clearly w,(-u) lies in P, and so it is the unique representative in P of U-1 E T. If t E P, by (1.84), it is clear that t + w,u lies in some alcove containing 0. Also t + w,u represents t + u E T. From the above, we get [t + u] = t + w,u.

(1.90)

The alcove which contains t+w,u is necessarily equal to wuP. Therefore wuu and wou both lie in wuP, and represent u E Z(G). By Proposition 1.24, we get (1.86). By (1.86) and (1.90), we obtain (1.85) and (1.87). Using Theorem 1.21 and (1.85), if t E P,

c(t + wuu) - wup = ct - pin t.

(1.91)

From (1.91), we get (1.88) and (1.89). The proof of our Theorem is completed.

1.8. Some properties of the characters of G. Put

CR = CR f1 F.

(1.92)

If A E Z RR+, let X, be the character of G which is the character of the irreducible

representation of G with highest weight A. By the Weyl character formula, if t E Treg, e2ia(w,\,t)

1 no (

XX

)

(t) -

TT (1 - e2iw(wa,t)) - wEW aER+

Equivalently

E ewe2iw(w(P+a),t) . XA(t) = 1 o(t) WEW

(1.94)

Recall that by [15, LemmaVl.1.2], if p E CR is not included in a Weyl chamber, Ewe2ir(wµ,t) = 0.

(1.95)

wEW

If p E GAR` lies in K, then by [15, Note V.4.14], there is A E

P=P+A_ If A E

+

such that

there exists a Weyl chamber K such that A E K. In general K is

not unique. PROPOSITION 1.27. The character X,\ does not depend on K.

PROOF. Assume that A E K, A E r. Then by [15, Lemma V.4.2], there is w' E W such that w'K = K'. Let R+, R. be the system of positive roots attached to K, K'. Clearly (1.96)

R+ = w'R+.

Also since A E K fl r, and w'K = r, then A E K, w'-'A E K. By [15, Theorem V.4.1], it follows that (1.97)

W/_1'\

= A.

114

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

Let xa (t), x\' (t) be the characters attached to K, K', of highest weight A. Using (1.93), (1.97) , we get (1.98)

Xf (t) = Xa ' (t)

Our Proposition follows.

For A E ZR`, we can then write X,\(t) without explicitly mentioning a Weyl chamber K. PROPOSITION 1.28. If A E U R-, W E W, (1.99)

Xw\ = Xa

PROOF. We may and we will assume that A E CR = GAR n K. Then wA E CR n wK. Equation (1.99) now follows from (1.93).

Recall that if A E CR , w E W, then wA - A E R It follows that if 91 i ... , 9a E

CR , then E ej E R if and only if, given (w',... , w') E W, then t w% ej E J=1

j=1 a

.

PROPOSITION 1.29. If E Oj E R, then 1I xe; descends to a function on the j=1

j=1

adjoint group G' = G/Z'(G).

PROOF. By Proposition 1.27 , we may and will assume that the ej's lie in UR+ = CR n K. By (1.93) , for t E Tres, 2ia(E wkek,t)

(1.100)

E

]a X0, (t) = j=1

8

e

k=1

(WI,...,w')EW 1111 (1

- e-2t1r(wfa.;))

j=1aER+ a

wjej

If E ej E R, then j-1

a

E R, and so by (1.100), j=1

xe, (t) descends to a function j=1

on the adjoint torus T' = t/R . The proof of our Proposition is completed. PROPOSITION 1.30. If E ej O R, then j=1

(1.101)

E ll

xe!(et+-u) = 0.

ijex*/Vxj=1 PROOF. This follows from (1.100).

PROPOSITION 1.31. If A E ."n, A 0 R, then (1.102)

XA(ev/`) = 0.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

115

PROOF. We may and we will assume that X E GTR+. If W E W, then (1.103)

XA(ep/c) =

Xa(ewp/c)

.

Take u E W*. Then by Theorem 1.22, (1.104)

wisp/c - p/c = u in T.

By (1.103), (1.104), (1.105)

XA(ep/c) =

Xa(ep/c+u)

.

Now since the representation associated to X is irreducible, the central element eu acts in the A representation like e2ir(a,u). From (1.105) , we get (1.106)

XA(ep/c) = e2iir(a,u)Xa(ep/c) .

If X V R, it is then clear that (1.102) holds. The proof of our Proposition is completed. Now we have the result of Kostant [361. THEOREM 1.32. If X E R, then

XA(e°/c) = 0, +1 or -1.

(1.107)

Take now p E N* and A E pP, so that A E 7R--+. Take u E Z(G) = R /CR. Then X/p+u E T is represented uniquely by an element [X/p+u] E wuP. Observe that by ((1.85),

[X/p+u] = X/p+wuu in t.

(1.108)

Also

p[X/p+u] - (X+pu) E ptR.

(1.109)

o(µ/(p+c)) 96 0, then

THEOREM 1.33. If IA E UR-/(p (1.110)

Xp(a/p+ul(ep/(P+0) =6WUe2ta(u,p)X,(eJA/(P+c)).

In particular

(1.111)

XPu(ep/(P+c))

= ew.eVw(u,p),

PROOF. By (1.94), (1.112)

1

XA(e)A/(P+c))

p(ep/(P+c))

=

1

Q(eP/(P+c))

EE Ewe

2ix(w(A+p),y

)

wEW E ewe2iir(w,\/Rp) wEW

e2hr(w(p-cX/P),7)

Also p[X/p + u] E wuP n CR`. When replacing P by wuP, p is replaced by wu p. When replacing A by p[X/p+u], using (1.109), we find that e26r(wa/P,p) is replaced by e21w(wa/p,,u),24w(wu'p). Also by equation (1.89) in Theorem 1.26, we find that when replacing X by P[X/p+u], p - cA/p E t is unchanged. Finally v(ep/(P+c)) is Equation (1.110) follows from the above arguments. changed into e,, Equation (1.111) is a consequence of (1.110). o(ep/(P+c)).

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

116

REMARK 1.34. If we use (1.110) with p = 0, and µ = p, we get 1 = ew es:n(v,u)

(1.113)

which is precisely equation (1.78). Theorem 1.33 will be used in Remark 8.4 as a consistency check on our index theoretic computations.

1.9. The elements whose centralizer is semisimple. We still assume that G is a connected and simply connected simple compact Lie group.

DEFINITION 1.35. Let C c t be the set of u E t such that {a E R; (a, u) E Z} spans V. Clearly

R C o.

(1.114) Since by (1.6), (1.12), VWR

(1.115)

= ZR, = At, then

Rc

C C.

In particular R C C acts by translations on C. Also the Weyl group W acts on C.

PROPOSITION 1.36. The set C/R is a finite subset of the adjoint torus T' _

t/R , which contains MIN*. Also W acts on C/R . PROOF. Let R' C R be such that the elements of R' span t*, and let R be the associated lattice. Clearly 17/7t is a finite set. Then (1.116)

or = U R "/R

.

R'

From (1.116) , its follows that C/R is finite. The proof of our Proposition is completed.

Let now K be a Weyl chamber in t. Let R.. be the corresponding system of positive roots so that R = R+ U (-R+). (1.117) DEFINITION 1.37. If u E G' = G/Z(G), let Z(u) C G be the centralizer of u. Recall that by [14, Corollaire 5.3.1], since G is connected and simply connected,

Z(u) is a connected Lie subgroup of G. Clearly if u E T' = t/R , T is a maximal torus in Z(u).

Also W acts like the identity on Z(G) = RlMR. Therefore, if u E T' _ t/R, w E W, then wu - u is well-defined in T = t/GAR. Put (1.118)

Wu={wEW;wu-u=0inT=t/CW}.

THEOREM 1.38. If u E T' = t/R, the root system Ru of Z(u) is given by Ru = {a E R, (a, u) E Z}.

(1.119)

Also &,+ = R. fl R+ is a system of positive roots for Z(u). If Z(Z(u)) C Z(u) is the center of Z(u), its Lie algebra 3(Z(u)) is given by (1.120)

;(Z(u)) = If E t, for any a E Ru, (a, f) = 0} .

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

117

Also the Weyl group Wz(u) of Z(u) is given by (1.121)

Wz(u) = W .

Finally (1.122)

C/R = {u E V, Z(u) is semisimple}.

PROOF. Let 3(u) C g be the Lie algebra of Z(u). Then (1.123)

b(u) _ { f E g, u f =f}.

As a T-space, g OR C splits as (1.124)

9®RC=(t®R C)®(®9a) aER

From (1.124), we get (1.125)

g(u) OR C = (t OR C) e (® 9Q) . .sER

From (1.125), it is clear that (1.119) holds. Since the forms a in R do not vanish on K, the same is true for elements in Ru, i.e. K is included in a Z(u) Weyl chamber K,,. If Ru,+ is the corresponding system of positive roots, then (1.126)

Ru,+ = R. n R+.

The identity (1.120) is trivial. Let N(T) C G be the normalizer of T. Then (1.127)

W = N(T)IT.

Similarly let N, (T) be the normalizer of T in Z(u). Then (1.128)

Wz(u) = Nu(T)/T.

Clearly (1.129)

NN(T) = N(T) n Z(u).

Therefore Wz(u) is a subgroup of W. Since u E Z(Z(u))/Z(G), if to E Wz(u), (1.130)

wu-u=O in T.

Conversely if w E W, let w' E N(T) represent to. If wu - u = 0 in T, then (1.131)

w'uw '1 = u in T,

i.e. w' E Z(u). Therefore w' E Nu(T), and to lies in Wz(u) By definition, Z(u) is semisimple if and only if 3(Z(u)) = 0. From (1.120), we get (1.122). The proof of our Theorem is completed. REMARK 1.39. Clearly, if we identify u to a corresponding element in t, then (1.132)

Wu={wEW,wu-uECR .

By Theorem 1.38, W. = Wzlul. Let Cl u be the lattice generated by the coroots of Z(u). Since u E Z(Z(u)), if to E Wz(u), wu - u E 7R-u- Therefore (1.133)

W. = {w E W, wu - u E CRu }.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

118

PROPOSUION 1.40. If n > 2 and G = SU(n), then

C = R*.

(1.134)

PROOF. By [15, Proposition V.6.3], it is clear that if al, ... , a, is a basis of t" over R, then a1, ... , a,. spans R. Equation (1.134) follows.

1.10. Some properties of semisimple centralizers. We still assume that G is a connected simply connected simple compact Lie group of rank r. Also we use the notation of Sections 1.1-1.3 and 1.9. Take u E C/W . To the Lie group Z(u), we can associate the objects we considered in Section 1.1 for G, with the reservation that since the action of W on t may be reducible, Z(u) is semisimple and in general not simple. However, we equip 3(u) with the scalar product induced from the scalar product of g, ( , ). Therefore t is equipped with the scalar product (,) , and the identification t = t' will still be the one we used for G. The objects we considered before which are attached to Z(u) will be denoted

with the index u. The lattices ru c t, A = r;, C t' are given by (1.135)

ru = V-R, Au = 7R-.

The roots R. have already been described in Theorem 1.38. Clearly (1.136)

CRu={ha,aERu}.

Note that in general (1.137)

rl(Z(u)) = UR--/VR;,

and so Z(u) is in general not simply connected. If U E C/W, put as in (1.35), (1.138)

p,,_- q-

a. aE&'+

Then P. EMU . THEOREM 1.41. For any u E C, (1.139)

2cu E R,

2p,, ER. If h E W/Z%, then 2c(u, h) E Z, 2(p - pu, h) E Z, and moreover (1.140) c(u, h) = (p - p,,, h) mod(Z). In particular, if h E 7rl (Z(u)) = GT/GL , then 2c(u, h) E Z, 2(pu, h) E Z, and (1.141)

c(u, h) = (pu, h) mod(Z).

PROOF. Let r : G' -+ End(g) be the adjoint representation. Then by (1.37), if

A,BEg (1.142)

(A, B)T = -2c(A, B) .

Also r induces a representation Z(u)/Z(G) -4 End(g). Then R C t is exactly the lattice of integral elements in t with respect to Z(u)/Z(G). By Proposition 1.5 and by (1.142), we then find that 2cu E R. Also 2pu E Ru C R.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

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If h E R , it follows that 2c(u, h) E Z, 2(pu, h) E Z. Also since CRu C if,,, and u E Ru, if h E CRu, (u, h) E Z, and, since pu E ZTRu, then (pu, h) E Z. Therefore mod (Z), if h E W% c(u, h) and (pu, h) only depend on the class of h in R /I.LLu. Now we establish (1.141). By [15, Proposition V.7.10], we may suppose that u E C is such that for a E R

No,u)I <1.

(1.143)

Take h E R . Since Z(u) is semisimple, and since the a E R are the weights of the restriction of r to Z(u), then by proceeding as in (1.29),

E[(a,u)](a,h) = E s( E a,h) =0.

(1.144)

aER

BE[0,1[

eea

lla.

Also,

E [(a, u)] (a, h) = E [(a, u)] (a, h)

(1.145)

aER

aER\R

((a,u)(a, h) - (1- (a,u))(a, h)) aER+\R,.,+

(a, u) (a, h) - ( aER\Re

a, h) .

aER+\R,.,+

If a E Ru, (a, u) E Z, and (a, h) E Z. Since roots in Ru come by pairs,by (1.40),(1.142), (1.144), (1.145),

2c(u, h) = ( E a, h)

(1.146)

mod (2Z) ,

aER+\R...+

which is equivalent to (1.140).

If h E UR, then (p, h) E Z. From (1.140), we get (1.141). The proof of our Theorem is completed. REMARK 1.42. Needless to say, the class of pu in t/17 does not depend on the choice of R+. This fact fits with (1.141).

1.11. The first homotopy group in a semisimple centralizer. Take u E

C/R.

Then Z(u) is a connected semisimple subgroup of G. Also by (1.135),

(1.137) and by [15, Theorem V.7.1], (1.147)

lr1(Z(u)) = CR/CRu,

Z(Z(u)) = Ru/GR. By (1.147), (1.148)

Z(Z(u)) Z(G)

- Ru/R

Let 1ru : 2(u) -i Z(u) be the universal cover of Z(u). Then by [15, Proposition V.7.16], (1.149)

Z(2(u)) = R,+/CRu,

and a1(Z(u)) is a subgroup of Z(2(u)).

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

120

Clearly (1.150)

ZZ(G)))

RJR C C/R

Take v E Z(Z(u))/Z(G). Then Z(v) > Z(u), CR,, D CRu. Therefore ai(Z(u)) = CR/CRu surjects on iri(Z(v)) = U-R/CR,. Let ru,v : CR/CRu -) CR/CR be this surjection. Clearly Ru/R maps into CMRu/R . Let h E iri(Z(u)) = 'R/CR.. Then h defines a character bh of Z(Z(u))/Z(G) given by (1.151)

Sh : v E Z(Z(u))/Z(G) i-+ exp(2i7r(h, v)).

Recall that by Proposition 1.36, CR /R C C/R . PROPOSITION 1.43. The element u E C/R is such that for any h E CR/CRu, dh(u) = 1 if and only if u E ZR"/R . PROOF. This is trivial.

PROPOSITION 1.44. Foranyu E C/R , v E

Z(Z(u)) = Ru/R , h E 7r1(Z(u)) _ Z(G)

CR/CRu, (1.152)

exp(2ia(h, v)) = exp(2ia(ru,t,h, v)).

PROOF. We have the exact sequence (1.153)

0 -> CRu/CRu -- ZrR/CRu '

CR/CR -+ 0.

Also if h' E CR,/CRu, exp(2iir(h', v)) = 1. Our Proposition follows. REMARK 1.45. Proposition 1.44 will be used in Remark 4.40.

1.12. Centralizers in a connected and non simply connected semisimple Lie group. Let G be a compact connected semisimple Lie group. We use otherwise the same notation as in Sections 1.1-1.3 and 1.11. Let iri(G) be the first homotopy group of G. Let G be the universal cover of G. Then G is a compact connected and simply connected semisimple Lie group, iri(G) is a subgroup of of Z(G) and (1.154)

G = G/7ri (G)

Let T be a maximal torus in G. Let u E T' = t/R . Let Z(u) C G be the centralizer of u.By [14, Corollaire 5.3.1], if G is simply connected, then Z(u) is connected. Let Z(u), be the connected component of the identity in Z(u). Then Z(u), is a normal subgroup of Z(u), and so Z(u)o\Z(u) is a finite group. Moreover T C Z(u)o, so that u E Z(u)0. If g E Z(u), let g E G be a lift of g, let u E G be a lift of u. Then [g, u] E iri (G) does not depend on g-, u. PROPOSITION 1.46. The map 9 E Z(u) H

u] E ai (G) induces an embedding

of Z(u)0\Z(u) into 7ri(G). In particular Z(u),\Z(u) is commutative.

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PROOF. The Lie group Z(E) C G is connected. Therefore Z(ii)/ir1(G) is a connected subgoup of Z(u).Since Z(u)/r1(G) and Z(u), have the same Lie algebra, it follows that Z(u), = Z(u)/x1(G).

(1.155)

Let g, g' E Z(u), let

E G lift g, g'. Let h, h' E 7r2 (G) be such that

gu9 I = uh,

ui'.

(1.156)

Then (1.157)

9'gug-19'_1 = g'uhg-1 = j'ug'-'h = uh'h.

If g E Z(u) if g E Z(u) lifts g, then (g,u] = 1.

(1.158)

Conversely if ( 1.158) holds, then g E Z(u) and g E Z(u),. The proof of our Proposition is completed.

Let N(T) C G be the normalizer of T in G. Then W = N(T)/T is the Weyl group. Put (1.159)

W. = {w E W, wu -u = O in T = t/r}.

PROPOSITION 1.47. The following identity holds (1.160)

W = (N(T) n Z(u))/T.

PROOF. The proof of our Proposition is the same as in (1.128)-(1.131).

Let Wz(u), be the Weyl group of Z(u),. Then Wz(u)a = Wz(u) Also Wz(u), is a normal subgroup of W. Then Wz(u),\Wu is a finite group. Let CR C t be the lattice in t spanned by the coroots of G. Let r C t be the lattice of integrals elements in t, i.e. whose exponential in T C G is equal to 1.

ThenC r, and (1.161)

l /GAR = a1 (G)

We define WW* as in Section 1.1 . Let V E t represent u E T' = t/W .Then w E W H wu - u E t /CR = r1 (G) is a well-defined map. Recall that Z(G) = R / ', and that a1 (G) = I'/GAR is a subgroup of Z(G). In particular, by Proposition 1.15, 7r, (G) embeds as a commutative subgroup of W.

THEOREM 1.48. The map w E Wu 1-1 wu - u E I'/GAR = Ir1(G) induces an embedding of Wz(u), \Wu into 1r1(G). In particular Wz(u), \Wu is commutative. If w E Wz(u), \Wu is identified to the corresponding element w' E ir1(G) C Z(G), the image of w' in W lies in W. and represents w in Wz(u),\Wu. PROOF. Let w, w' E Wu. Let h, h' E T be such that (1.162)

wu-u=h,w'u-u= h'.

Then (1.163)

w'wu-u=w'h+ h'.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

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Nowrc.tr,andso w'h - h E CR.

(1.164)

Therefore by (1.163), (1.164)

(1.165)

w'wU-U=h+h' in r/GAR.

If w E Wz(u)a, since u E Z(u), is in the center of Z(u)o, then (1.166)

wU- u = 0 in r/CR

Conversely let w E W be such that (1.167)

wu-2ECR.

Let g E G represent w.By (1.167),

(1.168)

[g, u.] = 1,

so that

maps into an element of Z(u),. In particular w E Wz(u), . The proof of our Proposition is completed. Observe that (1.169)

Wz(u)o\Wu = (N(T) n Z(u)o)\(N(T) n Z(u)).

Therefore Wz(u),\Wu embeds naturally into Z(u),\Z(u). THEOREM 1.49. We have the identity (1.170)

Wz(u)e\Wu

Z(u),\Z(u).

This identity is compatible with the embeddings of both groups into a1 (G) = r/CR.

PROOF. Let g E Z(u). Then gTg 1 is a maximal torus in Z(u),.Therefore there is h E Z(u)o such that (1.171)

(hg)T(hg)-1 = T,

so that hg E N(T) n Z(u). Therefore the embedding Wz(u). \Wu -> Z(u),\Z(u) is in fact one to one. It is trivial to verify that the above identification is compatible with the given embeddings into a1(G). The proof of our Theorem is completed.

1.13. The intersection of an adjoint orbit with a centralizer. We make the same assumptions as in Section 1.12. Let t E T. Let Ot be the adjoint orbit oft in G. By [15, Lemma IV.2.5], (1.172)

Ot nT = {wt}WEW

More generally, if H is a Lie subgroup of G, and t E H, let OH(t) be the adjoint orbit of tin H. In particular Ot = fla(t). TIIEOREM 1.50. If G is simply connected, if u E T', then (1.173)

Ot n Z(u) = U Oz(u) (wt). wEW \W

If t is regular, the above union is disjoint.

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123

If G is not necessarily simply connected, if t E T is very regular, (1.174)

Ot fl z(u) =

U

Oz(u), (wt),

wEWz(,, \W

Ot fl Z(u) =

U Oz(u) (wt), wEW,\W

and the above unions are disjoint. PROOF. If G is simply connected, then Z(u) is connected, and T is a maximal

torus in Z(u). Let g E Ot n Z(u). There is 9' E Z(u) such that g'gg -1 E Ot fl T. By [15, Lemma P/.2.5], there is to E W such that (1.175)

9,g9 -1 = wt,

so that (1.176)

9 E Oz(u) (wt).

Therefore (1.173) holds. If t E T is regular, since G is simply connected, t is very regular. Therefore the {wt} are distinct in T. Moreover two elements in T he in the same Z(u)-orbit if and only if they lie in the same W orbit. Using Theorem 1.38, it follows that when t E T is regular, the union in (1.173) is disjoint. If G is non necessarily simply connected, if g E Ot fl Z(u), then it E Z(g). If t is very regular, Z(g) is a maximal torus, which is included in Z(u)0. Therefore g E Z(u)o. The above argument shows that there is g' E Z(u),,, and to E W such

that (1.177)

9,99 -1 = wt,

which is equivalent to (1.178)

g E OZ(u), (wt).

So we have proved the first identity in (1.174). Since t is very regular in G, it is very regular in Z(u)0. So the union in the first identity of (1.174) is disjoint. Clearly

U Oz(u) (wt) c Ot fl Z(u),

(1.179)

and also (1.180)

U

(wt) c U

oz(,.) (wt).

wEWgl,ge\W

Therefore the second identity in (1.174) holds. If g E Z(u), to E W are such that gtg 1 = wt, if g' E N(T) represents to, then g -1gt(g'-1g)-1 = t. Since t is very regular, g'-1g E T, so that g E Z(u) fl N(T).It follows that to E W. Therefore the second union in (1.174) is also disjoint. 0 The proof of our Theorem is completed.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

124

1.14. The stabilizer of an element of the Lie algebra, and coadjoint orbits. Let G be a compact connected semisimple Lie group. We use otherwise the notation of Section 1.1. Recall that z : G -+ Aut(g) is the adjoint representation. DEFINITION 1.51. If P E t, put

Z(p) = {g E G ; r(g) p = p} .

(1.181)

By [15, Theorem IV.2.3], Z(p) is a connected Lie subgroup of G. Then T is a maximal torus in Z(p). Let 3(p) be the Lie algebra of Z(p). THEOREM 1.52. If p E t, then (1.182)

3(P) OR C = t ® ® 9a .GR (..P)=0

Also the root system R, of Z(p) is given by Rp = {a E R ; (a,p) = 01.

(1.183)

If Z(Z(p)) C Z(p) is the center of Z(p), its Lie algebra 3(Z(p)) is given by (1.184)

3(Z(p)) = If E t, for any a E Rp, (a, f) = 0} .

PROOF. The proof of these results is left to the reader. It is essentially the same as the proof of Theorem 1.38. DEFINITION 1.53. Let 7r : t OR C -+ C be the monomial (1.185)

7r(t) = 11 (2iva, t) . aER.(.

By [15, Corollary V.4.6 and Lemma V.4.10], if w E W

r(wt) = ewir(t) .

(1.186)

Also if t E t, one has the obvious (1.187)

det Ad(t) 1511 = zr2 (t/i) .

By (1.187), we find that a2(t/i) does not depend on K, and lifts to a G-invariant function on g. DEFINITION 1.54. Set (1.188)

greg = {p E g , Z(p) is amaximal torus}, treg

= It E t ; Z(t) = T} .

Clearly (1.189)

treg = 9reg n t.

PROPOSITION 1.55. The following identity holds (1.190)

9reg = {P E 9 , ir2(p/i) 4 01, treg = It E t ; ?ra(t/i) i4 0} .

PROOF. Take t E t. Since Z(t) is connected, t E tree if and only if (1.191)

3(t) = I.

Using Theorem 1.52, we get the second identity in (1.190). Also by [15, Theorem W.1.6], any G-orbit in g intersects t. Our Proposition follows.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

125

By (1.190), t E treg if and only if t lies in a Weyl chamber. The G-orbit oft in g intersects t at JWI distinct elements, which form the W-orbit of t. If p E 9, let Op be the G-orbit of p. Clearly, (1.192)

-7rp : 9 E GIZ(p) H gpg 1 E 0,

is a one to one map. Also by [6, Lemma 7.22], Op is equipped with a canonical symplectic form oo,. In fact G acts on the left on Op. If X E g, let X°P be the corresponding vector field on Op. Then if X,Y E 9, q E Op, (1.193)

uo,,q(X01,Y0)=(q,[X,Y])

Let fp be the left invariant 1-form on G (1.194) f p = (p, 9 1dg) Let 7rp be the projection G -- G/Z(p) Op.

PROPOSITION 1.56. The following identity holds (1.195)

dfp = -7ryop .

In particular the restriction of fp to Z(p) is a closed 1-form. PROOF. Clearly (1.196)

dfp = (p,-2[g-1dg,9 'dg])

from which (1.195) follows. Since ap maps Z(p) to a constant in G/Z(p), our Proposition follows.

PROPOSITION 1.57. If p E, fp is an integral closed 1-form on Z(p). PROOF. Clearly T is a maximal torus in Z(p). Then by [15, Proposition V.7.6], Z(p)/T is simply connected. Therefore r1(T) aurjects on zr1(Z(p)). To verify that fp is an integral 1-form, we only need to check that ifs E S1 i g, E T is smooth, the integral of fp on this loop lies in Z. Since p E GR , this is obvious. The proof of our Proposition is completed.

Now we assume that p E C7. Take g E Z(p). Let s E [0,1]' g, E Z(p) be a smooth path such that go = 1, 91 = 9. f1

PROPOSITION 1.58. The map g i exp (2iir J g, fpds) E S1 defines a repre-

sentation p, of Z(p). PROOF. This follows from Proposition 1.57. DEFINITION 1.59. Let Lp be the Hermitian line bundle on Op (1.197)

Lp=GxZ(p)C.

Clearly the connection (1.198)

d+2i7rfp

descends to a connection VLp on Lp. PROPOSITION 1.60. The following identity holds (1.199)

c1(Lp, V LP) = op

PROOF. This is obvious by (1.195).

0

126

JEAN-MICHEL BISMUT AND FRANQOIS LABOUEIE

2. Fourier analysis on the centralizers of semisimple Lie groups The purpose of this Section is to express certain Fourier series on T (which will later turn out to be the symplectic volumes of the stratas of the moduli space of flat G-bundles on the Riemann surface E, first computed by Witten [63, 64] ) as residues of certain holomorphic functions in several complex variables. The main point is that it is then possible to compute explicitly the action of certain differential operators on these Fourier series. The results of this Section will be used in Sections 5, 6 and 7. This Section is organized as follows. In Section 2.1, we make elementary constructions in linear algebra. In Section 2.2, we apply these constructions to the root system of G. In Section 2.3, given u E C/R and the corresponding semisimple centralizer Z(u), we consider an associated Fourier series Qu(t,x), which we express as a simple integral along the fibre of a torus fibration. In Section 2.4, we express Qu(t,x) in terms of iterated residues. In Sections 2.5-2.8, we introduce other related Fourier series, which are related in particular to the universal cover of semisimple centralizers. In Sections 2.9-2.11, we introduce our symplectic volume Fourier series, which are local polynomials on T. We express these Fourier series as residues. Finally, in Section 2.12, we compute the action of any power series of differential operators on these local polynomials. As explained in the Introduction, residue techniques have been developed by Szenes [53], [54] to handle the Witten Fourier series [63, 64]. The methods of Szenes are more conceptual than ours, which only usesimple linear algebra. It is probable that the results of this Section can be rephrased using Szenes's formalism.

2.1. Some linear algebra. Let V be a real vector space of dimension r. Let e1, ... , e, be a basis of V, let e1, ... , e' be the dual basis of V*.

For1
(ui,...,u)=

(2.2)

U1A...UiAe=+iA...Aer

eiA...Aer

Let fl,... , fr be another basis of V, let f I, ... , f' be the corresponding dual basis of V*.

Let E, E', F, F be the flags in V, (2.3)

E E'

0 C {ei} C {ei, e2} ... C {ei, ... , e,} = V,

F F'

OC{fi}C{fl,f2}...C{fi,...,fr}=V, {fi....,fr}J{f2i...,f,}D...{fr}D0.

We assume that F lies in the orbit of E. Equivalently for i, 1 < i < r, f1, ... , f{, ei+1, (2.4)

, e,. is a basis of E, i.e.

(f1, ... , f:) # 0.

Let A be the (r, r) matrix expressing fl,. . . , fr on the basis em,... , er. Then (2.4) says that the principal minors of A do not vanish.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

127

On V*, we can define the flags E*, E'*, F*, F'* associated to the basis e1, ... , er

,f and fl,... , f r. Now E* lies in the orbit of F*, i.e. for any i, e1, ... , ei, fi+1 is a basis of V*. For 0 < i < r - 1, let pi be the projection from V on {ei+1, ... , e,} with kernel

{f,... , fi }. Clearly (2.5)

Pi+1 = Pi+iPi

Also for 0 < i < r - 1, pifi+i,... pif, is a basis of {e;}1,... ,er}. More precisely for 0 < j < r - i, pifi+1.... I pifi+j+ ei+j+1, ,er is a basis of {ei+i,... ,er}. Clearly, if t E V, (2.6)

pit = t - (fi

e1))

fi.

By the above, there are similar formulas for p2, ... , pr. In particular, by (2.5),

(2.6),for1
i

(Pi-1t,e )

pit = pi-it -

ei)pi-ifi

(pi-if"

(Pj-1t,ej) pjifi. pit=t-E (pj-ifj,ej) j=1

Using (2.8) with i = r, we obtain

(Pj-1t, ej)

t=

(2.9)

j=1

(Pi-ifj, 0)

Pj-l fj

For 0 < i < r - 1, let qi be the projection from V* on If

f r) with

kernel {e1, .. ,e}. Then qi is the transpose of pi, i.e. (2.10)

qi = Pi.

Also, as in (2.6), if x E V*,

qlx=x - e',fi)el

(2.11)

Moreover the results which hold for the pi's also hold for the qj's. THEOREM 2.1. For 1 < i < r,

(pj-it,ei)(fj,gj-ix) = (t, x) - (pit,gix)

(2.12)

j=1

In particular, (2.13) i=1

(pi-1t,ei)(gi-1x,fi) = (t,x) (Pi-ifi, gi-lei)

PROOF. Equation (2.12) follows from (2.8), and equation (2.13) from (2.9).

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

128

PROPOSITION 2.2. If t E V, x E V", for 0< i< r,

ri (fhA...Afj-1AtAfi+1A...Afi,e1A...Aei) (fI A ... A fi, el A ... A ei) f"

(1 - pi)t

=1

(2.14)

(f, A...A fiAt,el A... Ae=Aei)

=

pit

S

(f1

ici+1 ,

j=i+1

A...A fi, e1 A ... A ei)

ej

((ts_

k=1

(flA...Afi,e1A...Aek-1AejAek+1A...Aei)(t,ek) v" (f A A f e1 A A ei)

e E(fiA...Afi,elA...Aei-1AxAei+1A...Aei) j (fiA...Afi,elA...Aei) j=1

(1 - qi)x

qix =

(f1 A ... A fj A fj, e1 A ... A e1 A x) fj (fl A ... A fi, el A ... A ei) jci+1

r r

i

E (fi, x) - k=1 E

j=i+1

(f1A...Afk-1AfjAfk+lA...Afi,elA...Aei)(fk,x))J fj (fhA...Afi,elA...Aei) PROOF. Clearly we only need to prove the first two series of identities in (2.14). The first identity and the first part of the second identity are standard linear algebra. Also for j > i + 1,

i

(fI A ... A fi A t, e1 A ... A ei A ei) _ (fI A ... A fi, el A ... A ei)(t, e3)

_E(_1)k i(f1A...Afi,e'A...Aek-1Aek}1A...Ae=Aej)(t,ek) k=1=1

(2.15) _ (f I A ... A fi, el A ... A ei) (t, ej) i

-E(fiA...Afi,e1A...Aek-1AejAek+'A...Aei)(t,ek). k=1

The proof of our Proposition is completed.

13

PROPOSITION 2.3. For any j, 1 < j < r (2.16)

(fj, qj-1x) = II. , x) - EI-1 k=1

iA...A _,,1Aei-i ',A...

f1A...Afi-i,e A...Ae

PROOF. This is a consequence of the last equality in (2.14). REMARX 2.4. From (2.16), we find that (fj, gj_1x) = (pj-1 fj, x) depends only

on the (fk,x), 1 < k < j.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

129

2.2. The linear algebra of the basis of a root system. Let G be a com-

pact connected and simply connected simple Lie group of rank r. Let T be a maximal torus in G. Let t be the Lie algebra of T. Let K be a Weyl chamber in t. Otherwise, we use the notation of Section 1.1. We will also use the notation of Section 2.1, with V = t'. We identify t and t'

by the scalar product (,) of Section 1.2. Then

r = dimt.

(2.17)

Let el,... , e, C R.E. be the simple basis of t` associated to K [15, Proposition V.4.5]. Any a E R+ is a linear combination with non negative integral coefficients of el,... , e,. Then el,... , e, generate R. Let e1, ... , e' be the corresponding dual

basis of R C t. Let (a,,... , at} be an ordering of R+, such that

ai = ei for 1 < i < r.

(2.18)

Recall that we use the notation of Section 2.1 . Clearly all the (ai..... , aij ) lie in Z. If G is not simply laced, m was defined in (1.8) and is equal to 2 or to 3. By convention, if G is simply laced, we take m = 1. By (1.12), (2.19)

mR C

A.

DEFINITION 2.5. Let d E N* be a common multiple of the mI (ai, A... A ai,) 1.

Observe that since e1, ... , e, E R.E., for any j, 1 < j < r, d is a multiple of m (ail .... , ai;) E Z. Also by (2.19), (2.20)

dR C CR.

Recall that C C t was defined in Definition 1.35. PROPOSITION 2.6. The following identity holds

dCCR.

(2.21)

PROOF. Let u E C, let a, , ... , ai, E R+ be a basis of t such that for 1 < j:5 r, (ai,, u) E Z. Since e1, ... , e, is a basis of R, we find that if H is the lattice

generated by ai .... , ai,, (2.22)

OFC H,

which is equivalent to (2.23)

dH` C R .

Since u E H*, from (2.23), we get (2.21). The proof of our Proposition is completed.

0

REMARK 2.7. Take n > 2. Since SU(n) is simply laced, m = 1. Also by [15, Proposition V.6.3], all the I (ai,..... a, )) are equal to 1. Therefore, for G = SU(n), we can take d = 1. Using (1.114) and (2.21), we recover Proposition 1.40.

DEFINITION 2.8. A family I = (ii,...,i,) of distinct indices in {1,... ,t} is said to be generic if (2.24)

(ai...... ai,) 0 0 for 1 < j < r.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

130

Given a generic family I = (i1,... , ir), we now use the notation in Section 2.1 associated to the given basis e 1 . . . . . er and ail, ... , ai, of t' = t. In particular the operators which appear in Section 2.1 will be denoted with the superscript I, to mark their dependence on I. Let ail .... , a4 be the basis of t* e t which is dual to ail, ... , ai,. Recall that (2.18) holds.

DEFINITION 2.9. A family I = (i1i... ,ij_1) of j - 1 distinct elements of {1,... , t} is said to be generic if Ii-1 = {i1,... , i j_1i j, j + 1,... , r} is generic.

If I = (i1,... ,i,.) is generic, Ij_1 = (i1i... ,ij_1) is also generic, and by construction (or by (2.14)), Ij_1 p{ -1 _ -Pj_1 ,

(2.25)

JIIj-1 = 4j_1 Ij_1

DEFINITION 2.10. If x E Ct, and if I = (i1i... ir) is generic, put r xI = xij aij E t = t* . (2.26) By (2.25),

1xI .

4f

(2.27)

1xI=41d_"1

Also by Proposition 2.3, (2.28)

(aij,Qj-1z') _ (Yj-1aij,xl) = j-1 aik_I,aii,IXik+l,... Xii

`j

[1

(ail,...

(ail,... ,aij_1)

k=11

Xi k

Note that the right hand side of (2.28) only depends on xi1,... ,xij. Assume that I = (i1i ... , ir) is generic. Then by Proposition 2.2, if t E t, p E t' (2.29)

(ail,... ,aij)

(pj_1ail, ej) _ (ail,... ,aij_1) (ail A... A aij, e1 A ... A ej-1 A t) (ail,... ,aij_1)

(pf-19/,ej)

(ail, a'i,11;)

In particular, by (2.29), we find `that d (2.3u)

,,[

Yj1aij, eJ)

EZ.

Also by (2.28), (2.29), if x E Zt, (2.31)

d

-1a`j,xi)

EZ.

(vj-,ai,F,ej)

DEFINITION 2.11. For u E C/R , let I C {1,... ,11 be given by (2.32)

I={i,l
SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

131

Put (2.33)

1u = IIuI

2.3. Fourier series and integration along the fibre. DEFINITION 2.12. For u E C/R , let RI' C R' be given by (2.34)

RI°={s= (s..... ,st)ERe, s'=0 foriIu}.

Let au : RI. -+ t*

t be given by au(s) _

(2.35)

alai.

iEI

The transpose au t

t* -> R. is given by

(2.36)

au(t) = ((ai,t))iv.

Set

Vu = kerau.

(2.37)

Then we have the exact sequence

0-a1;,- Rl°

(2.38)

0

We define ZIu C Zt, (R/Z)-r- C (R/Z)t as before. Then

au(ZI") = Ru.

(2.39)

Also au induces a surjection (R/Z)I* -4 t/R . Set Ku = kerau C (R/Z)I°

(2.40)

We have the exact sequence (2.41)

0

Ku + (R/Z)I*

0.

Set (2.42)

ryu = ZI* ft vu

Then Ku is a union of IR/RuI tori Vu/ryu

DEFINITION 2.13. For u E CI', let Hu C t/R be given by

(2.43) Hu = {t E t/-A, t = E tjaj, and {aj, j E ,7} does not span t* = t} . jE9C . Then H. is a finite union of hypertori in T. Put

H = Ho.

(2.44)

Clearly, (2.45)

H = {t E t/R, t =

E

tjaj, and {aj, j E ,7} does not span t* z t} .

jE9C {1,... ,P}

For any u E 01-R, (2.46)

H. C H.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

132

PROPOSITION 2.14. Let I = (i1, ... , ir) C I be generic. Then if t E (t/R) \H is represented by t E t, (pf_1t,el) ¢ Z.

(2.47)

PROOF. By (2.29),

(pj-lt, ej) _

(2.48)

aaij1l) ej, the expression (2.48) is equal to 1. Also (ail, ... , a41_1, t) vanishes Now for if and only if t is a linear combination of c%, ..., o%_,, ej+1, ... , e,.. Therefore the condition

(p_1t,ej) E Z

(2.49)

is equivalent to j-1

r

t=Eakai,,+bej+

(2.50)

ckek

,

a",ckER,bEZ.

k=j+1

k=1

Then,

j-1

r

t = E aka,,, + E ckek in t/R,

(2.51)

k=j+1

k=1

so that tEH. The proof of our Proposition is completed.

Let dt be the Lebesgue measure on t associated with (, ). We still denote by dt the Lebesgue measure on t/R. We map L1(t/R) into D'(t/R) by the map

fu

f dt

We use the same convention for other tori.

Volt/R) Clearly a induces a map a,,. from D'((R/Z)1") into V'(t/R).

PROPOSITION 2.15. If g E L1((R/Z)1°), then a,,.g E L1(t/R). More precisely, (2.52)

a,..9(t) =

Also if k E Z1°, then (2.53)

au.[e21w(k,*)] = e2i,r(A,t) if there is 1

=0

A E R such that ki = (A,ai),i E I,,,

otherwise .

PROOF. The proof of this result is trivial.

DEFINITION 2.16. For u E C/R , x E (C \ 2iirZ)l, t E t/R, set (2.54)

exp(2iir(A, t)) Q,.(t, x) = E7r flier. (2i1r(ai, A) - xi) )

Clearly as a function of t, Q (t, x) is a well-defined distribution on t/R. For M E N, we also consider the partial sums exp(2iir(A,t)) (2.55)

QUM(t,x) ae7i

Ia15M

11iE1. (2i7r(aj, A)

- xi)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

133

Note that Qu(t,x) depends only on the projection of x on CI". THEOREM 2.17. For any n E N", u E C/R", v E n, t E t/R, the following

identity holds

nt-_r E exp(2iir(v,t+h))Q,,(t nh,nx)

(2.56)

/

hER/nA

PROOF. Clearly, if A E R, v E W */n, (2.57)

n hER/nR

exp(2i7r(A +v,t+h)) = exp(2ia(X + v, t)) if A + v E

=0

otherwise.

From (2.54),(2.57) , we get (2.58)

nr"_r

1: exp(2iir(v,t+h))Qu(t+h,nx) n

hER/nlf

exp(2iir(A,t))

57

aEk. [I (2iir(ai, A) - (xi + 2i7r(ai, v))) iEI"

= Q. (t, x + 2iaauv) . The proof of our Theorem is completed.

0

THEOREM 2.18. The partial sums QM(t,x) converge uniformly together with their derivatives to Qu(t,x) on compact subsets of (t/R) \ Hu x (C \ 2i7rZ)e. The following identity of distributions holds (2.59)

Qa(t,x) = (-1)t"au.

(exp(xi) -1)

exp((x, s))1

in D'(t/R) .

In particular for x E (C\2iirZ)I, Qu(t, x) is a distribution in LO0(t/3h. Also (2.59) is an identity of smooth functions on (t/It \ Hu x (C \ 2iirZ)l. PROOF. For x e C \ 2itrZ, the Fburier series for ell (8 E [0,1]) is given by Oink& (2.60)

ell = es

1) kEz

Since ex' is smooth on [0,1], the partial sums in (2.60) converge uniformly together with their derivatives to ex* on compact subsets of R/Z \ {0} x C \ 2iirZ. From Proposition 2.15 and from (2.60) , we get (2.61)

au«[e(:'s)]

= rf (esi - 1) E

iE.

Acr 11 (-2iir(ai, A)+ xi) iel"

which coincides with (2.59).

Also for x 0 0, the wave front set of the distribution ell on S' is just {0} x R. By [26, Theorem 8.2.13], we see that (t,p) E t/R x Rl" \ {0} lies in the wave front set of Qu(t, x) only if t = 7. tiai, and (p, ai) = 0 when ti f Z. Therefore Q0(t, x) iE . is smooth on (t/R) \ H, .

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

134

The proof of our Theorem if completed.

2.4. Iterated residues and the series Q,, (t, x). Recall that by Theorem 2.18, for generic values of x, Q. (t, x) is smooth on (E/R) \ H. Therefore by (2.46),

Q.(t,x) is smooth on (t/R) \ H. Let x E R [x] E [0, if be the periodic function of period 1, such that [x] = x on (0, 1[.

In the sequel, if t E t/R, we represent t by a given element in t, which we also denote by t. V

Also the map (f 1, ... , fr) E {0,1, ... , d - 1}r 4 f =

f `ei E R defines a

one to one map into RIdR. In the sequel, we will always identify f E RIdR to the corresponding element in {0, 1,.. . , d - 1}r. THEOREM 2.19. For any u E C/R , for generic values of x E (C \ 2iirZ)e, for t E (t/R) \ H, if we still denote by t a representative in t,

Qu(t, x) _ (-1)r

(ai

1 , ai.)

r g.n..ic, JE

r

(2.62)

exp dE

tai ,xI) [13(pf-1(t+f),ei)1} 1,aif,e+)

((ailx!) - xi) 11

exp

d(pi_,a:;,xr)

iCI \I

PROOF. Take 1 < j < r, I = (i1 i ... , ii-1) C Iu such that (ail) # 0'... , r

(a1,... . ai;_,) 0 0. For k = (ki+1, ... , kr) E Zr-i, we identify k with E kiei E

t = t In the sequel, p!_I denotes the projection t -> {e1,...

fail,_

i=i+1

er} with kernel

In view of (2.28), for i E Iu, i 0 I, we will use the abusive

notation (2.63)

(pi-tai, zI) = ;-1

ai, air+s , ... , aii-. xi - [ (ait, ... , aik_ (a i I . .. . , aii_1 k=1 Equation (2.63) will in fact be a definition for the left-hand side. For s E R, if

1<j
e2iaks

(264)

QI = 7' kEZ 11 ((p;-tai, 2iir(kei + k) -

We claim that for at least one i E Iu \ I, (2.65)

(pp-1ai,eJ) 00.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

135

y 2( n + 1/2)i7c

n,+ x

0 An

r

-2(n+1/2)i7c FIGURE 2.1

In fact by (2.29),

/

)

(PJ-lai, e) =

(2.66)

Since the elements of R,,,+ span t* vanish.

(au,... aif_t, ai) (ait,...

.

t, for at least one i E Ia \ 1, (2.66) does not

Clearly

QI=

(2.67)

exp(2i7rd(s + f))

1

d o
(2in_lai, ei) d + (p,'_1ai, 2iik - xI )

Then for generic x E Ce, we can write (2.67) in the form

(2.68) QI =

1

EE

d 0
1

PLeSa=21rk l

r 1

(pI-1ai d

)a +

t-1ai, 2iak -

xl ))

\

}

I (ea _1 )

Assume that 8 f Z. Then for f E N, 0:5 f < d, (2.69)

0<[sdf]<1.

Also by (2.29) , if (ai...... aif_ ai) 96 0, then d (2.70)

- Z,

Va:+ eJ)

E

d(pJ1-1a" k) E Z .

(_1-ail ei)

For n E N, consider the contour rn = I',,,+ U rn,_ given in Figure 2.1, and its

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

136

interior A,,. For f E N, 0 < f < d, put (2711

n(a) = f

(

JjL1

ealILi ))1

'.i-lai, 2iak - xf

+

d

I (e° - 1)

By (2.69), as a E A,,, lad -+ +oo, (2.72)

gf (a) -+ 0.

r

r.

Also the integral / g f (a)da converges. We can then use the residue theorem to

r

evaluate f 9f(a)da. Finally by (2.65), (2.71), as n -+ +oo, r

Jr

gf(a)da - 0.

Then we find that for generic x E Ct,

1 Resagf(a) = 0.

(2.73)

OEC

Now for generic x E CI, the poles of 9f(a) other than {2ilrk}kEZ are simple (this follows in particular from (2.63)), and given by

tai ,2i7rk-xI) (e , i j E Iu \ I, (ail , ... , aii) # 0.

a _ -d -

(2.74)

(P9-l aii' e' ) In the sequel, we use the notation (2.75)

(I,ij) = (i1,... ,ij).

Observe that if (i1, ... , ij) is generic, if y r: t" for 1 < k < j -1, ,aii )

(Oil l ...

(2.76)

(ail, ... , aid 1

(ail,... ,aif_1) ((aii,... ,aiw-14,014}11... ,aij_,) (ail,... ,air,-I Iaii,ai.+1,... ,aif_1) (ail , ... , aij )

(ail,... ,aii_,,y))

In fact if y lies the vector space spanned by ail) ... I aitr_ 1 I ai f-I , aij e j}1, ... , e,. both sides of (2.76) vanish, and if y = ai,, both sides are equal to 1. So by (2.6), (2.63), (2.66), (2.68), (2.70), (2.73) and (2.76) with y = ai, i 0 (I,ij), we get (2.77)

(ail ... , a(i_, )

Qr = -

,aii) (-it

exp

C

dVfi0.0S1
(2.(p_1aj1,) (a+ f)+ d(-1ait,x) (s+f1\ (pl_aif>ei)

-laii,ej)

TT iElu \(I.i{ )

((PY u1)ai, 2iak - xf )))

l -d j)

(exp I ° 1- '

I -1 I

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

137

Now by (2.7), for j' > j + 1, (2.78) (Pj_laii ,e l

')

Using (2.77), (2.78), we obtain (2.79)

(a,,,... ,aif_t)

Qi = -

(a i,,...

,

(nit,... -jj)00,0
ail )

p-l_1a+iixl)

exp (2i7r(Y'is)(s+f)ej,k)+ d

ei ) 1

TT

W \(l,il)

(p(I s')a WA -xI))) (exp (

dam-

'

I-1I

Clearly (2.80)

exp(2iirE 1 ki(t,e'))

Q.(t,x) _ k=(kl,... ,k.)e2°

iEl

(2i1r(as, E kie') - xi) {.1

Also with the notation in (2.63), (2.81)

(ppai, xl) = xi

So using (2.79) with I = 0, 8 = (t,el), we find that for (t,el) 0 Z, tGy--++

Qu(t,x) _ -

1

0<J'
exp

x" (2ir(p(it)(t+flel),k)+d (ait) I

(t+flei1)l d

1)

1

( i

II

dx tt

((p('t)ai,2i7rk-xl))) (exP(_T) -1) t

Clearly, if t e (t/R) \ H, then (t,el) 0 Z, and so (2.82) holds. More generally,

if t e (t/R) \ H, f E R, by Proposition 2.14, (pp_1(t+f),ej) ix Z. Therefore using (2.79), (2.82), we can iterate the procedure. Finally observe that by (2.63), if r = (i1, ... , i,) is generic, for i ¢ I, r Ti aik-t, ON, air+t , ... , *0 (2.83) xik (Rlai, -r) = (ai ... ,air )

-

L_1

xi-(si,xl). Using (2.83), we get (2.62). The proof of our Theorem is completed.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

138

REMARK 2.20. If we had assumed instead that el,... er is a simple basis of R,,,+, we would have obtained a better equality in (2.62) , with t E (t/R) \ H,,. However, it is essential here that we have use the same simple basis e1, ... , e,. of R+ for all the u E C/R simultaneously. Finally observe that Theorem 2.19 can possibly be reformulated using the formalism of Szenes [54]. REMARK 2.21. Take n E N. Then we have the exact sequence

0 -+ R/dR n - R/dnR P, R/nR - 0

(2.84)

In (2.84), the map n is just multiplication by n, and p is the obvious projection. Now in (2.62),.we may replace d by nd. We get (2.85)

Q.(t, x) = fv(i3.... .:. )C ru ,r Qeneric

jdc. (p;-1aif,nxi)

1

s1,ei)

(ail ,... ,ai.) exp [Id

;_1

-1((t nh) +f) d)]

77r., n.1 ) - ) 1

(d(pf

(ijail

11je1 C

Comparing (2.62) and (2.85) gives the identity (2.56), with v = 0.

By definition, Q (t, x) is well-defined on t/R. However it is not entirely clear that the right-hand side of (2.62) is indeed well-defined on t/R. We will now check this fact directly. THEOREM 2.22. As a function oft E t, the right-hand side of (2.62) descends to a function on t/-R.

PROOF. By Proposition 2.2, (ai,,... ,ai,,)p1!ek is an integral linear combination of the (ep)p>k+1 Therefore for k < j - 1, dVj_lek is an integral linear combination of the (_1ep)p>k+1. So for k < j -1, d(pJ_1ek,e') is an integral linear combination of the (P _lek', e')(k + 1 < k' < j). To prove that the right-hand side of (2.62) is well-defined on (t/R} \ H, we only

need to show that if we add to t an element of dR, the total expression does not change.

Take f E R/dR. Then f is uniquely represented by an element we also note f ,

r

(2.86)

f=Efkek ,0
Clearly (2.87)

(j-1(t+f),e') = (-1(t+Efkek),e'). 1

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139

Put (2.88)

Ai = [!d(pf-i (t+

Since pIr_Ie = e

e3) = 1. Therefore if we add to fi an integral multiple

,

of d, Al is unchanged. By the above, if we add to f' an integral multiple of d, the right-hand side of (2.62) is unchanged. If we add to f*-1 an integral multiple of d, then only the term A,. is possibly affected. However as we saw before, d(pi_ier_I, e') is an integral multiple of (pf_Ier,e') = 1, i.e. (2.89)

,4 E Z .

dV,.-ier-i,e') =

Therefore adding to f r-i an integral multiple of d is equivalent to adding to f r an integer. This show the right-hand side of (2.62) is invariant under this change. A trivial downward recursion procedure shows that Q. (t, z) is indeed welldefined on t/R. The proof of our Theorem is completed.

2.5. The Fourier transform on quotient of lattices. Let A, A' be lattices in t, with A C A'. Then there is a projection t/A -4 t/d'. Let f E IY(t/0). For put iE (2.90)

fp(t) = o,

ICI keA,/A

e2i,,(,,,k)f(t+ k)

.

Then I E D'(t/0). Moreover if k E A'/0, (2.91)

fp(t + k) =

Also (2.92)

f(t) =

.fµ(t)

By (2.90) (2.93)

f (t + k) _ E exp(2irr (/c, k)) f (t) . FEA'/A"

Equation (2.93) is just an aspect of Fourier transform. Note here that if A C t/A

is such that f is C°° on (t/ A) \ A, then f is C°° on (t/0) \ U (A + k). Then kEI'/0 formula (2.93) only expresses f as a function which is smooth on (t/0) \ U (A+ kEa'/0 k), i.e. there is a loss of regularity in (2.93).

2.6. Fourier series on T. In the constructions of Section 2.3 , we may as will replace R by OR, R by UR, R by 7R-'. For u E C/R", we still define I,,, RI- as in (2.32), (2.34). However in (2.36), au : RI" -+ t' = t is replaced by bu : R- -> t = t' given by (2.94)

tih.; .

bu(t) iEtu

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140

Clearly

C'R C CR.

(2.95)

Therefore there is a a surjection b : (R/Z)I -> T = t/GAR. Then we have an exact sequence

0 a L -+ (R/Z)lu b T -+ 0

(2.96)

DEFINITION 2.23. For U E CI', let Su C T = t/GAR be given by (2.97)

Su = It E t/CR, t = E tihaj, and (h.,, j E 3) do not span t} . iE3CI.

Put S = Se.

(2.98)

Then (2.99)

S = It E T = t/Ck, t =

E

j E T} do not span t} .

tAh.i, and

jEJc{1,...,e}

As in (2.46),for any uEC/R Su C S.

(2.100)

`,aERt, 6ER,

By (1.12) , it is clear that if

(A, a) E Z,

(2.101)

m(A, ) E Z. In the sequel, when G is simply laced, we will make m = 1.

DEFINITION 2.24. For u E C/R, x E (C \ (2iirm)t, t E T = t/GAR, put exp(2iar(A,t))

(2.102) 1E

` 11iE u (2iir(ai, A) - xi)

Clearly, by (1.8), (1.10), (2.103)

Ru(t,x) = m IR-,+nR.1 E aE

exp(2iir(A,t)) IIiEI. (2iir(hat , A) -

IIh

2

II' x;)

Of course in (2.103) , Il 2.2 = 1 or m. From (2.103) , it should now be clear that Theorem 2.18 can be applied to R, (t, x). In particular on compact subsets of T \ S. x (C \ 2"'Z )1, R. (t, x) is a smooth function of (t, x). If G is simply laced, the objects we just constructed are the ones we already obtained in Section 2.3. Now we use the notation in (2.35). Put (2.104)

a=ae.

Then if s E Rt, t (2.105)

as = E8iai.

1

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

141

Let a : t = t* -+ Re be the transpose of a. Then (2.106)

at = ((al,t),... ,(at,t)),

and a maps R into V.

Now we will use (2.90)-(2.93), with A = GAR, A' = R. In the sequel, we view

Ru(t,x) as an element of D'(t/CR . If p E CR /R , we define (R.)µ(t,x) as in (2.90).

PROPOSITION 2.25. If p E R /R', if Ai, E UR- represents p, then (2.107)

(&)µ(t, x) = exp(2iir(Aµ, t))Qu (t, x - 2ia`aAi,) .

PROOF. If A E Z R , then (2.108)

exp(2i1r

t)) = exp(2ia(A,t)) if A E GR' maps top E Z R'/R , = 0 otherwise .

Then

(2.109)

(Ru)µ(t, x) _

L ie:r

exp(2iir(A + Aµ, t)) fl (2iir(ai,A) - (xi - 2ia(ai,Aµ))

From (2.109) , we get (2.107). The proof of our Proposition is completed.

REMARK 2.26. By Proposition 2.25 , we get the otherwise obvious fact that the right-hand side of (2.107) only depends on p and not on A,,. For any p E Z W /R", we choose Ai, E Z R` representing p. THEOREM 2.27. The following identity holds (2.110)

R. (t, x) _ E exp(2iir(aµ, t))Qu (t, x - 2iraAi,) . µEG''7F/R

PROOF. This follows from (2.92) and (2.107).

2.7. Iterated residues and the series R, (t, x). Let r : t/Z1R -3 t/R be the obvious projection. For u E C/R , put (2.111)

Su = T -'(H.),

S1-1 (H). Clearly r maps Su into Hu, S into H. Therefore (2.112)

Su C Su ,

SCS. If G is simply laced (2.113)

Recall that by (2.20), dR C TR.

Su=Su, S=9.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

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THEOREM 2.28. For any u E C/R, t E T\ (2.114)

R,L(t, x) =

I

MR

then 1

I (-1}r 1=(i,....,i,)Cr. I generic

(a;,,... ,ai,)

/ECCKIWR

pJ t(PI-1a ,x1)rl r1(t+f),e')J111 d,j=1 vi-1a;,e') d

1

((ai,xl) -xi) r

i-i

1

p( (p(

)

PROOF. We use Theorems 2.19 and 2.27. Also, if h E t,

(ah)I = h.

(2.115)

So we deduce from (2.115) that if i E I \ I, (2.116)

(ai,(aA,)I) - (-j, A,,) = 0.

By (2.19), (2.28), if a E

dWj-1aia.1) E Z. vi-laic , ej) From (2.13), (2.62), (2.110), (2.116), (2.117), we get (2.114). The proof of our (2.117)

Theorem is completed.

2.8. Fourier series for the universal cover of a semisimple centralizer. Take u E C/R . Recall that by (2.118) irI(Z(u)) Also

(2.119)

CR C RU*.

Let Z(u) be the universal cover of Z(u). Then by [15, Theorem V.7.1], (2.120)

Z(Z(u)) =

Therefore ir1(Z(u)) = CR/j is a subgroup of Z(Z(u)). Also ZWRu is the lattice of weights of Z(u).

DEFINITION 2.29. For u E CI- R, t E t/CRu, x E (c \ 2'-Z)', put (2.121)

Ru(t,x) = E

7Texp(2iir(a,t))

AEiru [f (2iir(A, ai) - xi)

iEI

Clearly (2.122)

lr1(Z(u))` =CRu/CR

Also Ru/R maps into Cru/CR, with kernel R° maps to an element of ir1(Z(u))'.

.

RCR

In particular u E C/R

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

143

DEFINITION 2.30. For u E C/R , q E Z, t E T = t/TR, put (2.123)

Ru,q(t,x) =

R E-

ICI hE

PROPOSITION 2.31. If u E C, the following identity holds

Ru,q(t, x) = Ru(t, x + 2iirgau) exp(-2ia(qu, t)) .

(2.124)

PROOF. Since u E, our identity follows from (2.123). THEOREM 2.32. For any u E 0/1r, q E Z, t E T \ 3`, then (2.125)

(ai 1 ,si,) exp(2iir((qu,f)))

Ru,v(t,x) = ICRI (-1)* I generic fEUR-/dN

PIdf

1

((a2,xl)-x.) r

-1(t+f),d)

_la,,,ef)

1

f_1 exp(

(p? 1

y7

PROOF. We use Theorem 2.28 and Proposition 2.31. Since for i E Iu, (u, ai) E Z, by (2.13), (2.29), (2.116), we get (2.125). The proof of our Theorem is completed.

0

2.9. Bernoulli polynomials and the Fourier series Pn(t). DEFINITION 2.33. For n E N, t E T = t/GAR, put

PP(t) = - E exp(2i(a,t)) [7r(AA)In

(2.126)

J1019"

x(a)#o

For M E Ne, we will consider the partial sums exp(2ir(A, t))

(2.127)

Pn (t) = -

AE-

[n(A)]n

Clearly Pn(t) is a well-defined distribution on T = t/CR. Recall that (2.128)

K = It E t, for a in R+, (a, t) > 0} .

Then by [15, Note V.4.14] and (1.186), ew exp(2in(w(a + p), t)) Pn(t) = (2.129) [a(a + p)]n

for n odd,

wEW AEZWR

=-E aE

nX

.6w

exp(2iir(w(A+p),t)) [ir(A + p)]n

n even

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

144

By (1.186), (2.126), for w E W,

P(wt) = EP(t) .

(2.130)

Since ir(t) is a polynomial, it

(p!) is a differential operator. Then we have the

identity of distributions on T = (2.131)

(2/8t

Ir

,

) Pn(t) = Pn_I(t) for n > 1.

DEFINITION 2.34. For t E [0,1], n > 0, put e2iakt (2.132)

p(t) = - 5 (2iirk)^ k0o

If G = SU(2), then T = S1 = R/Z. One verifies easily that the pn(t)'s are exactly the Pn's associated to G = SU(2). Then (2.131) is the equation of distribution on SI (2.133)

pn (t) = pn-1(t) , n > 1.

Also

po(t) = 1 - 5{o},

(2.134)

So by (2.133), (2.134), we get 1 (2.135)

(dt J n NO) = 1 - 8{0}

By (2.135) , it is clear that the` pn(t)'s are polynomials on Si \ {0}. Also for n > 2,

the series in (2.132) is absolutely convergent on [0,1]. For n > 1, the series in (2.132) converges uniformly together with its derivatives on compact sets of S1, not containing 0. By [45, Appendix B] , the pn's are exactly the Bernouilli polynomials. In the sequel, we will consider the pn's as polynomials on R, whose restriction to ]0, 1[ is given by (2.132). Recall that (2.136)

Td(x) =

i - e-s

Then (2.137)

Td(x) - Td(-x) = 1.

Finally by [51, p147] , if the Bk, k > 1, are the Bernoulli numbers,

+_ (2.138)

2k

Td(x) = 1 + 2 + E(-1)k+1Bk k=1

PROPOSITION 2.35. For n > 0, t E R, (2.139)

et6

1

pn(t) = Res [ an ° -1

1

(2k)I

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

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PROOF. For 0 < t < 1, put eta

1

ft,n(a) = an ea - 1

(2.140)

Clearly ft,n(a) has simple poles at a = 2iak, k E Z', and a pole of order n + 1 at

a=0. Then, forkEZ' e2inkt

Resa=2inkft,n(a) _ (2tiirk)n

(2.141)

Now we use the Cauchy residue theorem inside a circle of centre 0 and radius 27r(M+1/2), as M -+ +co. For n > 2, or for n = 1, t E)0,1[, the integral of ft,n(a) on the circle tends to 0 as M -+ +oo. So we find that 57 Resa=2iakft,n(a) = 0.

(2.142)

kEZ

From (2.142) , we get (2.139) for n > 2, or n = 1, 0 < t < 1. Then since pn(t) is a polynomial, (2.139) holds for any t E R. For n = 0, (2.139) is trivial. The proof of our Proposition is completed. PROPOSITION 2.36. For n > 0, t E R,

pn(t) = Td(-8/et)

(2.143)

n!,

P.+1 (t + 1) - Pn+1(t) =

n!'

PROOF. Clearly, ateot

(2.144)

By (2.144) , we find that for jal < 2a, (2.145)

Td(-8/8t)eat = Td(-a)eat

Using (2.139), (2.145), we get (2.146)

pn(t) = Resa=o Lan}1 Td(-a)] = Td(-8/8t)Resa0 I

+1,

= Td(-8/8t) to . By (2.139) to

eta

(2.147)

Pn+1(t + 1) - Pn+1(t) = Res

[a+1+1

I = to T! .

The proof of our Proposition is completed.

Recall that R,,+ denote the set of short positive roots.

THEOREM 2.37. For n > 1, as M -+ +oo, the partial sums PM(t) converge to Pn(t) uniformly on the compact subsets of T \ S. The following identity of distributions holds on T = tlVW, 1l

(2.148)

Pn(t) =

(-1)t+1,ntnlR,.+Ib. [nPn(ti)] i=1

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

146

In particular Pu(t) E Lao(T), Also (2.148) is an identity of smooth functions on T \ S. Finally Pu(t) is a polynomial on T \ S. PROOF. By proceeding as in the proof of Theorem 2.18 , we get (2.148). By the same argument as in Theorem 2.18, we find that Pn (t) is smooth on T \ S. Also the uniform convergence results for the p^n'f (t)'s and (2.148) imply the corresponding uniform convergence for PM(t). Now, we will show that Pn(t) is a polynomial on T \ S. In fact we will prove that for 1 < i < 1, for p large enough, (2.149) hay Pu (t) = 0 on T \ S.

Clearly b.8/8tI = ha,. Then using (2.135) , we obtain (2.150)

ha,b.[pn(t1)...pn(tt)] = b.[pn(t2)...Pn(te)] b.[8tl=opn(t2) ...pu(tt)]

Clearly

halb.[pn(t2) ...pu(tt)] = 0. Let V1 be the vector space in t spanned by ha, , ... , hae . If V1 is not equal to t, the support of b«[bt,=opn(t2) ... pu(tt)] is included in S. From (2.150) , we then get (2.151)

hn- 1b.[pn(t1)...pu(tt)] = 0 on T \ S. If VI = t, we can express ha, in the form t (2.152)

(2.153)

has = E aJhat i=2

By (2.135), (2.154)

ha,b.[Stt=opn(t2) ...pn(tt)] = b.[8tl=opn(t3) ... pn(tt)] - b. [Bt1=o,tx=opu(t3) ... pu(tt)] A.

Let V2 be the vector space spanned by ha ... , hae . If V2 # t, then by the analogue of (2.152), (2.155)

ha;14.[6tl=opn(t2) ...pn(tt)] = 0 onT \ S.

Now using (2.155) , and iterating the above argument, it should be clear that for p E N large enough, (2.156)

ha1Pn(t) = 0 on T \ S.

In (2.156), we may as well replace the index 1 by the index i, 1 < i < 1. Therefore we have established (2.149). The proof of our Theorem is completed.

2.10. The Fourier series Take u E C/R . Recall that 2(u) is the universal cover of Z(u). Then Z(u) is connected and simply connected. Therefore to Z(u), we can associate the objects we just constructed for G. Observe that t/emu is a maximal torus in Z(u), and t is its Lie algebra. DEFINITION 2.38. Let iru(t) be the function on t (2.157)

iru(t) = fl (2iira, t) .

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

147

Then 7ru(t) is the analogue of rr(t).

DEFINITION 2.39. For u E C/R ,

n E N, t E t/, put exp(2iir(A,t))

(2.158) AE

y

(rru(A))n

W-Noo

Then P.,,1(t) is the analogue of for Z(u). Take w E W. We identify w with a representative in N(T)/T. Then if u E C/R , W Z(u)w-1,

(2.159)

i'Rwu = wti[LU, Rz(wtt) = wRu, Rz(wu),+ = wRu n R+ . By (2.159), we get (2.160)

(-1)IR+nw(-R,..+)Inu(w-1t)

iwu(t) _

.

PROPOSITION 2.40. If u E GI-R, n E N, W E W, then (2.161)

Pu,n(t) =

('1)nlR+nw(-R,..+)IPwu,n(wt).

PROOF. Clearly by (2.159), (2.160), e2ia(a,wt)

(2.162)

u,n(t)

_ aEZ°Tw

(7ru(w_1A)]n

= -(-1) nIR+ nw(-R..+

e2ix(X,wt)

11

2-1 AEZSE,',

(lr wu (A) J n

-1)n[R+nw (-R».+) I Pwu,n (wt),

0

which coincides with (2.161).

Let Su c t/CR, be the analogue of S C t/GTR. Of course Su projects into Su. Then by Theorem 2.37 , A,. (t) is polynomial on Su. Let i1, ... , it. be the elements of Iu = {i E {1, ... , t}, ai E Ru,+} arranged in increasing order, i.e. i1 < i2 ... < it,,. If f (x) = f x E Riu , we denote by the expression (2.163)

Res.=of = Res.,,. --o... Res.,of (241, ... , xit,. )

It will be of fundamental importance that the order in (2.163) is fixed once and for

THEOREM 2.41. For U E C/f, n > 1, (2.164)

P, ,n (t) = -Res.'-=o (JJ1xi)n R (t, x) .

W.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

148

PROOF. Clearly (2.165)

Resz,_o

1

_

1

x; 2ia(ai, A) - xi

1

(2ii(ai, A))n =0

if (a A)

0,

if (ai, A) = 0.

Using (2.165) , we get (2.164). The proof of our Theorem is completed. REMARK 2.42. The identity (2.164) can be viewed as an identity of distributions on t/CRu, of smooth functions on (t/ZrR-.) \ SS,,, of elements of Also for the moment, the ordering in (2.164) is still irrelevant.

2.11. The function Pi,n,q(t). If t E t, we still denote by t the corresponding element in T = t/UR.

DEFINITION 2.43. ForuEC/Ih',gEZ,nEN,tEt/C ,put (2.166)

V'- E exp(2iir(qu,h))Pu,n(t + h).

Pu.n>q(t) = I

CI

Observe that the function exp(2iir(qu,t))Pu,n,q(t) descends to a well defined function on T = t/CR. Equivalently, we may consider Pu,n,q(t) as a section of the

flat line bundle Lu on T, associated to u E t'/Z°`. PROPOSITION 2.44. If u E C/1R`, q E Z, n E N, w E W, then (2.167)

(-1)"'A+nw(-R,,.+)IPwu,q,n(wt)

Pu,n,q(t) =

PROOF. Clearly w :

->

Proposition 2.40.

UN

Rwu

is one to one. Then (2.167) follows from

THEOREM 2.45. The section Pu,n,q (t) is a polynomial on T \ SS.

PROOF. Since Pu,n(t) is polynomial on (t/C) \ Su, it is clear from (2.166) that P,,,n,q is polynomial on T \ Su. THEOREM 2.46. For u E C177% n E N, q E Z, t E t/G%, the following identity holds

(2.168)

Pu,n,q (t) = -Resx_,

n &,q (t, x) .

1

Xi

PROOF. This follows from (2.123), (2.164), (2.166) .

DEFINITION 2.47. For u E C11 r, let Z be the set of generic I = (i1, ... , ir) C

Iu such that if yr E Sr is defined by i,r(l) < i,r(2) < ... < i,f(r), if j E Iu \I, either j < al (l), or if pf is defined by the condition (2.169)

ir(l) < ... < i,r(pi)
then (2.170)

aier(1)

Aaj 0 0.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

149

Condition (2.170) is equivalent to (2.171)

e j it {ai,,(1),

r

If I E Il1, we define Res.'=0 by the formula Res.'.'=o = Res:,,(,)=o ... R.es.,'(1)=o .

(2.172)

THEOREM 2.48. For u E C/R , n > 1, q E N, if t E t represents an element of T \ 3, then (2.173)

E

P.^9(t) = 1-11 (-1)r+1

1

\

(ail) ...

exp(2irr((qu, f)))

/erwIdw n j-1air,xI)

R.esl:=0

exPd

I

eJ)

aER, ,+

r

[!(P'I-1(t+f),e'7)]

d(pJ_l1

,xl) i=1 exp( (Pj--jam)

PROOF. By Theorems 2.32 and 2.46, for t E T \ S, we get

pu'ri's(t) ° ICI (-1)r+1

Js(11 g... iv)Cla

(ai1, 1 ,ai)

exp(2i7r((qu,f)))

/JEZ`JF/d1F

(2.174)

1

Rest=o (

r

1

W.

exP d E -1

(PJ_la'i r-jr

zf)

-

iEl,.\I

r

1

1

[(pi-i(t+f), ej)) JJ

d(pf_1al 21)

j=i exp

Now x1 depends only on xil,... ,xi,,. Therefore in (2.174), the dependence on the

(xi)iet \I is only via the term

1

II x. ((ai,xl) -xi) iEl.,\I

Take e > 0. Let v E C, Jul K e, w E C, Jwi > e. Then by the theorem of residues, for n > 1,

if

_

dx,

2iri f icc x?(V+w-xi) 1

i

r

27xi J Jec x° (V - xj)

0.

Take jE.T,,\I.If (2.176)

aj= k=1

ak ai'f (r) ,

1

(V+w)' '

JEAN-MICHEL BISMUT AND ERANQOIS LABOURIE

ISO

then

(aj, xt) _

(2.177)

akXi..,(r) k=1

Put Pi

r

v=E

(2.178)

w= E akxi>,(u) k=pi+1

L-1

Observe that to is identically 0 if and only if ak = 0 fork > pj + 1, i.e. if aj E {ai"(1),... aie,(pi)}.

I

To evaluate (2.174), we will use (2.175). Namely take a sequence et,... , et. in R+, with 0 < ei1 < ... < et,,. Then if f (xi.... , xieu) is a meromorphic function, by definition, (2.179)

;

Res:_of = 1

i1

f (xi1, ... , xie ) 2iir

...

doie 2iir

t5i<e If the sequenceel,... , et is enough decreasing, when taking the xi, as in (2.179), if aj V {aier(....... in (2.178), then jwj > ej. Using (2.175), (2.179), we find easily that for j E I. \ I, (2.180)

Resxi=ox

1

((axl) - x7)

=0

if c

E {a;,,(1),... aio;(pi)}

J,

(ay, xt)n the

if aj

{aiel(1),... ,aiv,(pi)}.

A related argument is as follows. Define v, w by (2.178). If to = 0, then since k < Pi have been made nearly equal to 0 before xj,

-r+1 +oo vk

1

(2.181)

v+w-x j

k

k=o xj

From (2.181) , we get 1

(2.182)

si=b [Xill v+w-xjJ

=0.

Ifw54 0, then

= +'

1

(2.183)

v + w - xj

(xj - v)k

wk+1 k-o

By (2.183) , we get (2.184)

Ressi=o

1

+00 Gk-1(-v)k_1+1

1

xJ v + w - xj

k ,n .1

wk+1

1

(v _+W)_

which gives another proof of (2.180). Finally observe that if j E I, (2.185)

(aj,x') = xj.

From (2.174), (2.180), (2.185), we get (2.173). The proof of our Theorem is completed.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

151

By Theorem 2.22, we already know that, as a section of a line bundle, P.,..,q (t) is polynomial on T\S. By Theorem 2.22, the right-hand side of (2.173) descends to a section of the same line bunlde . We will give a direct proof that the right-hand side of (2.173) is polynomial on T \ S.

THEOREM 2.49. The right-hand side of (2.173) is a polynomial on T \ S.

PROOF. In the right-hand side of (2.173), up to a locally constant factor on

T \'9, we may and will take off the brackets in [p_i(t + f ), e1)] . By (2.13),

E

(2.186)

(pj-iait,XI)

<

(t+ f, X-') _

Fj(t)x,'(j)

and the Fj (t) are affine functions oft E L Therefore

+w Fj t 1'x,r (i)) k

exp((t+f,xr))

(2.188)

/

j=1 k-0

Moreover

= Td

1

(2.189)

exp

Also by (2.28), (2.29), (Vj-jCtjj'XI)

VI-1aii,e'1)

(pJ_jJ

a

d(ptaiz')

d(p!_lat ,xt)

-

d(p'aJ z')

1

(ai,,...,ai,) j-1

((ail,... , a,_,)xi, -E(ai"...

aik}1,... Iaij_1)xi.

k=1

Finally the (aj, xl) are linear combinations of xi, , ... , xi, and consider one of the terms in (2.173) as a Freeze now xi,,(')'... function of xie,ill. Then it is clear that this term is a meromorphic function of xie,11I with a pole of finite order at xi,,(3) =0. When taking the residue at 0 in the variable xi.,(l) , by the above, it is clear this will introduce a finite power F1(t)k, i.e. a polynomial function of t. It is now clear that the procedure can be iterated. The proof of our Theorem 0 is completed.

2.12. The action of a differential operator on Pu,u,q(t). Let s E t' + F(s) E C[s] be a power series, which converges on a neighborhood of 0. Then F(818t) is a formal power series of differential operators. Since Pu,,,(t) is a polynomial on T \ Su, F(8/8t)Pu,,,(t) is a well-defined polynomial on T \ Su. Then if I = (ii,... ,i,) is generic, if Recall that we have identified t and

xECl,then x'Et=t.

JEAN-MICHEL BISMUT AND FRANCfOIS LABOURIE

152

THEoREM 2.50. For u E C/Jr, n E N*, q E N, if t E t represents an element

ofT\S, then (2.191)

R

F(8/8t)P.^v(t) = I

1

I (-1)r+1

exp (2ia((4u, f))) Resx,

(F(x1)

(a{10... ,ai.)

( (a .1))n AEI

P d(p;_la;;,ei)

t

(,.p

1

- 1`

5,

.

il-

p is

ei) /

PROOF. For y gE 0, [y} -y is locally constant. Using (2.186), if, E t is identified with the corresponding vector field, then _la:t,x r1 (2.192) s exp(dJ -lafi j) [ (i

(s, x') exp (d±

'Ial''xI) L1(Pi(t+f),e')JI d

From (2.173), (2.192), we get (2.191).

The proof of our Theorem is completed.

i=1-Ia;;,e1)

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153

3. Symplectic manifolds and moment maps In this Section, we recall known results on symplectic manifolds and moment maps. In particular we prove a form of the formula of Witten [64] Jeffrey-Kirwan [28], which expresses the integral of certain characteristic classes on symplectic reductions in terms of the action of differential operators on the symplectic volume. The results of this Section will be used in Section 5, where we will give a formula for the integral of certain characteristic classes on the strata of the moduli space. This Section is organized as follows. In Section 3.1, we recall elementary facts on orbifolds. In Section 3.2-3.4, we give elementary properties of moment maps. In Section 3.5, we give a direct simple proof that the image of the symplectic volume measure by the moment map can be evaluated in terms of the symplectic volume of the symplectic reductions. In Section 3.6, when 0 is a regular value of the moment map µ, we recall Jeffrey-Kirwan's expression of the volume of the neighbouring fibres in terms of integrals of characteristic classes on the symplectic reduction of µ-l (0). In Section 3.7, we prove the formula of Witten-Jeffrey-Kirwan. Finally in Section 3.8, we apply the above to the symplectic coadjoint orbits of G.

3.1. Orbifolds. Let X be a smooth compact manifold. Let G be a compact connected Lie group, and let g be its Lie algebra. We assume that G acts on X on the right. If Y E g, let YX E Vect(X) be the corresponding vector field. We assume that G acts locally freely on X, i.e. for any non zero Y E g, YX is a non vanishing vector field on X. Let gX be the subvector bundle of TX which is the image of g by I' -a YX. Then we have an exact sequence of G vector bundles 0 -+ ox -+ TX -> TX/gX -+ 0.

(3.1)

The above data define an orbifold X/G, and the G-bundle TX/CQX is also called the tangent bundle TX/G to X/G. If the G-bundle TX/gv is orientable (or equivalently if TX is orientable), we will say that the orbifold X/G is orientable. If G acts freely on X, then X/G is just the standard quotient.

If y E X, let Z(y) = {g E G; yg = y} be the stabilizer of y. Then Z(y) is a finite subgroup of G. By [24, Proposition 27.4] , there are finitely many conjugacy classes of finite subgroups of G, which occur as stabilizers. Inclusion induces a partial ordering on the set of conjugacy classes of finite

subgroup of G. On each connected component of X, there is a unique minimal conjugacy class of stabilizers S, called the generic conjugacy class of stabilizers. This minimal conjugacy class then acts as the identity on the considered connected component. The order IS] of a generic stabilizer is locally constant on X. Let Xreg be the set of y E X such that Z(y) lies in the minimal conjugacy class. Then Xreg is open in X, and Xrg/G is a smooth manifold included in the orbifold

X/G. DEFINITION 3.1. A 1-form B : TX -* g is said to be a connection form on

ir:X-+X/G if For Y E 91

0(YX)=Y.

(3.2)

For9EG,YE9, (3.3)

g"B = 9.g.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

154

One verifies trivially that G-connections exist. Then the curvature a of 0 is defined by

d9=-2[0,0]+e.

(3.4)

Also if YEg,9EG, iyx 8 = 0, g'e = e.g.

(3.5)

Put THX = {U E TX,9(U) = 0}.

(3.6)

Then THX is a G-invariant subbundle of TX such that

TX = THX ®gx. A G-invariant form a on X is said to be basic if for Y E g, iyxa = 0. From

(3.7)

now on, we suppose that X/G is an oriented orbifold. Then Xreg/G is an oriented manifold. By definition (3.8)

/r

JX/G

a=

a.

Let E1i... , E. be a basis of g, let E1, ... , En be the corresponding dual basis of g". We write the connection 0 in the form 0 = E1 B'E; .

(3.9)

Then E' A ... A En defines a volume form on G. Let Vol(G) be the corresponding volume of G.

We equip 9x with the orientation induced by the orientation of g. Then TX TX/gx ® gx is naturally oriented. If a is a G-invariant basic form on X, (3.10)

f

f aAO1 A...AOn.

le ( ) L/G a=VoIS1

If a is a G-invariant basic form on X, da is also G-invariant and basic. Then submanifolds of codimension > 2, since (X/G) \ (Xreg/G) is a union orfx/G Jda = 0.

(3.11)

In fact note that (3.12)

Jx

da A O' A ... A on _ (-1)deg(a+1)

r a A d(91 A ... A on). x

By (3.4), (3.13)

fxaAd(O'A...Aon)= fxaAE(-1);'101A...A0'-1A@'A...Aan=O, {=1 which provides another proof of (3.11).

Let E -i X be a complex G-vector bundle on E. Let VE be a G-invariant horizontal connection on X. Namely suppose first that G acts freely on X. Then Va is just a connection on the vector bundle X xG E over X/G. More generally, if G only acts locally freely on X, the above construction still makes sense. Let FE = VE,2 be the curvature of V. Then FE is a G-invariant basic 2 form on X with

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

L55 P(2FE

values in g. Let P be an ad-invariant function on g. Then P(E, DE) = ) is a G-invariant basic dosed form on X. If X/G is an oriented orbifold, the integral fx/a P(F, DE) is well-defined and does not depend on VE.

3.2. Symplectic manifolds and moment maps. Let (X, a) be a symplectic manifold, so that a is a nondegenerate closed 2-form. Let H : X -> R be a smooth function. The corresponding hamiltonian vector field YH is defined by the equation dH = iyxa, (3.14) which we rewrite in the form

(d - iy,,)(H + a) =0.

(3.15)

From (3.15), we get

LYXa=0.

(3.16)

If H, H' are two smooth functions, put

{H,H'} =

(3.17)

Then (3.18)

Y{H,H'} = [YH,YH')

Let G be a compact connected Lie group acting on the right on X, and preserving o. Let g be the Lie algebra of G. Let q : G -i Aut(g*) be the coadjoint representation. If Y E g, p E g", let {Y, p} E g` be the infinitesimal (left) action of Y on p. We say that IA: X -+ g* is a moment map if

ForxEX,9EG, (3.19)

µ(x9) =

If Y E g, if Yx is the corresponding vector field on X, then (p, Y) is a Hamiltonian for Yx. In particular, by (3.17),(3.19),

(3.20)

(µ, [Y, Y7) = -a(YX,Y X ).

Also by (3.19), (3.21)

YX µ = -{Y µ}.

Assume now that G acts locally freely on X. Of course, this never occurs if X is compact. Let 8 be a G-connection form on ir: X -> X/G. PROPOSITION 3.2. There is a unique closed 2-form 71 on X/G such that (3.22)

a = a'q - d(µ,8).

PROOF. By (3.3) , ( 3.19) , the 1-form (µ, 0) is G-invariant, so that if Y E g, (3.23)

Lyx (µ, 8) = 0.

Also by (3.16), (3.24)

Lyxa = 0.

JEAN-MICHEL BISMUT AND FRANC-OIS LABOURIE

188

Then by (3.14), (3.23), iyx (a + d(p, 9)) = iyxa + Ly.. (p, 9) - diyx (p, 9) = 0 (3.25)

By (3.25), we find that the G-invariant 2-form a + d(u, 9) is basic. Therefore it is of the form it i . The proof of our Proposition is completed.

Let alt be the restriction of or to TF1X. Since au is 0-invariant, it descends to a 2-form on X/G. The same is true for the 2-form (u, 0). THEOREM 3.3. The following identity of 2-forms holds on XIG, aH = lr«0 (3.26) - (u, 0) PROOF. By (3.4),

d(u, 0) = (du, 0) + (u, -2[9, 0] + 0).

(3.27)

From (3.27), we get (3.28)

[d(p, 9)]H = (u, 0) From (3.22) , (3.28), we get (3.26). REMARK 3.4. If X E g, by (3.15),

(d - iyx) (a + (u, Y)) = 0.

(3.29)

Classically [61], (3.29) shows that att + (p, 0) descends to a closed 2-form on X/G. Of course this also follows from Proposition 3.2 and Theorem 3.3. REMARK 3.5. Let (,) be a G- invariant scalar product on g. Then g and g« can be identified. Let H C G be a Lie subgroup of G, and let h C g be the corresponding

Lie algebra. Let ht- be the orthogonal space to h in g. Then g = 4 Q) 4J- is a Hinvariant splitting. Let PO : g -+ 4 be the corresponding projection. Put 9h = P4 0, 8

(3.30)

= P4 ''8.

Then 94 is a connection on X .- X/H, whose curvature eh is given by

Dry = Ph0 - 2Ph[9''1,9h1

(3.31)

Take u E g«, and suppose that Z(p) = H. Then (3.32)

(u, 2[010]) = (u, 2[9 L,9b

2Ph[9n1,9°1]>

Therefore by (3.31), (3.32), (3.33)

(p,-2[9,9]+0) _ (u,01).

So by (3.4), (3.22),(3.27), (3.33), if Z(u) = H, (3.34)

a = 7r"`r7 - (du, 9) - (u, 04).

Equation (3.34) is of special interest when u E t« ; t, and H = T.

Put (3.35)

E=X XGg«

Then E is an orbifold vector bundle over X/G. Moreover (3.19) says that u descends

to a section of E over X/G.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

157

3.3. Symplectic reduction. We make the same assumptions as in Section 3.2. We do no longer assume that G acts locally freely on X. If p E g*, put (3.36)

XP =

p1(p)

Let Z(p) = {g E G,g.p = p} be the stabilizer of pin G, and let 3(p) = {X E g, {X, p} = 0} be its Lie algebra. By [15, Theorem IV.2.3], Z(p) is a connected Lie

subgroup of G. Clearly Z(p) acts on X. PROPOSITION 3.6. The element p E g* is a regular value of A if and only if for x E X,, Y E g i-+ YX (x) E T:X is injective, or equivalently if and only if Z(p) acts locally freely on XP.

PROOF. By (3.29) if x E XP, Y E g lies in coker(dµ(x)) if and only if Yx(x) _ 0. Also by (3.21), if µ(x) = p, YX (x) = 0, then 11, p} = 0, i.e. Y E 3(p). The proof of our Proposition is completed. 0

Assume now that p is a regular value of p, and that Xp 0 0. Then X, is a smooth submanifold of X, on which Z(p) acts locally freely, so that XP/Z(p) is a Z(p)-orbifold. Let iP be the embedding X, -+ X. Then by (3.21), (3.29), for Y E 3(p), (3.37)

iyxiya = 0.

By (3.37) , we find that the Z(p) -invariant closed 2-form iya descends to a dosed 2-form aP on X IZ(p). Then (XP/Z(p), a,) is a symlplectic orbifold. If p E g*, g E G maps X. into XP.9. Also p E g* is regular if and only

if p.g is regular, and there is an obvious symplectomorphism (XP/Z(p),a,) -4 Let O C g* be a coadjoint orbit. If p E 0, Y, Z E g, put (3.38)

ao(Y, Z) = (p, [Y, Z])

By (1.193), ao is a G-invariant symplectic form on O. Moreover µo : p E O Hr -p E g* is a moment map for the right action of G on O.

Put (3.39)

Xo = IA-,(O).

Then G acts on the right on Xo C X. Let pi, p2 be the projections X xO -+ X,X xO -4 O. Then (X x O,pia+p2*ao) is a symplectic manifold on which G acts symplectically on the right, with moment map pile + pzµo.Then (3.40)

(X x 0)0 = { (x, p) E X x O, µ(x) - p = 0}.

So, if p E O, (3.41)

(X x OP)o _ Xo,, (X x Op)o/G = XPIZ(p)

Moreover p E g* is a regular value of u if and only if 0 is a regular value of p 1p + paµo. Then one finds easily that (3.41) identifies the symplectic forms on the corresponding orbifolds.

JEAN-MICHEL BISMUT AND FRANCOIS LABOURIE

168

3.4. Symplectic reduction with respect to a fixed stabilizer. Let G be a compact connected semisimple Lie group. We use the same notation as in Section 1.14.

Let (,) be a G-invariant scalar product on g. So we can identify g and g*, t and V. Let rl : G - Aut(g*) be the coadjoint representation. Of course we can now identify the representations r and q. Take po E g*. Put Z = Z(po),

(3.42)

3 = a(po). DEFINITION 3.7. Put (3.43)

9t = {p E t*, Z(p) = Z}.

Since the Z(p) are connected, (3.44)

91 = {p E t*,3(p) = 3}.

By (1.182), (3.44), 91 is the complement of a finite union of hyperplanes in {p E t*, if (po, h.) = 0, ( , h.) = 0}. Now we make the same assumptions as in Section 3.2. Suppose that G acts locally freely on X. Then by Proposition 3.6 , any p E g* is a regular value of p. DEFINITION 3.8. Put (3.45)

6 = µ 1(9t).

Then 6 is a submanifold of X, on which Z acts locally freely. Let ae be the restriction of a to S. Then ore is a closed 2-form on 6, which in general is not

symplectic. Let j : g* - a* be the obvious projection. Then jµ : 6 4 i* is a moment map for the action of Z on 6 with respect to ae. Finally 6/Z embeds into X/G. Let P be the orthogonal projection g -> g. Let 9 be a connection form on

X - X/G. Put B = P9.

(3.46)

Then 9 defines a Z-connection on X - X/Z. Let 6 be the curvature of 9. Clearly, over 6, since p E g, (3.47)

(p, 9) _ (p, B)

By (3.34), over 6, (3.48)

a = lr*n - ((dp, 9) + (p, ®))

In particular, if p E 9t, (3.49)

i;a =

i;(r*q

- (p, ®))

Recall that the vector bundle E was defined in (3.35). Let tE be the connection on Eke/Z induced by 9. Then over Xp/Z, (3.50)

O.&p

= 0.

Equation (3.50) explains why over Xp, a given by (3.49) is closed. In fact it is the pull-back of the symplectic form ap on Xp/Z(p).

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

159

PROPOSITION 3.9. Locally, over 5, the cohomology class of ay depends linearly

onpE9Z. PROOF. This is obvious by (3.49).

3.5. The image of the symplectic volume by the moment map. Let G be a compact connected semisimple Lie group. We use otherwise the notation of Section 1. Let (X, a) be a compact symplectic manifold, on which G acts symplectically, with moment map p : X -) g'. We make the assumption that Z(x) = 1 for a.e. x E X.

(3.51)

Then a.e., G acts locally freely on X. By Proposition 3.6, a.e., dp(x) surjects on g'. By Sard's theorem, a.e. p E g' is a regular value of p. Let dp, dt be the Lebesgue measures on g, t with respect to (, ). Let dg be the Lebesgue measure on G. If p E g, let dg, be the Lebesgue measure on Z(p). Clearly {e ° } °18x

(3.52)

dimx 2 = a

(dim X/2)!'

x/z

, is a dim X form on X, which does not vanish, and so defines an orientation of X. If f : X -+ R is a bounded measurable function, put

Then °°m

r f(x)Ie°l

(3.53)

=

r f(x)e°,

x

Ix

The notation (3.53) emphasizes the fact that the left-hand side is an integral with respect to a nonnegative measure. We will use a similar notation over XP/Z(p). Now we will compute the disintegration of the symplectic volume on X with

respect to the moment map A. The point of this proof is that it is parallel to a corresponding for moduli spaces given in Theorem 5.45. Recall that the monomial x : t -+ C was defined in Definition 1.53.

THEOREM/ 3.10. Let f X -> B. be a bounded measurable function. Then (3.54)

Jx f (x)I e°] = Jx f (x)le°l

=

f

/W Jir(t)I dt fx`/T le" I fG f

J9 11r(p)1

fx,/z(v) l e", I fz(v> f (x g)dgp dim x/7

PROOF. Since dµ(x) is a.e. surjective, it is clear that µ, ° m X/2)! is absolutely continuous with respect to dp. Also for a.e. x, dp(x) is surjective and p(x) E gm". Let K C t be a Weyl chamber. Let q be the projection greg -+ treg/W = K. If x E X is such that µ(x) E gres, we have a G-equivariant complex (3.55)

t-+0,

where the map g -* TTX is just Y -> YX(x). If d/e(x) is surjective, by Proposition 3.6, the cohomology of the complex (C,,8) is concentrated in degree 1. More precisely, one finds easily that in this case, (3.56)

H1(C,,8) =T,(x)Xgµ(x)IZ(gp(x))

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

160

In particular, by [35] and by (3.55), (3.56), we have a canonical isomorphism of real lines

(det C' )-I --

(3.57)

/Z(gp(x)))

Now det(TTX) is equipped with the volume form associated to v,,.Also g and t are equipped with the volume forms dp and dt. Therefore det(CC) is equipped with a natural metric. Let dvX9,.(m)/Z(qu(x)) be the corresponding volume form on TR(z)Xq (x)/Z(gµ(x)) via the isomorphism (3.57). By the formula of change fof variables,we get

1 f(x)Ie°I =

(3.58)

X

JI, /W

dtJ

X,/'

dvX,/T(x) f G

Take t E treg, x E X such that µ(x) = t. Consider the double complex (3.59)

0

0

I-0T

0

0 d(qp)

I

r

I

I

I

1

I

I

I

I

0

0

0

09T.X ED Tin Ad(t)

--0-0

0 -0 -> Im Ad(t) -'- Im Ad(t) -b- 0 In (3.59), the map g -+ Im Ad(t) is just f r- -Ad(t) f = -[t, f], the map TX -+ g is dµ(x). Also the columns and the lowest row are acyclic. In particular the cohomology groups of the two upper rows are isomorphic. Therefore they are concentrated in degree 1 and equal to HI(C.1,8). In the second column of (3.59), we equip Im Ad(t) ^-- TtOt with the 2-form or,, given in (3.38). In the third column of (3.59), we equip Im Ad(t) c TtOt with the metric induced by the metric of g. By (3.22), (3.27), the volume induced by l-,.7g , 1T)/2 the second row on its determinant is just the symplectic volume m e T /2 . on

detH1(C,,8). On the other hand, let R be the lowest row in (3.59). Since it is acyclic, det R = R has a canonical section n. Clearly (3.60)

1171 = 17(01-

Therefore dim(X,/T)/2I

(3.61)

dvx,/T = iWW01

(diim(Xt/T)/2)!.

By (3.58), (3.61), we get the first identity in (3.54) . By [14, Proposition 6.3.4], if h : g -+ R is a bounded measurable function, (3.62)

f h(p)dp = yo (T) 1

/w

I gr(t) I adt IG h(t.g)dg.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

161

From the first identity in (3.54) and from (3.62), we obtain

(3.63) J f(x)Ie°I = X

11W

Iir(t)Idt f,/T T\Ie°`I fG d 9 f f(x.t' 9)dt' = X

I.

T

I (P)I

fxr/Z(v) Ie''I f(P)

which is the second identity in (3.54). The proof of our Theorem heorem is completed.

DEFINrrioN 3.11. If t E t is a regular value of p, let IV(t)I be the absolute value of the symplectic volume of Xt/Z(t) with respect to at. THEOREM 3.12. Let f : g* -> R/ be a bounded measurable function. Then (3.64)

fX f(i (x))le°I =

f/w [f

f (t.9)d91 iir(t)IIV(t)Idt,

fx f(p(x))Ie°I = Vol(T) f f(p)IIr(p)I

dp

PROOF. Our Theorem follows from Theorem 3.10.

Clearly, if 0 is a regular value of µ, p*Ie°I has a smooth density with respect to dp near p = 0. In particular (3.65)

lim IV(p)I °yyp` Iir(p)I

exists and is the value of the above density at p = 0. A more precise statement is as follows.

PROPOSITION 3.13. If 0 E g* is a regular value of p, and if (3.66)

Z(x) =1 a.e. onXo,

then

(3.67)

IV(p)I

_ Vol(G)I`'(0)I Vol(T)

PROOF. By 3.12, one gets (3.67) easily.

3.6. The symplectic volume as a polynomial near the origin. In the sequel, we will assume that 0 is a regular value of µ, and also that (3.68)

Z(x) = 1 a.e. on Xo.

Let U be a W-invariant open neighborhood of 0 in t* consisting of regular values of µ. Let g*, C g* be the corresponding union of coadjoint orbits. Then g'p is an open neighborhood of 0 in g. Classically, U is small enough, there is a G-invariant open neighborhood V of Xo in X such that we have the identification of G-spaces (3.69)

V = Xo x g*p,

so that in the right hand-side of (3.69), IA is just the projection M x gU -+ gu By (3.69), if p E U, (3.70)

X0, = Xo x 0p,

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

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so that (3.71)

XoD/G = Xo xa On = XoIZ(p)

is a Op fibre bundle over the orbifold Xo/G. Now we construct an orientation over Xo/T. In fact if t E Tr g, then Ot and so

G/T,

Xo/T = Xo xG G/T. Let K C t be a Weyl chamber, and let t E K. Then the symplectic form ao, orients GIT = Ot, and the corresponding orientation on GIT does not depend on (3.72)

t E K. Also Xo/G carries the symplectic form ao. Therefore once K is fixed, Xo/T carries a canonical orientation. Take t E U. Let it : Xt -+ X be the obvious embedding. Then by (3.37), i,*a descends to a closed 2-form on Xt/T, which we denote at. Note that if t E t,.eg,

at = at. If t V t,eg, T C Z(t),T # Z(t), and at is in general not symplectic. By (3.69),

Xt = Xo x {t}.

(3.73)

We equip Xt/T = Xo/T with the given fixed orientation. DEFINITION 3.14. Put (3.74)

P(t) =

x,/T Then P(t) is a smooth function on U.

e$'.

THEOREM 3.15. One has the identity (3.75)

1P(t)j

= IV(t)I if t E tree, = O if t I trog.

IfwEW,tEU, (3.76)

P(wt) = e,,,P(t).

Also, near 0, P(t) is a polynomial. More precisely, if B is a connection form on it : Xo -> Xo/G, and if 6 is its curvature, (3.77)

P(t) = fxe/T exp (lr*co - (t, 6) + (t, 12[g, B]))

.

Near 0, P(t) and a(t/i) either vanish together or are nonzero, and then they have the same sign. In particular (3.78)

rr(t/i)P(t) > 0 near 0.

PROOF. By the considerations we made after (3.72), (3.75) holds. If w E W, let g E N(T) represent w E W = N(T)/T. Since G is connected, it acts on Xo, 8, g` by orientation preserving maps. Clearly (3.79)

TXo = rr*(TXo/G) ®gx.

Also (3.80)

g®RC=t®RG'®(go aER

)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

163

so that (3.81)

ON OR C = ®ga*ER

Then (g/t) OR C descends to the complex flied tangent space to the fibre G/T over Xo/G. Since g E N(T), g preserves t and its orthogonal t1 in g. Since g changes the orientation oft by the factor e,e, it changes the orientation of g/t by the same factor. So g changes the orientation of Xo/T by the factor e,e. Since a is G-invariant,

and g acts on Xo/T, (3.82)

ogt = at

From the above, we get (3.76). By (3.22),(3.27),

(3.83)

a = ir'rl - (du, 9) - (p, 8 - 2 [B, e]).

Clearly it7r'i is cohomologous to ip7r'7) = a'oo. So by (3.83), we get (3.77). From (3.77), it is clear that P(t) is a polynomial near 0. If t E U\t,eg, by the considerations after (3.72), of is an everywhere degenerate 2-form on Xt/T. Therefore P(t) = 0 on U\t,eg.

(3.84)

For t E U n tre$, of is a symplectic 2-form on Xt/T, so that P(t) # 0 on U fl treg.

(3.85)

By (1.190), (3.85), P(t) and 7r(t/i) have the same zeroes on U. By (3.38), on Xo/T, the form (t,1[9, 9]) is just the symplectic 2-form along the fibre G/T = Ot. If t E K fl U is close to 0, by (3.72), (3.77), P(t) > 0. Therefore, on K n U, P(t) and 7r(t/i) have the same sign. Using (1.186) and (3.76), we get the end of our Theorem, the proof of which is now completed. REMARK 3.16. Recall that we have a fibration Xo/T % X0/G. Then we can rewrite (3.77) in the form (3.86)

P(t) = r Xo/G

r e01

e-(t e)+(e,[e,e])

G/T

In (3.86), f0.... is an integral along the fibre. In Remark 3.26, we will reinterpret identity (3.86).

3.7. A formula of Witten-Jeffrey-Kirwan. We make the same assumptions as in Section 3.6. Also we suppose the set U to be bounded. Let (,) be a G-invariant scalar product on g. If f : g' -+ R is a bounded measurable function, let M f : g' --> R be given by (3.87)

Mf (P) = Vol(G) L f (p.g)dg

Recall that g and g' have been identified by (,), so that (3.87) also makes sense when f : g - R is bounded and measurable.

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DEFINITION 3.17. If f : p* -+ R is a bounded measurable function with support in g*,, for u E g, put (3.88)

4(u) = f f(p)e(u,v)dp. a

Clearly

Mfe = Mfg.

(3.89)

Also fe(u) is an analytic function of u. If f : t* -+ R is a bounded measurable function with support in U, for t E t, set

A(t) = f

(3.90)

f(p)e(t,v)dp.

t

If f is W-invariant, then ft is also W-invariant. The maps f H fe and f i-+ ft are injective. Also by Rossmann's formula [50], [60], if f : g* a R is taken as before and is G-invariant, then, (3.91)

Oft = n(2"')f6-

In the sequel, we orient X by the symplectic form a. Now we prove a formula related to a formula of Jeffrey-Kirwan [28] in a work where they prove a formula by Witten [64]. Our approach is closely related to Vergne [61] and especially to Liu [39, 40] who worked out similar formulas in the context of moduli spaces. THEOREM 3.18. Let f : g* -+ R be a bounded measurable function with support in 9t. Let 0 be a connection form on Xo -+ Xo/G, and let 0 be its curvature. Then (3.92)

fx f(µ)e° = Vol(G)

fr°/GMf0(-8)e°0.

PROOF. By Theorem 3.12,

fx f(p)e° = Vol(G)fyw Mf(t)jir(t)jIV(t)jdt.

(3.93)

By Theorem 3.15, (3.94)

1ir(t)IIV(t)l = 7r(t/i)P(t).

So (3.93) can be rewritten in the form (3.95)

Ix f (µ)e° = Vol(G) f /w(M f)(t)lr(t/i)P(t)dt.

By (3.77), (3.96)

P(t)

Xo/T

It follows from the above that we may as well assume that over Xo x g*, a is given by (3.97)

o = a*a6 - d(p, 9).

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165

Equivalently, (3.98)

a = 7r*ao - (dp, 0) - (p, - 2 [9, B] + 0).

Then (3.99)

i xf We' = lX°xo f (p)en o°-(P,e>-(dP,9>+# (Pde,el)

Since only the term of top degree in A(g*) contributes to the integral in the right hand-side of (3.99), we can rewrite (3.99) in the form (3.100)

f f(l+)e'

Y

= fXoxr lo".

Now recall that X is oriented by a, and XoIG by ao. In particular, near Xo C Xo x g*, Xo x g* is oriented by a. Also () is G-equiivariant. From (3.100), we get (3.101)

lx f (µ)e' = lxo/G e°p j a-(P,A) [JG f

dp,

which coincides with (3.92). The proof of our Theorem is completed. REMARK 3.19. In [28], Jeffrey-Kirwan use instead the coisotropic embedding theorem [Theorem 39.2] [24], which asserts there is a G-equivariant identification V = Xo x g,, so that a is exactly given by

a = piao - d(p, 0).

(3.102)

Recall that by Theorem 3.15, P(t) is a polynomial near t = 0. Therefore P(ft ) is a differential operator.

THEOREM 3.20. If f : t -i R is a bounded measurable W-invariant function with support in U, then (3.103)

VI G) [P(&)T(t/ii))f f(IA)e' =

lx

"(t)] J fe_o

PROOF. By (3.95), (3.104)

l

f(z)e' =

x

Vol(G) IWI

l If(t)ir(t/i)P(t)dt.

From (3.90), (3.104), we get (3.103). The proof of our Theorem is completed.

Let 8 be a G-connection form on Xo -4 Xo/G, and let 0 be its curvature. Let Q be a G- invariant C°° function defined on g with values in R. We define Q(-0) by its Taylor expansion, which only contains a finite number of terms. Then Q(-O) is a dosed form on X0/G, and fx0/G Q(-O)e°p does not depend on 0. Similarly, we define the differential operator Q(et) as the formal power series associated to the Taylor expansion of Q. When applying the power series Q(8)7r(eVir ) to the polynomial P(t), only a finite number of terms in the Taylor expansion contribute, so that Q(&)ir( a,A°)P(t) is a well-defined polynomial. THEOREM 3.21. The following identity holds, (3.105)

fx°/G Q(-e)e'0 =

010t IWI

[Q(8/8t)r(

few

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PROOF. Let f : g* -> R be a G- invariant bounded measurable function with support in g *u. By Theorems 3.18 and 3.20, (3.106) 1xe/G

fe(-e)e°p =ICI [P(8/8t)x(%g)

(0).

By Rossmann's formula (3.91) and by (3.106), we obtain (3.107)

(0).

lxe/Gfs(-6)e°p ='WI

Equivalently (3.108)

2i

IWI

lx°,Gfg(-9)e

i)P(t)l (0).

Then (3.108) is exactly (3.105) when Q = f9. Clearly, it is enough to verify (3.105) when Q is a polynomial. Then Q is the Fourier transform of a distribution whose support is {0}. Using (3.108) for f with support in g ,, and a simple limit procedure, we get (3.105) when Q is a polynomial. The proof of our Theorem is completed.

3.8. The volume of symplectic coadjoint orbits. Let t E t, and let Ot C g = g* be the G-orbit of t. Recall that vo, is the canonical symplectic form on Ot given in 1.193.

Let pt : G/T -+ Ot be given by

ptg = g.t.

(3.109)

Then pt is one to one if and only if t E treg. If t f (reg,

dim(G/T) > dim Ot.

(3.110)

We fix a positive Weyl chamber K C t. This defines an orientation on G/T, which is fixed once and for all.

DEFINITION 3.22. IftEt,XE9,put e(n,sX)+pi°o,

H(t, X)

(3.111)

LIT

Then since G acts on the right on Ot and preserves vo,, H(t,X) is a Ginvariant function of X E g*. Let K C t be a Weyl chamber, and let a(t) be the corresponding function on t defined in (1.185). PROPOSITION 3.23. If t 0 treg,

H(t,X) = 0.

(3.112)

If t E K, X E tre$f then (3.113)

H(t X) = '

1

E

7r(X/2i r) wEW

e(wt,x) 7U

PROOF. Using (3.110), we get (3.112). Recall that G acts on the right on Ot.

AlsoifXEg, (3.114)

d(p,X)+ixo,Uo, =0

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Then when t E K, we get (3.113) from the localization formula in equivariant cohomology of Duistermaat-Heckman[19], Berline-Vergne [5], [6, Theorem 7.11]. This equality obviously extends to the case where t E K. The proof of our Proposition is completed. REMARK 3.24. From (3.112), (3.113), we recover the well-known fact [15, Lemma

VI.1.2] that if t f treg, E ewe('°t") = 0.

(3.115)

WE W

Recall that H(t,X) is a 0- invariant function of X E U*. Also the function P(t) was defined in Definition 3.14. PROPOSITION 3.25. For to E U n k, 11

(3.116)

P(to) =

IWI

frito, a/et)ir (4 ) P(t)J It=0'

PROOF. By (3.113), if X E treg, (3.117)

2i1r -

Ewe(wto,X). WE W

Of course the identity extends to arbitrary X E I. In particular, by (3.117), we have the identity of formal power series of differential operators, (3.118)

H(to,8/8t)ir(2a&) =

Ewe(wto,&/et).

wEW

Now in view of (3.76), (3.119)

37 ewe(wto,a/&)P(t)1t.o = IWIP(to). wEW

From (3.118), (3.119), we get (3.116). The proof of our Proposition is completed.

0

REMARK 3.26. In view of (3.105), (3.116), we get (3.120)

P(to) =

xfo/a

H(to, -9)e°o

One then verifies that (3.120) is just a reformulation of (3.77).

Recall that we have identified g and g", t and t' by the scalar product Q. For t E t = t', the orbit Ot is equipped with the symplectic form ao,. THEOREM 3.27. The following identity holds (3.121)

fO/Te' 0 I = Vol(G) I'(t)I, Vol(T)

IJ

Vol(G) 1 7r(8/8t)7r(t/i) = 1. 2iia Vol(T) IWI

168

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

PROOF. If t E teeg, then Ot = G/T. So the first identity in (3.121) is trivial. Let T*G be the cotangent bundle of G. We identify X = T*G to G x g* via the left action of G on T*G. If 9 is the canonical left-invariant 1-form on G with values in g, if p E g*, (p, 9) is the canonical real 1-form on T*G G x g*, and -d(p,0) is the canonical symplectic forma on T*G. Also G acts on the right on T*G and preserves or. Then (g, p) E G x g* '-4 p E g* is a moment map for this action. Now we use the notation of Section 3.7. Take t E t. Then

Xt = G x {t},

(3.122)

and Xt/Z(t) = G/Z(t) can be identified with Ot by g E G/Z(t)

g.(t) E Ot. Then one verifies easily that the symplectic form at on Xt/Z(t) is just an,. By the first identity in (3.121), we get (3.123)

Vol T

x,/Tea'

so that

P(t)

(3.124)

_ Vol(G) Vol(T)

Finally observe that the moment map p : (g,p) E G x g* i p E g* is regular, that G acts freely on Xo = G, and Xo/G = 1. We apply Theorem 3.21 with Q = 1, use (3.124), and we get (3.125)

Vol(G)

8 8t

1

Vol(T) W

2s7r

1

= 1.

)7r(t/%)J

lt=o

Also 7r(18t)7r(t1i) is constant. It is now clear that the second equation in 3.121) follows from (3.125). The proof of our Theorem is completed. REMARK 3.28. It is of some interest to verify that as should be the case, the right-hand side in the second equation in (3.121) is positive. In fact (3.126)

7r(8/8t)ir(t/%) = (22r)t

E (al,a,ili) ... (a,,aoitl). oESe

Put (3.127)

(a1 ®... (D at)° = d E ao(1) 0 ... am(t) E Stt*. oESe

Then (3.126) can be written as (3.128)

1r(8/et)7r(t/i) = (27r)11!11(a1®... 0 at)v II2see*, 2i7r

which is indeed positive.

Recall that p E K, so that 7r(p/i) > 0. We now recover a well-known formula for Vol(G/T) [6, Corollary 7.27]. THEOREM 3.29. The following identity holds (3.129)

Vol(G) Vol(T)

-

1

7r(p/i)'

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

169

PROOF. We proceed as in [6]. Let A E A n F. Let Xa be the character of highest weight A. Then by Kirillov's formula [34], [5], [6, Theorem 8.4], for t E t, It] small enough, (3.130)

XA(et) _ ;(t) o(t)

f

e2ia(µ,t)toon+a,

oP+a

In particular (3.131)

e°0p+a.

Xx(1) = f op+,.

By Theorem 3.27, e,op+a

(3.132)

fop +x

Vol(G)7((P+A)/E) = Vo1(T)

Also by Weyl's dimension formula [15, Theorem VI.1.7], (3.133)

XA(1) _ ir(p + A) gy(p)

From (3.131)-(3.133), we get (3.129). The proof of our Theorem is completed.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

170

4. The affine space of connections In this Section, we construct a canonical line bundle L on the of ine space of connections on the trivial G-bundle on a Riemann surface E with marked points, and an action of a central extension of the gauge group EG on L. We compute the action of the stabilizers of certain connections on the line bundle L. We also construct a line bundle A,,, which, as we shall see in Section 6.3, will descend to the moduli space of flat G-bundles. The main purpose of this Section is to evaluate the angles of the action of the stabilizers on the line bundle ay, in order to apply the theorem of Riemann-Roch-Kawasald [32, 33] to this moduli space, which we will do in Section 6. Our Section is organized as follows. In Section 4.1, we briefly recall the construction of the central extension LG of the loop space LG. In Section 4.2, we consider the coadjoint orbits of LG, their symplectic form, and the corresponding line bundles. In Section 4.3, we construct a central extension of (LG)'. In Section 4.4, we give a formula for the holonomy of the canonical connection on the S1-bundle LG 4 LG. In Section 4.5, we describe the symplectic affine manifold A of G-connections on E, and a symplectic action of a central extension EG of the gauge group EG. In Section 4.6, we construct the line bundle L on A, and an action of EG on L. In Section 4.7, when G is not simply connected, we classify the G-bundles on E. In Section 4.8, we specialize the results of Section 4.7 when H is a connected subgroup of a simply connected group G. In Section 4.9, we compute the action of certain elements of EG on L. Finally, in Section 4.10, we define the line bundle Ap , on which EG acts, and we compute the action of certain elements

of EG on 4.

4.1. The central extension of the loop group LG. Let G be a compact connected and simply connected simple Lie group. We will use the notation of Section 1. In particular (,) denotes the basic scalar product on g. Let T be a maximal torus in G, let t C g be its Lie algebra. Let W be the corresponding Weyl group. Then W acts on UK. Let Wff be the afline Weyl group (4.1)

W = W x CR.

Let S' = R/Z, and let t E [0, 1[ be the canonical coordinate on S1. Then 8/cat trivalizes TS1 and dt trivializes T'S1. Let LG be the loop group of G, i.e. the group of smooth maps s E Sl H g, E G. Let Lg be the Lie algebra of LG, i.e. the set of smooth maps s E S' E g.

Ifa,bELg,put (4.2)

17(a, 0) =

Js1(a, dQ)

Then by [47, Section 4.2], q is a cocycle on Lg. By [47, Theorem 4.4.1], there is a

unique central extension p : LG SS-'- LG associated to rl. The Lie algebra of LG

is Lg = Lg ®R. If (a, a), (a', a') E Lg = Lg ®R, (4.3)

[(a, a), (a', a')] = ([as

f

s,

\

(a, da') 1

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

171

Observe that the Lie group G x S1 embeds as a Lie subgroup of LG, and that the Lie algebra of G x S1, g ®R (equipped with its standard Lie algebra structure) embeds correspondingly as a Lie subalgebra of Lg ®R.

Clearly a E S1 acts on LG by g E LG H keg = 9.+8 E LG. We can then form the semidirect product LG = S1 x LG. The Lie algebra Lg of LG is given by Zg = R ® Lg, and can be identified to the Lie algebra of differential operators

a&+ a,aER,aELg,so that [adt- + a, a'Td

(4.4)

+ a,I

=[a,a']+adta'-a'ata.

By (47, Theorem 4.4.1], the action of S1 on LG lifts toLG, so that we can form the semidirect product LG = S' x LG. Its Lie algebra Lg is given by

Lg = R ®Lg ® R

(4.5)

= ROLg LgeR. The Lie bracket in Lg is given by (4.6)

[(a, a, b), (a!, a', b')] _ (o, [a, a'] + ada'

l

f

11(a, da') 1

.

Also S1 x G x S1 embed as a Lie subgroup of Lg, and R ®g ®R (with its standard structure of Lie algebra) embeds correspondingly as a Lie subalgebra of R®Lg®R. Observe that we can reverse the orientation of S1. Namely let i,b : LG -+ LG

be the morphism ge H 9_,. Then ' lifts to a morphism LG -3 LG, so that if (x,y) E S1 x S1, (4.7)

O(x,y) = (x-', Y-,) -

By [47, Section 4.9], on Lg, there is a LO-invariant symmetric bilinear form (4.8)

((a, a, b), (a', a', b')) = f (a, a')dt - ab' - ba'. s'

By (4.8), we have the embedding (4.9)

R®LgC(LgOR)*.

Equivalently (4.10)

Lg C (Lg)' .

4.2. The coadjoint orbits of LG. The group LG has a coadjoint action on the right on (Lg)*. If we restrict this action to Lg C (Lg)*, for g E LG, we get (4.11)

(adt+a) g adt+ag_1dt+g trivial

lag.

G-bundle on S1. Let As' be the affine space of Let P = 81 x G be the G-connections on S1. A connection in As' can be written as (4.12)

d+A, AEfl1(S',g).

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

172

In the sequel, we identify d+A E AS' with the differential operator Lg. So AS' is an affine subspace of Lg. Clearly LG acts on the right on AS', so that if g E LG,

+A(d/et) E

=g-1(d+A)g

(4.13)

i.e.

1Ag.

(4.14)

By (4.11), (4.14), AS' embeds as an affine suspace of (Lg)* = Lg and the action of LG on (Lg)* induces the corresponding action of LG on AS' . Now we briefly develop the theory of coadjoint orbits for LG. DEFINITION 4.1. If A E Lg, let w E G be the holonomy of the operator dt +A on S1, i.e. if gt is the solution of (4.15)

dtgt1+At=0,

then

W = 91.

(4.16)

Then Floquet's theory of differential equations (Frenkel [21, Section 3.2], Segal

[47, Proposition 4.3.6]) shows that the orbits ofd + A in Lg can be expressed in terms of the adjoint orbits in G. Namely dt + A and dt + A' lie in the same LG-orbit if and only if the corresponding holonomies to and w' lie in the same 0orbit. In particular the LG orbits of the differential operator di + A always contain representatives of the form dt + A, A E t, and two d + A lie in the same LG-orbit if and only if \ and A' he in the same Was-orbit. If A E t, let Z(d/dt + A) C LG be the stabilizer of d/dt + A. PROPOSITION 4.2. If A E t, (4.17)

Z(d/dt + A) . Z(e a)

.

Namely if g E Z(e-a), the corresponding element of Z(d/dt + A) is t E S1 y e-tagetX E G. If a-' E TT.g, then (4.18)

Z(d/dt + A) = T C LG.

PROOF. The proof of this simple result is left to the reader.

0

Let Od/dt+A C Lg be the LG orbit of d/dt + A. Clearly the map (4.19)

9 E LG H g(d/dt +

A)g-1

E Lg

induces the identification (4.20)

LG/Z(d/dt + A) = Od/dt+a

In particular if A E P, (4.21)

LG/T = Od/dt+a

Also the theory of coadjoint orbits developed in Section 1.14 tells us that

Od/dt+a is equipped with a symplectic form

Namely let d/dt + A =

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173

DA E Od/dt+A Recall that LG acts on the right on Od/dt+,,. If a E Lg, let aOd/d`+A be the corresponding vector field on Od/dt+A Clearly and/d`+A = DAa.

(4.22)

Then formula (1.193) for the symplectic form

is

a(DAa, DA/3) = (d/dt + Al [a, a])

(4.23)

.

By (4.3), (4.8), (4.23), we get (4.24)

a(DAa, DAfi) = f (D.4a, i3) . sl Finally LG acts symplectically on the left and on the right Od/dt+A, and d/dt+A H (d/dt + A) E (Lg)* is a moment map for the left action of LG on Od,Idt+a

If p E R*, A E t, if pd/dt + A E Od/dt+A, then d/dt + A/p = DA/ P is a connection, and (4.25)

-0,d/d9+A (DAn, DA0) =P

(D.4/pa

J s=

, a)

=

Recall that T x S' C LG. DEFINITION 4.3. If (A,p) E CR x Z*, if p(A,_p) is the one dimensional representation of T x S1 of weight (A, -p), let H(A,p) be the line bundle on LG/(T x S1) _

LG/T, H(a,p) = LG xp(A,_v) C.

(4.26)

In the same way as the Weyl group W acts on the right on G/T, the affine Weyl group Was = W x Z.R acts on the right LG/T. In fact if w E W, w acts on LG/T by g)-+ gw and µ E 'R acts on LG/T by g )-> geut. Then Was acts on the left on M x Z* by the formula

w(A'P) _ (wA,P) p(A,P) _ (A + Pp,P)

(4.27)

We can then restrict (A,p) to vary in a fundamental domain of the action of lies. This just says that p E Z* , A E IPIP

(4.28)

Equivalently if p E Z*, (4.29)

for a E R+ , 0 < (a, A) < (ao, A) < p .

Let now p E Z*, A E IpIP n ME'. Then by (4.18), (4.20), (4.30)

Opd/dt+A = LG/T.

Therefore, the line bundle H(A,p) is well defined on Opd/dt+x Also by proceeding as in Section 1.14, the Hermitian line bundle H(A,p) is equipped with a unitary

connection V"('.') and (4.31)

c1(H(A,p), V('.')) = t1p,a/d`+A

If p E Z*, A E IpIP n', but A 0 pP, the theory is slightly more involved.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIB

174

Now, we will just assume that A E GAR ` and explain the construction of the line bundle H(a,p) on Opd/dt+a in full generality. By (4.17), Z(pd/dt + A) = Z(e-,\l P) .

(4.32)

DEFINITION 4.4. Let Z(pd/dt + A) be the stabilizer of pd/dt + A in LG.

Then Z(pd/dt+.1) is a central extension of Z(pd/dt+A). Let 3(e-,\/P), ;(pd/dt+ A) be the Lie algebras of Z(e-a/P), Z(pd/dt + A). Then

j(pd/dt + A) =

(4.33)

3(e-a/p) ® R.

PROPOSITION 4.5. If (X, a), (Y, b) E 3(pd/dt + A) = 3(e-111p) ® R, then (4.34)

[(X, a), (Y, b)] = ([X,1'], n (A, [X, Y])) .

PROOF. Clearly, if X E 3(e-a/P), the corresponding element in the Lie algebra 3(pd/dt+A) of Z(pd/dt+A) is just e'ta/PXetA`/P. In view of (4.3), (4.35)

[(e to/PXeta/P, 0) ,

0)]

(eta/P[X, Y]eta/p , (X, -[A/p, Y]))

``

Also (4.36) (X, -[A,1']) = (A, [X, Yl) . The proof of our Proposition is completed.

PROPOSITION 4.6. For any \ E C , (4.37)

(X, a) E 3(e "/P) ® R y (X, (p, X) + a) E j(pd/dt + A)

is an isomorphism of Lie algebras. PROOF. This is clear by (4.34).

REMARK 4.7. If A E pP l ZT, so that e-a/P E Treg, then 3(e-a/P) = t. In this case, in (4.34), (A, [X, Y]) = 0. Therefore (A, a) E 3(e a/P) ® R ,-> (X, a) E j(pd/dt + A) is also an isomorphism of Lie algebras.

Observe that T is a maximal torus in Z(e a/P) . By [14, Corollaire 5.3.1], Z(e--\/P)IT is simply connected. DEFINITION 4.8. Put (4.38)

Re_A,, = {a E R ; (a, .A/p) E Z} .

We define CRC-A,, as in (1.136). Let CRe-ai, C t be the lattice spanned by CRe-a/a By [15, Theorem V.7.1], (4.39)

ir1(Z(e-a/P))

=

n

URC-ain

Put (4.40)

It E t ; if a E Re_A,, , (a, t) E Z} .

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

175

By [15, Proposition V.7.16],

Z(Z(e-a/P)) = X -./P -A/p E Re ,/r CR Now (4.43)

maps into the group ir,(Z(e A/P))". In particular the map tifi(pd/dt+A) : h E 7r1(Z(e a/P))

e2:n(A/P.h)

EC

is well-defined. Since A E CR , e2ta(a/P'h) is a pth root of unity.

Recall that pd/dt is identified to the 1-form a E R H -pa E R. Also A E GR is a left invariant 1-form on LG. By Proposition 1.56, pd/dt + A is a closed 1-form on Z(pd/dt + A). Let Z(e--\/P) be the universal cover of Z(e'a/P). Then pd/dt is a closed form on Z(e'a/P) X,b(Pd/dt+)L) Sl

THEOREM 4.9. We have the identity of Lie groups (4.44)

Z(pd/dt + A) c 2(e a/P) Xlft(,d/d,+a) S1

In particular Z(pd/dt+A) contains a pth cover of Z(e-/P). Under the identification (4.44), (4.45) pd/dt + A ^ pd/dt, and the forms in (4.45) are closed and integral. PROOF. Clearly by (4.37), Z(pd/dt+A) is a locally trivial central extension of

Z(e-'/P). Let s E S' y t, E T be a smooth loop. The corresponding loop with values in Z(poldt + A) is given by s E S1 i-+ (e to/Pt,eta/P) E LG, i.e. by (4.46)

3ES'-+ t,EGCLG.

Now recall that G E LG, so that we have a loop (4.47)

s E S1 - t, E G n Z(pd/dt + A) .

By (4.37), the horizontal lift of 8 E S'- H t, E G C LG in Z(pd/dt+A) with respect to the flat connection on Z(pd/dt+A) -> Z(e'II/P) is given by 1dt,)`

(4.48)

8 E S1 H t, exp (2ilrj ( A/p, t g

1

From (4.48), we get (4.44). If SD denotes the group of pth roots of unity in C, Z(e'a/P) subgroup of Z(pd/dt + A), which is a p-covering of Z(e a/P). Observe that (4.49)

SI is a

(pd/dt + A, (X, (p , X) + a)) = -pa = (pd/dt, a).

From (4.49) we get (4.45). By Proposition 1.56, the 1-form poldt+A is closed on Z(pd/dt+A). It is even easier to see that pd/dt is closed on Z(e '/P) x+hcse/d,+a> S1. Finally it is trivial to

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

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verify that pd/dt is integral on 2(e -Vv) x,p(pd/dt+a) S'. The proof of our Theorem is completed.

By Theorem 4.9, as in Proposition 1.58, the integral 1-form pxd/dt + A defines a

representation p1 : Z(pd + A) -> S1. Also p2 : (g,x) E Z(e'/p) xd4pd/dl+a) S1 z'p E S' is also a representation. PROPOSITION 4.10. Under the isomorphism (4.44), P1 = py

(4.50)

PROOF. By (4.45), we get (4.50).

REMARK 4.11. Assume that A E pP f1 CR , so that Z(e a/p)

T. Then

2(e-'/p) = t.

(4.51)

Also the map (4.52)

(f,a) E t x S1 .+ (f,e2 (A/p,f)a) E L X S1

descends to an isomorphism T x S1 . t x+6(ad/dt+a) SI.

(4.53)

Then the 1-form pd/dt+ A on T x S1 corresponds to pd/dt on t x,/,,P,/d,+a) S1 In fact in this case,

Z(pd/dt + A) = T x S1,

(4.54)

and (4.53) is a special case of (4.44). Clearly (4.55)

Z(pd/dt + A)

Z(pd/dt + A) - npa/dt+A

DEFINITION 4.12. Let H(A,p) be the line bundle on npd/de+a,

H(A,p) = LG xP, S'.

(4.56)

Then H(a,p) is a Hermitian line bundle. As in (1.198), we can equip H(A,,p) with a unitary connection V'(a.o). By proceeding as in (1.199), we get (4.57)

c1(H(a,,),

=QOpd/dt.{a

4.3. A central extension of (LG)8. DEFINITION 4.13. Put (4.58)

Pe = (a,,...

A

a.) E (S1)' , A ai = 1). j=1

Then P, is a Lie subgroup of (S1)', and its Lie algebra p8 is given by (4.59)

P. = {(b1, ... , bg) E R8 , t b9 = 0}. j=1

Clearly P. C (S1)' is a Lie subgroup of (ZG).

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DEFINITION 4.14. Put

(LG)'

(4.60)

_ (LG)'/P

(Lg)5 = (Lg)slp8 Then (LG)' is a central extension of (LG)' and (Lg)' is its Lie algebra. Clearly (4.61)

(Lg)x = (Lg)8 ® R.

Also if (ai,... ,a,,a), (al,... a', a) E (Lg)', then (4.62)

[(a1,... a

,a,a )) = ([a1,a],... ,[as, ae), 7

Also G' X S' embeds as a subgroup of (G)', and g' E) R embeds as a Lie subalgebra of (q)'. Finally recall that in Section 4.1, we defined an action of S1 on LG. Therefore S' acts on (LG)8. DEFINITION 4.15. Put (4.63)

(W)' = S1 x (LG)', (L9)8 = R ®(Lg)' .

Then 4' is the Lie algebra of (LG)'. If we embed S',R into (S')", R, by the diagonal embeddings, then (LG)' C (LG)', and (L9)' is a Lie subalgebra of (Lg)'. Also by (4.10), (4.64)

(L9)` C (Lg)".

4.4. The holonomy of LG. Let rc be the closed left invariant 3-form on G, (4.65)

rc(X, Y, Z) = 2 (X, [Y, Z))

.

Then by [47, Proposition 4.4.5], (4.66)

rc E H3(G, Z).

Recall that n is the left-invariant 2-form on LG, (4.67)

Put (4.68)

77 =

if

1(9 1dg, a 9-1d9)dt.

2

A={ZEC, lzJ<1}.

Let g(s, t) : S1 x S1 - G be a smooth map. We identify g(s, t) with the smooth loop in LG s E S' '-+ g, = g(s, ) E LG. Clearly (4.69)

8D = Si .

Since 7ri(G) = 0, 0 < i < 2, g(s, t) extends to a smooth map g(z, t) : A x S' -* G. Therefore we get a map g: z E A H g(z, .) E LG.

JEAN-MICHEL BISMUT AND PRANQOIS LABOURIE

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THEOREM 4.16. The following identity holds

fs

(4.70)

S1 xs1

0g ag (9-1 8t 9_18x) dtds

- J xsl g'k+fr17=0. PROOF. Let h be the left-invariant 1-form on LG,

h=2 f (9 1 09 ,9-id9)dt.

(4.71)

A straightforward computation shows that (4.72)

dh = -17 + 2 f 1(g-18t

j[9 1d9, 9 1d9] )dt.

Using (4.72), we get (4.70).

Clearly

Lg=Lo ED R.

(4.73)

The splitting (4.73) defines a connection on the S1 bundle LG S LG . Let s E S1 H g, E LG be a horizontal lift of s E S1 - g, E LG. Since go = g1, then 91901 E S1 .

(4.74)

THEOREM 4.17. The following identity holds (4.75)

91901 = exp (2iirJr) exp (2ilr

(_j(o_1,fl_1)dsdt +f 2

ll

/ Oxs, PROOF. The first identity is a straightforward consequence of (4.2), (4.67). The second identity follows from Theorem 4.16.

4.5. The symplectic space of connections on E. Let X be a compact Riemann surface of genus g. Let x1, ... , x, be s distinct elements of X. Let A1, .... 0, be small non intersecting small open disks of center x1, ... , x,.

Put (4.76)

E = X\ &&j . j=1

Then E is a compact Riemann surface with boundary BE. Also E being oriented, BE is also naturally oriented. Let G be a compact connected and simply connected compact simple Lie group. We use otherwise the same notation as in Section 1. In particular g denotes the Lie algebra of G, and (,) denotes the basic scalar product on g defined in Section 1.2. Let P = E x G be the trivial G-bundle on E. Observe that since G is simply connected, a G-bundle on E is necessarily trivial. DEFINITION 4.18. Let A be the affine space of G-connections on P.

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Any element in A can be written in the form d+A, with A E 121(E, g). Therefore A is an affine space with underlying vector space fl' (E, g). Also

TA = A x fl1(E, g).

(4.77)

Let EG, Eg be the sets of smooth maps E -+ G, E a g. The Eg is a Lie algebra, which will be considered as the Lie algebra of EG.

DEFINITION 4.19. If U,V E fl'(E,9), put

o(U,V)=I--(UAV).

(4.78)

I-

Then u is a symplectic form on A. Observe that EG acts on the right on A, so that if g E EG, (4.79)

i.e.

1Ag+g-1dg.

(4.80)

Clearly EG preserves the symplectic form o. DEFINITION 4.20. If a E Eg, let aA be the corresponding vector field on A.

Let VA be the covariant derivative associated to the connection A E A. Then OA = V Aa .

(4.81)

For simplicity, we now fix an oriented parametrization of the s circles in 8E, so that

8E _ (S')'

(4.82)

DEFINITION 4.21. Let r be the restriction map (4.83)

g E EG '.4 91a£ E (LG)',

a E Eg H a1aE E (Lg)',

d+AEA H (d+ A)1en E (A')' All these maps are equivariant. By (4.83), we get a map (4.84)

d+AH

We still denote by p the projection (LG)' -+ (LG)'. DEFINITION 4.22. Put (4.85)

EG = {(g, g') E EG x (LG)' ; gleE = pg' in (LG)'} , Eg = {(a,a') E Eg x (Lg)' ; alaE = pa' in (Lg)'}.

Then Eg is the Lie algebra of the Lie group EG. More precisely (4.86)

E g = Eg ®R R.

Let r be the projection Eg = Eg ®R --- R. If (a, a), (a', a') E Eg, then (4.87)

[(a, a), (a', a')] =

([aa']j(ada')) s

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Also G x SI embeds a s a subgroup of EG by the map (q, t) H (g, (g, ... , g), t) C EG x (LG)8. Clearly the restriction map r extend to a map

Ed -> (id).

(4.88)

,

Eg - (Lo)e . In particular, there is a dual map (4.89)

(L9)°*

(Eg)* .

If A E A, then by (4.64), (4.90)

dt

+ A(818t) E (Lg-)a* .

So by (4.89), (4.90), we may view (d+A)18£ as an element of (Eg)*. As we just saw, EG acts on A and preserves the symplectic form o,. Therefore Ed also acts symplectically on A. Of course S1 C Ed acts trivially on A. If a E Eg, let ad be the associated vector field on A. If a = pa,

aA=aA=VAa.

(4.91)

Clearly (4.92)

Eg = R°(E,g)

Also (4.93)

112(E,g) C Sl°(E,g)*

DEFINITION 4.23. If A E A, let FA = dA+!,-[A, A] E fl2(E, g) be the curvature of A.

By (4.93), if A E A, FA C (Eg)* C (Eg)*. Also by (4.89), (4.90) if A E A, (d+A)18£ E (Eg)*. Recall that in Section 3.2, moment maps were defined with respect to symplectic actions of compact Lie groups on symplectic manifolds. Here we will use the same definition with respect to an action of EG on A. We now state an extension of a result of Atiyah-Bott [2, Section 9]. THEOREM 4.24. The map (4.94)

A E A H p(A) = FA - (d + A)18£ E (Eg)*

is a moment map for the action of Ed on A. PROOF. First we will prove that if a E Eg, (4.95)

d(FA - (d + A)IOE, a) = iaAa.

Clearly if 5 = (a, a1i ... , a,) E Eg®R' is identified with the corresponding element in Eg, (4.96)

(FA - (d +A), a) =

(FA, a) - / (A, a) + E

J£

8E

J=1

-j

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If B ETA = III (E, g), then (4.97)

r

(B, d(FA - (d + A)ISE, a))

(DAB, a) - JOE (B, a)

=

fE

r(VAa,B)

i o(B).

=

By (4.97), (FA - (d+ A)IaI;, &) is a Hamiltonian for the vector field V. Also by definition, if g E EG, FA'9 - (d + A A. g)1a1: = (FA - (d +.4)io) - g.

(4.98)

We have completed the proof of our Theorem.

4.6. The canonical line bundle on A. Now we describe the construction of the canonical line bundle L on A. In the case where there are no marked points, our construction follows closely earlier work by Ramadas, Singer and Weitsman [481. DEFINITION 4.25. Let (L, II IIL) be the trivial Hermitian line bundle on A. Let VL be the unitary connection (L, II IIL) :

VL=d-i1r r (., A).

(4.99)

J

PROPOSITION 4.26. The following identity holds

c1(L,VL) = a.

(4.100)

PROOF. This follows from (4.99).

DEFINITION 4.27. If a E Eg, put

La = Q,A - 2ia(Et(A), a) .

(4.101)

Clearly for a E Eg, La acts on COO (A, L). PROPOSITION 4.28. If a E Eg, then

[La,VL1= 0.

(4.102)

Also if a, 3 E Eg, [La, Lp-] = Lla,v'1 .

(4.103)

PROOF. The identities (4.102) and (4.103) are trivial consequences of (3.120) and of Theorem 4.24. PROPOSITION 4.29. If a = (a, a) E Erg = Eg ® R, then (4.104)

La = OVA - iir J (A, da) - 2i7ra. 1:

PROOF. By (4.99), (4.101), we get (4.105)

La = O,,A

-

i7r

JI;

(V Aa, A)

+2i7r

Je

- 2ilr f (FA, a) s

(A, a) - 2i7ra.

JEAN-MICHEL BISMUT AND FRANCOIS LABOURIE

182

Also (4.106)

- f E(A, da) - f E([A, A], a)

fE (VAa,A)

f (dA+![A,A],a)

f (FA,a) E

f (A,a)+f ((A,da)+2([A,A],a)). aE

E

From (4.100), (4.106), we get (4.104). The proof of our Theorem is completed. REMARK 4.30. By (4.104), if a,# E Eg, (4.107)

[La, La] = L1.,R1- 2ia f (a, dp). aE

In view of (4.87), (4.104), (4.107) fits with (4.103).

Now we will show that the action of Eg on L lifts to an action of EG. This result has been proved by Ramadas, Singer and Weitsman (48] in the case where there are no marked points, so that EG is just EG. THEOREM 4.31. The action of Eg on L defined in (4.101) lifts uniquely to a unitary action of Ed on L which preserves VL. In particular if (g, t) E G x S1 C EG, (g, t) acts on L by (4.108) f E LA H f . (9, t) = eliat f E LA.g .

PROOF. Let g E EG. Since ir;(G) = 0, 0 < i < 2, there is a smooth path s E (0,1] H g, E EG such that

go = 1, 91 = 9.

(4.109)

Recall that (4.110)

Eg=Eg®R. 1

The splitting (4.110) defines a left-invariant connection on the S1-bundle EG

EG

Let a E [0,1] H g, be the horizontal lift of a E [0,1] H g, E EG with respect to this connection, with go =1. Put

a, = 9,'dg' EE9, 9 = 91 Then (4.112)

ge 1

= a, E Eg .

Now we will show that if AE A, f ELA, 1

(4.113)

f 9 = exp (i,, f ds(A[j.98>]) f E LA.g

is an unambigous formula for the action of Ed on L. Clearly we may assume that

g1=go=1, sothat 8ES1=R/ZHg,EEG is a loop inEG.

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Recall that by(4.61), (4.114)

(L9)8 = (Lg)' ® R.

The splitting (4.114) defines a connection on the S' bundle (L'G)' -3 (LG). Then

by definition s E [0,1) H 4,IaE E (LG)' is the horizontal lift of s E S' H 9, E (LG)'. In particular 91les E S. By (4.104), we only need to verify that in this case,

1

exp Ci1r / ds (J (A 9e, da,)]) = 91.

(4.115)

lE

o

Observe that (4.116)

Let go, 9d be the g-valued left and right invariant forms on G, i.e. (4.117)

99 = 9 1d9,9d = d99 1.

Then

dog = -

(4.118)

From (4.111), (4.118), we get

[09, 09]

.

2

da,

(4.119)

dOd = 2 [od, od)

1491 9,18s(9;9d)9,

Therefore

(4.120)

8s(A,9;Bd)

Since go = 91 = 1, by (4.120), we obtain

r J de f (9,1A9,,do.) = 0.

(4.121)

o

s

By (4.121), (4.122)

I ds

J

(A g,, do.) = 4. ds j (9,1d9,, da,)

and so (4.122) does not depend on A. Observe that this last fact follows from general principles, and that the right-hand side of (4.122) is just the left-hand side evaluated at A = 0. Also (4.123)

j (g

1d9,, da,) = -

Jes(489, a,) -

a.).

In view of (4.65), (4.123), we get (4.124)

I

2 I ds JE(9, 1d9,,da,) _ -2

Jsl

dsJeE(g'89,a+)

- jxsl g...

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Recall that E is obtained from the Riemann surface X by deleting s disks O1 i ... , A, centered at x1,... , x,. Since 1r;(G) = 0, 0 < i < 2, the smooth map g : S1 x E -+ G extends to a smooth map g : S' x X -> G such that

g = 1 near S' x {x;} , 1 < y < s.

(4.125) From (4.124), we get (4.126)

l f ds J

1d9s, da,) _ 21 of ds f (g, E 2 SI

ar_

(9a89, a8) + E f 9'/ ;=l sI xo;

- fs' x01 .

Since tc E H3(G, Z),

f1xx g*ec E Z.

(4.127)

S

For0
h(z,t) = g(Izl,lzlt).

By (4.125), h is a smooth map from A x S' into G. Let h : A -4 (LG)n be the smooth map defined the way we did after (4.69). If we orient S1 as a component of OE, we get (4.129)

-12 f ds f (9,*09, a,) _ -12 s'

as

J=I

f

s' xSl

(h-1 h, h-18h)dtds.

at

as

Also for 1 < j < s, by orienting SJ as before, (4.130)

LxAj

g*rc = f

h*K.

Oxsf

Using Theorem 4.16 and (4.129)-(4.130), we get (4.131)

-2 fsds

f

aE

(9:69, a,) + E,=1 fsi xO19*k

=film. Using now Theorem 4.17 and (4.122), (4.126), (4.127), (4.131), we get (4.115). So formula (4.113) for f g is unambiguous. Also by (4.102), Ed preserves Vt.

Assume now that g E G. Let a E g be such that g = exp(a). Put (4.132)

g, = exp(sa).

Recall that G C EG. Then for s E [0,1], g, E EG. By (4.113), if f E LA, (4.133)

f g=f E LA.g.

Finally, by (4.104), if t E S', (4.134)

The proof of our Theorem is completed.

f,t=e2;Atf.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

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REMARK 4.32. By (4.120), in (4.124), we get

i

(4.135)

2 / ds / (A g,, 8) = 2 f (.d, d919i 1) _ 2 f ds j (ge id9> 9,18s ) sxlo,i]

g"IL .

So (4.135) makes more explicit the dependence of (4.113) on A. Also if f E C0O(A,L),g E EG, let g`f E C0O(A,L) be given by (4.136)

gf(A)=9

Then (4.136) defines a representation of EG on C°°(A, L). In this representation t E S1 acts like a-2"'t. Also if a E Ed, ±etaa , f = Law f .

(4.137)

4.7. The non simply connected case. Assume now that G is a connected semisimple compact Lie group, which is not necessarily simply connected. Let C be the universal cover of G. Then 7r,(G) C Z(G) and G = G/ai(G).

Let P9,3 be a 4g + 3s polygon covering E. The edges of P9,8 are denoted ,a91,bg1,ci,dl,C11,... ,c,,d.,d,'. For 1 < j < s, set wj = cjdjc 1. Let q be an element of E which is a common point to the ai,bi,wj. Then Tri (q, E) is generated by the circles a1, b1, ... , a9, b9, w1, ... , w,. The case g = 1, s = 1 is represented in Figure 5.1. Let P 0 . E be a G-bundle on E. Take z E E \ (8E U U =1(ai U b;)). Then

al,bl,a11,bi1,...

E\{x} retracts on (Ug-,at U b;) U(UJ'_1w1). Since G is connected, the restriction of P to the 1-skeleton (U,s iai U bi) U(UU_1wj) is trivial. Therefore PEA{,,} is trivial, i.e. (4.138)

PE\{x} = E \ {x} x C.

Let A be a small disk in E centered at x. We orient 8A as part of 8(E \ A). Then (4.139)

Po = A x G.

Let v : A \ {x} -+ G be the transition map describing the G-bundle P on E, i.e. (4.140)

(y,9) E Po = (y,Q9) E PE\{x} y E A \ {z}.

Then the homotopy classes of G-bundles are classified by the homotopy classes of maps : A \ {x} .4 G. Since A \ {z} retracts on 8A = S1, the homotopy classes of G bundles are specified by the element [P] of ai(G) associated to clan : S1 - G. Clearly [a] does not depend on the trivialization of a on A. The map 7r .- [P) E 1r1(G) depends explicitly on the trivialization of P on

E \ {x}. More precisely the homotopy classes of trivializations of P on the 1skeleton are given by iri(G)29+s This just says that if a trivialization of P on E \ {x} is given, all the other trivializations are given by the action of ir1(G)29+a on this trivialization.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

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Let p be the map : a

(4.141)

ECM E 7r1(G)

p : (a,, bl,... , as, b9, c1, ... , c9) E lrl (G)2D+B j=1

Then if given c E 7rI (G)29}8, we replace the given trivialization of P Oil E \ {x} by the action of c, (P] is changed into [P] - p(c).

Finally, with the above provisos, the map P -+ [P] E al (G) does not depend on the choice of x.

Clearly, if P G > E is trivial, P lifts to a G-bundle. Conversely, if p E , since Q is trivial, and P = Q/7rj (G), P is also trivlifts to a G-bundle Q ial. Therefore [P] E irl (G) describes the obstruction to a lifting of the G-bundle P to a 5-bundle Q. As before we fix a trivialization of P on E \ {x}.

Let VP be a connection on the G-bundle P a-P-E . Since (4.142)

PEA{:} = E \ {x} x G,

PjE\{:} lifts to the trivial G-bundle. (4.143)

Q=E\{x}xQ.

The connection VP lifts to a connection VQ on Q. Clearly, this construction of Q depends on the choice of the trivialization of P on E \ {x}. We will write that a family of circles A of center x tends to x if their radius tends to 0. Clearly as A. -+ x, the holonomy of VP around x tends to 1. PROPOSITION 4.33. As A -+ x, the holonomy of VQ around 8A C 8(E \ A) tends to [P] E irl(G) C Z(G). In particular if VP if flat, for A small enough, the holonomy of VQ around OE \ A is equal to [P].

PROOF. We fix an origin in SI = M. Let t E [0,1] -4 rt E G, ro = 1 be a horizontal lift of Sl C 8(E \ A) in Peo with respect to VP, in the trivialization Pro = A x G. Then t E [0,1] -+ atrtao 1 E G denotes the corresponding horizontal lift of S' in the trivialization PE\{:} = E \ {x} x G.

Let t E [0,1[x+ a`t E G, t E [0,1] i.4 Ft E G be lifts of t E [0, 1(H at E G, t E [0,1['-+ it E 0, with To = 1. Then t E [0,1] Hat Ttoo E G is a horizontal lift of Sl = 8A in QIeo in the trivialization QIE\{:} = E \ {x} x d, with respect to VQ. In particular, parallel transport along S' ^- 8A is given by alTl°a 1. Now recall that by definition (4.144)

ul = [P]ao , [P] E 7r,(G) C Z(G) .

Therefore the above parallel transport is just [P]ooTloo 1. Now as A -+ x, rl -+ 1. Therefore Ti -+ 1, and the parallel transport tends to

[P]

The proof of our Proposition is completed.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

18;

4.8. The case of a connected subgroup of a simply connected group.

Let now G be a connected and simply connected compact simple Lie group. Let H C G be a connected compact semisimple Lie subgroup of G.

Let p - ° - E be a G-bundle on E. Then (4.145)

P=ExG.

Reducing the G-bundle P to a H-bundle Q is equivalent to finding a section of P XGG/H = ExG/H. Let 1 E G/H be the image of 1 E G. Then if 7) E E(G/H), the corresponding H-bundle Q is given by (4.146)

Q={(y,g)EExG, g-lil=i}.

Therefore homotopy classes of H-reductions of P are just homotopy classes in E(G/H). Since 9ri(G) = 0, 0 < i < 2, we get (4.147)

ire (G/H)

r1(H),

iri (G/H) = 0, 0 < i < 1. Take x E E \ (8E U U 1(ai U bi)) as in Section 4.7. Recall that E \ {z} is homotopy equivalent to U9i=1(ai U bi) U UU_1wp Since it (G/H) = 0, 0 < i < 1, there is only one homotopy class of sections E \ {x) -+ G/H. If ri is a section of E x G/H, we may and we will assume that n = 1 on E \ A. In this case (4.148)

Ql£\o = E x H,

and Q has a canonical section over E \ A. Therefore homotopy classes of H-reductions of P are classified by homotopy classes of maps i : A -a G/H such that rllao = 1, i.e. by 7r2(G/H) = ir1(H). Take z E ir2(G/H) = ir1(H). Let Qz be the H-reduction of P associated to z. By the above, QI£\{x} has been canonically trivialized on E \ {z}. Since Q is a H-bundle, the class [Qz] E 7r1(H) is well-defined. PROPOSITION 4.34. The following identity holds (4.149)

[Q.] = z in 7r1(H).

PROOF. Let V E A -* h(y) E G/H be a smooth map such that h1av = 1, representing z in ir1(H) = ir2(G/H). Then (4.150)

QzIo = {(y,g) E E x G ; g'1h(y) = 1}.

To trivialize QZ on A, we fix a connection VQ on Q:. If (x, g) E Q.,., we parallel transport g along radial lines in A. This way, we obtain 91 : 0A -- H. By definition (4.151)

[91] = z in ir1(H).

Also a = g1 exactly defines the H-bundle Qz on E in the sense of (4.140). From 0 (4.151), we get (4.149). The proof of our Proposition is completed.

JEAN-MICHEL BISMUT AND FRANQ,,01S LABOURIE

188

4.9. The action of stabilizers on the line bundle L. Now we use the notation of Section 1, and more especially of Section 1.9. Let u E C/Z°R. We will specialize the results of Section 4.8 to the case H = Z(u). Clearly the map

g E G/Z(u) i-s gug'1 E Cu C G

(4.152)

is one to one. It maps i into u. If q is a section of E x 0,, over E, the corresponding Z(u)-bundle Q is given by Q = {(x, g) E E X G , 9-1179 = u} .

(4.153)

Since QJE\{x} = E \ {x} x Z(u), Q has a section over E \ {x}. Equivalently, there

isgEE\{x}xGsuch that q = gug-1 on E \ {X}.

(4.154)

Homotopy classes of trivializations of Q on E \ {x} are just homotopy classes of g E E \ {x} such that (4.154) holds. We have already classified these homotopy classes by ir1(Z(u))29 '. By (4.147), we know that nlE\{y} is homotopy equivalent to the constant q = i = u. In the sequel, we will assume that q = u on E \ A which corresponds to q = 1

onE\A.

Similarly

Quo = A X Z(u)

(4.155)

i.e. Quo has a section. Therefore there exists g E AG such that

q = gug 1 on A.

(4.156)

In particular by (4.156), gap takes its values in Z(u), so that 9lao E LZ(u). Also there is only one homotopy class of g E AG such that (4.156) holds. Then o = gigo E LZ(u) is exactly the a in (4.140) defining the Z(u)-bundle Q on E. GOR/CRU. We identify [a] to a given element Recall that [a] E 1r1(Z(u)) [a] E GAR. Let s E S' i b, E T be a loop which represents [a] in ri (Z (u)). Put

9° = (bg1)lao E LZ(u).

(4.157)

defines an element of LZ(u). Since 1r;(Z(u)) = 0, 0 < i < 2, g° extends to g° E EZ(u). By conjugation of q by g° E EZ(u), we may and we will assume in the sequel Then 910,6o is homotopic to 0. Equivalently

that

q=uonE\A,

(4.158)

7=

9u9-1 on A,

g E 0G, 91a,& E LT.

f

Clearly J

£\o g 1dg E GR is the homotopy class of g E LT.

Let now q E EG. Put (4.159)

A°={AEA; Arl=A}.

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189

Assume that (4.160)

An 0 0.

Put (4.161)

U = 0(x)

In the sequel we assume that U E C/GAR.

(4.162)

Then by (4.159), (4.160), it is clear that n is a section of the orbit 0 = 0.. In the sequel we assume that the conditions in (4.158) hold. Let B E t be such that (4.163)

u = exp(B).

For aE[0,1],xES\A,put I (x, s) = exp(sB) .

(4.164)

Then since 1ri(G) = 0, 0 < i < 2, the map !I: E \ -A x [0,1]

G extends to a map

q:Sx[0,1]-+Gsuch that 'gI£\nx[0,1l = exp(sB), ' 17lEx{1} _ 17 -

For s E [0, 1], put (4.166)

s) .

ii8(x)

Then s E [0,1] -+ 11. E EG is a smooth path. Let s E [0,1] E ff. EEG be the corresponding horizontal lift in EG. THEOREM 4.35. The following identity holds (4.167)

'71ILIA, = exp (-2ilr(/

Ja£\o

g'1dg, B))

PROOF. We consider the splitting (4.168)

E=E\AUA.

Now both E \ A and A are objects like E. They carry the trivial G-bundles PI£\a and P. Therefore to E \ A and A, we associate the spaces of G-connections A£\° and A°, the line bundles L£/A and L°, equipped with the actions of (E/A)G and G.

Clearly A embeds into A£\A x A°, and EG embeds into (E \ A)G x G. First we claim that (4.169)

L = (L£\° ®L°)IA

and the isomorphism (4.169) identifies the metrics and the connections. In fact this is clear by (4.99).

Put (4.170)

e = {(x, y) E S1 x S1 , xy = 1} .

Then ( ( E \ ) 0 x YG)/e acts naturally on L£\° 0 L.

JEAN-MICHEL BISMUT AND FRAN('OIS LABOURIE

190

We claim that Ed embeds as a subgroup of ((E \ z )G x &G)/s. In fact recall that we orient OE \ A as the boundary of E \ A. Also in Section 4.1, we defined

.v

the morphism f E LG H Of E LG. So if f E EG, let g E LG be such that p§- = flaE\a. Then to f EEG, we associate ((fj,_\AG, g), (ffo, r/-y)) E ((E \:,)G x Since as we saw in (4.7), if 8 E S1, 1/)s = s'1, we find that, this element of ((E \ A)G x iG)/s does not depend on the choice of g. Now Ed acts on L. Similarly, by the above embedding, EG also acts on (LE\° ® LA)JA. We claim that both actions coincide. In fact this is obvious by the explicit formula in (4.113). Let n, ma and %A be the restrictions of m to E \ A and A. Let s E [0,1] H i7; \a E E \ OG, s E [0, 1] > ij° E DG be the corresponding horizontal lifts. By the above, we get the equality in S', (4.171)

1L1\on1jL-%

.

Now we will compute both terms in (4.171). By (4.165),

?. "° = exp(sB).

(4.172)

Using (4.113), we obtain

-EA

771 LE\o

(4.173)

=

Consider the path s E [0,1] i-+ 08 E AG, with (4.174)

6, = 9

exp(sB)9-1

Clearly by (4.158), (4.165), (4.174),

rl° = Br .

(4.175)

Moreover since gloa takes its values in T, (4.176)

ao = Bean

Lets E [0,1] r+ W. E .G be the horizontal lift of s E[ 0 ,1 )0, E AG. By (4.175), (4.176), we get

il° =Br.

(4.177) Set

Ic, = exp(sB) E AG.

(4.178)

Let s E [0,1] -+ K, E Recall that

.G be the horizontal lift of s E [0,1]

(4.179)

Lg = Lg ®R R.

By [47, Proposition 4.3.2], we get (4.180)

r

1

9(B, 0)9 = (9B9-1, - J Q (9 1d9, B))

Using (4.180), we obtain (4.181)

919 =

exp

(_2iirf('d,B))

rc, E AG.

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191

Clearly

91 =U,

(4.182)

and so (4.183)

91 = 9919-1

If A E AA,sl, then A g E AAA1. By (4.181), we get (4.184)

1tIjLo = B11Lc exp (-2i7r10-11 (g-1d9,B))

Finally using (4.113) again, we find (4.185)

91IL, = 1.

From (4.177), (4.184), (4.185), we obtain (4.186)

il1ILo

= exp (2iir J (9_1d9, B))

e

= exp \-2i7r far-\A (9-'dg, B)JJJJ

8£\:

.

By (4.171), (4.173), (4.186), we get (4.167). The proof of our Theorem is completed.

0 4.10. The action of stabilizers on the line bundle Ap. Take now 81.... , 9 E ZW. Take p E Z. Let L1,... , L. be the canonical line bundles constructed in Sections 1.14 and 4.2, which are associated to the LG orbits of (_de + 81, ... DEFINITION 4.36. Put (4.187)

41(91,... ,9) _ {AE A,-p(d +A)Isi E G-Pfr+s, -p(dt+A)Is1 Ed_p +e,. 7-7 a

Let vp : Ap(91i... ,9,) H 1 10-Pa,+s; be given by f=1

(4.188)

A

vp(A) _ (- p(dt + Ais1), ... , -p(d + Alsa )) .

Equivalently, Ap(91i ... , 8,) is the set of A E A such that the holonomy of .4 on S11 lies in the G-orbit of eel/P.... the holonomy of A on S,' lies in the G-orbit of eea/P.

DEFINITION 4.37. Let A, be the line bundle on Ap(91,... ,81), (4.189)

ap=LP ®vp(®js=1Lj).

Clearly ZG acts on the right on A.($,,... , 9,). By Theorem 4.31, this action lifts to an action on the right of EG on ap. PROPosrrlON 4.38. The action of EG on Ap descends to an action of EG on Ap.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

192

PROOF. We only must show that S1 C EG acts trivially on AP. However by (4.108), if t E S1, t acts on the right on LP like Also t acts on the right on ®i_1Li like a2{aPt Our Proposition follows. eliwPt.

Let t E EG. Assume that An n A,(91, ... , 0,) # 0.

(4.190)

Put

u=r}(x).

(4.191)

By conjugating v7 by g E G, we may and we will assume that u E T. We identify u to a corresponding element in t, which we still denote by u. In the sequel, we assume that u E C/VtG,

(4.192)

so that Z(u) is semisimple. Now if A E An, the G-bundle P on E reduces to a Z(u)-bundle Q on E. Recall that by (4.138),

QIE\{s} = E \ {x} x Z(u).

(4.193)

Put E \ {x} x :7(U).

(4.194)

Then the connection A lifts to a Z(u) connection on

Still two different

trivializations of QIE\{x} produce two distinct Z(u) bundles Recall that by (1.173),

(4.195)

00,/, n Z(u) = U Oz(u)(WOj/P) wEW\W

To normalize QIn\{:}, we impose that the holonomy of the connection A along Sil

be conjugate in Z(u) to ewj'!/P E 2(u), with 0 E Wu\W. Let then [QJ E r1(Z(u)) be defined in Sections 4.7-4.9. THEOREM 4.39. The following identity holds (4.196)

rlha,IA"nA,(81,... '8,) = exp (_2i(Ew'oi +P[Q]u) l 9=1 J

PROoF. As we saw in (4.158), by conjugating , by an element of EG, we may and we will assume that (4.197)

17

= uonE\0, = gug lonOwithgEAG, gla&ET.

Then (4.198)

QlE\o = E \ A x Z(u).

Let VP be a connection on P which preserves r/. Then VP induces a connection V4, which, in the trivialization associated to (4.198), is given by (4.199)

VQ=d+A, AES11(E\A,a(u)).

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193

The connection V lifts to a connection 75 given by

V =d+A.

(4.200)

By our choice of Q, we may and we will assume that the holonomy of V 15 on 31 lies in the Z(u) conjugacy class of ew'e,/p By [21, Section 3.2], [47, Proposition 4.3.61 for 1 < j < a, there is hj E L2(u) such that

h3-1(d +Als)lhj=-wje' l P

(4.201)

Since Z(u) is simply connected, there is h E E2(u) such that (4.202)

h E EZ(u) be the image of h. Then h184 E LZ(u) is homotopic to the constant loop 1. By proceeding as in (4.157), we may and we will assume that hj8A E LT is homotopic to the constant loop 1 in Z(u).

Now we may as well replace n by h-lrlh and A E All by A h E Ah-'nh. By (4.197),

h-'nh)4 = h-'gug-1h,4,

(4.203)

h-1g184

E

T.

Finally the homotopy class of h'191sf E LZ(u) is the same as the homotopy class

of g1sjELZ(u),1<j<s. So basically, we may and we will assume that n verifies the assumptions in (4.197), but we also have the extra assumptions

(±+A)"±-wjpj

(4.204)

(-

, 1<j < s.

Now by definition, the right action of u E LG on Lj,_yd/dt+wJe, is given by exp(-2ia(u,wj9j)) E S1. Using (4.167), we see that (4.205)

t/jap = exp

je1

By Proposition 4.34, (4.206)

pJ 81

9 1d9, u)1 1

f

g-'dg = [Q] E lri (Z(u)) . 8E\4

From (4.205), (4.206), we get (4.196).

The proof of our Theorem is completed. REMARK 4.40. It should be pointed out that the expression (4.196) is natural. Also as we saw after (4.141), we know In fact it only depends on the uri E how (Q] changes if we change the trivialization of QjE\4. One verifies easily that (4.196) is compatible with this formula. Also by Proposition 1.44, one verifies easily that (4.196) is compatible with the fact that EG acts as a group on Ap.

194

JEAN-MICHEL BISMUT AND FRANCOIS LABOURIE

5. The moduli space of flat bundles on a Riemann surface The purpose of this Section is to construct the moduli space M/G of flat Gbundles over a Riemann surface E with marked points. We establish the Witten formula [63, 64] for the symplectic volume of M/G. Also we show that the formula of Witten [64] and Jeffrey-Kirwan [28] can be applied to M/G. In particular, we express the integrals of certain characteristic classes over ht/G in terms of the action of certain differential operators on local polynomials over the maximal torus T. Also we give a formula for cl (TM/G) . All these results will be needed in Section 6, when we apply the Theorem of Riemaun-Roch-Kawasaki [32, 33] to A1/G. As explained in the introduction, our derivation of the Witten formula is closely related to earlier work by Liu (39, 401.

This Section is organized as follows. In Section 5.1, we give the standard combinatorial description of E. In Section 5.2, we introduce the corresponding combinatorial complexes, which compute the absolute and relative cohomology of

flat vector bundles over E. In Section 5.3, we introduce the G-equivariant map X = Gas x OJ -+ G, such that M = {x E X, #I(x) = 1}. Also we relate the differential of 0 to the combinatorial complexes on E which compute the absolute and relative cohomology of the flat adjoint vector bundle E. In Section 5.4, we give natural conditions on the orbits Oi under which 1 is a regular value of 0, so that M/G is an orbifold. In Section 5.5, we give conditions under which the set of elements in the fibres of 0 or in X with non trivial stabilisers are of codimeusion > 2. In Section 5.6, we describe T M/G and the symplectic form w. In Section 5.7, we show that the symplectic volume form on M/G can be evaluated in terms of the corresponding combinatorial complexes. In Section 5.8, using the results of the previous subsections, we prove Witten's formula [63, 64] for the symplectic volume of the reductions of M/G. In Section 5.9, we define logarithms from certain subsets of G into g. In Section

5.10, we show that the invariant open sets of X where G acts locally freely are naturally equipped with a symplectic form, and that the action of G on these sets has a moment map. In Section 5.11, we compute the integrals of certain characteristic classes on the moduli spaces associated to the centralizers Z(u), u E C/ R . As we will see in Section 6.4, these moduli spaces correspond to the strata of M/G. In Section 5.12, we compute certain Euler characteristics. Finally, in Section 5.13, we give a formula for c1(TMIG).

5.1. Combinatorial description of the R.iemann surface E. If a, b lie in a group r, put (5.1)

[a, b] = aba-I b-1 .

Let g E N, 8 E N, g + s > 0. Let r be the discrete group generated by 1, ul,vl,... ,U9,v9,w1,... ,w8, and the relation 9

s

fl[ui, vi] A wj = 1. i=I

f=1

Let X be an oriented connected compact surface of genus g. Let xl,... , x8 be a distinct elements of X. Put (5.3)

X' = X \ {xl, ... , x8} .

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

195

FIGURE 5.1

Let O1, ... , A. be small non intersecting open disks in X, centered at xl,... , x,. Let E be the surface with boundary (5.4)

E=X\UA;. J=1

Let q E E. Then r can be canonically identified with ;rl(q, E) = In (q, X'). Let P9,8 be a 4g+3s polygon covering E. The edges of P9,3 are denoted a1, b1, ai ' , b ' . . . . . a;', by 1, cl, d1, ci 1, ... , c8 i de, c81. Then P9,8 induces a cell decomposition of E

with one two-cell, 2g + 2s one-cells, 1 + s zero-cells. The two-cell is the interior 0 P9,, of P 9, , . The 2g + 2s one-cells are the circles a1, b1, ... , a9, b9, dl , ... , d8, and the segments c1,...,c,. The 1 + s zero-cells consist of q which lies in the a;,bi, and is a boundary point for the c;, and also r1, ... , re which he respectively in c1, ... , c8 and in d1, ... , de. The case g = 1, s = 1 is represented in Figure 5.1, where q1, ... , q5 represent q, and ri, rj represent r1. In the description given above, the group 7rl (q, E) is generated by the circles ul = al, vl = b1, ... , u9 = a9 , 2v9 = b9, _I w1 = c1d1c1 ,... w, = ced,cs 1. Also the above decomposition induces a cell decomposition of 8E, with s 1-cells

d1, ... , d,, ands 0-cells r1, ... , r,. To the above cell-decompositions of E, we associate the corresponding complexes over Z, (C£, 8), (C8E, 0) which calculate the homology groups H. (E, Z), H. (81, Z). Let 0) be the quotient complex defined by the exact sequence (5.5)

0 -+ (C0E, 8) -4 (C£, 8) --> (CE ., 8) -e 0.

Then the homology of (CE.*, 8) is the relative homology H((E, 8E), Z).

Note in particular that C£2E'' is generated by P, C£" by the 2g + s one-cells al, b1, ... , a9, b9, c1, ... , c Cio 'r by the zero-cell q.

5.2. The combinatorial complexes on E. Let V be a finite dimensional complex vector space. Let r : r -+ Aut(V) be a represention of r. Equivalently let

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

196

ui, vi (1 < i < g), wj (1 < j < s) in Aut(V) such that 111ui, vi) ll wi = 1 i=1

Let ?r

j=1

E be the universal cover of E. Put

F=ExrV.

(5.7)

Then F is a flat complex vector bundle on E. Let VF be the flat connection on F. To the above cell decompositions of E, 8E, we associate the corresponding combinatorial complexes (CE (F), 8), (CE(F), 8), (CE,'(F), 8), whose homologies are respectively H.(E, F), H.(8E, F), H.((E, 8E), F), and which fit in the exact sequence of complexes (5.8)

0 -4 (Ce£ (F),.9) -* (CE (F), 8) --> (C£'' (F), 8) -> 0.

For s > 0, we have the associated long exact sequence (5.9)

0 -* H2((E, 8E), F) --> H1(8E, F) -+ H1(E, F) -+ HI ((E, 8E), F)

-+ Ho(8E, F) a H0(E, F) -3 0 (in (5.9), we used the fact that ifs > 0, H2 (E, F) = 0, Ho((E, 8E), F) = 0). O

Let p be the barycenter of the 2-cell P, let g1,...,g49}1,r1,ri,...Iry,rb be

the vertices of P (see Figure 5.1 for the case g = 1, s = 1).

Then C2 (F) is the space of flat sections of 1r*F on P, and can be identified to Fp. Similarly CIE (F) is the space of flat sections of F in the "interior" of the corresponding one-cells. We identify (7r*F)q, to it*Fp by parallel transport with respect to N*VR along a radial line connecting p to q1. Along 8P , we identify 7r*F to (1r*F)q, by clockwise parallel transport with respect to it*VF. Then we identify C1(F) over a1, with (7r*F)q, . over b1i with (7r*F)q2 . over a2, with (7r*F)qs . over 62, with (7r*F)qa . over c1, with (7r*F)q4,+i over d1, with (Rr*F),.i

Of course ultimately, (7r*F)q,, (7r*F)q2, (lr*F)qs, ... are identified to 7r*Fp as indicated before. Finally Co (F) is identified to (a*F)qt ® (®f..1 F,.,,), which itself is identified to a sum of 1 + s copies of (a*F)p.

So C2 (F) is just (lr*F)p, CE (F) is a sum of 2g + 3s copies of (ir*F)p, and Co (F) a sum of 1 + a copies of (a*F)p.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

197

Then one verifies easily that if f E C2 (F) = (1r'F)p, Of = (1 - U1V-1u11) flat + (1 - v1utv1-1)uI 1fle, + (1 - u2v2 1U 1)[vl, u1)fla2 + (1 - V2U2v2 1)u2 1[Vi ul]flb2 +... + (5.10)

(1- wl 1)[vg, u9] ... [v1, u1)fl" + [v9, u9) ... [v1, ul)fld, +(1 - w2 1)wj 1 [v9, u9]... [v1, u1)fle2 + wl i [v9, u9) ... [VI, u11 fjd2 + .. .

Similarly 8 : C11: (F) -+ Co (F) is such that if f E (,r'F)n, (5.11)

8(flai)

_ (1 - ui 1)flgl , vi 1)flgl

8(flbi)

,

flrj, 8(fidl) _ (1- wf 1) fl,j. 8(flc1)

=

In view of (5.6), (5.10), (5.11) one verifies that, as should be the case, 82 = 0. By restriction to 8E, we obtain the chain map 8: CfE(F) -+ Coy (F) given by

8fldi = (1- w 1)flj

(5.12)

The complex (CE"r(F), 8) is obtained by making formally flrj = 0, fld, = 0 in (5.10), (5.11). Note that for s > 0, we recover the fact that (5.13)

H2 (E, F) = 0 , Ho ((E, 8E), F) = 0.

PROPOSITION 5.1. The following identity holds (5.14)

H2((E, OE), F) ^' if E (x'F)a, (u1 -1)f = 0, (v1-1)f = 0,...

(u9-1)f =0,(v9-1)f =0, (wi -1)f = 0, ... , (w3 - 1)f = 01. PROOF. By (5.10), if f E CZ " (F), the condition Of = 0, can be written as (5.15)

ulvi 1u11f = f , [vi, ui)f = ullf , u2v21,u21[v1,u1)f

= [vl,ui}f ,

[v2, u2)[ul, ul)f = u2 1 [vi, ui)f,

-1-')Iv,, Us] ...[v1,u1]f = 0, (1- w21)w11[ug,ug1...[v1,v11f =0. (1

Ffom the first two equalities in (5.15), (5.16)

[vi,ui)f =Vi f=ullf,

so that (5.17)

vl1ullf=f.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURJE

198

From (5.16), (5.17), we find that the first two equalities in (5.15) are equivalent to (5.18)

u1 f = Vi f = f

By proceeding as before, we get from (5.15),

uif=f,vif=f,15i
(5.19)

so that

wjf=f, lEj<s.

(5.20)

The proof of our Proposition is completed.

Let F* be the flat vector bundle on E, which is dual to F. Then if V* is dual to V, F* is the flat bundle on E associated to the dual representation r* : r -a Aut(V*). Let (CE. (F), 0), (Can. (F), 8), (CE," (F), 8) be the complexes dual to (Can (F*),8), (CE,r(F*), 0). Then we have the exact sequence of complexes (5.21)

0),

0 -4 (Cr."-(F), 0) -* (C£-(F), 0) -+ (COE- (F), 8) -+ 0.

The associated cohomology groups are respectively H ((E, 8E), F), (H (E, F), H(8E, F). For s > 0, the corresponding long exact sequence (5.22)

0 -+ Ho (E, F) -+ H0(8E, F) -+ Hl ((E, 8E), F) -> Hl (E, F) -+ Hl (8E, F) - H2 ((E, 8E), F) -> 0

is dual to the exact sequence (5.9) for F*. Now we describe the chain map 0 in the complexes (5.21). We trivialize the flat vector bundle F as indicated above. In particular the complexes in (5.21) are now direct sums of copies of (7r*F)p. By (5.10), (5.11), we find that if f E a*Fp, 9: C£,o(F) .4 CE,1(F) is such that 9

(5.23)

0(f)q') = E((1- ui)f Iai + (1 - vi)f Ibi) + : Icj , i=1

j=1

0(flrr) = -flcj+(1-wj)fId,, and 9: CE-1(F) -+ C£'2(F) by (5.24)

8(f Iai) = 0(f Ibi) =

0(flcj)

.9(fldj) =

[ul,Vj]...[ui-l, vi-1](1 - uiviuil)f [ul,Vi] ...[ui-1,Vi-1]ui(1 - viui Ivy I)f ,

[ul, v1] ... [u9, v9]wl ... wj-1(1- wj )f [ul, Vl] ... [u9,v9]wl ... wj_l f .

The complex (CE,r, (F), 0) is obtained formally from (5.23), (5.24) by making the components indexed by dj or rj equal to 0. Finally 0: Co.aE(F) -} C1,eE(F) is given by (5.25)

8(f l r1) = (1 - wj)f idj

Observe that by (5.23), (5.24), it is dear that for s > 0, (5.26)

H2 (E, F) = 0 , Ho((E, 8E), F) = 0 .

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

199

PROPOSITION 5.2. The following identity holds, 8

(5.27)

H°(E, F) = { f (q1 +

f IrjI

, with f E x"Fp such that

j=1 (1 - ui)f = 0,(1 - vi)f = 0 (1 < i < g)

,

(1 - wj)f = 0 (1 < j < s)}.

0

PROOF. Our identity follows from (5.23),

REMARK 5.3. By Poincar duality

H°(E,F).

(5.28)

Using Propositions 5.1 and 5.2, the identification (5.28) has been made explicit. Another interpretation is to view the complex (CE. (F), 8) as the relative homology complex associated to the dual cell decomposition of E. Suppose that s > 1. Let 0) be the trivial complex concentrated in degree 1,

K1 = ®ker(1- wj) Icj = H°(8E, F).

(5.29)

j=1

Then by (5.24), 8) is a subcomplex of (C-I^' (F), 0). We have the obvious exact sequences

0 -i-ker(1 - w,i) -+-

(5.30)

Im(1 - wj) -. 0

The exact sequences (5.30) induce an exact sequence of complexes (5.31)

(K , 8) -> (C£ r (F), 8) -T (C (F), 8) - 0

0

By (5.23), (5.24), (5.32)

5£,k(F) = CE,r,k(F),k=0or2, 9

_

Im(1- wj),, , k = 1.

,*FPi,{ ® 7r'Fpi , ® i=1

j=1

is given by

By (5.23), (5.24), if f E it Fp, 8 : Cn ° (F) g

(5.33)

of =

s

((I - ui)flai + (1- vi)flb,) +,(1- wj)&; j=1

and 8: CEJ (F) -+ C£22(F) by (5.34)

8(flai) = [u1,v1]...[ui-l,vi-1](1 - uiviui 1)f ,

8(f lbi)

_ [u1, vi] ... [ui-livi-1]ui(1 - viui 1vi 1)f ,

8(fIcj)

_ [ul,vl]...[ug,vg]wl...wj_1f.

.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

200

Let H (E, F) be the cohomology of (C£ (F), 8). By (5.31), for s > 0, we have the long exact sequence (5.35)

F) -,,- KI = H° (8E, F) --r HI ((E, 8E), F)

0

H2((E,BE),F)---rf2(E,F)0.

T_

THEOREM 5.4. For s > 0, the following identities hold,

H°(E, F) = H°(E, F) ,

(5.36)

H2 (E, F) = H2 ((E, BE), F) .

Also Al (E, F) is the image of HI ((E, BE), F) in HI (E, F). In (5.35), the map H°(E, F) -+ KI is just the map H°(E, F) -r H°(8E, F), the map K' -+ HI ((E, BE), F) is minus the map H° (8E, F) -+ HI ((E, 8E), F), and T : HI ((E, BE), F) -a HI (E, F) is induced by the map HI ((E, BE), F) -- HI (E, F). PROOF. By (5.33), (5.37)

H°(E,F) = if E 7r'Fy,(1-ui)f = 0, (1
(1-v0 f =0,(1-wf)f =0,1 < j <s}. Using Proposition 5.2, and (5.37), we get the first equality in (5.36). The second equality in (5.36) follows from (5.24), (5.34).

Inspection of (5.23), (5.24), (5.37) shows that the map H°(E,F) -a KI is the

canonical H°(E, F) -+ H°(8E, F). Take f E ker(1 - wi). By (5.23), if f Irk is viewed as lying in C°(E, F), then (5.38)

Of Irr=-fIe,,.

Therefore the map KI -> HI ((E, 8E), F) is minus the map H° (8E, F) -> HI ((E, BE), F). By (5.35), we have the canonical isomorphism (5.39)

IHm H°'(0E'F))

By (5.22), (5.39), we find that ft' (E, F) is exactly the image of HI ((E, BE), F) in HI (E, F). The proof of our Theorem is completed. REMARK 5.5. Assume that F and F* are naturally isomorphic as flat bundles. This is the case when F is real and carries a flat scalar product. Then H° (E, F) and H2 ((E, OE), F) are Poincar dual, and the image of HI((E, BE), F) in HI (E, F) is equipped with a non degenerate 2-form, the intersection product. The conclusion is that in this case, the cohomology of (CE- (F),8) exhibits Poincar6 duality.

S.S. The map 0. Let G be a compact connected semisimple Lie group. Let Z(G) be the center of G. Then Z(G) is a finite subgroup of G. Let G' = G/Z(G) be the adjoint group. Then G' acts naturally on G by conjugation. If U E G', V E G, we will write (5.40)

I.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

201

Similarly G' acts on G29}8 by conjugation. We will use the same notation for the action of G on G or on G2g}e by conjugation.

Take 9,s EN, with 9+s>0. DEFINITION 5.6. Let 0: G29+a - G be the map 9

(5.41)

a

0(u1, v1, ... , u9, v9, w1i ... , wa) _ f[ui, vi) IN . i=1

j=1

Then ¢ is G'-equivariant, i.e. if g E G', x E G29+., (5.42)

0(9.x) = g - O(x)

Let g be the Lie algebra of G. If g E G, we identify TG to g = TeG via the right multiplication operator R9.. Then if z E G29+8, T_ZG29+8 = g29+e .

(5.43)

DEFINITION 5.7. Take x E G29}a. Let S : 9 -+ g29+a be the derivative at g = 1 of the map g E G'-+ g x E G29+8. Similarly let 5 : 92g+8 -+ g be the derivative at x E G29+8 of x' = G29+S ,.+ 4,(x') E G.

Clearly G acts on g by the adjoint action. PROPOSITION 5.8. Take x = (u1, v1, ... , u9, v9, 101,

, wa) E G29}8. Then if

f E9, (5.44)

Sf = ((1 - u1)f, (1- VOL ... , (1- u9)f, (1- VOL

(1 - w1)f,...,(1 - wa)f) Similarly, if (fl,.

, f2g+a) E g2s+a, 9

(5.45)

S(fi,... ,f2g+8) = E[u1iv1]...[ui-1,va-Ij i=1

((1 - uaviu{1)f2i-1 + ui(1 - viui 1vi 1)f2i)

+ )t[u1,v1)...[u9,v9]w1...wf_1f2g+j

i=1

PROOF. This is a trivial computation, which is left to the reader.

Let 0 C G be an orbit in G. More precisely, take g E G, and put O = {g'- g, g' E G}.

(5.46)

Of course g E O. If Z(g) C G is the centralizer of g,

0 = G/Z(g).

(5.47)

Clearly the tangent space T90 C T9G = g is given by (5.48)

Also T.Z(9) C 9 is given by (5.49)

T9O9 =

Im(1 - 9)

T. Z(9) = ker(1 - g).

Then the exact sequence (5.50)

0->T.Z(g)->T.G->T,O9-+ 0

JEAN-MICHEL BISMUT AND FRANcOIS LABOURIE

202

corresponds to the obvious

0 ---i-ker(1- g) -- 9 i_g Im(1- g) -- 0

(5.51)

Let 01,... , . be a adjoint orbits of Gin G. Put

X = G2g x ft 0, .

(5.52)

j=1

If x = (ul, 411, ... , B9, vg, w1, ... , w9) E X, S : 9 9 g2g+9 is in effect a linear map :

g - g2g ®®,=1 Im(1- w,). By restriction we obtain the linear maps

(D, d) : 0 -- 9 -

(5.53)

- 92g ®®1=1 Im(1- wj) a - 9 __,- 0

s

Note that in general S2 540.

(5.54)

Also observe that if g E G', g maps (D, 8)x into (D, 8)g..

DEFINITION 5.9. If x E X, let Z(x) C G, Z'(z) C G' be the stabilizers of z, and let 3(x) be the Lie algebra of Z(x) and Z'(x).

Of course Z'(x) = Z(z)/Z(G). PROPOSITION 5.10. For x E X, then

{fE0,Sf=0}=3(x).

(5.55)

PROOF. This is obvious by (5.44).

Let (, ) be a G-invariant scalar product on g. If l) is a vector subspace of g, let 41 be the orthogonal space to 4 in 9. PROPOSITION 5.11. For x E X, then

l1 [6(929

(5.56)

eIm(1-w,))1

=3(x).

j=1

Im(l - w,) be the adjoint of S

PROOF. Let S* : g -4 g2g

:

g2g

Im(1- w,) -4 g. Clearly e

S(929 ®® Im(1-w,))1 = ker 5* .

(5.57)

J=1

Let

PIm(1-w;) be the orthogonal projection operator

(5.58)

g -> Im(1- w,). Then

(1 - w3 1) pIm(1-wf) = (1- W 1),

and 1 - w.,-.1 is one to one from Im(1 - w,) into itself. Using 5.45, we obtain a Pm'(1-wf). By (5.58), to calculate kerb*, formula for S* involving the projection we may replace plm(1-wj) by (1- w,). By proceeding as in Proposition 5.1, we get (5.56).

Recall that o : X -+ G is said to be regular at x E X if di(z) : TTX is surjective.

To(x)G

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

THEOREM 5.12. The map

203

: X -3 G is regular at x E X if and only if

3(x) = 0. PROOF. Our Theorem is a trivial consequence of Proposition 5.11. DEFINITION 5.13. Put

M = {x E X, 4i(x) = 1).

(5.59)

If x = (ul,... ,w,) E M, then x defines a morphism r -* G. Also G acts on g via the adjoint representation. Let E be the flat real vector bundle on E constructed as in (5.7), i.e.

E=Exrg.

(5.60)

The following result was first obtained by Liu [39, 40].

THEOREM 5.14. If x E M, then in (5.53), 52 = 0, i.e. (D, 0)x is a complex. The complex (D, 5). can then be canonically identified to the complex (CE- (E),,9)..

PROOF. By (5.42), if x E M, 6(gx) = 1.

(5.61)

By (5.61), we find that 52 = 0. Comparing (5.33), (5.34) and (5.44), (5.45), our Theorem follows.

REMARK 5.15. By Theorem 5.4, if x E M,

ko(E, E) = H°(E, E)

(5.62)

H2(E, E) = H2((E, 8E), E) In view of Propositions 5.1 and 5.10, we find that if x E M,

H°(E,E) = 3(x),

(5.63)

H2(E,E) = l(x)'. So when x E M, Proposition 5.11 follows from (5.63).

5.4. The set of regular values of the map 0. Recall that G and G' act by the adjoint action on g. Then one verifies easily that if x E M, g E G,

(5 (E), 0) 8 (5

(5.64)

(E), 0)

is an isomorphism of complexes. In particular the induced map

ft(E) g FI9 (E)

(5.65)

is an isomorphism.

Let T be a maximal torus in G, let t be the Lie algebra of T. Let W be the Weyl group of G with respect to T. Let R = {a} C t' be the root system of G, let CR = {ha} C t be the corresponding coroots. Now we recall the definition of S C T given in Definition 2.23. DEFINITION 5.16. Let S C T be given by (5.66)

taiha ; the h, for which to 0 0 do not span t} .

S = It E T; t = aER

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

204

Let t1,... , t, E T. Let Oi, ... , O, be the orbits of tl, ... , t, in G. By [15, Lemma IV.2.5], (5.67)

O1fT=Wt3, 1<j<s.

DEFINITION 5.17. We will say that (ti, ... , t,) verify (A) if for any (wi, ... , w8) E Wa, s

twitfsfS.

(5.68)

1

THEOREM 5.18. For g > 1, the map ¢ : X -+ G is sui jective. For g > 1, t E T

is a regular value of 0 if and only if (ti, t2,... , t -t) E T'+I verify (A). For g = 0, t e T is a regular value of ¢ in the following two cases :

(5.69)

. t 9 0W. t E O(X), and (ti, ... , t -t) verify (A).

PROOF. By [14, Corollaire 4.5], since G is semisimple, the map (u, v) E G2 H [u, v] E G is surjective. Therefore, for g > 1, the map 0 is surjective. First we will prove the remainder of our Theorem for t = 1. By Theorem 5.12,

1 E G is a non regular value of ¢ if and only if there exists x E M such that 3(x) 0 0. Equivalently there exists p E g, p ,-+E 0 and x = (ui,... , w,) E M such that x E Z(p)2s+8. By [15, Theorem IV.2.3], Z(p) is a connected Lie subgroup of G, which is not semisimple, since Rp is contained in the Lie algebra of its center. Let To = Zo(Z(p)) be the connected component of the identity in Z(Z(p)). Then we have the exact sequence (5.70)

1-+ To --* Z(p) -* Z(p)/To - 1,

and Z(p)/To is semisimple. Put (5.71)

7 = iri (Z(p)/To) ,

and let Z(p)/To be the universal cover of Z(p)/To. Then (5.70) fits in the diagram (5.72)

1

1

1-.-1-----7

1

7-->1

J

_I

1

1

1

I

I

I

1

1

1

1 -s To

J

Z(p)/To _.> 1

1- To -> Z(p) -p- Z(p)fT0 -$1

In (5.72), 2(p) -, Z(p) is the obvious ry covering of Z(p).

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

205

Since Z(p)/To is semisimple and simply connected, any central extension of Z(p)/To by a torus is trivial, so that Z(p) = T. X Z(p)/To .

(5.73)

Therefore ry C Z(p)/To embeds as a finite subgroup of To x Z(p)/To. Let T(p) be a maximal torus in Z(p)/To. Then To x T'(p) is a maximal torus in 2(p). From (5.72), (5.73), we get the complex (5.74)

1

1

1

I

I

I

1

1

I

!

J

1

1

1

1

1->To- To xT(p) ->T(p) --> 1 1->To

i

Now we use the notation of Theorem 1.52. Let to = a(Z(p)) be the Lie algebra of To. Let t1 C t be the vector subspace oft spanned by the {ha}aert,. Clearly (5.75)

t = to ®t1.

Also t1 is the Lie algebra of TITO or of f (p). Let [Z(p), Z(p)] be the commutator subgroup Z(p), i.e. the group spanned by commutators in 2(p). By (5.73), since Z(p)/To is semisimple, (5.76)

[2(p), 2(p)] = Z()/To

Clearly [2(p), 2(p)] maps onto [Z(p), Z(p)]. Therefore [Z(p), Z(p)] is a compact connected subgroup of Z(p), and so it is a connected Lie subgroup of Z(p). Let now t E T fl [Z(p), Z(p)]. By the above, there is a E Z(p)/To, b = (bo, b1) E To x T (p) which map to t. Therefore, if c = ab-1, then c E Y. Clearly (5.77)

c = (bo I, abi 1).

By (5.74), (5.77), (5.78)

abi1E7CT(p).

From (5.77), we get (5.79)

a E T (p) .

We thus find that if t E T fl [Z(p), Z(p)], t is the projection of an element of T(p). Equivalently t can be represented by 1 E t1.

JEAN-MICHEL BISMUT AND FRANCOIS LABOURIE

206

s

9 7

IJ[ui,vi] R wj = 1 , j=1

i=1

and ui, vi E Z(p), 1 < i < g, wj E Z(p), 1 < j < s. Clearly T is a maximal torus in Z(p). So there is g E Z(p) such that gwjg-1 E T. Since gwjg-' E T no,,, by (5.67), there is wj E W such that gwjg 1 = w'tj .

(5.81)

Therefore (5.82)

to, =witj in Z(p)I[Z(p),Z(p)]

So by (5.80), (5.82), we get (5.83)

>2 wjtj E T n [Z(p), Z(p)] . j=1

By the above, we find that (5.84)

>2wjtj = >2 Saha in T. J=1

aER,

Now since p # 0, {ha, a E R)} does not span t, i.e. (ti, ... , t8) does not verify (A).

Conversely, suppose that g > 1, and (t1, ... , t8) does not verify (A). Then

there is (w1,... ,w') E W', such that E,=1 wjtj can be expressed as a linear combination of {ha} which do not span t. Let p E t, p # 0 be orthogonal to the corresponding {a}. By the above, B

(5.85)

>2 wjtj E [Z(p), Z(p)] . j=1

Using the fact that in a compact semisimple Lie group, any element is a commutator

[14, Corolla-ire 4.5], and also the considerations after (5.76), by (5.85), there is u9, v9 E Z(p) such that s

(5.86)

[u9,v9]nwitj=1. j=1

If x = (1,... ,1,u9,v9,w1t1,... ,w'te), then x E M, and p E 3(x), so that {0}. By Theorem 5.12, we find that 1 is not a regular value of 0.

Let t = to}1 E T. Let 08+1 C G be the G-orbit of t8 i. Put X+1 = G29 x rli'±; Oj. More generally we denote with the index +1 the objects we constructed above, which are associated to X}1. Clearly (5.87)

X+1 = X x 08+1.

Also if (x,w8+1) E X+1, (5.88)

cb+1(x,ws+1) = O(x)w.+1.

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207

Clearly ¢(x) = t if and only if (x,t,+1) E A41. By (5.88), if 0 is regular at x. 0+1 is regular at (x, t,+ I) Conversely, suppose that 6+1 is regular at (x, t,+1). By (5.88), (5.89)

Im(db+I)(x, to+1) = Im(db¢-1)x + (t8+1 - 1) (9) By (5.42), we find that (5.90)

(t8+1 -1)(9) C Im(dbb'')z.

From (5.89), (5.90), we find that 0 is regular at t if and only if 6+1 is regular at (x, t,+1). Now we use our Theorem for t = 1 and we obtain the stated result in full generality.

REMARK 5.19. Instead of studying the case t = 1 first, we could as well have used Theorem 5.12 and proceeded directly.

5.5. The stabilizers Z'(x). THEOREM 5.20. If G is simply connected, under one of the following three con-

ditions, if t E T is a regular value of 0, {x E 6-1{t},Z'(x) 0 1} is included in a union of submanifolds of {x E 6-'(t)} of codimension > 2

Forg>2. For g = 1, if one of the tj 's or t is regular.

For g = 0, s = 1, in which case 6-1{t} is empty, or if at least 3 of the elements It,,. . . , t t} are regular. If G is only connected, if t1, ... , t, are very regular, if t E T is a regular value of

0, {x E 6-1(t),Z'(x) # 1} is included in a union of submanifolds of {x E 0-'(t)} of codimension > 2 :

Forg>2. For g = 1, ifs > 1 or if t is very regular.

For g = 0, s = 1, in which case 6-1(t) is empty, ifs = 2 and t is very regular, or ifs > 3. PROOF. We will prove our Theorem in various stages. The case where G is simply connected and t = 1.

If 1 is a regular value of ¢, either M = ¢-1{1) is empty, or it is a smooth submanifold of X. By Theorem 5.12, when M is non empty, the group G acts locally freely on M. Therefore M/G is an orbifold. If dim(M/G) is the dimension of the maximal

statum of M/G, then (5.91)

dimM/G = (29 - 2) dim(g) + tdim(Oj) . j=1

Of course the same observation applies to the action of G' on Al. Clearly M/G = M/G'. In the sequel we will use the notation M/G or M/G' indifferently. However in Section 6.3, we will construct an orbifold G-line bundle on M/G, which may well not be an orbifold G'-line bundle.

Take now x E M such that Z'(x) 54 1. Let u E Z'(x), u # 1. If x =

(u1, ... , w,), then u1, ... , w, E Z(u). By conjugation, we may as well assume that u E T' =T/Z(G). Since G acts locally freely on M, the group Z(u) is semisimple. By Theorem 1.38, u E C/R . Of course here u 0 0.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

208

Since G is simply connected, Z(u) is connected. By Theorem 1.50, for any t E T,

o(t) n Z(u) =

(5.92)

U

oZ(u) (wt) .

wEW \W

For1<j<s,letw'EWsuchthat wj E OZ(u)(w'ti), 1 < j < s.

(5.93)

Put 8

Xu = Z(u)2g-2 x 11 Oz(u)(W't2) f=I

(5.94)

: Xu -+ Z(u) as in (5.41). Let Mu = ¢u1{1}. Then x E Mu. Moreover Z(u) acts locally freely on Mu. Therefore, by Theorem 5.12, 1 is a regular value of 4u, i.e. Mu is a smooth submanifold of M. Note that this also follows from the fact that M is a smooth manifold, G acts on M, and Mu is a We define qSu

component of the fixed point set of u in M. Then (5.95)

dimMu/Z(u)=(2g-2)dimg(u)+tdimOZ(u)(w't1). i=I

Now by Theorem 1.52,

dim;(u) < dim(g) - 2.

(5.96) Also (5.97)

dim OZ(u) (w' ti) < dim OO .

Now we consider 3 cases : If g > 2, by (5.91), (5.95)-(5.97),

dim Mu/Z(u) < dim M/G - 2.

(5.98)

By (5.98), Mu/Z(u) maps to a submanifold of codimension > 2 in M/G. Therefore the G-orbit of Mu in M is a submanifold of codimension > 2 in M. If g = 1, assume that tI E T is regular. Then (5.99)

dim Ot, = dim(g) - dim(t), dim OZ(u) (w'tI) = dim3(u) - dim(t).

By (5.91), (5.95), (5.96), (5.97), (5.99), we find again that

dim Mu/Z(u) < dim M/G - 2.

(5.100)

If g = O, if s = 2, then (5.101)

dim Ot, < dim(g) - dim(t) ; s = 1, 2,

so that using (5.91), (5.102)

i.e. M = 0.

dim M/G < 0,

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

209

If s > 3, assume that tl, t2, is are regular. Then (5.103)

dim M/G = dim(g) - 3 dim(t) + E dim(0j) , j=4

dim M'/Z(u) = dim 3(u) - 3 dim(t) +,E dim Oz(u) (wjtj) j=4

By (5.96), (5.97), (5.103) (5.104)

dim M"/Z(u) < dim M/G - 2.

So we have proved our Theorem in this case. The case where G is non simply connected and t = 1. First we proceed as above. In this case for u E C/R , u 0 1, Z(u) is in general non connected. However by (1.174), since t1, ... , t, are very regular,

®j fl z(u) =

(5.105)

U

(w'tj) .

WIEWZ( ),\W F o r 1 < j < s,let wj E W be such that

wj E 0Z(u),(uritj).

(5.106)

Again we define Xu by (5.94), and we define ¢u as before. Put Mu = ¢u'{1}. By the same argument as before, 1 is a regular value of ¢u. Then (5.95) still holds. If g > 2, (5.104) is still true. If g = 1, since t1 is very regular in G, then tl lies in Z(u)o and is very regular in Z(u), so that (5.100) still holds. If g = 0, under the stated assumptions, the argument in the proof above can still be reproduced. The proof of our Theorem is completed in this case. The case where t E T is an arbitrary regular value of 0. Let t = tj+1 E T. We use the notation in the proof of Theorem 5.18. Then t is a regular value of ¢ if and only if 1 is a regular value of ¢}1. By the above, we obtain our Theorem in full generality. 13 The proof of our Theorem is completed. THEOREM 5.21. If G is simply connected, under one of the following conditions, {x E X; Z(x) 96 1} is included in a union of submanifolds of codimension

>2: 9>1. g = 0, s > 2, and at least 2 of the tj's are regular.

If G is non necessarily simply connected, if all the tj's are very regular, then {x E X, Z(x) 96 1} is included in a union of submanifolds of codimension > 2 if

g>1. g = 0, s > 2, and at least 2 of the ti's are very regular. PROOF. By [24, Proposition 27.41, the set of conjugacy classes of stabilizers

Z'(x) is fnite. Take x E X and assume that Z'(x) 0- e. Let G'(x) be the orbit of G' through x. Then (5.107)

G'(x) = G'/Z'(x).

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Let Ny be the normal bundle to G'(x) at x. Then Z'(x) acts on N.. By [24, Proposition 27.2], there is a G'-invariant open neighborhood of G'(x) in X which can be identified as a G-space to a neighborhood of the zero section in N = G' x N.

Let Nx be the invariant part of N. under Z'(x). Then NI extends to a vector subbundle of the vector bundle Non G'/Z'(x), of the form G'/Z'(x) x N,f...

If y E N., then Z(y) C Z'(x) and Z'(y) = Z'(x) if and only if y E Nz. In particular Z'(y) is conjugate to Z'(x) if and only if y E N.I, in which case Z'(y) = Z'(x). It follows that near G'(x), the elements of X whose stabilizer is conjugate to Z'(x) form a neighborhood of the zero section of Nf. Let u E Z'(x), u 0 1. We may and we will assume that u E T. Let (G'/Z'(x))u

be the fixed point set of u in G'/Z'(x). Let codim(G'/Z'(x))u,G'/Z'(x)) be the codimension of (G'/Z'(x))u in G'IZ'(x). Clearly (5.108)

codim(G'/Z'(x))u,G'/Z'(x)) < dim(g) - dim(3(u)).

Let NN be the vector subspace of N. fixed under u. Then N;, C Ni. If Nu is the fixed point set of N under u, then Nu is a vector bundle over (G'/Z'(x))u, with fibre modelled on Nz. Let dim(Nf) = dim(G'/Z'(x))+dimNx be the dimension of the total space of Nf. Similarly let dim(Nu) be the dimension of the total space of Nu. By (5.108), we get (5.109)

dim(Nf) < dim(N") + dim(g) - dim 3(u) .

Clearly

dim X = 2g dim(g) +

(5.110)

dirn(O j) . j=1

Let Xu be the fixed point set of X under u. Then

Xu = Z(u)2° x 11(0, fl Z(u)),

(5.111)

j=1

so that (5.112)

dim(pi n Z(u))

dim(Xu) = 2g dim 3 (u) j=1

By (5.109), (5.112),

(5.113)

dim(Nf) < (2g - 1) dim 3(u) + dim g + t dim(Oi fl Z(u)). j=1

If g > 1, using (5.96), (5.110), (5.113), (5.114)

dim(Nf) < dim(X) - 2.

Assume now that g = 0, that s > 2, that G is simply connected and t1, t2 are regular, or more generally that t1, t2 are very regular. By Theorem 1.50, (5.115)

dim 0 j = dim(g) - dim(t) , dim 0j fl Z(u) = dim3(u) - dim t.

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So if g = 0, using (5.96), (5.110), (5.113), (5.115), we get

(5.116) dim(Nf) < dim3(u) + dim g - 2 dim(t) +,Edim(Oj fl Z(a)) j=3

< dim(X) - 2.

p

From (5.114), (5.116), our Theorem follows.

5.6. The tangent bundle to the moduli space and its symplectic form. We make the same assumptions as in Sections 5.1-5.4.

If x E X, let G'(x) be the orbit of G' at x, and let T,TG'(x) be the tangent

space to G'(x) at x. Recall that TTX C 9'9-B, so that T,,G'(x) C g29PROPOSITION 5.22. If x E M, (5.117)

TTG'(x) =.9(Ci'0(E)) C Cy'I(E)

If 0 is regular at x E M, then M is a submanifold of X near x, and (5.118)

TIM =

PROOF. By Theorem 5.14, the first part of our Proposition is clear. If q is regular at x E M, (5.119)

TIM = kerdo(x).

Using Theorem 5.14 and (5.119), we get (5.118).

0

REMARK 5.23. Of course, if ¢ is regular at x, then TIG'(x) C T.H. This fits with (5.117), (5.118), because 02 = 0. By Proposition 5.10 and Theorem 5.12, TIG'(x) has dimension dimg.

Let it : M -+ M/G be the obvious projection. If 0 is regular at x E M, Z'(x) is a finite group. THEOREM 5.24. If 0 is regular at x E M, M is smooth near x, and M/G is an orbifold near ir(x). More precisely, near 7r(x) E M/G, (5.120)

TM/G = M xG HE,I(E).

Also near ir(x), (5.121)

M/G = FIz (E, E)/Z' (x).

PROOF. These are classical facts from the theory of orbifolds (see [?, Proposition 27.7).

0

REMARK 5.25. By Theorems 5.4 and 5.24, T M/G is the image of HI ((E, 8S), E)

into HI(E,E). This fact was also observed in Guruprasad, Huebschammm,Jeffrey and Weinstein [25). Now we assume temporarily that G is simply connected. We use the notation of Section 4. DEFINITION 5.26. Put

(5.122) Amt (t..... , t,) = {A E A, FA = 0, -(fit +A)S) E O_*+t,},1 < j:5 s.

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Equivalently, A181(t1, ... , t,) is the set of flat connections A such that for 1 < j < s, the holonomy to, of A along 51 lies in the orbit O. Clearly EG acts on the right on Aflat (ti,... , t,). DEFINITION 5.27. Let EgG be the group of smooth maps g : E -a G such that

9g=1. It is easy to see that EqG acts freely on A"t(ti,... ,t,) . Also there is an obvious continuous map f : A"t(ti,... , t,) H M which, to A, associates (U12v12 ...Wi,.--

w,), the holonomies of A along the circles ai, bi, ... , cidici i, ... , c,d,c, i. We equip Aflet(ti,... ,t.)IE,G with the quotient topology. PROPOSITION 5.28. The map f induces an identification of compact spaces (5.123)

Aflet(ti,... ,t,)/EqG = M.

PROOF. Clearly f descends to a one to one map Aflat(ti,... ,t.)IE,G M. This map is continuous. By [18, Proposition 2.2.3], f is a pointwise identification. The fact that the topologies coincide follow for instance from the techniques of [18, Section 4.2], in a much simpler context. Assume now that (t1, ... , t,) verify (A). Then by Theorem 5.18, M is a smooth manifold, possibly empty if g = 0. Using the techniques of [18, Section 4.2], we find that AsBt (t1, ... , t,) is a smooth manifold. Note that here, we use explicitly the fact

that for any x E M, H2(E, E) = 0. By [18, Proposition 4.2.23], Aflat(ti,... t,)/EG is a smooth manifold. Also M -+ M/G is a G-orbifold. By the above, EG acts locally freely on Aflat(t1,... t,). PROPOSITION 5.29. We have the identification of orbifolds, (5.124)

Aaat(ti,... t,)/EG = M/G.

PROOF. By proceeding as in [18, Proposition 4.2.23], we find easily that the identification f in Proposition 5.28 is an identification of smooth manifolds. Our Proposition follows.

We do no longer assume G to be simply connected.

Let (,) be a G-invariant bilinear symmetric form on g. Then E is equipped with the corresponding flat bilinear symmetric form. Recall that by Theorem 5.4, H1(E, E) is the image of HI ((E, 8E), E) into HI (E, E). DEFINITION 5.30. If X E M, a, a' E Hxr(E, E), put (5.125)

wx(a,a') = f -(a,a'). I:

Then wx is an intersection form, so that it is non degenerate. THEOREM 5.31. The 2 form w is G-invariant. It descends to a symplectic 2form on the orbifold M/G.

PROOF. Suppose first that G is simply connected. By Proposition 5.28, the space A' (ti, ... t,)/EG is an orbifold. Then the results of Section 4.1 and Theorem 4.24 show that Aflat(t1.... t,)/EG is a symplectic reduction of the symplectic manifold A. Since it is a symplectic reduction of the symplectic affine space A, it

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.

carries a symplectic form. One verifies easily that w is just this form. Therefore is closed. If G is not simply connected, the homotopy types of G-bundles have been classified in Section 4.7. Let G be the universal cover of G. By introducing an extra holonomy h E Z(G), we can then replace G by G. The fact that w is closed is now a consequence of the corresponding result for G. The proof of our Theorem is completed. 0

Recall that G acts on the right and on the left on X. If x E AI, a E 17, then x(a) E G. We define a right action of r on X by the formula (5.126)

x.a = x(a).x, x E M, a E r

Then if aEr,9EG,xEM, (5.127)

(x.g).a =

(x.a).g,

i.e. the actions of r and G on M commute. Also r x G acts on the right on t (of course, the factor G acts trivially on E). Therefore r x G acts on the right on M x E. Also r x G acts locally freely on M, and r acts freely on E. Let V be a complex vector space. Let p : G -- Aut(V) be a representation of G. Then p induces a representation r x G -* Aut(V). Put (5.128)

F = (M x E) xrxG V.

Then F is an orbifold vector bundle on M/G x E. It is obtained via the identification

(5.129) (x, o, f) = (x.g, o.a, g'1x(a) 1 f) , (x, v, f) E M x E x V , (a, g) E r x 0 .

For a given x E M, the restriction of F to the fibre E is exactly the flat vector bundle considered in Section 5.2.

If o E E, then o*F is an orbifold vector bundle on M/G. If a, or' E E, if t E [0,11 .+ at E E is a smooth path, with ae = o, a1 = a', parallel transport with respect to the flat connection identifies o*F and a *F. Over E, there are two distinguished points p and q. Recall that G acts on g by conjugation. When V = g, let E be the corresponding real vector bundle on

M/G x E. DEFINITION 5.32. Let £ be the vector bundle on M/G, (5.130)

£ = P.E.

Also G acts by conjugation on G and on the 0.1's. Put (5.131)

G=MxGG,O1=Mxo03.

Then G -3 M/G is a G-bundle, and G,, -> M/G is a O,-bundle. Also u1, ... , v9 are sections of G and w1, ... , w, are sections of ®1, ... , 0,. Elements of G act naturally on £. As explained in Section 5.2, u1 can be considered as the parallel transport operator along the closed curve a1 ... For 1 < j < a, let T,,; 09l(M/G) be the relative tangent bundle to the fibre 0j As we saw in (5.48), (5.132)

T., O,, /M = Im(1- wj) C C.

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As explained in Theorem 5.14, on M/G, we have the bundle of complexes

0 -+ C£,0 (E) -+ C,1(E) -+ C£,=(E) -+ 0,

(5.133)

which, by (5.53), (5.132) can also be written as (5.134)

8 _ 0 - 6 _+,629 ®(®T 1(Oj/(M/G))) -i E -* 0.

j=1

THEoREM 5.33. On M/G, we have the identity (5.135)

TM/G = E29 $ (®Twf (5j/(M/G)) 8 62 in K(M/G). j=1

PRooF. By (5.120),

TM/G = S1(E, E).

(5.136)

Also over M, by (5.63),

Iij(E,E)=0,j=0,2.

(5.137)

Hence, over M/G, by (5.134), (5.136), (5.137), (5.138)

H1(E, E) = E2g ® (®Tw16j/(M/G)) 8 E2 inK(M/G). j=1

From (5.137), (5.138), we get (5.135).

THEOREM 5.34. If the orbits 01,... , Oe are very regular (resp. regular), then (5.139)

TM/G

=.62(9-2)+s 9 R8 dim t in K(M/G) (resp. in K (M/G) QQ Z.)

PROOF. Let t E T be very regular, let 0 be the orbit oft in G. Then Z(t) = T.

If9EG, (5.140)

Z(g.t) = g.T, 3(g.t) = g.t.

If w E 0, we have the splitting (5.141)

g=Im(1-w)®ker(1-w),

which corresponds to the splitting (5.142)

E = Two ® N01E.

By (5.140), the vector bundle 3(w) = ker(1-w) is trivial on O. The action of G on the orbit 0 lifts to an action on the vector bundle 3(w). From (5.141), (5.142), we find that the normal bundle N01E is equivariantly trivial. From (5.135), (5.142), we get (5.139).

If t E T is only regular, then 3(t) = t. Also 0 = G/Z(t). Then if w = g.t, g E G H g.3(t) = 3(w) induces a G-invariant flat connection on 3(w) = Note. Consider the splitting of vector bundles (5.143)

E=T.,(Ojl(M/G))a) ker(1-wj).

If the orbit Oj is very regular, ker(1 - wj) is a trivial vector bundle on M/G. Therefore (5.144)

Tws 0j/(M/G) = E e Rdim t in K(M/G).

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If the orbit OO is only regular, then one finds easily by the above that ker(1- w1) is equipped over M/G with a flat connection. We thus get (5.139) in this case. The proof of our Theorem is completed.

5.7. A metric on the determinant of T11/G. If A is a line, let A -I be the

dual line. If E is a vector space, set (5.145)

det(E) = A--(E).

If E _ ®'o E; is a Z graded vector space, set m

(5.146)

det(E) _ (det

E;)1-lit

i=0

Let G be a compact connected semisimple Lie group. DEFINITION 5.35. If x E M, let A be the real line (5.147)

A =(detAx(E,E))-'.

Let (,) be a G-invariant scalar product on g. Then E is equipped with a fibrewise flat scalar product. By Theorem 5.4 and Remark 5.5, (5.148)

Ii2(E,E) = (. °(E,E))', t'(E,E) ^_- (ft'(E,E))`.

Observe that in (5.148), the identifications depend explicitly on (,). By (5.148), for x E M, (5.149)

a2. = R.

Now R carries a canonical metric II IIIt such that IIIIIR = 1. DEFINITION 5.36. Let II II.\. be the metric on A such that the identification (5.149) is an isometry.

In the sequel, we assume that (t1, ... , t8) verify (A). PROPOSITION 5.37. If x E M, the metric II III. defines a volume element on A'°87Ts(M/G), which is exactly the volume associated to the symplectic form wx.

PROOF. If x E M, then Fiy (E, E) = 0, fix (E, E) = 0, so that (5.150)

A = det ul' (E, E) .

By (5.120), (5.151)

TXM/G = ft (E, E) .

Finally in (5.148), the identification ft.' (E, E) . (ft (E, E))' is done via the symplectic form wx. Our Proposition follows.

Now recall that ft (E, E) is the cohomology of the complex (Cs (E), 8). Put (5.152)

ax = (det(d. (E)))-'.

Then by [35], there is a canonical isomorphism (5.153)

J. = %.

Also, since E is equipped with a flat scalar product, % is also naturally equipped with a metric II Ill., which also depends on (,). Let II Iia. be the corresponding metric on A via the canonical isomorphism (5.153).

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DEFINITION 5.38. For 1 < j < s, put (5.154)

IQ°iI =Idet(1-wj)IIm(1-wi)I1/2,wj EOj.

As the notation indicates, I o°i I is a constant on Oj. We make the convention that if Im(1- wj) C g is reduced to 0, Ioo,I = 1

(5.155)

REMARK 5.39. Recall that the function o : G -+ C was defined in (1.42). If Oj is regular, if wj E Oj,

I-°il = Io(wj)I

(5.156)

The following result has been proved by Witten [63, Section 4], who exhibited the role of the Reidemeister torsion [49] in this context. THEOREM 5.40. For any x E M, s

(5.157)

jj l°C; III

IA

j=1

PROOF. By (5.31), (5.158)

det(CE(E)) = det(CE.t(E)) 0 (det(K))-',

which can also be written in the form, (5.159)

det(CE(E)) = det(CE,r'(E)) 0 det(H°(8E, E)).

By (5.29),

H°(8E, E) = ® ker(1- wj).

(5.160)

j=1

Let be the metric on det H°(8E, E) induced by the scalar product of E. By (5.31) , we find that under the isomorphism (5.159), (5.161)

II I1det51(B) =1I IldetC

r

l_I

I1detH0(&E,E)

det(1- wj)IIm(1-w;)

j=1

J

Also by (5.21), there is a canonical isomorphism (5.162)

det CE (E) = det CE.''(E) O det Ce£ (E) .

R II IldetcE(E) denote the obvious metric on det(CE(E)), by (5.21), (5.162), we get (5.163)

II IldetCE(R) = II IldetCE.*(E)II IldetCO£(E)

Also, we have a canonical isomorphism (5.164)

det(COE (E))

det(H(OE, E)).

By Poincar duality,

H'(BE,E) _ (H°(8E,E))*,

(5.165)

so that by (5.164), (5.165), (5.166)

det(C&E(E))

det(H°(49E,E))2.

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By (5.25), (5.166), 1 -1

8

(5.167)

II Ildet(Ce£(E)) = II IldotKe(8E,E) [II det(1- wj)jIn,(1-u;) j=1

J

Using (5.159)-(5.166),we get (5.168)

(det 6r(E))2 -- det C: (E) 0 det CE.r(E).

Also under the isomorphism (5.168), (5.169)

11

112

= II IIdetCE(E)II l1_1 rrI

j-I

det(1 - wj),Im(1-w )J

Now recall that by [35], there are canonical isomorphisms (5.170)

det(C£(E)) = det(H(E, E)) , det(C£,'(E)) = det(H((E, 8E), E) .

By Poincar6 duality, (5.171)

detH(E,E)0detH((E,OE),E) =R.

So by (5.170), (5.171), (5.172)

det CE (E) @ det CT-,r(E) = R.

Let II IIR be the trivial metric on R, such that hula = 1. We claim that under the Poincar duality isomorphism (5.172), (5.173)

II IldetCE(E)II

II IIR.

In fact (5.173) is a simple consequence of the existence of the Reidemeister torsion [49], or more precisely of the Reidemeister metric [13, Section 11 on detH(E,E)) and det H((E, 8E), E). In fact given any cell decomposition of the manifold with boundary E, one can construct Reidemeister metrics on det H(E, E) and det H((E, 8E), E) by the procedure indicated in [13]. The basic fact is that these metrics do not depend on the choice of the cell decomposition. By applying this argument to a cell decomposition and the corresponding dual decomposition, one deduces immediately that the Reidemeister metrics on det H(E, E) and det H((E, 8E), E) correspond by Poincare duality. As a consequence, (5.173) follows. From (5.169), (5.173), we get (5.157). The proof of our Theorem is completed. 11

From (5.156), (5.157), it follows that if d1,... 0Ce are regular, if x E B, (5.174)

Ia(wj)II IIa. .

II K. j=1

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5.8. The Witten formula for the symplectic volume distribution. Let G be a compact connected semisimple Lie group. Let (, ) be a G-invariant scalar product on V. Let dg be the Haar measure on G associated to (, ). We us otherwise the same notation as in Section 5.7. In particular T C G is a maximal torus in G. Let 01,... , 0. be s adjoint orbits of Gin G. Take wj E Oj. Let Z(wj) be the centralizer of wj. Then the map (5.175)

is an identification of smooth manifolds. The scalar product (, ) induces a volume form dvoi on Oj C G.

DEFINITION 5.41. Let dvx be the volume form on X = G29 x rjf_1 Oj, 2g

a

dvx(ul) vl,... ,U9,V9,w1,... ,ws) _ flduidvi [f dvo,(wj).

(5.176)

j=1

i=1

Let dt be the Lebesgue measure on t induced by (, ).

DEFINITION 5.42. If g E G, let Xo, C G29 x fl .. Oj x 09 be the set M in (5.59) associated to the orbits O1, ... , 0 09. More precisely e

Xo, = {(u1)v1,... ,w1,... ,w.,w) E G29 x AOj X 09,

(5.177)

j.1

H[ui,vi] A wjw =1} .

(5.178)

i=I

j=1

Clearly, Xo, can be identified with {x E X, 4(x) E 09-I}. Set

X9={zEX;¢(x)=g 1}.

(5.179)

Now G acts naturally on Xo,, and Z(g) acts on X9. Then one has the obvious

Xo,/G = X9/Z(g).

(5.180)

Clearly, if x E X, fn, f (x g)dg depends only on 1rx E X/G. Let Gvreg (resp. Greg ) be the set of very regular (reap. regular) elements in G. If g E Gvreg, let dt9 be the Haar measure on the maximal torus Z(g), which is

associated to (, ). By Sard's theorem, a.e. every g E G is a regular value of 0. For such a g E G,

Xo, is smooth, and G acts locally freely on Xo,. Also by Theorem 5.31, the orbifold Xo,IG is equipped with a symplectic 2 form w9. Let dvx,,Io be the corresponding volume element on Xo,IG. Since Xo,IG = X9/Z(9), let dvX,/z(9) be the associated volume on X9/Z(g). Observe here that dvX,/G is unsensitive to orientation. In particular the integral of a nonnegative function with respect to dvXop/p is nonnegative. In the sequel, we make the following assumptions: G is simply connected, and one of the following assumptions is verified :

g>2. g > 1, s > 1 and at least one of the tj's is regular. g = 0, s > 3, and at least 3 of the tj's are regular.

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219

or

G is connected, the t3's are very regular , and either

g>2.

9>l,s>1. g=0,s>3.

Now we will get an analogue of Theorem 3.10.

THEOREM 5.43. Let f : X -i R be a bounded measurable function. Then (5.181)

-I Iao;l

f f(x)dvx(x) _ x

T/w

Ia(t)Idt fx,/T "IT

IZ(G)I

f

M

IZ(G)i

f(s)

f

d9

, Ia(9)I

f

luxe/Z(9)(x)

xa/Z(9)

f (x t)dt9 .

PROOF. By Theorem 5.21, we know that a.e., Z'(x) = 1. By Theorem 5.12, for a.e. x E X, dd(x) is surjective. Therefore the image .dvx of dvx by ¢ is absolutely continuous with respect to dg. Also on the set {x E X, 0 is regular at x, Z'(x) = 1}, which has full measure and is stable by G, we can use the implicit function theorem, and also integrate along the fibres of the action of G , which are diffeomorphic to G. Let p : Greg -i Tvreg/W be the obvious projection. Since cb.dvx is absolutely continuous with respect to dg, is an open set in X, whose complement is dvx negligible. Then X E ¢-1(Gtreg) H po(x) E is a smooth map. Also if x E f-1(Gvreg), 9 E G, (5.182)

Wx 9) = POW

By (5.182), if x E 0-1 (0,,.g), the obvious analogue of (D, S) in (5.53) is the complex (5.183)

(C'=,s):o-,g a.gaD®®rm(1-ws) ;=1

By Proposition 5.10 and Theorem 5.12, if x E G is such that Z'(x) = 1, the cohourology of the complex (5.183) is concentrated in degree 1, and the first cohomology group of (5.183) is isomorphic to T (s)X O(x) IT. In particular, (detCC)-1 = det(TA(a)X, (z)/T) .

(5.184)

Now the complex Ca is equipped with a scalar product. Let II Ildet(c,) be the obvious metric on det(C' ), and let II Ildet(T,, ,lx,,c.>/T) be the corresponding metric on det(T,r(z)Xp0(x)/T). Then II Ildet(T (,Ix,.l,)/T) defines a smooth volume form on (X O(,)/T)re$, which will be denoted dvx, )/T. Then the formula of change of

variable asserts that (5.185)

Jxf(x)dvx(x)=IZ(G)I

T/wJx/TdvXfT(x) Gf(x9)d9

JEAN-MICHEL BISMUT AND FRAN¢OIS LABOURIE

220

If x E 4-1(Tvreg), put t = [O(x)]-1 E Tvreg. Then we have the double complex (5.186) 0

0

s

0

i

-b- t -- 0

Im(1 - wi)

92g

as

9 (3)

j-

t

0--'-O 0

Inl(1-t)0 T

t)

i

0

0

Let X E E\8E, and let A be a small disk of center x. Put E-I-1 = E\A. In (5.186), starting from below, the first row is trivial. The second row is just the complex (CC ij (E), 8) we constructed in (5.31) (with s replaced by s + 1). The third row is the complex (Cy, 8). Also the columns are acyclic. Let us now explain why the diagram commutes. By (5.34), if x = (ul, vi.... , ws), 9 (5.187)

8(,f1, ...

f29+s+1) - E([ul, vl]...[u{_l, vt-1](1 - uiviui I)f2t-1 i=1

+ut(1-viuivt 1)f2t)+E[ul, vi]...[u9, v9]w1... wj_1f29+j + j=1 9

7-a

fl[ut, vi] 11 wjf29+s+1 i=1

j=1

By construction, 9

(5.188)

jj[ui,vt] IJ wj = t_1 i-1

.

j=1

So in (5.187), (5.189)

8(/29+8+1) = t"1/29+s+1 .

So from (5.189), we find the diagram (5.186) commutes. We equip the vector spaces which appear in (5.186) with the scalar products

induced by the scalar product of g. The determinants of the columns of (5.186) are canonically trivial, and the norms of the associated canonical sections are equal to 1. Also since the first row is acyclic, the cohomology groups of the second and third rows are canonically identified, using the obvious long exact sequence. Finally

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

221

the first row is acyclic, and the norm of the canonical section of the determinant of the first row is equal to 1. By an obvious extension of [11, Theorem 1.101, it follows that if x E o-I (T%.,es) is regular, then the metric on det(H1(E+,, E)) (where HI (ET,, E) is the first cohomology group of the complexes (c , 8) or (0-'j (E), 8)) induced by the complexes

(Cz, 8) or (C j (E), 0) are identical. Using now (5.156) and Theorem 5.40, we get (5.190)

dvX,/T =

IIIoo1IIc(t)Idvx,1T.

J=1

By (5.185), (5.190), we obtain the first equality in (5.181).

The scalar product ( , ) induces a scalar product on g/t = t-'. Let dg be the corresponding volume element on T/G. Tautologically if k : G -a R is bounded and measurable 1 k(g)dg =

(5.191)

dg f k(tg)dt.

G

T

T\G

Also Weyl's integration formula [15, Theorem IV.1.11r] asserts that (5.192)

fGk(9)d9 = Vo1(T) IT/w I a1a(t)dt (G f(t g)dg.

Using the first identity in (5.181) and (5.191), (5.192), we obtain (5.193)

f f (x)dvx (x) = X

f

X,/T

dvx,/T (x)

rIt1

IZ(G)lo,

IZ(G)I

f Qf

I Gf I

r

Ili=1 Iao,, I

T\G

JT/4Y

I o (t) I dt

dg f f (x t'9)dt' T

dg

(9)I

dvxo,Z(9)(x) f

Xe/Z(9)

f(xt)dt9 .

Z(9)

The proof of our Theorem is completed. DEFmNrrrON 5.44. If t E T, let IV(t)I be the absolute value of volume of Xt/Z(t)

with respect to wt. Then V (t) is a W-invariant function on T, which extends to a central function on G. Now we prove a result which was already essentially proved by Liu [39, 40]. THEOREM 5.45. Let f : G -+ R be a bounded measurable function. Then (5.194)

fxf(4-1(x))dvx(x) _

f

f(6-1(x))dvx(x) =

fIIZ(G) ,I 41W Vol(TZnj_1Iao,I

fG PROOF. Our formula follows from Theorem 5.43. I

(G)I

f(9)IV(9)Id9 I0(9)I

REMARK 5.46. In [39, 40], Liu uses an essentially similar argument in his proof

of the Witten formula [63, 64]. In fact let pt(g) be the convolution heat kernel on G. Liu considers the quantity fX pt (0-1 (x))dvx, and following Witten [63], he studies its limit as t -+ 0. The arguments he uses to evaluate the limit as being (up

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

222

to a normalization constant) the absolute value of the symplectic volume of M/G are essentially the ones which are used in the proofs of Theorems 5.43 and 5.45.

Let A be the set of regular values of 0-1 in G. Then A is an open dense set, such that °A is negligible. Also on A, 4.dvx has a smooth density with respect to dg. By Theorem 5.45, it follows that o 9 is smooth on A n Gyreg. Assume temporarily that (t1, ... , t8) verify (A). By Theorem 5.18,1 is a regular value of ¢. By Theorem 5.20, Z'(x) = 1 a.e. on M = X1. Using Theorem 5.45, and proceeding as in the proof of Theorem 3.13, we get IV(g)] _ Vol(G)V(1).

In 'a s IQ(9)I

(5.195)

Vol(T)

Using (5.195) we find that

f

(5.196)

(x))dvx =

c

Vol(G) fl'- loo, I IZ(G)l

V(1)

which is a formula obtained by Liu [39, 40) by arguments essentially similar to the ones we gave in our proof of Theorems 5.43 and 5.45.

Let K be a positive Weyl chamber in t. Let A+ = A fl K be the set of nonnegative weights. Then the irreducible representations of G are indexed by A+. If A E A+, let Xa be the character of the corresponding representation of G. By Theorem 5.45, o s defines a LI G-invariant function on G. Therefore, it defines an invariant distribution on G, which can be expanded as a linear combination of the characters Xa of G. Let wj E Oj. Then Oj = G/Z(wj). Also Vol(Z(wj)) does not depend on the choice of wj. Finally Xa takes the complex value X,\(tj) on the orbit Oj. Now we prove a result of Witten [63, 64]. THEOREM 5.47. The following identity of G-invariant distributions holds on G, 2g+a-1

(5.197)

Io(9)I f9., lao,I -

(Z(G)IVol(T)fIG)1 Vol(Z(tj))

rlj'_1 Xa(tj)Xx(9) X,\(1)29+a-1

,EA+

PROOF. By Theorem5.45, we get (5.198)

IV(9)Id9

1

WO

V01(G) JG

Vol(G)Vol(T) rlf_1 Iooil

fxxa(.O(x))dvx(x)

To evaluate the integral in the right-hand side of (5.198), we follow Witten [63]. By [15, Theorem 11.4.56r], if a, b E G, (5.199)

1GXx(a9b9-1)dg =

(9-1b)dg

f Xa(a9)Xa'

= Vol(G)6a,A,

xa(ab)

.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

223

Finally, if h : Oj -+ R is a bounded measurable function, we have the easy formula (5.200)

10, h(9)dvo,(9) = `,ol(Z(I2))

h(9t19-I)dg.

1G

By (5.199), (5.200), we obtain (5.201)

f Xa(qS(x))dvx(x) =

Vol(G)2g+;n_I Iao,l2 Xa(29 s_Xa(ts)

IIj=I `ol(Z(t,))

X

(1)

)(A

So by (5.198), (5.201), we get (5.202)

lIV(g) I

Vol(G) G

z\ (g)

o'(9)I Vol(G)2g+s-I

IZ(G)I

d9 $

1

IQO, I

Vol(Z(tJ))

Vol(T)

Xa(ti)Xa(1)29+e-I

Now (5.202) is exactly the A-Fourier coefficient of the invariant distribution Li'7,#%is 0 The proof of our Theorem is completed.

Clearly the distribution ° gl in C°° at the regular values of the function q. Now we will make a crude analysis of the Sobolev regularity of IV(g)I.

PROPOSITION 5.48. If p < 2g - 1 - dim(t)/2, the invariant distribution in the right-hand side of (5.197) lies in HP(G). If 2g - 2 - dim(t) > 0, this distribution is continuous.

PROOF. Let p be the half-sum of the positive roots. By Weyl's dimension formula [15, Theorem VI.1.7], if R+ is the set of positive roots, (5.203)

X'0) = 11 aER+

(A + p,a)

( p, aj

'

By [15, Proposition V.4.12], if a E R+, (p, a) > 0. Therefore there is c > 0 such

that ifaER+,AEA+, (5.204)

(A + p, a) > sup(c,(A,a)).

By (5.203), (5.204), we find that since the a E R+ form a basis of t", there is C > 0

such that (5.205)

Xa(1) ? CIIAII

Also (5.206)

IXiI 5 Xa(1)

Let OG be the Casimir operator on G [47, Section 9.4]. By [47, Proposition 9.4.2], (5.207)

AGXa = 2 (IIA + PII2 - IIPII2)Xt

224

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

Using (5.207), we get (5.208)

fi=1 XA(tj)XA(g)

(pa)P/2

(X,\(1))28+8-1 .\r=A+

(II'+p112-IInll2)P/21

1

2a/2

XA(1)29+8-1

.EA+

11 XA(tj)XA(g)

j=1

By (5.205), (5.206),

(5.209)

I(I1A +pII2 - I1p112)"2111 XA(tj)I < C(1+Iap-(2g-1-P) Xa(1)2g+8-1

Also

d

(5.210d,\

t (1 +

if and only if p < 2g - 1 - d` 2 Proposition. Also, (5.211)

1

IA1)2(29-1-P) <

+oo

. From (5.209), (5.210), we get the first part of our

./ (1 + IXI)29-2 < +oo

if and only if 2g - 2 - dim(t) > 0. By (5.210), we obtain the second part of our

0

Proposition.

5.9. Logarithms. Let U be a nonempty open set in G, stable by the adjoint action of G. We assume that there is a well-defined logarithm log : U -+ g, i.e. a smooth function U -f g such that if g E U, g' E G, (5.212)

g = exp(log(g)),

log(g'g9 -1) = g' log(g) In particular by (5.212), log(g) is Z(g)-invariant. EXAMPLE 5.49. If U is a small ad-invariant open neighborhood of a central element in G, a logarithm is well-defined on U.

EXAMPLE 5.50. Let 0 C G be a very regular orbit. Then 0 n T consists of IWI distinct elements. Let t E C n T, and let h E t such that exp(h) = t. Then t H h extends into a well defined logarithm 0 -+ C. EXAMPLE 5.51. Suppose that G is simply connected. Let K be a Weyl Chamber in t, let P be the alcove in K whose closure contains 0. Then by [15, Proposition V.7.10] (5.213)

W\Treg = P.

Also by [15, Proposition V.7.11], the map (5.214)

(g, t) E G/T x P u+ g exp(t)g 1 E Greg

is one to one. Let log : Greg -+ g be the G-invariant function on Greg such that if

tEP,

(5.215)

log(exp(t)) = t E P.

Then log : Greg -+ 9 is a logarithm.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

725

We still assume that G is simply connected and simple. Let U E C/7 r. Let Z(u) C G be the centralizer of u. Let Z(u)vreg be the set of very regular elements in Z(u). Clearly (5.216)

Z(u) fl Greg C Z(u)\.reg .

Also the function log maps Z(u)flGreg into 3(u), and gives a logarithm on Z(u),.reg. Let Bu be the set of connections on the trivial G bundle over S1, whose holonomy w lies in U. Needless to say, in a given trivialization, any element of Bu can

be written in the form (5.217)

Tt

Wt

+ at , at E Lgt.

Let r° be the parallel transport operator along s E [0, t], so that w = 7-1. Set

wt = r°r°(r°)-I.

(5.218)

Then wt is just w under translation of the origin in S1 by t. Clearly log(wt) is well-defined, and (5.219)

log(wt) = r°log(w)(r°)-I.

Also

Dt log(wt) = 0.

(5.220)

By (5.219), (5.221)

et log(w,) a Ttoetlog(w)(ro)-1

Put (5.222)

°D

Dt

= e-t log(w,) Det log(w,).

Tt

By (5.220), (5.222), (5.223)

OD

D

Dt

Dt

+ 109(wt) .

From (5.220), (5.223), we get 0

(5.224)

log(wt) = 0.

The parallel transport operator °r° for Irt is given by (5.225)

NO = e-tlog(wt)'t

so that (5.226)

°r° = 1.

By (5.226), the parallel transport trivialization with respect to be is globally defined on SI, and in this trivialization, °D _ d (5.227)

Dt

Tt

JEAN-MICHEL BISMUT AND FRANCOIS LABOURIE

226

5.10. A symplectic structure on an open set in X. In the sequel, we assume that the assumptions before Theorem 5.43 are in force, and also that (t1, ... , t,) verify (A). Let U1,... , U. be ad-invariant open subsets of G on which a logarithm is well defined. Then G acts on the right on G2g x 11y=1 U1. Recall that x E 02g+a H ql(x) E G was defined in (5.41). Let x E G29 x jii_1 Uj

be such that O(x) = 1, and that G acts locally freely at x. Let U be an open neighbhorhood of 1 in G, such that a logarithm is still well-defined on U. Let V be an open neighbhorhood of x in G29 x Uj such that h(V) c U. Then we can find an open neighborhood A C G29 x ! 19=1 Uj of x, which is Ginvariant, such that h(A) C U, and on which G acts locally freely. Let 4+1 : Gz9+9+1 -+ G be given as in (5.41), with s replaced by s + 1. Set (5.228)

A' = {z' E

G2g+8+1. 0+1(x') = 1) .

Clearly x E G2g+8 H (x, 0-1(x)) E A' is a one to one G-equivariant map. Let A C A' be the image of A by this map.

Let E}1 be the Riemann surface E with s + 1 small disks deleted. Equivalently E+1 is obtained from E by deleting an extra small disk. Let E+1 be the universal cover of E}1. Put r+1 = irl(E}1). Then r+1 is generated by UI, v1, .. , U9, v9, w1, ... , w8+1, and the relation 9

+1

fl[ui, vi] 11 wj = 1.

(5.229)

i=1

j=1

As in (5.126), we find that r+1 and G both act on the right on A, and these actions commute. Also r+1 x G acts on the right on E+1 (and the action of G on E+1 is trivial). Then r+1 x G acts on the right on A x E+1. Since r+1 acts freely on E}1i r+1 x G acts locally freely on A x E+1. Set

C = (A X E+1)/r+1 .

(5.230)

Then C is a fibre bundle over E+1 with fibre A. Also G acts locally freely on A and this action descends to a locally free action on C. We can then form the orbifolds

A/G and C/G = A/G x E+i Let V be a complex vector space. Let p : G -+ Aut(V) be a representation of G. We still denote by p the corresponding representation of r+1 x G in Aut(V).

Put (5.231)

F=AxE+lxr+,xcV.

Then F is a vector bundle on A/G x E. Also,verifies easily that (5.232)

F=Cx0V.

Moreover F is obained by the identification in (5.129), i.e. (5.233)

(x, o,, f) = (xg,aa,g 1z(a 1)f) , z E A, aEE}1 i f EV, 9EG, aEr+1.

SYMPLECTIC GEOMETRY AND THE 16'ERLINDE FORMULAS

227

Recall that rl, ... , r,+i are the origin in S, , ... , S;+i. For 1 < j < s + 1, put (5.234)

Dt

Dt +log(wi,t)

OD

Then f is a connection on the G-bundle P on S with holonomy 1. For 1 < j < s+ 1, let V1 be a connection on the G-bundle A -+ A/G. Consider

the G-bundle A x E+i s'- A/G x E+t . Along the fibres r+I, we can equip this G-bundle with the trivial connection. This connection is r+1 invariant. Therefore it

descends to a G-connection on (A x E+i)/r+1 G X/G x E+1 along the fibres E}1. This connection along E+1 is exactly the flat G-connection associated to the given element of A.

Along SJ ,1 < i:5 s+1, we trivialize the G-bundle (Ax E+1)/r+1 G A/G x E+1

with respect to m. Then over A/G x SJ, 1 < j < s + 1, the connection Gi induces a G-connection on (A x E+1)/r+l

G-- A/G x E+1 along A/G. Since

the A x SJ are disjoint, we can extend this connection to a G-connection on

(A x E+i)/r+i

G

A1G x E+1 along A/G.

Ultimately the G-bundle (A x E+1)/r+i a A/G x Et1 is equipped with a G-connection V. Let F be the curvature of this connection. Then F is a Gequivariant basic 2-form on (A x E}1)/r+i with values in g. Let E be the orbifold vector bundle on A/G x E+1 associated to the adjoint

representation G -+ Aut(g). Let DE be the induced connection on E. Then F descends to a 2-form on A/G X E+1 with values in E. Set

E,,=rj*E; 1<j<s+1.

(5.235)

Then Ej is a vector bundle on A/G. Let V8' be the connection induced by CE (or V,,) on Ef. Observe that for 1 < j < a + 1, log(w,y) is a section of E,i.

Recall that C/G = A/G x E. Then A(T*C/G) = A(T*A/G)®A(T*E+1) .

(5.236)

If w E A(T*C), we can write w in the form w = Ew(P,a)

(5.237)

w(v,a) E A"(T*A/G)®A°(T*E}1) .

Let k be the embedding A/G X 8E}1 -+ C/G. Let e,, be the curvature of V,. THEOREM 5.52. The following identities hold (5.238)

F(°,2)

= 0,

k'F = e1 - DEf iog(wf)dtonA/G x S3

+ 1.

PROOF. By construction the first identity holds in (5.238). Also on A/G x Sj , (5.239)

k*F = k* (VEJ

+dt(Dt -log(wi)))

The proof of our Theorem is completed.

= ej - DEi log(w1)dt.

0

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

228

Let (, ) be a G-invariant bilinear symmetric form on g. Clearly F2 E A4 (T*C)O g ®g. Then (F2) E A4(T*C).

Let rr,rr' be the projections A/G x E+1 - A/G, A/G x 8E}1 -> A/G. Let p be the projection A -> A/G. For 1 < j < s + 1, let 0, be the connection 1-form on the G-bundle A -> A/G associated to the connection V1. Then Bj is a 1 form with values in g'. DEFINITION 5.53. Let a be the 2-form on Al 2

(5.240)

a = p'

1r.

[-j

8+1

-

+E(log(w,),ei) j=1

s+1

d(log(wi),05) j=1

THEOREM 5.54. The 2 -form a is G-invariant and closed. It does not depend on the choices made in its construction.

PROOF. By Chern-Weil theory, the form fQ is closed on A/G x E+,. Using Stokes formula, we get

(27r, da. [--] 2

(5.2 41)

[k'(F2)1 2

1

By (5.238), (5.241), we get 8+1

(5.242)

drr.(F2)=-(V 'log(wj),ej) [-i--]

j=1

Also by Bianchi's identity, (5.243)

d(log(wj), ej)) = (DE3 log(wj),

By (5.242), (5.243), we find that the form a is closed. Now we replace E+1 by E+1 x R. We still denote by rr the projection A x

E+1 x R -+ A/G x R with fibre E+1. The G- bundle A -+ A/G is replaced by A x R -i A/G x R. We consider a smooth family of data, which are used to construct the form at. In particular the connections V j depends on 1. We extend these fibrewise connections: V,j A -> A/G to a connection dt + V j on A x R -> A'/G x R. We will denote with a - the analogue of the above objects over A x E+1 x R. This way, we obtain a forma on A x R such that (5.244)

a` = ae + dt A Qt ,

where Qt is a 1-form on A/G. By the above arguments, a is closed. Now we will show that

0.

(5.245)

This will imply that (5.246)

aOcle

= 0.

t

So we will have established our claim that al does not depend on 1. Recall that along the fibres E}1, the connection on the considered G-bundle

does not depend on I E R, and is flat. It then follows easily that rr. [u] does not contain dl.

SYMPLECTIC GEOMETRY .AND THE VERLINDE FORMULAS

?29

Also the connection form 9j does not contain dt, i.e. it is of the form (5.247)

9j = Bi,e

Then (5.248)

0j = ©j,e +

Therefore (5.249)

(log(wj ), ®j) _ (log(wj), Oj,r + dC-2-0, j), d(log(wj),Bj)

(diog(wj),88,c)+(log(w ),d9j,t),

+(log(wj),dtj,t) From (5.142), we find the sum of the last terms in (5.240) does not contain d£ either. The proof of our Theorem is completed.

Now we fix tj ETnUl,...,t,ETf1U,. For1<j orbit of t1.

0 s,letOj CGbe the

Put (5.250)

X = {x E Awj E Oj , 1<-j:53}.

Then X C X is stable under G. Let m be one of the embeddings Y J ?1G -+ A/G. For 1 < j < s, let Oj C g be the G-orbit of log(tj). Since there is a well-defined logarithm on Uj, Oj and Oj are in one to one correspondance.

For 1 < j < s, let aj be the canonical symplectic form on the orbit Oj. If Y E g, let Y0t be the corresponding vector field associated to the right action of

GonOj. Then ifY,Y'E9,pEOj,asin(1.193), (5.251)

aj,r(Y,Y') = (p,

DEFINITION 5.55. Put (5.252)

or = m"a +t log(wj)"aj . j=1

Clearly a is a G-invariant closed 2-form on X. We will calculate a.

For 1 < j < s, the G-bundle X -+ X/G can be reduced to the Z(tj)-bundle tj } H {x E X,wj = tj}/Z(tj). Let Oj be a Z(tj)-connection on this last bundle. This connection lifts to a G-connection on k -+ Ii/G. We will make {x E

this choice of V j in our construction of or. Then for 1 < j < s, (5.253)

Vjwj = 0, Oj log(wj) = 0.

THEOREM 5.56. The following identity holds a

(5.254)

a = m p" (rr* (2 )1 + (log(wa+1), 0,+1)) -d(log(w,+1),J9,+1)

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

230

PROOF. The second identity in (5.253) can be written in the form dlog(w,,) + [B,,, log(w1)] = 0.

(5.255)

By (5.251), we get (5.256)

log(wi)*°i = (log(w1), 110j, 03])

.

Also

dOi=-2[Bi,Bi]+e3.

(5.257)

From (5.255)-(5.257), we obtain for 1 < j < a, (5.258)

p* (log(wj), AJ) - d(log(w,,),9,) + log(wj)*uj = -(d log(w,,), 9,) + (log w,,, [9,,, 0j]) = 0.

The proof of our Theorem is completed.

13

Set (5.259)

M={xE

Our notation in (5.259) is compatible with (5.59). By Propositions 5.10 and 5.11, we know that M is a submanifold of X, and that G acts locally freely on M. Let

i : M -a X, M/G - 91G be the obvious embeddings. By Theorem 5.31, M/G carries a symplectic form w. Recall that g is equipped with the scalar product Q. We identify g and g* by this scalar product.

THEOREM 5.57. If U is small enough, or is a symplectic form on X. Also x = (ul, vi.... , us, vs, w1, ... , w,+1) E X H log(w.+1) E g is a moment map for the action of G on X with respect to the 2 -form u. The associated symplectic form

on the symplectic reduction M/G coincides with i*x.(2) and with w. PROOF. By (5.254), (5.260)

i`a = i`plr.

(2z) \

\

Let OH, VI be the components of V along

//A1

/G, E+1. Since V is flat along

E+1,

(5.261)

F = V"2 + [7H, DI'].

From (5.261) we get (5.262)

".(2)))

=a*([VH,VV]2

Moreover by (5.238), (5.253),

k*F(1.I) = 0.

(5.263)

From (5.261)-(5.263), we get easily (5.264)

i*a ((2)) = P*w.

From (5.260), (5.264), we obtain (5.265)

i*a = p*w.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

231

Now by Theorem 5.31, w is a symplectic form. Therefore by (5.254), (5.265), if U is small enough, a is also a symplectic form. If Y E 9, let Y2 be the corresponding vector field on x . Then by (5.254). (5.266)

iyxo = -iyjrd(log(ws+1),8,+1) diy.$(lOg(w,+1), es+1)

d(log(w,+1), Y)

.

From (5.266), we find that x E X ,- log(w,+1) E g = g* is a moment map for the action of G on X. 0 The proof of our Theorem is completed. By Theorem 5.12, since G acts locally freely on X, the derivative of x E X H ws+, E G is surjective. Take t E U close enough to 0. Put

Xt={xEX,w,t1=t}.

(5.267)

Then Z(t) acts locally freely on Xt . So Xt/Z(t) can be equipped with the symplectic form at, the reduction of the symplectic form a. Also recall that by Theorem 5.31, Mt/Z(t) is equipped with a symplectic form wt. THEOREM 5.58. Fort E U, at = wt .

(5.268)

PROOF. Clearly the G-bundle A -* A/G reduces to Xt -+ Xt/Z(t). Let C,+1 be a Z(t) connection on this last bundle. Then V,+1 lifts to a G-connection on A -+ A/G. We will use V,+1 to calculate the restriction of a to Xt. On Xt, (5.269)

VE.+' log(w,+1) = 0, dlog(w,+1) + [e,+1, log(w,+1)] = 0 .

Over log(w,+l) _ -log(t), we get (5.270)

[9s+1, 109(w,+1)] = 0.

Then over log(w,+1) _ -log(t), using (5.257), (5.269), (5.270), 1

8,+1) = (log(w,+l ), Os+i) - (log(w,+1), 2 8s+1, 8,+1))

f

= (log(w.+I),

e,+1)

From (5.254), (5.271), we find that over Xt,

a = pair.

(5.272)

2

)

Using (5.272) and proceeding as in the proof of Theorem 5.57, our Theorem follows.

Takej,1<j<s. Put (5.273)

X f = {x E A;wi, E Oj,,j' 36 j,wj E Uj,w,+1 = 1}.

Let n j be the embedding 9j -+ A.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

232

DEFINITION 5.59. Put

Kj = n,a + E log(wj)'oj.

(5.274)

j'=1 , j'#j We will choose the connection V2., 1 S j < s, j' # j as in (5.253). Then by proceeding as in the proof of Theorem 5.56, we get (5.275)

rj = n;p (ir.

+(log(wj),6j))

-d(log(wj), 8j) .

THEOREM 5.60. If Uj is small enough, rj is a symplectic form on A. Also

x = (u1,vi,... ,us,vs,w1) ... ,w,) E Xj r log(wj) E g is a moment map for the action of G on Xj with respect to aj. Finally for tj E Uj, the symplectic form on the symplectic reduction {x E Xj,wj = tj}/Z(tj) is the symplectic form defined in Theorem 5.31.

PROOF. The proof of our Theorem is the same as the proof of Theorems 5.57 and 5.58.

REMARK 5.61. If ti,... , t, are restricted to be very regular, we may and will assume that B1, ... , 0, are T-connections. By Theorems 5.58 and 5.60, we find that restricts to the symplectic form of Theorem 5.31. Also by (5.275), the cohomology class of

w + t(log(wj), ®j)

(5.276)

j=1

is locally constant, which is a consequence of the theory of the moment map for torus actions obtained by Duistermaat-HeckmanDH.

5.11. The integral of certain characteristic classes on the strata of MIG. Let G be a compact connected and simply connected simple Lie group. We use the notation of Section 1. In particular, (,) denotes the basic scalar product on

Let ti,... , t, be regular elements in T. Then t1, ... , t, are very regular. We assume that s > 1, and that (ti,... , t,) verify (A).

Ifg=O,thens>3.

In particular (5.277)

2g - 2 + s > 1

Let u E C/R . Recall that 7ru : Z(u) -> Z(u) is the universal cover of Z(u), with fibre iri(Z(u)) ^' CR/VRu C Z(ZZ(u)). Also Tu = E/CRu is a maximal torus in Z(u). Remember that Treg is the set of regular elements in T with respect to G. Let t E Tres. Then Z(t) = T, t is very regular in Z(u), and Oz(u)(t) = Z(u)/T.

Let t E Tu be a lift of t in 2(u). Then E is still regular in Z(u).

Since

Z(t) = T, the centralizer Z(I of 1 in Z(u) is just Z(t) = t/CRu. Then O2(,,) (t) z

SYMPLECTIC GEOMETRY AND THE \'ERLINDE FORMULAS

233

Z(u)/(t/GRu) = Z(u)/T. Equivalently the projection O2,.)(E) -i Oz(l )(t) is one to one. As before, we identify tl,... , tb with corresponding elements of G-alcoves in t whose closure contains 0. This way, we get elements of T. = t/CR,,. which lift t1, ... J. unambiguously in Z(u). We still denote these elements by ti, ... , t,. Clearly (5.278)

Z'(u) = Z'(u).

Set

(5.279)

X = Z(u)29 X JJ 0zl,u(tj), j=1

Xu = 2('u)29 x 11 OZlui(tj). j=1

Then Z(u) acts on X and on Xu. Moreover (CF{/GRs)2g acts freely on .1 . 1....1 s) E Namely if f = (a,, 01.... a,, #g) E (GAR/CR.)", if .T = (u1, vl.... u,;

Xu put (5.280)

f.x = (ulal,0it1,qq ,llgag,ilg leg, 1D3>... t-v,)

The actions of Z(u) and of (VR1CRu)2g commute. Also the map ru : 2(u) -i Z(u) Tex,t-ends to a map Xu --, X. Clearly ru is Z(u)-equivariant. Also if f E (5.281)

ruf = ru.

More precisely ru is a (GR/CRu)2g cover.

DEFINITION 5.62. Let u : Xu -f Z(u) and ¢u : Xu --r Z(u) be the analogues of 0 defined in (5.41). Clearly (5.282)

Ouru =

Also u and ¢u are Z(u)- equivariant . Clearly (ti,...

t,) verify (A) with respect to Z(u) or Z(u).

Therefore by

Theorem 5.18, 1 is a regular value of 0u. Equivalently, by Theorem 5.18, 1 is a

regular value of ¢. By Theorem 5.12, G acts locally freely on Af = ¢-1(1), and so by Theorem 5.12, 1 is a regular value of ¢u. PROPOSITION 5.63. Any h E CR/CR is a regular value of Qu.

PROOF. Since 1 is a regular value of 0u, any h E r-'(1) is a regular value of ,,AA

Wu

Recall that the Lie algebra 3(u) is equipped with the scalar product induced by the scalar product (,) on g. Let U be a G- invariant open neighborhood of 1 in G, such that a logarithm log : U -3 g is defined, with (5.283)

1og(1) = 0.

JEAN-MICHEL BISMUT AND FRANcOIS LABOURIE

234

Put U. = U n Z(u), Uu = 7ru-I (Uu).

(5.284)

Clearly log maps Uu into ;(u). Then U is an open neighborhood of GAR/C& in Z(u), which only consists of regular values of iiu. In the sequel, we will view exp(1og(Ou(x)) as an element of Z(u). Observe that

if i E X is such that i E u 1(Uu), 7ru [¢u(a)exp(-log(¢.iq.(a))), =1.

(5.285)

Therefore

&(s)exp(-log(¢u?u)(!))) E Z7R/CRu.

(5.286)

Finally note that if x E (bu1(U), if i is such that iru(i) = x, then (bu(2) does not depend on z. Therefore (au(S) exp (-1og(Ouiru(i))) E GPI /CRu does not depend on g. DEFINITION 5.64. If h E

let O;I(U,)h C 0;I(U) be given by

(5.287)

Ou 1(Uu)h = {x E .u 1(Uu), if ir.(x) = x, iu (x) exp (- log(0uiru(f))) h = 1}.

Similarly, let k1(i7u)h C u1(Uu) be given by (5.288)

u 1(Uu)h = {2 E u 1(Uu), fin(,i) eXp (- log(4uiru (i))) h = 1}.

Then we have the disjoint union (5.289)

U

Oa1(Uu) =

4u1(vu)h,

IhEIA/GTR,

u1(Uu) = U hE

7

for any h E UR/URu,7ru :_fu'(Uu)h -+ 0uI(Uu)h is a (ZR/ L)2g-cover. Moreover Z(u) preserves ¢u'(Uu)h and 0;I(Uu)h, and acts freely on ¢uI(Uu)h If U is small enough, there is an open neighborhood f7. of 1 in 2(u) such that iru : (Vu) -3 Uu is one to one. Also if t E U, and t E Vu is the lift of t, then iru is one to one from O2(,)(t) into Oy(u)(t). DEFINITION 5.65. If h E Z R/Z°Ru, t e Uu, E E Vu, with iru ($) = t, put

(5.29O)Mt(Z(u),Oz(u)(t1),... ,OZ(u)(t,),h) = {x E4r1 (Uu)h,4u(x) =t-1}, Mt(Z(u), Og(u) (t1), ... , OZ(u) (t,), h) = {x E u 1(Uu)h, & (2) = t-Ih'I }. Then iru : Mt(Z (u), 02(u) (t1), ... , 01(u) (t,), h) -+ Mt (Z (u), OZ(u) (tI ), ... , OZ(u) (t,), h) is a BUR-/Z%129 cover. If Zu(t) is the centralizer of t in Z(u), iru is a Z. (t) equivariant map. By Theorem 5.12 and Proposition 5.63, Zu(t) acts locally freely on Mt (Z(u), OZ(u) (ti),. - - , 02(u) (t,), h) and on Mt(Z(u), Oz(u) (ti),... , Oz(u) (t,), h).

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

PROPOSITION 5.66. The map 7ru : Mt(Z(u),

02(,,) (t.). h)/ZZ(t) ->

is a jG /C 129 cover on the regular

part of the orbifold

Zu(t) be such that

O2(u)(ti),...

'235

Mt(Z(u), O2(U)(ti),... Of(u)(t,),h), andlet f E (C'1i/Zxu)25

(5.291)

f.i = i.g.

Then by (5.281), (5.291), (5.292) iru(i) = If x = lru(x) is in the regular part of

,Oz(u)(t,),h), from

(5.292), we get (5.293)

g = 1.

So by (5.291), (5.293),

f = 1.

(5.294)

p

The proof of our Proposition is completed.

Fort E Uu, let of be the symplectic form on Mt(Z(u), Oz(u) (ti ), ... , Oz(u) (t5), h)/Z,,(t), and let of be the symplectic form on Mt(Z(u),Og(u)(t1), , Oi(u)(ts)I

h)/Zu(t)

DEFINITION 5.67. For u E C/R , t E Uu, put (5.295)

VV(ti.... ts)t,h) = r

es:

mt, (z(u),ostu)(ti),...

(t)

°'

V.(ti,...ts,t,h) = IJ PROPOSITION 5.68. The following identity holds (5.296) of = a;,ot .

Moreover (5.297)

IVuR(ti.... ts,t,h) _'7R-/flag (Vuj(ti

PROOF. Equation (5.296) is trivial. Using (5.296), we get (5.297).

Let K c t be a Weyl chamber for G which is fixed once and for all. Then if u E C/R , K is included in a unique Weyl chamber K. for Z(u) or 2(u). Put (5.298)

:,+ = U-.. n Ku.

Then the irreducible representations of Z(u) are parametrized by

,+. If A E

Ccu +, let xf (u) be the representation of Z(u) with highest weight A. For t E Tu =

t/n , Put (5.299)

(eta(a,t)

oz(u)(T) _ aER, .+

`

- ecA(a,tf1lJ

Then loz(u)(t)I is well defined on t/R , and so is well defined on T = t/CSR.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

236

THEOREM 5.69. For any u E C/R , h E GAR/CRu, the following identity of Z(u)-invariant distributions in the variable t E U holds, (5.300)

h't)

CRu, .te' t = IZ(Z(u))IVol(t/ Io-Z(u)(t+h)n;=11 z(u)(,)I IV+I(tl,

Vo1(2(u)

)I,

2

29+s-i n;=1 Xa (u) (t:i )X2\(u) (t + h)

Vol(Tu)

PROOF. Recall that t E U is identified to the corresponding element in f". C Z(u). Since tj E Z(u) is very regular, the centralizer of tj in Z(u) is equal to T. Then by (5.156) and by Theorem 5.47, we get (5.300) . The proof of our Theorem is completed.

REMARK 5.70. Note that CR n Ku is exactly the set of nonuegative weights for Z(u). Also since h E Cl t/CRu lies in Z(Z(u)), (5.301)

X (u) (t + h) = e2ix(a,h) Xa (u) (t).

From (5.300), (5.301), we get easily, (5.302)

-

FhEiW/ice Ii I (ti , ... , t6, h, t) _

1

IUR

I17z(u) (t)II II;=I IaZ(u) (tj)l IZ(2(u))IIV0l(t/UA-s)I29-2ICR/CRu 1

'9

-

Vol(Zu)

29+s-1

Vol(T.,,)

n;_' XT U) (ti )Xa (u) (t) Z(u)(1)29+s-1 X.,

aECR`t1K

Clearly (5.303)

Z(Z(u)) = Z(Z'(u))/(CR/CRu),

Volt/C7,) = Vol(T)ICR/CR I, Vol(Z(u))

_ Vol(Z(u)) Vol(T)

Vol(Tu) From (5.302), (5.303), we get

(5.304)

1

IURR/URuI29

IaZ(u) (t)II II;_I I°Z(u) (t;)I

IZ(Z(u))IVoI(t/GR

2g-2

VOl(Z(u) Vol(T)

I29+s-1

n;_1 xa (u) (t; )Xr (u) (t) Z(u)

n7f

X'\

(1)29+s-1

Identity (5.304) fits with (5.197) and with (5.297).

Since tl,... , t, are very regular in Z(u), there are Z(u)-invariant open neighborhoods Ul , ... , U; of t1, ... , t8 in Z(u) on which a logarithm is well defined.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

237

Since Z(u) acts locally freely on Mrj(Z(u),Oz{uj(t1).... ,O2{u)(ts.),h). all the results of Section 5.10 can be used in this situation. In particular, by Theorem 5.57, there is a Z(u)-invariant open neighborhood Xu,h of Mo(Z(u), OZ(u)(tt),... ,OL(u)(t8), h) in ? ., equipped with a symplectic form au, such that the a, are the symplectic reductions of ou. So we may use the results of Section 3 in this situation. Let T,, = t/ be the obvious maximal torus in Z'(u) = Z(u)/Z(Z(v)). Now we will use the notation in Section 3.6. As in Section 3.6, the choice of a Weyl chamberK and of the corresponding Weyl chamber K. for Z(u) defines an orientation on the Mt(Z(u),02(u)(tl),... ,02(u)(t,),h)/Tu, fort E T n U`.u. DEFINITION 5.71. For t E T n U,, put (5.305)

H(u,9,8) (tI, ... t8, t) =

e

J Al,(Z(uJ,O2

.

Using Theorems 3.15, 5.57 and 5.60 , we know that H(u, y,8 I (t1, ... t8, t) is locally

a polynomial in t1, ... , t8i t. Also recall that Pu,2y+8-I (t), t E Tu was defined in Definition 2.39. By Theorem 2.37, Pu,29+e-1 (t) is a polynomial on Tu\Su. Finally remember that in (2.33) , we set lu = IR,,,+I THEOREM 5.72. The following identity of local polynomials in (ti,... t8, t) holds (5.306)

H(uy,8)(t1,... ,t8,h,t) _

(-1)e"(y-a+tlz(2('u))I

21 (v",h) j=1 8

e

Eu /pu,2y+8-I(t+h+Euiit1). 1=1

PROOF. Clearly, if t E Uu nTng, then Z'(t) = T', so that Z,(t) = T,'. By (5.295), (5.305), (5.307)

IH(u,9,8)(t1,...) t,,,h,t)I =tiu(tl.... to,h,t)on UunTreg.

Also (5.308)

Iaz(u)(t)I =sgn((-i)e"oz(u)(t))(-i)'"az(u)(t).

Moreover, by (1.43), if h E GAR/Mu, (5.309)

az(u)(t+h)

_e21x(p.,h)az(u)(t),

By Theorem 1.41, e2lx(v",h) = f1.

(5.310)

Moreover for t close enough to 0, (-i)e"oz(u)(t) and 7ru(t/i) have the same sign. Also by (3.129), (3.133), (5.311)

Xa(u)(1)

Vol(Z(u))/Vo1(Tu)

Pu+a

su (

i

)

JEAN-MICHEL BISMUT AND FRANCJOIS LABOURIE

238

Finally by Theorem 3.15, for t close enough to 0, H(,,,9,,)(t) and iru(t/i) either vanish together, or they are nonzero, and then they have the same sign. Using (5.300), (5.310), (5.311) , we get the identity of distributions (5.312)

(-1)1°(a-I)IZ(2(U))IIVoI(t/Uit: )129-2

H(u,9,e)(t1,... ,ts,h,t) = s

(tj))e-2ir(P,,,h)

fl sgn((-i)'az(u) j=1

)Zj=1 (oz(u)(tj)X"(u)(tj))oz(u)(t+h)Xa(u)(t+h) A))29+a-1

(au(pu +

aEC,,,+

By (1.94), Theorem 1.38 and by (5.312), we obtain

H(u,a,a)(t1,...

t ,h,t) =

(5.313)

(-1)t"(9-1)IZ(Z(u))IVo1(t/URu)12a-2

lli=1 sgn((-i)"-az(u)(tj))e 2i,(Py,h)

E

57

(u i, ..w ,w)E 6Vu+ 1

fl , E 'I

(5.314)

izp(2i,.(w(a+ e), t+h+Ej., w itt»

Now by (1.186), if w E Wu,

ru(w(Pu + A)) = Ewlru(Pu + A).

Also by using in particular [15, Note V.4.14], {w(pu + A)}E

(5.315)

.

= {A E

,rru(A) 0 0}.

Using (2.158), ((5.313)-(5.315), we get the identity of distributions on Uu, (_j)1.,(a-1)+1 H(u,a,s)(t1,...

II sgn((-i)I, oz(u) (tj

))e-2ir(p-h)

j=1

7-s

(5.316)

a

A,-1(t+h+t wjtj).

(w".....w °)Eb1'° 1=1

j=1

Now by Theorems 2.37, 3.15, 5.57, 5.60, we know that both sides of (5.316) are local polynomials of (ti,... , t t). Therefore (5.316) extends to an identity of local polynomials. The proof of our Theorem is completed.

Put (5.317)

M (Z(u), Oz(u) (ti),... , Oz(u) (ta), h) = Mo(Z(u), Oz(u) (ti),... , Oz(u) (ta), h) Then M(Z(u), Oz(u) (t1), ... , Oz(u) (t,), h) C M, and Z(u) acts locally freely on

M(Z(u),Oz(u)(t1),... Oz(u)(t.),h) Let 9 be a Z(u) connection form on the Z(u)-bundle M(Z(u),Oz(u)(t1),

...,Oz(u)(ta),h) - M(Z(u),Oz(u)(t1),...,Oz(u)(t,),h)/Z(u), and let a be its curvature. Let 81, ... , 9, be connection forms taken as in Section 5.10, and such

that (5.253) holds, and let e1, ... , e, be their curvatures. Then e1, ... , e, take their values in t.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

939

Let Q be a Z(u)-invariant 01 function on t(u), let QI, ... , QS be C" functions on t. Recall that w is the canonical sylnplectic form on .lI/G which is associated to the basic scalar product (,) on g. PROPOSITION 5.73. 1f P E R", the following identity holds (5.318) M(Z(uhOah.)(ti),...,Ozh.)lt.,),h) /Zi ui 1 p(9-1) dim3(u)+f dim(3(u)/1)

Q!-9) rl'=1 Q, (-©1)eP- _

IWLLI

Q(810t)AQi(8/ P

P

f=1

2i-,r

PROOF. Clearly (5.319)

Q(-0)ep- jlQ;(-0,) _

/r

P(dim M(Z(u),Oz(..)(tl),....Oz(.)(t, ),h)/Z(u) ), 2

Q(-0/p)

[JQ?(-eilp)ew

I

Also by Theorem 5.57 and Proposition 5.66, (5.320)

JM(z(u).oa(,.)(til ,... ,oa(v) (t. ),h)/Z(u)

Q(-e/p)e` = Q(-0/p)eoa .

1

M(Z(u),o$( )(h),...,0g( )(t.),h)/Z(u)Z(u)

IVW/CRu129

Finally our assumptions on the ti's, Theorems 5.20 and 5.21 guarantee that Z;, (x) _ 1 a.e. on M(Z(u), Og(u) (ti),... , OZ(u) (t,), h). We can then apply Theorem 3.21 to (5.320) and get (5.318). The proof of our Theorem is completed. REMARK 5.74. In [39, 40], Liu derived the above formulas for the intersection numbers of the corresponding moduli spaces.

5.12. An evaluation of certain Euler characteristics. Recall that x E R i-r [x] E [0,1[ is the periodic function of period 1 such that for x E [0, 1[, [x] = X.

PROPOSITION 5.75. Let m E N. Put z = e'm . Then for P E Z, 1 m-1 zkt _ 1 1 $

(5.321)

m k-1 T--;=k

2 - [m]

Fm-,

PROOF. Take pE]-1,+1[,andIwith 0
,

m-1 (5.322)

k=1

z 1-

pz

+00

+oo m-1

_kt

k=

E nO k=1

n=0

+°°

_ .=o E

1-P

mpe

1

=1-pm-1-p.

Pn

pnzk(n-f) =

C

k(n-e)

-1

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

240

By making p -+ 1 in (5.322), we get z-kt (5.323)

M-1

1 _ rk =

o

-1

k=1

which is equivalent to 1 'n -1

(5.321)

zk[

_

1

1

2-[rJ

_

1

2m

So we have established (5.321) for 0 < P < m. Similarly if -m < 1 < 0, using (5.324), we obtain

1 ,i-1 (5.325)

m

E

1 m-1 zk(m+f)

zkl

z-k _

1

1

2

[m]

m

T---Z-)L

_

1

2-

1 [i+_]

1

2m

1

2m,

The proof of our Proposition is completed.

Let r9 be a Riemann surface of genus g. Here, we have fixed a complex structure on E9. Let x1,... x, be s distinct points in Eg. Let D be the divisor e

D=:xj.

(5.326)

j=1

Let [D] be the corresponding holomorphic line bundle on E9. Let oD be the canonical section of [D]. Clearly

J c1 ([D]) = s.

(5.327)

Let m E N such that m divides s. Let A be a holomorphic line bundle on Eg such that A- = (D].

(5.328)

Put (5.329)

E9={tEA, oD=tm}.

Then P = E9 -+ E. is a branched covering of order m, with branching points x1i... , x,. Also by Hurwitz's formula, the genus g' of Ee is given by (5.330)

g'=mg+2(m-1)(s-2).

If 1 E Z/mZ, t E E9, put (5.331)

1(t) = est.

Then (5.331) defines an action of Z/mZ over E9 such that p1= p. Also if 1 E Z/mZ, 196 0, then x1 i .... x, E r9 are the only fixed points of 1.

Let r be the Riemann surface with boundary, which is obtained from E. by deleting s small disks centered at x1, ... , x,. Set (5.332)

Eb =p 1E.

SYMPLECTIC GEOMETRY AND THE CERLLNDE FOR IULAS

-,-

Then rb is also a Riemann surface with boundary, obtained from E by deleting. small disks ,t,... .fig centered at x1,... ,.r . Let 0 be a compact connected and simply connected simple Lie group.

DEFINITION 5.76. We will say that g E G is of order rn if g= 1. More generally if 0 C G is an adjoint orbit, 0 will be said to he of order m if for one ;ur

any) element gE 0, g' = 1. In the sequel, we assume that CSI.... Cps are of order rn..

Let x E M. Then the trivial G-bundle P over L' is equipped with the corresponding flat G-connection. Therefore the G-bundle p*P _, El is equipped with the corresponding flat connection. Moreover Z/rnZ acts naturally on this G-bundle and preserves the flat connection. Observe that for 1 < j < s, the holonomy of the flat connection over the circle P I(SJ) -which is a m cover of Set- is u!j = 1, i.e. r, `P has trivial ltolouonn around i I(S1). Therefore the flat connection on Ea extends to a flat connection on the trivial G-bundle p*P -3 r9. We claim the action of Z/mZ extends to the bundle p*P -4 Z". To define the flat bundle p*P near xj, we use the identification by parallel transport along the circle SS . In fact recall that for 1 < j < s. wl is the holonomy of the flat connection along SJ considered as lying in 8E. The holonomy along S, considered as the boundary of the disk Oj is w., L. If I E Z/mZ. t E E. f E P. then (5.333)

£(t, f) _ ('_2V t' f) .

However using the trivialization of parallel transport along p*Sj, in this trivialization (5.334)

£(t, f) = (e 6 t, w9 f) .

In particular the action of £ on p*Pz, is given by f E P ,-r wjf E P. Let V be a complex vector space. Let p : G -+ Aut(V) be a representation of G. Let F be the flat vector bundle on r (5.335)

F=PxGV.

Then by the above construction, p*F extends to a flat vector bundle on E9, on which

Z/mZ acts. In particular Z/mZ acts on Let [H(r9,p*F)]z/mz be the invariant part of H(E9, p*F) under the action of Z/mZ. Let (,) be a G-invariant bilinear symmetric form on V. Let (, )H,C F, and HI (Eb,p-F) be the coresponding intersection forms on RI (E, F) and H'( E9, p*F). PROPOSITION 5.77. The following identity holds (5.336)

FT (E, F) _ [H (E9, p*F)]z/mz

Under the identification (5.336), if a, a' E FII(E, F), (5.337)

(aAf1(E,F)=m(a,a')Hl(E.,p"F)

PROOF. Clearly, [Ho(E9, p*F)]z/mz consists of flat Z/mZ-invariant sections

of p*F on E. In particular by for 1 < j < s, wjfj=, = flyt. Therefore these

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

242

sections descend to flat sections of F on E. Using (5.27) and (5.36), (5.336) holds in degree 0. By Theorem 5.4, using Poincar6 duality, (5.336) holds in degree 2.

Let a be a Z/mZ-invariant closed form in Q'(E6,p*F) representing [a] E [H1(Eaa'

p*F)]z/rnz We may and will assume that a vanish near xl,... , x,. Then a descends to a smooth closed 1 form on E, which vanishes on 8E. Also a is defined up to the coboundary of a Z/mZ-invariant form in f1o(E9',p*F). Using Theorem 5.4, we find there is a well-defined map [HI(E9,p*F)]z/mz -+ I( F). Conaervely,

if /3 is closed in fl1(E, F) and vanishes near 8E, then p*,B is a smooth Z/mZinvariant closed 1 form in OI(E9,p*F) . So we have defined a map HI(E,F) -a It is now easy to verify these two maps are inverse to each [HI (E, p*F)]z/mz.

other. The identity (5.337) follows trivially. The proof of our Theorem is completed.

Let X(F) be the Euler characteristic of the complex (C£(F), 8). Then 2 _

X(F) _ E(-1)`dim(H{(E,F))

(5.338)

i=o 2

i=o So by (5.32), (5.338),

(5.339)

X(F) = (2 - 2g) dim F - E dim(1- wj)(F)) j=1

Let

Xz/mz(p*F) be the invariant Euler characteric of p*F on E9, i.e. 2

(5.340)

Xz/mz(p*F'') = E(-1) ' di 1[H`(Ep p*F)]z/mz. i=0

By Proposition 5.77, (5.341)

X(F) =

Xz/mz(p*F).

We will prove (5.341) again using the Theorem of Riemann-Roch-Kawasaki [32, 33], stated in Theorem 6.8. We get (5.342)

Xz/mz (p* F)

_ m [cim(F) / e(TE) + E Tr e

1

j=1

JE',

Now using (5.330), we get (5.343)

J b e(TE9) = 2 - 2g' = m(2 - 2g) - (m -1)s . a

Moreover (5.344)

m k=1

=-E w -m k-0

m

.

k=1 v4 ] 1

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

Now m m-1

243

wj is a projection on If E F.u: j f = f ). So by (5.344). M-1

Trp

(5.345)

] = dimker((u'j - 1),F) 1]

dim F m

1

By (5.342)-(5.345),

Xz/mz(pF)=(2-2g)dimF-E dim((l -j)(F))

(5.346)

j=1

which fits with (5.339), (5.341).

Now we assume that p : G - Aut(V) is a real representation of G. We will now give another proof of (5.346). Let Xh,z/mZ(p.F) be the holomorphic invariant Euler characteristic 1

Xh'Z/mz(p.F)

(5.347)

= Edim

[H°''(E9,p`F)]Z/mZ

i=O

By Hodge theory,

XZ/mz(P.F) = 2Xh,z/mZ(p.F)

(5.348)

Also by the Riemann-Roch-Kawasaki theorem [32, 33], (5.349)

[dim(F) fEa 2c1(TE)

Xh'Z/mZ(p*F) =

-

9

a m-1 j=1 k=1

-

1

m

TrF[wj].

By (5.343),

if g2c1(TE9)=1-g-2(1-m)s.

(5.350)

m

The eigenvalues of wily have absolute value 1. Since V is real, they are either f1, or they come as complex conjugate pairs. By Proposition 5.75, we get m-1 (5.351)

M

E 1-e

a:xk/mkv[wj]=-2mdimF+2dimker(wj-1)JF.

k=1

By (5.349)-(5.351), we obtain

(5.352)

Xh,z/mz(p*F)

= (1-g) dimF - I :L dim((1 - j)(F)), 1

2 which fits with (5.346).

5.13. Evaluation of c1(TM/G). In the sequel, we assume that G is a connected simply connected compact simple Lie group. Otherwise we use the notation

in Section 1. In particular (,) denotes the basic scalar product on g defined in Section 1.2.

JEAV-MICHEL BISMUT AND FRANQ-01S LABOURIE

244

^

As in Section 5.10, we construct a connection on the G-bundle -0 ?l1/G x E , such that the assumptions after (5.252) hold. In particular,

for1<j<s, (5.353)

= 0, = 0.

V1.w1

V1.log('w1)

n P by formula (1.35). Also since t1 lies in an alcove P, it determines p1 E Over M/G,t p1 descend to sections of E. By (5.353), for 1 < j < s, (5.354)

V1 P1 = 0.

Let Oj be the connection form on associated to V1, let e1 be its curvature. Then 81 can be considered as a t-connection, and 91 is a t-valued 2 form on M/G. Recall that TMIG is a symplectic vector bundle. Let JTM/G be any almost complex structure polarizing o, i.e. o is JTM/G invariant, and U, V E TM/G H v(JT M/GU, V) is a scalar product. Such JTnr/G exist and are homotopic. Therefore c1 (TM/G) is a well-defined element of H2(M/G, Q). Recall that c is the dual Coxeter number defined in Definition 1.7. THEOREM 5.78. The following identity hold, 9

(5.355)

c1 (TM/G) =2(cw+E(ct1-P1,®1)). 3=1

PROOF. First we assume that me N, that t1, ... , t, are of order m, and sum.

Then the G-bundle (f x ft)/P G M/G x E lifts to a G-bundle Q - M/G x Ell on which Z/mZ acts naturally. In fact, we only need to make the lifting constructions of Section 5.12 fibrewise. Recall that a complex structure has been fixed on E. and E9, and that Z/mZ acts holomorphically on Ey. By (5.120), we have the identity

T.M/G = I' (E, E).

(5.356)

Using Proposition 5.77 and (5.356), we get (5.357)

TRM/G = [H1 (E', p*E)]

z/mz

Now

(5.358) H1(E9,p*E) (DR C = H(1,0)(Es p*E) and the splitting (5.358) is Z/mZ invariant. By (5.356)-(5.359), we get (5.359)

[H(1,0)(E:,p*E)]z/mZ$

#'(E,E)®RC=

[H(°'1)(E°9,p*E)]Z/mZ

Let J be the complex structure on Hl (E9, p*E) which is ion H(1,0) (E9, p*E),

-i on H(0"1)(E96,p*E). We claim that J polarizes the symplectic form w on Hl (E, E) = TR,M/G. In fact by if a, a' E f11 (E, E) are represented by the forms n, ,7' E Sll (E,b, p* E) which are closed and Z/mZ invariant, then by Proposition 5.77, (5.360)

w(a, d) =

-I f£6 (17"o). 9

SYMPLECTIC GEOMETRY AND THE CERLINDE FOR:i;:L:1S

-w

It is now trivial to verify that J polarizes So by (5.356), (5.359), i 2'm2'

ct(TDI/G) = Cl

(5.361)

which is equivalent to

cl(TM/G) _ -cl

(5.362)

Let 9 be a Z/mZ invariant connection on the G-bundle Q G r x _1I!G . Observe that ({xj } x RI/G)I< j<s are exactly the fixed point of the action of Z/rnZ over E9 x M/G. Since the connection 9 is Z/rnZ-invariant, and since. for 1 < j < s. 1 E Z/mZ acts on Er5 . like wj, we find that over M/G

Cwj,*. = 0.

(5.363)

By (5.63), (5.364)

Ho (E, E) = 0.

By (5.336), (5.364), [HO(E9,p*E)]Z/mZ = 0.

(5.365)

Now we will use an equivariant version of the curvature theorem of Bismut and Freed [10]. Namely we equip TE9 with a Z/mZ-invariant metric. Recall that p*E is also equipped with a Z/mZ-invariant metric. Since Z/mZ is a finite group, the construction sof [9, Definition 2.21 provide us with a like metric on the line bundle det([H(Ov)(E°y, p*E)]z/siz). By proceeding as in [10], we also obtain a unitary connection on this line bundle. Incidently, observe that since all our data are holomorphic, we could instead use the holomorphic constructions of [11] in an equivariant context. The curvature of the connection on det([H(0'-)(E9, p`E)]z/n'z) is obtained by applying the techniques of [10] or [11]. An important technical point is to prove an equivariant version of the local families index theorem of [81. A large

part of the steps which are needed in extending the results of [8] is already done in [9]. By mixing the techniques used in Lefschetz fixed point theory and in the prooof of the local families index theorem as in [8], [6], [9], one finds easily that the curvature is given by a differential form version of the theorem of Riemann-RochKawasaki [32, 33]. Using (5.365) and the above considerations, we obtain (5.366)

c1

([Hoi)(E,E)] Z/mZl

rr

D m-1

1

Td(T E)ctl(p E, P

p_ O 1

+j=1Lk k=1 =1 1-e2ixk/m 1T- wj e7`

(') J

By (1.37) and by Theorems 5.55 and 5.56, (5.367)

I MJsb

(-!

(-)

Td(TEe)ch(p*E, VP E))

= 2cw.

)

246

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

Recall that

g OR C = t$ (® ga f

(5.368)

.

aER

Let Si be the operator acting on g OR C,

l

-1

Si =

(5.369)

m k=1 By Proposition 5.75, we find that 1

(5.370)

Silt Sits.

.

1

=2 =

1

1- e_2iak/m r(tf)

2m'

22m' -ER.

By conjugation by an element G, we may and we will assume that wi = t,. Then

(n-1 (5.371)

r

2

E 1- e 2irrk/m

e7/ 1) {wexp(_1\

)

_ -ZYe I Si oa,

Also because wi = ti, V'2 is a 2 form on M/G with values in t. From (5.370), we get easily (5.372)

®f= - L, - [(a,t2)]) (a, ®i) -Tr9 [s1] aER \2 = F'[(a,t1)1(a,03) aER

Now by Proposition 1.20, and by (5.372), we get VE,2 (5.373)

-Tr9 Si

2i7I

2(ctj - P1, 9J)

From (5.362), (5.366), (5.367), (5.371)-(5.373), we get s

(5.374)

c1(TM/G) = 2(cw + E(cti - p,, ®i)) . i=1

We claim now that in the special case when the orbits 01,... , Om are of order at, (5.374) is exactly (5.355). In spite of the formal simultanities, the objects introduced in both equation are not exactly of the same kind. However we leave to the reader the verification that they indeed coincide . Now we establish (5.355) in full generality. Clearly if (t1, ... , t,) are regular and verify (A), they can be approximated by a sequence of regular elements (tl , ... t;')

which verify (A) and are of order m. Let 8m > s be such that m1sm. Then we consider the above situation, with s replaced by sm. At xi,... , x,, we assume that the holonomies are w1 i ... , w, and at x.+1,... , x,., they are 1. Needless to say, since 1 is far from being regular, we do not impose any restriction on the connection

V,, j > s + 1. Since for j > s + 1, Tr9[Oi) = 0, it is clear that in the final formula, the j > s+1 do not contribute, so that (5.374) still hold. Then by the above (5.355) holds. A trivial limit procedure shows that (5.355) still holds in full generality.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

247

REMARK 5.79. By Remark 5.66, the cohomolo y class of :.;±E _ I (log(wj), OJ ) is locally constant. Observe that in (5.355). (p 0j) is a closed form on 11/G whose cohomology class does not depend locally on tI, .... ts. So Theorem 5.78 fits with the above considerations. Assume temporarily that some tj lie in Z(G) = R /CR. By (5.372). we find

that such tj do not contribute to formula (5.355) for ct(T IG). In other words if the t1 are either regular or central, formula (5.355) still holds, where in the right-hand side, the summation is limited to a sum over the regular orbits.

248

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

6. The Riemann-Roch-Kawasaki formula on the moduli space of flat bundles The purpose of this Section is to give formula for the index of a Dirac operator on the moduli space -41/G of flat vector G bundles, by using the theorem of Riemann-Roch-Kawasaki [32, 33]. To do this, we describe the strata of M/G and we express the contribution of each stratum as a residue in several variables, using the results of Sections 2 and 5. The results of this Section were already obtained by Szenes [53] for .G = SU(3) and Jeffrey-Kirwan [30] in the case G = SU(n), 8 = 1, with a central holonomy at the marked point for which M/G is smooth. This Section is organized as follows. In Section 6.1, we describe the strata of a general orbifold, and we introduce various associated characteristic classes. In Section 6.2, we state the theorem of Riemann-Roch-Kawasaki for almost complex orbifolds. In Section 6.3, we construct the orbifold line bundle AP on the orbifold M/G. In Section 6.4, we describe the strata of the moduli space M/G as moduli spaces associated to sernisimple centralizers in G. In Section 6.5, we compute the Atiyah-Bott-Lefschetz-Todd class of a given stratum. In Section 6.6, we compute the dimension of certain vector spaces which appear naturally in the evaluation of the Atiyah-Bott-Lefschetz class. Then we make a number of genericity assumptions on the t3's. In Section 6.7, we compute the contribution of a stratum to the Riemann-Roch-Kawasaki formula in terms of differential operators acting on symplectic volumes. In Section 6.8, we briefly show that under an obvious condition on the holonomies t,,,1 < j < s, the index of the considered Dirac operator vanishes identically. In Section 6.9, we give a residue formula for the index. In Section 6.10, we give another related asymptotic formula for jp) large.

Then we drop the genericity assumptions. In Section 6.11, we compute the index of a Dirac operator on a perturbed moduli space, for which the genericity assumptions hold. The point is that, as we shall see in Section 7, the index of the Dirac operator for the perturbed moduli space is exactly given by the Verlinde formula. Under genericity assumptions, this is only true asymptotically for the given moduli space M/G. 6.1. Almost complex orbifolds. Let M be a smooth compact manifold. Let G be a compact connected Lie group, and let g be its Lie algebra. We assume that G acts on M on the left. If X E g, let XM E Vect(M) be the corresponding vector field.

We assume that G acts locally freely on M, i.e. for any non zero X E g, X M is a non vanishing vector field on M.

We will use here the notation of Section 3.1, with X replaced by M. Then M/G is an orbifold. DEFINITION 6.1. If g E G, put (6.1)

M8 = {z E M , gx = x} .

Set (6.2)

H={gEG;M9&ci}.

Then H is a finite union of conjugacy classes in G. Let (H) be the corresponding finite set of conjugacy classes.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

249

If g E G, then Z(g) acts locally freely on M. We can then apply the above constructions to M9. Let H. C Z(g) be the generic stabilizer of M. Observe that if g' E G is conjugate to g. the above constructions correspond by conjugation. Take g E H. Let Nnfo1AM ^-- TM/TM-9 be the normal bundle to My in 31. Then we have the complex of Z(g) vector bundles over MY, 0

0

0

091131(9)-'NMa/Af--- -0 0 - gt -TAI - T 11/gA' -- 0 I

I

0 -------

I

-ATM'

T l1'/3'tf (9) - 0

1

1

1

0

0

0

and the rows and columns in (6.3) are acyclic. Clearly if 9' E G, g' maps M9 into M9 9.9 -' . Also g' acts on the complex (6.3).

Put (6.4)

NO =NAM/M/(9M13M(9)).

Then the third column in (6.3) is the exact sequence of Z(g) vector bundles on ?tf9, (6.5)

0 -4 TM'/Z(g) - TM/G --r NO -a 0.

Equivalently NO is the "normal bundle" to TM9/Z(g) into M/G. Clearly g acts on each of the vector bundles in (6.3). Then g acts like 1 on 3M (g), TM' and TM9/Z(g). In particular, there are locally constants 9, 0 < 0 < s on M9 such that (6.6)

NMa1M OR C = ® (Nn .1,%1 e Njge/M) e N'. 9
In (6.6), the B's are distinct, g acts on the left on NAOf,1Af, Nnf,l" by multiplication by ei9, e-{B, and on N' by multiplication by -1. Therefore NO OR C splits as (6.7)

NO OR C = ® (N9,s a NO,-') ® NO,'. 0<9«

DEFINITION 6.2. The orbifold M/G will be said to be almost complex if TM/G = TM/gM is equipped with a G-invariant almost complex structure ,7TMIc.

Since in (6.5), TM9/Z(g) is the +1 eigenspace of g on TM/G, we find that for any g E G, M'/Z(g) is an almost complex orbifold. Therefore NO is also equipped with an almost complex structure.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

250

Now we denote with the superscript (1, 0) the +i eigenspace of the given complex structure. In particular T(1AM9/Z(g) is well defined, and N°'(1'0) splits as

N9,0,0) = ® N'"°1'°.

(6.8)

8E]-a,n]\{0}

In the sequel, we will assume that the orbifold M/G is almost complex.

6.2. The Theorem of Riemann-Roch-Kawasaki. Let OT(''0)'No/Z(9),

,

VN9 (''0)" be Z(g) invariant horizontal connections on T(1"0)M9/Z(g), Let E be a complex G-vector bundle on M, equipped with a G-invariant horN9'(1,o),e.

izontal connection VE. Let FE be the curvature of VE. Then (E, DE)]Mg is a Z(g)-vector bundle equipped with a Z(g) invariant connection. DEFINITION 6.3. Let ch.(E,yE) be the closed form on M9/Z(g) (6.9)

FE

ch,(E,VE) = Tr [gexp (

afar

)]

.

In (6.9), g denotes the left action of the given element of G on on E. Let ch9 (E) be the cohomology class associated to the form ch9(E, DE).

DEFINITION 6.4. If B is a square matrix, put

Td(B) = det (1 e_B)

(6.10)

Td(B) = det ( 1-e1 B) A(B) = det

B

2 sinh(B/2)

Observe that

Td(B) = , (B)e ' "1B]

(6.11)

DEFINITION 6.5. Put (_FT(1o)M9/Z9

Td(T(10)M9/Z9,GTo)M/Z9))

(6.12)

= Td

liar

,

N9

T(N9(110), GN9(1.0)) =

TT

Td 8 - F

Then the forms in (6.12) are closed on Ms/Z(g). Let Td(T (1"0)Mg/Z9), Td(N9'(1,0)) be the corresponding cohomology classes. DEFINITION 6.6. Put (6.13)

L(g,E)=Td(T(1'0)M9/Z9)Td(N9,(1,0))chg(E)

L(g, E) depends only on the conjugacy class of g in G. J M°/Z(g) yE be Let V""'111,0 be G-invariant metrics on T(1,0)M,E. Let G-invariant unitary horizontal connections onT(1'0)M, E. Then AT*(1.0)M/G®E is VA2"('10) M/G®E. naturally equipped with a 0-invariant unitary horizontal connection Then

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

Recall that by [6, p 135] , if/G 2 E is a T_If/G Clifford module. If X E TM(G), let c(X) be the corresponding Clifford multiplication operator. Let dv be the volume element on llreg/G associated to hT ' 11.1.!.! G Let K = K+EDK_ be the vector space of G-invariant C'a sections of .1;T .11,,G ): E = (Aeven(T+(o,t)Af/G) 0 E) e, E). We equip H with the Hermitian product (6.14)

s, s' E H -* (s, a') =1

(s. s'):\(7'

G MEd(

Let e1 i ... , en be an orthonormal basis of T -111G.

DEFINITION 6.7. Let DM be the Dirac operator acting on K n

(6.15)

DA9=Ec(e4) I

Then DM is a formally self-adjoint operator which exchanges K.. and K_. Let D+ be the restriction of DM to K+. Then we write D'1 in matrix form as (6.16)

DA" = [DA'

0 ]

Then by [32, 33), D+M is a Fredholm operator. Its index Ind(D=tr) is given by (6.17)

Ind(D+M) = dim ker Dr*1 - dim ker D nr .

If y E (H), let g. E ry be any representative in G of the conjugacy class s. Now we state the theorem of Riemann-Roch-Kawasaki [32, 331. THEOREM 6.8. The following identity holds (6.18)

Ind(D+) = c(H)

L., /Z(9x) I ! L(9-:, E). I

6.3. The line bundle Xv. F1om now on, we suppose that all the assumptions of Section 5.11 are in force. Also we fix once and for all a positive Weyl chamber K and the corresponding alcove P C K whose closure contains 0. Finally we may and we will assume that (6.19)

t1EP,1<j<s.

DEFINITION 6.9. Let M E Z be given by (6.20)

M = {p E Z,pt1,... pt, E W}.

In the sequel we assume that M is not reduced to 0. Then there is po E N' such that M = poZ. (6.21) Let p E M. Put (6.22)

ej=ptj,1<s.

Recall that the set of connections Ap(81i ... , 0e) was defined in Definition 4.36. Also the Hermitian line bundle with unitary connection (ay, DIP) on A. (81, ... , d,) was

defined in Definition 4.37. By Proposition 4.38, EG acts on the right on ay and preserves DIP. Finally Aflat(t1,... ,t,) was defined in Definition 5.26.

252

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

Clearly if p E M, if B1, ... Be are given by (6.22), then (6.23)

ASat(t1,...

,t9) C AP(01,... ,B,).

Therefore the line bundle A, restricts to Aflat (tI , ... , t.). Also EG acts on Aflat (tl , ... , t,: Finally by Proposition 5.28, Aflat(tl,... ,t,)/EQG = M.

(6.24)

DEFINITION 6.10. Let AP, VAP be the Hermitian line bundle with unitary connection over M of the E9G-invariant sections of A, on Aflat(t1, ... 't')-

The notation for AP is justified by the fact that if p E M,p' E Z, then APP = (AP)eP.

(6.25)

It is then clear that the action of G on M lifts to Y. Recall that w is the canonical symplectic form on M/G which was defined in Definition 5.30, which is associated to the basic scalar product () on g. PROPOSITION 6.11. The following identity of closed 2 -forms holds on M, (6.26)

cl(AP, V") =pw.

PROOF. Let A E Aeat (ti, ... , ts). Let a, a' be 2 closed forms in fl1(E, E), which are exact on OE, i.e.. there are 13,i3' E fl°(OE,E) such that alaE _ VA/3,

(6.27)

CA13'. By (4.24),(4.78), (4.100), (4.189),

c1(A',CA')(a,a')=p f -(a,a')+p j£(,6,VA,l)

If [a], [a'] are the classes of a, a' in F11 (E, E), from (6.27), we get (6.28)

cj(A', V') (a, a') = pw([a], [a'])

The proof of our Theorem is completed.

Since w is a symplectic form, there is an almost complex structure Jon TM/G

which polarizes w, i.e. w(JX, Y) is a Riemannian metric on TM/G. Also J is unique up to homotopy. In the sequel, we will always equip T M/G with such a complex structure. Then we will apply the theorem of Riemann-Roch-Kawasaki [32, 33] the orbifold M/G and the orbifold line bundle AP. Up to now, we have made G act on M or on AP on the right. However to fit with the formalism of Sections 6.1 and 6.2 , we will now make G act on the left by setting gx = xg-1.

6.4. The Theorem of Riemann-Roch-Kawasaki on the moduli space of flat bundles. Recall that (6.29)

M={xEG29xII Oj,h(x)=1}. j=1

We will use the notation (6.30)

M=M(G,OI,...,O.).

Let r be projection T = t/GAR -+ T' = t/R .

SYMPLECTIC GEOMETRY AND THE 1"ERLtNDE FORMMLLAS

THEOREM 6.12. The following identity holds

(H) = W\C/G R.

(6.31)

Also if v E C/CR, if u = rv E C/R , :11" = 1Iu..1Moreover (6.32)

Mu=

U

M(Z(u),Oz(u)(w't1)....,UZcw(u'ytsi)-

and the union in (6.82) is disjoint-Finally, if v E C/fl, H,, = Z(Z(v)).

(6.33)

PROOF. Clearly, if u E G', then ugu-1 = g if and only if 9 E Z(u). It is then dear that if u E G', (6.34)

M° = {x E Z(u)29 x 11(Oj nZ(u)), h(x) = 1). j=1

Since G acts locally freely on M, if d"lu 0, Z(u) is semisimple. By Theorem 1.38. we get (6.31). If u E C/CR, using Theorem 1.50 and (6.34), we get (6.32). Finally by Theorem 5.20, we obtain (6.33). The proof of our Theorem is completed.

REMARK 6.13. By Proposition 1.40, if G = SI (n), n > 2, if u E C, then u E W*. From Theorem 6.12, it follows that G acts freely on 1f, so that 111/G is a smooth manifold.

Clearly Z(G) = R /GAR C T is fixed by W. Therefore if v E T. IFS depends only on u = rv E t/R . We will then write W,, instead of 14,, We use the notation of Section 1. In particular iru : Z(u) -r Z(u) is the universal cover of Z(u).

Let u E C/R , let t E T = t/UR be regular. Then Z(t) = T, and Oz(.) (t) Z(u)/T. Let T E t/CRu be a lift oft in Z(u). Then t is still regular in Z(u). Since Z(t) = T, the centralizer Z(t) in Z(u) is just Z(t) = t/CL. Then 02(u)(t) a" Z(u)/(t/emu) = Z(u)/T. Equivalently the projection 0i(u) (t) -i (t) is one to one.

Take u E C/R , (w',... ,w') E W'. Recall that here ti,.., t, are also considered as elements of t, so that w't1, ... , w't, E t . Ultimately, we may consider (W't, ) wltl, ... , w't, as element of 2(u). Then by the above, 02(u) (w'tl ), ... , lift Oz(u)(Wltl),... ,Oz(u)(w't,), and the projection ru identifies the corresponding orbits. Letx= (ul,vl,... ,u9,v9,wi,... ,w,) E M(Z(u),0z(u)(w't1),... ,Oz(u)('w'ts)). Let u1, vl, ... , ue, i 9 E Z(u) be lifts of ul, vl,... , us, v9 E Z(u). Also w1, ... , w, to E lift uniquely 01, ... , w, 0z(u) (w'tl ), ... , 0z(u) (w't,)

E

02w(wltl),... '0!(u)(w8t,). 9

PROPOSITION 6.14. The element fl(ui, vi] i=1

u1i... ,V9. It lies in ir1(Z(u))

ft wj E 2(u) does not depend on

jol

254

JEAN-MICHEL BISMUT AND FRANQOIS LABOUBJE

PROOF. Since ,r1(Z(u)) C Z(2(u)), the first part of the Proposition is trivial. Also, (6.35)

au

[us, vt] k0i = fl(us, vs) [j wi = 1. i=1

J

j=1

i=1

From (6.35), we get the second part of the Proposition. Using Proposition 6.14 , we can now define:

DEFINITION 6.15. Ifu E C/CR, (w1,... we) E W8, h E irl(Z(u)) =?7R/GR

put (6.36)

M(Z(u), OZ(u) (wltl), ... , OZ(u) (w8t8), h) _

{x = (u1, ... , U9, Vg, wl,

, ws) E M(Z(u), Oz(u) (w'tl), ... , Oz(u) (w8t$)),

n 1[ui,v,)j-jj_lw'jh=1}. Clearly, we have the disjoint union

M(Z(u),OZ(u)(w'tl),...

(6.37)

,oz(u)(wots)) _

UhEiW/UR-., M(Z(u), OZ(u) (wltl ), ... , OZ(u) (w8t8), h).

Also since CR/CRu C Z(2(u)), Z'(u) preserves each M(Z(u),OZ(u)(wltl), OZ(u) (w8ts), h)

By Definition 6.10, if p E M, there is a well-defined G-orbifold line bundle AP

on M/G. Let D. be the corresponding Dirac operator acting on smooth sections of A(T'(0'1)M/G) O XP over M/G. THEOREM 6.16. For P E M, the following identity holds (6.38)

I WI

Ind(DP,+) = uEC/CR

1

f

WI IZ(Z(u))I m-lz )

L(u, AP) .

PROOF. By Theorems 6.8 and 6.12, (6.39)

Ind(DP,+) _ uE W\C/VR

IZ(Z(u))I Ihru/2(u)

L(u, AP)

from which (6.38) follows. The proof of our Theorem is completed.

6.5. Evaluation of the Atiyah-Bott-Lefschetz Todd class on a stratum of the moduli space. Now we take u E C/R , x E Mu. Then u descends to a flat section of the G bundle G, which acts naturally on the left on the vector bundle E as a flat section of Aut(E). In particular, on Mu, the vector bundle 6 OR C splits as a direct sum of vector bundles (6.40)

E OR C = (D (6° ®E-°) ® E". 0<0<x

In (6.40), for -a < 9 < it, u acts on EB like et°.

For 0 < 0 < ir, 6B is a complex vector bundle on Mu/Z'(u). Let jI (eeri e V) (E9 be the corresponding characteristic class. Also ER is a real vector bun-

dle on Mu/Z'(u). Moreover 2i cosh(2) is an even function of x. Let II (eP -

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

255

e

= rl2icosh(;)(e") be the corresponding Pontryagin class of a". Needless to say, if e" has a complex structure , then this class is exactly r l 2i cosh(2) (e*, (l,o)) .

THEOREM 6.17. For u E C/R , the following identity of characteristic classes holds on Mu/Z'(u), (6.41)

Td(T(l,o)Mu/Z'(u)) = 29-2+s 1

Td(Nu(>,o)) =

,10<e<" 17 (e

- e-` 3- 1 (e8)

e#cl(n.,.0.0))e}1 EeEI-.,.1\{a} edim(N'(1.0)J)(-1)E-.<e
PROOF. Using Theorem 5.34 and (6.11), we get the first identity in (6.41). Clearly

(6.42) Td(Nu(1,0)) _ rleEl-","1\{0} (ej(x+i0)

-e j(x}49)) (Nu(i,0),e)1-i ..J\{0}edim(N"(i,o),e)

J

Also, for-7r<9<0, Nu(i,o),° = Nu(o,i),-e

(6.43)

From (6.43), we get (6.44)

11 0E)-","]\{0}

(e#(x+ie)

- e $(x+ie)) (Nu((,0)'8)

= 11 (e (x+ie) -

e 4(x+iei) (Nu(l,o),e)

(e}(x+ie)

- ei(w+i9)) (N"(oa),e)

Jf

0<e<" l

II

-e

}(x+ia)) (Nu'e) TT(ies/2 +ie. x/2)(N"(1'0),") (el(x+'°) 11 J 0<0<,r (-j)E_.<e
Also one finds easily that over Mu/Z(u), equality (5.139) of Theorem 5.34 can be split according to the values of 9. So we find that for 0 < 9 < ir, 11 (ej(x+{e) - e }(x+ie)) (N",e)

(6.45)

_ (n

(e}(=+ie)

- e }(x+ie))

Also 2i cosh(x/2) is an even function of x. Thenn (6.46)

(ee)129-2+3

j

r l (iex/2 + ie x/2) (Nu(1'0)''R) = IT (2i cosh(x/2)) (Nu''r) = [rl (2i C osh(2 ), 29-2+s (e")

= rl (e `ti=

-

e-'`=i) 2s-2+o (E")

By (6.42)-(6.46), we get the second identity in (6.41). The proof of our Theorem is completed.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

256

6.6. The dimensions of the splitting of the normal bundle to the strata. Let J be a G-invariant almost complex structure on T M/G which polarizes the symplectic form w. We know that J is unique up to homotopy. Take U E 0/1 7*. Let x E M". Then by (5.120),

(6.47)

T M/G

Recall that u defines a flat section of the bundle d - °- - E . Clearly (6.48)

9OR C=(3(u)(DRC)®( ® 9a) aER\R,.

Let x E R -r [x]' E] - 1/2,+1/2] be the function periodic of period 1, such that (6.49)

[x]' = x for x E] -1/2,+1/2] .

Clearly u acts on the left on 3(u) like the identity and on ga like e2,*(4." ). For -Tr <9 < 7r, put (6.50)

96 = ® *ER\R.

with the convention that if 8 = 0, g = 3(u). Then (6.48) can be written as (6.51)

9®RC= ® g°. -,r<6
Also (6.51) is a Z(u)-invariant splitting. It induces a corresponding flat splitting of the flat bundle B of the form (6.52)

E ® Be

.

-,r<e<x

In (6.52), E° is just the analogue of E when replacing G by Z(u). By (6.52), we get (6.53)

H'(E,E)®RC= ® H'(E,Ee). -,.<6<x

Also TM/G = (E, E), and J acts on H'(E,E). Since J is G-invariant, J commutes with u. In particular J acts on each H'(E,Ee). Let H('m(E,Ee), H(°"') (E, Be) be the +i, -i eigenspaces of J, so that (6.54)

H' (E, Be) = ft (',O) (E, Be) ®R(°') (E, Be) .

Observe that since ft' (E, B°) is equipped with the u-invariant symplectic form w, then (6.55)

[g(',°) (E, EO)]* = H(0'') (E, Be) , if 9 = 0, 7r,

[H(''0) (E, Eo)]* = H(0'') (E, E-e) , 9 E] - ir, 7r[.

Since J is unique up to homology, the dimensions of the vector spaces which appear in (6.55) do not depend on J.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

257

Since Z(u) is semisimple, for -7r < 9 < IT, (6.56)

0. aER

Therefore for any t E t,

E [(a, t)] E Z .

(6.57)

((a,u)P° THEOREM 6.18. For -7r < 9 < It, over M'u, (6.58)

dim H(0") (E, Es) = (g -1) dim gs + E E [(a, tj)] j=1

11 [(..:)ER

- Mt.

PROOF. First assume that m E N, that t1i... t, are of order m, and m1s. Then we make the construction in Section 5.12. In particular u lifts to a Z/mZinvariant parallel section over E. Then we have the Z/mZ invariant splitting (6.59)

H1(Es, E) ®R C = H(1,o) (E9 E) ®H(0'1) (Eg, E)

.

By arguing as in the proof of Theorem 5.78, when taking the Z/mZ invariant part

of (6.59), we get a complex structure on fi1(E,E) which polarizes a. It is then feasible to take (6.60)

H(0'1)(E, E°) = [H(',') (E, Eo)]z/mZ .

By construction, (6.61)

[H°(E, E°)] = 0.

Therefore (6.62)

dim[Ho,1(E,Es)]z/mZ = _Xh,Z/mZ(Eg,p"EO)

Using (5.321) and the theorem of Riemann-Roch-Kawasaki [32, 33] as in (5.349), (5.350), and also (6.62), we get (6.58). Let us now consider the general case. As in the proof of Theorem 5.78, we

approximate (tj,... , t,) by regular (tm,... , te) which are of order m and such that (A) holds. Recall that over Eg, we still have a Z(u) bundle, so that we may and we will assume that at xj (j > s+1), we have in fact a Z(u) connection. Since Z(u) is semisimple, for j > s + 1, (6.63)

TrEe[9j] = 0.

Again the extra points x34.1, ... , x3m do not contribute to the computation, so that

(6.58) still holds. Since for 1 < j <

[(a,tj)], we get (6.58) in full

generality. The proof of our Theorem is completed. THEOREM 6.19. For 9 E] - It, in, (6.64)

dins t (',0) (E, Es) = (g -1) dim ge + t

(1- [(a, tj)]) dER

lM

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

PRooF. By (6.55), for B E] - r, r[, dim H1,0(E, E°) = dim H°'1(E, E-0).

(6.65)

So by Theorem 6.18, we get (6.66)

dim Hl,o(E, E°) = (9 - 1) dim g-A + t E [(a, 0 1 aER

j=1

I(a,4)J,°5x

Now for B E] - r, 7r[,

dim g° = dim g'A.

(6.67)

Also since for 1 < j < s, a E R, then (a, tj) 0 Z,

F,

(6.68)

[(a,tj)] = E [(-a,t1)] = QER

oER

(1- [(a, tj)]) aER

From (6.66), (6.68), we get (6.64) when B E] - r, r[. Also E" is a real vector bundle. Then (6.69)

dim H(1,0) (E, E") = dim .H(011) (E, E-).

By Theorem 6.18, we get (6.70)

dim H(1,°) (E,

(g - 1) dim g'+

E [(a, tj )] .ER

j=1

Observe that if a E R, then [(a, u)]' = i if and only if [- (a, u)]' = 3 . Also when changing a into -a, [(a, tj)] is changed into 1- [(a, tj)]. So for B = r, we find that (6.70) is still equivalent to (6.64). The proof of our Theorem is completed.

THEoREM 6.20. For u E C/R , B, B E] - r, r]\{0}, (6.71)

dim N°(l,o),° = (g - 1) dim g° + E F, j=1

(1 - [(a, t1)]) .

.ER

PROOF. Recall that (6.72)

TM/G = Sl(E,E) .

Also (6.72) is an identification of u-spaces. Then our Theorem is just another formulation of Theorem 6.19.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

259

6.7. The contribution of a stratum of the moduli space. Recall that by Theorem 1.41,

h E CR/CRu H exp(2ia(pu, h)) E S1

(6.73)

is a character with values in ±1. By Theorem 5.18 and by Proposition 5.63, for g > 1, M(Z(u), Oz(u) (w't1), ... , Oz(u) (wst,), h) is a non empty smooth manifold, and for g = 0, M(Z(u), Oz(u) (wits),... , Oz(u) (w'te), h) is either non empty and smooth, or it is empty. In the sequel, it will be understood that any geometric statement about the empty set is empty. Recall that by [34], [6, Lemma 7.22] and (1.193), the orbits O(wjtj) carry a natural complex structure, which polarizes the canonical symplectic form ao(wlt1) on O(wj tj ). Let N0$°)) (wJLf)/o(wfLi) be the holomorphic normal bundle to Oz(u) (w1 tj )

in O(wjtj). As we saw after (5.305), HH,g,,(t1,... , t t) is locally a polynomial of (t1, ... , t3, t).

THEOREM 6.21. For u E CIGAR, p E M, p $ -c, (w', .. , , w8) E W', h E GK CR,. '

L(u, )t') = (p+C)(g-1)dim3(u)+f dim3(u)/t M(Z(u),Oalu) (w'Li ),... ,Oa(>) (w't' ),h)/Z(u) (din(g)-dim(3(u)) s (=1) 7-r eir EoES+I(+of a,u))+2{a(wiptf+PLt,u) (WU

jl_ll II

(6.74){

11 A

((0"

P+c)) 2g-2+s 1

2sinh (a ((a,

aER+\Ru,+

e-y

((P-eLi

P+C

) +2isr[(a,u)])) Hu,9,s(wlt1,...

1

"Tu (0/0')

2ri

j=1

X29 I

Also

{Td(N

(6.75) 7r7

)int(z(u),os(.)(w t,),...,oz( )(w't.))/z'(u)} 9-1rr$ 1 1

(dsin r a,u

))

llj=1 et 1-u-

(0)

=

N(io) of

PROOF. First we consider the case where g = 0, and M(Z(u), Oz(u) (wits ), ... , Oz(u) (wet.), h) = 0. By definition, the left-hand side of (6.74) is 0. Also fort E T

close enough to 1, Mt(Z(u),Oz(u)(w1t1,... ,Oz(.)(w8t.),h) is also empty. By (5.305), it follows that H.,9,, (ti, ... , t t) vanishes identically near t = 1. Then (6.74) is just the identity 0 = 0. Now we assume that M(Z(u),Oz(u)(wit1),... ,Oz(u)(w8t,),h) is non empty.

Ifw=(w1,...,ws) E W8, put (6.76)

Mw,h =

M(Z(u),Oz(u)(w'ts),... ,Oz(u)(w8ts),h)

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

260

We use the notation of Section 4.7. If X E MM,h, the corresponding flat connection

reduces the G-bundle P to a Z(u)-bundle we still denote by P. This Z(u)-bundle lifts to a Z(u)-bundle Q. By Proposition 4.33, and by (6.36), (6.77)

[P] = eh.

Wi y+h,u) By Theorem 4.39, the right action of u on A' over Mw,h is given by widi+h,u) By (6.41) , and The corresponding left action is then given by e2iwp(Ej:t

by the above result,we get

fm.. z(U) L(u, gyp) =

'4(&0)

f i t rW ,./Z(++)

11 II e "! e-

Ee 1W)

0<egw jci(T".'M1G)+Pwe

(6.78)

E9EJ-,, J\{U} edim(N'("o).e)

(-1)E_.<e
e2lx(wipti,u)e2ia(pu,h) llfs-1

Now we use Theorem 5.78, which gives us a formula for c1(TM/G). By Proposition 5.73 and by (6.78), eiEee)-..*!\(o) edim(N..(ao),e)

L(u,AP) =

(6.79)

1WuI

MW.,,/Z(u)

(-1)E_.<e
'910t

ll

Leiftu.+ A

.es

2sinh(t((a,

dea+

2

A ,+. 1

11_1

24 cosh( (a, P+C)P+C

Hu,9,.(wltl,... w"tsjh t) I

Fic.

I

((a, P+c )

TTe2ix(w°ptiu)e2t*(ph,u)

iol

Observe that if a E R\Ru, when changing a into -a, 2 sink (2 ((a, P+ct)+ 2iir[(a,u)])) is unchanged. Therefore (6.80)

TI

1

0
(2 ((a, 4) + 2iir[(a, u)J))

«aE

=

1

2i cosh 2 ((a, tee)

j 1

aER+\R.,+ 2siah (2 ((a,

P+C

) + 2i7,[(., u)])

By Theorem 6.20, (6.81)

exp(i/2

0dim(Nu(l,0),6))

_

eel -w,wl \{0}

exp(i/2( E 9(dimge(9-1)+E E

(1-[(a,wit3))))).

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

261

Now the 9 E] - ir, 7r[\{0} come by opposite pairs. Therefore

E

(6.82)

0 dim g6 = it dim g"

,

9E1-x,,rl\{0}

and so ei£a dim ge(9-1) =

(6.83)

(-1)x(9-1)

For1<j<s,put tj

(6.84)

= wjtj

Rj,+ = w1 R+

Then R j,+ is the positive root system associated to tj. For 1 < j < s, E8EI-112,1121sEi(nbE

(6.85)

_ E.ER[(a, U)]'(1 - [(a, tj')])

= EaERi.+ ([(a, u)]'(1- (a, tj)) + [-(a, UT (a, t;)) _ - EaER[(a, u)l'(a, t j ) + E Ri.+[(a, u)l' . Clearly, (6.86)

a,t;) aER

$

lfn,

Now for a given s, the {a E R, [(a, u)]' = a} are exactly the weights of the representation of Z(u) on if E g OR C, U f = e2in' f }. Since Z(u) is semisimple,

E a = 0.

(6.87)

n@R

From (6.81)-(6.87) , we get (6.88)

exp(2

E

9dim(Nu(1A),6)) = (-i) d-

"(9-1) fl exp($a E [(w1a, u)]') OER+

j=1

eel-W,al\{0}

Also by Theorem 6.20, using (6.87) , we get (6.89)

E-A<0<0 dimN" (1,0),6)

= (g - 1) E-x
(1 - [(a, tj)])

= (g-1)j{aER,-2 < [(a,u)]' <0)I a,tj)+I{a'E R, -a <[(a,u)]' <0,(a,tj) > 0}1) (g -1)]{a E R, -a < [(a,u)]' < 0}l + E,=1 I{a E R, -L < [(a, u)]' < 0, (a, tj) > 0} 1

_ (g - 1)1{a E R, -a < [(a, u)]' < 0}I +E;=1 I{a E R+, -1 < [(wja, u)]' < 0} .

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

262

So by (6.88),(6.89), (6.90)

0dim(Nu(1,O),O))-1)E_.<e
exp(' OE]-,r,7r]\{0}

1)(9-1)(I(aER,0<[(a,u)]'<' }I+il{aER,0<((a,u)]'= }p a

II;=1 exp(da E [(w''-, u)]) &ER+ Moreover

(6.91)

I{a E R,0 < [(a, u)]' < 2)I+ 21{a E R, [(a,u)]' = 2}1 = 2(dim(g) - diln3(u)). From (6.79), (6.80), (6.90), (6.91) , we get (6.74). Using (6.74) , or by proceeding directly, we get {Ta(Nu(l,0))IMM,n/Z1(u)}(0) =

(-1)(9-1)(dimg-dimi(u))/2lLjexp(iir E [(wia,u)J)

t

aER+ Jr

(6.92)

ilaER+\R,,.+ 2smh

ia,u )

a9-2+s

Using the invariance of sinh(aa[(a, u)]) when a E R\Ru is changed into -a, from (6.92), we get (6.93)

9-t

{Td(Nu(l,0))IM:,,,/Z,(u)}0 =

ii,=1 f aQw! eR+1R

4 sinx(a,u ))

2sinh in a, w) - u

-

9-1

ik i a °E - LtsER+\R,,,+ [4sin (,r a,u ] 9-1 s 1 - 11aER+\R,,,+ [4sin a a,u), lij=1 et 1-u-

i I

oa(,)(,ui ei)/o(wi ei

which is just (6.75). The proof of our Theorem is completed.

Recall that (6.94)

B = IR+I 1u = IRu,+I,

r = dimt.

Our assumptions on g, s guarantee that 2g - 2 + s > 1. Also by (2.100) and by Theorem 2.45, the function Pu,I,+.,29-2+e(t) is a polynomial on T \ S.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

263

THEOREM 6.22. For u E C/ZR, p E M, p 54 -c, then fM"/2(u) L(u, AP) = Vol(T)29 2 Z Z(uffl (p + (-1)1(9_1)+1

(6.95)

{A(@ER\R.2siah

C)(9-1)dim(3(U))+A-dim(3(U)/()

! lLER,,.+

(a,2PLf')+2ex[(a,u)J [sgn((-i)t°GZ(u)(wit7))e('e,°(P-eti),e.; e`)+

+iIr E.ER+ ((w') a, IL)3 + 2iir(wipt9, u)] Pu,29- 2+s,P+c (t)It=z, w& ti

PROOF. By (6.32), (6.37), (6.96)

fM°/z(u) L(u, AP) _ L(u, AP)

W....

.

We use Theorem 6.21 to calculate the term in the right-hand side of (6.96) . Also by (2.131) and by Theorem 5.72,

vu (61i8) Hu,9,3(wltl,... ,wlt.,h,t) = (-1)t, (9-1)+1

(6.97)

IZi (Z'(u))IVOl(t/v+tiu/Ia9-2 E(w11,...,w")E W,

7 llj=l

2ia(Pn,h)Pu 28-2+e(t +

h + E;=1 w'iw'ti)

Clearly

Z(Z(u)) = Z(Z(u))

(6.98)

and so

IZ(Z(u))I = IZ(Z(u)l ICI .

(6.99)

Moreover

Vol(t/[,Ru = Vol(t/C )I OR I = Vol(T)I VXI .

(6.100)

By (6.97)-(6.100) , we get (6.101)

1ru

($. )

"

' g (wltl,... , wets, h, t)

JZ(Z(u))jVol(T)2g-2E(u

1,...,u

)EW,(u)ll.=letii((-i)tuv2(u))(w'ti)

exp(-2iir(Pwh))p Also by Theorem 1.41 , for h E CR/CRu, (6.102)

exp(-2i7r(pu, h)) = exp(2iir(cu, h)).

264

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

Now observe that if W 'j E Wu, by (1.45),

etij= QZ (u) (W'jw''tj ) ,1<j<s.

(6.103)

aZ(u) (wjtj )

Therefore (6.104)

sgn ((-i)t"QZ(u) (wjtj)1 Ew'j = sgn ((-i)11az(u) (w'jw'tj))

Also

fu + (dim(g) - dim(3(u))) /2 =1.

(6.105)

From (2.166), (6.74), (6.96), (6.101)-(6.105), we get (6.95). The proof of our Theorem is completed. 0 REMARK 6.23. As shown in Section 2.11, the function exp(2ia(qu, t))Pu,29_2+.,,,+ (t) descends to a function which is well-defined on T = t/Z R. Ultimately, this explains why equation (6.95) is unambiguous.

t7 Recall that the function 0(t) = 11 (eir(a,t) - e-ix(,t)) is well-defined on T = e2i,r(v,t)

t/VR, and the function

a(t)

i=1

J

is well-defined on T' = t/R .

PROPOSITION 6.24. For any u E C/R", for any s E treg, x E P, w E W, (6.106)

sgn((-i)az(u) (wx)) arj

1

2 sinh(.((a, s) + 2iir[(a, u)]))

[(wa,46)])=Ew 11

exp(ilr aER+

1

eR+

2 31nh(a (a,, 8 +

2i7rue2ia(wv,u)

))

PROOF. As we just saw, both sides are unchanged when replacing u by u+y, 'y E CR. By [15, Proposition V.7.10], we may as well assume that for any a E R,

Ka,u)I < 1.

(6.107)

If a E R\Ru, sinh(y

((a, s) + 2ia[(a, u)])) is unchanged when a is replaced by -a. Therefore using (1.45), we get (6.108)

ilceR+ 2sinh(((,a)+24t (a,u)])) _ (_1)IR+nti `(-R".+)1

-rr 1 _ 11aEWR+ 2sinh( ((a,a)+2ia (a,u) )) (_ 1) IR+nw-' (-R",+) I+I { aER+,-1 <(wa,u)
faER+ 291ah( a,w-'(s+2ii u))) l(-R",+)I+I{aER+,-1G(wa,u)<0}]+I{aER+,(wa,u)=t1}I --rr

11aER+ 2ainh

(a,s+2i,ru )

Moreover, for X E P, (6.109)

sgn((-i)t"7Z(u)(wx))

_ (-1)Inti 1(-R",+)1

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

265

Also by (6.107), (6.110)

exp(iir F, [(wa,u))) _ aER+

exp(ilr(w C` a,tL))(-1) aER+

e2ia(wp,u) -1)lfoER+,-l<(wa,u)<0}I+I faER+,(wa,u)=tlll

From (6.108)-(6.110) , we get (6.106). The proof of our Proposition is completed.

0

REMARK 6.25. It is clear that the right-hand side of (6.106) only depends on

the class of u in C/R . Observe that the function II (ars)e2ia(wP,u)

aER.,+ (6.111)

EC

s E ti-+

2 sinh(2 (a s + 2i7ru) ) aER+

is a well-defined holomorphic function near s = 0.

THEOREM 6.26. For any u E CI GTR, p E M, p 0 -c, the following identity holds,

fm./Z(u) L(u, A') = Vol(T)2g-2

2 Z(

/,.. + )(#-1)dI-(3(u))+srdIM(s(u)/t)

!'

2g-2+s

8/8t (6.112)

IOeRu,+(a

(-1)t(9-1)+1

2sinh aER+

p+ c)

1

t

\2 (a, n+ c +

2ixu))

P(('+ta(P-

1)e P+C )

+2i7r(WI(P+ptj),u))Pu,2g-2+s,p+e(t)It=E;,j wits PROOF. Clearly, since 2g - 2 is even, (6.113)

L 11 A((a, L P+c E R.,+ aERl R,,,+ 2 sinh(2 ((a,

/

a/at ) + 2iir[(a, u)J))

8+

11 (a, P

)

aER,;,+

11 2 sinh( (a, aER+

2

p+

c+ c + 2i7ru))

2g-2

.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

266

Also, since tT E P, by Proposition 6.24, we get

MER,.,+ A((., '1,1+8C1)) 11aER+\R,,.+

1

2 sinh(2 ((a, (6.114)

ita

s

urit

V/Vt c) + 2iir[(a, u)])) p+

ei*E°ER+1(w'a,u))

(a,e/8t)

II

p+C

&ER,.,+

Ew

2 sinh(2 (a, aER+

p+c

e20(w'P,u)

+ 2iau) )

By Theorem 6.22 and by (6.113),(6.114), we get (6.112). The proof of our Theorem is completed. O

REMARK 6.27. Suppose temporarily that t1, ... , t3_1 verify assumption (A), and that e > 0 is small enough so that if It,I < e, (t1,... ,t,) still verify (A). Then (6.112) can be written in the form

(6.115)

J ro/Z(u)

M

L(u,AP)

Vol(T)2g-2IZ(Z(u))I (p

+ e)(s-1)dim(a(u))+fdim(a(u)/t)

IW'I 2g-2+s aER,..+

(_1)1(9-1)+1

IV

0& + 2iiru)) 11 2 sinh 2(1(a,P+C

IaER+

E Ewe(w(P+Pt.),

wE W

+2(xu)

E

e-1 Ewre(w'(v-qtr), a$)+2+A(w'(P+pt'),u)pu

f =1

2g-2+HP+c(t)It=Etai witj

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

267

Using (1.94), (2.131), since pt, = Bs E GAR+, we may rewrite (6.115) in the form (6.116)

JM /Z(u)

L(u, AP) _

Vol(T)2g-2IZ(Z(u))I (p+C)(9-1)dim3(u)+I%Udim(3(u)/()

IWuI 2g-2+s-1

Blot p+C

(-1)t(9-1)+1

a/at 2 sinh (2 (a,

p+c

aER+ e-1

I

d 8e

Xpt. (e- ° p+° +u)

+ 2i,,ru))

v+e )+2%a(wt(p+pti),u)

(w1. ,w"-1)EW°-19=1

Pu,29-2+s-1,p+c(t))t=E._i wit, With respect to (6.112) , s has been replaced by s - 1, and we have the extra

+u)

differential operator Xpt, (e

Let M' be the manifold attached to G, 01,... , 0,-1. Let F be the vector bundle over M'/G associated to the representation of highest weight pt, C Z`R of G. Then if we still denote by L(u, AP) the corresponding class (6.13) over M'u/Z(u), in view of (5.318), we can rewrite (6.115),(6.116) in the form (6.117)

fm/Z(u)

L(u, AP) =

JM'°/Z(u) L(u, AP)c11u(F) .

Also by Theorem 5.57, for It,l small enough, the orbifold M/G fibres over M'/G with fibre G/T. Let p be the projection M/G -+ M'IG. Then one verifies easily

that (6.118)

p.L(u, AP) = L(u, AP)chu(E),

which makes (6.117) tautological. REMARK 6.28. If u E CIGAR, w E W, one should have the equality L(wu' AP) .

L(u, AP) =

(6.119)

fm'/Z(u)

JW" /Z(wu)

We will briefly explain why the right-hand side of (6.112) is unchanged when replacing u by wu. Put

(6.120) C =

Vol(T)29-2IZ((ti))I (p+c)(g-1)dim3(,.))++/2dim(3(a)/1)(_1)t{g-1)+1

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

268

Then using (1.45), (2.167), (6.112), we get 2g-2+s fAfwU1z(wu) L(wu AP) = C

(ew

ll*ER

nQEg+2sinh j(wa,--

(_ 1)IR+f1w(-R.+)I)' / (wi....

ae

+2iwwu)

' ew Rj=1

Pwu,2g-2+8,P+c (t)It=Et wtt/ = C

IIaER

2g-2+s

oy at

IIuER+ 2sinh J(a,2ixu)

E(w.....

EW flee=1 ewr

(6.121)

°v

e(w' (v-ctt ),)++2ix (,si (P+Pti ),u)

L(u, AP).

wit

Pu,2 -2+s,v+c t

So we find that (6.112) is compatible with (6.119).

Recall that in Section 1.8, we saw that 6.8. The case where E,=1 ptj if Ej=1 ptj E A, for any (wr, ... , w') E W', then Ej=1 wiptj E 11.

THEOREM 6.29. If Ei Pt j 0 R, if v E C/R , then (6.122)

J

L(u, AP) = 0.

In particular, if p E M,p # -c, Ind(DP,+) = 0

(6.123)

PROOF. To establish (6.122), we use Theorem 6.26. In fact, in the right handside of formula (6.112), we observe that the various terms depend on the image u = ru, with the exception of n'=1 exp(2iir(wfptj, u)). Also 8

8

(6.124)

E exp (2iar(E w1 pt j, v)) Vex /iW

=0

j-1

if F, wi pt j ¢ j=1

IR 'a

if E wjptj E R . j=1

(6.125)

It is now clear that (6.122) holds. Equation (6.123) follows from Theorem 6.16 and 0 from (6.122).

6.9. A residue formula for the index. In view of Theorem 6.29, we may and we will assume that (6.126)

Eptj E j=1

We now use the notation of Section 2. Recall that 3` C t/GAR has been defined in (2.111).

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

269

DEFINITION 6.30. We will say that (t,,... , t,) verify assumption (A) if for any

(w', ...,ws)EWa,

d

(6.127)

E w'ti i=1

Clearly assumption (A) is stronger than assumption (A) of Definition 5.17. In

the sequel, we assume that (t,,... , t,) verify (A).

THEOREM 6.31. For any u E C/R , p E M,p # -c, the following identity holds

fM /z(u) L(u, AP) = I Z I Vol(T)4-2 z Zu (p + c)(9-i)*(_1)t(9-i)+,

[[

1

Res..,ER+

2sinh (z (a,

_ +2inu

!EZ°Ir/dTF 1

(6.128)

+2ia(Elwl(p+pt9)+(p+c)f,u))exp(dr_j'=1 Z'(-1(Ek=, wktk +f),e') n,

ai,1_,e

PI-la{f ;z

)

... > i.

exp

PROOF. If (W',... , ws) E W 8, E,'=l w1 t f 0 S. Then we use Theorems 2.50 and 6.26 and we get (6.128).

THEOREM 6.32. If p E M,p # -c, then (6.129)

Ind(DP,+) =ICI

(G)I (p+c)(9-i)r(_1)t(9-i>+r Vo1(T)2g-2IZIWI

E

uEC/R`

r

-129-2+s

I f<<:3 .)EX, IE'Z7C/a7r

(ai, , 1 , ai.)

: X11+

2 sink (z (a, P

-,

+ 2ilru))l

I EwI...Ew.e)Cp((5- w'(P-ct.),p+c)

+2iaLwd(P+pti)+(p+c)f,u))exp(dE J=1

i=1

i-tai ,e') s

[d(Pj. i(Lwktk+f),e')]) '-1 f e p (d(Pr_, tai PROOF. Recall that Z(G) = R/ZR. Using Theorems 6.16, (6.124) and 6.31, we get (6.129). The proof of our Theorem is completed. REMARK 6.33. As we saw in Proposition 1.40 and Remark 2.7, for n > 2 and G = SU(n), then C/N* = 0, and we can choose d = 1. Then formula (6.129) has an especially simple form.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

270

6.10. A formula for the index for large p. THEOREM 6.34. For p E M, and IpI large enough, the following identity holds Vol(T)2,-2I I (I)I (p+c)(9-1)r(_1)e(9-1)+,

Ind(DP,+) =ICI

-

r=c+,.)Ex (ac,,... ,a+.) !EZ Fe/dA

E

uc-C/t£'

Res'

LaER+ 2sih (a(a, L

1

,

8

ewi...ew.exp((Ew;

(6.130) E

j=1

p

-c"')

+2i7r(t uri (p + ptj + (p + c) f, u)) j=1 exP

(d

I) [1(PJ'-1(L w_ptk +f),e1)])

j=1 ""7_1aa,,ej) d

p+c

k=1

1

wic

PROOF. Clearly, for p -4 +eo, Fj8_1 wjt7 V 3`, if .f

-> 0. Also by Proposition 2.14, since

ER, 5

(p9-1( wktk + f), ej)

(6.131)

Z.

k-i

Therefore by (2.13) , for IpI large enough,

(6.132) d

j =1

1(pf 1(:t W P1tA

jj-lai1 e9)

d 9-

kk=1W"p+c

+ f) '

d5lYj-1a ef) d(Pj-1 (t wktk+f),e'), k=1

- ( c ti, XI) j=1

By (2.115), J=1 (6.132) , we find that for Ip( large enough, e

(6.133)

CEjol

[

\

-1 (Lkel pi-c

f)

)]

= exp ((-Eje=1 p+ ,xr))

r

exp d

!-1 E j=1

I_ ( tw ktk + .f ), ej 1 k-1 From (6.129), (6.133) , we get (6.130). The proof of our Theorem is completed.

REMARx 6.35. The formulas in Theorems 6.32 and 6.34 are essentially identical.The point is that, as we will see in Theorems 7.23 and 8.1, the right-hand side of (6.130) is just Verlinde's formula.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

271

6.11. A perturbation of the index problem. We still assume that a > 1 and that if g = 0, then s > 3, so that 2g - 2 + s > 1. Note that these assumptions are almost irrelevant, since by adding as many marked points as one wishes with holonomy equal to 1, one can make s arbitrarily large.

Recall that, by Proposition 1.23, P embeds into T. Also the orbit under the Weyl group W of any element in T always intersects P.

Let t1, ... , t, be s elements in T. We may and we will assume that they all lie in P. In the sequel we will consider tl,... , t, as elements of t. We still define M C Z as in (6.20), and we assume that M is not reduced to 0. For p E M, set

8j=p9j,l<j<s.

(6.134)

Then 01ER,1<j
(a). Put

(6.135)

X6 = G29 x 11 Otj+61 j=1

and let M6 correspond to XS as in (5.59). Then M6 is a smooth submanifold of V.

We will briefly show how to equip M6 with an orbifold line bundle )1p. We use the notation of Section 4.10. For 1 < j < s, consider the orbit G_pd/dt+p(t,+a;) C Zg. By (4.30),

(6.136)

0-pd/dt+p(ts+6;) = LG/T.

Also as in Definition 4.3, L0/(T x S1) = LG/T can be equipped with the line bundle

of weight (0j, p). Therefore 0_pd/dt+p(ti+6f) can be equipped with

the corresponding line bundle Lj. Recall that ptj = 8j. Then we define the line bundle )p as in (4.189), and the line bundle )' on M6 as in Definition 6.10. We can then define a Dirac operator Dpt'+6f) in the same way as in Sections 6.3 and 6.4.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

272

THEOREM 6.36. For p E M,p 34 -C,

Ind(D(t+ of )) =

(6.137)

I

R Vol(T)29-2 I 1 I (p + W

c)(1-1)''

I

E

(_1)1(0-1)+r E

1

(Oil' ... a{. )

few/=

"°=0

1

+ 2iau))

Ert+ 2 sinh (a (a, a

swl...C..exp((:bi(p+ j=1

(wt.... ,w )EW

+2ia(twk(p+ptj)+(p+c)f,u)) j=1 dr

P

j=1

lai ) 1

(p_1ai,ei)

'' (twk(tk+6k)+fJ

11

k=I

1

7r

j=

exp (

(-1a+,,ej)) -1

PROOF. The proof of our Theorem follows the same steps as the proof of Theorem 6.32, where the case 6 = 0 was considered. We briefly describe the main steps where the proof of our more general result differs. Instead of (6.26) in Proposition 5.11, we have (6.138)

cl (aP, VaP) = p(w + t (5j, ej)) j=1

This follows from an easy computation which is left to the reader. In formula (6.74) in Theorem 6.21, the main point is in fact that 11;=1 e2ix(w;Ptj+Pn.,t)

is unchanged. The argument is in fact the same as in the proof of Theorem 6.21. Also using (6.138) instead of Theorem 5.78, one finds that in formula (6.74) in Theorem 6.21, j'j3_1 e

((P-0tf),e1att) should be replaced

By proceeding otherwise as before, we get (6.137). The proof of our Theorem is completed.

REMARK 6.37. Using (6.137) and proceeding as in the proof of Theorem 6.29, we find that (6.123) is still true. So in the sequel, we may and we will assume that (6.126) holds.

Now we will slightly modify the statement of Theorem 6.36. Recall that

t1, ... , to all lie in P. Therefore for p E M, p > 0, for 1 < j < s, pti /(p + c) still Use in P.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

273

THEOREM 6.38. For P E K P > 0, if ei...... C. E t are such that e1,

>

+e, E P, and that, for any (w',... ,w') E W',

f_I wj (

P+C

+

+ej) f{

then

+`t)) =

Ind(DP ,+

Vl(T)2,-2I (G I (p+c)(9-1)r

I

FEW`

(_ 1)L(g-1)+r

IWI 1

E (ai, ... , c ,,) uEC/R 1 ° 0 ....t,.)EZ /E71/79

(6 139) Res'x=0 f

1 TT I aER+ 2sinh (2 (a, + 2iru))

ewe...e,,.exp((Ew1(

P -e)),X1) p -CC

j=1

+2i7r(Ewj(p+ptj)+(P+c)f,u)) j=1 dr

exp

-1aii,xl) (pt_1aif,ej)

,

1 Ld(r'j-1

wk((p kckc+ek)+f1 , ej

J=I 1

`

TT (exp (d(°i-1ai3+x') \

111

`

'

j-tai,,ei) J

)

J

11 .

)

PROOF. In Theorem 6.36, we take

bj = Z-CtL +ej ,

(6.140)

P+c

1<j<s. D

Then we get (6.139).

PROPOSITION 6.39. For p E M, p > 0, there is el,... , e, E t such that for 1 E +Lej E T is regular, and for any (w1, ... , w') E W', I E]0,1], P+C

]0,1], 1 -<j:5 s,

Ej=1 wj (V+ + lej) ¢ 5. PROOF. Recall that 5-` is a union of hypertori in T. Therefore 3{fl P is the intersection of P C t with a union of hyperplanes in t. Our Proposition now follows easily-

13

Let [x]+, [x]_ be the functions defined on R with values in (0,1], which are periodic of period 1 and such that [x]+ = x forx E [0, 1[, [x]_ = x forx E]0,1] .

(6.141)

Observe that by 2.14 , for I E]]0,1), 1 E R/dd, 1 < j < r, (6.142)

(p,-1 f

wk 1

\

+.eek)

+f) ,eiO Z.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

274

From (6.142) , it follows that either

( p+c ') +f) Iej) 0 Z,

(6.143)

or

to (+nc) (P+f) ,ej) E Z (P"'-1

(6.144)

a

(PI-1

gWid =1

, ej)

54 0.

DEFINITION 6.40. Given (w1,... ,wa) E Wa, for f E R/dR, I = (i1,... ,ir),

put

+if (p3_I

(6.145)

rk=1 wk

or if (pi-1 1

`kk=1

(Pj-1

(P+c) +f) ,e') V Z,

wk (Ptk I + f) , ej) E Z, P+c

wkek

,ej) > 0

=1

(Ewk(p+kC\+f),e')EZ,

if (i _1 k=1 s

(P_1

(wks) ,e') <0. k=1

r

Observe that if f = E fjej k=1

(6.146) k=1

wkP+ (kc +

By (6.146), it is clear that

eJ) _ f-1

( k=1

Ptk wk p+ C

+ k=1 fkek) , ej)

(f, I) depend only on fl,... ,11,i1,... f J, , ij-1

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

THEOREM 6.41. Given p E M, p > 0, the integer Ind pend on E E]O,1]. More precisely Dp++tr:>

Ind

(6.147)

DpP`+tr, )

275

does not de-

= ICWIVol(T)"1-2IZW I)I(P+c)(e-1)r

rElrk/eff

ll aER+ 2 sink (1(a,

Resxr=,

2

1

r

p+c

+ 2iau))

e

Ewi...Ew.exp((Ew1( j=1

(w1.... .w')EW

P +C),zr)

+2ia(t w1(p+ptj)+ (p+c)f,u)) 1

r

exp

(dt j=1

_1a11,xr)

(

r

j-lai,, ej)

P _i (t 1

{d

wk(p+c)+f)

k=1

f

1

9

d(P9_la,j xI)

11 (

j-laii+e'i)

1 )- ).

PROOF. Clearly, for I E]0,1],

[(P_iw(c++f>e]

(6.148)

k=1

depends continuously on t E]0,1], and moreover as t -* 0, it converges to 1 (6.149)

r [dlPj-1

r

wjtj c+f) p+

eI) I

v(.1....."" (f,I)

DP++trJ)

depends continuously on t E]O,1], and By (6.139) and the above, Ind so remains constant. This last fact should also be clear by the construction of Py+e+trt)

D(P.+ Using (6.139), (6.149) , we get (6.147). The proof of our Theorem is completed.

0

REMARK 6.42. Observe that in our direct index theoretic computations, we have avoided introducing central holonomies, because they are non generic. We dealt with this case by a perturbation argument. However we may assume that one of the tj, say t is equal to ho E Z(G), while the other holonomies are generic.

276

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

By Proposition 1.24, we can identify h,, to a unique element in P, which we still denote h so that h, E ]5n r. When the moduli space M/G is an orbifold (which is the case when condition (A-) holds), all the computations we did in Sections 6.1-6.9 can be done directly. By proceeding as in Remark 6.27, it is obvious that in both cases, we compute the same index. This is less obvious at the level of explicit computations. Using in particular equations (1.78) and (1.140), we find that equation (6.112) is now replaced by fm.lz(u) L(u, a1') = e,,,n, Vol(T)2g-2

zzy

(p) c)(e-1)a1m(a(u))+aim(I(++)/A

(6.150)

I

0ERa,+

(_111(9-1)+1

29-2+s-1

8/8) p

l

2 sinh (2 (a, --- + 2ilru)) AER+

p+c

lljol E,,i exp((wJ(P - cti), " ) +2iir(vri (p + ptj), u)) exp(2iir((p + c)ho, u)) pu,29-2+a-1,p+c(t)lt=r_j'mi witi+h, The fact that ultimately, the two explicit index formulas coincide will be verified via the Verlinde formulas in Remark 8.4, by using in particular Theorem 1.33.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

277

7. Residues and the Verlinde formula In this Section, we apply residues techniques to the finite Verlinde sum, we express it as a residue in several complex variables, and we prove that for p large enough, the Riemann-Roch number of M/G is given by the Verlinde formula. Without any condition on p, the Verlinde formula is the Riemann-Roch number of some perturbation of the moduli space of M/G. As indicated in the introduction, higher cohomology groups on M/G may well

not vanish. This would account for the discrepancy between the index and the Verlinde formula for small p. This Section is organized as follows. In Section 7.1, we give a residue formula

for a Fourier series over W */4r , Lq(t,x). In Section 7.2, we consider a related series Mg(t, x), which we also evaluate as a sum of residues, the sum being indexed by the semisimple centralizers Z(u). In Section 7.3, we introduce the Verlinde sums V,,, (01, ... , B,). In Section 7.4, we express the Verlinde sums as residues. Finally in Sections 7.5-7.7, we "localize" the Verlinde formulas to put them a

form which is very similar to the form we obtained in Section 6 for Ind(D,,+) This is done using the techniques we already developed in Section 2.

7.1. A residue formula for L,(t, x). DEFINITION 7.1. For q E N*, t E R/qR , x E (C\2ilrZ)t, put

exp(2iir(t,t/q))

Lq(t,x) =

(7.1)

AEI /9

2i,r(at,A)-a; l Ii=1 2q tanh 2q

If t E R/qR, then t/q E (R/q)/R C t/R. Recall that the subset H of t/R was defined in (2.44), (2.45). Then by (2.45), (7.2)

E

{t E R/qR,t/q E H} _ {t E R/qR,t =

t1aj in t/qR,

jEJC{1,... ,r}

t} .

and the {aj, j E J} do not span t'

As in Section 2.4 , we identify R/dR to {0, 1,... , d - 1}r, i.e. f E R/dR is

r identified to E f iei, f i E {0, ... , d -1 }.

By (2.14), if I = (i1, ... , ir) is generic, if x E Zt, then d(ej, xf) E Z, 1 <

j < r, and so for 1 < i < t, d(ai, xf) E Z.Therefore the function x E Ct H -z) is periodic of period (2iagd, ... , 2iirgd). tanh (a' sq Similarly if t E R, we claim that the function 1ais,xf)[ xECtHexp(dE(p -1NJIef) d _1t,..),) q

(7.3)

j=1

is periodic of period (2iagd, ... , 2iagd). In fact by (2.13), (2.28), (2.29), if x E (2iirZ)t, (7.4)

exp

(dE -laii,x')ei)[1d (Pj-I f t j=1

),) = P q

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

278

Also, as we just saw, if t E R, z E Ze, d(t,xr) E Z. From (7.4), it follows that if x E (2i7rgdZ)e, t E R, (4.4) is equal to 1, which proves the periodicity of (7.3).

let gf C Ce be such

If g = (gl,... , gr) E Zr, if I = (i1, ... , ir) C that (gr)si

= g1,

1<j
(gr)i = 0, i 0 I. THEOREM 7.2. For generic values of x E (C\2i7rZ)e, fort E R/qR, t/q ¢ H, if we still denote by t a representative of tin R, then r

Lq(t, x) -

E

dr

S p n ((a;, , ... ' a0))

JEH/dW.9e(Z/dZ)I

r

1 I ej ej) [d(Pj-1(t+qf), 4 )]

(p'-1a,f xt)

d= (p' j=I

nie`r 2q tanh (

2q

)J

(x + 2i7rggr) r j=1 exp

1

_

PROOF. Take 1 < j < r, I = (i1i... ,ij_i) C {1,... ,f} such that (ai,) 0 0, (ai, ... , aif_,) # 0. We use otherwise the notation and the conventions in the proof of Theorem 2.19, in particular in equation (2.63). For

r

For a E C,

k'ei E

(kj+,, ... , kr) C Z'-j, we identify k to i=j+1

x E Ce, t E R, put (7.7)

Sr(a,x) _

j-1

(n-iaiq,x!)

_, ej {exP(oit>+dE (P-ia, 09,<,

r1

e,l

r en)LP'it'

)]

n=1

1
((P;_Ini,aei+2i,rI_z1)) Me`r 2q tanh 1\ JJ 2q

}\x+2i7rq(gi,...

gj-1)i).

}, with We claim that as a function of x E C', each of the expressions { (gl,... , gj-i) = 0, is periodic of period 2iirgd,... , 2irrgd. In fact by (2.63), if

xEZt, (7.8)

d(pjl-1,ai,xr) E Z.

Also by (2.28), (2.63), for 1 < n < j -1, x E Ze (7.9)

(t-lai,., en)

E Z,

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

279

so that if x E ZI, using (2.12), 1-1 ,MnI-1a;"' xn-Iain,e

exp 2i7rd

( 7.10)

r Id

i-l

exp

V.-lain,xI) LIT ri-l /_f-lash g

I

exp (2q

((t,xr) - (pil-, t' X')) Also by (2.14), (2.63), if t E R, X E ZI,

d(t,xl) E Z, d(pp-lt,xr) E Z.

(7.11)

By (7.11), it follows that if x E gdZe, (7.10) is equal to 1. Ultimately, we find that indeed, each of the expression { } in (7.7) is periodic in x E Ct of period 2i7rgd,... , 2iirgd. Therefore, we will write (7.7) in the form

Sr(a,x) = E

(7.12)

{exP(a(P_it,

) +d

9E(Z/dZ)I-i ri

n-lt'

et) Ld

4

q

1

(x+2i7rggi)

77.-7r

i1iE`12gtanh(

2q

In particular Sr(a, x) is periodic in x E Ct of period 2i7rq,... , 2ilrq. Now we claim that SI(a,x) is periodic in a E C, with period 2iirq. In fact, since of = (aef)t, (7.13)

e') _ (Pi-1ai,(aei)r)-

Also by (2.12),

-1 (7.14)

eq) = (t, e7) n=1

(Pn'-lain , (Qej)r) ! (p.1-lt, e n) (P,-1ain,en)

Since t E R, then (t, e') E Z and so, by (7.14), we get (715 )

(pil

Pj-1 t, ej )

f-1 (/-1ai,,(ae1)') (Pn I- 1t, ,en)) mod (Z ) F, n-I (PI -lain en)

From (7.12), (7.13), (7.15), we find that (7.16)

Sr(a+2iirq,x) = St(a,x-2iirgae') Sr(a,x),

i.e. SI(a,x) is periodic in a with period 2iirg.

Put (7.17)

LI = E SI (2iirk, x) . kEZ/qZ

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

280

Clearly

LI=

(7.18)

1 St(2iirk,x). d

kEZ/dqZ

From (7.18), we obtain C24aa kl

LI = I E pxp

(7.19)

J

kEZ/d2gZ

Sf

(2dk,a)

.

By (7.12), for generic x, we rewrite (7.19) in the form

(7.20) Lr = d2 E 5 kEZ/d2gZ l

F_

Resa=2jak

[ex (a _-I (t + d ei )ei l) j I

\L

off
J

a-Iai*,x n [P_it,etV)]) n--1

n-taia,e

i

I lie°I 2q tanh r` (pf at,aei29/d+2iak-x1) eat 1 J 7

Now we just saw that the function of a,

}(x + 2iirg91) ..., which appears in the

0<1
right-hand side of(7.20), is periodic of 2iirgd.

We claim that for f, 0 < f < d, g E (Z/dZ)J-1, the function of a,

(alp!i,_/q)1N (7.21)

hf ,g(a) =

11iEeI2q 77''77

1

f

2q

ea - 1

is periodic of period 2iird2q. This follows from the fact that since t E

d(p'_1(t + gfef,ef) E Z ,

(7.22)

d(_1ai, ef) E Z . By Proposition 2.14, if t/q if H, 0<

(7.23)

[(Pi_l(t+Qf51)i5u/)]

< 1.

d

Then for given f, 0 < f < d, g E

we will apply the theorem of residues to the function hf,e(a) on the domain given in Figure 7.1. By (2.65), for at least one if ¢ I, -Iail,ef) # 0. For generic x, mod (2iird2q), the poles of the function hf,g(a) other than the 2iak are simple and given by (7.24)

a=

d

J-tail, ef)

I

((Pf_I

0 < 91 < dI(pj-Iail,e'))

ai1, x' - 2iir) + 2tiag'q) , ii 0 I, (pi-Iail,

0 0.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

281

FIGURE 7.1

Needless to say, in (7.24), d(_1aii,e') E Z. In view of (2.28) , we rewrite (7.24) in the form (7.25)

J

d

a=

J)

-2iirk),

0 < g' < dl(pjL1ai,, e')1. By periodicity, the integrals of h9,1(a) over 6+ and 6- cancel each other. Using the theorem of residues and (2.70), (2.76), (2.78), (7.20), (7.23), (7.25), we get (7.26)

LJ = -d i:(nil,...,a;i>#o 1

aii, PT)

(2i."ei

e

-1

+dE (N,-iai,,,x') (-1aia,e) [1V.-1(t+gfe,,),e°V4)1 d n=1

1

((pif'ci)at:2ixk-xt))

77

lli¢(J.ii) 2gtanh

29

1

d(p{ai x')

exp((v,_

} x+2iir4(9. g7)J,ii) JJJ

)-

1

(

JEAN-MICHEL BISMUT AND FRANCOIS LABOURIE

282

We claim that as functions of xj,, the terms in the right-hand side of (7.26) are periodic of period 2ixgd. In fact this follows from what we saw after (7.7), by replacing j -1 by j. We also claim these functions, are periodic as functions of x,, , with period 2i&rgd(pj1'_Ia, , e'). In fact, using (2.29),

d___Ia{e_)

(7.27)

(ai...... a,,)

- (aj......d aj-I)-E Z.

In view of (2.63), (7.27) takes care of the terms tank

((p'I Ii ai2 sak - xf)) 2q

Also

among the term (pr,_la;,,,xr), n < j, only (pf_Iaj,,xr) depends on xi,. More precisely by (2.63), a

(7.28)

ax,,

-

I aif,xr)=1

Clearly, (7.29)

qd

(PI

a, e'')

1

[d(t';-I(t+gfei),e'lg)J =

s-Iaf,,eJ)d d(pp_I(t+gfe1),ef) mod (Z).

Using (2.29), we find that (7.29) is 0 mod (Z), i.e. the left-hand side of (7.29) lies in Z. Ultimately, this guarantees the periodicity of the functions appearing in (7.26) in xi,, with period 2i7rgd(pf_Ial,,e1). Now by (2.29), (7.30)

d(pf_Ia,,,ej) =

d(af,,... ,a,,) (a,,,... ,ai,_,)

From (7.30), we find that (7.31)

d(Vj-Iai,,e>) I d2.

Since the functions in (7.26) are periodic in xg, of period 2i7rgd(_Ia;,, e'), using (7.31), in (7.26), we may and we will replace the condition 0 < g1 <

''-I

by the condition 0 < g1 < d2, and this introduces a correcting factor I d in the right-hand side of (7.26). Using the periodicity in x{, of period 2i7rgd, we may finally replace the condition 0 < g' < d2 by 0 < g' < d, with a new correcting

factor d. Ultimately in (7.26), we replace the condition 0:5 g' < dj(_Ia,,,d)j by 0 < g' < d, with a correcting factor I(_Ia;,, ei)1. Also by (7.30), (7.32)

sgn(Pp-la;,,e') =sgn(a;,,... ,ae,)sgn(a{..... cq-I).

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

283

So, from (7.26), (7.32), we get (7.33)

Lt =

d-1

{ exp

(22rYif)ei,k)(P-I(t+gfei),ei)+d-laZ'xl)

t Ld

sgn((ai11... (xi/_1)(ai..... ,c

))

net `rn-iaiw , en)

(Pn-1(t + gfei), en/q)J

J

1

1

17(x +2£7rQg(,,ii))

ig(I,if) 2qtanh

2q

1

exp (/ d(f_1ac ,x1)11

y Clearly (7.34)

Lq(t, x) =

exp(2iir Ej=1 ki (t, ei/q)) 7 kjo)-mi ) k=(k1,...,k.)E(2/q2)'It, lli=2qtanh (Vw(ai,E 2q

J

Also with the notation in (2.63), (7.35)

(Po ai, xr) = xi.

Then we use (7.33), so that if t E 1/97R, t/q it H, (7.36)

E

Lq(t,z) = d

0
sgn(ai1)exp

(2i1r(P.1)(t+gfle1),k)+d(ai1) f 1

1

` Moil 2gtanh ( (Pi+1)ai,2i,rk-xf) 2q 77

(x + 2iirgg ,)

)

e-p ((si1)) - 1

Using (7.33), (7.36) and an obvious iteration, in view of (2.83), we obtain (7.6). The proof of our Theorem is completed. O

7.2. A residue formula for M8(t,x). DEFINITION 7.3. For q e N*, t E'A/q .R, x E (C\2iirZ/m)t, put exp(2iir(A,t/q)) Mq(t,x) _ E (7.37) L ITi=1 2qtanh ( 2ia(a ,a)-st

,\Eir/qR

29

Now we will adapt the formalism of Sections 2.5 and 2.6, with t/Gk replaced by Wt/qZ°R. If f (t) is a function on Rt/gUR-, if p E G R `/R*, put (7.38)

fµ(t) =

Ee

IN/ M vElt/ix

2iT(;4,V) f(t + qv)

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

284 Then by (2.92),

f(t) _ E fu(t)

(7.39)

µECr/RR*

/R , if A, E Z'R represents µ, fort E R/qCR,

PROPOSITION 7.4. If p E

(Mq)v(t, x) = exp(2iir(ar+, t/q))Lq(t, x - 2iiraAp) .

(7.40)

PROOF. If A E CIA*/qR , then

exp(2ia (,\, t/q)), = exp(2i7r(a, t/q)) if A maps to p

in VR `/R

(7.41)

otherwise,

0

Then r

exp(2iir(A + A,,t/q)

(g)µ(t,x) =

(7.42)

R`

aE

raCa(ai,a)-(

lli-12gtanh `

aix(at,a >))

,

g

which is equivalent to (7.40).

J

13

PROPOSITION 7.5. For q E N*, t E R/gc,R, x E (C\2irrZ/m)r,

M5(t, x) = E exp(2ia(A , t/q))Lq(t, x - 2iiraAµ) .

(7.43)

µECiZ /R'

PROOF. This follows from (7.39), (7.40).

Recall that r is the projection t/CR -+ t/R. Then r induces the projection WlgVR -I R/qR. Also recall that the set S c t/GR was defined in (2.111). By (2.45), (2.111),

{t E R/qGR, t/q E 3} = {t E R/qR rt = E tjaj ,

(7.44)

j5.7

{aj, j E , T} does not span t* } . THEOREM 7.6. For generic values of x E (C\2i7rZ/m)e, t E R/qZ°R, t/q ¢ Y, if we still denote by t a representative oft in R, then (7.45) Mq(t,x) =

(-1)r

E

sgn((a;1,...,aa.))exp(2iir(Aµ,t/q)) /7r`

f E7r/d$,pEf

[Id(pj(t't gf),ealq)]/ jL-Ji

I.

1

77

fl

, 2q tanh (

ai 2 -xi l (x + 2iirggt - 2iira,\,) q

1

77- n;=1

JJ

/(aj..iaii.')) - 1

(x

-

.

1

PROOF. This` follows from Theorem 7.2 and Proposition 7.5.

13

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

286

DEFINITION 7.7. If I = (i1, ... , i,.) is generic, let HI be the lattice in t gener-

ated bya",...,a'°.

Recall that C C R was defined in Definition 1.35. PROPOSITION 7.8. The following identity holds

C= U HI.

(7.46)

I generic

In particular for I generic,

dHICR .

(7.47)

PROOF. The identity (7.46) follows from the definition of C. Then (7.47) follows from (2.21), (7.46). The proof of our Proposition is completed. By Proposition 7.8, we have a natural projection

hI : HI /dHI - HI /R ,

(7.48)

whose kernel is just

ker hI = R /dHI .

(7.49)

PROPOSITION 7.9. For any generic I,

d'

IR /dHII =

(7.50)

..,a:.)I

a

PROOF. Recall that e1,... , er span R, and el, ... , e' span R*. Therefore (7.51)

IR /dHII = d'I(a"A...Aa",elA...Aer)I dr

I(a:, A...Aai,)I The proof of our Proposition is completed.

THEOREM 7.10. For generic values of x E (C\2iirZ/m)r, for t E R/gZW, t/q 0 3', if we still denote by t a representative of tin R, then (7.52)

M5(t, X)

=I

I (-1)r E

uEC/r

E f Et'W/ dr

1 exp(2iir((u,t+qf))) (a{,,... >ai) l1iE°/2gtr 2q

-1a`f,zI)

P (d 1-1

1

I1(pj-1(t-hgf),-jlq)]

I d eiL ) (1_1ai,{, 1

111=1 (ex

n..r

(d(j_i:iJ)) - 1)

)

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

286

PROOF. Clearly the map g E (Z/dZ)e ,-f gl E H'/dHI is one to one. Also if t E R, f EX, by (2.13), (2.28), (2.29), (7.53)

exp

(d1:

(_1a,,,eJ)

[d!Vj 1(t+4f),e1l4)J

= exp(2iir(t + qf, gi)) = exp(2ia(t + qf, hlgi)) Moreover by (2.115), (7.54)

(a,, (4A,)') - (aA )i = 0.

Also if i ¢ I, using (7.47), (7.55)

(ail 911) - (91)t = (at,9i),

q(at, 9i) = q(ai, ht 9i) in Z/qZ. Finally by Proposition 7.8, the map hl : Hr/dH' - C/R` surjects on {u E C elements. and by (7.49), (7.50), the fibre has C/R', I C

-

+

From (2.13), (2.119A), (7.45), (7.53)-(7.55), and proceeding as in the proof of Theorem 2.28, we get (7.52). The proof of our Theorem is completed. 0

7.3. The Verlinde sums. Recall that by Proposition 1.27, if A E G R , the character XA of G does not depend on the choice of a Weyl chamber K such that A E F. Also by Proposition 1.28, if to E W (7.56)

Xwa = XA

Moreover the function (4eO)2(t) is well defined on T' = t/R'. Now we introduce the Verlinde sums [62j, [3].

DEFINITION 7.11. For9EN,gEN",01....,9,EGR,put (7.57)

V9,q (01, ... , B,) _

lilt

9-1

qq

1

IW' aeZ71`/aalr

aP/a)fo

1

(ita(X/4))29

X0

By Proposition 1.12, we get

*91

1

Aegpnur

1

7,3=1

Recall that in Section 1.8, it was shown that Ej=1 Oj E R, if and only if for

(W"..., w`) E we, Ef=1 wjef E R By Proposition 1.29, observe that if E =1 Bj E R, then nj*=1 Xet (e'19) is a well-defined function of A E CR°/qR' THEOREM 7.12. If E fa18t ¢ R, then (7.59)

V9,q(91,... ,B,) = 0.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

If Ef_1 0j E (7.60)

287

then 9-1

CR

V9,q(91,... ,0,)

IZ(G)I IWI

AEZ71 /9k

7-

1

X9J(ea/4).

(tea(A/Q))29-2 11

PROOF. Clearly (7.61)

V9,q(Oi,... ,9,) _

91

(i LL

9uI

1

IWI v E

r

1

/ate.

(i'a(./9))29-2

II XBi (ea/q+u).

µEr/z j=1

Using Propositions 1.29 and 1.30, (7.61), and the fact that Z(G) = R 1UR-, we get our Theorem. THEOREM 7.13. The following identity holds, (7.62)

V9,0(91,...,9,) =

11 XeJ(ePIC). j=1

If one of the Bj's does not lie in W,

'08) =0.

(7.63)

More generally (7.64)

V9,o(91,...,9,)=0 , +1 or -1.

PROOF. By Proposition 1.10 and 1.11, we get (7.62). By Proposition 1.31 or by Theorem 7.12 when a = 1 and by (7.62), we get (7.63). By (7.62) and by Theorem 1.32, (7.64) follows. The proof of our Theorem is completed.

Let K be a Weyl chamber. DEFINITION 7.14. If 9 E (7.65)

put

B(t) = 1 E ewe2br(w(p+8),t) wEW

By (3.115) , if p + 9 does not he in a Weyl chamber, (7.66)

'(t) = 0.

If p + 9 lies in the Weyl chamber woK, then (7.67)

Also if to E W, (7.68)

Xe (t) = Xp+e-wop(t)

{

Xwe (t) = X9 (t).

288

JEAN-MICHEL BISMUT AND FRANCIOIS LABOURIE

DEFINITION 7.15. For g E N, q E Z*, 01,... ,08,08+1 E GTR`, put g-1

-K

(7.69)

1

GAR

V9,q(01) ... 08,08+1)

qCR

E

IWI aez'/qzF 2a181

1

X9; (e' q)Xe +i (eA/q) .

(itO'(A/q))9-1

THEOREM 7.16. For 9 E N, q E N", the following identity holds,

t (7.70)

E 6,16,2

V9,q(01,... ,6a) _ (-'

I9M (u1,v2)EW2

K

9+1,q (01, ... 108,V1 p + v2P)

PROOF. Recall that (7.71)

Xo = 1.

By (7.57), (7.71),

y-1

ZW

(7.72)

V9,q(01,... ,6) =

Ti

1

1

qCR

IWI

E aE4z

3(e'/q) .

X9;

(iCa(A/q))29-2

By (7.56), we may and we will assume that 01i ... , Oa E E. = CR n K. Using (1.94), (7.72) we obtain (7.73)

E

V9,9(01,... ,0,) = W

r!r

J

1

1 (ita(A1q))29-2 (o,(AIq))'+3

88+3

11 Ewi (w1....,w')EW'+3 jot

}

+3

exp 2i1r(Ew1(p+Oj)+ E wIP,,\Iq) j=,+1

J=1

_G R /qVR1e-1 ` IWI

r

L.

1

(itor(A/q))29 (or(A/q))'+1 app/a)fo

EvlEve

(vl,v2)EW1

exp

1

rG

,+1

IEwi

(w1,...,w'+1)EW+1 je1

(2i1r(t wI(p+Oj)+ws+1(p+v'p+v2P),,\Iq)

which is just (7.70).

j.l

,

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

289

The proof of our Theorem is completed. REMARK 7.17. If w E W, p+wp E R. Therefore (7.59) and (7.70) are compatible. Also observe that the fusion rules [31,1621 express Vg,a (.) in terms of V9-1,q (.).

Here equation (7.70) goes the opposite way.

7.4. A residue formula for the Verlinde sums. In the sequel, we fix a Weyl chamber K C t, and we use the notation in Section 2. Also by (7.56) and (7.59), we may and we will assume that

8jE GR't=-crn7, 1<j<s,

(7.74)

8

E of E R. j=1

DEFINITION 7.18. If w E W, let ow E Se, let £w(1),... ,ew(I) E {-1,+1} be such that for 1 < i < 1, W-1 ai = Ew(i)avv(i)

(7.75)

By [15, Corollary V.4.6 and Lemma V.4.10] , we know that e

C. = 11Ew(i)

(7.76)

i=1

DEFINITION 7.19. For x E Cc, w E W, put (7.77)

WWW i=1

Equivalently, by (7.76), Ww(x) = Ew

(7.78)

e l;i ew(i)x:

1 x

i=' =1

THEOREM 7.20. For g > 2, q E N', the following identity holds, (7.79)

Vol(T)2g-2q(s-1)rm

Vg,9(01,... ,0.) =

IWI

(_1)c(9-1)ss1`o ,c)

1

2g-2

[n;=l2sinh(2q), M9

For g (7.80)

(

11 Vwr (xfq) (wl,...,w')EW'j=1

wj8j,x1I

f

0, for a even, 2g - 2 + s > 1 and q E N*, the following identity holds, Vol(T)2,-2q(9-1)r _UJ

V9,9(81, ... , o) = 1

(-1)

11

1

2g-2ta

x=0 [ni=l2sinh( Q)} s

MQ(E wJ(p+8j), x). j=1

7

II Ewi

(wi,...,w')EW'j=1

RK

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

FIGURE 7.2

PROOF. First we assume that g > 2. For A E M 14R , w = (w',... w°) E W°, x E (C\2irZ/m)1, put (7.81)

1

Ua,w (x) =

I

'

l 19_2 1 pwi (xl q)

1

H 2q tank (2iw(X,ai)-xc 1 zq

J

Then Ua,w (x) is a meromorphic function of x1i ... , x4t, which is periodic of period

2irq, ... , 2irq. For e > 0, let A, be the domain in C given in Figure 7.2. For e > 0 small enough, as a function of xi E A., the function UA,w(z) has poles at 0, and also at 2iirq aq ]. Here q [21211] is the real number in [0, q[ which represents (A, ai) [ Fore > 0 small enough, the poles do not meet the boundary 80s = d_ US+. mod q. Assume first that [ aQ ' ] 36 0. Then 2irq [f1] is a simple pole of UA,w(x). We now use the theorem of residues on A,. Since g > 2, as xi -+ -+oo inside A,, the function UA,w(x) tends to 0. Also since UA,w(x) is periodic of period 2iirq in the variable xi, the boundary integrals cancel each other. Therefore (7.82)

Resx,oUA,w(x)+Resx{=2iw4r ac ;''1Ua,w(x)=0.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

291

Clearly (7.83)

Resx:=2iw4

((a'4 )1 UA,w (x) _ 2g-2

1

2i sin

q

xt Wwt

1

nj#{ 2 sink ()

1

q q ... xi_I

a{)

q

xj 1 ,

1

4 J j#i 2qtanh

J

2q

Assume now that 11&99111 = 0. Then as a function of x{, the function UA,w(x) has a single pole at xi = 0. By the theorem of residues used as before, we get (7.84)

Res.,=oU.\,w(x) = 0.

By (7.83), (7.84), we obtain

(7.85)

(-1)e(g-1)

sl,..."

x

1

Pwj (-') 9

12g-2

x=o

[]Zi2sinh( 9)J a 1

Mq

j=1Wjej'

x

(L i o(a/q))29-2

l

(m ..

II cP&j C2iir j=1

Q 1)

, ... , 2ia (A t)1 exp (2i r(vrj9j, al q))

4

J

Now by (1.93), (7.75), (7.78), for 1 < j < s, (7.86)

E (Pwj

q

>...

a, )l ,2ia(q ) exp (2iir(S9j,A/q))

wt EW

= X9{ W/9 . FYom (1.17), (7.85), (7.86), we get (7.79).

Now we consider the case where g > 0, s is even and 2g - 2 + a > 1. Put 1

1

(7.87) Ha w(x)

)}2g

-2+s

[ni=12 sinh(2q

2in(A,x4)-x; i=1 2q tanh (

\.

Then Hk,w (x) is a meromorphic function of xl,... , xe. Since s is even, it is periodic

of period 2iaq,... , 2iirq in z1, ... , xl. 0. Since 29 - 2 + a > 1, as xi tends to too inside Assume that [ aq ; , Of, Ha,,,(x) tends to 0. We thus find that the analogue of (7.82) holds, with U,.,,, replaced by HA w. The analogue of (7.83) is now 2g-2+a

R8x;=2iwq[$

(7.88)

i

J.]HA,w(x) _ - [2i sin m E-T llioi

nj#{ 2gtanh

Y

-=i

2sinh

4

292

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

Also the analogue of (7.84) still holds. So we get the analogue of (7.85),

(7.89)

x

(-l)'(.9-I)Res'

1

l2q /1]

li'fq

(t 0

Ew i.. . E w +

29-2ts

[W, 2 sink r

(w1....

E

1

1 (i,a(a/4))29-2 (Q(A/4))'

(.ut.. .W')EW+ 8

ew, exp (2iir (wj (p + 61), A/4)) j=1 1

Now by (1.94), for 1 < j < s,

(7.90)

ewj exp(2iir(w,(p+03),A/4)) =

a(AIq)

Xej(eA/9).

w/EW

From (1.17), (7.89), (7.90), we get (7.80). The proof of our Theorem is completed.

0

REMARK 7.21. It is crucial that in (7.80), a is even. In fact under the given conditions on the OjIs, E;=1 wjOj E R, but we know that E,_3 Op E R only if a is even.

7.5. The generic case with g > 2. DEFINrrION 7.22. We will say that (9ki... ,0,) verify assumption (Aq) if for any (w1, ... , w') E W', a Ej=1 wj9 j 0

In the sequel, if u E C/R , we choose once and for all a representative of u in C, which we still denote by u.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

THEOREM 7.23. If q E N", g > 2, if (01i ... , Be) verify -AQ), then

V9,9(91i... ,9)

I CRjVoI(T)2s-2IZ(G)I q(9-1)r(_1)e(g-1)+r IWI 1

Gr uE

(aii,... ,ai.)

feuw/d

R,esIx=o

2g-2+a

H

IjER+j 2 sinh (12(a, xI /q + 2ilru)) 1

ewr ... ew. exp (t w p, xI /q)

(7.91)

j=1

1

+2iir(Evi (p+91)+qf,u)) f=1 P

a

laij,ZI) r1 dE (pr j-1a:'s ef)

ejU

j=1

k-1 1

exp

PROOF. Take t E R. By Theorem 7.10, if t E R, t/q ¢ S, then ,e)

x=O

(7.92)

1

[u2s()]

Res(1,...,e)

H, 2q taxlh

2q

1

exp(2i7r(u, t + qf))

,ai.) Prr

((a:,xr+2iagu)-xi

j-1a"'xI)

Cd`1 (p;-lair,eI) 1

TTr (ex(;))

j-1

128-2 11(PwI

ni=12 sinh ( c /j 1

(a{, , ...

Jj(Pwi ()M9(t, x)

X87

1

x-o

2g-2

)

(x/q)

293

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

294

Observe that in the right-hand side of (7.92), the expressions starting after exp(2iir(u, t+

q f )) does not depend on the xi, i ¢ I. Now we explain how to evaluate the residue in (7.92). We use the notation in Definition 2.47.

1. The case i¢ ori¢I,iEZu,a,¢{a,r(1),...,a,i(y,)}. Note that the above condition just says that i I, and (ai, u) E Z, ai {a,t(1),... ,aot(pt)}, or (ai,u) 0 Z. If (ai,u) E Z, ai 19 {a,'(1),... ,a,,(,,)}, for at least one k E I, k > i, then (ai,ak) 34 0. So since the {xk)k>i have not yet been made "small", for generic x, (a,, x1) is "large". So we find that under the assumptions we made in 1., for generic x, mod (2iirgZ), (ai, x1 + 2iaqu) should be considered as different of 0. Then by proceeding as in (7.82) and using the fact that g > 2, we get (7.93)

Resx,=o

1

[2sinh (2q zq

2g-2+e

)]

1

2qtanh

3a Z;.1 r.i (i)xi

= (ai.x`+2iaqu)-xi) j

2q-

29-2+s 1

`2 sink ( (a; ,xf+2i_gu)) 2q

e (Ei.t S ,j (i)at,x'+2ixqu)

2. The case where i ¢ I, i E Zu and ai E {a,z(1), ... , a,r(,,)}. In this case (ai, u) E Z, and so

_

1

(7.94)

((a:,x++2iaqu)-xt)

2qtanh

2q

1

2qtanh (a{

2q

1 - tank ( ) tank ( ac qX'))

- - 2q

tanh()(1-

tank ( of2gx' ) )

tanh(2q)

Now since ai E {a01(1),... ,a,r(,,)}, (ai,xr) is a linear combination of the x,r(k), k < pi, which have been made "small". Therefore we get the expression

tanh

tank (a`'xf) _

_

1 (7.95)

a;x' ) -1 2q

+

tanh2 (a:,x' ) 2g

tanh(xi/2q) + tanh2(xi/2q) + ....

tanh(xi/29)

In view of (7.92), (7.94), (7.95), for k > 1, we should evaluate (7.96)

Resx,.o [

[2sinh(xi/2q)]2s-2+s

tanhk(xi/29)

The function of xi which appears in (7.96) is periodic of period 2iirq. Recall that the contour A. was defined in the proof of Theorem 7.20. For e > 0, 0 is the only

pole inside 4. By using the theorem of residues as in (7.84), and the fact that g > 2, we find that (7.96) vanishes.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

From (7.94)-(7.96), we see that if i (7.97)

m:.o

I, i E Z,., ai E {a,,(1),... ,a,r(pt)}, then 1

L LL [2 sinh(xi/2q)]

2i--2+s

1

A E;-, E,.i (i)xi

0.

(at,zr+aiaq aqu)-xt

2gtanh

295

Using (7.79) in Theorem 7.20 and (7.92), (7.93), (7.97), we get

V"'(01>... ,o,)

(7.98)

E

R

-IMEI

IWI

1

L

uEC/

Vol(T)2,-2 IZ(G)Iq(s-1)*(_1)r(g-1)+,

ai,.)

(ail>...

leux/dw

7

f

a-o

7-

1

1

iE12 sinh(ay) g12 sinh

E

" ((E i4t

Ewi ... $w. eXp

(w',..,w)EwI

J

9

E ewi (i)xi ) + iEl

Ewi (i)ai>

4

4

7=1 (i)ai

exp (2ia(u, t(Eigr

j=1

+ wjoj) + qf))

2

r exp(dt (-lai,xl) d1 j-,(twkok+q}),e'/q)1/ ll 1C1iieej)

j=1

k=1

1

ll1 77r

(a(_io..

x1)1 j_yati,ei) 1

'

1 1

Now observe that if i E I, (7.99)

(at,x!) = xi

Therefore

E (Emi (i)a,, xr) + E Ewi (i)xi =

(7.100)

iEl

i(f7

I

I

(ESwi(i)ai,x'l) = (w'E ai,xl) = 2(uIp,xl). i=1

i=1

Also if i E I C Z,., then (ai, u) E Z, and so 2g-2

(7.101)

.

xr+2i,rqu)l= [2s(a: 1\ 29

I 2 sinh

(at mr Q2iA

ia)

exP

\

[2]

2g-2

'

2iir(u,Ewi(i)ai) 2

.

2 sink (s)

296

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

From (7.98)-(7.101), we get (7.91). The proof of our Theorem is completed.

0

REMARK 7.24. Clearly, 1 (7.102)

2 sin ((a. x:/q+2iru) z 111

does not depend on the representative u in C. Also

II

(7.103)

1

2 Sink ((ai, xr/q + 2i7ru))

-1w; exp ((wip, xr/q + 2iiru)) _

1

sER+ 1- exp (- (wi a, xl /q + 2i1ru)) does not depend either on the representative u E C. This explains why the terms in the right-hand side of (7.91) do not depend on the choice of the representatives of U.

7.6. The non generic case for g > 2. Recall that the functions [x]+, [x]_ were defined in (6.141).

Let rh,... ,,7, be r functions from R/dR x {1, Igeneric} into {+, -} such r

that if f = E fje1 E R/dR, f = (i1, ... , i,), then ri, (f, I) depends only on

f l,... ,fail,... ,ij_l. First, we will extend Theorem 7.2 to the case where t E R/qR is arbitrary. THEOREM 7.25. For any t E R/qR, there are meromorphic functions W1 (t,x2, , xi-1, xi+1, , xe) ... which vanish identically when t/q V 'XI).... , wj (t, xl, H, such that (7.104)

sgn ((ai,,... ,ai.))

L9(t,x) _ (,tlr)r

j_l 7j

1

"'j) [d

-1(t+4f),

Ci q

r 1

r

2g t8II11

1

(x + 2iirggr)

(

2q

)

11 exp

t +

l 1

(_(pL.icij,ei)

,xi-l,xi+l,...,xe).

PROOF. We use the notation in the proof of Theorem 7.2. Let rll, ... , r)f_l E _, t, 1,1)) is replaced by

{+, -}. We define Sr(a, x) as in (7.7), except that ['a

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

297

y

x

FIGURE 7.3

Take ,,(f): f%... , d-11 -+ {+, -}. Then we rewrite (7.20) in the form (7.105)

LI =

E

.Cry

O
kEZ/d2qZ

Res°=2iak [exp

+d f_

(aa [(PS_i(t+gfe1),S/q)1 d

PIn-lain,xl)

1

7 11iE`I 2q tanh ( (Pi

1

ns(f)

(pl -it, aniq)

(x + 2i qgr) ai,aei/d+2i,4-°1) e° 1 11 J

)

2y

Instead of (7.21), we now set exp (a (7.106)

[P_i(t+fei4->l

1

J

ns(f))

hf,9(a) =

rL,1I2gtwh

``

1

2

The key point is that we do no longer assume that (7.23) holds. Equivalently may now well be an integer. For n E N, we now replace the contour in Figure 7.1 by the contour of Figure 7.3. We apply the theorem of residues to h f,g (a) on this contour, and let n -+ +oo. (i4_1(t+

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

298

(t+dei),`i/4) is an i nt eger, if ,,, (f)

Clear ly, if (7.107)

n

hf,9(a) =0,

a

hf,g(a) _

= f , then 1

t ank

H

r_lai, 2i7rk - 2t) `

2q

ta;_le;..i)ao

1

(2q)I°II(_

1 I{iE

Observe here that by (2.65), {i E CJ; (i4_lai,ef) 36 0} is non empty. Then by (2.63), the right-hand side of the second equation in (7.107) does not depend on

[

So by proceeding as in (7.21)-(7.25), we find that e,)ei/4) is not an integer, (7.26) still holds, with [ ] replaced by if (P1-'(t+i lip' ' 'I (Pi_y(t+gfei),eil9) integer, writing n; instead of n;(f), then (7.26) is d

replaced by

(7.108)

LI

d

0$J
l_laii' e,)

exp(2iir(pyr'ii)e;'k)(is-)(t+gfe;),e'/q)

+dE 1

llig(I,di) 2qtanh (

tl.y) a;,2irkx r) 2q

1

d(l1ai pf7i' xt)

exp

fexp

Ir

+4 l

e) / /

o
1

OE(Z/dx)g

(d [ (Pn-isi..,xr) (1

n/g) a,.

at, 2i7rk - xl)) J

tank iE'r

(x+2iag(g,9;)%,ii)

)

1\

2q

(x + 2iirggr)

1

(2g)I °II

/I

(_1)I{iE`I,(Pf_la{'ei)#0. sgn(Pf_la:,ei)=ni(f)}I

the key point being that the last sum in (7.108) is a sum of functions which do not

depend on the xi, i E e1 , (-jai, e;) $ 0). Using (7.108) and proceeding by recursion as in the proof of Theorem 7.2, we get (7.104). The proof of our Theorem is completed.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

299

REMARK 7.26. The last term is the right-hand side of (7.108) depends in a non trivial way on the choice of nj. More precisely, it is a sum of terms depending on nj up to sign. As a corollary, we find that the function tpi (t, xl, ... , xi-1, xi+1 , xp) also depends on the choice of 'qi,... , n, in a non trivial way. >

Now we extend Theorem 7.6 to the general case. We still take Theorem 7.25.

as in

THEOREM 7.27. For any t E R/qR, there are meromorphic functions , x[),

+01(t, x7726s,

, Iki(t, xl,

, xi-1, xi+l,

.

, xt) which vanish identically when

t/q 0 J, such that

MQ(t,x) =

(7.109)

(-1)*

P

sgn((ai...... ai.))exp(2i?r(Av,t/q))

"_lai),xl)

d r

j.l 1

{E`l 2q tank (ai, 29 -xt )J ,.

ff 1

(x + 2iag9t - 2iiaAµ)

1

(z - 2i7rq5A,) +

I

E wi(t,xl,... ,xi-1, xi+1,... ,xl). i=1

PROOF. Using Proposition 7.5 and Theorem 7.25, we get (7.109).

Given (w', ... , we) E W', let 711"1.... a , ... , r); I....." be s functions from R/ddx {I, I generic) into {+, -} having the properties listed before Theorem 7.25.

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

THEOREM 7.28. If q E N*, g > 2, then V gQ(B1,..., 98)=17

(7.110)

VI(T)2,-2

JZ(G)' (g-1)r

,W, q

ICRI

(- 1)e(g+1)+r

1

+e(+1+)E=v (ail, ... , ai, )

uE

Resy=IO

1

11

aeR+ 2 sinh (2 (ai, x! /q + 2ilru)) Ew. exp

((t w1P xl /q)

+2ilr(EUr (P+0')+qf,u) exp

-laii'x!)

Cdr

1

d

j=1

j

Wkk + f)' el)]

-1 k=1

q

1

nr9=1 (exp (..i:aiz1) (pi_lati,ai PROOF. We use (7.79) in Theorem 7.20 and 7.27, and also the arguments in the proof of Theorem 7.23. We will show that for 1 < i < 1, (7.111) a

1

Resy'=61

e

[

Wwi(xlq)Wi(x1e... ,xi-1, xi+1e... xt) _0

2 sinh2q()J 2g-2 j=1 11

1

from which (7.110) follows. To establish (7.111), we only need to show that

(7.112)

7-

1

Res.,--o

(2Q)]2g

[2sinh

-2 1l'Pwi 7_=1 (x/q) = 0 .

The proof of (7.112) is the same as the proof of (7.84). We have thus established our Theorem.

7.7. The general case. We will now establish another form of Theorem 7.28.

'I

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

301

THEOREM 7.29. If g > 0, if 8 is even and if 2g - 2 + s > 1, the following identity holds (7.113)

Vg,q(01, ...

Vo1(T)2g-s

, 8.) = I CR I

l (G)I (g-1)*(-1)t(s+1)+r 1

1

DEC/ 1-(il

Ie

Res

)Ex., /dw

(ail,...

ER, 2 sinh (a (ai, xI /q + 2inu))

E

fwi ... ew exp(2aa(> wI (P + BI) + qf, u)) j=1

r

exp

(d E;= ( -,ae , ) a(PJI-1( wk(p ek) +f) 8

'

?

k=1

l

eJ)J

1

j=1 (exp (/'1:.1;>) PROOF. We proceed as in the proof of Theorems 7.23 and 7.28. Instead of (7.79) in Theorem 7.20, we use (7.80) and we still use Theorem 7.27. In particular the obvious analogues of (7.93) and (7.97) still hold because s is even and 2g-2+s > 1. For the same reason, the analogue of (7.111), (7.112), with the owj replaced by 1 still holds. The proof of our Theorem is completed.

JEAN-MICHEL BISMUT AND FRANQOIS LABOUR.IE

302

8. The Verlinde formulas In this Section, we prove the Verlinde formulas, in the restricted sense which was described in the Introduction. Namely, we show essentially that for p large enough, the R.iemann-Roch number of M/G is given by the Verlinde formulas. Also we show that the Riemann-Roch numbers of suitable perturbations of M/G are given by the Verlinde formulas. This Section is organized as follows. In Section 8.1, we establish our results under a suitable genericity assumption on the holonomies. In Sections 8.2 and 8.3, we consider suitable perturbations of the moduli space.

8.1. The generic case. Here we will assume that s > 1, 2g - 2 + s > 1, that tI, ... , t, are regular and lie in the alcove P, and that (t1,... , t,) verify assumption of Section 6.9. Namely we assume that for any (w',... , w') E W', E9_1 wjtj SS.

Recall that M C Z was defined in (6.20). Here we assume that M is not reduced to 0.

THEOREM 8.1. For p E M, p > 0 large enough,

(8.1)

Ind(DD,+) = V9,n+o(ptl,... ,pt,).

PROOF. By Theorem 6.29 and by Theorem 7.12, we know that if pti ¢ R, then both sides of (8.1) are equal to 0. So we may as well assume that pt1 E R. First we consider the case where g > 2. Observe that for p E M large enough,

(ptl,... ,pt.) verify (A,+.). By comparing formula (6.130) in Theorem 6.34 and formula (7.91) in Theorem 7.23, we get (8.1). Now we consider the case g = 1. By Theorem 7.16, we get

(i)t I

'

(p F.)m

E (,I,,2)EW2

2,p+c(?rtl,...

eviCV2VyjFK

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

303

Observe that for p large enough, (ptl,... , pta, vlp + v2 p) verify (A,+,). Using (1.17), Theorem 7.23 with g = 2 and (8.2), we obtain

Vi,p+c(pti,... pts) _ IWI)I(-1)''

(8.3)

1

UEC/r r°c.i,)ex. (oil I... ,ail) Iek/dR.pe

/k'

s+3

I

1

x=0

.ER+ 2sinh (2(a,xl/4+2ilru))

exp(8+1

wp,xl/(p+c))+2i1r(L, ra

j=1

a+s

w1 11tj1(P+c)+

j=1 s+3

8

+2iir(twi(p+ptj)+ j=1

vrip+(p+c)f,u) j=sId +1

(P! la=/,xI) i(pf-1( j=1 (Pj-1aiei)

Sexp (dE l

j=s+2

l

vppl(p+c),ar,)

/

wkptk/(p+c)+

l

s+3

k=s+2

----- (d:I)) l 1

n31

Ewe ... f,s.+a

(w= ...,

ex

(x - 2iira.\,).

-1J

Since (t1,... ,t8) verify (A), by Proposition 2.14,

(pjI-lEwktk+f,ej) f Z. k=1

By proceeding as in (6.132) and using (8.4), we get for p E M large enough, ,+3

s

(8.5)

[a(pi-1(Ewkptkl(p+c)+ E wkP/(p+c)+f),ei)] = k=s+2

k=1

s+3

s

v>kp/(1>+c),e')

-1 k=s+2

k=1 Also by (2.13),

*+3 4111

(8.6)

j=1

-1aixI)!P

j-1ai" ej)

s+3

wkpl(p+c),e?) _ (E wkp/(p+c),XI). k=s+2

k=s+2

304

JEAN-MICHEL BISMUT AND FEAN¢OIS LABOUR.IE

So by (8.3), (8.5), (8.6), we get

(8.7)Vi,p+o(ptl,... pte) = IZ(0) (-1)r 1

.)ex.. +-(i' lexA.F *Gux*/x'

,+E

(ail, ... , ai. ) 8+3 1

Res 0

Ewi ...8,,,+s

aER+ 2sinh ((a,x'/(p+c)+2iiru)) d+3

d

exp & urJ p,xl/(p+c)) + E wjP'

(w' ,...,+')EW

+3

I

x

p+c+2i'ru)+

( J=d+1

.i=l

2ia(Ew'ptj/(p+c),A)+2i7r(>w''(p+pt')+(p+c)f,u)) 3=l

.i=1

lexp

'PI-1""'x j=l (-la=,'e'') Ld 1

llj=1 Cexp

(

k=I

11

/.,

(x - 2i'raa.1µ) .

pf_la: ,sf)

-1)

Now by (1.94), (7.71),

(8.8)

xI

1 77-- rr

C+2i'ru))wEE,u

p((w/,p+c+2i'ru))=1.

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305

By (8.7), (8.8), we obtain (8.9)

vl,p+c

neC/R

I,... ,pt,) = I In(-1)r

1E7F/dA.PEZ°r /3F

(ail, 1 , a. 8

I

1

a=0

exp

fw,...ew.

aeR+ 2sinh (2(a,x'/(p+c) +2iau))

iw,,..EiEW,

((t wjp,x'/(p+c)) +2iir(Ewjptj/(p+c),As,) j=t

j=1

+2iir(E wj (p + ptl) + (P + c) f, u)

j=I

fe,p (d

j=1

J

(pj-t

ptk

-t

d

1

c

+f,)

}(x-2itrnal,).

r

j=1 (exp 1\ (pl_,

JJJ

1

By comparing (6.130) and (8.9), we get (8.1) for g = 1. A similar proof can be given for the case g = 0. The proof of our Theorem is 0 completed.

8.2. The perturbed case at P+C . Now we make the same assumptions as in Section 6.11. Namely take p E M, p > 0. We assume that e1 i ... , ed E t s, A + tej E T is regular, and for any

are such that form I E]0,1], 1 < (wt, ... , w5) E W', I E]0,1],

wj P+C + 8ej

THEOREM 8.2. For g > 2, (8.10)

Ind(Dy+ +t`;i)

,Pt8)

For g = 0 or g = 1, if (t1, ... , t,) verify (a), for p E M and p > 0 large enough, equation (8.10) still holds. PROOF. By Remark 6.37, we find that if Ej8=1 ptj ¢ R, then the left-hand side of (8.10) vanishes. By (7.59) in Theorem 7.12, the right hand side also vanishes. As in the proof of Theorem 8.1, we may and we will assume that E3=1 ptj E R. First we consider the case g > 2. Then (8.10) follows by comparing (6.147) in Theorem 6.41, and (7.110) in Theorem 7.28. In the case 0 < g < 1, we proceed as in the proof of Theorem 8.1, and use (8.10) in the case g > 2. 0 The proof of our Theorem is completed.

308

JEAN-MICHEL BISMUT AND FRANQOIS LABOURIE

8.3. The perturbed case at et2 E P, i.e.

P+C

. Observe that if t E P, for p E N,

is regular. This follows follows from (1.33), (1.34), (1.36).

The same argument as in the proof of Proposition 6.39 shows that if s > 1, given p > O,p E M, there exist el, ... , es E t such that for l E]O,1),1 < j < s, +Iei E T is regular, and for any (wl, ... , w°) E W, I E]O,1], Ep=1 wi ("Is +

lei)

In Theorem 6.36 we now take

(8.11)

6i= P

c3+lei.

By the construction of Section 6.11, we get a Dirac operator Dp. THEOREM 8.3. For g > 0, and 2g - 2 + s > 1, for l E]0,1], (8.12)

(° Ind(Dp,+

+1") )

=V9,P+C(Pti,... 'Pt.).

PROOF. As in the proof of Theorems 8.1 and 8.2, we may restrict ourselves to the case where Ejj_1 pti E R. First assume that s is even. Then (8.12) follows from (6.137) in Theorem 6.36, with 6i given by (8.11), and from (7.113) in Theorem

7.29. When s is odd, first we perturb the a holonomies as indicated before. Then we add an extra marked point x,+1, with holonomy equal to 1. We perturb the holonomy 1 to a generic holonomy, and we use the result we just proved with s + 1 marked points. We can then gently make our holonomy t,+1 tend to 1. This is in fact possible because the perturbation of the first s holonomies are generic and verify condition R. The proof of our Theorem is completed. REMARK 8.4. Suppose temporarily that the assumptions of Remark 6.42 are satisfied. In particular we assume the holonomy t, to be central and represented by h0 E P. A direct treatment would show that an analogue of (8.12) would still hold, where, in the Verlinde sum (7.57), the term Xph,(e"/(p+°) should apparently be replaced by ewh, exp(2ilr(h0, A)). However Theorem 1.33 tells us that indeed, this is just Xph, (ea/ 0), so that we recover the standard Verlinde formula. REMARK 8.5. By Teleman's vanishing results [55, 56], if (t1, ... , t,) verify (A), for any p E M, one should have Ind(D,,+) = dimH°(M, )L'). (8.13)

Since dim H°(M, A") is given by Verlinde's formula, then one should have for any

pEM, Ind(Dp,+) = Vg,p+.(ptI, ... ,pt,). The arguments of Teleman also show that more generally, if dl, ... , 6, are taken as in Theorem 6.36, for any p E M, (8.14)

(8.15)

Ind(DP+6't) = Vg,p+0(Ptl,

,pt,)

Here, Theorem 8.1 only asserts that if (t,,... , t,) verify (A), for p large enough, (8.13) holds. For a given p E M, Theorems 8.2 and 8.3 give a corresponding equality

for a suitable perturbation of the moduli space, depending on p, the perturbation being smaller as p -1 +oo. A proof of (8.14), (8.15) for any p E M would be possible if one could modify Theorem 7.20, so that in (7.79) or (7.80), the term where the

function M. appears would be instead M. (P E j=, w195, x). To establish such a

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

307

result, one would need to prove the vanishing of a certain residue. This vanishing result is true for 9 large enough, but not obvious for small values of g.

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308

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[43] Meinrenken E., Woodward C. : A symplectic proof of Verlinde factorization. dgga/9612018 (1996). (44] Meinrealcen E., Woodward C.: Fusion of Hamiltonian loop group manifolds and cobor-

dism. Preprint, July 1997. [45] Milnor J.M., Stasheff J.D.: Characteristic classes. Annals of Mathematics Studies 76. Princeton: Princeton University Press 1974. [46] Narasimhan M.S., Seshadri C.S. : Stable and unitary vector bundles on a compact Riemann surface. Ann. Math., 82, 540-567 (1965). (47) Pressley A., Segal G. : Loop groups. Oxford Mathematical Monographs. Oxford: Clarendon Press 1986. [48) Ramadas T.R., Singer I.M., Weitzman J.: Some comments on Chern-Simons theory. Comm. Math. Phys. 126, 409-420 (1989). [49] Reidemeister K.: Homotopieringe and Linsenraume. Hamburger Abhandl.,11,102-109 (1935).

(50] Rossmann W.: Kirillov's character formula for reductive Lie groups. Invent. Math., 48, 207-220 (1978). [51] Serre J.P.: Cours d'arithmdtique. Paris: Presses Universitaires de France 1970. [52] Szenes A.: Hilbert polynomials of moduli spaces of rank 2 vector bundles. I. Topology, 32, 587-597 (1993). [53] Szenes A.: The combinatorics of the Verlinde formula. In Vector bundles in Algebraic geometry, 241-263. LMS Lecture Notes Series 208. Cambridge: Cambridge University Press 1995. [54] Szenes A.: Iterated residues and multiple Bernoulli polynomials. Preprint MIT 1998. (55] Teleman C.: Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve. Invent. Math., 134, 1-57 (1998). [56] Teleman C.: The quantization conjecture revisited. 4-geom/9808029 v2 (1998). [57] Thaddeus M.: Stable pairs, linear systems and the Verlinde formula. Inv. Math. 117, 317-353 (1994).

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(64] Witten E.: Two dimensional gauge theories revisited. J. Geom. Phys. 9, 303-368 (1992).

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DiPARTEMENT DE MATHEMATIQUE, UNIVERSITE PARTS-SUD, BATIMENT 425, 91405 ORSAY, FRANCE E-mail address: bismutAtopo.Math .u-paud.tr

DFPARTEMENT DE MATHEMATIQUE, UNIVERSITE PARIS-SUD, B9TIMENT 425, 91405 ORSAY,

FRANCE B-mail address: labourisdtopo.math.n-pand.tr

Surveys in Differential Geometry, vol. 6

Counting curves on irrational surfaces Jim Bryan Naichung Conan Leung

ABSTRACT. In this paper we survey recent results and conjectures concerning enumeration problems on irrational surfaces.

CONTENTS

Introduction Counting curves via the method of Yau and Zaslow 3. Problems and conjectures 4. Gromov-Witten invariants 5. Modified invariants and the case of K3 and Abelian surfaces 1.

2.

6.

Acknowledgments

References

313 314 316 319 325 335 335

1. Introduction Problems in enumerative algebraic geometry associated to counting curves on projective varieties have been heavily influenced in recent years by the introduction of stable maps, Gromov-Witten invariants, and quantum cohomology. Using these

new ideas as well as classical methods, much progress has been made for many enumeration problems on rational surfaces. In this survey paper, we focus on the situation for irrational surfaces. The first breakthrough in this direction was the work of Yau and Zaslow [91] in 1995 which gave a formula for the number of rational curves on a K3 surface. Their formula expresses a generating function for the number of curves as a modular form. Their method is motivated from mirror symmetry considerations and it is strikingly different both the "classical" algebro-geometric approach and the methods using quantum cohomology or Gromov-Witten invariants. Methods from quantum cohomology and Gromov-Witten theory are not very useful in solving enumeration problems on most irrational surfaces. For instance, a generic K3 surface has no curves at all and so its ordinary Gromov-Witten invariants 01999 International Press

313

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Bryan and Leung

are zero. Nevertheless, some modifications of Gromov-Witten theory can be made and have been recently applied with success in the case of K3 and Abelian surfaces. In [34], GSttsche gave an intriguing generalization of the Yau-Zaslow formula which conjecturally applies to any surface and any genus. The Gottsche-YauZaslow formula has been verified to order eight by the work of Vainsencher [82] and Kleiman-Piene [51] and it has also been proved in the case of K3 and Abelian surfaces [15][17]. The conjecture is also consistant with various recursive computations on rational surfaces [18][83]. The Gottsche-Yau-Zaslow formula conjecturally provides a very nice answer to a fairly general set of enumeration problems. Our survey is organized as follows. We begin in Section 2 by describing the method of counting rational curves due to Yau and Zaslow which led to their discovery of the presence of modular forms in enumeration problems on surfaces. In Section 3 we formulate the various kinds of enumeration problems on surfaces and then we focus on the primary problem of interest: counting curves in a linear system passing through a fixed number of points. We describe in detail Gottsche's generalization of the Yau-Zaslow formula. In Section 4 we start with a short general exposition of the Gromov-Witten invariants including a general enough version to include families of symplectic structures. We then discuss the problems with the Gromov-Witten invariants on surfaces whose geometric genus and/or irregularity is non-zero. We discuss Taubes' "Seiberg-Witten equals Gromov-Witten" theorem and describe its relation to enumerative geometry. Finally, we discuss when the Gromov-Witten invariants are "enumerative". In Section 5 we give an expository account of our computation of modified Gromov-Witten invariants to prove the GSttsche-Yau-Zaslow formula for K3 and Abelian surfaces. We include a description of how to use a "matching technique" to compute the contribution of multiple covers of nodal rational curves to the invariants. We end the section with a brief description of recent work of Behrend and Fantechi who give a purely algebraic modification of the Gromov-Witten invariants that generalize the (non-algebraic) modifications used in the K3 and Abelian surface case.

2. Counting curves via the method of Yau and Zaslow The first big breakthrough for counting curves on irrational surfaces came in the 1995 paper of Yau and Zaslow [91] who discovered the unexpected link between modular forms and enumerating curves. In [91], Yau and Zaslow describe a method to count rational curves on K3 surfaces (for an exposition see Beauville [8]). They found that the numbers are given by the coefficients of the series

- (I_ 9m)-24

700

(1)

0q9) _ II

They prove this formula under the assumption that all the rational curves are reduced, irreducible, and nodal. More generally, if the curves are all irreducible and reduced (but possibly having complicated singularities), their argument counts

Counting curves on irrational surfaces

315

curves with multiplicities that are shown by Fantechi-Gottsche-van Straten [28) to be related to Gromov-Witten multiplicities (see also Chen [19]). The formula was later generalized by Gottsche [34] to a conjectural formula that applies to all surfaces and any genus (see Section 3). Although it is not clear if Yau and Zaslow's argument can be generalizedto other situations, it is so beautiful and strikingly different that we feel it is worthwhile to describe it here. Let pr be an r-dimensional linear system on a surface X with a finite number

of rational curves (for example P' could be the sublinear system of a complete linear system Iobtained Cl by imposing the appropriate number of point conditions). Consider the compactified family of Jacobians:

a:9 -3 P' so that tr''(p) is Jac(C) if p is a point representing a smooth divisor C and if C is singular then 7r'' (p) is Jac(C), a compactification of the Jacobian of C (such a family exists by [4][3], c.f. [1][2]). Yau and Zaslow show that if we assume all the rational curves in the linear system are nodal, then

# of rational curves in the linear system P' = Euler characteristic(,?). The crucial observation is that the fibers Jac(C) have Euler characteristic zero unless C is rational'. We denote the Euler characteristic of a space M by e(M). Recall that the Euler characteristic of a fiber bundle is the product of the Euler characteristics of the fiber and the base and if X = U U U` is a disjoint union of an algebraic space

X into a Zariski open set U and its compliment, then e(X) = e(U)+e(Ue). The linear system Pr has a natural stratification given by the geometric genus of the corresponding divisor. The map a : ,7 -+ P' is a fiber bundle restricted to each strata so that e(J) is given as the sum over all the strata of the product of the Euler characteristic of the fiber times the Euler characteristic of the corresponding strata. We see that only the strata corresponding to rational curves contribute to the Euler characteristic. Consequently, we get

e(J) =

E

e Jac(C)).

rational curves C in P"

If C is a nodal rational curve, then e(.7ac(C)) = 1 and so e(J) is exactly the number of rational curves. In general, e(J) counts the rational curves with multiplicities given by e(J-ac(C)). Fantechi-Gottache-van Straten [26] show that if C is irreducible, then e(Jac(C)) coincides with the length of the (zero dimensional) moduli space of genus 0 stable maps with image C and so the Yau-Zaslow multiplicities agree with the multiplicities arising in Gromov-Witten theory (in the language of Subsection 4.4 we can thus say e(J) is weakly enumerative).

In general, there is no easy way to compute e(9), but for K3 surfaces we can utilize some very special properties of J. A r-dimensional complete linear system ICI on a K3 surface X has a finite number of rational curves. If we assume that the linear system consists of only reduced and irreducible curves (for example, if O(C) generates Pic(X)), then the associated family of compactified Jacobians 'This is because T(C) fibers over Jac(C), the Jacobian of the normalization and e(Jac(C)) _ 0 if g(C) 0 0, i.e. C is not rational (see section 2 in Beauville [81).

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Bryan and Leung

.7 -3 ICI is a smooth hyperkdhler manifold (see Mukai [68] ex. 0.5). Furthermore,

it is birational2 to X [r], The Hilbert scheme of r points on X. X [r) is also a smooth hyperknhler manifold and by a result of Batyrev (7), the Betti numbers (and hence the Euler characteristics) of two birationally equivalent, smooth hyperkahler manifolds agree. The Euler characteristic of the Hilbert scheme X[rl was determined in [35] by Gottsche using Deligne's proof of the Well conjectures. Equation 1 then follows from his calculation.

3. Problems and conjectures In this section we describe the general set up for enumeration problems on surfaces and we explain the Gottsche-Yau-Zaslow formula which conjecturally gives

the answer to a very general set of enumerative problems. The discussion of this section for the most part applies equally well to rational surfaces as well as irrational.

3.1. Formulation of the problem. Since the only interesting subvarieties of a fixed algebraic surface X are curves, the general enumeration problem for X is to count the number of curves on X satisfying some set of prescribed properties. It is natural to begin by fixing the geometric genus g of the curves to be counted and to fix the homology class [C] E H2 (X, Z) of their image3. The set of curves with fixed geometric genus and homology class will in general form a positive dimensional family and so to get a well defined counting problem one imposes additional conditions. Some typical conditions on a curve C are given below:

Point: Require C to pass through a prescribed set of fixed points. The condition that a curve pass through a single fixed point is a codimension one condition on the family of all curves. FLS: Require C to He in a fixed linear system (FLS), by considering C as a divisor.

Equivalently, fix the holomorphic structure on the line bundle O(C). For irregular surfaces (H1(X,O) 54 0), this imposes a non-trivial constraint of codimension equal to dim HI (X, 0). Loop: Require C to pass through fixed loops in X representing non-trivial elements in H'(X; Z) (this also is a non-trivial constraint only on irregular surfaces). Each loop imposes a real codimension one condition on the family of curves. In Section 4 we will show that these loop constraints are directly related to the FLS constraint (see Theorem 4.1).

Multi-point: Require C to pass through a fixed set of points with a prescribed set of multiplicities. This can be reformulated as a homological condition on the blown-up manifold; i.e. curves on X in the class [C] passing through fixed 2The birational morphism between .7 and X[rl can be seen as follows. The generic point of 9 corresponds to a smooth genus r curve C C X and an element of Jac(C). Since Jac(C) is generically isomorphic to Sym'(C) the generic point in .7 gives us r (unordered) points on C and hence r points in X. Conversely, r generic points in X are contained in a unique smooth genus r curve C in JCJ; the points on C then further determine an element of Jac(C) via the map

Sym'(C) -i Jac(C). 3Notation: if C is a curve in X, then we denote its homology class in Ha(X,Z) by [C], its Poincard dual In H2(X,Z) by [C]v, the corresponding line bundle O(C), and the linear system ICJ := P(H°(X,O(C))). We will also use the shorthand C2 := [C] [C] and KC := c1(TX)QCJ). In subscripts for moduli spaces and invariants we will just write C for the homology class, e.g. Ms,e(X) := Ms.[Cl(X)

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points xi., ... , xi with prescribed multiplicities ai are in one-to-one correspondence with curves on 131{:,,...,:,}X in the class [C] - E;_, ai[Ei] where Bl{sI .. x,1(X) is the blow-up of X at the points xl,... , xi and El,... , Ea are the the exceptional divisors. Tangency: Fix an auxiliary smooth curve D and require that C meet D with a prescribed degree of tangency. This can be reformulated in terms of curves on a certain family of blow-ups (see [16]). See also [24][25] where the problem was considered on P2 and attacked using certain stable maps to the incidence variety. CX structure: One can also impose conditions on the complex structure of C itself. For example, one can count only curves with a fixed complex structure; for g > 2, this imposes 3g-3 constraints (for example see [42],[69]). For another example, one can count only those curves that are hyper-elliptic. The expected

number of constraints this imposes is g - 2 since the codimension of the hyper-elliptic locus in the moduli space of curves is g - 2 (for example see [37]).

For the most part, we will focus on the most straight forward problem: counting genus g curves in a fixed linear system passing through a fixed set of points (with multiplicity one), although we will comment on the other kinds of constraints as they come up. We formalize the problem below: Let C be a curve on a surface X and let ICI denote the corresponding complete

linear system and let K denote the canonical class. We assume that O(C - K) is ample so that the dimension of ]C] is determined by the Riemann-Roch formula. Define the Severi variety V9(C) to be the closure of the set of curves in 101 with geometric genus g. The condition that a divisor passes through a fixed point imposes a linear condition on ]C[. We can thus interpret the degree of V9(C) C ICI

as the number of genus g curves in ]CI passing through r points where r is the expected dimension of V. (C). The expected dimension of V. (C) is given by

dimV9(C) = r

_ -KC+g-1+p9-q -KC+g-2+X(Ox) where p9 = dim Ha (X, O), q = dim Hl (X, O), and x(Ox) = 1 - q + p9 is the holomorphic Euler characteristic of Ox. Main Enumeration Problem: The main problem we consider is computing N9(X,C) which is defined to be the number (when finite) of genus g curves in the linear system ICI passing through r = -KC+g-1+p9 - q generic points. Equivalently, N9(X,C) is the degree of 119(C) (when V9(C) is of the expected dimension).

3.2. The Gottsche-Yau-Zaslow formula. A priori, N9 (X, C) depends heavily on the complex structure of X and the choice of the linear system 1C1. Under sufficient ampleness conditions on C it is conjectured (essentially in Vainsencher

[82]) that N9(X,C) only depends on the numbers g, C2, CK, K2, and c2(X), where c2(X) is the Euler characteristic of X. Gottsche has found a remarkable generalization of the Yau-Zaslow formula. The formula is a generating function for the numbers N9(X, C) in terms of five universal

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power series and the numbers g, C2, CK, K2, and c2(X). Three of the five universal power series are describes explicitly as (quasi-)modular forms while the remaining two series have coefficients that can be determined recursively.

CONJECTURE 1 ([34]). Let C be a sufficiently ample' divisor on X. Then Ns(X,C), the number of genus g curves in ICI passing through

r = -KC + g - 2 + X(Ox) points, is given as the coefficient of q}C(C-K) in the following power series in q: z

B1K'B2CK (DG2)r(A(D2GzGX(ox)/z

(2)

where D = q7, Gz is the Bisenstein series: Ga(q) = -24 +

d)gk, k>O dlk

A is the discriminant:

A(q)=gII (1-q')24 k>O

and B; (q) are universal power series whose first terms are B1(q)

=

Ba(q) =

1 - q - 5qz + 3983 - 34584 + 1 + 5q + 2qz + 3583 - 140g4 + ..

(see [34] for the coefficients of Bi to order 20).

Note that when K is numerically trivial (X is a K3, Abelian, Enriques, or hyperelliptic surface), then the power series (2) does not depend on B1 or B2 and so is given by an explicit (quasi-)modular form. In the case when X is a K3 or Abelian surface, the conjecture was proved in [15] and [17] to hold for all C representing a primitive homology class (using a slightly modified definition of Ng (X, C), see section 5). The impetus for Gottsche's generalization stemmed largely from two sources: the work of Vainsencher [82] (and its subsequent generalization by Kleiman and Piene [51]), and the formula of Yau and Zaslow [91]. In [82], Vainsencher gives universal polynomials that count the number of curves

with 6 or fewer nodes, passing through the appropriate number of points, in a sufficiently ample linear system on any surface. In other words, he computes Ns (X, C) for any g, X, and C provided that C is sufficiently ample and

2

(C+K)-g<5.

Kleiman and Piene have refine the methods of Vainsencher to extend his results up to eight nodes. They also provide explicit bounds for the power of the ample class

required to guarantee that the formulas hold. The methods used to obtain these results are classical in the sense that they do not use physics, quantum cohomology, or even stable maps. They also provide precise enumerative information as oppose to 4For the precise formulation of the ampleness condition see [34]. Depending on X, the conjecture is expected to hold under weaker assumptions on C.

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the "virtual" or "weakly enumerative" count sometimes determined by the Gromov-

Witten invariants (c.f. subsection 4.4), in particular, they show that when the ampleness conditions are satisfied, all the curves passing through a generic choice of points are nodal. We refer the reader to [82] and [51] for more details. Gottsche's crucial observation is that if there exist universal polynomial formulas for N9 (.X, C) that apply to any surface, then they must satisfy very strong multiplicative properties. The reason is that they should also apply to disconnected surfaces, and Gottsche shows that the obvious relationship N9(X1 H X2, C1 JJC2)= E N91 (X1, C1)N9a(X2, C2) 91+92=9

forces the formulas to take on a very special form. In particular, the polynomials for N9 (X, C) must be determined by five universal power series. On the other hand, it was the work of Yau and Zaslow that suggested that the universal power series may be related to modular forms. Using ideas from physics, Yau and Zaslow gave predictions for the number of rational curves with n nodes on a K3 surface (see Section 2). The numbers appear as the coefficients of the Fourier expansion of a well known modular form, the discriminant. This work is in a sense complimentary to the Vainsencher work; while Vainsencher's formulas apply to any surface but for only a small number of nodes, the Yau-Zaslow formula applies to only genus 0 and a K3 surface, but to any number of nodes. Thus, while Vainsencher's formulas determine the first few coefficients of each of the five power series, the Yau-Zaslow formula provides a closed form for a certain product of three of the power series. By building on this knowledge along with other known results (particularly the recursive formulas of Caporaso-Harris [18] and Vakil [83]) and using some remarkable pattern recognition, Gottsche arrived at Conjecture I and verified it to a fairly high degree of redundancy. Closed formulas for the series B1 and B2 are unknown, but a recursive scheme for the coefficients can be derived from the Caporaso-Harris or Vakil formulas.

One intriguing aspect of the conjecture is the appearance of modular forms. The underlying "reason" for the modularity is currently a mystery.

4. Gromov-Witten invariants In this section we begin with a short general exposition of the Gromov-Witten invariants including a general enough version to include families of symplectic structures. We then discuss the problems with the Gromov-Witten invariants on surfaces whose geometric genus and/or irregularity is non-zero. We discuss Taubes' "Seiberg-Witten equals Gromov-Witten" theorem and describe its relation to enumerative geometry. Finally, we discuss when the Gromov-Witten invariants are "enumerative".

Many papers have been written on Gromov-Witten theory, for the reader's convenience we give an extensive list in the bibliography: [63] [61] [62] [43] [12] [6] [73] (85) [691 [50] [52) [87] [67] [40] [30] [ill [65] [27] [89) (49] [21] [9] [33] [66] [39] [86] [14] [57] [48] [46] [70] [54) [32] [29) [74] [231 [76] [581 (561 [38] [311 [751 [20] [13] [47] [45] [41] [53] [64] [72] [711 [55].

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4.1. Gromov-Witten invariants. Gromov-Witten invariants have their origins in symplectic geometry and conformal field theory but have been recently defined purely algebro-geometrically (11][62]. The basic object of study is the moduli space M9,n,c(X) of stable maps of n-marked, genus g curves to X in the class C. In general, Gromov-Witten invariants are certain intersection numbers of cycles on M9,a,c(X) which are shown to be invariant under deformations of the (almost) Kiihler structure of X. We briefly outline the framework of the Gromov-Witten invariants in order to fix notations and we refer the reader to (for example) [5] or [28) for complete accounts. We include here a straight forward generalization of the usual framework to include families of (almost) Kahler structures (see [15][17]). Let (X, w) be any compact (almost) KAhler manifold. Recall that an n-marked, genus g stable map of degree [C] E H2 (X, Z) is a (pseudo)-holomorphic map f

E -> X from an n-marked nodal curve (E, z1, ... , xn) of geometric genus g to X with f.([E]) = [C] that has no infinitesimal automorphisms. Two stable maps f : E -> X and f': E' -+ X are equivalent if there is a biholomorphism h : E -+ E' such that f = f' o h. We write M9,n,c(X, w) for the moduli space of equivalence classes of genus g, n-marked, stable maps of degree [C) to X. We will often drop the w or X from the notation if they are understood and we sometimes will drop the n from the notation when it is 0. If B is a family of almost Kiihler structures, we denote parameterized version of the moduli space: Mg,n,c(X, B) =

J1 Mg,n,c(X, kit)-

tEB

If B is a compact, connected, oriented manifold then M9,n,C(X, B) has a fiduciary cycle [M9,n,c(X, B)]°ir called the virtual fundamental cycle (see [15] and the fundamental papers of Li and Tian [62][59][60], or alternatively Behrend-Fantechi and Siebert [11][77]). The dimension of the cycle is

dimR[M9,,,,c(X, B)]vir = -2KC + (6 - dimR X)(g -1) + 2n + dims B. The invariants are defined by evaluating cohomology classes of M9,n,c on the virtual fundamental cycle. The cohomology classes are defined via incidence relations of the maps with cycles in X. The framework is as follows. There are maps

M,,1,c =v + X 1ft

M9,c called the evaluation and forgetful maps defined by ev({ f : (E, xl) -+ X}) = f (xi )

and f t({ f : (E, xl) -+ X J) = if : E -+ X).5 The diagram should be regarded as the universal map over M,,c Given geometric cycles al, ... , ai in X representing classes [al],... , [ai] E H, (X, Z) with Poincare duals [al ]", ... , [at]", we can define the Gromov-Witten 5There is some subtlety to making this definition rigorous since forgetting the point may make a stable map unstable, but it can be done.

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invariant (3)

§g"") (a1,... ,at) =

J lM..c(x,B)]v:,

ft.ev*([a1]v) U ... U ft.ev'([at]v)

(x B) a

a counts the number of genus degree C maps which are seudoholomorphic with respect to some almost Kahler structure in B and such that the image of the map intersects each of the cycles a1, ... , al.' The Gromov-Witten invariants are multi-linear in the a's and they are symmetric for a's of even degree and skew symmetric for a's of odd degree. If pi,... , pk are points in a pathconnected X, we will use the shorthand (x,B)k (pt. , aktl, ... , a{) §9,C §(X' U+1, ... Pk, ak+I- - , at). Now suppose that X is a Kahler surface. The only non-trivial constraints arising from intersecting with cycles are those coming from zero and one dimensional cycles, i.e. points and loops. The constraints imposed by intersecting two dimensional cycles (divisors) are determined purely homologically (the so-called "divisor equation") and cycles of dimension three and four impose no constraints. The loop constraints (C passes through a fixed set of loops) are related to the FLS constraint (fixing the linear system of C) by the following: THEOREM 4.1 (Thm. 2.1 of [17]). Let y', ... , ryb, be loops representing an ori-

ented basis of H1(X,Z). Then the invariant: 9,C(71,... ,'Ybl, counts the number of genus g maps whose image lies in the fixed linear system ICI and passes through I points.

We can formulate a more precise version of this. In order to count curves in a fixed linear system ICI one would like to restrict the integral of Equation 3 to the cycle defined by 'E-o'u (0) where 'FEo is the map

TEo : ,M9,C(X,w) -+ Pic°(X) given by

f -* d(Im(f) - Eo) where Eo E ICI is a fixed divisor. Dually, one can add the pullback by 'Ee of the volume form on Pic°(X) to the integrand defining the invariant: 'pso ([pt.]') U ft.ev'([a1] ') U ... U ft.ev`([at]v).

The class 'IEo([pt.]v) can be expressed in terms of classes arising from the constraints imposed by loops: THEOREM 4.2. Let X be a KOhler surface and let [y] E H1 (X, Z) and letry be the corresponding class in H1(Pic°(X),Z) induced by the identification Pic°(X) 5M

H1(X,R)/H1(X,Z). Then TEo(7) = ft.ev`(['Y]v)

COROLLARY 4.3. Fj,([it.]v) = ft.ev'([71]V) U - U ft.ev*([ryb1]v) 6The integral is defined to be 0 if the integrand is not a form of the correct degree.

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PROOF: Theorem 4.2 is proved in the appendix of [17]. Roughly the idea is this: We view classes in HI(M9,C(X);Z) as homotopy classes of circle valued functions on M9,c(X). The values of a certain circle valued function on M9,c(X) representing the class f t.ev`([y]") are given as integrals of a form representing [y]" over 3-cycles that are obtained by sweeping out a path of curves in X. On the other hand, it is shown that the values of a circle valued function representing 'I' (y-) are given by an integral over X of a certain 1-form wedged with [y]". The equality of the two circle valued functions is then established with essentially a residue calculation. The corollary follows immediately and is essentially a restatement of Theorem 4.1.

4.2. Difficulties with ordinary Gromov-Witten invariants. Gromov-Witten invariants have been remarkably effective in answering many questions in enumerative geometry for rational surfaces (see for example [84][83][36][56][24][37] and many others). Rational surfaces all have p9 = q = 0; however, the ordinary GromovWitten invariants are not very effective for counting curves when p9 or q are not zero7. One basic reason is that the moduli space of stable maps fails to be a good model for a linear system (and the corresponding Severi varieties) for dimensional

reasons: For an effective divisor C such that C - K is ample, the dimension of the Severi variety V9(C) (the closure of the set of geometric genus g curves in the complete linear system ICI) is

dimcV9(C)=-KC+g-1+p9-q. On the other hand, the virtual dimension of the moduli space Mg,C(X) of stable maps of genus g in the class [C] is

virdimc M9,c(X) = -KC + g -1. If the virtual dimension of the stable maps doesn't match the number of constraints required for Ng(X, C), the corresponding Gromov-Witten invariant must be zero.

The discrepancy p9 - q arises from two sources. Since the image of maps in M9,c(X) are divisors not only in ICS but also potentially in every linear system in Picl°t (X ), one would expect dim M9,c (X) to exceed dim V9 (C) by q = dim PiclCl (X) 8 As we discussed in the previous section, this discrepancy can be accounted for within the framework of the usual Gromov-Witten invariants using loop constraints (see Theorem 4.1). However, even if we consider M9,C as a model for the parameterized Severi varieties V9([C]) _

11

V9(C'),

O(C')EPicl°i(X)

there is still a p9 dimensional discrepancy (Donaldson also discusses this in detail [22]). Not much is known about the enumerative geometry of irrational surfaces with p9 = q = 0 even with quantum cohomology and the usual Gromov-Witten invariants at our disposal. However,

it may be possible to adapt the methods of Subsections 5.1 and 5.2 to the case of the Enriques surface (which is irrational and has p, = 9 = 0). $we use Pic(OI(X) to denote the component of Pic(X) corresponding to line bundles with first Chern class [C]".

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The reason is the following. The virtual dimension of M9,C(X) is the dimension of the space of curves that persist as pseudo-holomorphic curves when we perturb the Kiihler structure to a generic almost Ki hler structure. The difference of p9 in the dimensions of M9,C and V9([C]) means that only a codimension p9 subspace of V9([C]) persists as pseudo-holomorphic curves when we perturb the

Kihler structure. Algebraically, this arises as the obstruction to those infinitesimal deformations of the map that deform the image of the map in the direction (see Subsection 5.3). One way to rectify this situation is to find a compact p,-dimensional' family of almost Kahler structures that has the property that the only almost Kahler structure in the family that supports pseudo-holomorphic curves in the class [C] is the original Kiihler structure. If T is such a family, then the moduli space Mg,C(X,T) of stable maps for the family T is a better model for the space V, ([C]) in the sense that its dimension is stable under generic perturbations

of the family Ti-* T'. Given the existence of a p9-dimensional family as described above, these invariants can be used to answer enumerative geometry questions for the corresponding surface and linear system. In general, it is not clear when such a family will exist; however, if X has a hyperkahler metric g (i.e. X is an Abelian or K3 surface), then there is a natural candidate for T, namely the hyperkahler family of Kahler structures. We call this family the twistor family associated to the metric g and we denote it T9. It is parameterized by a 2-sphere and so dimf T9 = 2 = 2p9 as it should. Furthermore, the property that all the curves in Mg,C(X,T9) are holomorphic for the original complex structure can be proved with Hodge theory (of course this need no longer be the case for a perturbation of T. to a generic family of almost Kahler structure. We will discuss this case in further detail in Section 5 and in Subsection 5.3 we discuss recent work of Behrend and Fantechi that in many cases fixes the "p9 discrepancy" purely algebraically.

4.3. Relationship between the Seiberg-Witten and Gromov-Witten invariants. In a series of papers [78][79)[80][81] in 1994 and 1995, Taubes proved that the Seiberg-Witten invariants of a symplectic 4manifold are given by a certain set of Gromov-Witten invariants. This work has had a profound impact on symplectic and smooth topology and has found numerous applications; however, from the point of view of enumerative algebraic geometry, it gives us no information. The Seiberg-Witten invariants are topological invariants

only when bg+(X) > 1; for a Miler surface this is equivalent to p9(X) > 0. As we explained in the last section, the Gromov-Witten invariants are zero in any class [C] with O(C - K) ample, so there is only a small range of possible classes with non-zero invariants. In fact, in the Kibler case Taubes' theorem reduced to a fact already explained in Witten's original paper [88]: the Seiberg-Witten invariants only "count" connected components of the canonical divisor. That is, the only spine structures with non-zero solutions to the Seiberg-Witten equations correspond to the classes in H2 (X, Z) that are the various sums of the components of the canonical 9By this we really mean a real 2p9 dimensional family; the parameter space for the family need not have a complex structure or even an almost complex structure.

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divisorlo. In particular, on a minimal surface of general type, the Seiberg-Witten and Gromov-Witten invariants are non-trivial only for the canonical class (and the zero class).

Taubes shows in general that the Seiberg-Witten invariants count pseudoholomorphic curves that are always smoothly embedded (though possibly disconnected). From the point of view of enumerative geometry, smoothly embedded curves are uninteresting (the corresponding Severi variety is a linear system and so

Ns(X,C) = 1, i.e. there is exactly one smooth curve in JCJ passing through the appropriate number of points). It is worth noting here that although the enumerative information in the SeibergWitten/Gromov-Witten invariants is trivial, the actual Seiberg-Witten/GromovWitten multiplicities can be somewhat subtle. Non-trivial multiplicities for curves with a smooth image occur when the curve is a multiple of a square-zero curve of genus one. Multiplicities for the ordinary Gromov-Witten invariants arise because of the possibility of many different maps multiply covering the same image. Taubes' defines his own version of Gromov-Witten invariants which count embedded curves (rather than maps) and allow for the possibility of many components. His multiplicities are defined using the spectral flow of a certain operator that arises naturally in the Seiberg-Witten context. The exact relationship between the two definitions of Gromov-Witten invariants was clarified by Ionel and Parker [43] who showed that the invariants contain equivalent information.

4.4. When are Gromov-Witten invariants "enumerative"? By counting maps to X rather than subvarieties of X, the Gromov-Witten invariants acquire many advantageous properties such as deformation invariance and numerous relations. The disadvantage to enumerative applications is that the maps may contract or multiply-cover some of their components. Thus Gromov-Witten invariants may count genus g maps whose image does not have geometric genus g. Furthermore, a given curve in X may be the image of many different maps, possibly even a family of maps. Thus the Gromov-Witten invariants may count a given isolated curve with a non-trivial multiplicity that may be negative and/or non-integral. For these reasons, the Gromov-Witten invariants are said to give a virtual count of curves. When the count defined by a Gromov-Witten invariant coincides with the actual number of curves, the invariant is said to be enumerative. Here we introduce an intermediate notion: DEFINITION 4.4. A Gromov-Witten invariant is said to be weakly enumerative if it counts only curves with geometric genus g each with positive, integral multiplicity that is one for curves with (at worst) nodal singularities.

This notion is particularly applicable to surfaces where one can often rule out maps that collapse or multiply-cover components by dimensional arguments. The 100n a symplectic manifold there are two different ways to identify spin' structures with elements of H2(X, Z). One can always take the dual to the first Chern class of the bundle of positive spinors, i.e. c1 (W+)v E H2(X, Z); alternatively, on a symplectic 4-manifold every spin' structure can be obtained by twisting the canonical spin' structure Wo by a line bundle L and so we can consider the class cl(L)v E H2(X, Z). The latter correspondence is more natural from the point of view of Gromov-Witten theory since on a Kibler surface, a smooth curve C corresponds to a solution of the Seiberg-Witten equations for the spin' structure O(C) 0 Wo .

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basic reason for this is the dependence of the dimension of the space of stable maps

on dimX and g. When X is complex dimension 2 or less, the dimension of the space of stable maps grows linearly with g.

For surfaces, a Gromov-Witten invariant will be weakly invariant if all the maps counted by the invariant are birational isomorphisms onto their image. For example, GSttsche and Pandharipande [36] show that all the genus 0 GromovWitten invariants of p2 blown-up at n generic points are enumerative for n < 10 but their arguments also imply that the invariants are weakly enumerative for all n. If the location of the blow-up points are not generic, then the invariants may fail to even be weakly enumerative (c.f. Subsection 5.2). Other examples include the modified Gromov-Witten invariants for K3 and Abelian surfaces discussed in Section 5 which are weakly enumerative for generic choices of the K3 or Abelian surface. If a genus 0 Gromov-Witten invariant on a surface is weakly enumerative, then the work of Fantechi-Gottache-Van Staten [26] shows that the multiplicities of the

irreducible curves are determined solely by the type and number of singularities (see also Section 2).

5. Modified invariants and the case of K3 and Abelian surfaces In this section we give an expository account of our use of modified GromovWitten invariants to prove the Gottsche-Yau-Zaslow formula for K3 and Abelian surfaces. We include a description of how to use a "matching technique" to compute the contribution of multiple covers of nodal rational curves to the invariants. We end the section with a brief description of recent work of Behrend and Fantechi who give a purely algebraic modification of the Gromov-Witten invariants that generalizes the (non-algebraic) modifications used in the case of K3 and Abelian surfaces.

5.1. The K3 and Abelian surface case. In this section we explain the proof of the Gottsche-Yau-Zaslow formula for primitive classes in K3 and Abelian surfaces [15][17]. Let X be a K3 or Abelian surface and let C C X be a curve representing a primitive homology class. To verify the conjecture we need to show first that the numbers N9(X, C) only depend on g and [C]2 (and whether X is a K3 or Abelian surface) and then show the numbers are given as the coefficients of the predicted modular forms. Since p9(X) = 1, there are difficulties with the ordinary Gromov-Witten invariants (see Subsection 4.2). In fact, it is easy to see the following:

LEMMA 5.1. All the (ordinary) Gromov-Witten invariants of X are zero.

PROOF: Gromov-Witten invariants are invariant under deformations of the (almost) Kahler structure. Any K3 or Abelian surface can be deformed to a (nonalgebraic) Kiihler surface that has no holomorphic curves at all. q.e.d. However, this problem can be rectified and a proof of the Gottsche-Yau-Zaslow formula can be obtained. The following two theorems are the main theorems:

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THEOREM 5.2. There exists a family Gromov-Witten invariant that computes Ng(X,C). Furthermore, the invariant only depends on C2 and g (and possibly the divisibility of [C)). The invariant is weakly enumerative for generic X.11

The following theorem computes the invariant of Theorem 5.2 thus verifying the Gottsche-Yau-Zaslow formula: THEOREM 5.3. The Gottsche-Yau-Zaslow formula holds for all C C X representing a primitive homology class. That is, if X is a K3 surface then N. (X, C) is the coefficient of qiC2 in the series g

00

(1 - Qm)-24 (t kQ(k)g4

q-1 m=1

kLG=1

= A-1(DG2)9 and if X is an Abelian surface then Ng(X, C) is the coefcient of gJC2 in the series g-2 00 ll

k2Q(k)gkka(k)gkD2G2(DG2)9-2

(9=2

where a(k) = EdIk d is the sum of the divisors of k. REMARK 5.4. There is also a formula for the number of genus g curves on an Abelian surface passing through g points (without imposing the FLS condition). The numbers are the coefficients of the series g(DG2)9-1 (see [17]). We first outline the proof of Theorem 5.2. The family of Kahler structures used

in Theorem 5.2 is provided by the existence of hyperkahier metrics on X. Since c1(T X) = 0, Yau's proof of the Calabi conjecture [90] provides a Ricci flat Ki hler Einstein metric, which for surfaces is a hyperkahler metric. A hyperkahler metric is characterized by a 2-sphere's worth of Kahler structures amt + bwj + cwx where

S2={(a,b,c)ER3:a2+b2+c2=l} and the associated complex structures I, J, and K satisfy the algebra of the imaginary quaternions. We call this family the twistor family associated to a hyperkahler metric. The following theorem lists the key properties of the twistor family which allow us to use it with the family version of Gromov-Witten invariants to compute N9(X, C). THEOREM 5.5. Denote the twistor family associated to a hyperkahler metric h by Th. Then, 1. For any two hyperkahler metrics It and h', Th is deformation equivalent to Th,. We denote the deformation equivalence class by simply T. 2. For any orientation preserving diffeomorphism f the family f*(T) is deformation equivalent to T. 11 By generic in this setting we mean that X Is generic among those K3 surfaces that admit a curve in the class [C]. See subsection 4.4 for the definition of weakly enumerative.

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3. For any class [C] E H2(X;Z) with [C]' > -2, there is exactly one member of the family T,, which admits holomorphic curves in the class [C].

If C C X is a curve on a K3 surface, choose a hyperk5hler metric h on X and consider the invariant fig CT"1(pt.9). Property 1 of the previous theorem shows that this invariant is independent of the chosen hyperkhhler structure on X. The orientation preserving diffeomorphisms act transitively on elements of H2 (X; Z) with the same square and divisibility and so by property 2, the invariant 'Pg CT) (pt 9) is a universal number that only depends on 9, C2, and the divisibility of [Cl. Finally, property 3 shows that 9 CT) (pt.9) only counts curves that are holomorphic with respect to a single complex structure on X. With a little further work, one can show that 419 CTl (pt 9) is weakly enumerative for generic X. That is, for a generic K3 surface X, N9 (X, C) = 4(X .T) (pt 9)

as long as we understand that curves should be counted with certain positive integral multiplicities if the singularities are other than nodes. A similar discussion applies to an Abelian surface X with the addition complication due to non-trivial q = dim Hl (X, Ox). Since we wish to count curves in a fixed linear system passing through g - 2 points, we combine Theorem 4.1 and Theorem 5.2 to show that for a generic X and loops 7j, ... , 74 giving an oriented basis of Hi(X, Z), (4)

N9(X,C)

=.k96C)(71i...

2)

(again, in the weakly enumerative sense). This explains the proof of Theorem 5.2. Then with the invariants in hand, it suffices to compute the invariant for some particular choice of X and C to prove Theorem 5.3. We choose X to be elliptically fibered with a section and we take the class [C] to be the section class plus a multiple of the fiber class. This is a primitive homology class and so our computation will give us a proof of the Gottsche-YauZaslow formula for all primitive classes (Theorem 5.3). We do not know how to make the analogous computation for multiples of this class (or any other non-primitive class), but we make a brief remark about this case below:

REMARK 5.6. To even formulate the enumeration problem for non-primitive classes on a K3 or Abelian surface, one must always deal with non-reduced curves. On these surfaces, there is a curve in a class d[C] if and only if there is a curve in the class [C]. So for example, when counting rational curves in the class d[C] one must decide how to count those rational curves in the class [C] with their non-reduced structure. Our definition of N9(X,dC) using the family Gromov-Witten invariant applies in the non-primitive case as well, but its enumerative significance is less clear. When we show that our Gromov-Witten invariant is weakly enumerative for generic X and primitive [C] we use the fact that all the curves will be irreducible and reduced. A priori, the family Gromov-Witten invariant may assign a negative and/or non-integral multiplicity to non-reduced curves.

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We illustrate the computation of Theorem 5.3 with the K3 case and afterwards we discuss the new issues involved for the Abelian surface calculation. A elliptically

fibered K3 surface with a section generically has 24 singular fibers consisting of nodal rational curves N1, ... , N24 and a section S which is a rational curve of square -2. We choose our g points x1, ... , x9 away from the section and lying on g distinct, generic, smooth fibers F1,... , F9:

F1

...

Fe

N1

...

N24

l

P1

Fix n > 0 and let [Cn] = [S] + (n + g)[F]. Note that [Cn]2 = 2g - 2 + 2n and so to verify the GSttsche-Yau-Zaslow formula we need to show:

E tPCT'(pt p) .g-1+n = q-1 7 00

00

n=O

m1=11

(1 _ gm)-24

(E ko(k)g k k=1

The advantage of our choice of X and Cn is that we can really see all the curves in the linear system 10n]. They are all reducible and their components consists of

the section S along with n + g fiber curves (possibly non-reduced). In order for a curve in Al to pass through the points x1, ... , xy and have geometric genus no more than g (as it must in order to be the image of a genus g map) it must consist of the section, the g fibers F1, ... , F9 (possibly with multiplicities), and some number of the rational fibers N1i... , N24 (again possibly with multiplicities). In other words, the curve must be 24

S+Eb;F;+EajNj {=1

j=1

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where the g-tuple b = (b,,... , bg) and the 24-tuple a= (a1 i ... , a24) satisfy b; > 1,

a j 10, a n d E 1

=9b+g.

To compute the invariant §g CT) (pt 9) we must compute the number of maps with the various images determined by a and b. What we mean by this, strictly speaking, is that we must compute the virtual fundamental class of the moduli space of stable maps with image given by a and b. The virtual dimension of the moduli space is zero and so the virtual fundamental class is a number, but the moduli space may actually be higher dimensional. In the case at hand, we show that the moduli space splits as a product of other moduli spaces and the virtual fundamental class splits into a product of virtual classes coming from each of the factors. The factors in this product can be identified with the moduli spaces of maps whose images multiple cover a single fiber F= or Nj and the corresponding factor of the virtual class is (essentially") the usual virtual class of each factor.

In this way we show that the contribution to the invariant from maps with image corresponding to a and b is a product of "local" contributions from multiple covers of the fibers Fl,.. . , Fg and N1i... , N24. The invariant is thus of the form 24

llr(b,)11p(aj)

9 CT)(Pt °) =

j=1

i=1

(a,b):

E, a1+Et b;=n+g

where r(b) and p(a) are the "local contributions" of b-fold covers of a smooth fiber and a-fold covers of a nodal rational fiber respectively. Multiplying both sides of the equation by qg-l+n we get 24

7g

,Po.CT) (Pt9)g9-1+n = q-1

E (a,b):

!lr(bt)gb' llp(aj)gad i=1

j=1

E1 of+r; b;=n+9

and then summing over n: 0o

n-0

24

g

,Pg CT 1(pt g)qo-1+n

= q 1 E 11 r(bj)gb' II p(aj)gas j.1

(a,b) i=1

,o

g

00

24

= q- (Erbh1) (Ep(a)a) 6=1

4=0

To prove the theorem then, it remains to be shown (which we will do in the next subsection) that 1. the local contribution r(b) of b-fold covers of a smooth fiber is given by bo(b),

and 2. the local contribution p(a) of a-fold covers of a nodal fiber is given by the number of partitions of a since the generating function for the number of 12The only difference is the role that the section plays. It turns out that the obstruction to deforming the section is one dimensional and exactly cancels the obstruction to deforming the complex structure on X in the direction of the twistor family. The end result is the same as if we pretend the section is a square -1 curve and there is no family.

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partitions is given by 00

II (1-qm)-1. M=1

The computation for an Abelian surface follows by similar methods by again assuming that the Abelian surface is elliptically fibered, i.e. it is a product of two elliptic curves E1 x Z. The only new complication is the use of loops as geometric constraints (see Equation 4). Using a careful choice of loops, all the possible images of the genus g maps satisfying the geometric constraints are easily identified. The invariant is then again computed by calculating the contribution of the various multiple covers of a given image.

5.2. Local contributions. In this subsection we compute the local contributions r(b) and p(a) of multiple covers of smooth and nodal fibers. The contribution r(b) is easy to understand. It is the virtual fundamental class of the moduli space of genus 1, 2-marked, degree b maps to a fixed smooth genus

1, 2-marked curve that send the marked points to the marked points. The two marked points correspond to the intersection with the section and the point xi. By declaring one of the marked points to be the origin, we give the domain and range the structure of elliptic curves and then the map must be a homomorphism. Thus r(b) is the number of elliptic curves with a non-zero marked point that admit a degree b homomorphism onto a fixed elliptic curve with a non-zero marked point

mapping the marked point to the marked point. The number of elliptic curves admitting a degree b homomorphism to a fixed elliptic curve is the number of index b sublattices of Z e Z which is classically known to be or(b). The number of choices for the location of the marked point in the domain is then just b and so the moduli space of b-fold covers of a smooth fiber is a discrete space consisting of ba(b) points.

It is more difficult to directly see the contribution p(a) of a-fold covers of a nodal fiber. This moduli space does not have the expected dimension zero so we need to compute its virtual fundamental class. Define M (S + aN) to be the moduli space of genus 0 maps to X in the class [S] + a[N] with image S + aNj for some fixed Nj (it doesn't matter which onetheir neighborhoods are all biholomorphic). Denote its virtual fundamental class

by [M(S + aN)]91'. The moduli space M(S + aN) has a number of different path components arising from the possible "jumping" behavior of the map at the node. The basic phenomenon is illustrated below for degree two maps. The moduli space M(S+2N) has different components depending on whether the map factors through the normalization or not. In the figure below, points labeled by A and B are mapped to corresponding points labeled A and B and the normalization map identifies A and B to the nodal point. Notice that the bottom map cannot be factored through the normalization (the dashed map doesn't exist) even though the map to the nodal curve is well defined.

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In fact, the space M (S + 2N) has three connected components: one for maps that factor through the normalization and the two maps that consist of the bottom map in the figure along with the section attached (it can attach to either of the components). For higher degree, maps that factor through the bottom map of the figure will lie in a different component from maps that factor through the normalization. In general, the components of M (S+aN) are labeled by sequences... , 3-2, 8-1, so, 81.... of non-negative integers with E si = a. This can be seen as follows. Consider the nodal rational curve E whose dual graph is:

O S

O

O

£-a

£-a+t

O £o

E1

£a-1

E.

The curve E maps to S + N by sending the component S isomorphically to S and each component Ei by a degree one map to N in such a way that the nodes connecting E; to Ej.1 are mapped by a local isomorphism onto the node of N. It can be shown that every map in M(S+aN) factors uniquely through a map to E. The components of M(S+aN) are determined by the various ways the degree of the map can distribute among the Ei components of E. We denote by M(S+E; s{E,) the moduli space of genus 0 maps to E in the class [S] + E; s; [E11.

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The preceding discussion leads to

p(a)

_ [M (S + aN)]°i' [M(S+ESiEi)]vsr

{8, j:E, 8{=6

1

where by [M(S + Ei siEi)]°='' we mean the virtual fundamental class pulled back from M(S + aN). This fundamental class is induced by assigning normal bundles O(-2) to the Ei components and O(-1) to the S component. We can now compute [M(S + Ei siEi)]°ww indirectly by identifying it with another moduli-obstruction problem arising from certain blow-ups of p2. It will turn out that [M(S+Ei siEi)]°=* is always zero or one depending on whether the sequence {si} satisfies a certain property or not. To identify the moduli-obstruction problem with one coming from a blow-up of p2' we must blow-up p2 in such a way that it contains a configuration of rational curves isomorphic to E with normal bundles O(-2) on the Ei curves and O(-1) on the S curve. To see that this can be done, begin by blowing up a line at three points. Its proper transform is a -2 curve which we identify with Eo; it meets three -1 curves, one of which we identify with S. The remaining two -1 curves can be made into -2 curves by blowing up a point on each of them. We identify their proper transforms with E_1 and E1 and we repeat this process with the new -1 curves, continuing until we have the configuration E. We then consider the Gromov-Witten invariants of this blow-up of p2 in the class corresponding to S + Ei siEi. Assuming that all the rational curves in the blown up p2 in the class [S] + Fi si[Ei] actually lie in the configuration E,13 the number [M(S + Ei siEi)]°' is given by the corresponding invariant on the blown up P2. Genus 0 Gromov-Witten invariants of blow ups of p2 have been thoroughly studied in general by Gottsche and Pandharipande [36] but the particular invariants arising in this moduli-obstruction problem can be computed from elementary properties. The key property we use is the invariance of the Gromov-Witten invariants under Cremona transformations. By successive applications of the Cremona transformation, one shows that the invariant corresponding to [M (S + Ei siEi)]vir is either zero or equivalent to the number of lines in the plane through two points, i.e. one. The latter case occurs if and only if the sequence {si} is such that for each i > 0, si+1 is either si or si -1 and for each i < 0, si_1 is either si or si - 1. Following [15] we call a sequence with this property 1-admissible. By the preceding arguments we conclude that p(a) _ # of 1-admissable sequences {si} with

si = a

and we simply need to see that 1-admissible sequences of total sum a are in one-toone correspondence with partitions of a. This is achieved by exhibiting a bijection between 1-admissible sequences of total sum a and Young diagrams of size a (which '3Some extra work is required to justify this assumption. One can perform the blow-ups in such a way that the resulting surface has an action of C" preserving the configuration E. This action provides the tool needed to show that there are no curves outside E in the relevant homology class. One assumes that such a curve exists and studies its limits under the C` action; the desired contradiction is then arrived at by a homological argument.

Counting curves on irrational surfaces

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are well known to correspond bijectively to partitions). Given a Young diagram define a 1-admissible sequence {s;} by setting so equal to the number of blocks on the diagonal, sl equal to the number of blocks on the first lower diagonal, 82 equal to the number of blocks on the second lower diagonal, and so on, doing the same for

3-1,8-2.... with the upper diagonals. It is easily seen that this defines a bijection and thus concludes the proof of Theorem 5.3.

5.3. Other modifications via algebraic means. Although the results of the previous section are purely algebraic, the use of the twistor family to modify the usual Gromov-Witten invariants is a non-algebraic tool. Behrend and Fantechi have recently announced [10] a purely algebraic modification of the Gromov-Witten invariants that apply to any smooth algebraic variety with H°'22(X) > 0, and in particular to surfaces with p9 > 0. For K3 and Abelian surfaces, their modification is equivalent to the twistor family invariant. In this section we outline their modification for surfaces in general. In the usual algebraic definition of Gromov-Witten invariants, one defines a virtual fundamental class on the space of stable maps that is invariant under deformations. The ingredients in Behrend and Fantechi's approach to this [11] are the intrinsic normal cone and the obstruction complex of the moduli space of stable maps M9,,,,c(X). Behrend and Fantechi modify the usual obstruction complex so that the resulting virtual fundamental class has its dimension larger than the usual dimension by p9 and is invariant under deformations of X preserving the (1,1) type of [C].

The tangent-obstruction complex for M9,n,c(X) is built from the tangentobstruction complex of the Deligne-Mumford moduli space of stable curves M9,,, and the relative tangent-obstruction complex of M9,n,c(X) -+.M9,,,.

The relative tangent and obstruction spaces at the map If : C -+ X) can be identified with H°(C, f'TX) and Ht(C, f*TX) respectively. An infinitesimal deformation of X together with f is automatically obstructed unless the class [Im f] remains type (1, 1). For such deformations, the obstructions always lie in the kernel of

H1(C, f*TX) -i H2(X,O). We can define this map via its dual map H°(X, Bz) -+ H°(C, f -n 1X 0 wc)

which is induced by the composition of H°(X,Il ) -3 H°(C, f*n') and the map induced by

f*StX -> f*flx ®f*Stx -> f*f1X'' c - f'SIX ®wc. Behrend and Fantechi modify the usual tangent-obstruction complex by replacing the relative obstruction space by the kernel of HI(C, f *TX) -+ H2(X, O). In order for their machinery to work, the obstruction complex must be "perfect", that is, equivalent in the derived category to a two term complex of vector bundles.

Thus, if H'(C, f*TX) -a H2(X,O) is surjective (or of constant rank) for all f, then the modification leads to an invariant. The theorem is as follows:

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THEOREM 5.7. Let X be a surface and [C] a class of type (1,1). There is a (modified) virtual class [M9,c]"i d of dimension -KC + g - 1 + ps. If the map

Hl(C, f *TX) -a H2(X,O)

is surjective for every f f : C -> X) in M9,c, then [Mg,c] nrod defines modified Gromov- Witten invariants that are invariant under deformations of X preserving the (1,1) type of [C].

The hypotheses of the theorem can be shown to hold for K3 and Abelian surfaces and also for more general surfaces with appropriate ampleness conditions on C. REMARK 5.8. Using these invariants and Theorem 4.2, one can define Ns (X, C) using Gromov-Witten theory and perhaps use them to devise a proof of the GottscheYau-Zaslow formula.

REMARK 5.9. As with the families version of Gromov-Witten invariants, it is

not clear what the analogues of the composition law and quantum cohomology should be.

REMARK 5.10. Except for the cases where X admits a hyperkahler structure, it is not clear if there is a symplectic version of Behrend and Fantechi's modified invariants.

The computations for K3 and Abelian surfaces described in section 5 easily extend to compute the modified invariants of any elliptic surface with a section in [S] + n[F] where S is the section and [F] is the class of the fiber. the class For example, let E(m) be a generic elliptic surface over PI with a section and Euler characteristic 12m.14 With this convention, E(1) is a rational elliptic surface, E(2)

is a K3 surface, and etcetera so that ps(E(m)) = m - 1. Let 4' be the modified invariants of Behrend -Fant echi. Then the methods of Subsections 5.1 and 5.2 can M) be used to computesI Cn (pt e): 00

(5)

14Q

C)(pts)gs+n = (4 )m/a (DG2)9.

I%=:o

There are a few remarks worth making about this formula. REMARK 5.11. The formula is different from the Gottsche-Yau-Zaslow formula

except in the case m = 2 (the K3 case). In general, the classes [S] + n[F] do not satisfy Gottsche's ampleness conditions so there is no contradiction with the conjecture. It does show that the ampleness conditions cannot be removed. REMARK 5.12. The invariants

g c l only have enumerative significance for

E(1) and E(2). For m > 2, one cannot deform away the elliptic fibration and so 4g c l always gives the virtual count done in the computation of Subsections 5.1 and 5.2. For E(1) and E(2) (i.e. P2 blown up at nine points and K3) a generic deformation will no longer be elliptically fibered and 4e a l will be weakly enumerative. In fact, for E(1), 3g c is just the ordinary Gromov-Witten invariants for p2 blown-up at nine points. For genus zero they are determined recursively from 14The classes [Cal are exactly characterized as those classes [C] such that [C] K = m - 2.

Counting curves on irrational surfaces

335

the quantum cohomology and are enumerative in the strong sense (for a generic choice of blow-up points). Furthermore, in [15] it is shown that the Gromov-Witten invariant of E(1) for any class C with CK = -1 is equivalent to the invariant for some Cn. The fact that there is a closed formula (different from the Gottsche-Yau-Zaslow

formula!) for these invariants of E(1) in terms of modular forms is somewhat of a surprise. Although the genus zero numbers are determined recursively, it is not clear how to obtain the closed formula from the recursion or how the modularity is reflected in the structure of the quantum cohomology. lonel and Parker [44] have recently outlined a new proof of the genus zero formula for E(l) that is perhaps the most transparent. They use a formula for the invariants of a fiber sum along with a topological recursion relation to show that the left hand side of Equation 5 satisfies a differential equation that is solved by the modular form on the right hand side.

6. Acknowledgments We would like to thank the many people with whom we have had valuable conversations about Gromov-Witten theory and enumerative geometry. In particular

we thank K. Behrend, A. Bertram, H. Clemens, 0. DeBarre, R. Donagi, S. Donaldson, B. Fantechi, E. Getzler, A. Givental, L. Gdttsche, T. Graber, E. Ionel, A. K. Liu, S. Katz, S. Kleiman, J. Li, T. Parker, R. Pandharipande, Y: B. Ruan, B. Siebert, C. Taubes, M. Thaddeus, G. Tian, A. Todorov, C. Vafa, I. Vainsencher, R. Vakil, S: T. Yau, and E. Zaslow. We would also like to thank the organizers of the conferences that made it possible to talk to many of the above people. In particular, Y. Eliashberg and L. Traynor for organizing symplectic geometry program in IAS/Park City summer institute (1997); H. Clemens, E. Arbarello, and J. Rarer for organizing the workshop on "Hodge theory, mirror symmetry, and quantum cohomology" in Pisa/Cortona,

and A. Bertram and Y: B. Ruan for organizing the AMS summer workshop on Quantum Cohomology.

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[681 Shigeru Mukai. Symplectic structure of the moduli space of sheaves on an Abelian or 1(3 surface. Inventions Mathematicae, 77(1):101-116, 1984. [69] Rahui Pandharipaade. Counting elliptic plane curves with fixed j-invariant. Proc. Amer. Math. Soc., 125(12):3471-3479, 1997. [70] L. Polterovich. Gromov's K-area and symplectic rigidity. Geom. Funct. Anal., 6(4):726-739, 1996.

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[72) Yong Bin Ruan and Gang Tian. A mathematical theory of quantum cohomology. Math. Res. Lett., 1(2):269-278, 1994. [73] Yongbin Roan. Symplectic topology and complex surfaces. In Geometry and analysis on complex manifolds, pages 171-197. World Sci. Publishing, River Edge, NJ, 1994. [74] Yongbin Ruan. Topological sigma model and Donaldson-type invariants in Gromov theory. Duke Math. J., 83(2):461-500, 1996. [75] Yongbin Ruan and Gang Tian. A mathematical theory of quantum cohomology. J. Differential Geom., 42(2):259-367, 1995. [76] Yongbin Ruan and Gang Tian. Higher genus symplectic invariants and sigma model coupled with gravity. Turkish J. Math., 20(1):75-83, 1996.

(77] Bernd Siebert. Algebraic and symplectic Gromov-Witten invariants coincide. Preprint: math.AG/9804108, 1998. [78] C. H. Taubes. The Seiberg-Witten invariants and symplectic forms. Mathematical Research Letters, 1:809-822, 1994. [79) C. H. Taubes. The Seiberg-Witten and Gromov invariants. Mathematical Research Letters, 2(2), 1995.

[80] C. H. Taubes. Counting pseudo-holomorphic submanifolds in dimension 4. Journal of Differential Geometry, 44(4):818-893, 1996. [811 C. H. Taubes. SWn.Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. Journal of the American Mathematical Society, 9(3), 1996. [821 Isreal Vainsencher. Enumeration of n-fold tangent hyperplanes to a surface. Journal of Algebraic Geometry, 4(3):503-526, 1995. [83] Ravi Valdl. Genus g Gromov-Witten invariants of Del Peso surfaces: Counting plane curves with fixed multiple points. preprint: alg-geom/9709004, 1997. [84) Ravi Vakil. Enumerative geometry of plane curves of low genus. preprint: math.AG/9803007, 1998.

(851 Claire Voisin. A mathematical proof of a formula of Aspinwall and Morrison. Compositio Math., 104(2):135-151, 1996.

[86] Claire Voisin. Sym4trie mirror, volume 2 of Panoramas et Synthisea [Panoramas and Syntheses]. Socidtd Mathtunatlque de France, Paris, 1996. [87] P. M. H. Wilson. The role of c2 in Calabi-Yau classification-a preliminary survey. In Mirror symmetry, II, volume 1 of AMS/1P Stud. Ads. Math., pages 381-392. Amer. Math. Soc., Providence, RI, 1997. [88] E. Witten. Monopoles and four-manifolds. Mathematical Research Letters, 1:769-796, 1994.

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Addresses: DEPARTMENT OF MATHEMATICS TULANE UNIVERSITY

6823 ST. CHARLES AVENUE NEW ORLEANS, LA 70118 SCHOOL OF MATHEMATICS UNIVERSITY OF MINNESOTA

MINNEAPOLIS, MN 55455

Special Lagrangian Fibrations II: Geometry. A Survey of Techniques in the Study of Special Lagrangian Fibrations

s Mark Gross* Mathematics Institute University of Warwick Coventry, CV4 7AL mgrossOmaths.warwick.ac.uk

§0. Introduction. This paper is a progress report on work surrounding the Strominger-Yau-Zaslow mirror symmetry conjecture [28]. Roughly put, this conjecture suggests the following program for attacking an appropriate form of the mirror conjecture. Let X be a Calabi-Yau n-fold, with a large complex structure limit point pin a compactification of the complex moduli space of X. One expects mirrors of X to be associated to such boundary points of the complex moduli space of X. For complex structures

on X in an open neighborhood of the boundary point p and suitable choice of a Ricci flat metric on X, one attempts to construct the mirror of X via the following program: (1) There is an n-torus representing a homology class in HH(X, Z) which is invariant under all monodromy transformations about the discriminant locus passing through the point p. (See [14], §3 for details of this representative). The first task is to find an homologous n-torus which is special Lagrangian.

(2) Having found one special Lagrangian torus, show that it deforms to yield a fibration f : X -+ B all of whose fibres are special Lagrangian and whose general fibre is an n-torus. (3) Construct the dual n-torus fibration as follows. Let Bo C B be the complement of the discriminant locus of f, fo : Xo -+ Bo the restriction of f to Xo = f -' (Bo). The dual n-torus fibration over Bo is Xo = R1 fo.R/R' fo.Z -+ A. Find a suitable compactification of Xo to a manifold X along with a fibration

/:X -a B. (4) Show that X and X satisfy a topological form of mirror symmetry. It is not clear what this means in arbitrary dimension, but for threefolds, this will be Supported in part by NSF grant DMS-9700761 341

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Special Lagrangian Fibrations II: Geometry an isomorphism Heven(X Q) flt Hodd(5,Q) and Hodd(X Q) ae Heven(X Q). This in particular implies the usual interchange of Hodge numbers for threefolds. One might also dare to hope these isomorphisms also hold over Z; we will see this will often be the case in Theorem 3.10.

(5) Put a complex and KAhler structure on X. The choice of such structures determines the mirror map. One expects that the complex structure on X should entirely determine the Kahler structure on X, while the Kiihler structure on X along with a choice of the B-field, a cohomology class in H2(X,R/Z), or a related cohomology group, will determine the complex structure on X. In [16], a somewhat more precise conjecture was given as to how this interchange of structures should look on the level of cohomology. Specifically, let W, 1 be the Ki hler form and holomorphic n-form on X with 1 normalised so that fx6 St = 1. In addition, one is given a choice of B-field, which right now we'll take to be a cohomology class B E H'(B, Rl f«!R) (The B-field will always be denoted by a bold-face B to differentiate it typographically from the base B of the fibration.) The choice of Kiihler and complex structures on the mirror, determined by forms Co and CZ, should satisfy the following relationship: using the identifications H' (B, R' f.R) _t H' (B, Rn-1 f.R) and HI (B, Rn-1 f.R) °-` Hl (B, RI f.R) which conjecturally hold, the following identities should hold in these cohomology groupsP) = [Im Il] (Im CI] = [w] Ift]

where ao is a chosen zero-section of f : X -+ B. (6) Show that the above procedure yields the correct enumerative predictions for Gromov-Witten invariants of X and X. This program is still a long way from completion, and this paper represents only one small step in this direction. Not much is known about items (1) and (2) yet; this may well prove to be the hardest part of the program. In [16], we gave examples of special Lagrangian Ts-fibrations with a degenerate metric. In [32], Tnfibrations are constructed on Calabi-Yau hypersurfaces in smooth toric varieties. These fibrations are constructed as deformations of the natural Tn-fibration on the large complex structure limit given by the moment map. Unfortunately these tori are neither special nor Lagrangian, but this fibration may be sufficient for a purely topological version of mirror symmetry and provides evidence for the SYZ conjecture. We will not address issues (1) and (2) further in this paper. Instead, throughout

this paper, we assume the existence of a special Lagrangian fibration f : X -+ B on X, much as we did in [14]. However, we wish to delve more deeply into the properties of such fibrations, and to do so, we need to make reasonable guesses as to what kind of regularity properties such fibrations will possess. We discuss these issues and assumptions in §1. In [14], we did not make real use of the fact that f : X -4 B was special Lagrangian, but rather only used the topological fact that f was a torus fibration, along with a technical condition we called simplicity

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to control the cohomological contributions of the singular fibres. In this paper, we try to make more serious use of the special Lagrangian condition. We find that we make very heavy use of the Lagrangian condition, but as yet we do not use the additional special condition in a profound way. We do however use the fact that special Lagrangian submanifolds are in fact volume minimizing. This allows us to provide moral guidelines as to what we might expect of special Lagrangian fibrations, by drawing on the wealth of material known about volume minimizing rectifiable currents. Some of these ideas are discussed in §1.

Of course, the study of Lagrangian torus fibrations is a familiar one in the subject of completely integrable Hamiltonian systems. In §2 of this paper we review and generalise slightly for our purposes Duistermaat's work on global action-angle coordinates [12]. Simplifying a bit, if f : X -+ B is a Lagrangian T" fibration with

a Lagrangian section and if X# is the complement of the critical locus of f in X, then one can canonically write X# as a quotient of the cotangent bundle of B by a possibly degenerating family of lattices suitably embedded in T. Furthermore, the canonical symplectic form on TB descends to the symplectic form on X#. Thus if X is a Calabi-Yau manifold, the existence of a special Lagrangian fibration gives us coordinates on a large open subset of X on which it is easy to write the symplectic form. I think of these coordinates as special Lagrangian coordinates. These can

be compared with traditional complex coordinates, where it is easy to write the holomorphic n-form (dzr A ... A dz ), but very difficult to write down the Kahler form of a Fticci flat metric. Thus we should expect that in the coordinates given by a special Lagrangian fibration, the difficulty will be to write down the holomorphic n-form, or equivalently, the complex structure.

Taking this analogy further, we note that complex coordinates give a filtration on de Rham cohomology, namely the Dolbeault cohomology groups. Similarly, special Lagrangian coordinates can be thought of as giving rise to a filtration on cohomology, namely that given by the Leray spectral sequence associated to the special Lagrangian fibration. Just as the Dolbeault cohomology groups give rise to the Hodge filtration, we saw in [14] that the Leray filtration should, modulo some conjectures about monodromy, give rise to the monodromy weight filtration associated to the large complex structure limit point.

In [14], we studied the Leray spectral sequence of f with coefficients in Q. We now show in §3, modulo suitable regularity assumptions stated in §1, that three-dimensional special Lagrangian fibrations satisfy the condition of Z-simplicity introduced in [14]. This is a stronger hypothesis then was used in [14], and as

a result, we can analyse the Leray spectral sequence over Z. This leads to some interesting results as to the role torsion in cohomology plays in mirror symmetry. In particular, it provides an explanation of the phenomenon proposed in [5] of the role

of "discrete torsion" (torsion in HI(X,Z)) in mirror symmetry. This is discussed in §3. In addition, the analysis of the Leray spectral sequence over Z sheds light on item (4) of the program proposed above. Moving on, we next address the subject of putting symplectic and complex structures on the mirror X. This subject has already been discussed to some extent by Hitchin in [18]. Our approach is inspired by Hitchin's, and is a generalisation of that approach. In fact, one of the goals accomplished is to restate Hitchin's

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constructions in a more coordinate independent form, so as to allow us to understand the cohomological ramifications of these constructions.

One should in fact consider any special Lagrangian submanifold M C X, and consider the moduli space of deformations of this submanifold. Calling this moduli

space B, the D-brane moduli space of M is then the set of pairs (M', a) where a is a flat U(1) connection modulo gauge equivalence on M' a deformation of M.

This is a T'-bundle over B, where a = b1(M). Specifically, if f : U -3 B is the universal family of special Lagrangian submanifolds parametrized by B, U C B x X,

then the D-brane moduli space is M = R' f.R/Rl f.Z i B. In addition, Mclean [22] gives us a canonical isomorphism between the tangent bundle of B, TB, and Rl f.R®CO0(B). This gives a canonical embedding of Rl f.Z in T. However, TB of course does not carry a canonical symplectic form; rather, it is TB which does. So to find a symplectic form on M, we need to reembed R' f.Z in TB*. There are two ways of doing this. One is to use periods integrals related to Im 12, the imaginary part of the holomorphic n-form on X; the other is to use a canonical metric introduced by McLean on Ta to identify TB with TB. In fact these two methods give the same embedding of Rl f.Z in Ti. This allows us to define a symplectic form on M

by writing M as 7 /Rl f.Z and taking the form on M induced by the canonical symplectic form on T. This is the same method as proposed by Hitchin, recast in a slightly more invariant way, which makes it easy to see that the cohomology class of the symplectic form defined in this manner is as predicted by the conjecture of [16]. Thus, if we can understand how to extend this symplectic form to the compactification of the D-brane moduli space, we will solve the first half of item (5). This circle of ideas is discussed in §4. Because it is easy to write the symplectic form in special Lagrangian coordinates, we expect it to be very difficult to write the complex structure. We take up this issue in §§5 and 6. First we explore what data is necessary to place a complex structure on a Lagrangian torus fibration in order to make the fibration special Lagrangian and the induced metric Ricci-flat. A moment's thought shows that given knowledge of w, to specify an almost complex structure compatible with w, it is enough to give for each point b E B a metric g on the fibre Xb, and for each point x E Xb the Lagrangian subspace J(Tx,,x) S T. Then J is completely determined by the requirement that g(v, w) = w(Jv, w). The collection of subspaces J(Tx,,,) can be thought of as the horizontal subspaces of an Ehresmann connection

on f : X* 4 B. We then need to ask when this data determines an integrable complex structure which induces a Ricci-flat metric in which the fibres of f are special Lagrangian.

Following [18], it is easier to determine this by describing the holomorphic n-form 11. In local coordinates yl, ... , yn on the base, and canonical coordinates xl, ..., xn on the fibres of TB, we can write the general form of fZ as n

n

SZ = v A(dxi + E Nijdyj). i=1

j=1

where ,8,j is a complex-valued function on r and V is a real-valued function. The Ehresmann connection is in fact encoded in Re#, while Im,6 is just the inverse of

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the metric on the fibres. The integrability and Ricci-flatness conditions are then easy to write down. In fact, we show that one needs the conditions: (1) The matrix ,8 = (f;,) is symmetric, Im,9 is positive definite, and V = 1/ et m;i. (2) dfl = 0. The first condition is of course easily achieved, but the second condition is a quite subtle condition. The second condition is what requires real effort, and understanding it is really at the heart of the SYZ program. It turns out that (2) is equivalent to the following three conditions:

(1) The almost complex structure is integrable. (Note that dfl = 0 is a much stronger condition than integrability. Given any Lagrangian fibration on a Calabi-Yau manifold with a holomorphic n-form fl, by replacing fl by e'0l1i0 for a suitable function 6(x), one can ensure that Imfl restricted to each fibre is zero. However now fl is no longer holomorphic.)

(2) dxl,... , dx are harmonic 1-forms on X6 for each b E B. (That this is a necessary condition follows from Mclean's results.)

(3) The volume form Vdxi A ... A dx,, on fibres is parallel under translation via the Ehresmann connection. In [18], Hitchin constructed a complex structure on the D-brane moduli space by specifying the holomorphic n-form Q. The form he constructed had very special

properties: he used Rep = 0 and Imf constant along fibres. As a result, the integrability condition dfl = 0 was much easier to check. In the general case, it requires a great deal more effort to analyse this equation, and the results of §5 and 6 represent only a beginning. The conditions (1)-(3) above should perhaps be thought of as mirror equations to the usual complex Monge-Ampere equations which arise in the study of Ricci-flat metrics. Understanding their solution should be one of the key steps in the Strominger-Yau-Zaslow program. We analyse the solution in several simple cases. For example, we show that if the Ehresmann connection is in fact flat, then the metric on each fibre must be flat. This explains why in Hitchin's

situation, where the connection was trivial, it would be impossible to obtain any solutions which are not flat along the fibres. In particular, the connection clearly cannot be fiat if the fibration possesses singular fibres. We also give a whole family of solutions related to Hitchin's, but which take the B-field into account. This gives some hint as to the role the B-field plays and leads us to a refined form of the mirror symmetry conjecture (Conjecture 6.6). We will argue that the B-field should not be thought of as an element of Ha (X, R/Z) but rather as an element of Hi (B, RI f.R/Z). Thus the group the B-field takes values in should depend not just on X but on the fibration f. These two groups can in fact be different, so this suggestion is quite a serious modification of previous interpretations of the B-field (for example that of [5]). In §7 we apply much of what we have done to the situation of K3 surfaces, by way of an extended example. Here we know that special Lagrangian fibrations do exist, and we know the precise construction of the mirror map. We are able to show that the various recipes and constructions given in earlier sections of the paper can be carried out completely for K3 surfaces. In particular, we obtain an almost purely differential geometric description of mirror symmetry for K3 surfaces.

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Finally, in §8, we give a brief discussion of results about the Strominger-YauZaslow conjecture obtained since the initial version of this paper was prepared. I would like to thank P. Aspinwall, D. Calderbank, N. Hitchin, M. Micallef, A. Todorov, P. Wilson, S: T. Yau, and E. Zaslow for useful discussions. In addition, I would like to thank S: T. Yau for his hospitality at Harvard, where some of this work was carried out.

Convention. For a p-form a and tangent vectors Vi.... ,vq, c(vt,...,vq)a denotes the p - q-form a(vi, ... , vq, )-

§1. Special Lagrangian Fibrations. In what follows, X will denote a complex Calabi-Yau manifold with a Ricci-flat Kihler metric g, Kibler form w, and holomorphic n-form (1. The form fl is always normalised to be of unit length, i.e.

w"/nl = (-1)°("-l)/2(i/2)°f1 A fl. This only fixes ft up to a phase factor. As the metric is Ricci-flat, we can take the metric to be real analytic. Recall

Definition-Proposition 1.1. (17] Ref! is a calibration, called the special Lagrangian calibration. An n-dimensional real submanifold M C X is special Lagrangian if Re IIM = Vol(M). Modulo orientation, a submanifold M C X of real dimension n is special Lagrangian if and only if wIM = 0 and Im fllM = 0. It is natural to extend the notion of special Lagrangian submanifolds to special Lagrangian integral currents, as done in [171. For an introduction to integral and rectifiable currents and geometric measure theory, see [23], and for more rigourous treatments see [13] and [27]. We will not make much use of the language of geometric

measure theory here except to justify some of the assumptions made on special Lagrangian fibrations. The reader unfamiliar with this language should just keep in mind that passing from submanifolds to integral currents means extending the class of submanifolds to subsets with integer multiplicities attached, over which one can still integrate forms. There are natural compactness theorems for integral currents, enabling one to easily construct volume minimizing currents. One can then try to control the singularities of such currents. The biggest regularity result of this nature is Almgren's monumental

Theorem 1.2. [1,2] Suppose N is an m + I dimensional submanifold of R'"+" of class k + 2 and that T is an m-dimensional rectifiable current in R-+n which is absolutely area minimizing with respect to N. Then there is an open subset U of R"'+^ such that Supp(T) f1 U is an m-dimensional minimal submanifold of N of class k and the Hausdorff dimension of Supp(T) - (U U Supp(8T)) does not exceed m - 2. The proof [2] is unpublished and is over 1500 pages long in preprint form. Thus an urgent question is

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Question I.S. Are there nice regularity theorems for special Lagrangian currents? For example, [19] shows that holomorphic integral currents are obtained by integration over complex analytic subvarieties, and of course the singular locus of such varieties is well-behaved. This is much stronger than Almgren's result, which doesn't even guarantee finite m - 2 dimensional Hausdorff measure of the singular set. We do not want the theory of special Lagrangian currents to have to depend on Ahngren's result. Next we should consider what is a reasonable definition of a special Lagrangian fibration. It might be reasonable to say that

Definition 1.4. If B is a topological space and f : X i B is a continuous map, we say f is a special Lagrangian fibration if for all b E B, Xb f-1(b) is the support of a special Lagrangian integral current T with OT = 0.

Even this might be too strong; one might insist that the fibres be special Lagrangian on a dense subset of B. This would allow some fibres to jump dimension. Nevertheless, we do not expect fibres to decrease in dimension as this would suggest the cohomology class of the general fibre Xb was trivial, which contradicts '& n 0 0.

I do not want to give a rigorous argument of this sort here as it requires being clearer about concepts such as the dimension of the fibre. However it is clear that special Lagrangian fibrations cannot behave as many completely integrable Hamiltonian systems do, in which some fibres are tori of smaller dimension. On the other hand, it is not as clear that we want to rule out the possibility of fibres jumping up in dimension, something which often happens in algebro-geometric contexts. Nevertheless, we will stick to Definition 1.4. We are interested in very specific sorts of special Lagrangian T" fibrations. As argued in [14], §3, we are looking for special Lagrangian fibrations on Calabi-Yau

manifolds near a specific large complex structure limit point in the boundary of complex moduli space, and the homology class of a fibre should be represented by a specific vanishing cycle associated to the boundary point. It was argued in [14], Observation 3.4 that, in the three dimensional case, if this homology class is primitive, then for general choice of complex moduli near the large complex structure limit point, all fibres off : X -+ B must be irreducible. Let us be more precise. There is a notion of indecomposable integral current ([13], 4.2.25) which is analagous to the notion of irreducibility in algebraic geometry, and we will say a fibre f-1(b) is irreducible if the current Tb obtained by integrating forms over f -1(b) (with orientation induced by fl) is indecomposable. Definition 1.5. A special Lagrangian fibration is integral if each fibre is irreducible and the currents {TbIb E B} all represent the same integral homology class.

This last condition rules out fibres which need to be thought of as multiple fibres. However, it is difficult to define multiplicity of a fibre in the context of a continuous map. Here the term "integral" is being used in its algebro-geometric sense: each fibre is irreducible and "reduced". Now a special Lagrangian fibration need not be integral any more than an elliptic fibration need only have integral fibres. But this is the generic behaviour of

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special Lagrangian fibration in dimensions < 3, and the assumption of integrality vastly decreases the range of potential singular fibres. Without assumptions of integrality, it would be very difficult to relate the cohomology of a T"-fibration and its dual. But one should keep in mind that even in dimension 3, integrality should fail for special values of the complex structure. Furthermore, if the homology class of a fibre is not primitive, then there is still the possibility that integrality will fail, and perhaps even multiple fibres appear. I have no argument to rule this out, but as we shall see in later sections, it is special Lagrangian fibrations with sections which are most important, and for these the homology class of a fibre is primitive. The next natural question is whether we expect B itself to be a manifold. I would like to give a rough argument that this is a natural expectation if the fibration f : X -- B is integral. Indeed, given a fibre Xb, Xb will be smooth at a general point x E Xb, and in a neighborhood of x, using the exponential map, the deformations of Xb can be identified with deformations of Xb inside its normal bundle near x. Thus locally the fibre of the normal bundle of Xb at x yields a natural local section for f : X -+ B, and hence gives a manifold structure on B. This construction hopefully in addtion yields a COO structure on B. Thus we will feel justified in making the following assumption, to be in force throughout the remainder of the paper.

Assumption 1.6. All special Lagrangian fibrations f : X -+ B will be C°° maps of C°° manifolds. Furthermore, f will be assumed to have a local section at each point b E B. In addition, if f is assumed to be integral, we will assume that for any point x E Xb - Sing(Xb), there is a local C°O section passing through x.

Again, if the fibration is not integral, I would not be surprised if singular B can arise naturally. Also, since the metric on X is real analytic, we can hope that with suitable coordinates on B, f will in fact be real analytic. We will not assume that here, however. Next, we pass to the nature of the discriminant locus. Given f integral as in Assumption 1.6, we can now consider the rank of the differential f.: Tx,: - Ta,f(.) at various points x E X. At points with a local C°O section, rank f. = n. It will be shown in Proposition 2.2 that if rank f. = v, then the fibre in fact contains a submanifold of dimension v on which rank f. = v. Thus the existence of points

x E X for which rank f. = n - 1 contradicts Almgren's theorem. Even if one doesn't accept Almgren's theorem, it is not difficult to rule out the existence of such points in low dimension using regularity results for minimizing hypersurfaces. Thus it is quite safe to assume that rank f. n -1 for any point x E X. This gives a stratification of the discriminant locus

A. = f ({x E X1 rank f.: TV,. -3 TB, f(s) < v},

with ho C

C 0n_2 = A, the discriminant locus. Then Federer's generalization of Sard's Theorem, [13], 3.4.3, states that 3l"+(2n-")/k(A.) = 0, where 3ln denotes the n-dimensional Hausdorff measure. In particular, if k = = 0 for all e > 0, so 0 is of Hausdorff dimension < v. We need however stronger information than Hausdorff dimension to reach any cohomological conclusions. Thus the following assumptions will be used at various points of this paper; unlike Assumption 1.6, we will assume these only when we need them, mentioning them specifically.

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Assumption 1.7. For an integral special Lagrangian fibration f : X -+ B, b E

p-

i there is a dense subset .C of the real Grassmannian of n - v dimensional subspaces of TB,b such that one can find locally, for each L EC, a submanifold B' of B passing through b such that L and B' n D, = {b}.

This requires reasonable regularity results about the discriminant, which certainly hold if f : X -+ B is real analytic, so that 0 is a sub-analytic set. A stronger form of this assumption which we will need in §3 and will comment on there is

Assumption 1.7'. In addition in Assumption 1.7, B' can be found so that f -1(B') is a submanifold in a neighborhood of the fibre Xb. Finally, we will require some assumptions on Sing( b). Almgren's theorem that the Hausdorff dimension of Sing(Xb) is no more than n - 2 is not sufficiently strong for most purposes, and at the very least, we will frequently need to use

Assumption 1.8. If f : X -+ B is an integral special Lagrangian fibration then

H'(Sing(Xb),Z) = 0

fori>n-2, forallbEB. This is a restriction on homological dimension. If f is real analytic, then this should hold given Almgren's result.

§2. Action-angle Coordinates. There is a standard theory of global action-angle coordinates due to Duistermaat [12]. We will extend this slightly so as to include information about the smooth part of the singular fibres. In this section, f : X -+ B denotes any special Lagrangian fibration satisfying Assumption 1.6. However, many of the results in this section apply to C' Lagrangian fibrations f : X -+ B with B a manifold and f having a local section in a neighborhood of each point b E B. The first observation is that there is an action of Ts, the cotangent bundle of B, on X.

Proposition-Definition 2.1. There is a C°O action ofr(U,TB) on f-1(U) for any U C B, which we write for any a e r(U, TB) as a map Ta : f -1(U) -+ f -1(U). This satisfies the following properties: (1) If da = 0, then Ta is a symplectomorphism. (2) Ta acts fibrewise, and Tai f-i(b) : f'1(b) -> f'1(b) only depends on the value of a at b. (3) T. o To = Ta+p Proof. This is standard: see for example [12], §1, or [3], Chap. 10. We review the definition of these maps however. If a is a compactly supported 1-form on Y, then f *a is a compactly supported 1-form on f -1(U). There is then a vector field

va on f'1(U) with t(va)w = f *a. This generates a flow qt : /'(U) -+ f -1(U) for all t, and we take T. = 01. It is then standard that if da = 0, v is locally Hamiltonian and ¢t then is a 1-parameter family of symplectomorphisms.

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Special Lagrangian Fibrations II: Geometry

If locally da = 0, we can write a = dH for some function H on B, and if G is any other function on B, {G o f, H o f } = 0 since f : X --4 B is a Lagrangian fibration. But then Ot is the Hamiltonian flow associated to H, and

0 ={Go f,Ho f}(x)=

dtlt_o(GOf)(Ot(x))

so Go f is a constant on ¢t(x) for any x. Thus 4t acts on fibres. Clearly valf_1(b) depends only on the value of a at b, so the action of Ot on f -1(b) depends only on the value of a at b. In particular ¢t acts on fibres for arbitrary a, not just compactly

supported a. . Next, a standard analysis of the orbits of this action.

Proposition 2.2. H b E B, x E f'1(b), then the orbit {T0(x)la E TB b} is diffeomorphic to R! x T8 and I + a coincides with rank(f* : TB,b -> TX z) = rank(f* TX,. -1 TB,b)

Proof. If a E TB,b is in the kernel of f *, then va is zero at x, so T. (x) = x. Thus for any a E TB,b, Ta(x) depends only on a modulo ker f *, and the orbit of x is homeomorphic to a quotient of V = TB b/ ker f * by a subgroup r, via the map a E V Ta(x). The differential of this map at 0 E V is injective, and hence the map V -+ X is a diffeomorphism of an open neighborhood of 0 E V with its image in X. Thus r is a discrete subgroup of V, and the orbit of x is diffeomorphic to

V/r. .

Suppose first that f : X -4 B is equipped with a C°° section oo : B -> X. We define a C°O map 7r : TB -) X by, for a E TB b, 7r(a) = Ta(oo(b)). The image

of the zero section of TB is ao(B). Let A C TB be given by A = v 1(ao(B)). Then Ab C TB,b is the discrete subgroup of the vector space TB,b given by Ab =

it 1(ao(b)). Let Xo be the image of the map r, f# : Xa i B the projection. Then clearly Ab is canonically isomorphic to H1((f #)-1(b), Z), which is isomorphic

to Hn'1((f#)-l(b),Z) by Poincard duality. Here we use the fact that f is special Lagrangian to give a canonical orientation on the fibres of f #. Also, A, as a subset

of the total space TB, is closed as it is continuous. Since the map it is a local isomorphism, A is also dtale over B. Thus we can think of A as the dspace dtald of R',''1 f #Z, and in particular, we obtain an exact sequence of sheaves of abelian groups

-a T8 -+ X00 -i0,

where this now defines the group structure on Xo . Observe also that since A is dtale over B, Rn'1 f#Z has no sections with support in proper closed subsets of B. In particular, ff. (B, f #Z) = 0. We recall the notion of canonical coordinates on the total space of the bundle TB. Given U C B an open set with coordinates y1, ..., y,,, canonical coordinates on Tr; are yl, ... , Yn, x1, ... , x,,, where (Y1.... , yn) x10... , xn) is the coordinate repWe will use canonical resentation of the differential form Exidyi E

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coordinates consistently throughout this paper, whenever we use local coordinates to perform calculations. Note that TB always carries a standard symplectic form, which in canonical coordinates can be written as E 1 dx, A dy,. (This is the opposite of some sign conventions).

Proposition 2.3. With notation as above, let yl, ... , y,, be local coordinates on a neighborhood U C B. Then on 7U,

a'w = E dxi A dpi + 1: aij dyi A dyj i,j where the a,j are functions depending only on yl, ... , y,,. Furthermore, if ao : B --> X is a Lagrangian section, then Tr*w =

dxi A dyi

on U, and thus it*w is the standard symplectic form on 7B . Finally, if H2 (B, R) = 0,

then every section off is homotopic to a Lagrangian section. Proof. Since the fibres of TB -3 B are Lagrangian with respect to ir'w, we can locally write

n*wdxiAO,+Ea,jdyjAdyj with the B, 1-forms not involving the dx,'s and ail functions on 7B. Now the function y, induces on X a Hamiltonian vector field which, by definition of the map

ir, must be it 8/8xi. Thus c(8/8x,)(a'w) = dyi, from which we see that B; = dyi. The condition d(a*w) = 0 then implies that the functions a,j are independent of xi, ... , x,,. If ao is Lagrangian with respect tow, then the zero section of T is Lagrangian, from which we see that a,j = 0. If ao is not Lagrangian, let w' = E; dx, A dyi (locally) be the standard symplectic form on TB. Then a'w - w' is a closed 2-form locally given by Zi j aildyi Ady j, and hence is the pull-back of a closed 2-form on B. Thus if H2 (B, R) = 0, there

exists a one-form 0 on B with dO = n*w - w'. Then -0 defines a section of TB which is Lagrangian with respect to ir'w, and this maps to a Lagrangian section of f homotopic to ao. . This allows us to prove a result stated in [14). Recall that X# is the complement

of the critical locus off in X. Theorem 2.4. (Theorem 3.6 of [14]) Let X be a Calabi-Yau n-fold, B a smooth real n-dimensional manifold, with f : X --* B a C°° special Lagrangian torus fibratfon such that R"f*Q = QB and such that the singular locus of each singular fibre has cohomological dimension < n - 2. Suppose furthermore that f has a C0O section so. Then X# has the structure of a fibre space of groups with so the zero section. In fact there is an exact sequence of sheaves of abelian groups

0-R"-'f*Z-7B-4X#-0.

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Special Lagrangian Fibrations II: Geometry

Given a section a E r(U,X#), one obtains a C°° diffeomorphism T, : f-1 (U) f1 X# -+ f-1(U) f1 X# given by x P-r x+a(f(x)), and this diffeomorphism extends to a diffeomorphism T, : f -1(U) -a f -1(U). Proof. This follows from the previous discussion if we can show that the hypotheses imply two things:

(1) X# = Xo

f#Z ° R"-1f.Z.

(2)

To show (1) and (2), let Xb be a fibre, and let Z C Xb be the singular locus of Supp(Xb). We then have an exact sequence Hn-2(Z,Q) ..} He _1(Xb - Z,Q) -+ H'n-1(Xb,Q)

-+H"-1(Z,Q)=0-+Hn(Xb-Z,Q)-fH"(Xb,Q)--*0. Here Hn-1 (Z, Q) = 0 by the assumption on the cohomological dimension of Z. Thus, since H"(Xb, Q) = Q by assumption, Xb - Z can have only one connected component. Since the TB,6-orbit of a°(b) is already one connected component of

Xb - Z, we see that X6 - Z = (Xo )b. But as Xb C Xb - Z, we must have Xo = X#. We also see from the above exact sequence that there is a surjection

R,-1f#Z - R"'1f.Z -+0. This is an isomorphism outside of A. But since this surjection is in fact an isomorphism.

p° (B, R . 1 f # Z) = 0, we conclude

Note that the hypotheses of Theorem 2.4 hold if f : X -+ B is integral and satisfies Assumption 1.8. Another useful observation about the topology of singular fibres:

Lemma 2.5. If f : X -> B is integral and satisfies Assumption 1.8, and b E B with Z = Sing(X6), then Z is connected.

Proof. Let U = X6 - Z. As argued above, U °° Ra x

T"_k for

some k.

Assuming Z is non-empty, we have an exact sequence 0 = H°(U, Z) -+ H°(Xb, Z) -* H°(Z, Z) -+ H,,(U, Z) -> H'(Xb, Z)

and we wish to show H°(Z, Z) = Z, which is equivalent to the injectivity of H' (U, Z) -* Hl (X6, Z). If HH (U, Z) = 0, then there is no problem, so the only

possiblity is that k = 1 and H' (U, Z) = Z. In this case, Hr'(U, Z) = Z"'1 and thus in a small neighborhood V C B of b, the sheaf Rn-1 f#Z contains Z"-1 as a subsheaf. Thus over V, there is a T"-1 bundle T -+ B, T C X#. The fundamental class of each fibre yields a non-zero section a of RC'f # Z over B, and under the map RR f #Z -a Rl f.Z, a maps to a section of Rl f.Z which is non-zero on V - A. But the map H,1 (U, Z) -> H1(X6i Z) is the induced map on stalks, and hence is non-zero, thus injective.

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We now address the situation when f : X -+ B does not have a Lagrangian section but we assume f is integral. We recall though that Assumption 1.6 is always in force, giving the existence of a local section. This theory was developed in Duistermaat's paper [12], and is completely analagous to Kodaira's theory of elliptic surfaces, or to Ogg-Shafarevich theory.

Let X# = X - Crit(f) as usual. We now obtain a map -3 7e

-0 :

as follows. Given a fibre Xb, y E Hl (X,*, Z) 25 Hn-1(,Xb , Z), map y to the differential

V H - J 1(v)w 7

where we choose any lifting of v E Ta,a to X#. Now in the case f did have a section, we previously constructed an embedding R,-1 f#Z -s T. We compare these two constructions. Let U C B be an open set where f -1(U) -4 U possesses a section, which we can take to be Lagrangian. Using this section as the zero section we obtain an exact sequence

0 -+ A =

f #Zju-"4Th_+ f-1(U)# -a 0

as before. Now if A E Ab = Hn-1(X6 , z) = H1(X6 , z) C 7s,b via the map ti', then in local coordinates (pi,. .. , y,,) on B, A = E A dy;. Now

- f t(8/8y,)w = - f(o,...,0)

i(8/8yi)n'w

= da; (o,...,0) = at

so the two maps +fi,vb' : f#ZIu -l 77 coincide. Since A C TU is Lagrangian, we conclude that the image of R',1-If! Z in TB under v' is Lagrangian. Thus there is an exact sequence

defining J# in which J# inherits the standard symplectic form from TB. Since f : X# -a B is locally isomorphic to J# -3 B, a standard argument [12] shows that one can obtain X# -> B from J# -3 B by regluing using a Cech 1-cocycle {(U{, a;)) where o is a Lagrangian section of J# -> B over U. We call j : J# -3 B the Jacobian fabration off : X# -> B, in analogy with the theory of elliptic curves. This gives a one-to-one correspondence between the group H1(B,A(J#)), where A(J#) is the sheaf of Lagrangian sections of J#, and the set (f : Y# -+ B a Lagrangian fibration with local section and Jacobian j : J# -4 B}/ ? . In fact, this can be extended to the compactifications. We phrase this more generally for Lagrangian fibrations.

Special Lagrangian Fibrations II: Geometry

354

Theorem 2.6. Let f : X -+ B be a proper Lagrangian fibration with connected fibres, with a local section everywhere. Then there is a symplectic manifold J, called

the Jacobian of f, and a (proper) Lagrangian fibration j : J -+ B which is locally isomorphic to f X -* B, and which has a Lagrangian section. Furthermore, there is a one-to-one correspondence between the sets if : Y -+ B a Lagrangian fibration with local section and Jacobian j : J -> B}/

and H'(B,A(J#)). Proof. To construct J, choose an open covering {U;} such that f-1(U;) -> U; has a Lagrangian section of for each i. Then f'1(U{f1 U') has a symplectomorphism we will write as T;,j obtained by treating of as the zero-section and then translating

by oj, so that T;,j takes o; into oj. We construct J by identifying f-1(U;) and f'1(Uj) along f-1(Uf fl Uj), using T;,j to identify f'1(U; fl Uj) C f-1(U;) and f-1(U; fl Uj) C f(Uj). These identifications are compatible over U; fl Uj fl Uk because T j,k o T;, = T;,k . Thus we obtain a Lagrangian fibration j : J -+ B with a section, as desired. The usual regluing construction gives the 1-1 correspondence. 11

For j : J -+ B special Lagrangian and integral, one computes H1 (B, A(J#)) by using the exact sequence

A(TB) is just the kernel of exterior differentiation acting on 78', so H'(B, A(7;)) 111+1 (13, R) for > 1. From this we obtain the sequence

H2(B,R) -4 H1(B,A(J#)) -+ H2(B,Rn-'j#Z) -4 H3(B,R). In any dimension, if H2(B,R) = 0, then H2(B,Rn'1j#Z),or. C Hl(B,A(J#)). Duistermaat [12] observed that if an element a E H'(B,A(J#)) comes from an element [a'] E H2 (B, R), then the corresponding f : X -+ B can be obtained by choosing a 2-form a' on B representing (a'], and taking X = J with symplectic form w + f'a'. Any two choices of a' can be related by translation by a section. Example 2.7. If n = 3 and H2(B, R) = 0, then we have

H2(B,le'f#Z):.", S H'(B,A(

)) s

If n = 2 and J is a K3 surface, then H2(B, RR f #Z) = 0 and so there is a sequence

H1(B,RRf#Z) -> H2(B,R) -3 H1(B,A(J#)) -- 0. Remark 2.8. If U C B - A is a simply connected set, then there is no monodromy in the local system (R"'1 f.Z)I u C Th. Thus, if Al, ... , A" are sections of 7u generating (Rn-'f.Z)Ju, the fact that dA; = 0 shows there are functions u{ such that this = A; on U. The ul,... , u" form local coordinates on U, since dul,... , du" are independent. This is the standard construction of action coordinates on U. We

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will say that ul, ... , u are action coordinates for f : X -+ B on U. Canonical coordinates ui,... , u", xi, ... , x" are called action-angle coordinates. These are also the coordinates that Hitchin introduces after [18], Prop. 1. The advantage of working in this coordinate system is that the periods are now just the constant periods dui, ... , dun.

§3. Simplicity and the Leray spectral sequence revisited. Recall from [14] that a special Lagrangian T"-fibration f : X -+ B was said to be G-simple if Rp f.G

for all p, where i : B - A -+ B is the inclusion, fo = f If -tiB_ol, and G is an abelian group. This condition was crucial for getting a handle on the topology of X and the relationship between the topology of X and its dual. In [14], we only made use of Q-simplicity, while here, we will go further. We are interested in a broader range of groups G. In particular, G = R, Q, Z, Z/mZ, or R/Z will be of relevance for us. Clearly Z-simplicity implies Q-simplicity or R-simplicity, but Z/rnZ or R/Z-simplicity provides extra information about monodromy modulo in which may be valuable in trying to classify possible monodromy transformations about the discriminant locus. If f is integral, then f has connected fibres, and G = f.G = i. fo.G. We have seen in §2 that if Assumption 1.8 holds then R" f.Z = R, f.#Z = Z, and thus by the universal coefficient theorem, R"f.G = G. Thus for integral fibrations satisfying Assumption 1.8, the simplicity condition holds for p = 0 and n and any abelian group G. We now prove it holds for p = 1, under the additional Assumption I.V.

Definition 3.1. We say a point b E B is a rank k point if k = min rank f.: TX,. -+ T2,6. XEXb

In particular b is rank n if and only if f is smooth over b. We first comment on when Assumption 1.7' might hold. Lemma 3.2. If b is a rank k point, k O 0, then Assumption 1.7 implies Assumption 1.7' at b if Xb contains only a finite number of orbits of the action of TB b. Note Assumption 1.7 automatically implies Assumption 1.7' at a rank 0 point.

Proof. Consider the set Z' = {Im f.: Tx,. -- TB,bI x E Xb} of subspaces of TB,b Note that if x,y E Xb with T, (x) = y for some a, then f.,. = f.,, o (T,). where f.,x denotes the pushforward of tangent vectors at x. Thus Im f.,. = Im f.,,. By the assumption that Xb is a union of a finite number of orbits, we see then that C' is finite, and minLEL, dim L = k. Because .C' is finite we can choose a subspace T C TB,b of dimension n - k in the set C given by Assumption 1.7 which intersects every element of G' in the expected dimension. It then follows that dim f.-'(T) = 2n - k for all x E Xb. Now if B' is taken to be a submanifold of B with tangent space T at b, we see that the implicit function theorem implies that f-1 (B') is a manifold in a neighborhood of Xb.

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This makes Assumption 1.7' appear quite reasonable, at least in the threedimensional case, as the finiteness of the number of orbits for rank 1 fibres would follow from finiteness of the 7i1-measure of the singular locus. (As mentioned in §1 however, such a result is not actually known.) Theorem 3.3. If f : X -- B is an integral special Lagrangian fibration satisfying Assumption 1.7, then i.R1fo*G = R'f*G. Proof. Let

AO 9Al be the stratification of 2 given in §1. Let ik : B - Ak -+ B - AL.-1 be the inclusion. We will show using descending induction that

ik*(Rlf*G)Is-ob = (R'f*G)IB-nk_ for each k. One always has a functorial map

(R1f*G)IB-4k_, -+ ik*ik(R'f*G)IB-o,,_, = ik.(R'f*G)IB-&,, and we just need to show this is an isomorphism on the level of stalks at each point b E At,, b It Ak-1. We can choose a B' through the point b using Assumption 1.7', so that Ok n

B' = {b} and X' = f -'(B) is a manifold. This gives a diagram

B'- {b} 1i' B - Ak

B'

-.K+

ji

14 B - ik-1

i', j', j the inclusions. Let f : X' -> B' be the restriction of f. Then j*Rlf*G = Rl f.'G and in particular the stalks of Rl f.G and Rl f.G at b are the same. On the other hand, there is a natural map (ik*ik*R1 f.G)b -> (i,i1*R1 f.G)b. Indeed, an element of (ik.ikR1 f*G)b represented by a germ (U, a), a E r(U - Ak, R' f.G), is mapped to (U n B', al(anB')-{b}), aI (uns-)-{b} E r((U n B') - {b}, Rl f.'G). This map is in fact injective. Indeed, by descending induction, (R1 f.G) I s-A. has no sections over any open subset of B - Ak supported on a proper closed subset. Thus the restriction maps of the sheaf (R1 f.G)I B-o,, are injective, and it follows that the map (ik.ikR1 f*G)b _ (i*i'*Rl f*G)b is injective. We then have a diagram (R1f.G)b 1

(ik*i**R'f*G)b

(R1f*G)b

y

1

(i*i"R1f*G)b

so (Rl f*G)b -> (ik*ikR1f*G)b is an isomorphism if (R1f*'G)b - (i:i1*R1f.G)b is. Thus we need to show that limn

H1(f'-1(U), G) -> lim r(U - {b}, R1 f.G)

bEUs,9,

bEUCa'

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is an isomorphism. We can take the direct limit over contractible U. We will need to know what H2y6(X',G) is. By the universal coefficient theorem. there is an exact sequence

0 - HX6 (X', Z) ®z G -a HX6 (X', G) - Torz(H' bl (X', Z), G) -+ 0. Now by a suitable form of Poincare duality ([10], V 9.3), HX6 (X', Z) S5 H2n-k-i (Xb, Z),

where the latter group is Borel-Moore homology. This is computed via the exact sequence ([10] V, §3, (9))

0 -; Extl

(HHn-k-i+l

(Xb, Z), Z) -+ Hen-k

(X6, Z) -+

Hom(HHn-k

'(Xb, Z), Z) -+ 0.

Given that HH (Xb, Z) = Z and Hn-1(Xb, Z) = Hn-1(X6 , Z) is free, we see that HX6 (X' Z) __

1Z

0

if k = n - 2;

ifk
and HX6 (X', Z) is free, so that

X6(X, 'G' _

fG ifk=n-2; 0 ifk
Of course HX6 (X', G) = 0 for i < 2. There are two cases: Case 1: k < n - 2. For b E U C B', H1 (f)-1 (U), G) 9-1 HI (f'-'(U - {b}), G), by the relative cohomology long exact sequence. On the other hand, the Leray

spectral sequence for f : X* = f'-'(U - {b}) -+ B* = U - {b} yields the exact sequence

0 = H1(B*,G) -a H1(X*,G) -* Ho(B*,R1f;G) -+ H2(B*,G) -> H2(X*,G). This last map is injective as X* -> B* can be assumed to have a section. Thus we obtain the isomorphism

H1(f'-1(U),G) -+ He(B*,R1f.G), and taking direct limits gives the desired isomorphism. Case 2. k = n - 2. In this case, the relative cohomology exact sequence gives f o r b E U C B', X * = f'-1(U - {b}), B* = U - {b},

0 - H1(f'-'(U), G) -3 H'(X*,G) -+ HX6(f'-1(U),G) -4 H2(f'-1(U),G) In this case HX6 (f'-1(U), G) = G. We have a commutative square H1(B*, G)

H{b} (U, G)

I1

H1(X*,G) - 4 H2X6(f'-1(U),G)

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358

showing the map H' (X *, G) -3 Hxl, (f'-1(U), G) is surjective, yielding

0-+H1(f'-1(U),G) 4Hl(X*,G) -+G-}0. On the other hand, the Leray spectral sequence for f' : X* -> B* gives 0 -i H1(B*, G) °° G -* H1(X *, G) -* Ho(B*, R1f;G) -+ 0. Putting these two sequences together one finds that the map

H1(f'-'(U), G) -+ Ho(B*,R1f*G) is an isomorphism, and hence

lim H1(f'-1(U), G) -- lim r(U - {b}, R1 f*G) is an isomorphism. We next try to understand R"-1 f*Z. In any event, in §2 we have seen that if J : X -4 B is integral and satisfies Assumption 1.8 then RI-1 f #Z 9 R"-1 f*Z and j_i°o (B, Rn-1 f #Z) = 0. Since there is an exact sequence 0 -->

(B, R"-1 f*Z) --> R"-1 f*Z -+ i*R"-1 fo*Z - HI (B, R"-1 f*Z) -+ 0,

we have already shown at least that the natural map R"-1f*Z - i*R"-1fo*Z is injective. However, to show surjectivity, we need a more delicate understanding of the inductive structure of the singular locus. If b E B is a rank k point, it would, for example, be sufficient to show that there is a fixed-point-free Hamiltonian Tk action in a neighborhood of X6 in order to achieve a sufficiently strong inductive description of the singular fibres. However failing to prove such a result, we will make an ad hoc argument in the n = 3 case. Nevertheless, in any dimension we have

Lemma 3.4. If f is integral and satisfies Assumptions 1.7 and 1.8, Ao the set of rank 0 points of B, io : B - Ao " B the inclusion, then io*iO*R"-1 f*Z = R"-1 f*Z.

Proof. If U is a contractible open neighborhood of b E Ao, we have an exact sequence

0 -4 H"-1(f -1(U), Z) -+ H"-1(f -1(U - {b}), Z) -4 HX, (f -1(U), Z) °f Z -+ 0. In addition, the map

ilm H"-1(f-1(U), Z) -+ lim r(U - {b}, R"-1 f*Z) AEU

AEU

is injective. We just need to show this map is surjective. jFYom the Leray spectral

sequence for f : X* = f-'(U - {b}) -> B* = U - {b}, we obtain a map w :

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359

H"-'(X*,Z) -+ r(B*,R"-1f.Z). We first show this map is surjective. Indeed,

given a section a E r(B*, R"-' f*Z) C r(B*, T;), let M C X* be the circle bundle

over B* whose fibre at b E B* is Ro(b)/Za(b) C X6 = Ta,b/(R"''f*Z)b. Let [M] E H"-' (X *, Z) be the image of 1 E HM ' (X', Z) 4 H° (M, Z) in H"-' (X * , Z). It is then clear that cp([M]) = a. Thus p is surjective. We now have a diagram

1

0 - lim.H"-'(f '(U),Z) --i

0-

ltim-.,H"-'(X* , Z)

-p

limes r(B*1, R"-' f.Z)

-->

1W

Im p o $

Z

-0

coker a -a 0

I

0

I.

and we wish to show that coker a = 0, so that a o cp' = W o,6 is surjective as desired. By the snake lemma, kerv kerr". By the Leray spectral sequence for f : X* -a B*, the image of H"-' (B*, Z) .+ H"'' (X *, Z) is contained in ker y. On the other hand, by the commutativity of

Hn-'(B',Z) -+

H{b}(U,Z)

I H"-'(X*,Z) -> Hxb(f-'(U),Z) it is then clear that ker W surjects onto HXb (f (U), Z) = Z, so ker cp" = Z and coker a = 0.

Theorem 3.5. If dim X = 3, f : X -, B an integral special Lagrangian fibration satisfying Assumptions 1.7 and 1.8, and if the fibres Xb for rank 1 points b are a union of a finite number of T;,6-orbits, then f is Z-simple. Proof. The hypothesis of this theorem implies the hypothesis of Theorem 3.3, so the simplicity condition holds for p = 0,1 and 3. In the notation of the proof of Theorem 3.3, we need to show i1*ii(R2f*Z)IB_o. = (R2f.Z)IB_oa, as Lemma 3.4 then allows us to complete the proof of simplicity. Choose b E'i, b 0 Ao, and choose a 2-dimensional disk B' passing through b as in the proof of Theorem 3.3,

f: X = f-' (B') - B'. As in that proof, we just need to show that H2(Xb,Z)L- lim H2(f'-'(U),Z)-3 lim r(U-{b},R2f.Z) b6UCaB'

bEUcB'

is an isomorphism, and we already know this map is injective. Note also that the cokernel of this map is torsion-free: since R2 f; Z C TB* IB', if an integer multiple of a section of Ref.'Z over U - {b} extends to a section over U, the section itself extends. Now B* = B' - {b} is a punctured disk, and hence (R2f.,Z)IB. is a local system determined by a single monodromy transformation T. By Poincard duality,

(R' f.'Z)IB. has monodromy tT. Since r(B*,R2f/Z) = ker(T -1), we see that

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Special Lagrangian Fibrations II: Geometry

rank r(B*, R2 f; Z) = rank r(B*, Rl f; Z) = rank H' (Xb, Z) by simplicity for p = 1. Thus it will be sufficient to show that rank H1(Xb, Z) = rankH2(Xb, Z) to show the above map is an isomorphism. Since b is a rank 1 point, the singular locus of Xb is a union of circles, each being a closed orbit of the action of 7;,b on Xb. Since there are assumed to be only a finite number of such orbits, each S1 is a connected component of Sing(Xb). But since Sing(Xb) is connected by Corollary 2.5, Z Sing(Xb) S1. Now we also have exact sequences

0iHH(Xb-Z,Z)->H$(Xb,Z)->H'(Z,Z)_*0 for all i, the exactness for i = 1 shown in the proof of Corollary 2.5 and for i = 2 shown in the proof of Theorem 2.4. From this we see that if H2 (X6 - Z, Z) = Z, then H1 (X6, Z) = H2(X6, Z) = Z, while if H,2: (Xb - Z, Z) = Z2, then H' (Xb, Z) _ H2(Xb, Z) = Z2, completing the proof in these cases. Finally, suppose HH (Xb - Z, Z) = 0. Then H,-(Xb, Z) = 0 and it follows as in the proof of Theorem 3.3 that HX6 (f-1(U), Z) = 0, so one obtains from the relative cohomology sequence a surjection

H2(f'-1(U), Z) -+ H2(f'-1(U - {b}), Z) -4 0. The argument of the proof of Lemma 3.4 shows that H2(f'-1(U- {b}), Z) -+ r(U-

{b},R2f;Z) is surjective, and hence so is H2(f'-1(U),Z) -+ r(U - {b},R2f;Z). Taking direct limits, one concludes that H2(Xb, Z) = ker(T- I) as desired, showing simplicity. But notice in fact this case can't occur, since as H2(X6, Z) = 0, we also have H' (Xb, Z) = ker(tT - I) = 0, contradicting H' (Xb, Z) = H' (Z, Z) = Z. . Remark 3.6. If f : X -* B is a Z-simple special Lagrangian T3 fibration, we obtain some restrictions on the cohomology of a singular fibre Xb. Clearly H°(Xb, Z) = H3(X6, Z) = Z, so if b{ = rankH'(Xb, Z), we will say for the duration of this remark that Xb is of type (b1i b2). Clearly bl, b2 < 3, and if b2 = 3, then Xb is non-singular. If b1 = 3, then b2 = 3, since A2 H' (Xb, Z) C H2 (X6, Z), and so Xb is non-singular. This also shows that if b, = 2 then b2 > 1, and a similar argument shows that if b2 = 2, then b1 > 1. Thus the possible values for (b1, b2) are (2, 2), (2,1), (1, 2), (1, 1), (0,1), (1, 0) or (0, 0). We describe the probable topology of an integral singular fibre with each of the above possible cohomology groups. (2,2) Such cohomology is realised by a fibre of the form II x S', where II denotes a Kodaira type 11-fibre. (1,2) S' x T2/{pt} x T2. This was seen in the example in §1 of [14].

(2,1) This was described in [14), Remark 1.4. Identify T3 with the solid cube [0,1]3 with opposite sides identified. Then take T3/ -, where (xi, x2, x3) (xi,x'2,xs) if and only if (x1,x2,x3) = (xi,x2,xs) or (XI, X2) = (xl,x2) E 8([0,1])2. This fibre is singular along a figure eight. (1,1) There are two possibilities here. The fibre could be II x S1, where II denotes a Kodaira type II fibre, or it could be T3/ _, where (xi, x2, xs) ^' (xi, x2, xs) if and only if (x1,x2,xs) _ (Xi,x2,x'3) or (XI, X2) _ (Xi,x'2) E {0,1} x [0,1] or (xl, x2), (x'1, x2) E [0,1] x {0,1}. (This is the same as contracting one loop of the singular figure eight in the (2,1) case to a point).

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361

(0,1) T3/ ..., where (xl,x2,x3) (xi,x3,xs) if and only if (xl,x2,x3) = (xi,x3,x3) or (xx, x2), (x',, x2) E 8[0, i]2. (This is equivalent to contracting the singular figure eight of the (1, 2) case to a point). (1,0) T3/..., Where (21,x2,x3)'" (xl,x2,x3) if and only if (x1,x2,x3) _ (xl,xg,x3) or (xl, x2), (xl, x2) E 8[0,1]2 and xs = x3 or xs, x3 E {0,1}.

(0,0) T3/..., where (XI,x2,x3) - (x1,--2,=3) if and only if (x1,x2,x3) _ (xl,x2,x3) or (xl, x2, x3), (x'1, x2, x3) E L9[0' 1]3. (We are contracting the boundary of [0,1]3 to a point. This is topologically a sphere). One notes that for each fibre of type (m, n), there is a fibre of type (n, m) which should then be its dual. (In particular, the fibres of type (2, 2) and (1,1) should be self-dual).

I cannot prove yet that this provides a complete classfication of integral threedimensional singular fibres, but it seems to be a reasonable conjecture. In addition, these types of examples extend to higher dimensions, and one finds a much wider range of possible topologies, which nonetheless exhibit the desired duality. Having proven Z-simplicity in some cases for special Lagrangian fibrations, we wish now to return to a more careful study of the Leray spectral sequence for special Lagrangian fibrations, with special consideration of the role torsion plays. First we make some observations on the Leray spectral sequence in any dimension.

Lemma 3.7. If f : X -+ B is a Z-simple special Lagrangian T"-Sbration with a section, then in the Leray spectral sequence for f , El,I = Ej and El"-1 = Ei '1. In addition, the Leray filtration yields a surjection H"(X, Z)t,r9 -+ (El,n-1)tors. Proof. The only possible non-zero differential to or from E? 1 is d2 : Ei,l -+

E3,0 = H3(B, Z). But since f has a section, the map H3(B, Z) -* H3(X, Z) is injective, and thus d2 = 0. Thus E?, = E;" . Next, E12,"-1 = Hl (B, R"-l f *Z), and recall from [14] that that Hl (B, R"-l f *Z)

is the group of sections of f modulo homotopy, with a fixed section, say s0, the zero section. Then for any section o, the cohomology class [a] - [ao] E H" (X, Z) and the element [a] E Hl (B, R"-1 f * Z) representing the section coincide up to sign in En-1 by (14], Theorem 4.1 (which holds with Z coefficients if f is Z-simple). Thus Ei°n-1 =,"-1, and if H" (X, Z) = F° D Fl D is the Leray filtration

on H"(X,Z), then F1/F2

EiSince F°/Fl -- Z, H"(X,Z)t,,.8 = F,,,, and

thus there is a map H"(X, Z)t,r, -+ (912,"_1 )tore. To see this map is surjective, suppose a is a torsion section of f. Then a must be disjoint from ao. Indeed, if x E a n ao, let U C B be a small open neighborhood of f (x) E B in which or is represented by a section & E r(U,TB), such that &(f (z)) = 0. Then if m is the order of the torsion section a, m& is a non-zero section of RI'1 f #Z which is zero for at least one point, which is impossible. Thus a and ao are disjoint. By [14), Theorem 4.1, TT : H* (X, Q) -+ H* (X, Q) is a unipotent operator, but on the other hand T,' = I since a is m-torsion. Thus T; = I, so [a] = [ao] in H"(X,Q) and [a] - [ao] is in fact a torsion element of H"(X, Z). This shows that the map H11 (X, Z)t,,.8 -- (Ei,"-1)tora is surjective.

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Special Lagrangian Fibrations II: Geometry

Even if in general the Leray spectral sequence for f does not degenerate, the above result might be sufficient for many applications; as we will see in later sections,

H' (B, Rl f.R) and Hl (B, R"-1 f.R) play important roles in mirror symmetry. Note that if Z-simplicity fails because f has reducible fibres, we expect the second part of the above result to fail. We now focus on the three dimensional case.

Proposition 3.8. Let f : X -+ B and I : X -+ B be dual Z-simple special Lagrangian Ts fibrations, and suppose that H' (B, Z) = 0. Then Hl (X, Z) = 0 if and only if Hl (X, Z) = 0. Proof. Since Hl (B, Z) = 0, H2 (B, Z) is torsion, so

rank(H'(X, Z)) = rank(H°(B, R' f.Z)). But since X is Ki hler, the first betti number of X is even, so if H'(X, Z) i4 0 then rank(H°(B, RI f. Z)) > 2. The wedge of two independent sections of R' f.Z yields a section of R2/,Z, and R2 f.Z °-° R'/Z, so H° (B, Rl f.Z) # 0, hence H'(9, Z) # 0. Repeating the same argument interchanging X and X gives the result.

Theorem 3.9. Let f : X -3 B, f : X -4 B be dual Z-simple special Lagrangian T3-fibrations with sections, and assume H'(X, Z) = 0. Then the Leray spectral sequences for f and f with coefficients in Z degenerate at the E2-term, and

rankzH'(B,R1 f.Z) = rankzH3-'(B,

R'-9 f.Z).

If in addition f and f are R/Z-simple, then Tors(H'(B, R' f.Z)) LM Tors(H4-'(B, Rs-i f.Z)).

Note that the additional assumption of R/Z-simplicity is not a particularly strong one. Indeed, we have seen that given suitable regularity hypotheses, we have only failed to show the G-simplicity condition for p = 2. But Z-simplicity implies

R2f.Z 9. R' f.Z and R2 f.R

Rl f.R, whence R2f.R/Z L- Rl f.R/Z. Hence the

existence of the dual fibration f and the R/Z-simplicity condition for p = 1 implies

itforp=2.

Proof. The E2-term of the Leray spectral sequence, by the arguments of [14], Lemma 2.4, looks like

(3 . 1)

Z 0 T2'3 Zh"2 2 0 e Tl Z"' 1 a T2,2 0 Zh' ' ' a Ti,' Zh ' 2 a T2,1 Z T2'0 0

Z T3,2 T3,1

Z

with a similar diagram for 1. Here T''1 = H' (B, Rf f. Z)t.,... We are using Hl (X, Z) = Hl (X, Z) = 0 to obtain the zeroes on the left column and the top and bottom rows. Clearly the desired statement on ranks follows.

Mark Gross

363

Recalling from [14], §2 that Z-simplicity implies R` f.Z

R1-4 f.Z, it follows

that if 2'",i = H'(B, R' f.Z)tor,, then T',J -- TO-1. Since f has a section, the argument of [14], Lemma 2.4, combined with the

degeneration statement of Proposition 3.7, shows that the above spectral sequence degenerates. The same holds for f. As for the statements about torsion, clearly T2,3 -- T2,0 -- H2(B, Z). To show the rest, we use Poincare-Verdier duality (see for example [9]). In any dimension, applying duality to the map s : B - pt, we obtain isomorphisms

RHom(RI'(R' f.Z), Z) H-1

Applying

R.l'R

n (R' f. Z, Z[n]).

to both sides, we obtain

H-1(RRom(RF(R'f.Z), Z))

(3.2)

Ext"-i (R' f.Z, Z).

The left hand side is easily computed by choosing a complex of projective Z-modules quasi-isomorphic to Rr(R' f.Z) and applying the algebraic universal coefficient theorem, which yields exact sequences 0-+ Ext' (Hi+t (B, R' f. Z), Z) - H-J (RHom(Rr(R' f. Z), Z)) -> Hom(H' (B, R' f.Z), Z) -0.

The difficulty in applying Poincard-Verdier duality for non-locally constant sheaves is the difficulty of comparing the Ext's and the cohomology groups. We will do this

forn=3,i=2. First on Bo one has i (R2 fo.Z, Z) Rl fo.Z by Poincard duality, and if i : Bo - B is the inclusion, the natural map

i.I of (R2fo.Z, Z) - p.w(i.R2fo.Z,i.Z) is an isomorphism. Thus by Z-simplicity, Hom(R2 f.Z, Z) Rl f.Z. Also, R/Zsimplicity implies Z/mZ-simplicity for any m, and so a similar argument shows j m,(R2 f.Z, Z/mZ) "R1 f.Z/mZ. So by the local-global Ext spectral sequence one has a five-term sequence (3.3)

0-;H'(B,R'f.Z).- Ext'(R2f.Z,Z) ->HO(B,E 11(R2f.Z,Z)) - H2(B,R'f.Z) -> Ext2(R2f.Z,Z).

I claim that y+,a.1 (R2 f.Z, Z) is a torsion-free sheaf. Indeed, apply to the exact sequence

om(R2 f.Z, )

We obtain an exact sequence

0--tR1f.Z T4R1 f.Z--+R1 f.Z/mZ--3E 1(R2 f.Z, Z)-E 1(R2f.Z, Z). But in fact Rl f.(Z/mZ) Rl f.Z/mR' f.Z since R2 f.Z is torsion-free, so we see the multiplication by m map is injective on 9291(R2f.Z,Z). Thus this sheaf is torsion free.

364

Special Lagrangian Fibrations II: Geometry Now the left hand side of (3.2) is ZVI' ® T2,2 for j = 1 and is Zh"' ® T3'2 for

j = 2. Thus by (3.2), rankzExt' (R2 f.Z, Z) = h1'1 = rankzH' (B, R' f.Z), so in (3.3) the fact that H° (B, E1(R2 f. Z, Z)) is torsion-free shows that the map

Ext1(R2f.Z,Z) -+ H°(B, xt1(R2f.Z,Z)) is zero. Thus we have

HL(B,R1f.Z) °` Ext1(R2f.Z,Z) and

0 - H°(B, Ext1(R2f.Z, Z)) - H2(B, Rl f.Z) -* Ext2(R2f=Z, Z) exact. Thus (3.2) implies T1,1 ?, T3,2, and by using the same argument for T1,2 c T3,1. In addition, T2'1 = H2(B,R'f*Z)tors C Ext2(R2f.Z,Z)tors = T2'2. On the other hand, from (3.1) and Proposition 3.7 there are exact sequences (see the beginning of the proof of Theorem 3.10 for details of the first sequence) 0 -+ T2,1 -3 H3(X, Z)tors

T1,2 -+0'

0 --> T3'1 i H4(X, Z)tors

T2'2 _y 0,

and in addition for any oriented 6-manifold H3 (X, Z)torg Q' H4(X, Z)tors by Poincar6

duality and the universal coefficient theorem. So #T2'2 = #T2'1 and this implies T2'2 g T2'1. .

Theorem 3.10. Let f : X -> B, f : X -* B be as in Theorem 3.9, and assume in addition that B is simply connected. Then there are non-canonical isomorphisms Heven(X Z[1/2]) Hodd(X, Z[1/2j)

Hodd(X,Z[1/2]) Heven(X, Z[1/2])

In general, there are short exact sequences

0 -i H2(B, RL f.Z)tors .4 H3(X, Z)tors -) H' (B, R2f.Z)tors -+ 0 0 --> H2(B, RI 1*Z)tors

H3(X, Z)tors

H1(B, R2!.Z)tors .4 0

and if they split, the above isomorphisms hold over Z. This happens, for example, if both X and X are simply connected. In any event, #Hevea(71r Z)tors = #Hodd(X, Z)tors

#Hodd(X, Z)tors = ##Heven(X> Z)tors

Proof. The Leray filtration on H3(X,Z) is

OCFo=Z[T3] CF1CF2CFs =H3(X,Z).

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365

Since the cohomology class [T3] is primitive in H3(X, Z), Fo C F1 is a primitive em-

bedding and (Fi)tore = (Fi/Fo)tor, = T'2,' in the notation of the proof of Theorem 3.9. It then follows from Proposition 3.7 that there is an exact sequence 0 -+ T2'1 -+ H3(X, Z)tor, -+ T1'2 -+ 0.

First assume this sequence for X and X splits, so H3(X, Z)tor, = T2,1 e T1,2. Now H3(X, Z)tor, °-` H4(X, Z)tor so H4(X, Z)tor, = T2,1 s T1"22 also. Putting this together we see that Heven (X Z)tor, = T 1,1 s T2,1 e T' 2 and

Ho`td(X,

Z)tor, =Ts,2 a T2,1 ED T' 2

=T',' ,D T2,1 EDT' 2

On the other hand =T32 ®T2,2 s T1,1.

=Heven(X Z)tor,

Since T2,2 -- T2,1 by Theorem 3.9, we are done. Note that if X and X are simply connected, 0 = T1,1 cg T1,2 and 0 = Tl,1 T1,2, so the sequences trivially split. If the sequences don't split, then it is still clear

that the numerical equalities hold. Finally, we finish the proof of the theorem by showing the sequences do split over Z[1/2]. We define a map dt : T1,2 OZ Z[1/2] -, HI (X, Z[1/2])t,,.,. Indeed,

for a a torsion section off : X -} B, given by v E H1(B,R2f.Z)tor we can set 0(a) = (logT,)([ao]) E H3(X,Z[1/6]) where [so] E H3(X,Z) is the cohomology class of the zero section. Here log T,* = (TQ - I) - (T; - 1)2 + (T, - I)3, as (TQ - I}4 = 0 by [14] Theorem 4.1. As observed in the proof of 3Theorem 3.7, (T; - I)([ao]) _ [a] - [ao] is torsion and represents the class a E H1(B, R2 f.Z)tor,

Thus (T, - I)3([ao]) E H3(B,Z) C H3(X,Z) must be zero as this element is also torsion. So (1ogT;)Qaol) _ ((T; - I) - 2(T; - I))([ao]) E H3(X,Z[1/2]). Furthermore, ¢(a +T) = (logTQ+,)([ao])

= (log T; oTT)([ao1) = (log T; +logTr)([ao])

= 4(a) + 4(r), so ¢ is a group homomorphism. Thus ¢ gives the desired splitting over Z[1/2]. The problem that arises with two-torsion in the above theorem seems at the moment to be unavoidable, and does not make the statement very aesthetically pleasing. The heart of this issue is the following: given a two-torsion element in H2(X, Z), is the square of this element non-zero in H4(X, Z)? If it is non-zero, then it follows from [14], Theorem 4.1, that if a is the corresponding torsion section

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Special Lagrangian Fibrations II: Geometry

of f, then [o] - [ao] is not two-torsion, making it unlikely that the exact sequences in Theorem 3.10 split over Z. Example 3.11. The only example personally known to me of a Calabi-Yau threefold X with H3(X, Z)t,,.5 non-zero is the "Enriques threefold", obtained by dividing out K3 x E with the involution (t, -1), where t is the Enriques involution on the K3 surface. (See [4] for calculations of the cohomology of this threefold.) This possesses a special Lagrangian Ts fibration f in much the same way as the examples in [16]. In fact H3(X, Z)tor, = Z/2Z, and the fibration f : X --> B is seen to have a torsion section, so H1(B,R2 f.Z)tor, = Z/2Z. B Remark 8.12. If f : X -3 B does not have a section, then

does, as well as the Jacobian j : J -> B of f . Then j and f are dual, and

Theorem 3.11 applies to this pair. In addition, R f. Z = R' j. Z, so the E2 term of the Leray spectral sequence for f is still given by (3.1), and H2 (B, Rl f.Z)tor, Lv H2(B,R2f.Z)tor,. However now the spectral sequence won't degenerate. Since there is no class in H3(X, Z) restricting to the generator of H3(Xb, Z), one of the differentials from H°(B, R3 f.Z) must be non-zero. This now gives an explanation for the speculations of [5) of the role that H3(X,Z)tor, should play in mirror symmetry. There it was argued on physical grounds that the Kahler moduli space of a Calabi-yau threefold was in fact H2(X,C/Z), so in fact it had one component for each element of H3(X,Z)tor,. Thus the complex moduli space of the mirror X should have a similar number of components. It was not clear what this meant. But in our current context it is clear: if H2(B,R1f.Z)tor, # 0, f : X -> B will have many dual fibrations, only one of which will have a section. All the other duals are obtained by twisting the one with a section. The set of such dual fibrations is classified by H2(B,R1f.Z)tors = H2(B,R2f.Z)tor, and we have seen that these groups are related to H3 (X, Z)to,,. However, they do not necessarily coincide with H3 (X, Z)tor and this will lead us to modify the definition of B-field in Conjecture 6.6. We end this section with some comments concerning the de Rham realisation of the Leray spectral sequence, which we will need later. In general, let f : X -> S be a smooth map (i.e., f. always surjective) of differentiable manifolds. Then one has an exact sequence of vector bundles

0 -+ f*AlS -4 flx -> fZXIS - 0. This gives rise to a filtration FPf

on the de Rham complex,

n- = F°Sl, such that

F11ZX

P

... 9

FPstx a/FP+1fIX ° = A \ f"0s ® A \ flX,sThis filtration gives rise to a spectral sequence with P

E

9

®A HXIS)

Mark Gross

367

with differential d° being exterior differentiation along fibres of the map f. Then

EPq = r(S, as 0 R4f.R), and d' is the Gauss-Manin connection VOM. Here, given a form a E FPfl 7 with d°a = 0, the element represented in EPq is given as follows. For v, ... , v, E The, the form (e(v1, ... , vp)a)I x, defines a well-defined cohomology class in HQ (Xb, R). This yields an R9 f.Ii-valued p-form in EPq. Next EPq = HP(S, RQ f.R),

which coincides with the E2-term of the Leray spectral sequence for f . Now let us specialise to the case that f : X -+ B is a special Lagrangian T"fibration, fo : Xo -4 B0 = B - A the smooth part of the fibration. On Bo, McLean's result gives a natural isomorphism Tan °` R1fo.R 0 Coo (Bo).

Mclean also defines an n-form on the a base. This is given by

e(v1i...,vn) =

JXe

(-t(vl)w) A... A (-t(vn)w),

In canonical coordinates, this is A, O(o/8y1,...,8/O1yn)=J

1r

dr1 A...Adxn.

X,

Here Xb is oriented canonically by fl as it is special Lagrangian. Of course, this form goes to infinity at singular fibres. Another way to think about this form is via integration along fibres of w"/n!. Since =(-I)n(n+1)/2dy1 A ... A dyn A dxl A ... A dxn, w"/n! f.(wn/n!) = (-1)n(n+1)/2e.. Thus in particular,

/ w"/n! = (_I)n(n+1)/2

x

We can identify isomorphisms

/fin-q

/r e. B

TBo With ABo by contracting with 0, and so obtain

A'

Thus the EP n_q term of the de Rham realisation of the Leray spectral sequence for fo is r(Bo,1Bo (& CB° ). is symmetric if Definition 3.13. A cohomology class [ao] E

there is a representative ao E Eli n_1 = r(Bo, flgo ® ABI.) of [ao] which is invariant words, ao E under the involution of f1B0 ®flB' given by a ®b H b ®a. In other

r(Bo,S2flB0) C r(Bo,flg0 ®ci 5. A cohomology class a E H'(B,R"-'f.R) is said to be symmetric if its restriction to H'(Bo, Rn-1 fo.R) is symmetric. The following result, which identifies the symmetric cohomology classes, will be useful in §6 in studying the role of the B-field.

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Special Lagrangian Fibrations II: Geometry

Theorem 3.14. A cohomology class [ao] E Hl (Bo, R"-1 fo.R) is symmetric if and only if [ao] A [w) E H2 (Bo, R"fo.R) is zero. In particular, if H3(B, R) = 0, then all elements of H1(B, R"-1 f. R) are symmetric. Proof. We work in action angle coordinates, on a neighborhood U, so that the Gauss-Manin connection is the trivial connection. Thus if y,,.. . , y" are the action coordinates,

VGM(E fJJdyr ® dyj) = E d(fijdyj) ® dyj In addition, in suitably ordered action-angle coordinates e = dy1 A .. A dy",from (-1)i-1 dxi A' A dxi A A dx" which one sees easily that the cohomology class of on a fibre Xb is identified with dyj E fab. Choose a representative ao E r(Bo, n' ® Il') for [ao], so on U we write

ao = E aijdyi ® dyj,

(3.4)

i,j with ao symmetric if and only if aij = aji. Locally, this class can also be represented

by the n-form on f -'(U)

cio = E(-1)j_'aijdyi A dx, A ... A dij A ... A dx",

ij

from which we see that

ao A w= E(-1)j-laijdyj A dxj A ... A dxj A ... A dx" A dx j A dyj i, j

= E(aji - aij )dyj A dyj A dx, A ... A dx,,. i<j

Thus we see ao is a symmetric representative for [ao] if and only if ceo A w = 0 in r(Bo, n 2B, 0 f ). Now ao A w represents the cohomology class [ao] A [w] E H2(Bo,R"fo.R). This is zero if ao is symmetric. Conversely, suppose [ao] A [w] is the zero cohomology class. Then there exists a 6 E r(Bo, flgo 0 Los,) such that VGMI3 = ao A w. ,Q also gives rise to an element ,6' E r(Bo, noo ®St1) by using the map a ® b i-; b ® a. The following claim then proves the theorem. Claim. ao +VGMR' is a symmetric representative for [ao]. Proof. Locally write /3 = Ei f3idyi ®1, so 0' = Ej /3j1® dyj. Now VGM(/3) _ ao A w means that 8yi

8yj

aji - aij.

Also,

al) +VGMI9' = E(aij + 'flj)dyj 0 dyj.

ij

ON

But

- (aji+ 19A) aij+aA 8yj Oyj

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so ao + Vom,Q' is a symmetric representative for [ao]. Remark 3.15. There is a related map which partly explains the interest in symmetric cohomology classes. If f : X -3 B is an integral, Z-simple special Lagrangian fibration with a section, then there is a natural inclusion R"-1 f. R/Z

'

A(X#). This is induced from the inclusion R"-1 f Z " A(TB) and the inclusion obtained from this by tensoring with R: R"-1 f.R y A(78). Thus one obtains a natural map H1(B, R"-1 f.R/Z) -4 H1(B, A(X#)). To analyse this map, consider instead the map H1(B, R"-1 f.R) -+ H1(B, A(Tl )) S5 H2 (B, R). This in fact coincides with the map H1(B,R"'1 f.R) -4 H2(B,R"f.R) °-° HZ(B,R) obtained by cup-product with -w. It is easiest to see this over Bo: in this case, we have resolutions 0 -+ R"-1 fo.R -i StBo ®12Bp and 0 -i A(TB0) - Hao . The map R"-1 fo.R -+ A(TB.) extends to a map of complexes (l 0 illBO --* fI BO given by a ® Q H a A fl. Thus ao as in (3.4) is mapped to E{
-

which by (3.5) can be identified with -ao Aw. Thus the map HI (BO, R"-1 fo.R) HI (Bo, A(TB0 )) coincides with the map A(-w). It is not difficult to show this holds over B also, but we omit the cohomological argument. Thus we see that the symmetric cohomology classes are those that map to zero

in H1(B,A(X#)).

§4. The symplectic form on D-brane moduli space. Let B be a moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold X of dimension n, along with a universal family

U if C- XxB B

Let p : U -+ X be the projection. For the moment we assume f is smooth, so that points in B are parametrizing only smooth special Lagrangian submanifolds. We do not assume these submanifolds are tori. Here dim B = dim H' (Ub, R) =: s. The D-brane moduli space is the space of special Lagrangian submanifolds along with a choice of flat U(1) connection modulo gauge equivalence, i.e. an element of H1(Ub, R/Z). Thus the D-brane moduli space M should be R1 f.(R/Z). The prediction from string theory is that M should be a complex Kahler manifold, so

we need to understand how to put these structures on M. As long as there are no singular fibres to deal with, Hitchin has described how to put a complex and Kahler structure on M. Here, we will describe a more coordinate independent way of describing the same Kahler form (i.e. a symplectic form in the absence of a complex structure). This both allows us to compute the cohomology class represented by this symplectic form and in principle should allow one to extend this construction to singular fibres. In what follows, we assume the fibres off are special Lagrangian with respect to the symplectic form w and holomorphic n-form Il, with the standard normalization wn/n! = (-1)n("-1)/2(1/2)"SEAS. We also use a holomorphic n-form fl" normalised by S2

H"=fTU. p-Q

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Special Lagrangian Fibrations II: Geometry

We use fl,, instead of fl for several reasons. First, we do not want the symplectic structure on M to depend on w, but only on A. If we multiply w by a constant, we must also rescale fl. If we rescale 11 and use fl instead of fl" in the construction below, then the symplectic form Co we construct on M also changes. Secondly, this normalization fits with the usual form of the mirror map as described in item (5) of the introduction. To obtain a symplectic form on M, we define a map

Rlf*Z-4 r in such a way that the canonical symplectic form on TB descends to a symplectic form on TB/Rl f*Z. We follow Hitchin's suggestion of computing the periods of Imp*fl". Now (R' f*Z)b °t Hl (Ub, Z) °-` Hn-1(Ub, Z), so for y E (Ub, Z), map y to the differential V H - 1, c(v)lmp*f

where again we choose an arbitrary lifting of v to U.

Lemma 4.1. The image of Rl f*Z in TB is Lagrangian with respect to the standard symplectic form on 78. Thus M = TB/Rl f*Z inherits this symplectic form, which we will call w.

Proof. See Hitchin's paper [18]. His proof is as follows: in a small open set of B, choose r C U a family of n -1-dimensional submanifolds representing a section

of R'f*Z over U, 7r : r -+ B the projection. Then the section of TB obtained by taking periods with respect to r is just the 1-form -7r*((p* Imfl)Jr). Since Cl is a closed form, so is this push-down, and hence -7r*((p* Im flJr) is a Lagrangian section of TB with respect to the standard symplectic form. We will now clarify what the cohomology class of w is. To do so, we will

compare the Leray spectral sequences for f and f : M -* B, but will use the de Rham realisation of these spectral sequences discussed in §3, which we can do as f and f are smooth. Our construction yields a canonical isomorphism Z) HI (Mb, Z) and hence a canonical isomorphism H"-1(Ub, Z) 111 (Mb, Z), which yields a canonical isomorphism (4.1)

Rn-'f*RL" R'f*R.

Since Imp*fl" restricts to zero on the fibres of f, Imp*fl" E FPAU, and since dfl" = 0, Imp*fl" in particular gives rise to a ,class [Imp*fl"] in Ell,.-, = r(B, 01 R"'1 f*R). Now on M, w E Fl(f1J, as Co restricts to zero on the fibres of f, and thus w determines an element [w] of Ei 1 = r(B, rlB®R1 f*R). Via the isomorphism (4.1) we can identify Ell,n-, and t1',1.

Proposition 4.2. Under this identification, [w] _ [Imp*fl"]. Thus in particular, they represent the same class in Hl (B, Rl/*R) '_ H1 (B, Rn-1 f*R). Proof. A section of r(B, uZB ® R"'1 f*R) associates, to any tangent vector v E Ta,b, an element of Hn-'(Ub,R). Specifically, [Imp*fl"] associates to a tangent

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vector v E Ta,b the cohomology class represented by t(v)(Imp"Stn). To determine

what cohomology class this is, we choose a basis yl...... , of calculate the periods

J

Z) and

t(v) Imp'Stn.

On M, [w] E r(B, f)B 0 R'j.R) is similarly represented by V r-y t(V)w,

a n d y , ... , 7. also form a basis for HI (Mb, Z) by construction. Recall that we embedded Hn-1(Ub, Z) = Hi (Ma, Z) in TB,b by mapping Ys to the differential

v H - J (vImpfn. Now choosing local coordinates yl,... , y, on the base, x1, ... , xs, yl, ... , y, canonical coordinates on TB,

7.

1'(8/8yi)(D _ - I dx = f t(8/0Yi) Imp'fln 7:

7i

by construction. Thus [w] = [Imp"Stn].

We recall here for future use: Observation 4.3. (McLean (221) BecauseTB,b is naturally isomorphic to the space of harmonic 1-forms on Ub, there is a metric h on TB coming from the Hodge metric. Precisely, for v E Ta,b, -(t(v)p*w)Ix, is the corresponding harmonic one-form, and -* (r.(v)Imp*(St))Ix and we define, forv,w E Ta,b,

h(v, w) = -

Jx,

(t(v)p'w) A (c(v) Im p'1).

This is a Riemannian metric on B.

Specialising down to the case that f : X -+ B is a special Lagrangian T nfibration with possible degenerate fibres, the above method gives us a way of constructing an open subset of the dual fibration along with a symplectic form on that

open set. On B0 = B - 0, we have defined an embedding Rl fo.Z -+ r,. This allows us to define Xo via the exact sequence

0-+R1fo.Z-+ r,-+.Yo-30 and X'o acquires a symplectic form w inherited from the canonical symplectic form on TB,. Next we need to prove

Special Lagrangian Fibrations H: Geometry

372

Conjecture 4.4. The embedding Rl fo.Z '-p TBo extends to an embedding

R'f.Z-*TB. If k# is defined as TB/R' f.Z, then X# is a manifold with symplectic form w inherited from the standard symplectic form on TB. Furthermore, X# can be B extending !#, on which w compactified to a manifold X with a map extends to a symplectic form on X.

This involves first understanding the asymptotic behaviour of the periods as one approaches 0, as well as understanding the issue of compactification.

If this conjecture holds and f and f are both R; simple, then it is easy to see that [CO] E Hl (B, R3 f.R) coincides with [Im SZn] E Hl (B, R"-1 f.R). Indeed, Hl (Bo, R"-1 f.R). by Proposition 4.2, these classes agree in H' (Bo, R' f.R) However, since f and f were assumed to be simple, Ho (B, Ri f.R) = 0 for i = 0 and Hl (B - 0, R' f.R), 1, and ditto for !. Thus there is an injection H3 (B, R f.R) and so the classes [w] and [Im f l] agree also in H' (B, R' f.R) H3 (B, R"-1 f.R).

f : X -+ B is not the only possible Lagrangian fibration we might construct. This fibration possesses a Lagrangian section by construction. By Theorem 2.6, any element of H1(B, A(X#)) gives rise to another, locally isomorphic Lagrangian fibration g : Y -> B. See also Remark 3.12 and Conjecture 6.6. Remark 4.5. Having constructed a symplectic form w on X, or on an open subset of X, w"/n! defines an orientation on X. Thus we can check that this agrees with the choice of orientation on X made in [14], Convention 4.3. First, note that in having fixed fl, we have fixed an orientation on the fibres off : X -+ B. If we have fixed canonical coordinates y,,. .. , yn, xl.... , xn, then 13X6 = Vdxl A . Adz" with V a real function and either V > 0 or V < 0. By changing the order of the variables y-, we can ensure V > 0, and then dyl A-. Adyn yields a canonical orientation on B. We will always assume our coordinates are so oriented. Note that this orientation on B is the same as that induced by the n-form © on Bo. Now let us check Convention 4.3 of [14] is correct. Recall that the convention of [14] for the cohomology class of a fibre, [X6], was that

a=J aA[X6]. JX6

X

With a = [11], we then have 0<111=11ZA[Xb].

X6

X

We can take [Xb) to be the pull-back of a nowhere zero n-form o

A dyn. Then 11 A [X6] = V f dxl A ... A dxn A dyl A ... A dyn, while ,"/n! =

(-1)"("-l)/2dzl A ... A dxn A dyl A ... A dyn. Since V > 0, we need sign(f) = (-1)n(n-1)/2. Thus we take [X6] locally to be of the form (-1)"("-1)/2I f idyl A .A dy,,. Now the dual class of (-1)"[X6] in H"(B,R"f.R) will be locally represented by something like (-1)n(n+r)/2gdy, A ... A dyn A dxl A- . A dxn, g> 0. This is the

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same sign as w"/n! = (-1)"(n+1)/2dy, A. Ady Adxt A- Adxn, so the orientations agree.

In the case of torus fibrations, we now describe an alternative way of putting a symplectic form on lo. We do this by providing an alternative description of the embedding R1 fo. Z -+ Tge, using the Riemannian metric h on the base defined in Observation 4.3. Since we obtained in §2 an embedding R"-1fo.Z -r TBo, we obtain dually an embedding R1 fo.Z -+ Tae. We will then use a normalised form of

h to identify Ta, and Tee. First, given that there is an identification between Ht(_Xb,R) and Te,b, via v E Te,b i-+ -t(v)w, we have also a canonical identification of H"_,( b,R) with Tm,b, via Poincare duality. In fact, write Xb = I//A, V = TB b, A = Ht (X6, Z) C 1', V" = Tn,b, A" = {cy E V"lw(A) C Z} C V". There is a canonical identification of

A" with Ht (X6, Z). On the other hand, the identification An A a Z determined by the orientation on Xb gives us a natural identification of with A" via the perfect pairing n-1

Ax

(X6, Z) =

A"-1

A

n

AA-+AA? AA-24Z.

(Note: Whenever we use Poinca 6 duality, there is an arbitrary choice of order in this pairing which may affect the signs of the isomorphisms. This was seen in [14[, where certain conventions were chosen. Here we also make a choice, and keep in mind that we could just as well have chosen the pairing A"-1 A x A -+ Z.)

Proposition 4.6. If a E H` 1(Xb,R), ry E Hn_1(.X6,Z)

A" C Te,b via the

above identification, then

a=-I

6(7)w n U.

Jy

Proof. We compute both sides using local action-angle coordinates. Let , yn be action coordinates as in Remark 2.8. Then the lattice A C V = Ti ,b form a dual basis for is generated by et,... , en, e; = dyt, and then A" C Ta,b, e; = 8/ft- Suppose a = dxt A ... dxi A ... A dxn. Then y1i

-!

A

6(e; )w A a= J I dxi A a Xb

Xb

_ (-1)i_tati On the other hand, the isomorphism between A" and A"-1 A identifies e, with A en. The latter defines an oriented n - 1-torus in X6,

(-1)1 'e, A ... A et A

namely the quotient of the subspace of V spanned by e1..... et-t, e+,,.. . , en by the lattice generated by these vectors, and a is just the integral of a over this torus, which is clearly (-I)i-16ij. Thus

f,a=e,

l Cb

(ei').Aa

Special Lagrangian Fibrations II: Geometry

374

and the result follows from linearity.

This now gives us the opportunity to rephrase the embedding Hl (Xb, Z) y TB,b. Let h be the normalised metric on Bo given by h (v, W) = h(v, w) .fX,

Then we have

Proposition 4.7. For -y E Hl (Xb, Z) _u

Z) = A" C TB,b, the 1-form

v H - t(v) ImC coincides with the 1-form -hn(y, ). Thus the embedding R1 fo.Z -+ TBo previously defined coincides up to sign with the embedding R1 fo.Z --> Teo composed with the isomorphism Teo °-` TBo induced by the Riemannian metric h,,.

Proof. By Proposition 4.6,

r

- f &(v) Im Sl = XJ

(y)w A (v) Im a

7

(-

c(-y). A I(,) Imfl

1

JX6

Xb

h(-f, v) fX, Cl

.

In fact, we see that h also describes the class [Im

E Ej1,,,-1 = r(Bo, SlB0

R"-1fo.R): Proposition 4.8. Under the isomorphism El,,,-1 = r(Bo, n, ® R,-1fo.R) °C r(Bo, flB0 ® Cl 0) given in §3, [Im

coincides with h E r(Bo, Szn'

C r(B0, nB0 ®flBo

(Xb, Z) Proof. First note that for a point b E B, ei,... , e;, a basis of A" C TB,b, [Im fln] associates to a vector v E TB,b the class of t(v) Im fl, E H"-i (Xb, R), and in terms of the periods,

f t(v) Im fl = -

JXy

t(e{)wA t(v) Im Z

= h.(ef,v) Thus, in action-angle coordinates as used in the proof of Theorem 3.14, corresponds to the element of r(Bo, CB0 ® R"-1 fo.R) given as

ij

h (8/8flt, 8/8yj)dyi 0 (-1)i-1 dx1 A ... A cdx f A ... A dxn

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which coincides with

E

8/bpi)dpi ®dy, E r(Bo, s2ctep)

i,3

as desired.

§5. Complex structures on special Lagrangian torus fibrations. Recall from [18] the following: (for K3 surfaces, this was noticed in [31]; see also [30].)

Theorem 5.1. Let X be a real 2n-dimensional manifold. If 0 is a complex-valued C0O n-form on X satisfying the three properties (1) d11= 0;

(2) Al is locally decomposable (i.e. can be written locally as 01 A are 1-forms); (-1)n(n-1)/2(i/2)"fl A A > 0 everywhere on X, (3)

A On where

then fl determines a complex structure on X for which Il is a holomorphic n -form.

Theorem 5.2. Let X be a real 2n-dimensional manifold. Suppose w is a symplectic form on X and SZ is a complex-valued n-form on X such that (1) 1) satisfies the conditions of Theorem 5.1; (2) w is a positive (1,1) form in the complex structure of Theorem 5.1; (3) (-1)"("-1)/2(i/2)1'11 A fl = w"/n!. Then 11 induces a complex structure on X such that w is a Kdhler form on X whose corresponding metric is Ricci-flat.

Now let f : X -- B be an integral special Lagrangian fibration with a Lagrangian section, so that in local coordinates, w takes the standard form. Let 11 be the holomorphic n-form on X normalised so that Im f ix, = 0 for all b E B, w"/n! = (-1)"("-1)/2 (i/2)"11 A fl, and fxb 11= Voi(Xs) with respect to the metric induced by the Kahler form w. As before, we set On =11/ fx, f1. Following the suggestion of [18], since Cl is locally decomposable, we can write,

for local coordinates yl, ... , yn on B as usual,

0 = V f\(dxi + E oiidyi), i

where V is a real function of yl,...,xn and Vlx,dxl A - A dx,, is the volume form on Xb, while Nij is a complex valued function. Using Remark 4.5, we will always assume that V > 0. The forms of type (1,0) are spanned by the 1-forms Bi := dxi + Ei,i ti fdyi. Thus the entire complex structure is encoded in the matrix (fij). We now look to see how the conditions of Theorems 5.1 and 5.2 translate into conditions for the functions V and fii:

Calculation 5.3. (-1)"("-1)/2(4/2)"11 A f1= V2 det(Im f )w"/n!.

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376

Proof. Write

(=VB1A...A8,,. Now

- Qij)dxi A dyj + .. .

Bi A Bi

(ImsidYi) + .. .

_ -2idxi A Thus

( ASt = V2(-2)ni"det(Im/3)dxl Adx2 A...Adyn so (-1)n(n-1)/2(i/2)n% A 12

= (-1)n(n-1)/2V2 det(Im/3)dxi A dx2 A ... A dyn

On the other hand, wn/nl = (_1)n(n-1)/2dx1 A ... A dyn,

hence the result.

Calculation 5.4. w is a positive form of type (1,1) if and only if ,Q is symmetric and Im,Q is positive definite.

Proof. We first examine the condition that w is of type (1, 1), i.e. we can write w = 1 E h,8, A ej

i,j

Now

6 ABj =dxj Adxj +dxi A (>!JL.dYk) - dxj A (EtkciYk) k

+ E /3ik#j/dyk A dyl k,i

and

hijIJjkdyk - Ehji)3jkdyk

EhijOiABj=E'(hi*j-hji)dxiAdxj+EdxiA(

i<j

i

j,k

j,k

+ E hjj/3jkRjidyk A dyl i,j,k,i

Thus in order for w = E dx j A dyj = E hi j9j ABj, we must have, in particular,

hij = hji, i.e. h is symmetric. On the other hand, since w is real, hij = hjj, and thus the matrix h is real. Also,

I= 2(h,3-h/3) = hIm,B.

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Thus we have h = (Im 1)-1, and Imp is symmetric. To ensure the last term vanishes, we need

E hij,3 j, = E hjajjk,

ij

i,j

or equivalently, the matrix tph/ is symmetric. Now

°,6ha = (Re t/3 + i Im,8)(Im f3)-1(Re $ - i Im p)

= (Ret13)(Js)-'(ReO) +iRe/3 - iRetf + Imp, while

9h/3 = (Re to - i Imp) (IMO)-'(Re a + i Imp) = (Ret/3)(Im0)-1(Re/3) +iRetf3 - iRe/3+Im/3, so symmetry of t/3h4 is equivalent to Re t0 = Rep. Thus w is of type (1,1) if and only if 0 is symmetric. In addition, to ensure w is a positive (1,1) form, h = (Imp)-' must be positive definite, so Im fl must be positive definite. The real problem is understanding the condition do = 0. This is the heart of the difficulty, and we will return to this shortly. We first connect CI to the description of the choice of almost complex structure given in the introduction, namely as a choice of horizontal subspaces of an Ehresmann connection and the choice of a metric on the fibres.

Proposition 5.5. The matrix

is the matrix of the metric (gij) on the fibres of f. For a point x E X6*, J(Tx,,x) is spanned by the tangent vectors (Im)9)-1

{e/8y j - E Re pijO/8xi11 < j:5 n}, i

where J : Tx m -a Tx* is the almost complex structure induced by ft.

Proof. Since w = 2 Ei j hijBi n Bj with h = (Im p)-1 is the Kahler form of the metric, the Kahler metric itself is g = EhijBi ®Bj. Thus gij = g(8/8xi, 8/8xj) _ hij, giving the interpretation of Imp. Next, let J be the almost complex structure on Txr induced by St, and tJ the almost complex structure on TX*. Since the space spanned b y 01, ... , On is the +i eigenspace oft J at a point x E X#, the cotangent space 7'x,. decomposes as VI ®V2 with VI = V2 =

since Imp is with tJ(Vi) = V2 and tJ(V2) = V1. Note that V2 = span invertible. Thus also Tx,,, = V1 ®V2 , Vi the annihilator of Vj, with J interchanging

378

Special Lagrangian Fibrations II: Geometry

Vi and V,-. Now

VZ = span(8/8x1, ... , V1 = span({dxi +

V1 = span({8/8yi -

Re,Qiidyi})

Re,0ii8/8xi}).

Thus we see that Rep determines J(Txb,,,) = V1 as claimed. Summarizing, we now have

Theorem 5.6. Specifying an n-form ) on X# satisfying properties (2) and (3) of Theorem 5.1 and properties (2) and (3) of Theorem 5.2 and such that f# is a special Lagrangian fibration with respect to 11 is equivalent to specifying (1) A metric (gij) on each fibre of f # : X# -+ B.

(2) A splitting Tx# = Tx#/B ® Y, Where Tx#/B is the subbundle of TX, with Tx#/B,: = Tx#,=, and .T is a Lagrangian subbundle of Tx# . Proof. Proposition 5.5 shows that 11 specifies the metric on the fibres and a splitting as desired with 2 = J(Tx#/B) This is clearly a Lagrangian subbundle since Tx#/B isConversely, giving a splitting of the exact sequence

0--->Tx#/B- 4Tx#--r.F->0 determines Re,B for us. Indeed, at a point x E X#, with local coordinates as usual, such a splitting gives a map a :.F.. Tx#,x, and there is a matrix (bii) such that

8(p(8/8vi)) = 8/89.i - 'E bii8/8xi. We take Re,l i = big. Note that the symmetry of the matrix bi j is equivalent to s(F,) being Lagrangian. Thus specifying (1) and (2) is equivalent, in local coordinates, to giving f , = b21 + igii, where gii is the metric on the fibre. Then we must have, with

Bi = dxi + F ,3iidJi,

for some real function V. Calculation 5.4 then tells us that w is a positive (1, 1)form in the corresponding almost complex structure, since by construction ,5 is symmetric and Imp is positive definite. Calculation 5.3 shows that wn/n! = (-1)n(n-1)/2(i/2)nflAf2 if and only if V = det(gii) since ( det(gi'))adet(gii) = 1.

We have now seen how the n-form fl can be determined by choosing a matrix _ (Qii) = (big + igi'), so that

n=

det(gij) A(dxi + E fjiidvi)

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If Q is chosen to be symmetric and Im,Q positive definite, then we have seen that dfl = 0 implies both the integrability of the almost complex structure induced by it and the R.icci-flatness of the metric induced by this complex structure and the standard symplectic form w. At first sight, the condition dfl = 0 looks very complicated if one proceeds by brute force and tries to compute the exterior derivative of Sl. In fact, this condition can be simplified, and we wish to examine this here. We note that the calculation below of dfl is quite similar to calculations carried out in [29] and [30] for an analagous situation in the study of deformations of complex structures on Calabi-Yau manifolds. However, we will introduce a formalism using differential operators to make the calculation easier. To begin, first note that the integrability of the almost complex structure de-

termined by fl is a weaker condition than dfl = 0. Let us first understand this weaker condition.

Theorem 5.7. The almost complex structure induced by fl is integrable if and only if BalkQ _ 801j 8131k _ 813, 8xy

Nij

qq

8x{ N`i`)

8yk

8yj

for all j, k and 1.

Proof. Writing as before

ft=V01A...A0,,, the almost complex structure is determined by the fact that should span the space of (1, 0) forms. To show that the almost complex structure induced by 81,...,8, is integrable we need to show that d9i is of type (2,0) + (1,1) for all i. Since

n

p Bi=dxi+E,3ijdyj, j=1

01, ... , 8,,, dyj, ... , dy form a basis for the space of 1-forms, and thus we can write

dB{ _ E 1A,10, A Oj +>B{j0i Adyj +E 1C jdyj Adyj. id

ij

+,j

The almost complex structure is integrable if and only if C{j = 0 for all i, j and I. Here A! and C! are skew-symmetric matrices. Note d81=

ij

861j dxi A dyj

8xi

+

ij

01611

8y{

dyi A dyj

Since dOi contains no dxi A dxj terms, we must have Alij = 0, and then B,j = 8(3ij/8xi. Then the almost complex structure is integrable if and only if dO1 =

Bi50i A dyj

is '96" dxi A dyj + i 8xi

L1LQikdyk A dyj

8xi

Special Lagrangian Fibrations II: Geometry

380

which holds if and only aplk {

8,81j ax. Qij - ax Pik =

8131k

+

-

8I31j

,

This is the desired condition. We wish to rephrase this condition. We are going work locally for the moment,

fixing coordinates yl,...,y,, on an open subset U C B, and consider all forms as living on U x R^ with coordinates yl,... , yn, xi ,... , x". We then write

f3 _E,Bijdyj ®8x{ E r(U xR",f`fa®?r/B)

ij

(This is not a coordinate independent expression. The correct coordinate independent expression would be 8 dx{ ® +,B E r(.f-J(U),nz (& Tx/B), 8x but for practical purposes it is more convenient to work with the above expression.)

Definition 5.8. For expressions of the type v = Ej dyj ®vj, w = Ej dyj ®wj with vj, wj vector fields, vi = E7 { vu8/Ox{, wj = Ei w{j818x{, we define [v, w] = E[vl, wm)dyl A dym I'm

where

(vu

IV,, W-I = i,t

awjm

Svjl

8x{ - wim ex{

8 ax j

is the usual Lie bracket of vector fields. We also set

dyv =ijk E 8?k dyk A dyj ® 8x{ We then obtain

Theorem 5.9. The almost complex structure induced by O is integrable if and only if

40 -2x,81=0. Proof.

\

dy,8 = ,

r
8yjt - a-8 p.,j J dyi n dyj 8x1

while 2

'a. 80t1 ax{ a `-

dy1 n dy. xi

Comparing with the formula of Theorem 5.7, we see that the two formulae are equivalent. Our next goal is to prove

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Theorem 5.10. dfl = 0 if and only if the almost complex structure induced by f is integrable and dfl C F2nn 1. Thus, locally, one just needs to check that the coefficients of dyi n dxl A- Adz,,, 1 < i < n in dSl are zero, and also check the integrability condition. To prove this theorem, we must introduce some additional algebraic structure to accomplish the calculation. We continue to use a choice of local coordinates, and work on the space U x R'2, f : U x R" -+ U the projection. Let T. be the subbundle of the tangent bundle of U x R" generated by 8/8xi i ... , 8/8x,,, and let 52y be the subbundle of the cotangent bundle generated by dyl, ..., dy,,. For q > 0,p:5 0, set 9

-P

OPU'4=r(Afla®nT:) and set flo = dx1 A ... A dxc. Then there is an isomorphism P

I$ n,

U R-)

where OPU.R., denotes the sheaf of C°° p-forms on U x R". This isomorphism sends

8 (D v to 8 A t(v)1 o. In particular, r(1<) °-` ®y-o

OuP'P

For a 0/3 E SlUq, a' 0 /3'pE flU'° , we can define the product

(aAa')0(/3A/3')E1GtjP'R+9'. This satisfies the commutation relations

(a ®N) ' (a' ®N) = (-1)PV +q9' (a' 0 $') . (a 0 $). fl, This gives us a bigraded ring structure on ® JU°. Note that the subring ®P= o n-AP U is in fact a commutative ring with 1, and 1 E A' corresponds to no under this isomorphism.

Lemma 5.11. Under the isomorphism (5.1),

n (dxi+E/3iidvi) j

i=1

where ,0 =

i,j

=exP(P):=EO1IPi P=O

8 Oiidili ®8xi

in the notation introduced above.

Proof. This is a straightforward though slightly tedious calculation. Here is where all the signs must be dealt with correctly. For this and subsequent calculations it is convenient to keep in mind that t(8/8xI)flo = (-l)Mdxl., where

T`={1,...,n}-IandM=#{(i,j)IiEI,j EI*,i>j}.

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382

The next step is to turn f1X into a double complex. We have exterior differentiation d : F(fl'uxa^) -+ r(12j 1Rn),

and under the isomorphism (5.1), it is clear that d(fU°) C 0U 1'9 ®0UQ+1. Thus we can write d = d,, + dy, so that (flu', d, dd) defines a bicomplex, with dx : flU°

S2U 1'9

dy.fWdliftUV+1 One checks that

dyr 0 (ax Ldx;).

d.(dya ® a)

(5.2)

i=1

(Here, (8/8xi A8/8x{)Ldx; = 8/8XI, and 8/8nxLdxi = 0 if i 0 1.) Let D : ftu -> flu be a graded endomorphism of flu := O u. We now recall what it means for D to be a differential operator of order < r. Put on fnu 0 flu the anti-commutative algebra structure given by (a ® b) (a' ®b') =

(_1)(dega') (desb)aa' ®bb'.

Here the dot is the standard dot product, keeping in mind flu is bigraded. This turns flu ® flu into a bigraded anti-commutative algebra. Let A : flu -> flu ® flu be given by A(a) = a ®1- 1®a. We define

CD:fru -rflu by

r obD(a1, ... , ar) = m o (D (9 idpu) (I A(ai)) i-1

Here m(a ® b) = ab. We say D is a differential operator of order < r if f" 1

is

identically zero. Note that

02 (a, b) = D(ab) - D(a)b -

(-1)(dega)-(degb)D(b)a

+ D(1)ab

and

-FD(a, b, c) = 4D (a, bc) - (1D2 (a, b)c - (_l)(deg

c) 4D (a, c)6.

(See [20], §1.) As usual, the composition of differential operators of orders < r and 8 is order < r + 8.

Definition 5.12. We set dx : flu -+ flu to be the operator acting on f2U° by (_1)p+9+1d=,

Lemma 5.13. dz is a differential operator of order < 2 and dy is a differential operator of order < 1. Proof. That dy is a differential operator of order < 1 follows immediately from the definition, while (5.2) shows that d'z can be written as a sum of a composition of

Mark Gross

383

two operators: differentiation in the direction 8/8xi and a®Q H ao (-1)r*',3Ldx2. One checks easily that these are each first order operators, and hence d. is second order. Now define, for a,)3 E flx,

so (5.3)

d. W) =

Since 1d, = 0, we obtain [a,Q7] =

(5.4)

y]Q,

and from Lemma 5.13,

dy(a/) = dy (a)Q + We note that this definition of bracket is an extension of Definition 5.8:

Proposition 5.14. If a,# E fl-1'1, then [a, Q] as defined above agrees with the earlier definition.

Proof. By linearity, we can assume that a = fdyi8/8x; and Q = gdyj8/8xk. Then dx(a#) + d.(a)# + dx(Q)a

d . (f 9 d y i A dyl

8 -exi8 A 8 ) + d. (f dyi

88x;

xk

xk )9dyi 88

88xk)fdyi 8 +dx(9dyl a axi

(8

9 + f i99

dyi A dyt

8x; + (8x; g+4 8x;) dv' A dyi ftk

x8x;

A dyi 8xk + f8kdyi A dyi - 9dy, 8x;

=f 8x- dyi A dyi 8xk - 8xk 9dyi A dyl ex;

proving the desired equality.

Proof of Theorem 5.10. First suppose that d0 = 0. Then the almost complex structure is integrable by Theorem 5.1, and obviously dfE C F'If1X 1. Conversely, suppose dfl C F2 nn1 and the almost complex structure is integrable. We use the isomorphism (5.1) and Lemma 5.11 to write ft = V exp(Q) for

Special Lagrangian Fibrations H: Geometry

384

some Q E ft-1,1, V E 000. To show that d51= 0, we need to show that each graded piece of d) is zero, i.e.

dy(V/3")+(n+I)!d.

(VQ"+1)=0,

for all n>0.

This is equivalent to

4(v#")

=+

nldx(VQ"+1)_

The fact that dfl C F2f2X 1 is equivalent to (5.6) for n = 0. Note that this states (5.7)

dy(V) = d:(VQ) = [01 V) +4[/3)V

by (5.3), since d'z (V) = 0. Observe that since [B, ] acts as an ordinary (non-graded) derivation on the commutative ring S. fl-P.P, we have [8,,6n] = n[,B,

We prove by induction that (5.7) implies (5.8)

d',(VQ"+1) = (n+1)dy(V)Q" + n(n2 1)[Q,Q]VQ"-1.

Indeed this is true for n = 0, by (5.7). Then dz(VQn}1) = d':(Q VQ")

_ V#"] +dx(Q)VQ" +d'z(V/3' )Q _ [Q, V]Q" + n[i6, #]V,6"-1 + dx (Q)l'Q" + dx (vQ")Q dy(V)Q" + n[Q, #]V#"-1 + dz (V/3")Q

by (5.7), so by induction the desired result holds. Next, we note using the integrability condition dy,B =

and (5.8) that

dy(VQ") = d,(V)Q" + dy(Q")V = dy(V)Q" + n4(Q)VQ"-1

n+1 Z This proves (5.6), and hence the theorem. Next I would like to reinterpret the equations we've seen above so that they may look more natural. As we have seen earlier, b = Re,6 defines an Ehresmann connection whose horizontal subspaces, given by the subbundle .F, are Lagrangian. In local coordinates, this connection is determined by

b = E bit dy 8x; f ®ReO. to

Mark Gross

385

It makes sense to define the covariant derivative with respect to this connection.

This will be an operator Vb : r(f*1lB ®TX*/B) -> r(f *ms 1 (9 TY*/B) defined by

Vba:=da-[b,a].

It is easy to check that this definition is now independent of the choice of coordinates. The curvature tensor of V6 is then Fb E r(f *S2B TY*/B) given by

F6:=d,b- 2(b,b]. It is easy to check that Fb = 0 if and only if the horizontal distribution .7 is integrable. Of course, an Ehresmann connection gives rise to parallel transport along a path contained in Bo; we say a family of p-forms on the fibres of fo : Xo -* B0 is parallel if it is invariant under parallel transport. If in local coordinates over

U C B0 this family of forms is written as a = Er frdxr, fr a function on f-1(U), a is parallel if duct -Gba=0; by this we mean A8a

.'ef - 4E, bij8/esia = 0 for each j. (For a similar treatment of Ehresmann connections, see [21]). We can now rephrase the integrability conditions in a more invariant way.

Corollary 5.15. Let (fur) = (b{,, +ig'f), so we write Q = b+ig-1, V =

etg.

Then dS! = 0 if and only if

F6+2[g-1,91]=0

(5.10)

Vbg-1 = 0

(5.11)

dxl,..., dx are harmonic forms on each fibre

(5.12)

Vdzl A ... A dx is parallel.

Proof. The first two equations are the real and imaginary parts of dy$ z [/3, i3] = 0. The last two are (5.7) broken up again into its real and imaginary parts. We remark here that in the study of Rica curvature in the context of Riemanaian submersions, some similar structures arise. See [8], Chapter 9.

Special Lagrangian Fibrations II: Geometry

386

Corollary 5.16. Suppose dIl = 0 and Vb is Bat. Then the metric g is Bat along the fibres.

Proof. By (5.9) we have [g-1,9-1] = 0. We work on one fixed fibre Xb with coordinates x1 i . . . , xn. Let gi be the vector field E; 94J 8/8x;, so g', ... , gn form a basis for TX, at each point of X6, and [gi,gi] = 0. Let w1 i ... , wa be the dual basis of one-forms. Then dwi(j,gk) = gu(wi(gk)) - gk(wi(gi)) - wi([gJ,gk])

=0 since w; (gk) = 6ik is constant. Thus w1, ... , w are closed 1-forms, and so w; = dcp, for a function cp, on RI, the universal cover of X6. cp, will be a linear function

plus a periodic function. Since wi = Ei gijdxj, we see that g,j = 8cp,18xj = gji = 8cpj/8xi, so there exists a function cp on R" such that Vi = 8V/8x,, and gij = 82cP/8x,8xj. cp satisfies the real inhomogeneous Monge-Ampere equation det(82(p/8x,8xj) =Va. Next we prove V is constant. Note that V-1w1 A Awn = Vdx1 A. Adxn, the volume f o r m on Xb,so *dx,=c(g')(V-1wjn...Awn)=fV-lw1A .A Awn, and so d(*dxi) = ±gi(V-1)wi A. -Awn. Thus, by (5.11), we must have gi(V-1) = 0 for all i so V-1, hence V, is constant. Thus cp is a solution to the equation

8p det (8

x,8xj)__C,

C a constant, and cp is a function on Rn, with cp = cpquad + Vii. + cpper, the decomposition into a quadratic, linear, and periodic part. We can of course assume that coin = 0 and f Xb cp,,,,.dx1 A ... A dxn = 0. Then I claim co = and so

gij = 82w/8x,8xj is constant. To see this, one applies a standard technique for non-linear elliptic partial differential equations. Let cot = tcp + (1 Let mij(xk,) denote the ijth cofactor of the matrix (xkj), so in particular

mii(xki) =

Put

8det(xk1)

8x,i

1

oij = f mij(82Wt/8xkOxi)dt. 0

Now (82cp/8x,8xj) is positive definite. So in fact is

note that if we put hij = h(8/8y,,8/8yj), we calculate

hi j = -L. f (8/Oyi)w A i(8/8yj) Im A

= f X. dx, A CV r`(-l)k-1 gkjdxl A ... A dik A ... A dxn k

= f Vgiidxl A ... A dxn. X,

To see this,

Mark Gross

387

Since V is constant, hii = V yfXb dxl A... Adxn. Thus since (h,3) is positive definite, so is (82'pquad/82;8x3). Thus (82cpt/8xi8xD is positive definite for all 0 < t < 1, so the matrix (aij) is also positive definite. Claim:

["` aii _(_ -squad) = constant. i.i

8xi8xi

Proof.

a rat

(49-T02,,,

(det

k8xi>)

_:

a1,,t

0,,P,

Caxkaxi) 8xi8xi8t

_

ij

mia ( 82,Pt ) 82(W - coquaa) 8xk8xi /1 8xiOX J

Integrating with respect to t gives the desired result.

Now w - coquad is a periodic function, so applying the maximum principal (or minimum principal, depending on the sign of the constant), [p - lpquad is Constant. Hence ip = cquad This explains why in [18], the integrable complex structures constructed on torus fibrations had to have flat metric on the fibres given that Re,6 was taken to

be zero in that paper.

§6. The complex structure on the mirror. Having understood, at least to some extent, how one describes complex structures on torus fibrations, we now wish to explain how one should put a complex structure on the D-brane moduli space. We cannot solve this problem at present due to the complexity of the equation dfl = 0; however, here we will give guidelines as to where to look for the correct solutions. We continue with an integral special Lagrangian torus fibration f : X - B with a Lagrangian section, along with forms w and 11. In §4, we have seen how to put a symplectic form w on f : Xo -+ B which has the property that [w] and [Ira Sln] agree in r(Bo, flB0 ®R1 fo.R) °-! r(B0, BO 01 0 Rn-1 fo.R). To specify the complex structure on X, we need to construct the form fl. This should, according to the appropriate conjectures, be determined by the B-field, i.e. something like

an element B E H2(X, R/Z), and w the Kahler form on X. Certainly the first requirement for ll should be that [w] and [Im s)n] should agree on r(Bo, S1Ba R1fo.R) °-° r(B0, Sl, a R-1 10.1t), so that the double dual brings us back to X. The second requirement should involve Re An and is much less precise at this point. We can only be guided by item (5) of the introduction, but will try to be more precise later.

Proposition 6.1. If 6n is a normalised holomorphic n-form on Xo making lo : to -+ B special Lagrangian and if An is the induced normalised Riemannian metric on the base, then [w] = [Im iln] in r(Bo, flea 0 R1fo.R) °_' r(B0, fl11 a Rn-1fo.R) if and only if hn = hn.

388

Special Lagrangian Fibrations II: Geometry

Proof. Let y,. .. , yn be action coordinates for f . (We now have two different special Lagrangian fibrations, f and 1, hence two different sets of action coordi-

nates.) Fixing b E B, Xb = TB 6/A, A is generated by el,... , en with ei = dyi and A' C TB,6 is generated by ei,... , en, e; = 8/8yi. Now Xb is identified with TB b/A", where e; is identified with the 1-form -hn(ei , ) _ - E1(hn)ijdyj, and thus ei E A is identified with - Fi h;; 8/8y, E TB,b. Thus

hn(J:h;'8/8yi, 8/8yk) _ - f t( E h;; 8/8yf)(:J A t(8/8Yk) Ira (in 6

i

f t(8/8yk) Im Sln e{

by Proposition 4.6, where e; E A = Hn_1(X6i Z) _I H1(Xb, Z). If [w] = [Im f ln], this latter integral coincides with

f

- L.'

= aik-

Thus hn(8/8yi, 8/8y,) = (hn)ij, so hn = hn. The argument reverses to prove the converse.

Moral 6.2. fIn must be chosen on k so that An = hn. Remark 6.3. While this moral was deduced by beginning with a special Lagrangian torus fibration and applying the principal that double dualising should bring one back to the initial fibration, there is no reason this can't then be generalised to provide a guide for putting complex structures on more general D-brane moduli spaces. In the situation of §4, given a family u -+ B, one has the metric hn on B. Then a holomorphic n-form should be chosen on M so that M -+ B is special Lagrangian and the induced metric on B is hn. It is more difficult to say exactly what role the B-field plays. According to the conjecture originally stated in [16], and restated in the introduction, ft. should be chosen so that [f ln] - [ao] E H1 (B, R' ' f.R) should coincide with the choice of the B-field B E H1 (B, R' f.R). This provides little guidance, but we will see an example of this below which may point in the correct direction for interpreting the B-field. Example 6.4. (The Hitchin solution.) Hitchin [18] gives a choice of !ftn which

under certain assumptions about the metric hn satisfies dfin = 0. He expresses it locally in terms of action-angle coordinates, but it can be written down in arbitrary

coordinates on the base in a natural way. One takes for be the Gauss-Mania connection. This is a linear connection VGM on TB'o = (Rn-1fo.R) 0 C-(Bo), whose flat sections are the sections of Rn-1 fo.R. Taking the horizontal subspaces of this connection, it is easy to see that these descend to give a fiat Ehresmann connection on Xo, which one takes to define Ob. Note that in action coordinates 1)1,. .. , vn for fo, the Gauss-Mania connection is trivial, so in these coordinates one

Mark Gross

389

takes b = 0. Since now V6 is flat, we must have g constant along fibres by Corollary 5.16. Thus

hn(8/8y , 8/8gj) = I V4''dxl A ... A dxn/ JCe

=0,

J L.

'dxi A... A dxn

and since we want hn = h,,, we have no choice but to take 0 = (hn)jj, giving rise to a choice of fn. To check to see if dSln = 0, one checks conditions (5.9)-(5.12). (5.9) and (5.11) are immediate, while (5.10) can be checked. (5.12) is then equivalent, in this case, to fXb f)n being independent of b. But

fin=i

J L.

det§ijdx,A...Adxn

L. 1

4 dx1A...Adxn.

det(hn) Xb

Hence this quantity must be independent of b for (5.12) to be satisfied. In particular, if ul,... , un are action coordinates for fo and v1,. .. , vn are action coordinates on

the same open subset for fo, and VI ...... n,xi,...,x are canonical coordinates, then fXb dxi A A dxn is a constant independent of b, so in these coordinates, d(In = 0 is equivalent to det(hn) = constant.

A simple calculation now shows that if this is the case, then ftn coincides with Hitchin's f1' (up to a constant factor) in [18], §6, where

A' = A(dxi* +

dui).

Thus one recovers [18], Proposition 5. Of course, W = E dx, A dvi, and Hitchin views mirror symmetry as an exchanging of the roles of the two sets of action coordinates {u;} and {vi I. By [18], Proposition 3, the condition det(hn) = constant

(in coordinates v1,...,v,) is equivalent to the condition det(hn) = constant (in coordinates ui,... , u, ). This holds in particular if the metric g ii is constant on fibres.

Example 6.5. (The Hitchin solution twisted by the B-field.) Continuing with the above example, assume df1n = 0. Now choose a symmetric cohomology class

B E H'(B,R'-' ffR), with a symmetric representative b E r(Bo,flB, 0 f1l ); in action-angle coordinates for f, this will be of the form Eb jdv; 0 dvj. Now take, in this coordinate system, the n-form fln,b to be given by the matrix (,6,j) = This in fact gives a well-defined n-form on all of Xo. Note that (bj + since (bbj) is symmetric, so is fl, and thus we just need to show that dfl.,b = 0 in order to show that f'Zn,b induces a complex structure with a Ricci-flat metric. This closedness can be seen to be true as follows. If U C Bo is a sufficiently small open z1,.. . , xn for f, we can find an element set with action-angle coordinates a E r(U, no oft,) such that tGM(a) = b, since OGM(b) = 0. Here a = F, a; lOdv;,

390

Special Lagrangian Fibrations II: Geometry

withOai/Ovj =byj. Thus by symmetry ofb;j, (vl,...,vn)'- (vl)

.... vn,al,...,an)

gives a Lagrangian section or of 1-1 (U) -* U, and it is easy to see that

n,b =Ta(In, where fln is the Hitchin solution of Example 6.4. Since d(ln = 0, dfln,b = 0 also. Note also that if b' = E b' jdy; ® dyj is a different symmetric representative for B, then there exists an a E I'(B0, f0Bo ®SlE) such that VGM(a) = b' - b. As before, a gives a Lagrangian section a of fo, and To n,b = fln,b'

Finally, one sees that (In 6 - (1n represents the class B E H1(B,Rn-1 f«R), and fln,b and fln yield the same complex structure if B E,xH' (B, Rn-1 f.Z), for then there is a global section a off : X -3 B with TT)n = Ln,bOne can in fact go further. We have observed that for small open sets U C B0,

the complex and Kiihler structures on Xo induced by (ln and (n,b on f-1(U) are isomorphic, so one should think of the Kiihler structure induced by (n,b as a torseur over that induced by Zn. Specifically, fixing the complex structure An on Xo, note that the sheaf A on B0 defined by

A(U) = {sections a : U -> J(U) such that T;w = w and T, (n = (In} coincides with RI-1 f.R/Z. In fact, writing these conditions in the coordinates ui and x{ of Example 6.4, one sees that the condition T, fln = fln guarantees that the

section a is constant with respect to the Gauss-Manin connection. Thus the set of all special Lagrangian fibrations over Bo obtained from f : lo -> B0 by regluing using these translations is HI (Bo, Rn-1 fo.R/Z). Because W and &In are preserved by these translations, they glue to give forms on the twisted fibrations. Thus each element B E H' (Bo, Rn-1 fo.R/Z) gives rise to a fibration fB : Xo,B -+ Bo with symplectic form WB and holomorphic n-form (n,B. This is a potentially wider class of examples than were constructed above using symmetric cohomology classes in W (B, Rn-1 f.R); if B does not come from a symmetric class, then fB will not possess a Lagrangian section, and may not even possess a topological section. Note also that if f and f are R/Z-simple, then

H1(B,Rn-1f.R/Z) °_° H1(B,R1 f.R/Z).

This leads us to conjecture that the correct group for the B-field to live in is H1(B, R1 f.R/Z). This new proposed definition for the B-field is dependent not just on X but on the topology of the fibration, and even in the threefold case does not necessarily coincide with H2 (X, R/Z), as we saw in Example 3.11. Nevertheless, I believe this is the correct interpretation of the B-field. This now leads us to a refined mirror symmetry conjecture.

Mark Gross

391

(1) for each open set U C Bo = B - A on which both f and f have sections, f-1(U) -+ U and f'1(U) -+ U are topologically dual fibrations. (2) For b E Bo, ry E HI(Xb, Z) RI

Jy

Z) and v E Ts,b, L(v)(i, = 1 L(v) Im Sl". y

(3) For y E H.-i(X6, Z) sx- Hi (X6, Z),

1y L(v)w=

L(v) Im A

.

(4) f possesses a topological section if B E Hl (B, R1 f.R)/H1(B, R1 f.Z) C H' (B, R1 f.R/Z). In this case, if o is a topological section, then [Re sL] - [v] defines a class in Hl (B, R"-1 f.R) which is well-defined modulo H'(B, R"-1 f.Z), and agrees

with Bin H' (B, R1f.R)/Hl (B, Rl f.Z). (5) If J is the Jacobian of-k, then k is obtained from J as a s}anplectic manifold via the image ofB under the composed map Hl (B, RI f.R/Z) 91 Hl (B, R"-1 f.R/Z) -r HI (13, A(J#)) of Remark 3.15. (6) Once X and Co are fixed, A. is unique up to translation by a Lagrangian section

of J acting on X. I do not however suggest that f : X -> B is obtained as a Kohler manifold as a torseur over some basic J --* B. While this occurred in Example 6.5, there is no reason to suspect this works when the metric on the fibres is not flat. We simply don't expect there to be isometries given by translation by a section. However, in some sense this might provide an initial approximation to the correct answer. Remark 6.7. We can show a local form of the conjectured uniqueness. Suppose It is a family of holomorphic n-forms on X with respect to which a fixed symplectic form w induces a Ricci-flat metric and f : X -a B is special Lagrangian. In addition assume [Slt] E H"(X, C) is a fixed cohomology class. Then by local Torelli all S1t induce the same complex structure, so there exists diffeomorphisms ¢t : X -+ X such that Ot I1t = flo, ¢o = id. Now Oe w is a symplectic form on X inducing a Ricciflat metric in the complex structure induced by ¢t Sit = no, and represents the same

cohomology class as w, so by uniqueness of Ricci-fiat metrics, ¢t w = w. Thus ¢t is a family of symplectomorphisms. Assuming Hl (X, R) = 0, differentiating this family of diffeomorphisms at t = 0 yields a Hamiltonian vector field v induced by a Hamiltonian function H on X: L(v)w = dH. Then ImS1tIx, = 0 for all b implies that (G Im Sto)Ix, = 0 for all b. But G Im S1o = d(t(v) Im i1o), and if flo is given as usual by a matrix ,0 = (,Q,,, ), fi = b; f + igti, then a simple calculation shows

that (d(L(v)Imfo))Ix, _

Vg''60 'dx, A...Adx,,. La

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Special Lagrangian Fibrations II: Geometry

Here we have used the fact that i(8/8y4) Im no is closed. Thus we see that on each fibre, H satisfies the second order elliptic partial differential equation

OH = 0.

E

8x48xj

4,i

By the maximum principal, H cannot have a local maximum on each non-singular fibre unless H is constant on the fibre. Since the set of non-singular fibres is dense, we conclude that H is the pullback of a function on B. In particular, it follows from §2 that 4t must be translation by a Lagrangian section. Remark 6.8. One natural question is to determine the relationship between Vol(Xb) = fX, Cl and Vol(Xb) = fgb Cl, since knowledge of the latter allows us to reconstruct (1 from ft,,. We can describe this relationship if the metric is constant along the fibres off : X -+ B and f : X -+ B. Let yl,... , yn be action coordinates for f. As in Example 6.4,

hn = 90 Then

Vol(Xb) = X. f

det(y4j)dx1 A ... A dan

det(Q). On the other hand, for a fixed b E B, Xb can be written canonically as TB b/A" as in Proposition 4.7, where A" is generated by the one-forms hn(8/8y4, ), i.e. the one-forms Ej(hn)4jdyj. The metric on the fibre Xb is still given by §4j = (hn)I,

since hn = J. Thus Vol(Xb) = f

det(hi )dxl A ... A dxn 6

=

det(h;,')det((hn);,) 1

Vol (Xb)

This is the familiar "R i-+ 1/R" relationship of T-duality. If the metric is not flat, we expect some corrections to the volume, and this may affect this relationship. However, as we shall see in §7, this relationship continues to hold for K3 surfaces. We end this section with a brief discussion of the Yukawa coupling. Mirror symmetry instructs us that given a Calabi-Yau X, the Yukawa coupling on X contains information about the genus 0 Gromov-Witten invariants on X, and in particular in the three dimensional case, these Gromov-Witten invariants can be completely recovered from the Yukawa coupling on X. How do we see the Yukawa coupling in the context of special Lagrangian fibrations? Suppose that Conjecture 6.6 holds. Assume for simplicity that we only consider values of the B-field B E Hl (B, R' 1 f.R/Z) which come from symmetric classes in Hl (B, Rn'1 f.R), so that although B varies, we can fix the underlying manifold X

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393

and symplectic form and simply let i)n vary; we denote the dependence on B by writing Sln,B. Of course, we will not have SIn,B+a = Sln,s for of E H'(B, Rn-i f Z),

but merely expect that there exists a Lagrangian section u of f : X -+ B with Sln,B+n The (1,n - 1)-Yukawa coupling of interest is then, for tangent directions 8/8b,,. .. , 8/8bn E Hl (B, Rn-, f:R), the tangent space of the torus Hl (B, Rn-1 f»R/Z), 8 8 8b, , ... , 8bn

r

8n

t

11n

n B A 8b1... 8bn n,s

In local coordinates, we write

hn,B = Vn,BO1(B) I A ... A 0.(B). Note that in taking the n derivatives of 1n,B by using the product rule, all terms still containing any undifferentiated O will disappear after we wedge with SIn,B. Thus

ff

on

ffr Sln,sA 8b, ...8bn 6 . , B

a V881A...ABn/

a

VES

86,

A.../

89,

86o(n)

Writing 6i = dxi + E, A5(B)dy 8if

86i

8b,(i) -

8bo(i)

dy!

so the above integral is

I V.B E det(8 oES.

8bQ(i)

So far, this is not particularly illuminating in the general case. However, for the twisted Hitchin solutions, it is an elementary calculation to show this integral can be evaluated in terms of the topological coupling on X, as expected. This comes then in action-angle coordinates, where from observing that if 88, /8bi = Eh Vn = 1, the integrand above coincides with n

(-1)n(n'1)/2

nbikdxf A dyk

§7. K3 Surfaces. Mirror symmetry for K3 surfaces has been completely understood using Torelli theorems for K3 surfaces. We will now show that the previous material of this paper gives us a differential geometric construction of mirror symmetry for K3 surfaces,

and in doing so, we will show Conjecture 6.6 holds in two dimensions. In other words, given a special Lagrangian T2-fibration on a K3 surface, and a choice of

Special Lagrangian Fibrations II: Geometry

394

B-field, we will construct the mirror in the sense made explicit in Conjecture 6.6. This will prove to be a variant of the mirror symmetry for K3 surfaces described in [6] and [11].

To begin, let S be a K3 surface with holomorphic 2-form ft and Kibler form w corresponding to a Ricci-flat metric. We insist on the usual normalisation and this implies in particular that (Re f1)2 = (Im fl)2 = w2 > 0 and (Re Q) A (Im S1) = w A (Re Sl) = wA (Im f2) = 0. In order for S to possess a special Lagrangian fibration

there must be a cohomology class E E H2(S, Z) such that E2 = 0 and w.E = 0. We assume such a class exists, and we fix it. We take E to be primitive. Now one constructs a special Lagrangian fibration on S by the usual hyperkiihler trick, as originally suggested in [28]. First, multiply 51 by a phase e'9 to ensure Im 51.E = 0. Following the notation of [16], there is a complex structure K compatible with the Ricci-flat metric in which flX = Im 11+iw and wK = Re fl. Then special Lagrangian submanifolds on S are complex submanifolds in the K complex structure. Denote the K3 surface in the K complex structure by SK. Then by construction ERIK =

0 so E E Pic(SK). It is then standard that SK possesses an elliptic fibration. However, the class of the fibre need not be E; E might be represented by a fibre plus a sum of -2 divisors. In any event, replace E by the class of the fibre, positively oriented. E will now remain fixed, and we have an elliptic fibration f : SK -+ P1,

which we identify with a special Lagrangian T2 fibration f : S -+ B = S2. We will assume f is integral, which is true if and only if there is no 6 E Pic(SK) with 52 = -2 and 6.E = 0. This will certainly hold for general S. We now consider the spectral sequence for f over Z. Since there exists a class v such that a.E = 1, the sequence in fact degenerates and takes the form H°(B,Z) 0 H2(B,Z) 0 H1(B,R1f.Z) 0 H°(B, Z) 0 H2 (B, Z) It is also clear that H1 (B, R'f.Z) is canonically isomorphic to E-L/E. We now wish to construct a mirror fibration given the data

f : S -4 B as above and a choice of B-field B EE-/E0 R/Z _` H1 (B, R1 f.R/Z).

We first recall some facts about elliptic fibrations. See [7] for general facts about the analytic theory of elliptic surfaces. The fibration f : SK -+ P1 in general does not possess a holomorphic section; in fact for general choice of S, PicSK = ZE. However, there is a Jacobian fibration

j : JK -> P1 of f which is locally isomorphic to f, and which does possess a holomorphic section.

Proposition 7.1. There is a diffeomorphism ¢ : JK -+ SK over P1 which is holomorphic when restricted to each fibre. Fbrthermore, if U C P1 is an open subset on which there exists a biholomorphic map 1; : j-1 (U) -> f-1 (U) over U, then 0-1 o : j-1(U) -> j -'(U) is given by translation by a (not necessarily holomorphic) section of j-1 (U). Proof. This follows easily from the fact that the "CO* Tate-Shafarevich group," i.e. the first cohomology group of the sheaf of CO° sections of j : JK'4 P1, is zero.

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Thus f : SK -> P1 possesses a C°° section, and ¢ can be taken to identify this COO section of f with a holomorphic section of j, such that ¢ is holomorphic on each fibre.

We fix one holomorphic section o0 of j : J -r P1, and identify oo with the topological section ¢(ao) of f : S -+ B. Having chosen this section, we can take it to be the zero section of i : Jx -- P1 and obtain a standard exact sequence

0-aRlf.Z-4 R'f Os 04JK-40 where J# denotes the sheaf of holomorphic sections of j : JK -> P1. Here R1 f. Os" can be identified with the normal bundle of the zero section, and the map t is just the fibre-wise exponential map. For K3 surfaces, R1 f*OsK L wpi . The underlying r e a l bundle is 7', 2 . This also gives a map 7r : 752 -+ SK with 7r _ 0 o ii.

Just as in the real case, the total space of wp,, the holomorphic cotangent bundle of P1, has a canonical holomorphic symplectic form fly. In local coordinates, if z is a coordinate on P1 and w the canonical coordinate on the cotangent bundle, then S2i, = dwAdz. Furthermore, any holomorphic symplectic form on the cotangent

bundle of P1 is proportional to 0.Proposition 7.2. There is a map X : Ts, - TSa given by fibrewise multiplication by a complex constant so that, for 7r' = 'r o X, 7r'* (IlK) = Slc + f *a, where a is a 2-form on 82. Proof. First note that there are two different complex structures on the total space of Ts, in this picture: one is the standard complex structure coming from being the holomorphic line bundle Opt (-2), while the other is induced by ?r* Slx. To distinguish between these two complex structures, let J be the total space of Ts, with the standard complex structure, and let S denote the total space of Ts, with the complex structure induced by x*SlK. Let 3 S2, f : S -a S2 be the projections. Now let U C S2 be a sufficiently small open set so that there exists a biholomorphic map l; : j'1(U) -- f'1(U) over U. By Proposition 7.1, ¢-1 o f = T,

for some section a of j'1(U) -> U. and if U is small enough, a can be lifted to a section o of j'1(U) -+ U. We let denote translation by the section o (with the zero-section of TS, taken to be the origin). We then have a commutative diagram 9`1(U)

f-1(U)

j'1(U) -> f-1N) since 7r o f = O o O o T; = ¢ o T, o o =

o ,. In particular, since OK is a holomorphic 2-form on f-1(U), *rr*flK is a holomorphic 2-form on 31(U), and

hence can be written as gdwndz in local coordinates, for some holomorphic function g. This function g must be constant along the fibres of j since g must descend to a holomorphic function on the. compact fibres of j. Thus on f-1(U),

Ir*f1K = (f-1)*(gdwndz) = T"; (gdw n dz) = (g o T_a)dw A dz + hdz n df

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Special Lagrangian Fibrations II: Geometry

for some function h. Of course, g o T_; = g since g is constant on fibres. Let i : J -4 S be the identity map; this is of course non-holomorphic. The above equation shows that the (2, 0) part of i*ir*flK, being locally of the form gdz A dw, is in fact a holomorphic 2-form. In addition, this holomorphic 2-form is nowhere vanishing: if g vanishes then OK A AK = 0 at that point. Thus the (2, 0) part of i*lr*flK is proportional to fl,,, say Cfle. In addition, we then see from dSlh = 0

that h must be constant along fibres and hence lr*flK - M, is the pullback of a (1,1) form a on Pi, i.e. V*OK = CS2c + Pa.

Now let X : TS, -+ TS, be given by w y C-Lw. Then X*lr*CIK = ft,; + Pa as desired.

To sum up, replacing n by ir', we now have a map 7r : TS, --3 Sh with kernel Ri f*Z and such that lr*11K = Sty + Pa. This gives, identifying the underlying topological spaces SK and S#, lr*w = Im(I. + Pa) and a* (Im ft) = Re(I, + f *a). A local description of these forms are as follows: given complex canonical coordinates z, w on 7, z a coordinate on U, write z = y, - iy2i w = x, + ix2. The signs are chosen so that with real coordinates yi,y2 on the base, yl,y2,x1,x2 are canonical coordinates on TS,. Then

lr*w=lm(dwAdz+f'*a)

=dy2Adxi+dx2Adyi+f*Ima and

a*lint) =Re(dwAdz+f*a) =dxiAdyi+dx2Ady2+f*Rea. This completes our first goal of finding coordinates on S# in which w and Imfl have simple forms. Note that our map it : Ts, 4 S# is not the same as defined in §2 because the symplectic form is not the standard one. We will see why we have made this choice in the proof of the following theorem. Theorem 7.3. Conjecture 6.6 holds for the general integral special Lagrangian

fibration f:S-+B=S2. Proof. Choose a B-field B E (El/E) ® R/Z. Lift this to a representative B E El/E ® R. At times, we will also further lift B to an element B E El 0 R chosen so that B.[ao] = 0, where ao is the fixed topological section of f : S -3 B chosen previously. We are now trying to construct a special Lagrangian fibration f : S -> B satisfying the properties of Conjecture 6.6. Because this fibration may not have a Lagrangian section, we first construct the Jacobian J : J -) B (in the sense of §2) as a symplectic manifold. This Jacobian should be the dual fibration with a symplectic form Oij as constructed in §4. To do so, we reembed Ri f*Z ' Ts, using the periods given by Im ft,,. Now fl.= 01 611, and this embedding takes a cycle Y E Hi (Se, Z) = Hl (Sb, Z) C TS.

Mark Gross

to the one-form

v+- (c(v)Imf2 ry

1

VOl(Sb)

/ 71(v)( d.Ti A

+dx2 A

which yields the 1-form ry/Vol(Sb).

The moral is: The dual lattice is simply the original lattice scaled by a factor of 1/Vol(Sb). We have in fact chosen the map 7r so that this would happen and so make it transparent that dualising does not change the topology of the fibration.

Instead of resealing the lattice, it is easier to identify J with S topologically, and rescale the symplectic form. Since in local coordinates we want w3 = dz1 A dy1 + dx2 A dy2, by (7.2) we take on J = S w3

= (Im 0 - f* Re a)/Vol(Sb).

How do we obtain S as a symplectic manifold? Item (5) of Conjecture 6.6 instructs

us to proceed as follows. B E H1(B,Rlf*R/Z) a H'(B,Rlj*R/Z) maps to an element of HI(B,A(J#)). Having lifted B to an element of HI(B,RIf*R), we obtain in this way an element -B A [W3] E H2(B, R), via Remark 3.15, which then

maps to the appropriate element of HI(B,A(J#)). As remarked before Example 2.7, this means w = wj + j*al, where al is a form on B such that fB al = -B.[Cj3].

Now [tD3] = [Im fl,.] by construction, so it would appear that we take fe al = -B.[Imfln]. There is a slight subtlety in this: here we are representing B E

H'(B,R' f*R) as an element of El/E 0 R, but that does not mean that if we reinterpret B as a class in HI(B,Rlj*R), this class will coincide with the original B in S under our identification of S with J. In fact, the correct choice is in al = B.[Iln Stn].

We will be vague about this here, but will see this sign change more explicitly shortly. We then set Cw=iDf+j*al. Note that the choice of a1 is not important; any two choices representing the same cohomology class can be identified by translation by a section. Let S be the symplectic manifold obtained with underlying manifold S and symplectic form ', and let f : S -4 B be the same map as f : S -+ B. We note that the cohomology class of w satisfies the relation

[6] = [

Stn] + (Im fln.(B - [ao)))E.

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Special Lagrangian Fibrations H: Geometry

Next we construct the form Im Stn. The first observation is that t(v) Im fl,, must be harmonic for any v E 79,b On the other hand, the same is true of t(v)w, and t(v)CD = adz, + bdx2 for a and b constant. Thus we already know, in the 2dimensional case, the harmonic n - 1 forms. This is a crucial point in dimension 2 which fails in higher dimensions. Applying item (3) of Conjecture 6.6, in the form given in Proposition 6.1, we can now determine Stn as follows. First 6ij hnWON, 8/8yj) = Vol(Sb)

dx1 Adx2 S,

at a point b E B, as is easily computed from the definition and (7.1), (7.2). Then in order for h1 = hn, we must have -J t(8/Oy)C. A 1,(8/Oyj) Im 1Ln = j' j(S'b) fb dx1 Ads2, and a quick calculation shows this implies we locally can write Im Stn = -w + hdy, A dye,

where h is a function. But the condition that d Im Cl,, = 0 implies h is constant on fibres, so

ImCn=-W+f*a2 for some form a2 on the base. Here we see the sign reversal explicitly. It might be a bit surprising that we have obtained -w instead of w. But this is the fault of the identification we have chosen. We want [Im Stn] = [w] as classes in H1 (B, R1 f.R) HI (B, R; f.R); it

is only an accident of dimension that we have been able to identify S and S as manifolds and then compare cohomology classes directly. In fact, this sign change must occur if we want to identify S and S without changing the orientation of the fibres.

Condition (3) of Conjecture 6.6 does not tell us what a2 must be; a2 is not determined until one knows something about Re Stn. In fact, condition (4) tells us that we require [Re Stn] = [ao] - B mod E, where again we are making use of the sign reversal observed above. Now our form Im Stn constructed above satisfies [Im Stn] = [-w] mod E,

which tells us that in order for Cl, = 0, we must have (Cl,,] satisfying (7.3)

(Cl,,] = [o'o]

- (B + iw) + (1- (B + iw)2/2 + i(w.ao))E.

Here we have chosen a representative of B E El such that B.ao = 0. Thus, in particular, we need to choose a2 so that JB f a2 = -B.w + ao.w.

Mark Gross

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Again, we need to ask how much freedom we have to choose a2, given that we have fixed al. We had seen that al could be chosen to be any representative of its cohomology class, as any choice could be obtained from any other by translating w by a section of T;2. Once we have fixed the form al, however, we can only translate by sections corresponding to 1-forms a with da = 0. Let a = oldyl +a2di,2 Then

Adxl+dx2Ad 1

d2Adx1+dx2Ad,-(!+!)dYiAdY2. (2a &Y1

-9Y2

Thus T; (w) - w = - f * (d * a), where * denotes the Hodge *-operator in, say, the Fubini-Study metric on B = Pl. Thus if a2, ag are two 2-forms on B representing the same cohomology class, we just need to find a 1-forma on B such that da = 0 and d * a = a2 - c4, and then To (w + f'a2) = w + f *o'2. By the Hodge theorem, such a a can always be found. Thus we have complete freedom to choose a2i and any two choices are related by translation by a Lagrangian section. This will be the last remaining choice in the construction which is not forced on us by any items in Conjecture 6.6; thus the uniqueness of item (6) of Conjecture 6.6 will hold, given that the lack of uniqueness in the lifting of B can be rectified by changing the choice of the zero section ao. Finally, we set ImfE = (Im(ln)IVo1(Sb). It now follows immediately from (7.1) and (7.2) that as forms,

(Im1))A(Im))=wAD >0, and

(Iml )Aw=0. Thus lK = Im (1 + iw is a 2-form which satisfies the conditions of Theorem 5.1 and hence determines a complex structure on We call 9 with this complex structure As observed above, [(I.] must satisfy (7.3), so we need to look for a form Re fI such that

[Re Cl] =

V

oll(Sb) ([ao] - B - (B2 - w2 - 2)E/2).

If [Re Cl] is a Kiihler class on 9K, then by Yau's theorem, there exists a unique Kahler form Re Cl whose metric is Ricci-flat, so we only need to ensure [Re Cl} is a Kiihler lass. We first note that [Re C].[Irn fl] = [Re A).[6)] = 0, so [Re Cl] is a (1,1) class. Next we observe it is a positive class. Indeed, I 9K - B is still a holomorphic elliptic fibration, and the complex structure on each fibre .K,b is the same as that of SK,b Since [Re fl].E > 0, this shows then that [Re Cl] is in fact positive on E. As long as Pic(SK) contains no -2 classes, this shows that R,e l] is a Kiihler class. Hence we obtain a Kahler form Re fl as desired, and set

Cl=ReSl+ilmC. We make a few closing comments. First, in the above proof we can write [fi] = V o1(Sb)([ao] + 13 + i[w]) mod E

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Special Lagrangian Fibrations II: Geometry

and

[h] = Vol(S6)

([-o] - (B + i[w])) mod E.

Thus we see that Vol(b) = 1/Vol(Sb), conforming with Remark 6.8. However this does not quite look like mirror symmetry: if we repeat the process we appear to get [A] = Vol(Sb)([iro] - (B + i[w])) mod E.

This again is the fault of our identifications, essentially having put in a 90° twist in r during each dualising. This is rectified on the double mirror by pulling back all forms by the fibrewise negation map on S. This acts trivially on H°(B, R2 f.R) and H2(B, f.R), but by negation on H1(B, Rl f.R). Finally we note that the construction of the mirror K3 surface given in this proof is of a different nature from previous constructions of mirror symmetry for K3 surfaces. Normally one appeals to the Torelli theorem to construct the mirror. Here, once we have produced a special Lagrangian fibration on S, we produce the mirror without an appeal to Torelli. Instead, we are essentially applying Yau's Theorem to solve the equations of Corollary 5.15. However we are still aided by some key points which don't hold in higher dimensions. These are: (1) We know the harmonic n - 1-forms on fibres, since n - 1 = 1. (2) We know the cohomology lass of a holomorphic 2-form f2 if we know its class modulo E; this is completely determined by the requirement [fl]3 = 0. (3) We can use the hyperkiihler trick.

§8. Postscript. Since the initial version of this paper was prepared, there have been a number of new results which have greatly increased the evidence for the Zaslow conjecture. The first is a result of W: D. Ruan [25] constructing a Lagrangian Ts-fibration on the quintic threefold. The basic idea is to deform the Lagrangian T3-fibration on the degenerate Calabi-Yau threefold zoziz2z3z,y = 0 in p4 induced by the moment map lL : P4 -+ (where ? is the polytope corresponding to P4) as suggested in [16]. Such a procedure was also carried out in [32]. But Ruan introduces a new technique, using a symplectic flow, to move Lagrangian tori from the singular Calabi-Yau to the non-singular ones. Of course, this gives toil which are Lagrangian with respect to the Fubini-Study Kiihler form, but by Moser's theorem, such a Calabi-Yau is symplectomorphic to the one whose Kahler form is given by the Ricci-flat metric. Because this fibration constructed is not differentiable, the structure of Lagrangian fibrations studied in §2 does not apply, and in particular these fibrations look quite bad from the point of view of mirror symmetry. The discriminant locus is codimension one, which rules out the possibility of G-simplicity, and this construction has to be modified to obtain a fibration suitable for dualizing. In my own paper [15], I addressed this issue in a completely different way. I worked in a completely topological setting, dealing with T3-fibration whose topological properties resemble those one expects of special Lagrangian fibrations as

Mark Gross

401

discussed in §1 and 2 here. I then showed how to use Z and Z/nZ-simplicity to classify monodromy near semi-stable fibres, i.e. fibres where the local monodromy group is unipotent. By construction of examples of such topological fibrations, one in fact proves this gives a complete classification of possible monodromy. In particular, using this classification, one accomplishes step (3) of the program outlined in the introduction for those topological T3-fibrations with only semistable fibres. This gives one (but not necessarily the only!) solution to the compactification problem. Next, I applied these ideas to an explicit example. In [25], Ruan had calculated the monodromy around his codimension one discriminant locus. Using this, one can explictly construct a torus fibre bundle with the same monodromy, and then glue in a fibration with suitable discriminant locus to match the same monodromy. This

yields a fibration f : X -4 53, and one can then prove that X is diffeomorphic to the quintic. The best test of the Strominger-Yau-Zaslow conjecture is then to dualize this fibration using the general compactification method that we have developed, to obtain f : X -> S3. One can then show that X is indeed diffeomorphic to the mirror quintic, or rather to one particular minimal model of the mirror quintic. Different choices made for the fibration f : X --3 B yield different minimal models of the mirror. This now shows that at least on the topological level, the StromingerYau-Zaslow conjecture gives a satisfactory explanation of mirror symmetry. More recently, Ruan has released another paper [26], announcing a construction of Lagrangian fibrations on the quintic and its mirror with the correct properties (in particular, these fibrations will be simple), and he shows that these fibrations are dual. The method is via a modification of his symplectic flow technique, though details of the construction are postponed until future papers. The fibrations constructed are, on the topological level, the same as those constructed in [26].

There still remains the hardest question: that of obtaining special Lagrangian fibrations. This remains a very difficult problem, despite this recent progress.

Bibliography [1] Almgren, F. J., Jr. "Q Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area Minimizing Rectifiable Currents up to Codimension Two," Bull. Amer. Math. Soc. (N.S.), 8 (1983), 327-328. [2] Almgren, F. J., Jr., "Q Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area Minimizing Rectifiable Currents up to Codimension Two," preprint. [3] Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd edition, SpringerVerlag, 1989.

[4] Aspinwall, P., "An N = 2 Dual Pair and a Phase Transition," Nucl. Phys. B, 460 (1996), 57-76. [5] Aspinwall, P., and Morrison, D., "Stable Singularities in String Theory," with an appendix by M. Gross, Comm. Math. Phys. 178, (1996) 115-134. [6] Aspinwall, P., and Morrison, D., "String Theory on K3 surfaces," in Essays on Mirror Manifolds II, Greene, B.R., Yau, S.-T. (eds.) Hong Kong, International Press 1996, 703-716.

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[7] Barth, W., Peters, C., and van de Ven, A., Compact Complex Surfaces, SpriugerVerlag, 1984. [8] Besse, A., Einstein Manifolds, Springer-Verlag, 1987. [9] Borel, A. et al, Intersection Cohomology, Birkhiiuser, 1984. [10] Bredon, G., Sheaf Theory, 2nd edition, Springer-Verlag, 1997. [11] Dolgachev, I., "Mirror Symmetry for Lattice Polarized K3 surfaces'," Algebraic Geometry, 4, J. Math. Sci. 81, (1996) 2599-2630. [12] Duistermaat, J., "On Global Action-Angle Coordinates," Comm. Pure. Appl. Math., 33 (1980) 687-706. [13] Federer, H., Geometric Measure Theory, Springer-Verlag, 1969. [14] Gross, M., "Special Lagrangian Fibrations I: Topology," to appear in the Pro-

ceedings of the Taniguchi Symposium on Integrable Systems and Algebraic Geometry. [15] Gross, M., "Topological Mirror Symmetry," preprint, math.AG/9909015 (1.999).

[16] Gross, M., and Wilson, P.M.H., "Mirror Symmetry via 3-tori for a Class of Calabi-Yau Threefolds," Math. Ann., 309, (1997) 505-531. [17] Harvey, R., and Lawson, H.B. Jr., "Calibrated Geometries," Acta Math. 148, 47-157 (1982). [18) Hitchin, N., "The Moduli Space of Special Lagrangian Submanifolds," preprint, dg-ga/9711002.

[19] King, J., "The Currents Defined by Analytic Varieties," Acta Math., 127 (1971), 185-220. [20] Koszul, J.-L., "Crochet de Schouten-Nijenhuis at Cohomologic," in The Mathematical Heritage of Elie Carton (Lyon, 1984), Asterisque, Nurnero Hors Serie, (1985) 257-271. [21] Mangiarotti, L., and Modugno, M., "Graded Lie Algebras and Connections on a Fibered Space," J. Math. Pures. et Appl., 63, (1984), 111-120. [22] McLean, R.C., " Deformations of Calibrated Submanifolds," Duke University Preprint, 1996. [23] Morgan, F., Geometric Measure Theory. A Beginner's Guide, 2nd Edition, Academic Press, Inc., 1995. [24] Morrison, D.R., "The Geometry Underlying Mirror Symmetry," Preprint 1996. [25] Ruan, W: D., "Lagrangian Tori Fibration of Toric Calabi-Yau Manifold I," preprint, math.DG/9904012 (1999). [26] Ruan, W: D., "Lagrangian Tori Fibration of Toric Calabi-Yau Manifold III: Symplectic Topological SYZ Mirror Construction for General Quiutics," preprint, math.DG/9909126 (1999). [27] Simon, L., Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. [28] Strominger, A., Yau, S: T., and Zaslow, E., "Mirror Symmetry is T-Duality," Nucl. Phys. B479, (1996) 243-259.

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[29] Tian, G., "Smoothness of the Universal Deformation Space of Compact Calabi-

Yau Manifolds and its Petersson-Weil Metric," in Mathematical Aspects of String Theory, (San Diego, California, 1986), 629-646, Adv. Ser. Math. Phys. 1, World Scientific Publishing. [30] Todorov, A., "The Weil-Petersson Geometry of the Moduli Space of SU(n > 3) (Calabi-Yau) Manifolds I," Commun. Math. Phys 126, (1989) 325-346. [31] Well, A., "Final Report on Contract AF 18(603)-57," Oeuvres Scientifiques, Vol. II, 390-395. (32] Zharkov, I., "Torus Fibrations of Calabi-Yau Hypersurfaces in Toric Varieties and Mirror Symmetry," preprint, alg-geom/9806091.

Mirror Principle I

Bong H. Lian,' Kefeng Liu,2 and Shing-Tung Yau3

Abstract. We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced by any concavex

bundles - including any direct sum of line bundles - on P". This includes proving the formula of Candelas-de la Ossa-Green-Parkes for the instanton prepotential function for quintic in p4. We derive, among many other examples, the so-called multiple cover formula for GW invariants of P1. We also prove a formula for enumerating Euler classes which arise in the so-called local mirror symmetry for some noncompact Calabi-Yau manifolds. At the end we interprete an infinite dimensional transformation group, called the mirror group, acting on Euler data, as a certain duality group of the linear sigma model.

Appeared in Asian Journ. Math. Vol 1, No. 4 (1997). 'Department of Mathematics, Brandeis University, Waltham, MA 02154.

2Department of Mathematics, Stanford University, Stanford, CA 94305.

3Department of Mathematics, Harvard University, Cambridge, MA 02138.

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Contents 1. Introduction . . . . . . . . . . . . . . . . . 1.1. The Mirror Principle . . . . . . . . . . . . 1.2. Enumerative problems and the Mirror Conjecture

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2.3. Concavex bundles . . . . . . . . . . . . . 2.4. Linked Euler data . . . . . . . . . . . . . 2.5. The Lagrange map and mirror transformations . 3. Applications . . . . . . 3.1. The first convex example: The Mirror Conjecture 3.2. First concave example: multiple-cover formula . 3.3. Second concave example: Kp2 . . . . . . . . 3.4. A concavex bundle on P3 . . . . . . . . . . . 3.5. A concavex bundle on p4 . . . . . . . . . . . 3.6. General concavex bundles . . . . . . . . . . . . . . . . . . 3.7. Equivariant total Chern class 3.8. Concluding remarks . . . . . . . . . . . .

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1.3. Acknowledgements

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2. Euler Data . . . . . . . . 2.1. Preliminaries and notations 2.2. Eulerity . . . . . . . .

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406 406 409 412 413 413 415 421 428 431 437 440 443 444 445 446 447 448 449

1. Introduction 1.1. The Mirror Principle In section 2, we develop a general theory of Euler data, and give many examples. In particular we introduce the notions of a convex and a concave bundles on P", and show that they naturally give rise to Euler data. In section 3 we apply our method to compute the equivariant Euler classes, and their nonequivariant limits, of obstruction bundles induced by a convex or a concave bundle. We briefly outline our approach for computing multiplicative equivariant characteristic classes on stable map moduli. This outline also fixes some notations used later. Our approach is partly inspired by the idea of mirror transformations (Candelas et al), the idea of the linear sigma model (Witten, Morrison-Plesser, JinzenjiNagura,..), and the idea of using the torus action and equivariant techniques (Ellinsrud-Stromme, Kontsevich, Givental,..). All these ideas are synthesized with what we call the Mirror Principle, which we now explain.

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Let M be a projective manifold and 3 E H2(M, Z). Let Mg,k(0, M) be the stable map moduli space of degree /3, arithmetic genus g, with k marked points. For a good introduction to stable maps, see [16]. Throughout this paper, we shall only deal with the case with g = 0. We begin by analyzing two distinguished types of fixed points under an induced torus T action on Mo,o(d, PI). Both types of fixed points reflect the structure of the stable map moduli space. A smooth fixed point we consider is a degree d cover of a T-invariant P1 joining two fixed points pi, pj in Pn. A singular fixed point we consider is in the compactification divisor. It is given by gluing together two 1pointed maps (fl, Cl, xi) E .Mo,l(r, Pn) and (fi, 02, X2) E Mo,l(d -

r,Pn) at the marked points with fl(xl) = f2(x2) = pi E Pn (pi being a T-fixed in Pn), resulting in a degree d stable map (f, C). We consider two types of T-equivariant bundles V on Pn, which we called convex and concave respectively (Definition 2.7). To be brief, we consider the convex case in this outline. A convex bundle V -4 Pn induces on Mo,o(d, Pn) an obstruction bundle Ud whose fiber at (f, C) is the section space H°(C, f *V). First we have the exact sequence over C

0-+ f*v-+ flV®f2V->VIP,-*0.

Passing to cohomology, we have

0-aH°(C,f*V)->H°(Cl,fiV)®H°(C2,f2V)-4VIp, -i0. Hence we obtain a similar exact sequence for the Ud restricted to a suitable fixed point set. Let bT be any multiplicative T-equivariant characteristic class [23] for vector bundles. The exact sequence on the fixed point set above gives rise to the identity, which we call the gluing identity:

bT(V) . bT(Ud) = bT(Ur) . bT(Ud-r)

Let Md :_ X0,0((1, d), Pl x Pn). This space has a G = S' x T action. There is a natural equivariant contracting map it : Md -> Mo,o(d,Pn) given by it : (f, C) E Md

(9r2 o

f, C') E .Mo,o(d, Pn)

where 7r2 denotes the projection onto the second factor of Pl X P'. Here C' = C if (7r2 o f, C) is still stable; and if (ire o f, C) is unstable, this is the case when C is of the form C = C, U CO U C2 with ir2 o f (Co)

Mirror Symmetry I

408

a point in P', then C is obtained from C by contracting the unstable component Co. Now via -7r, we pull back to Md all the information obtained above

on Mo,o(d, P'). The reason is that there is an collapsing map

:

Md -* Nd = p(n+l)d+n which then allows us to perform computations on the linear object Nd. We call Nd the linear sigma model and Md the nonlinear sigma model. There is a natural G action on Nd such

that iP is G-equivariant. For example, to determine an equivariant cohomology class w on Nd, we only need to know its restrictions i,*i, (w)

to the (n + 1) (d + 1) G-fixed points {Pir } in Nd. The corresponding weights of the G action on Nd are Xi+ra. Let Qd be the push-forward of bT(Vd) = 7.*bT(Ud) into Nd, ie. Qd = tp! bT(Vd). Then the classes Qd inherit the gluing identity (Theorem 2.8)

/

tpi,p(Qo) - tp*ir(Qd) = t;, (Qr) - tPir(Qd--r)

which is an identity in the ring HG(pt) = H*(BG). The sequence Qd is an example of what we call an Euler data (Definition 2.3). We summarize the various ingredients used, now and later, in our constructions: Vd = lr* Ud

Nd 4-

Md

Ud

P* Ud

Mo,o(d, P") 4 Mo,i(d, Pn)

-14

where p forgets, and ev evaluates at, the marked point of a 1-pointed

stable map. Also note that Ud = p!(ev*V), ie. the direct image of ev*V.

The gluing identity is not enough to determine all Qd. In order to get further information we localize the Qd to a fixed point in Nd whose inverse image in Md is a smooth fixed point, and compute Qd at the special values of a = (Ai - ),)/d. The property of bT(Ud) at a smooth fixed point comes into play here. For simplicity in this outline, let's consider for example the case when bT is the equivariant Euler

class eT and V = 0(l). In this case we find that, at a = (Ai -)1j)/d, 6d

Pi - m(Ai - A1)/d)

bpi,p(Qd) = M=0

This immediately tells us that the Qd should be compared with the sequence of classes Id

P: Pd= fJ (lic - ma)

Pn

Lian, Liu, and Yau

409

-

(Theorem 2.10), which has tp, o (Pd) = t;, ,,(Qd) at a = (A We find that P is another example of an Euler data. This Euler data will then naturally give rise to a generating series of hypergeometric type, hence explains the very origin of these functions in enumerative problems on stable moduli! The same holds true for a large class of

vector bundles V on P.

Finally, under a suitable bound on c1 (V), we can completely determine the Qd, in terms of Pd by means of a mirror transformation argument (Theorem 3.9). We can in turn use Qd to compute eT(Ud) and their nonequivariant limits (Theorem 3.2). Our approach, thus, makes the roles of three objects and their relationships quite transparent: certain fixed point sets, equivariant multiplicative characteristic classes, and series of hypergeometric type. Our method works well for many characteristic classes such as the Euler class and the total Chern class. We summarize this Mirror Principle. Let bT be a multiplicative equivariant characteristic class, V an equivariant vector bundle on P" which induces a sequences of vector bundles Ud -+ Mo,o(d, PI). 1. (Euler Data) The behaviour of the Ud at a singular fixed point gives rise to the gluing identity. In turn this defines an Euler data Qd on the linear sigma model Nd. 2. (Linking) The behaviour of the Ud at a smooth fixed point allows us to read off the restrictions of Qd at certain fixed points of Nd for special values of weights. The restriction values determine a distinguished Euler data Pd to be compared with the Qd. 3. (Mirror Transformations) Compute the Qd and the bT(Ud) in terms of the Pd explicitly by means of a mirror transformation argument.

Our approach outlined here can be applied to a rather broad range of cases by replacing P" by other manifolds. They include manifolds with torus action and their submanifolds. We will study the cases of toric varieties and homogeneous manifolds in a forth-coming

paper [37]. On the other hand, we can even go beyond equivariant multiplicative characteristic classes. In a future paper, we will study a sequence of equivariant classes Td of geometric origin on stable moduli satisfying our gluing identity.

1.2. Enumerative problems and the Mirror Conjecture

For the remarkable history of the Mirror Conjecture, see [14]. In 1990, Candelas, de la Ossa, Green, and Parkes [8] conjectured a formula for counting the number nd of rational curves in every degree d on a general quintic in p4. The conjectured formula gives a

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Mirror Symmetry I

generating function for the nd explicitly in terms of certain hypergeometric functions. Their computation is based on the existence of "mirror manifolds", conjectured by a number of physicists including Dixon and Lerche, Vafa and Warner on the basis of physical intuition.

Mirror symmetry took a dramatic turn upon the appearance of the papers of Greene-Plesser [21] and of Candelas et al [8]. In [21], they established the existence of mirror manifolds for a particular class of Calabi-Yau manifolds including the quintic.

On the mathematical side, it has been conjectured earlier by Clemens that the number of rational curves in every degree in a general quintic is finite. The conjectured formula agrees with a classical

result in degree 1, an earlier computation by S. Katz in degree 2, and has been verified in degree 3 by Ellingsrud-Stromme. In 1994, using results of Uhlenbeck-Sacks, Gromov, Parker-Wolfson, and Wit-

ten, Ruan-Tian introduced the notions of J-holomorphic maps and symplectic Gromov-Witten (GW) invariants. Motivated by the same works, Kontsevich proposed an algebraic geometric analogue of GW invariants and stable maps. The latter is the algebraic geometric counterpart of J-holomorphic maps. Generalizations have been given by Li-Tian and Behrend-Fentachi. A recent paper of Li-Tian shows that the symplectic version and the algebraic geometric version of the GW theory are essentially the same in the projective category. Beautiful applications of ideas from quantum cohomology and GW theory have recently been done by Caporaso-Harris [10], Crauder-Miranda [11], DiFrancesco-Itzykson [12], Bryan-Leung [2], and others solving many important enumerative problems. Closer to mirror symmetry, it is known that the degree k GW invariants for Pl (the so-called multiple-cover contribution) is given by k-3. This was conjectured in [8], justified in [1)[44] using a different compactification, and in [38] using the stable map compactification. According to [34], the number

Kd = E nd/kk-3 kid

is the degree of the Euler class ct,,(Ud) for Ud -+ Mo,o(d, p4) induced by 0(5) --* P4. By means of the torus action on P4, Ellingsrud-Stromme [15] and Kontsevich [34] have verified that n3 and n4 respectively agree with the conjectured formula. In some recent papers [18][19], Givental introduces some ideas which emphasize the use of equivariant version of quantum cohomology (see also Kim [27]). Unfortunately, the proof in [18]for the formula of Candelas et al, which has been read by many

Lian, Liu, and Yau

411

prominent experts, is incomplete. In light of our present paper, it is now clear that essentially all ingredients (equivariant cohomology theory, the Atiyah-Bott formula, equivariant Euler classes, obstruction bundles, fixed point structures, the linear sigma model, the graph construction, hypergeometric series, mirror transformations, etc) required to prove the formula of Candelas et al have all been well-known to experts since at least 1995. Thus the remaining nontrivial task, is to set up and put together in details all the pieces in a correct way to give a complete proof. In our present paper, we have introduced the substantial new ideas (beyond the known ingredients) needed to give the first complete proof. It is conceivable that our new ideas here can furnish the necessary parts missing in the paper [18]. More recently, fixed point method has also been applied by Graber-Pandharipande

[20] to study GW invariants of P'. We emphasize that the formula of Candelas et al, albeit conjectural, not only computes the Kd in terms of hypergeometric functions, but also provides a crucial guide for putting together various technical inputs in our proof. While applying the Atiyah-Bott localization formula on the stable moduli and on the linear sigma model yield important pieces of information about the cohomology classes in question, it is the formula of Candelas et al which provides the clues for how to put the pieces together - in terms of hypergeometric functions (which are periods of a holomorphic 3-form on a mirror manifold). Earlier attempts on the conjectured formula using quantum cohomology seems to have confused the issue considerably. We believe that the machinery introduced in this paper will be useful for many other enumerative problems, aside from proving the formula of Candelas et al. In fact we have applied our machinery to problems in local mirror symmetry proposed by C.Vafa, S. Katz, and others. 4 We now formulate one of our main theorems in this paper. Let

Kd =

.Mo,o(d,P4)

Ctr ,(Ud),

F(T) =5T+EKdeaT 6

d>0

Consider the fourth order hypergeometric differential operator [8]:

L:=(d)4_5et(5dt+1)...(5 t+4). S. Katz has informed us that A. Elezi has also studied a similar problem.

Mirror Symmetry I

412

By the Frobenius method, it is easy to show that

fs - a! 1(d dH

ed(t+H)

llm=1(5H+ m)

i=0,1,2,3,

+m)`

d>o

form a basis of solutions to the differential equation L f = 0. Let

T

X(T)

fo'

fo).

2(fo fo

Theorem 1.1. (The Mirror Conjecture) F(T) _ .F(T). The transformation on the functions fi given by the normalization

fi H

f0

and the change of variables

t + T (t) =

fo

are known as the mirror transformation. By the construction of Candelas et al, the functions fo,.., f3 are periods of a family of Calabi-Yau threefolds. By the theorem of Bogomolov-Tian-Todorov, these periods in fact determine the complex structure of the threefold. A similar Mirror Conjecture formula holds true for a three dimensional Calabi-Yau complete intersection in a. toric Fano manifold [37]. This will turn out to agree with the many beautiful mirror symmetry computations [40][36][5][9][4][30][31].

In this paper, to make the ideas clear we restrict ourselves to the simplest case, genus 0 stable moduli space for P'. In a forth-coming paper [37], we extend our discussions to toric varieties, homogeneous manifolds, and general projective manifolds. We also hope to eventually understand from our point of view the far reaching results for higher genus of [6], and the beautiful computations [17][13] for elliptic GW invariants. 1.3. Acknowledgements

We thank A. Todorov, A. Strominger, C. Vafa for helpful discussions. Our special thanks are due to J. Li who has been very helpful throughout our project.

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413

2. Euler Data One of the key ingredients in our approach is the linear sigma model, first introduced by Witten [45], and later used to study mirror symmetry by Morrison-Plesser [29], Jinzenji-Nagura [25], and others, resulting in new insights into the origin of hypergeometric series. In this paper, we consider the S1 X T-equivariant cohomology of the linear sigma model. 2.1. Preliminaries and notations Let T be an r-dimensional real torus with a complex linear representation on CN+1. Let /30, .., ON be the weights of this action. We consider the induced action of T on PN, and the T-equivariant cohomology with coefficients in Q, which we shall denote by HH(-). Now HT(pt) is a polynomial algebra in r-variables, and /3i may be regarded as elements of HT (pt). Throughout this paper, we shall follow the convention that such generators have degree 1. It is known that the equivariant cohomology of PN is given by [32]

HT(PN) = HT(pt)[]I

-,6s) I (ft(( i=0

Here t;, which we shall call the equivariant hyperplane class, is a fixed

lifting of the hyperplane class of PN. Each one-dimensional weight space in CN+1 becomes a fixed point pi in PN. We shall identify the rings HT* (pi) and HT(pt) = H* (BT). There are N + 1 canonical

restriction maps i , : HT(PN) -- HT(pt), given by S H /3i, i = 0,.., N. There is also a push-forward map HT(PN) -+ HT(pt) given by integration over PN. By the localization formula, it is given by

w- /PN w=ResC N

W

r Ii= N 0(S-pi).

Two situations arise frequently in this paper. First consider the

standard action of T = (Sl)"+1 on C"+1, and let A = (Ao,.., An) denote the weights. On P", there are n + 1 isolated fixed points po,.., p". We shall denote the equivariant hyperplane class by p, the canonical restrictions by tp, : w t;,(w) = w(ai), and the pushforward by

pf : HT(P") -+ HT(pt) = Q[A]. 0 on the ring HT(P"), and We shall use the evaluation map A shall call this the nonequivariant limit. In this limit, p becomes the ordinary hyperplane class H E H* (P").

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Mirror Symmetry I

We now consider the second situation. For each d = 0, 1, 2, .., consider the following complex linear action of the group G := S' x T on C(n+i)(d+l) We let the group act on the (ir)-th coordinate line in C(n+l)(d+l) by the weights .1i+ra (Ai being the weights of T as before, and a being a weight of Sl ), i = 0, .., n, r = 0,.., d. Thus there are (n+ 1)(d + 1) isolated fixed points pi, on the projective space P(n+l)d+n given by those coordinate lines. In this case, we shall denote the equivariant hyperplane class by rc, the canonical restrictions by t;. w I-+ tp;r (w) = w(Ai + ra), and the push-forward by pfd : HG(P(n+l)d+n)

HG(Pt) = Q[a, A].

Here we have abused the notation rc, using it to represent a class in HG(p(n+l)d+n) for every d. But it should present no confusion in the context it arises. Let Nd be the space of nonzero (n + 1)-tuple of degree d homogeneous polynomials in two variables wo, wi, modulo scalar. There is a canonical way to identify Nd with P(n+l)d+n Namely, a point x E P(n+l)d+n corresponds to the polynomial tuple [E ZOrv,owi-r, xnrwowi-''] E Nd. This identification will be used throughout this paper. The natural T-action on (n + 1)-tuples together with the S1action on [WO, wl] E Pl by weights (a, 0), induces a S' x T-action

on Nd given by [fo(wo, wl), , fn(wo, w1)] y [e'0fo(eawo, wl), er`^ fn(eawo, wl)]. This coincides with the Sl x T-action on P(n+l)d+n described earlier.

Definition 2.1. (Notations) We call the sequence of projective spaces {Nd} the linear sigma model for Pn. Here are some frequently used

notations: G := SixT, R := Q(A)[a], R'l := Q(A, a), RHG(Nd) HG(Nd) ®Q A,a) R, R-lHH(Nd) := H7(Nd) ®4[.,a] R-l, and deg,, w means the degree in a of w E R.

Obviously the maps tt,, , pfd defined linearly over Q[A, a], can be extended R- or R-1-linearly. There are two natural equivariant maps between the Nd, given by I : Nd_l -+ Nd,

- : Nd --* Nd,

[fo,

fn]

[wlfo,

,

wlfn]

[fo(wo, wl), , fn(wo, w1)]'-* [fo(wl, wo), , fn(wl, wo)]

The second map induces on equivariant cohomology R''HG(Nd), i =

rc - da, a = -a, ai = Ai. In particular any x E R has the form

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415

x = x_ + x+ with xt = Ex±. We also extend - to the power series ring R[[et]] by leaving t invariant. Composing a chain of d I's, we get a canonical inclusion No =

P"-4Nd. Note that the image of the fixed point pi is pi,o. For W E R-1HG(Nd), we shall denote by II(w) E R-'HH(No) the restriction of w to No. Since S1 acts trivially on P", we can write

He (No) = H ,(P")[a] In particular note that HT (P") is invariant under - , and that Id (r.) _ P.

Obviously the set of classes w E HT.(P") with tpi(w) 0 0 for all i, is closed under multiplication. We localize the ring HT(P") by allowing to invert such elements w. We denote the resulting ring by

HT(P")-1.

Definition 2.2.

(Notations) The degree in a of a class w E

Hz .(P")-i[a] will be denoted by dega w. A class fl E HT(P")'1 with t. (fl) # 0 for all i will be called invertible. Throughout this paper, fZ will denote a fixed but arbitrary invertible class. S denotes the set of sequences of cohomology classes Q :

Qd E R-'HG(Nd), d=1,2,...

2.2. Eulerity

Definition 2.3.

We call a sequence Q :

Qd E 71H6(Nd),

d=

1, 2,..., an Il-Euler data if for all d, and r = 0, .., d, i = 0,.-, n, (*)

Pi(cl) Pi.r(W:d) = L;1,0(Qr) tps.0(Qd-r),

where Qo := n. We denote by An the set of fl-Euler data. When dealing only with one fixed class Cl at a time, we shall say Euler rather than ft-Euler, and shall write A for the set of Euler data. More explicitly, condition (*) can be written as f2(Ai)Qd(ai + ra) = Qr(Ai)Qd_r(Ai). Applying this at r = d, we find that

Qd(,\i + da) = Qd(Ai)

(2.1)

Putting a = (A, - \i)/d, we see that Qd(Aj) at a = (a5 - Ai)/d coincides with Qd(Ai) at a = (.1i -,\j)ld. Applying both (*) and (2.1) at a = (A5 - Ai)/r (hence \j = \i + ra), we get 0('\i)Qd(\j) = Qr(A.7)Qd-r(Ai) at a = (A5 -.1i)/r.

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416

Lemma 2.4. (Reciprocity Lemma) If Q is an Euler data, then for i, j n, r = 0, 1,.., d, d = 0, 1, 2,.., we have (r) Qd(A + da) = Qd(A ) (ii) Qd(Aj) at a = (A - Ai)/d coincides with Qd(Ai) at a = (Ai

\j)/d ford # 0.

(iii) l(Ai)Qd(A1) = Qr(Aj)Qd-r(Ai) at a = (Aj - )ti)/r for r

-

0.

Example 1. Let l be a positive integer. Put Id

P : Pd = ]I (llc - ma) E HG(Nd). M=O

That this is an lp-Euler data follows from the identity: Id

t(d-r)

Sr

lai 11 (lai + (r - m)a) = fl (lai + ma) x H (lai - ma). m=0

M=0

m=0

This will arise naturally in the problem of computing the equivariant Euler classes of the obstruction bundles induced by 0(l) -> P. (See below.)

Example 2. Put SZ = p-2, and d-i

P : Pd = 11 (' - ma)2 E HG(Nd). M=1

This Euler data will arise in the problem of computing the so-called multiple cover contributions, ie. the GW invariants for Pi. Example 3. Put Sl = (-3p)-1, and 3d-i

P : Pd = 11 (-3ic + ma) E HG(Nd). m=i

This Euler data will arise in the problem of computing the equivariant Euler classes of the obstruction bundles induced by the canonical bundle of P2.

Example 4. Put fl = -1, and 2d

2d-i

P : Pd = 11 (2k - ma) x fl (-2K + ma) E HG(Nd). M=O

m=1

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This Euler data will arise in the problem of computing the equivariant Euler classes of the obstruction bundles induced by 0(2) ® O(-2) on

Ps Example 5. It is easy to show that if Q is SZ-Euler and a E Q(.A) is any nonzero element, then the data Q' : Qd = a Qd is ant-Euler. Similarly, the data Q' : Q'd = (-1)d+1Qd is also -S2-Euler. Example 6. We observe that the set of Euler data is a monoid, ie. it is closed under the product QdQd, and has the unit given by Qd = 1 for all d. Hence the product of an S2-Euler with an Q'-Euler data is an S2Sf'-Euler data. In the geometrically setting, this multiplicative property sometimes corresponds to taking intersection of two suitable projective manifolds. In this case, the class S2 E H .(P") plays the role of the equivariant Thom class of the normal bundle of such a projective manifold.

Example 7. Let Qd = s (r. - da) E HG(Nd). Then it is again trivial to check that Q is n,2-Euler.

Example 8. Let Q be an fl-Euler data, and Q' be an S2'-Euler data. Suppose Qd/Qd E RHH(Nd) for all d > 0. Then it is immediate that they form a sequence, denoted by Q/Q', which is S2/S2'- Euler.

As a special case, let Qd = l2ic(i - da) as in Example 7, and let P as in Example 1. Then P/Q is a (1p)-1-Euler data. Example 2 is obtained by squaring this. Examples 3 and 4 can also be obtained in a similar way. Example 9. Introduce a formal variable x. We can extend everything above by adjoining x, ie. by replacing the ground field Q by Q(x). For example, it is easy to show that !d

P : Pd = fl (x + In - ma) E HG (Nd) (x) M=O

satisfies the gluing identity as in Example 1, thus is an Euler data in a more general sense. Such Euler data will appear in the computations of equivariant total Chern classes. Example 10. Let Md := Mo,o((1, d), P1 x P") be the mod-

uli space of holomorphic maps P1 -+ P1 x P" of bidegree (1, d). Recall that Md is the stable map compactification of Md . Each

map f E Md can be represented by f

: [wo, w1] [w1, wo] x [fo (wo, w1), .., f"(wo, w1)], where ff are degree d homogeneous poly-

nomials. So there is an obvious map

p:Md -4Nd, f which is G = S' x T equivariant. For convenience we define MO

No = P". With a bit of work (see below), it can be shown that

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Mirror Symmetry I

the map cp has an equivariant regular extension to cp : Md -> Nd.

Let (f, C) E Md. Then C is an arithmetic genus 0 curve of the form C = Co U C1 U U Crr such that it1 o f : Co3P1, where In, 7r2 are projections from P1 x P" to the first and second factors.

Each C;, j > 0, is glued to Co at some point x3 E Co. The map ire o f : Cj -3 P" is of degree dj with Ej dj = d, and ir1 o f : Cj -3

P1 is constant map with image in1 o f (xj) E P1. If we denote by , ar,,] the degree do polynomials representing 7r2 o f : Co -+ P", then cp : (f, C) [vog, , ong], where g = H, (ajwo - bawl )di with irl o f (xj) = [a;, bj]. Thus cp collapses all but one component of C. The idea of using a collapsing map relating two moduli problems is not new. The map cp was known to Tian [43], and a similar map also appeared in [35] in which a collapsing map was used to relate two moduli spaces. The map cp was also used in [18]. Similar maps have also been studied in [24][26].

[oo,

Let V = 0(l) for l > 0, and consider the induced bundle Ud -3 Mo,o(d, Pn). Pulling this back via the projection in, we get a bundle Vd -+ Md (see Introduction). Let eT(Vd) be the equivariant Euler class of Vd, and let co eT(Vd) be its push-forward via W. The following theorem will be a special case of a general theorem proved in the next subsection.

Theorem 2.5. The sequence cp!eT(Vd) E HG(Nd) above is an lpEuler data. We now return to the map V. The reader who wishes to skip technical details can safely omit the proof.

Lemma 2.6.

There exists a morphism cp : Md -+ Nd. Moreover cp is equivariant with respect to the induced action of S' x T. Proof: The following proof is given by J. Li. Let Md be the moduli space of stable morphisms f : C -+ P1 x Pn from arithmetic genus 0 curves to pi X Pn of bi-degree (1, d), and let Nd be the space of equivalence classes of (n+1)-tuples (fo, ..., fn), where fi are degree d homogeneous polynomials in two variables, and (fo, ... , fn) - (fo..... fn)

if there is a constant c 96 0 such that f; = c f; for all i. We first

define the morphism cp : Md -4 Nd. For convenience, we let S be the category of all schemes of finite type (over C) and let .F : S --> (Set)

be the the contra-variant functor that send any S E S to the set of families of stable morphisms

F:X--*P1xPnxS

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over S, where X are families of connected arithmetic genus 0 curves, modulo the obvious equivalence relation. Note that F is represented

by the moduli stack Md. Hence to define W it suffices to define a transformation

:F--*Mor(-,Nd).

We now define such a transformation. Let S E S and let C E i(S) be represented by (X, F). We let pi be the composite of F with the i-th projection of P1 x P" x S and let pij be the composite of F with the projection from P1 x P" x S to the product of its i-th and i-th components. We consider the sheaf p*Opn (1) on X and its direct image sheaf £4 =1 13*P2OPn (1).

We claim that L is flat over S. Indeed, by argument in the proof of Theorem 9.9 in [22], it suffices to show that irs*(C 0 7r4l Opi (m)) are locally free sheaves of Os-modules for m >> 0, where 7rpi and irs are the first and the second projections of PI x S. Clearly, this sheaf is isomorphic to p3*(p2Opn (1) O p*Opi(m)), which is locally free because

Rsp3*(p2OPn(1) ®p1OP1(m)) = 0

for i > 0 and m >> 0. For the same reasoning, the sheaves G4 satisfy the following base change property: let p : T --* S be any base change and let E .17(T) be the pull back of . Then there is a canonical isomorphism of sheaves of OT-modules 'ca'(C)

(ip1 x p)*.Cg.

(2.2)

Since G£ is flat over S, we can define the determinant line bundle

of Lt, denoted by det(Ct) [28]. The sheaf det(L4) is an invertible sheaf over P1 x S. Using the Riemann-Roch theorem, one computes that its degree along fibers over S are d. Further, because ,C£ has rank one, there is a canonical homomorphism CC --3 det(C£),

(2.3)

so that its kernel is the torsion subsheaf of Lt, denoted by Tor (C£). Let w°, ... , w,, be the homogeneous coordinate of P" chosen before.

too,. .. , w form a basis of H°(P", Opn (1)). Then their pull backs provide a collection of canonical sections of CC, and hence a collection of canonical sections 07t,o, - - , 0C,n

E H°(S, irs*det(,C8))

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Mirror Symmetry I

based on (2.3). Then after fixing an isomorphism det(G4) - 4r*M ®irpl Opi (d)

(2.4)

for some invertible sheaf M of OS-modules, we obtain a section of

irs* (7rPi Op (d)) ®o M = HP, (Opl (d)) ®c M. Finally, we let wo, w1 be the homogeneous coordinate of Pl. Then the space H°, (Opi (d)) is the space of degree d homogeneous polynomials in variables wo and wl. This way, we obtain a morphism

it (S):S-3Nd that is independent of the isomorphisms (2.4). It follows from the base change property (2.2) that the collection W(S) defines a transformation

T:.F-*Mor(-,Nd), thus defines a morphism co as desired. It remains to check that for any w E S, the sections a4,o(w), ... , aC,n(w) E H°(P1, det(G£) (Do, k,,,)

has the described vanishing property. Because of the base change property of Ge, it suffices to check the statement when S is a point and E id(S) is the stable map f :C -* PI x Pre. Let X 1 ,-- . , xN be the set of points in P1 so that p1:C -3 P1, where pr = 7rpi o f, is not flat over these points. Let Ci be pi 1(xi) and let mg be the degree of f ([C;]) E H2(P°). Then G£ = pl*p2Op. (1) is locally free away from x1,. .. , xN and has torsion of length -m; at xi. Then G4/Tor (J) is locally free of degree k - E m; . It is direct to check that the canonical inclusion

Gt/Tor (G{) -+ det(Ct)

Opi (d)

has cokernel supported on the union of xl, ... , XN whose length at xE is exactly m;. The statement about the vanishing of o ,o(w), ... , at,n (w) follows immediately. The fact that cp : Md -+ Nd is S' x T-equivariant as stated follows

immediately from the fact that s° is induced by the transformation' of functors. This completes the proof.

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2.3. Concavex bundles

Definition 2.7. We call a T-equivariant vector bundle V -+ P" convex (resp. concave) if the T-equivariant Euler class eT(V) is invertible

and if HI(C, f*V) = 0 (resp. H°(C, f*V) = 0) for every 0-pointed genus 0 stable map f : C -+ P". We call V a concavex bundle if it is a direct sum of a convex and a concave bundles. We denote by V# the convex and concave summands of V. By convention, we consider the zero bundle to be both convex and concave so that concavexity includes both convexity and concavity.

The convexity of a bundle is analogous to the notion of convexity of a projective manifold introduced by Behrend-Manin. For example 0(l) -+ P" is convex if 1 > 0, and concave if l < 0. Given any concavex bundle V -+ Pn, we have a sequence of induced bundles Ud -* .Mo,o(d, P")

whose fiber at (f, C) is the space H°(C, f *V+) ® H'(C, f *V-). Pulling back Ud via the contracting map it : Md -4 Mo,o(d, P"), we get a sequence of bundles

Vd:=it*Ud-+ M. We denote by eT(Vd) the equivariant Euler class of Vd. We also introduce the notations: S1V := eT (V +)

eT(V-)

Qd := coleT(Vd), QO =

12V.

By convention, if V is the zero bundle, we set eT(Vd) = 1, eT(V) = 1,

OV=1.

Theorem 2.8. The sequence cp!eT(Vd) E H,(Nd) is an SZV -Euler data.

Proof: We first discuss some preliminaries. Let M and N be two compact smooth manifolds with the action of a torus T, and c : M -+ N be an equivariant map. Let F be one component of the fixed

submanifold in N and iF be the inclusion map F in N. Let ¢F = iF!(1) E HT(N) denote the equivariant Thom class of the normal bundle of F in N. We then have, for any w E HT (M)

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Mirror Symmetry I

wcp*(OF)

fM

=

IN

W!(w)OF =

fP

ZF(W!(w))-

On the other hand, let {P} be the components of the fixed submanifold contained in cp'1(F). By the Atiyah-Bott [7][3] localization formula on M, we get

EP fP

= fF

Here eT(P/M) denotes the equivariant Euler class of the normal bundle of P in M. The reason is that the contribution of the fixed point sets not contained in cp-1(F) is clearly zero. Actually if Q is a component not contained inside cp'1(F), its contribution to the localization is given by IQ i (wco*(cF))/eT(Q/M)

But by the naturality of the pull-backs, we have i*Q W*(OF)

= coaE(OF) = 0

where E = cp(Q) is a fixed submanifold in N and cpo denotes the restriction of co to Q. Note that if F is an isolated point, then ipcp*(OF) can be pulled out of the integral. The above formula will be applied to the collapsing map cp Md -3 Nd. All manifolds involved here are at worst orbifolds with finite quotient singularities, so the localization formula remains valid without any change as long as we consider the corresponding integrals in the orbifold sense.

We consider the S1-action on the P1 factor in P1 x P" with weights a, 0. Combining with the natural T-action on P", we get the naturally induced G = S1 x T-actions on Md and Nd, with respect to which the collapsing map co is equivariant. As described in section 2.1, the G-fixed points in Nd are all of the form

Pir = [0, ... 0, wrwi'r 0, ...

I

0]

in which the only nonzero term is in the i-th position. For each r > 0, let {F} C Mo,1(r, P") denote the T-fixed point components in f40,1 (r, P") with the marked point mapped onto the fixed point pi in P". Let N(Fr) = NF./.c o.i(r,pn) denote the normal bundle of Fr in Mo,1(r, Pn)

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423

Let ir1i 1r2 be the projections from Pl x P" onto the first and second factors. From the construction of gyp, we see that the G-invariant submanifold that is mapped to Pir consists of the following degree

(1, d) stable maps f : C -> P1 X P" with C = C1 U Co U C2. Here

Co^--PIand

12 o f (C°) = [p, ... , p,1, p ... ,

O] = pi E P"

where 1 is at the i-th position. The map ir1 o f : Co -+ P1 is an isomorphism and maps x1 = Co n Ci and X2= Con C2 to 0 and oo respectively. Actually

irlof(Ci)=0, 7rio.f(C2)=oo in P1, ie. the curves C1 and C2 are respectively mapped to the points 0 and 00 of Pi. The maps ire o f restricted to C1 for j = 1, 2 are stable maps in Mo,i(r, P") and Mo,1(d- r, P") respectively. We consider Fr x Fd-r as a G-fixed submanifold of Md by gluing each pair to Co at x1 and x2 respectively as above. It is easy to see that {Fr x Fd_r} are the G-fixed point sets in Md whose image under V is the fixed point pi,. We first consider a convex bundle V on P", and the case r # 0, d. Then we have Qd(Ai + ra) = fN cb piQwd = a

'Ma

*() eT(Vd).

(2.5)

Here OPir denote the equivariant Thom class of the G-fixed point pi, in Nd. We will apply the localization formula to compute the right hand side of (2.5). First we need to know the normal bundle of the

fixed points, which is, in the equivariant K-group of Fr x Fd_r [34] [20],

NFrXFd_r/Ma = N(F,) + N(Fd-r) + [H°(Co, (it1 o f)*TPl)] + [Lr ® Ty1Co] + [Ld-r 0 Tx2CO] - [Tp,Pf1] - [AC0].

Here Lr denotes the line bundle on .Mo,i(r, P") whose fiber at (fl, 01, x1) is the tangent line at xi. Likewise for Ld_r. Their contributions correspond to the deformation of the nodal points xi and x2. The term H°(Co, (ir1 o f)*TP1) corresponds to the deformation of f restricted to Co. The term [Ago] is the bundle representing the infinitesimal automorphism of Co fixing the two points xi, x2. The term -[TpiP"] comes from gluing Fr and Fd_r onto Co and the property that 1r2 o f (CO) = pi-

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Mirror Symmetry I

This gives the following formula for the corresponding equivariant Euler classes: eT(Fr X

eT(N(Fr))eT(N(Fd-r))eT(Lr 0T,,Co) X eT(Ld_r 0 Tx2Co)eT

(TpiP')-1

x eT(H°(Co, (xi o.f)*TP1))eTl(Aco)

Each term in this formula can be explicitly calculated. We clearly have eT(Tp,P") = fly#$(A1- Aj); the weights of T., Co and T2;ZCo are

a and -a respectively, therefore eT(Lr (a T..1Co) = a+ci(Lr),

eT(Ld-r ®TT2Co) _ -a+cl(Ld-r)

where ci (Lr), cl (Ld_r) are the restriction to Fr and Fd_,. of the equivariant Chern classes of the line bundles L,. and Ld_,. with respect

to the induced T actions on Mo,i(r, P") and .Mo,,(d - r, P91). To compute eT(H°(CO, (7r, o f)*TP1)) and eT(Aco), first note that we have the standard exact sequence

0-30-*0(1)®C2-4 TP1 -4 0, with 0 being the trivial bundle. From this we get 0-+0_+ H°(Co,0(1))®C2-3H°(Co,(7rio f)*(TP1))-4 0. The weights of H° (CO, 0(1)) are a, 0, the weights of C2 are -a, 0

and the weight of 0 is 0. Therefore one finds that the weights of H°(Co, (al o f)*(TP1)) are a, -a, 0. For [Ac0], we have the exact sequence

0-4 Ac0 -+HO (Co,(lriof)*(TP1)) -4Ty1CoED TT2CO -0. The weights of and Tr22Co are a and -a respectively. So [Aco] contributes a 0 weight space which cancels with the 0 weight space of [H°(Co, (its, o f)*(TP'))]. Therefore we will ignore the zero weights in our formulas and write as

eT(H°(Co, (7r, o f)*(TP1))eTl(A0) = -a2.

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When Vd is restricted to Fr X Fd_r considered as a fixed point set of Md as before, we have the exact sequence:

0-4Vd-+ VrIF,.®Vd-r1Fd-,-9VIP: ->0. Note that Vr I F,. and Vd-r I Fa_,. is the same as p* Ur I F, and p* Ud-r I Fd_ r

which respectively are the restrictions to Fr and Fd_r of the pull-backs

to Mo,l(r, Pn) and Mo,1(d - r, PI) of the corresponding bundles on Mo,o(r, PI) and Mo,o (d - r, PI). Here VIP; denotes the fiber of V at Pi E

P.

Here comes the important point. The multiplicativity of equivariant Euler classes gives us Qv(1i) - eT(Vd) = eT(V,) . eT(Vd-r) = p*eT(Ur) - p*eT(Ud-r),

when restricted to Fr x Fd_r C Md.

Here p : .Mo,1,(d, Pn) -4 Mo,o(d, Pn) is the forgetting map (same notation for all d). Note that the above equality is just the pull-back via ir from ,Mo,o(d, Pn) of the gluing identity discussed in the Introduction. For the case of r = 0 or d, there is only one of the curves Cl or C2, ie. C is of the form Co U C2 or Cl U Co. In this case we identify Fd as the fixed point set in Md by gluing its marked point to Co at xl or x2. The normal bundle in these two cases are respectively given by

NFd/Md = N(Fd) + [H°(Co, (Ii o f)*TPI)] + [Ld 0T.iCo] - [A00]

in the K-group of Fd. Here Ld is the restriction to Fd of the line bundle on Mo,i(d, Pn) whose fiber is the tangent line at the marked point. For simplicity we write L as Ld in the following. For j = 1, 2, T.,iC0 is the tangent line of Co at the corresponding marked point x1. In these two cases, one easily shows in the same way as above that the term eT(AC0), except the 0 weight, contributes one nonzero weight -a or a respectively. Its 0 weight space still cancels with the 0 weight space of [H°(C0, (irk, o f)*TP )]. By putting all of the above computations together and combining

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Mirror Symmetry I

with (2.5), we get, for r # 0, d, nV (Ai)Qd(Ai + ra) = SZV(Ai) 1r W*(Opir) eT(Vd) JJMd

//,, = -a-2 fl(Ai - 1j)eT(PirlNd)

jai p*eT (Ur)

x X

Fr eT(N(F'))(a+cl(L,)) P*eT(Vd-r)

Fd_r F'_' eT(N(Fd-r))(-a + Cl (Ld-r)) (2.6)

Here eT (pir/Nd) = tpir (O.,p:r) Note that cp* restricted to F,. x Fd-r is the same as eT(pir/Nd) which is a polynomial only in a and A as given below. Similarly for r = d, we have p*eT(Ud)

Qd(Ai + da) = a-leT (Pid/Nd) E 1I Pd JFd eT(N(Fd))(a+cl(L))

(2.7)

and for r = 0 Qd(Ai) =

-a-leT(Pi0/Nd)Efp,.

p*eT(U) eT(N(Fd))(-a + cl(L))

Pd

We can easily compute eT(pir/Nd): n

d

(Ai - Aj + (r - m)a).

eT(pir/Nd) = ]I ft j=O m=O(j,m)O(i,r)

For r = 0 and d we have n

d

(Ai - Aj - ma)

eT(biO/Nd) = 11 ft j=0 m=Ou'mm)#(i,0) 7d7

eT (pid/Nd) =

ft 11

j=0 m=O(j,m) n

(A1 - Aj + (d - m)a) (i,d)

d

=H11 j=0 m=O(j,m)#(i,0)

(Ai-Aj+ma).

(2.8)

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427

The last two identities together with (2.7), (2.8) clearly gives us Qd(Ai + da) = Qd(A )

(2.9)

where a = -a, ai = Ai. Finally our asserted quadratic relation: 0v(Ai)Qd(Ai + ra) = Qr(Ai)Qd-r(Ai)

(2.10)

follows from (2.6), (2.7), (2.8), (2.9), and the following elementary identity: n

Inn

(ai-Aj) x 11

11

d

H

(A -A,+(r-m)a)

i=ojoi n

r

=11 H

(Ai-.j+ma)

A=O m=0(A>m)36(i,0)

n d-r x fl 11

(Ai - aj - ma)

9=0 m=O(.9,m)#(i,0)

Note that the last identity is just the interesting identity

eT(Tp,Pn) eT(Pir/Nd) = eT(pir/Nr) . eT(piOINd-r)

When V is concave, the fiber at (f, C) E Ao,o(d, Pn) of the bundle Ud is H'(C, f *V), and we need only one change in the above argument. The gluing exact sequence in the concave case is

0-+VIpi -+ Vd-+VrIF,9Vd_rIFd_r->0 instead. Therefore the gluing identity for equivariant Euler classes becomes eT(Vd) = eT(Vr) - eT(Vd-r) 6PieT(V)

Since ii' = 1/eT(V) for concave V, the quadratic relation (2.10) remains valid in this case. The case when V is a direct sum of a convex and a concave bundle

is also similar. D

Mirror Symmetry I

428

2.4. Linked Euler data

Definition 2.9. Two sequences P, Q E S are said to be linked if LP*;,. (Pd - Qd) E

R-1 vanish at a = (Ai - Aj)/d for all j 54 i, d > 0.

Let V be a T-equivariant concavex bundle on P", and C = P1 be any T-invariant line in P". By Grothendieck's principle, we have the form Via = ®a 10(la) ®®b_1O(-kb) for some positive integers la, kb. (0 cannot occurs because eT(V) is invertible, by definitiion.) Assume that {la} and {kb} are independent of C. This is the case, for example, if V is uniform [41]. We call the numbers (11i .., dN+; k1, .., kN-) the splitting type of V. With this notations, we have Theorem 2.10. Let Qd = cp!eT(Vd) as before. At a = (Aj - Ai)/d, i # j, we have kbd-1

lad

bpj.o(Qd) = H fi (loAj-m(Aj-Ai)/d)xfl fi (-kaAj+m(Aj-Ai)/d) a m=0

b

m=1

In particular the Euler data Q is linked to !ad

P: Pd

11 (laic a m=0

kbd-1

- ma)xH 11 (-kbh+nia). b m=1

Proof: Since we shall evaluate the class (Qd) at a = (Aj - Ai)/d, we introduce the notation Qd(s;, a) = Qd, and denote the value of the class above by Qd(Aj, (Aj - Ai)/d). First consider the case V = 0(l) on P". We consider a smooth point in (f, C) E Md with C = P1 and in coordinates

f : C -> P' X P",

[wo, w1] -+ [w1, wo] x [0' ... , wa, ... , v d,... 2 0}

where in the last term, tug is in the i-th position, wi is in the j-th position, and all of the other components are 0. The image of (f, C) in Nd under cp is the smooth point

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429

It is easy to see that, if the weight of S' in the group G = S' x T is a = (A5 - Ai)/d, then P, is fixed by the action of the subgroup of G with a = (A5 - A;)/d. So (f, C) is a smooth point in Md fixed by the subgroup in G = Sl x T with a = (A - Ai)/d. The class Qd(rc, a) restricted to P,i is just Qd(Oj, (A, - ) j/d). At the points (f, C) E Md and Pij E Nd, the map ii is a canonical identification. From definition, Qd(rc, a) restricted to Pq is the same as eT(Vd) restricted to cp 1(P5) = (f, C) E Md, which by definition, is the same as eT(Ud) restricted to the T-fixed point (9r2 o f, C) in M0,0(d, P"). Note that (ir2 o f, C) is the degree d cover of the Tinvariant line joining pi and pj in P". Explicitly 7f2 o f : [wo, wl]

(0, ... , wDdi ... , wdl, ... 0)

So the induced action of T on C = Pl has weights {a;/d, A3/d}. Now let us compute eT(Ud) restricted to (ir2 o f, C). When restricted (7r2 o f, C) E Mo,o(d, P") the fiber of Ud is just the section space H°(C, (7r2 o f)*O(d)) = H°(C,O(ld)). It has an explicit basis {wo wrd-k } with k = 0, , id. Since the T-weight of wo wid-k is k)t /d + Id- k) A, /d, by multiplying them together we get

Qd(Ai, (A5 - A)/d) = Id11 (la; - m(A; - Ai)ld) m=o

For a general convex vector bundle V on P", the fiber of the induced bundle Vd restricted to V'1(P5) = (f, C) E Md is H°(P', (7r2 o f)*V). By Grothendieck's principle, V restricted to the line spanning pi, pj splits into direct sum of line bundles {O(la)}. Pulling them back to C via the degree d map 7r2 o f, we get the direct sum of {O(lad)}. Since V is convex, each la > 0. By applying the same computation to each summand, we get 14d

Qd(Aj, (A3 - a:)ld) _ fl [I (laAj - m(Ag - A )/d) a M=0

For a concave bundle V, we need only one minor change in the above argument. We leave to the reader as an exercise to check that

for V = O(-k), k > 0, kd-1

Qd(A,, (A5 - A )ld) _ [J (-kA5 +m(ai - a;)/d), m=1

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Mirror Symmetry I

by using either the Atiyah-Bott fixed point formula or by writing down

an explicit basis for H1(P1, O(-kd)). So for an arbitrary concave bundle V, we have the form kbd-1

Qd(t,, (Aj - As)ld) = II II (-kbAj + m(Aj - Ai)/d) b

m=1

Similarly for an arbitrary concavex bundle V, we have the form tod

Qd(A

, (Aj - Ai)/d) = II [J (la.j - m(Aj - Ai)/d)

a m=0 kbd-1

X fl H (-kbAj +m(Aj - ai)1d) b

0

m=1

Theorem 2.11. Suppose P, Q are any linked 1-Euler data. If deg. LP:.(, (Pd - Qd) <- (n + 1)d - 2

for all i = 0,.., n and d = 1, 2,.., then P = Q. Proof: By definition, PO = Qo = St. We will show that Pd = Qd, assuming that

Pr=Qr, r=0,..,d-1.

(2.11)

Since the R-valued pairing pfd(u v) on RHG(Nd) is nondegenerate, it suffices to show that

L. := Pfd('' . (Pd - Qd)) is zero for all s = 0, 1, 2, .... By the localization formula for pfd, we get n

d

La = E E(ai + i=0 r=0

Par (Pd

L

n

7- d

- Qd)

11k=0llm=0(k,m) (i,r)(Ai -.k + (r - m)a)

Since P is an Euler data, it follows that t . (Pd), for each r = 1,.., d -1, is expressible in terms of P1, .., Pd-1. Likewise for Q. Thus

Lian, Liu, and Yau

431

by the inductive hypothesis (2.11), the sum over r above receives con-

tributions only from the r = 0, d terms. Applying the Reciprocity Lemma (i), we further simplify Le to L8

- E (A; Ai(a) + (Ai + da)8 Ai(-a) (-a)d / i=01 ad (-i)d tP1.o (Pd - Qd)

A,(a) : =

d!

Ak) llk#i

l lk#i(Ai

(2.12)

l lm=1(Ai - Ak - ma)

Since P, Q are linked Euler data, we have (2.13)

tP..o (Pd - Qd) = 0

at a = (Ai - Ak)/d, k 76 i. By the inductive hypothesis (2.11) and the Reciprocity Lemma (iii), (2.13) holds at a = (A1 - Ak)/m for m = 1,.., d as well. This shows that Ai E R = Q(A) [a] for all i. By assumption deg.Ai < (n + 1)d - 1 - nd = d - 1. But since L, E R ie. polynomial in a, for all s, it follows easily that the Ai must be identically zero. D 2.5. The Lagrange map and mirror transformations Throughout this subsection, we fix an invertible class fl and shall

denote by A = A° the set of a-Euler data.

Definition 2.12. An invertible map A : A -* A is called a mirror transformation if for any P E A, µ(P) is linked to P. We call A(P) a mirror transform of P. (Notations) So denotes the set of sequences Bd E R-1H6(No), d = 1, 2,... We define the map 2 : S -+ So, P H 1(P) = B where Bd = Id (Pd).

Definition 2.13. B:

Recall that any equivariant cohomology class w E R`1H, (Nd) is determined by its restrictions tpjr (w) E R-1, i = 0, .., n, r = 0, .., d. Conversely given any collection wir E R-1, there exists a unique class w E R-1HG(Nd) such that tp., (w) = wir for all i, r. In fact, n

d

w=EEwir i=0 r=0

X - Aj - ma

11

(j,m)(i,r)

Ai-Ai-(m-r)a*

Mirror Symmetry I

432

In particular given a sequence B E So, then for each d there is a

unique class Pd E R-1HG(Nd) such that tAir (Pd) = 1.;. (

)-1

tA. (Br) a;. (Bd-r), i = 0, .., n, r = 0, .., d, (2.14)

where we have set B0 := Q. This defines a sequence P E S, hence a map

Gg:So-aS,

We shall call C = Gn the Lagrange map. By (2.14) at r = 0, we get

t;i(Bd) = tP, 0(Pd) = tpiId(Pd), i = 0,..,n.

(2.15)

First, this implies that the two classes Bd, Id (Pd) on No = P" coincide for each d. Thus

B = I(P) = I o G(B). (2.16) Thus C : So -3 S is a section of the onto map I : S -4 So. Second, substituting (2.15) into (2.14), we get aPi (Q)aPir

(Pd) =

(Pr) tpi o (Pd_,.).

(2.17)

If, furthermore, we have Pd E RHG(Nd) rather than in R-1HG(Nd), then eqn. (2.17) says that P is an Euler data. (A) The image P = C(B) of a given B E So under the Lagrange map is an Euler data if Pd E RHG(Nd), d > 0.

On the other hand, it is trivial to show that if Q E A C S then

Q=GoI(Q).

(2.18)

Now using L we can lift any map µo : So - So to a map

µ=Loiool:S---S, which we shall call the Lagrange lift of Mo. Thus from eqns. (2.16) and (2.18), we have

(B) Let µo : So -+ So be invertible with inverse vo, and let p, v be their respective Lagrange lifts. Then µ o v = v o µ = idA when restricted to Euler data. We now discuss the relationship between Euler data and series of hypergeometric type.

Lian, Liu, and Yau

Definition 2.14.

433

Given any B E So, define edt

HG[B](t) := a-"/' (S2 + E

Bd

7d>0 l lk=O l im=1(P - Ak _ ma)

)

where p E HG (N0) is the equivariant hyperplane class of No = Pn.

Note that HG[B](t) is a cohomology valued formal series. If P : Pd = Him+old(tic - ma) as in Example 1, it is obvious that in the limit A -* 0, we have

HG[Z(P)](t) = e Ht/a E !l M=O ((n + 1)H - ma) edt

[I'=1( H - ma n+l

d>0

)

where H E H*(Pn) on the right hand side is the hyperplane class of Pn. The coefficients of (-:K)' )' for i = 1, .., n, are exactly solutions to a hypergeometric differential equation discussed in the Introduction. We now consider a construction of mirror transformations. Let B E So, and set Bo := S2. Given any power series g E etR[[et]], there is a unique b E So such that

HG[B](t+g) = HG[B](t). In fact, since

HG[B](t + g) =

Bd edted9

e-Pt/°e-P9/'

n

d>0 fk=0 11m=1(p - Ak - ma)

if we write edg = Es>0 9d,se8t, gd,s E R and a-P9/" = Es>0 g' g's E R[p/a], then it is straightforward to find that d-1

eat,

d

Inn 77'7 Bd= Bd+>9a_rB;.ll 11 (p - Aj - ma)

r=0

d-1

j=O m=r+1 n

(2.19)

d

Bd := Bd + > gr,d-rBr H fJ (P/ - Aj - ma). r=0

j=O M=r-I-1

Thus we have an invertible transformation to : So -4 So, B H B. Similarly, given any power series f E etR[[et]] we have an invertible transformation µo : So -+ So, B i-+ B, such that

ef/a HG[B](t) = HG[B](t).

Mirror Symmetry I

434

Again if we write of /I = E810 d-1

fseat, fs n

E R[a-1], then

d

Bd = Bd + > fd-rBr 11 rl (p - Aj - ma). r=0

(2.20)

j=O m=r+1

We now make an important observation about the transformation

po in each case above. For d > 0 the class fl 0(p - Aj - da) always vanishes when restricted to the fixed points pi E P", at a = (Ai Aj)/d. It follows immediately from (2.19) and (2.20) that t,, (Bd), t*i (Bd) always agree (whenever defined for all d) at a = (Ai for j 5A i. To summarize: (C) Given g, f E et1Z[[et]], let po : So -> So, B H B, be the invertible transformation defined by

eff" HG[B](t+g) = HG[B] (t). Suppose B is such that all values of t7,(Bd) are well-defined at a = (Ai - Aj)/d, j i4 i. Then these values are preserved under Fto

Obviously if P is an Euler data and B = I(P), then the restrictions t,*, (Bd) = tP, o (Pd) E R are polynomial in a. Hence they are always well-defined at a = (.Ai - \ j) /d, j 54 i.

Lemma 2.15. Let.i be the Lagrange lift of the above transformation po : So -+ So, B B. Then p is a mirror transformation. In particular, if P is an Euler data, then P = it(P) is an Euler data with

e" HG[I(P)](t + g) = HG[I(P)](t)

Proof: The second assertion follows from the first assertion and the

fact that lo p=ToCo1.iool=l-tool.

It suffices to consider the two cases f = 0, g = 0, separately. In each case we let P be an Euler data, and denote

P = p(P),

B = I(P), B = po(B)

Since,u := Co uo ol, we have P = G(B). In each case we will show that

P is an Euler data. We claim that this suffices. First, by statement

Lian, Liu, and Yau

435

(B) above, p is invertible as a transformation on the set A of Euler data. Second, by statement (C) above, the restrictions tp; (Bd) = (Pd) and L;, (Bd) = t*, i. P o (Pd) agree at a = (Ai - .1 j) /d, o L Thus P is linked to P. So, by definition, p is a mirror transformation. P*

We now proceed to checking Eulerity of P. Since P = G(B), (2.14) holds. Multiply both sides of eqn. (2.14) by the respective

sides of the following identity: e(a;+ra)(t-r)/a edr n d j=0 Hm=O(j,m)#(i,r)(Ai + ra - Aj - ma) 1

A1)

x e-air/a

ert

x ea;t/a

Al + ma)

11'= e(d-r)r 77d-r

H,=011m=1('ri -'`j - ma) and then sum over i = 0,.., n, and r = 0,.., d. The result is edr pfd(Pd e1.(t-r)/a) d

pf rr=0 [e-pr/a

-1

[e

-P tla

r Br(Pert ]1 n3=0 11m=1 - Al

rIx n7dr

Bd-r

- ma) ]

e(d-r)r

11 j=0 11,=1(p - A j - ma) ] Now summing this over d = 0, 1, 2,.., we get:

Eedr pfd(Pd

e"('-r)/«)

= pf (52-1 HG[B](t) HG[B](T))

.

(2.21)

d>0

Likewise, of course, for P and B. First case:

HG[B](t) = HG[B](t+g(et))

(2.22)

By (2.21), we have

Eedr Pfd(Pd e" (t-r)la) pf (SZ-1 HG[B](t) HG[B](r)) = d>0

Pf (0-1 HG[B](t + g(et)) HG[B](r+g(er)))

_ ` ea(r+s(c')) pfd (Pd d>0

Mirror Symmetry I

436

By (2.22), we can equate the two right hand sides above. Setting

q = e', _ (t - r)/a, we get

qd pfd(Pd e'K) = T gded9(9) Pfd

x (Pd

(2.23)

d>o

d>0

ek(9+(qes")-g+(q))/a e--(.9-(9eS")+g-(q))/a

where g = g+ + g_ with gt = fgt. Obviously for any y(q) E R[[q]],

g+(qel*) - g+(q) E a R[[q,c]]. Since the involution w '-+ w on R simply changes the sign of a, the fact that g_ is odd shows

that g_(q) E a

R[[q]]. Likewise for g_(gesa). We know that Pd E RHG(Nd) (since P is Euler), and that pfd maps RH* (Nd) to R. So the right hand side of (2.23) now clearly lies in Rfq,

So, likewise for the left hand side of (2.23). It follows that Pfd(Pd ice) E R, s = 0,1, 2,...

(2.24)

A priori Pd E R-1H6(Nd) has the form

pd=aNn.N+...+ao, a;ER-1, N=(n+1)d+n. Since pfd(, N) = 1, it follows from (2.24) that aN,.., ao E R. Hence

Pd E RHG(Nd) (rather than in R-1HG(Nd)). By statement (A) above, P is an Euler data. Second case:

HG[B](t) = of/a HG[B](t). Again applying (2.21) and writing f E etR[[et]] as f = f+ + f_ with f± = ±f±, we get

Eqd pfd(Pd eS') = e-f(et)/* ef(e')/a pf (0-1 HG[B](t) HG[B](T)) d>0

= e-(f+(qes")-f+(9))/a e(f-(qes")+f-(q))/a x E qd p.fd(Pd el ). d>O

The right hand side lies in R[[q, C]] as before, implying that P is Euler.

0

Lian, Liu, and Yau

437

3. Applications Definition 3.1. A concavex bundle V on P" is called a critical bundle if the induced bundle Ud -4 Mo,o(d, Pn) has rank dim Mo,o(d, PI) = (n+1)d+n-3. We denote the nonequivariant Euler class by ct0P(Ud). Recall the notation that given a concavex bundle V = V+ ® V-,

we have

SZv

eT(V+)IeT(V ) Q: Qd=(P!eT(Vd), d>0. By Theorem 2.8 the sequence Q is an Qv-Euler data. If V is a critical bundle, introduce

Kd:=

ctoP(Ud),

( :_ EKdedt d>O

Theorem 3.2. Let V be a concavex bundle on Pn. (i) The restrictions Id (Qd) E HG(PT) has degaII (Qd) < (n+1)d-2. (ii) If V is critical, then in the nonequivariant limit A -3 0,

f e-Ht/a J

f

n

n

d

I Qd

a)n

+1 = a-3(2 - d t)Kd

rim=1(

(HGZ()](t) - e-t/a)V) = a-3(2-

Proof: The second equality in (ii) follows trivially from the first equality.

By eqn. (2.8) in the proof of Theorem 2.8, we have P`eT(Ud)

Qd(Ai) = Opj (,\i)'YPn (Ai) E f Fd eT(N(Fd))[a(a - ci(L))] Fd

(3.1)

rlj,.i(p OP_ where OP, := - A1), !!j Olim=1(p - Aj - ma) E HT(Pn). From the localization formula, we deduce that

438

Mirror Symmetry I «(«-c 3(L))ev'OPi=,fpn

p"eT(Ud)

a°ae cy L

ev

cp$ [OP. /ev! ( P*eT(Ud) i(L) ) ]

pi ev' t/(a(

!ud"4) )

Th'as(3.1) canbewrit (tq,

i = 0,.., n.It f ollowsthatld (Qd) _

-

Opn ev! 1 «(«e ci(L))) .(3.2)Thisshowsthatdeggld(Qd) < degacbp,. 2 = (n +\ 1)d - 2, proving (i). Since Qd = cp!eT(Vd) E HG(Nd), their nonequivariant limit A -7 0 exist. In this limit (3.2) gives

A := f e-Ht/« P"

_

Id (Qd)

rJm=1(H - ma)n+1

e-ev-Ht/« P*etop(Ud)

a(a - c1(L))

Mo,i(d,Pn)

=

fo,o(d,P) ctop(Ua) P!

(

e-ev'Ht/«

a(a - cl(L)))

Now ctop(Ud) has degree the same as the dimension of M0,o(d, Pn).

The second factor in the last integrand contributes a scalar factor given by integration over a generic fiber E (which is a P') of p. So we e -ev"HO/a cy(3 pick out the degree 1 term in «(«_el(L)), which is just -ev"Ht «::

+ ;,

Restricting to the generic fiber E, say over (f, C) E M0,o(d, P'L)

the evaluation map ev is equal to f, which is a degree d map E = P1 - P. It follows that .[Er

Moreover, since cl(L) restricted to E is just the first Chern class of the tangent bundle to E, it follows that JE

c1 (L) = 2.

So we have

A = (- as +

a )Kd

It is easy to work out the complete list of critical concavex vector bundles V on PI which are direct sums of line bundles. Such a V is of the form

V=V+®V-

V+ = ®a 0(la) V = ®b=10(-kb)

Lian, Liu, and Yau where

439

IN+, k1, .., kN- are positive integers. By Riemann-Roch,

the bundles Ud that V induces on Mo,o(d,P") has rank Ud = d (E 1a + E kb) + N+ - N', which must be (n + 1)d + n - 3 for

all d if V is critical. Thus we must have

1.+Ekb=n+1

N+-N-=n-3. The complete list of critical bundles that are also direct sums of line bundles is: PI: P2:

P3

P4 P5: ps P7:

Note that we have excluded the critical bundles in which the hyperplane bundle 0(1) occurs because in the nonequivariant limit it only

reduces a given case of P" to Pn-'. For example, even though the bundle 0(1) ® O(-1) ® 0(-1) on P2 is certainly critical, computing the Kd for the induced bundles is equivalent to doing the same with 0(-1) ® 0(-1) on P1. It is curious to note that the numerical conditions (3.3) is rather similar to the condition for having a projective complete intersection Calabi-Yau threefold. In fact five of the examples on P4 through P7 above involving only positive bundles are exactly the cases in which each critical bundle cuts out a complete intersection Calabi-Yau threefold. We also note that, with the exception of the P1 case, the three examples which involve negative bundles in fact correspond to noncompact Calabi-Yau threefolds. The

total space of 0(-3) -3 P2, the total space of 1,*0(-2) -+ X, where V) :X P3 is a quadric, and the total space of i/i*O(-1) --- P4 where : X -4 p4 is the intersection of two quadrics, all three are noncompact Calabi-Yau. These examples arise in the so-called local mirror symmetry. In the next subsection, we shall compute the Euler classes of the induced bundles for the list above.

Mirror Symmetry I

440

3.1. The first convex example: The Mirror Conjecture Throughout this subsection, we set d = n+1, consider the convex bundle V = 0(l) on Pn, and fix Sl' = lp. P, Q shall denote the following two linked Euler data (cf. Theorems 2.8, 2.10.): Id

P: Pd= H(ln - ma) M=O

Q : Qd = ipieT(Vd) Consider the hypergeometric differential equation

((y - let(l at + 1) ... (lat +n) ) h(t) = 0. We have seen that a basis fi, i = 0, .., n-1, of solutions can be read off from the hypergeometric series (cf. Introduction) in the limit A -4 0:

HG[T(P)](t) = e at1a

E

d>0

l lm=o(lH - ma) edt m=1 (H

H

- ma)-+'

H2

=lHfo-f1 a +f2 a2 Recall that

T(t) := fl = t+ fo is the mirror map of Candelas et al, where ,dµ)l

!

f0 d>0

(d

)n+l edt,

91

( d>1

Id

_cdt E / 7n=d+1

Lemma 3.3. In the limit A -* 0, we have HG[T(Q)](T(t)) _ kHG[T(P)](t) Proof: By expanding in powers of a-1 and using the assumption that

l=n+l, we get

HG[T(P)](t) = e-Pt/'

ma)

77-77n

d>O I1k=011m=1( - Ak

= lp [fo -

edt

- ma)

a-l (P f1 + 92 E Ak) + .. . k=e

J

Lian, Liu, and Yau Id)!

441

d

1 at where 92 = Ed>1 (d! >m=1 me Put f :_ (l09 fo)a + Ofz, Et=0 Al, E etR[[et]]. By Lemma 2.15, we have a mirror transformation p such that

HG[Z(P)] (t) = of /u HG[Z(P)] (t)

(3.7)

where P = µ(P). Substituting (3.6) into (3.7), we get n

Ak + ...)

HG[Z(P)](t) = (1 + a-192

f0

x fo 1 lP [fo

k=O

- a 1(P f1 +92 T Ak) + ...J

(3.8)

k=o

lp - a1lp2f1

fo+...,

By Lemma 2.15 again, we have a mirror transformation v such

that HG[Z(Q)](t) = HG[Z(Q)] (t + fu )

where Q = v(Q). Since, by Theorem 3.2 (i), Id (Qd) = it is straightforward to find that

O(a(n-!-1)d-2),

HG[Z(Q)](t) = e-n(t+'7o)/cz(lp+...) = lp-a-11p2(t+ fo)+.... (3.9) From (3.8) and (3.9), we conclude that for d > 0, rjk=0

Id (Pd - Qd)

=0

fm=1(p - Ak - Ma)

modulo order a-2, and hence deg,,, cni.0 (Pd - Qd) 5 (n + 1)d - 2.

But AP = µ(P) is linked to P, and Q = v(Q) is linked to Q. Since P and Q are linked, it follows that P and Q are also linked. By Theorem 2.11, we have P = Q. In particular, we have HG[Z(Q)](T(t)) = HG[Z(Q)](t) = HG[Z(P)](t) = oft z HG[Z(P)](t)

Mirror Symmetry I

442

Since both Pd,Qd lie in HH(Nd), their nonequivariant limit exist. Taking A - 0 yields our assertion. 0 Throughout the rest of this subsection, we set 1 = n + 1 = 5 and consider the critical bundle 0(5) -3 P4. We assume that we have taken the nonequivariant limit A -3 0. Recall that 56 3

F(T) :

Kde', + >2 d>0

).

.F(T)

:

2 fob o

fu

Theorem 3.4. (The Mirror Conjecture) F =.F. Proof: Since

HG[ Z( P )J() 77 t= 5H (fo -

H2

flaH+f2«2 -f3 H3 /

J

(3.10)

C93

we will prove that (cf. [8])

5,a2-TF'5

2F H3)

(3.11)

Eqns. (3.10), (3.11) and the preceding lemma imply F =.F. Denote the right hand side of (3.11) by R. Then H2 24i H3 eHT/aR = SH (1 + T4?'-a-2 + 5 ) = 5H + O(eT )

where

:= F -

56

= O(eT ). Similarly eHT/aHG[Z(Q)] (T) = 5H + O(eT), which also has no polynomial dependence on T. So (3.11) is equivalent to e-HTIo (eHT/aHG[Z(Q)])

fP 4

= fps a-HT/a (eHT/aR)

By Theorem 3.2(ii), this left hand side is fP4 HG[I(Q)] = a 3 (2(1 - T'1') +

which coincides with fp, R. D

f

4

e-HT/a5H = a-3 (2F - TF'),

Lian, Liu, and Yau

443

It is straightforward to generalize Theorem 3.4 to all other critical convex bundles V in the list (3.4). In each case, nv becomes rja lap and the Euler data P to be linked with Q : Qd := co!eT(Vd) is given by Pd = rja = o(lan - ma). In the nonequivariant limit, the hypergeometric series HG[I(P)](t) will produce some hypergeometric functions fo, .., f3 defining the function F. The generating function F for the Kd is modified by simply replacing the term ss by s fX V)*H3, where 0 : X -- P" is the Calabi-Yau cut out by V. With these minor modifications in each case, Theorem 3.4 holds. We leave the details as an exercise for the reader. 3.2. First concave example: multiple-cover formula

Let V be the bundle O(-, ®O(-1) on Pl. Ford > 1, V induces a rank 2d - 2 bundle Ud -+ Mo,o(d, P') whose fiber at (f, C) is the space H'(C, f*V), thus V is a critical concave bundle on Pl. We set 11V = 1/eT(V) = p. We shall compute the equivariant classes Qd := SPIeT(Vd), and the numbers Kd for this critical bundle. Note that by definition Q1 = 1 and K1 = 1. As a consequence of Theorem 2.10, d-1

tP:,o1(Qd) = fi (A{ - m(A; - \j)/d)2 m=1

at a = (A1- A j) /d, j # i. Thus Q is linked to d-1

P: Pd:= 11 (r. -ma)2, m=1

which is a p-2-Euler data. Obviously degald (Pd) = 2d - 2. It follows from Theorem 3.2 (i) that degald (Pd - Qd) < 2d - 2, implying Q = P by Theorem 2.11.

Corollary 3.5. Kd = d-3. Proof: By Theorem 3.2(u) in the limit A -+ 0, we have

f e-Ht/« Pl

Ii(Qd) = a-3 (2 - d t)Kd. rjm=1(H - ma)2

Since Q = P, we have Id (Qd) = Id (Pd) =11m 11(H - ma)2, giving

I

e-Ht/a l

Id(Qd) = a-3d-3 t(2 - d t). 11m=1 (H - ma)2 d

Mirror Symmetry I

444

3.3. Second concave example: Kp2

Let V be the canonical bundle O(-3) -+ P2. For d > 0, this bundle induces a rank 3d - 1 bundle Ud -3 Mo,o(d, P2). Thus V is a critical concave bundle. We set n V = 1/eT(V) = (-3p)-1. We shall compute the equivariant classes Qd := co!eT(Vd), and the numbers Kd for this critical bundle. As a consequence of Theorem 2.10, we have 3d-1

tp;,a(Qd) = II (-3.Ai+m(ai - \j) 1d). M=1

at a = (,\i - \j) /d, j 36 i. Thus Q is linked to 3d-1

P : Pd := 11 (-3n + ma). m=1 (-3p)-1-Euler data.

which is a

Corollary 3.6. FId>0

HG[2(Q)](t + g) = HG[I(P)](t) where g

(_i)d d I e dt d

d

Proof: By expanding in powers of a- we get HG[T(P)](t) = x

((-3p)-1 + E 1

3

d=1 (-3p m d /m

+ ma)

e dt

-lt+g (3.12)

As before, it is now straightforward to show that HG[T(Q)] (t + g) _- HG[-T(P)l(t)

modulo order a 2. Once again by Theorem 2.11, the two sides are equal identically. Using Theorem 3.2(ii) and the preceding corollary, we obtain the

Lian, Liu, and Yau

445

Kd, for d = 1,.., 10: Kd

d

3

1

s

45

2 3

244 9 - 12333 64 211878 125 102365

4 5

6 7 8 9

10

6

64639725 343 - 1140830253 512 6742982701 243 36001193817

_

100

3.4. A concavex bundle on p3.

Let V = O(2)®O(-2) on p3. This is a direct sum of a convex and a concave bundle. The induced bundle Ud -+ Mo,o(d, p3), with fiber at (f, C) being H°(C, f *O(2)) ®Hl (C, f *0(-2)), has rank 4d. We set OV = eT(0(2))/eT(O(-2)) = -1. We shall compute the equivariant classes Qd := cpleT(Vd), and the numbers Kd for this critical bundle. As a consequence of Theorem 2.10, we have

p,,o

2d

2dd--1

m=0

m=1

(Qd) = jI (2ai - m(ai -.j)/d) x 11 (-2Ai + m(.i - Aj)/d)

at a = (Ai - Aj)/d, j # i. Thus Q is linked to 2d

2d-1

P : Pd:= ll (2rc - ma) x fl (-2sc + ma). m=0

m=1

which is a -1-Euler data.

Corollary 3.7.

HG[2(Q)](t + g) = HG[T(P)](t) where g :_

Mirror Symmetry I

446 ,2

Ed>o a a! ' eat Proof: By expanding in powers of a-1, we get

HG[T(P)](t) = e -Pt/' x

x m_i (-2p + ma) edt

(-1 + d>0

nk=0

llm=1(p

- Ak - ma)

=-1+a-lAt +g)+..., (3.13)

which, as in the previous examples, agrees with HG[T(Q)] (t + g) up to order a-2 . Hence Theorem 2.11 yields our assertion. 0 Using Theorem 3.2(ii) and the preceding corollary, we obtain the Kd, for d = 1,.., 10: d 1

2

3 4

Kd -4 _92

_ 328

27 777 16

5

_ 30004

6

4073 3

7

_ 2890808 343 _ 7168777

8

125

128

9 10

_ 285797488 729 - 714787509 250

3.5. A concavex bundle on p4.

Consider now the critical bundle V = 0(2) ® 0(2) ® 0(-1) on P4. The induced bundle Ud -4 Mo,o(d, p3), has rank 5d + 1. We set fl' = eT(O(2))2/eT(O(-1)) = -4p. We shall compute the equivariant classes Qd := coleT(Vd), and the numbers Kd for this critical bundle. As a consequence of Theorem 2.10, we have 2d

tp,,o(Rd)

d-1

= II (2a; - m(A1- A,)/d)2 X [J (-A + m(A - A,i)/d) M=O

m=1

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447

at a = (A - A,)/d, j # i. Thus Q is linked to

d-i

2d

P : Pd :_ fl (2rc - ma)2 x 11 (-rc + ma). m=i

M=0

which is a -4p-Euler data.

Corollary 3.8. Ed>O

(-i)d 2d !Z edt d

HG[Z(Q)](t + g) = HG[Z(P)](t) where g

d!

Proof: By expanding in powers of a-1, we get HG[2(P))(t) = e -P'10

C_4p+E d>O

nm o(ZP - ma)2 x nm 11(-P+ma) dt 7-ik.O IImda1(P e L 4 - Ak - ma)

= -4p+a-14p2(t+g) +... (3.14)

which, as in the previous examples, agrees with HG[I(Q)] (t + g) up to order a-2. Hence Theorem 2.11 yields our assertion. p We can work out the Kd here as we did before. The Kd here can be obtained by taking Kd from the preceding example on P3, and multiply it by 4(-1)d. This is so because in the nonequivariant limit, the hypergeometric series HG[I(P)](t) (cf. (3.13) and (3.14)) in this example on P4 and the preceding example on p3 are related by first a multiplication of 4p followed by a change of variable edt + (-1)dedt. 3.6. General concavex bundles In fact the examples above are representative of the most general

concavex bundle. Let V = V+ a V- be a concavex bundle on P", and let Q, P be as defined in Theorem 2.10, and assume that V has splitting type (li, .., IN+; ki,.., kN- ). Note that E la + E kb is the value of the class ci(V+) - ci(V-) on a T-invariant Pi in P".

Theorem 3.9. If d (E la + E kb) - N- < (n + 1)d - 2 for all d > 0, then Q =P. If d (E la + E kb) -N- < (n + 1)d for all d > 0, then there exists a mirror transformation la, depending only on the la, kb,

such that Q =,u(P). Proof: By definition of P in Theorem 2.10,

degala (Pd) = d ( Ia + E kb) - N-.

Mirror Symmetry I

448

Consider the first case, where this is bounded above by (n + 1)d - 2 for all d. Then by Theorem 3.2,

deg.(Id(Pd) - Id(Qd)) S (n+1)d - 2, implying Q = P by Theorem 2.11. Consider now the second case. Obviously our assumption implies that E la+E kb < (n+1). It is trivial to show that the only possibilities not covered by the first case are: (1) N- = 0 and E la = n + 1;

(2)N'=1andEla+k1=n+1. (3)N-=0and Ela=n; In

each of these cases, a mirror transformation can be constructed by immitating the previous examples in a straightforward way. Case (1) immitates the example 0(5) -+ P", while cases (2), (3) immitate the

example 0(-3) --3 P2. It is obvious that in each case, the mirror transformation depends only on the data la, kb.

Corollary 3.10.

Under the same hypotheses as in the preceding theorem, the Euler data Q : Qd = (p!eT(Vd) depends only on the splitting type, ie. the numbers la, kb, of the concavex bundle V on P".

Note that not every concavex bundle on P" is a direct sum of line bundles. For example the tangent bundle is convex, but is not a direct sum of line bundles. 3.7. Equivariant total Chern class

For simplicity, we restrict to convex bundles. Let V be a rank r convex bundle on P", and let CT(V)=x''+xr-lcl(V)+...+c,.(V)

be the T-equivariant Chern polynomial of V. Similarly we denote by cT(Ud) the equivariant Chern polynomial for Ud. As explained in Example 9, we can extend the notion of Euler data Q allowing Qd to depend on x polynomially, simply by replacing the ground field

Q by Q(x). Then a similar argument as in Theorem 2.8 shows

that the sequence Qd := Sp!(rr`cT(Ud)) is also an Euler data in the generalized sense. Moreover, the analogue of Theorem 2.10 holds, ie. at c = (A1- )t;)/d, we have the form lad

L*j,. (Qd) = II 11 (x + la - m(aj - i)l d) a m-0

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449

Hence Q is linked to the Euler data 1,d

P: Pd=fl H(x+lars-ma). a M=O

Again, under a suitable bound on cl(V), one can easily relate P, Q by a generalized (depending on x) mirror transformation.

For example, by taking the bundle 0(4) on P3, and applying the above result, we can compute the nonequivariant limits of all fJ. 0 o(d P3) c4d(Ud). They are expected to count rational curves in a Pl-family of K3 hypersurfaces in P3. Similarly we can take 0(3) on p2 and compute fMo,o(d,P2) C3d-1(Ud) which should count the number of rarional curves in a P2-family of elliptic curves in p2. Details will be reported in full in our forth-coming paper [37].

3.8. Concluding remarks The most important result we establised in this paper is the Mirror Principle. For simplicity we have restricted our examples in this paper, to studying only Euler classes and total Chern classes. As mentioned in the Introduction, the Mirror Principle works well for any multiplicative equivariant characteristic classes. We shall study in details more examples of the total Chern class in our forthcoming paper [37]. Generalization to manifolds with torus action will also be dealt with in details there. Finally, we make a tantalizing observation which might be of both physical and mathematical significance. As we have seen, the set of linked Euler data has an infinite dimensional transformation

group - the mirror group. For suitable concavex bundle V -t P", two special linked Euler data (cf. Theorem 2.10) Q : Qd = cp!eT(Vd) arising from the nonlinear sigma model (the stable map moduli), and P the corresponding Euler data of hypergeometric type, are related

by a mirror transformation. Since the mirror group is so big, there are many other Euler data which are linked to P and can be obtained simply by acting on P by the mirror group. From the physical point of view, P arises from type IIB string theory while Q arises from type IIA string theory, and mirror symmetry is a duality between the two. This relationship manifests itself on the linear sigma model as a duality transformation. This suggests that some other Euler data linked to P may arise from some other string theories which are dual to type IIA and IIB, via more general mirror transformations. From the point of view of moduli theory, P is associated to the linear sigma model compatification Nd of the moduli space Md we discussed in

450

Mirror Symmetry I

Example 10. Whereas Q is associated to the nonlinear sigma model Md, which is the stable map compactification of Md . This suggests

that some other Euler data linked to P may correspond to other compactifications of Md. If true, we will have an association between string theories, linked Euler data, and compactifications of moduli

space of maps, all in the same picture, whereby there is a duality in each kind which one sees in the linear sigma model. It would be interesting to understand this duality more precisely.

Lian, Liu, and Yau

451

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[16] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, alg-geom/9608011. [17]

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F. Hirzebruch, Topological methods in algebraic geometry, SpringerVerlag, Berlin 1995, 3rd Ed. [24] Y. Hu, Moduli spaces of stable polygons and symplectic structures [23]

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on on .Mo,n, Comp. Math. (to appear), alg-geom/9701011. M. Jinzenji and M. Nagura, Mirror symmetry and exact calculation of N - 2 point correlation function on Calabi- Yau manifold embedded in CPN-i, hep-th/9409029. M. Kapranov, Chow quotients of Grassmannian, I.M. Gel'fand Seminar, AMS (1993). B. Kim, On equivariant quantum cohomology, IMRN 17 (1996) 841.

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Mirror Principle II

Bong H. Lian,l Kefeng Liu,2 and Shing-'11mg Yau3

Dedicated to Professor Michael Atiyah.

Abstract. We generalize our theorems in Mirror Principle I to a class of balloon manifolds. Many of the results are proved for convex projective manifolds. In a subsequent paper, Mirror Principle III, we will extend the results to projective manifolds without the convexity assumption.

First appeared in Asian Journ. Math. Vol 3, No 1, ( March, 1999) 1

Department of Mathematics, Brandeis University, Waltham, MA

02154. 2

Department of Mathematics, Stanford University, Stanford, CA

94305. 3

02138.

Department of Mathematics, Harvard University, Cambridge, MA

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Mirror Symmetry II

Contents . . . . 1. Introduction . . . . . . . . . 1.1. Main Ideas . . . . . . . . . . . . 2. Set-up . . . . . . . . . . . . . . . . 2.1. Equivariant localization . . . . . . . 2.2. Functorial localization formula . . . . . . . . . . . . . 2.3. Balloon manifolds 2.4. Sigma models . . . . . . . . . . . 2.5. Regularity of the collapsing map . . . 3. The Gluing Identity . . . . . . . . . . 4. Euler Data . . . . . . . . . . . . . . 4.1. An algebraic property . . . . . . . . . . . . . . . . 5. Linking and Uniqueness 6. Mirror Transformations . . . . . . . . . 7. From stable map moduli to Euler data . . 7.1. The Euler data Q . . . . . . . . . 7.2. Linking theorem for A . . . . . . . . . . . . . . . . . . . . 8. Applications . . . . . . . . . . 8.1. Toric manifolds 8.2. Chern polynomials for mixed bundles . 8.3. Convex bundle . . . . . . . . . . . 8.4. A complete intersection in P1 x P2 X P2

8.5. V = 01(-2) 0 C2(-2) on P1 x P1

.

9. Generalizations and Concluding Remarks 9.1. A weighted projective space . . . . 9.2. General projective balloon manifolds 9.3. A General Mirror Formula . . . . . 9.4. Formulas without T-action . . . .

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456 457 460 460 461 463 466 471 474 479 481 482 486 491 491 493 495 495 496 499 500 501 502 502 502 503 504

1. Introduction For the long history of mirror symmetry, consult [17]. For a brief description of more recent development, see the introduction in [37][38]. The present paper is a sequel to Mirror Principle 1 [37]. Here, we generalize all the results there to a class of T-manifolds which we call balloon manifolds. These results were announced in [38]. Let X be a projective n-fold, and d E H2(X,Z). Let Mo,k(d,X)

denote the moduli space of k-pointed, genus 0, degree d, stable maps (C, f, xi, .., xk) on X [32]. Note that our notation is without the bar. By the work of [34](cf. [7][19]), each nonempty

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M°,k (d, X) admits a cycle class LT°,k (d, X) in the Chow group of degree dim X + (ci (X), d) + n - 3. This cycle plays the role of the fundamental class in topology, hence LT°,k (d, X) is called the virtual fundamental class. Let V be a convex vector bundle on X. (ie. H1(P 1, f *V) = 0 for every holomorphic map f : P1 --+ X.) Then V induces on each M°,k (d, X) a vector bundle Vd, with fiber at (C, f, x1i .., xk) given by the section space H°(C, f*V). Let b be any multiplicative characteristic class [26]. (ie. if 0 -4 E' -> E -+ E" --4 0 is an exact sequence of vector bundles, then b(E) = b(E')b(E").) The problem we study here is to compute the characteristic numbers Kd :=

f

b(Vd)

LT0,o(d,X)

and their generating function:

fi(t) := 2 Kd ed't. There is a similar and equally important problem if one starts from a concave vector bundle V [37]. (ie. H (P1, f "V) = 0 for every holomorphic map f : P1 -+ X.) More generally, V can be a direct sum of a convex and a concave bundle. Important progress made on these problems has come from mirror symmetry. All of it seems to point toward the following general phenomenon [12], which we call the Mirror Principle. Roughly, it says that the function 1)(t) can be computed by a change of variables in terms of certain explicit special functions, loosely called generalized hypergeometric functions. When X is a toric manifold with cl (X) > 0, b is the Euler class,

and V is a sum of line bundles, there is a general formula derived in [29] from mirror symmetry. This formula was later studied in [21] based on a series of axioms.

Acknowledgements. We thank B. Cui, J. Li, T: J. Li, and G. Tian, for numerous helpful discussions with us during the course of this project. B.H.L.'s research is supported by NSF grant DMS9619884. K.L.'s research is supported by NSF grant DMS-9803234 and the Terman fellowship. S.T.Y.'s research is supported by DOE grant DE-FG02-88ER25065 and NSF grant DMS-9803347. 1.1. Main Ideas

We now sketch our main ideas for computing the classes b(Vd). Step 1. Localization on the linear sigma model. Consider the

moduli spaces Md(X) := M°,o((1,d),Pl x X). The projection Pl x

458

Mirror Symmetry II

X -+ X induces a map 7r : Md(X) -+ Mo,o(d, X). Moreover, the standard action of Si on P1 induces an S' action on Md(X). We first study a slightly different problem. Namely consider the classes 7r*b(Vd) on Md(X), instead of b(Vd) on M0,o(d,X). First, there is a canonical way to embed fiber products (see below)

Fr = Mo,i(r, X) xx Mo,i(d - r, X)

each as an S' fixed point component into Md(X). Let it : F,. Md(X) be the inclusion map. Second, there is an evaluation map e : Fr -+ X for each r. Third, suppose that there is a projective manifold Wd with S' action, that there is an equivariant map c : Md(X) -3 Wd, and embeddings jr : X -3 Wd, such that the diagram Fr

-L+ Md(X)

eJ,

X =4

cp Wd

commutes. Let a denotes the weight of the standard S' action on Pl. Then applying the localization formula [3], this diagram allows us to recast our problem to one of studying the Si-equivariant classes Qd := V!ir*b(Vd)

defined on Wd. Moreover we can expand the class

Ad =

ioQd eSl (XO/Wd)

on X in powers of a'1, and find that it is of order a-2. The spaces Wd in the commutative diagram above are called the linear sigma model of X. They have been introduced in [39] following [45] when X is a toric manifold, Step 2. Gluing identity. Consider the vector bundle Ud 7.*Vd -* Md(X), restricted to the fixed point components Fr. A point in (C, f) in Fr is a pair (CI, fl, x1) x (C2i f2i x2) of 1-pointed stable

maps glued together at the marked points, ie. fi(xi) = f2(x2). From this, we get an exact sequence of bundles on Fr: 0 -4 ir*Ud -} U, ED Ud_r -4 e*V -+ 0.

Here i*Ud is the restriction to Fr of the bundle Ud -+ Md(X). And Ur is the pullback of the bundle U,. -+ M0,1(d, X) induced by V, and

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similarly for Ui_r. Taking the multiplicative characteristic class b, we

get the identity on Fr: e*b(V)b(i*Ud) = b(U*)b(Ud-r)

This is what we call the gluing identity. This may be translated to a similar quadratic identity, via Step 1, for Qd in the equivariant cohomology groups Hs, (Wd). The new identity is called the Euler data identity. Step 3. Linking theorem. The construction above is functorial, so that if X comes equipped with a torus T action, then the entire construction becomes G = Sl x T equivariant and not just Sl equivariant. In particular, the Euler data identity is an identity of G-equivariant classes on Wd. Our problem is to first compute the Gequivariant classes Qd on Wd satisfying the Euler data identity, and with the property that Ad - a-2. Note that the restrictions QdIp to the T fixed points p in Xo C Wd are polynomial functions on the Lie algebra of G. Suppose that X is a balloon manifold. Then it can be shown that (with a nondegeneracy assumption on eG(Xo/Wd)) the classes Qd are uniquely determined by the values of the QdIp, when a is some scalar multiple of a weight on the tangent space TpX. These values of QdIp can be computed explicitly by exploiting the structure of a balloon manifold.

Once these values are known, it is often easy to manufacture explicit G-equivariant classes Qd with the restrictions QdIp having the above same values, and satisfying the Euler data identity. In this case, we say that the data Qd are linked to the data Qd. By a suitable ood , c2. By change of variables, one can also arrange that e 1 g s o /Wd the preceding discussion, we get Qd = Qd. Step 4. Computing fi(t). Once the Classes Qd = co 1r*b(Vd) are determined, we can unwind the many maps used in Step 1. The preceding computations can be done simply in the form of power series. This finally computes the generating function fi(t). The answer for fi(t) is given in the form of Conjecture 9.1. In this paper, for clarity, we restrict ourselves to the case when the tangent bundle of X is convex. We prove that Conjecture 9.1 holds whenever X is a balloon manifold having a linear sigma model Wd such that eG(Xo/Wd) satisfies a nondegeneracy condition. In the nonconvex case, we must replace Mo,k (d, X) by Li-Tian's virtual fundamental cycle [34] for the purpose of localization and integration. The sequel, Mirror Principle III, to this paper will be devoted entirely to dealing with the added technicality arising from this replacement. All the results in this paper will generalize with

Mirror Symmetry II

460

only slight modifications as a result of this replacement, but with no change to the overall conceptual framework. By the equivalence, established in [35], of symplectic GW theory and algebraic GW theory for projective manifolds, we also expect that the results in this paper can be readily generalized to the symplectic case [43][44].

2. Set-up 2.1. Equivariant localization We first discuss some basic facts about localization. The key technique of our proof is the equivariant localization formula, due to Atiyah-Bott [2][10][3], and Berline-Vergne [9]. For an orbifold version of the localization formula, see [31]. The spirit of the localization we'll use is closer to the Bott residue formula. We first explain this formula.

Let X and Y be two spaces, by which we mean compact manifolds or orbifolds, with a torus T-action. When an orbifold is involved, the integral and localization formulas should be taken in the orbifold sense. Let {F} be the components of the fixed point set. Let denote the equivariant cohomology group with complex coefficient, and

iF : F -- X the inclusion map. We say that equivariant localization holds on X, if the two maps iF : HT* (X) -* HT(F), iF! : HT(F) -> HT(X).

which are respectively the pull-back, and the Gysin map, are such that the following formulas holds: given any equivariant cohomology class w on X, we have ZF.w

w = EiF! F

keT(F/X) )

This formula is equivalent to the integral version of the localization formula 2F W

X

F eT(FIX)

An important fact about equivariant theory is that, if V is an equivariant vector bundle on an orbifold X, then any characteristic class of V has an equivariant extension. Let T = Sl for simplicity. If c2k is a characteristic class of degree 2k, then its equivariant extension can be represented by the form C= C2k + C2k_2 + ... + CO

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in the equivariant cohomology of X. Here is a way to calculate the terms in the localization formula.

Assume that each fixed point component F is smooth. If c2k is a Chern class, then by using splitting principle, it can be expressed as a symmetric function of the Chern roots: P(xl, , xt) where I = rank V. When V can be decomposed into a direct sum of line bundles on F:

VIF=LiED ...eLi with the T-action on Lj given by, say the character fian'1-1net, then the restriction of its equivariant counterpart c to F is

i.c = P(ci (Li) + nit, ..., ci(Lt) + ntt). The computation of the equivariant Euler class of F in X is similar. When the restriction of TX to F has a decompostion into line bundles

TXIF=ElED ...®E.

where T acts on Ej by the character e2 rv' m t, then

eT(F/X) = fl(ci(Ej) + mjt).

j In the above, the n j's and m j's are integers for if X is a manifold, and are ratonal numbers if X is an orbifold. 2.2. Rtnctorial localization formula In this subsection, we derive two formulas which are often used in our work. Let X, Y be two T-spaces, and

f: X-*Y be an equivariant map. Let E be a fixed point component in Y, and F := f'1(E) be a fixed component in X. Let g be the restriction of

f to F, and jE : E -+ Y, $F : F -1 X be the inclusion maps. Thus we have the commutative diagram:

F `i X .f

g4.

E -14

Y.

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Lemma 2.1.

E:

Given any class w E HT(X), we have the equality on 9Ef! (w) ZFw eT(E/Y) =9! eT(F/X)

Proof: Let us consider localization of w f * jE! (1) on X, w

f*jE!/1) = iF! ZF(wf*jE!(1)) eT (FIX )

ll

Note the contributions from fixed components other than F vanish. Applying the push-forward fj to both sides, we get

(j/jo)).

f!(w)jE!(1) = fiiF! Now f o iF = jE o g which, implies

f!5F! = 2E!9!, ZFf* = 9*3E Thus we get

f!(w)7E!(1) =3E!9!

(

i;(w) 9*eT(ElY) eT(F/X)

Applying jE to both sides, we then arrive at ?Ef!(w) eT(E/Y) = eT(ElY) 91

(i ,(w) 9*eT(E/Y)

= eT(E/Y)2 9!

eT (F/X )

iF(w)

(e1C))

Since eT(E/Y) is invertible, our assertion follows.

The same argument applies to the case when E and F are Tinvariant subspaces. A slightly different argument for the proof of the above lemma will be given in our subsequent paper. We will also need the following formula, which is actually a special case of [18], Theorem 6.2. Here we include a proof for the convenience of the reader. The spaces involved are T-spaces, that is, T-manifolds or orbifolds.

Lemma 2.2. Suppose we have a T-equivariant commutative diagram

V -4 W g4.

if

Z

Y

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such that f *j!(1) = i!(1). Then for any class w on HZ.(W), we have the following equality on Z: j* f! (w) = g!i* (w).

Proof: By assumption, we have

w f*j!(1) =w i!(1) =i!i*(w). Applying j*f! to both sides, on the one hand we get

i*f!(w f*j!(1)) =j*(f!(w) j!(1)) =j*f!(w) eT(Z/Y) On the other hand, we get j*f!(i!i*(w)) = j*j!g!i*(w) = g!i*(w) eT(Z/Y)

Thus our assertion follows. 0 The case we will use in this paper is when Z and V = f -1(Z) are both T invariant submanifolds of same codimension, in which the condition in the Lemma clearly holds. 2.3. Balloon manifolds By a balloon manifold, we mean a complex projective T-manifold

X with the following properties. There are only finite number of Tfixed points. At each fixed point p, the T-weights on the isotropic representation TpX are pairwise linearly independent. This class of manifolds were introduced by Goresky-Kottwitz-MacPherson [22]. (We refer the reader to [24] for an excellent exposition.) Throughout this

paper, we assume that X is convex, ie. H'(Pl, f*TX) = 0 for any holomorphic map f : P1 -+ X. One important property of a balloon n-fold is that at each fixed point p, there are exactly n balloons, ie. T-invariant P1, each balloon connecting p to one other fixed point q. The induced action on each balloon is the standard rotation with two fixed points p and q. (see [22][25]). We denote by pq the balloon connecting the fixed points p, q. Toric manifolds, complex C-spaces and spherical manifolds are examples of balloon manifolds. We fix a T equivariant embedding of X into the product of projective spaces

p(n) := pn, x ... X pnm

such that the pull-backs of the hyperplane classes H = (Hl, , H,") generate H2(X, Q). We use the same notations for the corresponding

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equivariant classes of the H's, and their restrictions to X. For W E H2(X) and d E H2(X), we denote their pairing by (w, d). For convenience, we introduce the following notations:

H = (H1,

,

Hm) =H1(1+...+Hm(m

H(p) _ (Hi(p),..,Hm(p)) HC(p) = H1(P)(1 Here C

((1, .., Cm) are formal variables. We denote by K" C H2(X)

the set of points in H2(X, Z) free in the dual of the closure of the Kahler cone of X. Since K" is a semigroup in H2(X), it defines a partial ordering >- on the lattice H2(X, Z) f,.,,. That is, d >- r if d - r E K". Let {Hj" } be the basis dual to the {H3 } in H2 (X). If d ?- r for two classes d, r E H2 (X), then d - r = d 1 Hl + + dmH,"16 for nonnegative integers d1i

-

, d,,,.

We also consider a balloon manifold as a symplectic manifold with a symplectic structure given by w = HS for some generic C. By the convexity theorem of Atiyah [2] and Guillemin-Sternberg [23], the image of the moment map AS in the dual Lie algebra T* is a convex polytope, known as the moment polytope. When X is a toric manifold, the moment polytope is known as a Delzant polytope [15]. In this case, it is well-known that the normal fan of this polytope is the defining fan of X. We say X a multiplicity-free manifold, if for each point p in T*, the inverse image jus 1(p) is connected.

Lemma 2.3.

Let X be a multiplicity free balloon manifold, then H(p) # H(q) for any two distinct fixed points p and q in X. Proof: Let pC denote the moment map of the T-action on X with respect to the symplectic form H( _ (H, () for a generic choice of C E Ck. Then the image of µs : X -+ T* is a convex polytope whose vertices are given the images of the fixed points {p}. The weights of HS at the fixed points, up to an over all translation, are the same as {I&S(p)} which are all different. Since X is multiplicity-free, the inverse image of each vertex contains only one fixed point in X. Since C is generic, this implies that the H(p)'s are distinct at different fixed

point. 0

We shall assume throughout this paper that H(p) 0 H(q) for

all distinct fixed points p, q in X. Equivalently, if c(p) = c(q) for all c E HT (X), then p = q. This condition is also equivalent to the statement that the moment map with respect tow = HS and the T

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action is injective to the set of vertices of the moment polytope, when restricted to the fixed points XT. By the above lemma we know that toric manifolds and compact homogeneous manifolds all satisfy this condition.

When X is a toric n-fold, we have N = m + n T-invariant divisors in X. Let D. = cl (La), a = 1, .., N, be the equivariant first

Chern classes of the corresponding equivariant line bundles. These T divisors correspond 1-1 with the one-cones of the defining fan of X [40]. Moreover the fixed points correspond 1-1 with the n-cones. Labelling the n-cones by p E X' P, we have a balloon [40] pq in X if the n-cones p, q intersect in a codimension one subcone. Since X is smooth, hence the n-cones are regular, there are exactly n balloons pq for each fixed q. One can give a dual description of all these by using the Delzant polytope. Returning to the general case, suppose that X is a balloon manifold, and that we have equivariant classes {Da} in HT (X) with the following property. At every fixed point p, Da(p) is either zero or it is a weight on TPX. Let pq be a balloon in X. The induced T-action on pq is the standard rotation with fixed points p, q. By applying the localization formula on pq = P' and the integral (c, [pq]), we have c(q) = c(p) + (c, [pq])A

for all c E H* (X), where A is the weight on the tangent line TQ(pq).

Let A = D0(g). Specializing to c = Da, we get Da(q) = Da(p) + (Da, [pq])D0(q). This shows that (Da, [pq]) # 0. For otherwise we would have Da(q) = Da(p) 4 0, and this would mean that Da(p) is a weight on Tp (po) for some edge po running in the direction of Da(q) = Da(p) from p to o. So we had three vertices lying joined in a line from q to p to o in the moment graph. This would mean that there is a pair of linearly dependent weights on the tangent space TPX, which can't happen in a balloon manifold. A similar argument shows that (Da, [pq]) = 1. Lemma 2.4. Let w = HS and p, q E XT, r >- 0 and A be a weight on TqX. If w(q) = w(p) + (w, r)A for generic (, then p, q are joined by a balloon, r = [pq], and A is the weight on the tangent line TQ(pq).

Proof: It suffices to prove that p, q are joined by a balloon. The last two conclusions then follow immediately. Then under the correspond-

ing moment map p, p, q are mapped to w(p), w(q) (up to an overall affine transformation), which are distinct because w(q) - w(p) = (w, r)A 0 0.

(2.1)

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Since A is a weight on TqX, there is an edge emanating from the point w(q) in the direction of A, ending at some other vertex w(o), where qo is a balloon in X. If w(p) # w(o), we would have three distinct vertices of the moment polytope lying on a single line. Thus w(p) = w(o), which implies that p = o.

Lemma 2.5. The zero class w = 0 is the only class in HT(X) with the property that fX w

e1

=0

for all generic S E C. Proof: Suppose

fX

weH(=0.

By localization, we have

PE

eT

X)eHS(P) = 0.

But since the vectors H(p) are all distinct, those exponential functions

in S are linearly independent over the field Q(T*), implying that

w(p)=0 forallp. Thus w=0. D 2.4. Sigma models

Let X be balloon manifold with a fixed T-equivariant embedding X -4 P(n), as discussed above. We write

Md(X) := Mo,o((l, d), P' x X). Since X is assumed to be convex, Md(X) is an orbifold. The standard S1 action on Pl together with the T action on X induce a G = Sl x T action on Md(X). Here is a description of some Sl fixed point components Fr, labelled by 0 r -< d, inside of Md(X). Let Fr be the fiber product

F, =Mo,i(r,X)xxMo,i(d-r,X) More precisely, consider the map

ev,. x evd_,.: Mo,i(r, X) x Mo,l(d - r, X) -- X x X

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given by evaluations at the marked points; and

A: X-.XxX the diagonal map. Then

Fr = (evr x evd_r)-'A(X). Note that Fd = Mo,1(d, X) by convention. The set Fr can be identified with an S' fixed point component of Md(X) as follows. Consider the case r ; 0, d first. Given a point (C1, fl, x1) x (C2, f2, x2) in Fr, we get a new curve C by gluing C1, C2 to P1 with x1, x2 glued to 0, oo E P1 respectively. The new curve C is mapped into P1 x X as follows. Map P1 C C identically onto P1, and collapse C1, C2 to 0, oo respectively; then map C1, C2 into X with f1, f2 respectively, and collapse the PI

to f (XI) = A X2). This defines a point (C, f) in Md(X). For r = 0, we glue (Cl, fl, x1) to P1 at x1 and 0. For r = d, we glue (C2i f2, x2) to P1 at x2 and oo. We will identify Fr as a subset of Md(X) as above, and let

ir:F,.4Md(X)

denotes the inclusion map. Clearly, we also have an evaluation map

er:Fr -}X which sends a pair in Fr to the value at the marked point. In the following, we will simply write er as e without causing any coonfusion. We call a compact manifold or orbifold Wd with G = S' xT action a linear sigma model of degree d for X, if the following conditions are satisfied:

1. The S' action on Wd has fixed point components given by X,., labelled by 0 -{ r -< d, and each Xr is T-equivariantly isomorphic

to X. 2. There is a G-equivariant birational map W from Md(X) to Wd, such that cp1F, = e, and cp 1(Xr) = Fr. 3. All equivariant cohomology classes in HG (Wd) are lifted from

HT(X), and the lift b E HG(Wd) of D E HT(X) restricts to D + (D, r)a on Xr. 4. The G-equivariant Euler class of the normal bundle of Xo in Wd has the form

ec(Xol Wd) = H H(Da - maa) a ma where the ma's are positive integers and the Da's are classes in HT (X), such that at a given T fixed point p in X, the nonzero D0(p)'s are multiples of distinct weights of TX.

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Here a birational map, in algebraic geometry language, is a regular morphism which is an isomorphism when restricted to a Zariski open set in Md(X). Note Wd need not be unique. We identify X, with X by assumption 1, and denote by

j,.:Xr 4Wd the inclusion map. We call a balloon manifold X admissible if it has a linear sigma model Wd for each d, and that Hc(p) H, (q) for any two distinct fixed points p, q in X. The main result in this paper is to show that the mirror principle holds for any admissible balloon manifold.

Remark 2.6. Condition 4 is actually assuming more than what we need. This condition can be replaced by the following weaker, but more technical condition. For each fixed point p and for any d, as a function of a, eG(Xo/Wd)lp has possible zero only at either 0 or a multiple of a weight A on TpX. In addition if [pq] is a balloon and d = 8[pq], then A/S is at worse a simple zero. For example, the following form would

meet this criterion: eG(XO/Wd) =

Haa Ima(Da - maa) 11b r!nb (D6 - nba)

where the ma, nb are nonzero scalars.

Example 1: Projective space Pn with Wd = p(n+1)d+n is admissible. The existence of V was proved in [37], which is the so-called Li-Tian map. The lifted hyperplane class s; has the required property

that

jrx = H + (H, r)a = H + ra. The equivariant Euler class 7-d

eG(P0/Wd) = 11 11 (H - Ai - ma) I7nn-

i=O m=1

where Ai's denote the weights of the torus T action on Pn. Clearly the equivariant classes {H - Ai } has the required property. Example 21 More generally for P(n), we can take Nd(P(n)) to be Wd. In fact the S' fixed point components on Nk,i are exactly k + 1 copies P;., r = 0,.., k, of PI. Each P. consists of 1 + 1 tuples of monomials, each being a scalar multiple of wowi-''. Similarly the Si fixed point components on Nd(P(n)) are copies P(n)r, 0 r -< d, of

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P(n). All equivariant cohomology classes in HG(Nd(P(n)) are lifted from Hz2,(P") (cf [37]). Let ni be the lift of the hyperplane class H of the ith factor P"= . Then

j,M=Hi +(Hifr)a where j, denotes the inclusion of P(n), in Nd(P(n)). By using the formula in [37], it is easy to show the equivariant Euler class eG(P(n)o/Nd(P(n))), which is a product of eG's in last example, has the required property.

Example 3: Let NA;,! be the space of 1+1 tuples [ fo, .., fl] of degree

k polynomials f;(wo,wl), modulo scalar. Thus Nk,i p(l+i)k+t. It is called the linear sigma model for P'. (See (37].) Let Nd(P(n))

X ... X Nd,,, "m.

Recall that we have a collapsing map cp : Mk(PI) --> Nk,l, which is G := S1 x T equivariant. By taking composite with the projection from Md(P(n)) to each Md, (P"i ), we obtain a G-equivariant map Md(P(n)) .4 Nd(P(n)) which we also denote by W. Note that Md(X) can be viewed as a cycle

in Md(P(n)). We denote the image cycle cc'(Md(X)) in Nd(P(n)) cp by Nd(X). If Nd(X) is a manifold or an orbifold, then Properties 1-3 are automatically satisfied, if furthermore eG(Xo/Wd) has property 4, then we can simply take Wd = Nd(X) as the linear sigma model. Example .4: Convex toric varieties. In this case Wd is a toric nfold, as introduced by Witten [45]and used first by Morrison Pleaser [39]to study quantum cohomology. Recall that a toric manifold X

can be realized as the GIT quotient CN//Tc where TC is a mdimensional complex torus acting on CN. Here m = rank H2(X, Z), N = n + m. Let [zi, , zN] denote the coordinates on CN. Then each zj can be viewed as a section of a line bundle L3 on X [14] [39]. Modulo the induced action by TC from CN, a map from P' into X is uniquely represented by an N-tuple of polynomials [fi(wo, wi), ... , fiv(wo, wi)]

where ff is a section of the line bundle O(1,,) over P' with 1 = (ci (Lj), d). Let CN (d) be the vector space of N-tuple of polynomials

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of degree (d1i - - -, dN) as above. Then as described in [39], Wd is the

GIT quotient by the induced action of To on it:

Wd = CN(d)//Tc Let Md(X) denote the set of points (f, C) in Md(X) such that C = P'. We call MS(X) the smooth part of Md(X). We can define a map Wa from Md (X) to Wd in the following way: each (f, C) gives

a map from P1 to X, and modulo the induced TT action, uniquely determines N-tuple of polynomials as above, therefore gives a point in Wd, which we define to be the image of (f, C) under V,,. This is clearly a canonical identification. It is not difficult to see that the S1-fixed components in Wd can be described as GIT quotient, Xr ,., {[a1w0(Do,r)w(Do,d-r)' ..., aNW(DN,r)W(DN,d-r)fla E CN}//TC . The equivariant Euler class of its normal bundle in Wd is N

eG(Xr/Wd) = 11

(D.,d)

n

(Da + (Da, r)a - ka).

a=1 k=0,k&(D.,r)

Here Da = c1(La) is the equivariant first Chern class of the line bundle L. corresponding to the ath component in the coordinates of X. The lift Da of Da to Wd clearly has the property

j,Da = D. + (Da, r)a. As pointed out in [39], the cohomology of Wd are generated by the Da. Thus Wd has properties 1, 3, and 4. In the next subsection, we establish that the co extends to a regular G-equivariant map

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It is known that Pa is equivalent to Pn/Za where Za is a finite group. The space Md(Pa) for Pa is equal to Md(Pn)/Za. On the other hand we can also take Nd(Pn)/Za as the linear sigma model Wd. In this case Wd is an orbifold, a weighted projective space. Since the action of Za commutes with the action of torus T, we see that the induced collapsing map tp :

Md(Pa) -+ Nd(Pa) = Wd

is clearly a regular map. The corresponding equivariant Euler class has the expression: 77n daj

ec(F'a/Wd) = 11 H (ajH - aj - ma) j=0 m=1

with H the T-equivariant hyperplane class and \j's the weights of the T-action. Examples of singular toric varieties will be discussed again in our subsequent paper, in which resolution of singularities will be used to

reduce to the smooth case. The above example was motivated by a question of Mazur, who suggested that the situation of counting rational curves in orbifolds is similar to certain Diophantine problem in number theory. Example 6: For a general projective manifold X embedded in

P(n), assume it is defined by a system of polynomial equations P(z1, , zn) = 0 where zj = (4, , z,-7,,) denotes the coordinate of Pni. Assume the variety defined by the induced equation p(f1, ... , fn) = 0 in Nd(P(n)) where fj = (fi (wo, wi), ... ) fn; (wo, w1)) is the tuple of polynomials, the coordinates for the linear sigma model Nd(Pni ), is an orbifold. Then we can take it to be our linear sigma model Wd. Note that Nd(X) in Example 3 is embedded inside this Wd. Very likely they are the same. Though we don't know whether this variety is an orbifold or not, it is clear that the fixed point compoenents in the above variety are given by Xr's. In fact, we only need to assume that the localization formula holds on it. This is the case if the fixed point components embedded into Wd as local complete intersection subvarieties. We conjecture that this is the case for any convex projective manifold. Later, we will state a general conjectural Mirror Formula in terms of this Wd.

2.5. Regularity of the collapsing map

For a toric manifold X, the following lemma show that Wd described in Example 4 is a linear sigma model of X.

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Lemma 2.7. For tonic manifold X, there is a regular extension cp : Md(X) -+ Wd of the map W,, in Example 4 above.

Proof: We simply follow the argument in [37], together with the construction in [14]. We will define a morphism W : Md(X) -3 Wd. Let S be the category of all schemes of finite type (over C) and let

.F: S -+ (Set)

be the the contra-variant functor that send any S E S to the set of families of stable morphisms

F:X --+ P'xXxS over S, where X are families of connected arithmetic genus 0 curves, modulo the obvious equivalence relation. Note that F is represented by the moduli stack Md(X). Hence to define the rnorphism gyp, it suffices to define a transformation

%F :F-4Mor(-,Wd). We now define such a transformation. Let S E S and let t; E F(S)

be represented by (X, F). We let pi be the composite of F with the i-th projection of P1 x X x S and let pij be the composite of F with the projection from P1 x X x S to the product of its i-th and j-th components. We consider the sheaf p2Ox (Lj) on X and its direct image sheaf L,i,f = p13*p2*Ox (L3 )

Here the Lj are the line bundles on X, as defined in Example 4. As in [37], one can show that £j,4 is flat in a standard way. For the same reasoning, the sheaves G,i,4 satisfy the following base

change property: let p : T -+ S be any base change and let p* E F(T) be the pull back of . Then there is a canonical isomorphism of sheaves of OT-modules z,i,p`(E)

= (1P1 x p)*G.i,c

Since £M,f is flat over S, we can define the determinant line bundle of 4j,C, denoted by det(G4,£) which is an invertible sheaf over P' x S.

Using the Riemann-Roch theorem, one finds that its degree along

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fibers over S is lj = (cl(Lj), d). Furthermore, because Cj,t has rank one, there is a canonical homomorphism Gj,g --+ det(,Cj,g),

(2.3)

so that its kernel is the torsion subsheaf of Gj C.

Let zj be the j-th homogeneous coordinate of X (Example 4). Then zj is a section in H°(X, Lj). Its pull-back is a section of Gj,g, which induces a section aj,t E H°(S,1rs*det(,Cj,t)) based on (2.3). Then after fixing an isomorphism det(Gj,g) ^_' 7r;M ® xp, Opi(lj)

(2.4)

for some invertible sheaf M of Os-modules, where lj = (ci(Lj),d). We then obtain a section in 1rS* (7rP' Op' (1j)) ®Os M - Hp' (Op' (lj)) ®C M.

So o j,g is determined up to certain constant Aj coming from M. Now apply the above argument to each L j, j = 1, , N, we get N sections [vl,g, , 6N,g]. Let wo, wl be the homogeneous coordinate of P', we will write 6j,g = f j (wo, wi) as a homogeneous polynomial of degree lj. In this way we get a point in CN(d). The constants Aj from M in choosing o j,g must satisfy the relation ri, 1. Here the nj are vectors in an integral lattice, which generate the 1-cones in the defining fan of X, and m is any element in the dual lattice. (See [14].)

For such Aj's we can then find an element g in TV such that , fN] by g, therefore they , ANfN] is transformed to [fl, represent the same point in Wd. In this way, after taking GIT quo[A1!1,

tient by TC , the induced action on [fi(wo, w1), , fN(wo, wi)] from the action on CN, for each map (f, C) E Md(X), we have obtained canonically a point in Wd, therefore a morphism

T(S): S -+ Wd that is independent of the isomorphisms (2.4). It follows from the base change property (2.2) that the collection i'(S) defines a transformation thus defines the morphism p as desired.

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The fact that V : Md(X) -* Wd is S' x T N -equivariant follows immediately from the fact that cp is induced by the transformation if of functors. This completes the proof. For another proof of the above lemma we can proceed as follows. We use the notations as in the above Example 3. We show that the regularity of the collapsing map for P(n) induces the regularity of the collapsing map for X. For this we only need to prove that Nd (X ), the image cp(Md(X)) in Nd(P(n)) of the collapsing map for P(n), lies in Wd-

First, we show that Wd lies in Nd(P(n)). Note that both Wd and Nd(P(n)) are toric manifolds, and a Zariski open subset Wd in Wd is embedded G-equivariantly in Nd(P(n)). Also the G-fixed points in Wd are all in X therefore in P(n), and Nd(P(n)). Any point in Wd is in the closure of a generic GC orbit in Wd passing through two G fixed points in Xr's. By the equivariance, this GC orbit is also in Nd(P(n)), therefore the closure of this orbit lies in Nd(P(n)). Second, we show that Nd(X) lies in Wd. For this we note that coo extends to Md(X), since it is actually the restriction of the corresponding map on Md(P(n)). Now by taking closure of the inclusion w(Md(X)) C Wd, which is induced from the canonical identification, we get

'P(Md(X)) = W(Md(X)) 9 Wd = Wd, since Wd is itself closed.

3. The Gluing Identity Returning to the general case, we let X be an admissible balloon manifold from now on. In this section, we apply the functorial localization formula to the linear sigma model. The argument used here is modelled on the one used in [37], except that the T-action is not used here. Thus all the results in this section hold for manifolds without T action. We will have more to say about the mirror principle without T action later. Recall that we have the commutative diagram:

F,. - Md(X)

el Xr

,l.cP

-14

Wd.

We also have the natural forgetting map p : Mo,1(d, X) -+ Mo,o(d, X ),

and the projection map it : Md(X) -} M0,o(d,X). Note that we have

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a commutative diagram

Md(X) 7r 4.

\ i0

Mo,o(d, X) - Mo,1(d, X). Let V : Md(X) -+ Wd, e : Fr --+ Xr play the respective roles of f : X -> Y, g : F -+ E in the functorial localization formula. Then it follows that

Lemma 3.1.

Given any G-equivariant cohomology class W on Md(X), we have the following equality on Xr for 0 - r -< d: r W! W)

eG(X,/Wd) = e,

$r (w)

eG(FrlMd(X))/

Actually this lemma may be viewed as an equivariant version of the so-called excess intersection formula of [18], Theorem 6.3.

Let Lr denote the line bundle on Mo,1(r,X) whose fiber at (f, C; x) is the tangent line at the marked point x E C. Let 7r1 denote the projection from P1 x X to P1. The normal bundle of F,. in Md(X) can be computed just as in [37]. For r 0- 0, d, we have N(Fr/Md(X)) = H°(Co, (7r1o f)*TP1)+TT1Co®Lr+Tx2Co®Ld_r-A00.

Here we have used the notations as in [37]: a point (fl,C1,x1) in Mo,1(r, X) and a point (f2i C2, X2) in Mo,1(d - r, X) is glued to Co =

P1 at 0 and oo respectively to get the point (f, C) in Md(X) with C ^_- C1 U Co U C2. Since x1 and x2 are mapped to the same point in X

under the projection 7r2: P1 xX -+ X, so this point can be considered as a point in Fr by gluing together (fl, C1, x1) and (f2, C2, x2) at the marked points. Similarly, for r = 0, d, we have

N(Fo/Md(X )) = H°(Co, (r1 o f)*TPi) +TT1Co ®Ld - Ac0 and

N(FdlMd(X )) = H°(Co, (x1 o f)*TP1) +

®Ld - Aco.

In the above H°(Co, (7r1 o f)*TP1) corresponds to the deformation of Co; T.l Co ® L,. and Ty2Co ® Ld_r correspond respectively to the

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deformations of the nodal points xl and x2; Are, denotes the automorphism group to be quotiented out. The equivariant Euler classes of the normal bundles above are computed as in [37], to which we refer the readers for details. For r # 0, d, the equivariant Euler classes are:

eG(Fr/Md(X)) _ -a(-a +cl(Ld-r))' a(a+cl(L,.)) where the two factors on the right hand side are pulibacked to F,. from MO,l (d - r, X), Mo,l (r, X) respectively. For r = 0, d, we have

ea(Fo/Md(X)) = -a(-a+cl(Ld)), eG(Fd/Md(X)) =a(ce+ci(Ld)) respectively. Combining this with the preceding lemma, we get the following equality on X = Xo:

i (w) ev! (a(a - rl(Ld))

eG(Xo/Wd)

l

Here we have dropped the subscript from evd. In particular, if ii is a class on M0,o(d, X), then for w =1r , we get i*(w) = i*(ir*il)) = This yields

Lemma 3.2.

Given any T-equivariant cohomology class Mo,o(d, X), we have the following equality on X : jo*w!(1r*O)

_

eG(Xo/Wd) -

Lemma 3.3. For 0

ev!

on

a(a - cl (Ld))

r -< d, we have the following equality on X :

eG(X,IWd) = eG(XOIW,.)eG(XOIWd-r)

In particular, we have eG(XdlWd) = eG(XoIWd)

Proof: Consider the commutative diagram F,. e 4.

X -°

Mo,I(r,X) x Mo,1(d-r,X) evr x evd-r

XxX

(3.1)

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where 0 is the diagonal map, and Do is the inclusion induced by 0. In particular, by definition we have (evr x evd-r)*0,(1) = (Ao)!(1). So we have

0*(evr x evd_r)!(w) = e,A*(w)

(3.2)

for any class w on Mo,l(r, X) x Mo,i(d - r, X). Now put w = a(a+

Lr

x a(a-cl (Ld_r)) Then (3.2) becomes

(evr)!eG(Fo/Mr(X)) (evd r)'eG(Fo/Md-r(X)) = e!eG(F*IMd(X)) Since cp : Md(X) --3 Wd is an isomorphism on a Zariski open set, we see that w!(1) = 1. In fact, by Prop. (5.3.3) [1],
on which cp is an isomorphism, we find
By taking 0 = 1 in the preceding lemma, we get 1

1

(evr)!eG(Fo/Mr(X)) (evd

eG(XOIWr)'

1

1

r)!eG(FOIMd-r(X))

eG(XOIWd-r)

By taking w = 1 in Lemma 3.1, we get e,

_

1

eG(F /Md(X))

1

eG(Xr/Wd)

Combining the last four equations yields our assertion. 0 Fix a T-equivariant multiplicative class bT. Fix a T-equivariant

bundle of the form V = V+ ® V-, where V: are respectively the convex/concave bundles. (cf. [37].) We call such a V a mixed bundle. We assume that SZ := bT (V +)

bT(V-)

is a well-defined invertible class on X. By convention, if V = V} is purely convex/concave, then a = bT(V±)±1. Recall that the bundle V -+ X induces the bundles

Vd->Mo,o(d,X), Ud-aMo,l(d,X), Ud-+Md(X) Moreover, they are related by Ud = p*Vd, Ud = lr*Vd, Throughout this section, we denote Q:

Qd :_ v!(7r*bT(Vd))

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If w is a class on Wd, we write Z*wV :_

* jrw

eG(Xr/Wd)

which is a class on X = X,..

Lemma 3.4. For0-< r-< d, n ir'4d = ZoQr i0Qd-r Proof: For simplicity, let's consider the case V = V+. The general case is entirely analogous. Recall that a point (f, C) in Fr C Md comes from gluing together a pair of stable maps (fl, C1, x1), (f2, C2, x2) with f1(xl) = f2 (x2) _ p E X. From this, we get an exact sequence over C:

0-+ f*V-> f1*VED f2V->VIP-_0. Passing to cohomology, we have

0 -+ H°(C, f*V) -4 H°(C1i fiV) ®H°(C2, f2V) -+ VIP -+ 0. Hence we obtain an exact sequence of bundles on Fr: 0-+Z,,*Ud-+U*ED Ud-r -+e*V -30.

Here Z,Ud is the restriction to Fr of the bundle Ud -* Md(X). And U: is the pullback of the bundle Ur -- Mo,1(d,X), and similarly for Ed-r. Taking the multiplicative characteristic class bT, we get the identity on Fr: e*bT(V)bT(ir*Ud) =

This is what we call the gluing identity. Now put w

bT(Ur)

bT(Ud-r)

e0(Fr/Mr(X)) x eG(Fo/Md-r(X))

From the commutative diagram (3.1), we have the identity:

0*(evr x evd_r)i(W) = el0o(w).

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On one hand is bT(Ur)

A*(evr x evd-r)!(w) _ (evr)!

eG (Fr /MM (X))

_ (evr)!

_____Vr)

eG(FrIMr(X))

(evd-r)! (evd_r)!

bT(Ud-r) eG (Fo /Md-r (X) )

p*bT (Vd-r)

eG(FolMd-r(X))

= ioQr i0 d-r)

the last equality being a consequence of Lemma 3.2. On the other hand, applying the gluing identity, we have bT(UU-r) ) =- e! ( a(abT(U,) + cl (Lr )) a(a - c1(Ld-r))

e! mo o*( W

)

= eie*bT(V)i*bT(Ud) eG(Fr/Md(X )) = bT (V) e!

i bT (Ud)

eG(Fr/Md(X ))

= bT(V)i*Qd.

the last equality being a consequence of Lemma 3.1. This proves our assertion.

Remark 3.5.

(a) If we take V to be the trivial line bundle, and bT to be the total Chern class, then the preceding lemma reduces to Lemma S.S.

(b) All the lemmas in this section, in fact, holds for a general projective manifold X without T-action, provided that we still have the S1-equivariant map cp : Md(X) -+ Wd, with properties 1.-2. stated in section 2. All G-equivariant classes above are then replaced by their S1-equivariant counterparts.

4. Euler Data Notations: We denote by ,c,, the G-equivariant class on Wd with the property that j,*, is = Hi + (Hi, r)a. By the localization theorem,

xi is determined by these restriction conditions, and is a class in

the localized equivariant cohomology of Wd. More generally a class 0 E HT(X) has a G-equivariant lift ¢ E HH(Wd) determined by j*q5 _ 0 + (4), r)a. We denote by (HT (X)) the ring generated by HT (X), and by Rd the ring generated by their lifts . We put R = Q(T*)[a],

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480

where Q(T*) is the rational function field on the Lie algebra of T. For convenience, we introduce the notations h ' S = hS :=

I ncSni

rw

iswv :_ r

eG(Xr/Wd)

where w is a class on Wd.

It is often necessary to work over a larger field than C for coefficients of cohomology groups. For example when we consider the case of the equivariant Chern polynomial cT, a formal variable x is introduced. In this case we replace everywhere the scalars C by C(x). This will be implicit in all of the discussion below. Recall the localization formula:

W=

jr* (w)

0-
Wd

We shall often apply the following version:

f

f w e" c Wd

Definition 4.1.

0-
ir*wv eH<+(H( r)«

fix

Fix an invertible class 1 E HT(X)-1. A list

P : Pd E HH(Wd)-1, d >- 0, is a fZ-Euler data if on X, SL ZTPd = i0Pr' ioja r

(called Euler data identity) for all r -< d, and the fWd Pd - w are polynomial in a for all w E Rd. By convention we set Po = Q. Example 0. In the last section we have proved, using the gluing identity, that the data Q : Qd = cp!(R*bT(Vd)) associated with a mixed bundle V and a multiplicative class bT satisfies the Euler data identity. This indicates that the gluing identity is really the geometric origin of Euler data. This is what motivates our definition of Euler data. Note that since Qd is the equivariant push-forward of a class in HG(Md(X)), the polynomial condition on Q is automatic. This condition will be needed when mirror transformation is discussed. Example 1. Let L be any equivariant line bundle with cl(L) > 0. Let L be the G-equivariant lift of cl(L). (ci(L),d)

Pd = fl (L - ka) k=0

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is an St-Euler data where St = ci(L). Example 2. Let L be any equivariant line bundle with cl(L) < 0. Let L be the G-equivariant lift of c1(L). -(cl (L),d)-i Pd =

11

(L + ka)

A;=1

is an l-Euler data where Il = c1(L)-1. Example 3. If P, P are 0, SZ'-Euler data respectively, then P P. P' is a flIl'-Euler data as shown in [37]. Example 4. Let L be as in Example 1, and x be a formal variable. Then (ci(L),d)

Pd=

II (x+L-ka) k=o

is an S2-Euler data where Il = CT(L) denotes the Chern polynomial.

In each of Examples 1-4 above, the Euler data identity follows immediately from the algebraic identity Il j,*,Pd = jo*P,. jo*Pd-,., and Lemma 3.3.

Strictly speaking, in the examples above, we must require that c1(L) be an invertible class. This requirement can be easily met by twisting L by a trivial line bundle on which T acts by a suitable weight.

In the end, we will only be interested in the nonequivariant limit of an Euler data. Thus the choice of twisting is of no consequence at the end. Alternatively, we can consider the Chern polynomial or the total Chern class (which is automatically invertible) instead of the first Chern class. 4.1. An algebraic property Let S denotes the set of sequences B : Bd E HG (X) -1, d ?- 0. By convention, we set Bo =!Q.

Definition 4.2.

Given any B E S, define the formal series

HG[B](t) :=

Bd

C0+ d>-O

J

(t) takes value in the ring HG(X)-1[[Kv]] Note that E R}. We (Notations: if R is a ring, hen IZ[[K"]] := {&6A e(H,, interchangeably.) use the notations ed-t =

Mirror Symmetry II

482

Let P be an Euler data, and let B be the list with Bd := ioP' . By the localization formula and the Euler data identity, we have

f

f f rid

Pd ex c

Wa

r-{d

irPd eHS (Ht,r)a

Jx

1-1 [e-Ht1 ai*PT,

[e-H,

r e`d

Here t = (a + r. Note that S = -C, a = -a, and all other variables are invariant under the "bar" operation. Now multiply both sides by ed'T and sum over d E K", we get the formula: ed'T d

JWa

Pd e"C =

Jx

0-1 HG[B] ((a + r) HG[B](T).

(4.1)

By definition, the coefficient of ed'T on the right hand side is a power series in C with coefficients which are polynomial in a, ie. the series lies in 1 [[eT, C]]

Conversely, given B E S such that

Jx

n-1 HG[B] (Ca +T) HG[B](r) E R[[er, C]],

there exists a unique Euler data P : Pd satisfying (4.1). Naunely, Pd is defined by the conditions

jrPd = fl-1eG(XrIWd) Br Bd-r. Thus an Euler data P gives rise to a list B E S in a canonical way. Abusing the terminology, we shall call such a B an Euler data.

5. Linking and Uniqueness Lemma 5.1. Let w E HZ.(X)-1(a). Suppose that (a) for each q E XT, wq(a) := w(a)jq is a Laurent polynomial in a with degaw(a) < -2; (b) the power series in C: fx (w(a)eHc + w(-a)eHc+(Hc,d) a) has coefficients which are polynomial in a.

T ).T J

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Then w=0. Proof: Suppose w 34 0, and we will get a contradiction. By assumption

(a), we can write (q/X) - aqa k +

bqa-k+1 +

.. .

eT

which is a finite sum, with aq independent of a and k > 2. By supposition, not all the aq are zero. By localization, we get

(w(a)e'S +

_

w(-a)eHC+(HC,d) a)

((aqa-k + bqa-k+l +...)eHC(q) q

+ (aq(-a)-k +bq(-a)-k+l + ...)eHC(q)+(H(,d)a By assumption (b), order by order in S, this expression is polynomial in a. Since k > 2, the polar term with a -k must vanish, and so ageHC(q)(1

+ (-1)k) = 0.

Since not all aq are zero and the functions eHC (q) are linearly indepen-

dent, it follows that k is odd. Now the coefficient of a-k+' becomes eHC (q) (2bq

- aq(H(, d)) = 0.

Again by linear independence of the exponential functions, it follows that aq = 0 = bq for all q, which is a contradiction. 0

Lemma 5.2. Suppose A, B are Euler data with A,. = B, for all r -< d. Suppose that the (Ad - Bd) 1q, q E XT, are Laurent polynomial in a. Suppose also that dega(Ad - Bd) < -2. Then Ad = Bd.

Proof It suffices to show that

w(a) := Ad - B has property (b) of the preceding lemma.

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484

Let Ad be the coefficient of ed'T in the series

Ix 0-'HG[A] ((a + r)HG[A] (r).

Likewise for the B. Since A, B are Euler data, A,13' are power series in ( with coefficients which are polynomial in a. Explicitly,

Ad '= EJ SZ-leH
r-d X and likewise for the Bd. Using that Ar = B,., r < d, and that AO = B0 = SZ, we see that A'd - Bd is a sum over r with only two surviving terms, corresponding to r = 0, d. That is,

Ad - Bd = f (w(a)eHt +

w(-a)eHS+(H,;,d)

Since both Ad, Bd have coefficients which are polynomial in a, this shows that the class w(a) has property (b) of the preceding leimna.

0

Definition 5.3. Two Euler data A, B are linked if for even/ balloon pq in X and every d = S[pq] >- 0, (Ad - Bd)Iq

is regular at a = a where A is the weight on the tangent line TT,(pq).

Suppose A, B both come from Euler data Q, P respectively, ie. Ad = ioQd and Bd = ioPd. Suppose also that

j0(Pd)Iq=j0(Qd)Iq

ata=A/S.

(5.1)

whenever d = S[pq] >- 0 as above. Recall that a = A/S is at worst a simple pole of 1/eG(Xo/Wd)Iq. It follows that (Ad - Bd)Iq is regular at this value. This shows that the conditions (5.1) guarantee that A, B are linked.

Theorem 5.4. Suppose A, B are linked Euler data satisfying the following properties: for d }- 0, (i) If q E X", the only possible poles of (Ad - Bd) I q are scalar mul-

tiples of a weight on TX.

(ii) deg.(Ad - Bd) < -2.

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Then A = B.

Proof: We will prove, by induction, the assertion that Ad = Bd for all d. If d = 0, there is nothing to prove. Suppose the assertion holds for all r -< d. Set wq(a) := (Ad - Bd)lq as before. We will show, under assumption (i), that the wq(a) are Laurent polynomial in a. It follows then, from the preceding lemma and assumption (ii), that Ad = Bd. Let A E T* - 0. We will show that each wq (a) is regular at a = A. Recall the power series in (: Ad, Bd, with coefficients polynomial in a as in the preceding proof. Thus for any integers k,1 > 0,

Resa=A ((a - A)k(a+ A)'(A'd -B')) = 0. Also recall that A'd - Bd' =

Jx

(w(a)e'< + w(-a)eH<+(H<M

_

1

qEXT

eT(a/X)

(wq(a) eH<(q)+wq(_a) eH<(q)+(H(,d)a

J1

From the preceding two equations, we get

0=

i CT( IIX) 9 Ex7

(eHS(q) Resa=A,(a-A)k(a+A)iwq(a)

+eH<(q)+(H<,d)aResa=a(a

(5.2)

- A)k(a + A)twg(-a))

.

If Res,,\(a - \)k(a + A)iwq(-a) = 0 for all q, then the preceding equation shows that Resa=a(a - A)k(a + A)'wq(a) = 0 for all q, because the vectors HS (q) are distinct. Similarly if Resa_a (a - A) k (a + A)'wq(a) = 0 for all q, then we have Resa=A,(a-A)k(a+A)lwq(-a) = 0

for all q. In either case, we conclude that each wq(a) is regular at a = A. So if a = A is a pole of a wq(a), then we necessarily have Resa=A(a - A)k(a + A)'wq(a) 0 0 and Resa=,\(a - A)k(a + A)'wp(-a) # 0 for some p, q and some 1, k, such that HC (q) = Hc(p) + (H(, d) A,

to ensure cancellation of the exponential functions in (5.2). Note that since d >- 0 and A ¢ 0, we have p # q. By our assumption (i), the pole a = A of wq(a) must be of the form A = :A 0 for some weight A' on TqX, and some scalar 8 ¢ 0. By Lemma 2.4, p, q must be joined by a balloon, d = 8[pq], and A' is the weight on the tangent line Tq(pq).

Mirror Symmetry II

486

Thus if d is not a multiple of [pq], then we have shown that wq(a) is regular away from a = 0. Now suppose that d = 6[pq], and consider the only possible pole

of wq(a) at a = 6 # 0, as above. By hypothesis, A, B are linked. But this means that wq(a) is regular at a = a 0 0. Remark 5.5. In our applications later, the situation is better then the conditions (i)-(ii) demand. We will have two Euler data A, B such that Ad, Bd separately, rather than just Ad - Bd, will satisfy both conditions (i)-(ii) at the outset. In this situation, to prove that A = B, it suffices to prove that they are linked.

6. Mirror Transformations Throughout this section, we fix an invertible class Sl on X, and will denote by A the set of f2-Euler data.

Definition 6.1. A map p : A -+ A is called a mirror transformation if it preserves linking. In other words, p(A) and A are linked for any A E A. We call p(A) a mirror transform of A. We now consider a construction of mirror transformations, as motivated by the classic example of [12]. Consider a transformation

p : S -3 S, B - B, of the type

Bd=Bd+Ead,rBr r{d

where the ad,,. E HH(X)-i are a given set of coefficients. This transformation is obviously invertible, and preserves Bo = Q. Lemma 6.2. Suppose that B, B are both Euler data. Let d = 6[pq] >0 for some balloon pq in X. Suppose that the coefficients in (6.1) are such that their restrictions ad,,-(q), r -< d, to the fixed point q are regular at a = A16 where A is the weight on Tq(pq). Then (td - Bd) I q is regular at a = A/6. Proof: From (6.1), it suffices to show that the functions Brlq, 0 -< r -f d, are regular at a = a/6. Suppose the contrary that some Br I q = 0 has a pole of order k + 1 there. Since B is a Euler data, we know that S2- iaHtB8e(H(,s)a Br_e

Bi B. := = E s-
X

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487

is a power series in [; with coefficients polynomial in a. By the localization formula,

Br _

eT(o/X) eHC(0)+(HC $)9)(o)-'B8(O)Br-s(o) sir OEXT

Now multiply both sides by (a - A/S)" and take residue at a = A/S. We get

8
eT(o/X) e

HC (o)+(H< ,s)A/6Res_ a_a/6(a - A/S)kSt(o)'1Ba(o)Br _s

(o) .

By assumption, the summand above with s = 0 o = q is nonzero. Observe that this term has an exponential factor etc W. Thus in order to cancel this term, any other term contributing to this cancellation must have an identical exponential factor. This means that H((q) = Hc(o) + (HS, s)A/S for some s with s -< r, and some o E XT. By Lemma 2.4, this implies that s = S[pq], contradicting that s -< r -< d. p

Definition 6.3. The transformation (6.1) is said to have the regularity property if for every balloon pq in X and d = 6[pq], the coefficients are such that their restrictions ad,r (q), r -< d, are regular at a = A/S where A is the weight on TT(pq).

Thus the preceding lemma says that transformation (6.1) having the regularity property preserves linking. Again, motivated by [12] and [28], we consider the following special types of transformations. Given a power series f E R[[K" ]] with no constant term, we have an invertible transformation p f : S -3

S, B H B, such that of/a HG[B](t) = HG[B](t). In fact, we have

Bd = Bd + >.fd-rBr r{d

This is clearly a transformation of type (6.1) having the regularity property. (In fact, all the coefficients fd-r are regular away from a = 0.)

where ef/a = E$ro fses-t, fs E

Mirror Symmetry II

488

Given power series g = (gl,.., g,,,,), gj E R[[K" ]] with no constant

term, we have an invertible transformation v9 : S -a S, B ti b, such that HG[B](t + g) = HG[B](t). In fact since

E Bd

HG[B] (t + g) =

er,

ID-0

9d,s E R and e-'1' _

if we write eds = E.>-o 98 E R[H/a], then

Bd = Bd +

gye,

ad,,-B, r-(d

where the ad,,. E HG(X)-' are quadratic expressions in the g,y. Thus we obtain another transformation S -> S of type (6.1), again having the regularity property.

Theorem 6.4. The transformations p,, v9 : B H B above each defines a mirror transformation. That is, if B is a Euler data then pf(B) and v9(B) are both Euler data linked to B. Proof: Let B be a given Euler data. We have seen that the preceding lemma guarantees that µf (B), v9(B) are linked to B. So it suffices to show that they are Euler data. First case: set f3 = v9(B), ie.

HG[B](t) = HG[B](t+g(et)).

(6.2)

(Here et means the variables (etl,.., et-).) Set t = (a + r, % = erj.

On the one hand, we have

f

X

0-1 HG[B](t) HG[B](,r) ==qd rm d,m

Bi,m(a)

(6.3)

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for some Bd,,1, E R. Now compare (*)

fx f 2-1 HG[B](t) HG[B](r) Bded red Sa x "`

_ f f-leHC

fx

x Q-1 HG[B](t+g(et))

fx 0

F

-leHt

x x

This shows that the series (**) can be obtained from (*) by the replacements ,r H r+g(q), Thus combining (6.2) and (6.3), we get

fx n-1 HG[B](t) HG[B](r)

=

(C + (g(gesa) d,m

(6.4)

- g(q))/a)m Bd,m(a)

Now write g = g.f. + g- with gf = ±gf. Obviously for any g(q) E R[[q]], g+(geCa) - g+ (q) E a R[[q,C]]. Since the involution w H w on R simply changes the sign of a, the fact that g_ is odd shows that g_ (q) E a R[[q]]. Likewise for g_ (geCa). This shows that (6.4) lies in R[[q, (]]. This completes our proof in this case. Second case: set b = p f(B), ie.

HG[B](t) = ef/a HG[B](t).

Again writing f E R[[et]] as f = f+ + f- with ft = ±f±, we get

fx

0-1 HG[B](t) HG[B](r) = e-f(et)/a ef(e)/a x

Jx

0-1 HG[B](t) HG[B](r)

= er-(f+(qe(°)-f+(q))la e(f-(geCO1)+f-(q))/a

x J D-1 HG[B](t) HG[B](r).

x

490

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The right hand side lies in R[[q, (]] as before.

Remark 6.5. All mirror transformations we will use later will be of the type pf, v9 as above. Moreover, all Euler data we will encounter will have property (i) of Theorem 5.4. The transformations p f, v9 clearly preserve this property.

Theorem 6.6. Suppose that A, B have property (i) of Theorem

5.4, and that A, B are linked. Suppose that A is an Euler data with deg"Ad < -2 for all d -< 0, and that there exists power series f E R[[K"]], g = (ga, ..,gm), gf E R[[K"]], all without constant term, such that efl"HG[B](t) = SZ - SZH

(t +g)

+ O(a-2)

(6.5)

when expanded in powers of a-1. Then

HG[A](t+g) = of"" HG[B](t). Proof: By Theorem 6.4, f,g define two mirror transformations lcf, v9, with

HG[B](t) = efl"HG[B](t) HG[A](t) = HG[A](t + g)

(6.6)

where ,& = w f(B), A = vg (A). Now both B, A have property (i) of Theorem 5.4. (See remark after Theorem 6.4.) Since deg"Ad < -2, HG[A](t) has the same asymtotic form as HG[B](t) in eqn. (6.5) mod O(c 2). It follows that

eH'tl" HG[A -A(t) =

O(a_2),

or equivalently deg"(Ad - Bd) < -2. Thus A, .i§ satisfy condition (ii) of Theorem 5.4. Since A is linked to B, it follows that A is linked to B. By Theorem 5.4, we conclude that A = B. Now our assertion follows from eqns. (6.6).

Remark 6.7. The preceding theorem says that one way to compute A (or Q) is by first finding an explicit Euler data B linked to A, and then relate A and B via mirror transformations.

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491

7. From stable map moduli to Euler data Fix an admissible balloon manifold with ci(X) > 0. Fix a Tequivariant multiplicative class bT. Its nonequivariant limit is denoted by b. Fix a T-equivariant bundle of the form V = V+ ® V-, where Vt are respectively the convex/concave bundles. As before, we write bT(V+) S2

bT(V_).

Let Vd be the bundle induced by V on the 0-pointed degree d stable map moduli of X. Throughout this section, we denote Q:

Qd

cQ!(lr*bT(Vd))

f

Kd

b(Vd)

Mo.o(d,x)

:= EKd A: Ad:=io*Qa Note that all these objects depend on the choice of bT and V, though the notations do not reflect this.

7.1. The Euler data Q

Theorem 7.1. (i) degaAd < -2. (ii) If for each d the class bT(Vd) has homogeneous degree the same as the degree of M0,0(d,X), then in the nonequivariant limit we have

fx e Jx (HG[A) (t) - e-H'tl a1i) =a 3(M-E t;

). i

Proof: Earlier we have proved that

Ad = iQd=ev! Ca(a -

c(L))

where L = Ld is the line bundle on Mo,1(d,X) whose fiber at a point (f, C; x) is the tangent line at x.

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492

Assertion (i) now follows immediately from this formula. The second equality in assertion (ii) follows from the fil:st equality. By the above formula again,

I :=

Jx

e-H't/aAd P*b(tid)

- JMo 1(d,x) = fM0,o

a(a - ci (L) ) b(Vd) P!

( d,x)

\ a(a - el(L)) /

Now b(Vd) has homogeneous degree the same as the dimension Mo,o(d, X). The second factor in the last integrand contributes a scalar factor given by integration over a generic fiber E (which is a which PI) of p. So we pick out the degree 1 term in is just -ea tt -I- C0L . Restricting to the generic: fiber E, say over (f, C) E Mo,o(d, X), the evaluation map ev is equal to f, which is a degree d map E ?' P1 -4 X. It follows that

L Moreover, since cl(L) restricted to E is just the first Chern class of the tangent bundle to E, it follows that

JE

cl (L) = 2.

So we have

I=

(_a3 t

+ 23 )Kd.

D

Theorem 7.2. More generally suppose bT is an equivariant inultiplicative class of the form

rkV=r where x is a formal variable, bs is a characteristic class of degree i. Suppose s := rk Vd - dim Mo,o(d, X) > 0 is independent of d >- 0.

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493

Then

r a-H t/°`Ad = a-3x-s(2 - d- t)Kd

$

s!

(dx) Ix -°Jx

lI( dl =0 s. TX a

x

(HG[A](t) - e-H . t,c

= a-3x 9 (M - E tit ). Proof: The proof is entirely analogous to (ii) above. 7.2. Linking theorem for A Now consider a mixed bundle V = V+®V- on X. Fixed a choice

of equivariant multiplicative class bT. We assume that V has the following property: there exists nontrivial T-equivariant line bundles Li , .., LN+; Li , .., LN_ on X with cl (Lt) > 0 and cl (L1) < 0, such

that for any balloon pq = P' in X we have V}lpq

- ®j 1L:%q'

Note that N* = rk V±. We also require that

bT(V+)/bT(V ) =11bT(L= )l flbT(La ). j

i

In this case we call the list (Li , .., L,+v+; LT, .., LN_) the splitting type

of V. Note that V is not assumed to split over X. Given such a bundle V and a choice of multiplicative class bT, we obtain an Euler data Q : Qd = 0(1r*bT(Vd)) (or A) as before.

Theorem 7.3. Let bT = eT be the equivariant Euler class. Let pq

be a balloon, d = a[pq] .- 0, and A be the weight on the tangent line Tq(pq). Then at a = A/b, we have (ci(L; ),d)

.A0 (Qd) I q=

11

H

i

k=0

(cl (Li) I q- kA/5)

-(ci(Li ),d)-1

X 11

j

11 k=1

(cl (L;) I q+ kA/b)

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494

In particular Q is linked to

-(c,(L ),d)-1

(ci(L±),d)

fi (L - ka) x ll

P : Pd = fl i

j

k=0

H

(L + ka).

k=1

Proof: We first consider one positive line bundle L. As in [37], we consider a point (f, C) E Md(X) where f is 6-cover from C = P1 to the balloon pq = P1. For a = A/8, this map can be written as

f: C--4P1xpgcPlxX where the second map is the inclusion. In terms of coordinates we can write the first map as

f.

[wo, w1] -+ [w1, wo] x [w0, w1].

Note that the T-action induces standard rotation on pq P1 with the weights A,, A2 and A = A, - a2. It is now easy to see that this point (f, C) is fixed by the subgroup of G with a = A/8. On the other hand as argued in [37], (7r2o f, C) is then a smooth fixed point in Mo,o(d, X)

under the T-action. The restriction joQdlp with a = A/6 is equal to the value of eT (Ud) at (f, C). This, in turn, is equal to the restriction of eT(Vd) at (1r2 o f, C) in Mo,o(d, X).

Assume the restriction of L to pq = P1 is 0(l) with l =

(c1(L), [pq]). We compute that the equivariant Euler class restricted to this point (1r2 o f, C). As in [37], we get to

ll(1A1-mi).

eT(Ud)=

M=O

Also note that c1(L)(p) = IA1 and d = 8[pq], this implies that Qd = cp!(ir*eT(Vd)) is linked to

(ci (L),d)

Pd = [I

(c1(L)

- ma).

m=0

Similarly for a concave line bundle L, if its restriction to the balloon pq is O(-l) with -l = (c1(L), [pq]), then to-1

eT(Ud) _ II (-lA1 +ma) m_-

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which implies the formula that in this case Qd is linked to -(cx(L),d)-1

Pd =

11

(C1(L) + ma).

m=1

The general case is just a product of these cases. o Similarly we can prove the following formula for the Chern poly-

nomial.

Theorem 7.4. Let br = cT be the equivariant Chern

polynomial. Let pq be a balloon, d = S[pq] >- 0, and A be the weight on the tangent

line Tq(pq). Then at a = A/S, we have (ci (L; ),d)

?o (Qd)Iq =11

11

x

fl

(x + cl(Li )Iq - ka/S)

k=0

i

-(cl(L,y ),d)-1

j

fl

(x +cl(Lf )Iq+kA/S)

.

k=1

In particular Q is linked to (c, (Lt ),d)

P : Pd = fl i

fl

k=0

-(c1(L f ),d)-1

(x+Li - ka) x 11 j

11

(x+L9 + ka).

k=1

8. Applications 8.1. Toric manifolds We call a toric manifold X reflexive if its defining fan satisfies the

following combinatorial condition: the convex hull of the primitive generators of the 1-cones in the fan is a reflexive polytope. It has been shown [4][41] that a pair of polar reflexive polytopes gives rise to a pair of mirror (in the sense of Hodge numbers) Calabi-Yau varieties, by taking anti-canonical hypersurfaces in the corresponding reflexive

toric manifolds. It has been conjectured that [5] a similar statement holds for complete intersections in toric manifolds. It is known that [29] a toric manifold X is reflexive if cl(X) > 0. We shall assume

Mirror Symmetry II

496

that X is reflexive. Recall that for a (convex) tonic manifold X, we have (D ,d)

ea(XO/Wd) _ 1111 (Da - ka) a

k=1

where each D. is the T-equivariant first Chern classes of the line bundles corresponding to a T-invariant hypersurfaces in X. 8.2. Chern polynomials for mixed bundles

To proceed, we make two further choices: let bT be the Tequivariant Chern polynomial CT, and let V = V+ ® V- be a mixed bundle with splitting type (L+,,.., L+; L-,.., LN_ ). Here the L's are T-equivariant line bundles on X with cl(Li) > 0,

cl(L.) < 0,

SE:= cT(V+)/eT(V ) =fl(x+c1(Lt ))/fl(x+c1(La )) i

Ecl (L;) - E cl (L,-) = c1(X ). a

i

From this, we get an f -Euler data Q : Qd = co,(7r*CT(Vd)) as before. By the Linking Theorem, Q is linked to the Euler data (c,(Li ),d)

P:

Pd=11 i

II

(x+Ls - ka) x fl

k=0

-(cl(L.i ),d)-1

j

11

(x+L1 +ka).

k=1

As before, we set

B: Bd=i*P7,

A: Ad=i*Qa

We consider three separate cases. We will be using the elementary formula

H( k=1

- k) = (-1)MM!(1-

E)

(8.1)

k=1

where " _" here means equal mod 0(a-2), to compute the leading

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497

terms of (c (L±),d)

Bd = fl i

11

(x + c1(Ls) - ka)

k=O

x 11

JJ

j

(x + cl (Li) + ka)

k=1

a

(Da - ka)

k=1

eel (Lt I d)

Sl cT(V ) a-N fl i

-(ci(L. ),d)-1

xHH j

k=1

fl k=1

(x+c11(Li+)

-k)

(x+c1(Li) +k) a

1

x

77(D-,d) Du

[Ia l ik=1

(a - k )

First suppose that rk V- = N- > 2. In this case we have

degc,Bd = -rk V- < -2 and hence

HG[B](t) _- SZ - S1

at

By Theorem 6.6, we conclude that A = B and Q = P. This completes the computation of A and Q in this case.

Now consider the case rk V- = N- = 1, hence V- is a line bundle. In this case we have Bd

a-1 Il (x+cl(V-)) (-1)(ci(V-),d)

(-(cl(V ),d)

1)!rji(c,(a d,)d)!

x

a-1 1 (E HiOd,i + Od) i

where the ¢d,i E Q, did E Q[7-*, x], are determined uniquely by the writing c1 (V-) E HT (X) in the last equality, according to the decomposition HT (X) = Sm 1QHi ® T*. Hence we get e-H-t/a Bd =_ 11 (a-1H . Od + a-laid).

Mirror Symmetry II

498

Summing over d E Kv, we get

HG[B](t) = 11(1- a-1H (t + F) + a-'G)

F:= -E cbd G := >2'bd ed't. From this we get e-G/a HG[B] (t) = SZ - 0

H H. (t + F)

a

By Theorem 6.6, we conclude that

e-G/aHG[B] (t) = HG[A] (t + F).

(8.3)

This completes our computation of A and Q in this case. Recall that

dim Mo,o(d, X) = (cl(X), d) + n - 3

(Lt), d) - F (cl(LT), d) + N+ - N_

rk Vd = s

j

_ (cl(X), d) + rk V+ - rk V-. To applied Theorem 7.2, we assume that rk V+ - rk V- > n - 3, and we determine all Kd immediately. Explicitly (in the nonequivariant limit T* -+ 0):

( a!

8

dx) I=o Jx (e-G1aHG[B](t) - e-H't/aSl) = a-3x-$(2-11( -

(8.4)

t)

tt aat

).

where s := rk V+-rk V- -(n-3), !:= t+F(t). Note that this same formula applies also when rk V- > 2, whereby we put G, F = 0. We now consider the case when V is purely convex: N- = 0.

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499

8.3. Convex bundle

We will denote the Lt simply by Li. Using formulas (8.1) and (8.2), we get

Bd - nfi(cl(Li),d)! fla(Da, d)!

(D.,d)

a

(cj(L.),d)

1

x (1+a-1EDa k=1

-a'1E(x+c1(Li)) E i

)

k=1

OAd + a-1 > HiOd,i + a-1Vid i

Here the Ad, cbd,i E Q, Od E Q[T*, x] are determined uniquely by the writing each Da, c1(Li) E H .(X) in the last equality, according to the decomposition HT (X) = e r 1QHi®T*. Since we get

e-H-t1a Bd = 1l(Ad - a-1H (Adt - Od) + a-10d) Summing over d E K", we get

HG[B](t) = c (Fo - a"1H (Fot + F) + a-1G) Fo := 1 + E Ad

ed-t

F := - E Od G := E'Pd ed-t. Put f := a log FO - G. . Then we get efla HG[B](t) = SZ - Il

a

By Theorem 6.6, we conclude that

of /a HG[B] (t) = HG[A] (t + i-).

(8.5)

This completes our computation of A and Q in this case. Again to apply Theorem 7.2, we assume that rk V > n - 3, and determine all Kd immediately. Explicitly:

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500

s

!

Ix-o

C

l -

f(f

HG[B](t) -

(8.6)

a-3 x -8 (21 (i) - E t , ___ i

where s: =

r kV-(n -3) , t

:

=t+Fu(t

dti

)

)

Let us now specialize to the case rk V = n - 3 (ie. s = 0), and V = ®iLi. We can then set x = 0, so that bT = cT becomes the equivariant Euler class eT, and the Kd is just the intersection numbers for e(Vd). Then the formula (8.6) yields the general formula derived in [28] and in [27], on the basis of the conjectural mirror correspondence. Note that D'0 = C` l li (cl (Li), d)! ed.t

11. (D., d)!

is an example of a hypergeometric function [20]. It has been proved in [29] that Fo is the unique holomorphic period of Calabi-Yau hypersurfaces near the so-called large radius limit. For the purpose of comparison, we should mention that the definition of ' here differs from the prepotential in [28][27] by a degree three polynomial in t', and the definition of the hypergeometric series HG[B](t) here differs from that denoted by wo(x, p) in [28][27] by an irrelevant overall constant factor. Precursors to the above general formula have been many examples [27][6][13][8]. We now specialize to a few numerical examples which have been frequently studied by both physicists and mathematicians alike. 8.4. A complete intersection in P1 X P2 X P2

The complete intersection of degrees (1, 3, 0), (1,0,3) in this 5dimensional toric manifold X has been studied in [27] using mirror symmetry, and in [30] computing some of the intersection numbers Kd for the Euler class b = e in terms of modular forms. From our point of view, that complete intersection correspond to the following choice of convex bundle:

V = 01(1) 0 O2(3) a 01 (1) 0 03(3)

where Oi(l) denotes the pullback of 0(l) from the ith factor. The Kahler cone of X is abviously generated by the hyperplanes

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501

H1i H2, H3 from the three factors of X, and hence Kv can be identified with the set of d = (d1, d2, d3) E Zoo. We consider intersection numbers Kd for the Euler class b = e as before. Thus we set fZ = e(V) = (H1 + 3H2)(H1 + 3H3). The Euler data P we need in eqn. (8.6) is given by 4,

di+3d3

Pd = 11 (hl + 3rti2 - ka) x 11 (!v"1 + 3rc3 - ka) k=0

k=0 di+3d2

d1+3d2

jo (Pd) = 11 (H1 + 3H2 - ka) x 11 (Hl + 3H3 - ka). k=o

k=O

The linear sigma model is Wd = Nd(P(n)) = Nd1,1 XNd2,2 xNdy,2

The equivariant Euler class, after taking nonequivariant limit with respect to the T action, is given by d,

eG(Xo/Wd) = H 711=1

d2 (H1 -

ds

ma)2 11 (H2 - ma)3 11 (H3 - ma)3. m=1

7n=1

Now we can easily write down the hypregeometric series and all the Kd can be computed by our formula (8.6) at once using the obvious intersection form on X, given by the relations:

L Once we have the hypergeometric series, the corresponding Picard-Fachs equation can be easily written down as given in [27].

8.5. V = 01 (-2) & O2(-2) on P1 x P1 Here we denote by Oi(l) the pullback of 0(1) from the ith factor

ofX =P1 xP1. Our bundle V has rk V+ - rk V- = n - 3 = -1. Thus we can apply our formula (8.4) with x = 0. We put 1 = H>-3H2-'

The Euler data P in eqn. (8.4) that compute the Kd is now given by: 2d1-1

2d2-1

Pd = H (-2rc1 + ka) x II (-2r-2 + ka). k=1

k=1

The corresponding equivariant Euler class, after taking the nonequivariant limit with respext to the T-action is

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502

d2

d1

eG(XOIWd) = 11 (H1 - ma)2 fJ (H2 - ma) 2. m=1

M=1

Again one can immediately write down the hypergeometric series as well as the corresponding Picard-Fuchs equation by using our mirror principle.

9. Generalizations and Concluding Remarks 9.1. A weighted projective space

Consider the following example: the concave bundle V = O(-6) over P3,2,1, fl = i This example will be studied in our subsequent paper by using resolution of singularities. This is an example of "local mirror symmetry" studied in physics [33]. The mirror formula there can derived as a special case of our general result. In fact, the Euler data which computes the Kd in this case is determined by 6d-1

joPd = [J (-6H + ma). M=1

The corresponding equivariant Euler class, after taking nonequivariant limit with respect to the T action, is: d

2d

3d

eG(Xo/Wd) = ft (H - ma) 11 (2H - ma) fi (3H - ma). m=1

m=1

na=1

The corresponding hypergeometric series and Picard-Fuchs equation can be immediately written down. It turns out that the hypergeometric series gives the periods of a meromorphic 1-form for a family of elliptic curves [33].

9.2. General projective balloon manifolds

Let X be a projective manifold embedded in P(n), with a system of homogeneous polynomial defining equations P(z1, , zn) =

0, where zz = (zi,

,

z;,f ).

For each P, by taking the coeffi-

cients of each monomial wo wi in P(f 1, , fn) = 0, where f p _ [f1(wo, w1), , f, (wo, w1)] for j = 1,. , k is the tuple of polynomials that define the coordinates of Nd(P(n)), we get several equations

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of the same degree as P. These equations together define a projective variety, which we denote by Nd (X ), in Nd (P (n) ). As discussed earlier, we see that the Sl fixed point components in Nd(X) are given by the X,'s which are copies of X. We do not know whether the localization formula holds on Nd(X). The localization formula holds if the fixed point components embedded into Wd as local complete intersection subvarieties. It is likely that this is the case for any convex projective manifold. If this is true, then we can take Nd(X) to be the linear sigma model Wd for X. Then our mirror principle may apply readily to compute multiplicative characteristic numbers on M0,0(d, X) in terms of the hypergeometric series.

9.3. A General Mirror Formula

Many of our results so far are proved for projective manifolds without T-action. Here we first discuss a formula for computing the numbers Kd =

f

b(Vd)

M0,0 (d,X)

for a general convex projective n-fold X without T-action. For sim-

plicity, let's focus on the case when the multiplicative class b is the Chern polynomial c, and V is a direct sum of line bundles on X. There is a similar formulation in the general case. We fix a projective embedding X -+ P(n), as before. Note that the map : Md(X) -+ Nd(P(n)) is now only Sl-equivariant. Recall that the subvariety Wd := w(Md(X)) c Nd(P(n)) contains as Sl fixed point components copies of X: X 0 r -< d. We assume that the localization formula holds on it. We denote by est (Xo/Wd) the equivariant Euler class of the normal bundle of X0 in Wd. Let

V = V+ ®V-, V+:=W, V- :_ ®L; satisfying cl(V+) -cl(V-) = cl(X) and rk(V+) -rk(V-) - (n-3) >-

0, where the Lt are respectively convex/concave line bundles on X.

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504

Let

SZ = Bo := c(V+)/c(V ) = [J(x + c1(L, ))/ H(x + cr(L-))

j

i (c, (Lt),d)

Bd := e st (Xo/Wd)

x fl

xfli

(x + c1(LE) - ka)

-(ci(L.i ),d)-1

II

j

HG[B](t)

11

k=O

(x + ci (L,-) + kcx).

k=1

Bded't

c (t)

Conjecture 9.1. There exist unique power series G(t), F(t) such that the following formula holds: 8

s!

Ix=o

TX Cd/

JX

`

a G/aHG[B](t) -

where s := rk V+-rk V- - (n-3), i:= t+F(t). Moreover G, F are determined by the condition that the integrand on the left hand side is of order 0(a-2).

9.4. Formulas without T-action

One of our key ingredient, the functorial localization formula plays an important role in relating the data on Md(X) and those on Wd. It turns out that similar formula holds in K-theory. It holds even when X has no group action. This indicates that our method may be extended to compute K-theory multiplicative type characteristic classes on Md(X) (and ultimately on Mo,o(d, X)), in terms of certain q-hypergeometric series, even for projective manifolds without group action. We now write down the relevant localization formulas for con-

vex X without torus action, both in cohomology and in K-theory. The notations and proofs are basically the same as before. Given a manifold X, let's assume that there is a linear sigma model Wd.

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Lemma 9.2.

For any equivariant cohomology class w on Md(X), the following equality holds on Xr for any 0 - r -< d: j, p! (W)

= e![esl

esl (Xr/Wd) -

irw

(Fr/Md(X

Here esL denotes the Sl-equivariant Euler class. As in the cases we have studied earlier, the left hand side of the above formula indicates that when V = L is a line bundle, we should compare the Euler data Qd = cppr*e(Vd), to the Euler data given by (ct(L),d)

Pd = 11 (ci(L) - ma). M=0

What is left is to develope uniqueness and mirror transformations, which we are unable to achieve at this moment, though they can be easily axiomized. Now let us look at K-theory formula, which can be proved by using equivariant localization in K-theory. First following the same idea, we get the explicit formula as follows: given any equivariant element V in K(Wd), we' have

j*V V = Ejr, EG(Xr/Wd) r

where EG(XlWd) is the equivariant Euler class of the normal bundle of Xr in Nd(X). Here the push-forward and pull-backs by j,. denote the corresponding operations in K-theory. By taking V = 1, we get 1

e![EG(Fr/Md(X))l

__

j, cp!(1)

EG(Xr/Wd)*

Second, we have the following lemma; If W! (1) = 1, which is the case if X = P", this formula gives explicit formulas for some K-theoretic characteristic numbers of the moduli spaces.

Lemma 9.3. Given any equivariant element V in KG(Md(X)), then we have formula e![EG(FrlMd(X)))

EG(Xr/Wd)

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506

denotes the equivariant Euler class of the corresponding normal bundle in K-group. where

In particular we have explicit expressions from the decomposition of the normal bundles:

EG(F*IMd(X)) = (1 - e")(1 - e-")(1 - e"L,.)(1- e-"Ld-r) and similarly

EG(FolMd(X)) = (1 - e")(1- e"Ld), EG(Fd/Md(X))

= (1 - e-")(1 - e-"Ld) For a toric manifold X, we also have the explicit class in K-group, (D6,d)

EG(XolWd) = fJ fi (1 - em"[Da]) a

m=1

where [Da] are the equivariant line bundle corresponding to the T divisors Da. If V is a multipicative type K-theory characteristic class, then we

can develope a similar theory of Euler data and uniqueness. These result can also be extended to the nonconvex case without a hitch.

Lian, Liu, and Yau

507

References [1]

[2]

[3]

[4]

[5]

[6]

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C. Ailday, V. Puppe,Cohomology methods in transformation groups, Cambridge University Press, 1993. M. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982) 1-15. M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1-28. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi- Yau hypersurfaces in tonic varieties, J. Alg. Geom. 3 (1994) 493-535. V. Batyrev and L. Borisov, On Calabi-Yau complete intersections in tonic varieties, alg-geom/9412017. V. Batyrev and D. van Straten, Generalized hypergeometric functions and rational curves on Calabi- Yau complete intersections in toric varieties, Comm. Math. Phys. 168 (1995) 495-533. K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45-88.

P. Berglund, S. Katz and A. Klemm, Mirror Symmetry and the Moduli Space for Generic Hypersurfaces in Tonic Varieties, hepth/9506091, Nucl. Phys. B456 (1995) 153. N. Berlin and M. Vergne, Classes caracteeristiques equivariantes, Formula de localisation en cohomologie equivariante. 1982. R. Bott, A residue formula for holomorphic vector fields, J. Diff. Geom. 1 (1967) 311-330. M. Brion, Equivariant cohomology and equivariant intersection theory, math.AG/9802063.

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P. Candelas, X. de la Ossa, P. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal the-

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P. Candelas, X. de la Ossa, A. Font, S. Katz and D. Morrison, Mirror symmetry for two parameter models I, hep-th/9308083,

ory, Nucl. Phys. B359 (1991) 21-74.

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Nucl. Phys. B416 (1994) 481-538. D. Cox, The functor of a smooth tonic variety, alg-geom/9312001. T. Delzant, Hamiltoniens periodiques et image convex de l'application moment, Bull. Soc. Math. France 116 (1988) 315-339.

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D. Edidin and W. Graham,Equivariant intersection theory, alg-

geom/9609018. S.T. Yau, ed.,Essays on Mirror Manifolds I, International Press, Hong Kong 1992. [18] W. Fulton, Intersection Theory, Springer-Verlag, 2nd Ed., 1998. [17]

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K. Fukaya and K. Ono, Arnold conjecture and Grornov-Witten invariants, preprint, 1996. I. Gel'fand, M. Kapranov and A. Zelevinsky, Hypergeometric functions and Loral manifolds, Funct. Anal. Appl. 23 (1989) 94106.

[21]

A. Givental, A mirror theorem for toric complete intersections, alg-geom/9701016.

[22] M. Goresky, R. Kottwitz and R. MacPherson, Equivariant cohomology, Koszul duality and the localization theorem, Invent. [23] [24]

Math. 131 (1998) 25-83. V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982) 491-513. V. Guillemin and S. Sternberg, Supersymmetry and equivariant de Rham theory, preprint.

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V. Guillemin and C. Zara, Equivariant de Rham Theonj and

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Graphs, math.DG/9808135. F. Hirzebruch, Topological methods in algebraic geometry, SpringerVerlag, Berlin 1995, 3rd Ed.

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S. Hosono, A. Klemm, S. Theisen and S.T. Yau, Mirror symmetry, mirror map and applications to complete intersection CalabiYau spaces, hep-th/9406055, Nucl. Phys. B433 (1995) 501-554. S. Hosono, B.H. Lian and S.T. Yau, GKZ-generalized hypergeometric systems and mirror symmetry of Calabi-Yau hypersurfaces, alg-geom/9511001, Commun. Math. Phys. 182 (1996) 535578.

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S. Hosono, B.H. Lian and S.T. Yau, Maximal Degeneracy Points of GKZ Systems, alg-geom/9603014, Journ. Am. Math. Soc., Vol. 10, No. 2 (1997) 427-443.

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S. Hosono, M. Saito and J. Stienstra, On mirror conjecture for Schoen's Calabi-Yau 3 -folds, preprint 1997.

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A. Kresch, Cycle groups for Artin Stacks, math. AG/9810166. [32] M. Kontsevich, Enumeration of rational curves via torus actions. [31]

In: The Moduli Space of Curves, ed. by RDijkgraaf, CFaber, Gvan der Geer, Progress in Mathvo1129, Birkhauser, 1995, 335368.

[33] W. Lerche, P. Mayr, N. Warner,Noncritical Strings, Del Pezzo [34]

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Singularities and Seiberg- Witten Curves, hep-th/9612085. J. Li and G. Tian, Virtual moduli cycle and Gromov-Witten invariants of algebraic varieties, J. of Amer. math. Soc. 11, no. 1, (1998) 119-174. J. Li and G. Tian, Virtual moduli cycle and Gromov-Witten invariants of general symplectic manifolds. Topics in symplectic 4manifolds (Irvine, CA, 1996), .47-83, First Int. Press Lect. Ser. I, Internat. Press, Cambridge, MA, 1998. J. Li and G. Tian, Comparison of algebraic and symplectic GWinvariants, To appear in Asian Journal of Mathematics, 1998. B. Lian, K. Liu and S.T. Yau, Mirror Principle I, Asian J. Math. Vol. 1, No. 4 (1997) 729-763.

B. Lian, K. Liu and S.T. Yau, Mirror Principle, A Survey, to appear. D. Morrison and R. Plesser, Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, alg-

geom/9412236. [40] T. Oda, Convex Bodies and Algebraic Geometry, Series of Modern Surveys in Mathematics 15, Springer Verlag (1985). [41] S: S. Roan, The mirror of Calabi- Yau orbifolds, Intern. J. Math. 2 (1991), no. 4, 439-455. [42] Y.B. Ruan, Virtual neighborhoods and pseudo-holomorphic curves, math.AG/9611021. [43]

Y.B. Ruan and G. Tian, A mathematical theory of quantum cohomology, Journ. Diff. Geom. Vol. 42, No. 2 (1995) 259-367.

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E. Witten, Phases of N=2 theories in two dimension, hepth/9301042.

Differential Equations from Mirror Symmetry Bong H. Lian,1 and Shing-Tong Yau2 Abstract: We discuss a method for deriving differential equations for the prepotential and the mirror map arising from a pair of CalabiYau manifold. Examples with one Kahler modulus are given. Here we find that the differential equations we derive almost entirely characterize the prepotential and the mirror map in question. Some of the results here have been announced previously in [3].

July 99 1

Department of Mathematics, Brandeis University, Waltham, MA

02154.

2 Department of Mathematics, Harvard University, Cambridge, MA 02138.

Lian and Yau

511

1. Introduction We construct a new type of differential equations which govern the B-model prepotential and the mirror map which arise from a pair of mirror manifolds. By this we mean a pair of Calabi-Yau threefolds whereby the B-model prepotential of one threefold agrees with the A-model prepotential of the other threefold. The existence of those equations are in fact closedly related to the special geometry of the moduli spaces of Calabi-Yau manifolds.

In this paper we will restrict ourselves to the case where the mirror pair X, Y have h''1(Y) = 1 = h2'1(X) = 1. Our construction can be generalized to the multi-moduli cases as well.

2. The mirror map and the prepotential Let i : X -* P1 be a smooth algebraic family over Pl. We assume that the fibers X,. := 1(z), except at finitely many points, are Calabi-Yau threefolds with h2"1(X,z) = 1, and that the singular locus is a divisor with normal crossing. In a neighborhood of a smooth fiber Xz, we have a map from this into a universal family of CalabiYau threefolds, which by the theorem of Bogomolov-Tian-Todorov, is

a smooth family over a disk in H2"1(X,s). We assume that this map is locally 1-1. We shall fix a smooth base point z° E P1. Away from the singularity, there is a natural line bundle over the base whose fiber over z is H3'0(Xz). There is a section St whose value Q(z) at z is a nowhere vanishing holomorphic 3-form on X. It is clear that St is unique up to a choice of a holomorphic function. For a given symplectic base c, , a2042 of H3(X,to) with (aj, fy) = b; , we can consider the integrals of 11(z) over them near z°. Put together, they form a 4-vector of (multivalued) holomorphic function which we call a period vector ir. It can be shown that (see [2][4]) that the period vector ec 8{i ac) where G is a homogeneous function of takes the form (/t0 fa' 15 it follows that degree 2 of C1. By a change of variable, t F = q2G is a function oft only, and we can write

i = X0(1, t, F', 2F - tF').

(2.1)

It is well-known that the components of the period vectors are flat section of a bundle equipped with a natural flat connection. Concretely, the normalization of 1 can be choosen so that the components of the period vectors are solutions to a fourth order ordinary differential equations on P1 with regular singularities. It is known as a

Differential Equations from Mirror Symmetry

512

Picard-Fuchs equation. In particular it is an ODE with polynomial coefficients.

Under suitable conditions, Mirror Symmetry interpreter the func-

tion F as a holomorphic function defined on the tube domain T = HIJ(Y, R) + vr-_1K(Y), where Y is another Calabi-Yau threefold and K(Y) is its Kahler cone. The change of variable t = is interpreted as a mapping from P' into T. Note that as given, this mapping is only defined near a based point zo. Inverting the mapping locally,

we get a holomorphic function z(t) which we call a mirror map. In Physics, F' is known as a Yukawa coupling. Note that both z(t) and F(t) are both defined at least as functions from a domain in C, independent of the above Mirror Symmetry interpretation. In the context of mirror symmetry, many physicists have posted the following question (we thank M. Bershadsky and C. Vafa for communicating this question to us): Is the function F governed by a differential equation? In this paper, we answer this question in the affirmative. We shall construct a (rather complicated) 7th order polynomial differential equation for the function

K:= F"' with constant coefficients. More generally, we have

Theorem 2.1. Let L be a 4th order linear differential operator on P' with polynomial coefficients. Suppose t o, c1, eoF'( 40 ), eo(2F(f) -

LF'( )) are linearly independent solutions to Lf = 0 near 0 for

some holomorphic function F on a domain. Then K(t) := F"'(t) satisfies a 7th order polynomial differential equation with constant coefficients. Likewise for z(t) as defined above.

The result is really local in nature and can be formulated entirely in a disk. The details of the construction of the 7th order differential equation for z(t) was done in [3) and shall not be repeated here. We shall first prove the theorem, and then return to examples in the last section. The hypotheses of the theorem will be in force throughout this paper. The basic idea of the proof is this: ODEs with solutions above can be viewed as the differential algebraic analogue of polynomial equations in two variables. The functions F and z play the roles of the two variables. In the algebraic case, one can use eliminating theory to construct algebraic equations for each variables by "separating" the variables. The strategy is to try to do the same in the differential algebraic case. We achieve this by developing and using some elementary tools in differential graded algebra.

Lian and Yau

513

Acknowledgements. We thank A. Klemm and S.S. Roan for participating in the initial stage of this project a few years ago. Special thanks to C. Doran for proofreading this paper. B.H.L. is supported by NSF grant DMS-9619884. S.T.Y.is supported by DOE grant DEFG02-88ER25065 and NSF grant DMS-9803347.

3. Change of variables Note that the functions z(t) and p(t) depend only on the ratio of two solutions 6, i;o, hence remain unchanged if we perform a change of variables f (z) = A(z)g(z) on the equation Lf = 0. We can choose A so that the new equation reads: 9""(z) + a2(z)9"(z) + al(z)9'(z) + ao(z)g(z) = 0

(3.1)

where the as are rational functions.

Lemma 3.1. a,(z) = a2 (z). For the proof of this, see [2]. Under a change of variable z H t,

(3.1) becomes

L9 := 9""('(f)) + c29"(z(t)) + c29'(z(t))

+ co9(z(t)) = 0

(3.2)

where

c2 =a2z'" -

15 z" x(3) (' )2 + 5 = a2z'2 + 5{z(t), t} z

2

3 dal ,2 ,, 3 n2 135z"4 co =aoz + z z - 4a2z - 16z'4 2 dz 3a2z'z(3) +

75z"2x(3) 4zj3

15x1312

4z'2

_

15z"x(4)

2z'2

+

3z(5)

2x'

The prune. here means Uj. Note that Co, c2 are differential rational functions of z(t), ie. a rational function in z, z', z", ... Now (3.2)has four linear independent solutions of the form u(1, t, F, 2F-tF') where u = &o/A. We may now view (3.2) together with its four special solutions as a system of polynomial ODEs in z(t), F(t), u(t). Our polynomial ODEs for z(t), F(t) with constant coefficients will be obtained by separating the variables.

Differential Equations from Mirror Symmetry

514

First we claim that

K = coast. u-2 = coast.

0

A

.

(3.4)

To

Substituting the 4 special solutions above in (3.2), we get U(4) + C2U" + C'U' + Cou =0

4u"'+2c2u'+c ,',, u=0 K(6u" + c2u) + 4K'u + K"u =0

(3.5)

2Ku'+K'u=0. Our claim follows from solving the 4th equation above. Note that given this K, the 2nd equation in (3.5)cau be obtained

by differentiating the 3th equation. From now on we want to view the system (3.5)of equations as a system of ODEs in the two dependent variables K(t), z(t) and one independent variable t. The system (3.5)can be written as c2 (z; t) =r2 (K; t)

co(z; t) =ro(K; t)

where c2, co are rational function of z, z', .. defined in (3.3). The r2, r° are rational function of K, K',.. given by r2

2K"

K

5K'2 k) 2

5 K'2 K" _ 5 K12 _ 2 K' K(3) K(4) 16 K4 + K3 K2 + 2 IC 4K2

_ -35 K'4 r°

(3.7)

Note that the system (3.6)of equations are derived with no special assumption about the coordinate t other than that it is the ratio of two linearly independent solutions: t = Thus the form of the system (3.6)must remain the same if we use another ratio 1. Thus the new equations will read:

c2(z(t); ) =r2(K(t); t) co(z(t); t) =ro (K(t); t)

Lian and Yau

515

4. Some differential algebra If we assign the weight 1 to each t differentiation (ie each prime), then both C20'2 in (3.6)are expressions of weight 2. Similarly both co, ro are expression of weight 4. This motivates the following considerations. Let x, x', .., y, Y',.. be formal variables. We assume that wt x(k) = wt y(k.) = k. We may also freely use some other formal variables a, a', .., j3, /3', .. which will bear the analogous meaning. A rational function f (x, x', .., y, y', ..) in these variables will be called a differential rational function. We usually denote it as f [x, y]. If it depends only on the x, we write f [x]. Given two such functions, say f [x], g[y], it makes sense to compose them. Namely we do so in an obvious way f [g[x]] according to the usual rule of differentiation. Let X [x], Y[x], A[y], B[y] be differential rational functions, homogeneuos with positive weights IXI = IAI, IYI = IBI. We would like to study the system of ODEs: X [x] =A[y]

Y[x] =B[y}.

(4.1)

We write

I(A[y], B[y])

{P[a, b]I P[A[y], B[y]] = 0}.

(4.2)

This is obviously a differential ideal in the ring of differential polynomials in two variables a,/3. If a, f3 are assign the weight IAI, IBI respectively, then I (A[y], B[y]) is in fact a graded ideal. Note that

if x(t), y(t) are solutions to the system (4.1), then every element P[a, b] E I (A[y], B[y]) gives a differential equation

P[X [x(t)], Y[x(t)]] = 0

(4.3)

for the function x(t). This equation can of course be trivial. We shall construct a nontrivial differential equation for the function K(t) by first studying the differential graded ideal I (c2, co) where c2, co are given in (3.3). We will prove that there exists an element of minimal weight in this ideal. In this sense, this will be the simplest polynomial differential equation for K(t). Let's consider the effect under the formal substitution law t

t=

, x H xi, y H y' on a homogeneous differential rational

function of the form P[X [x], Y[x]] of weight 1. It is called covariant if it transforms like P[X [x], Y[x]] H y21P[X [x], Y[x]l. Suppose the

516

Differential Equations from Mirror Symmetry

differential rational functions X [x], Y[x] transform as follows: n

X[z] r-+ 1: ,Y2n_jc1Qj[X[x],Y[z]] j=0

Y[z] H E'Y2m-'c'Rj [X [i], Y[z]] j=0

where -y = ct + d, n, m are the respective weights of X, Y, and the Q1, Rj are some fixed differential polynomials of respective weights n - j, m - j. We also assume the same transformation law for the differential polynomials A[y], B[y], but only with x, x replaced by y, respectively. Note that the leading term must be given by Q0[X, Y] _

X and R0 [X, Y] = Y. The assumption above simply says that the differential polynomial C(t)-algebra generated by X, Y is closed under the transformations from SL(2, C).

Lemma 4.1. Suppose that P[a, /3] is an element in I(A[y], B[y]) of minimal weight. Then P[X [x], Y[x]] is covariant.

Proof: It is enough to show that P[a, /3] y y21P[a,,6] under formal substitution: n a H C` y2n-jo'Qj[a,,3] jj=0 M

E ,),2m-j C7 R.7

Q

[a, N]

j=O

Obvioiusly under (4.5), we have t

P[a, /3] H E yet-jCj Sj [a, a] j=0

for some differential polynomials Sj of weights 1- j. We must show that Sj = 0 for all j > 0. By assumption A, B have similar transformation law (4.5). More precisely we have 21-jCjSj

P[A[y], B[y]]

Ely

[A[j], B[9!]]

j=0

Since P[a, /3] is an element of the ideal I(A[y], B[y]), it follows that P[A[y], B[y]] is identically zero. Hence the right hand side of (4.7)must

l.iau and Yau

517

also be identically zero. Thus we have Sj[A[y], B[y]] = 0 for all j. Since it is ,just, it I anal vtn'iable, the same holds true with y replaced by y. This utcaus that. the -111i la, ,tj] are in the ideal I(A[y], B[y]). But

S1 has weight l - j ;rntf sc, by the minimality of P[a,Q], we have 0 for all j > 0. Thus (4.6)now says exactly that P is covariant of weight I. This completes the proof. 4.1. SL(2) a ctimu uu sr,lvfion set

V Te now return to our original system of equations (3.6), and diagress slightly to eons tiler its general solutions. Given a complex parameter /. let, i. = ;`! l w here ad - be = 1. Viewing t -> t as a change of coordinates. we have the formal relations: c2(z(t):1)

(4.8) tr t : t i 1: 1) + 3'y7cc2 (z(t); t) + 6rysc2c2(z(t); t).

co(z(l); i)

Proposition 4.2. Lrt ;: it ). K(t) be solutions to (S.6). Then so are

at+b

(ct+d) {i {t)

:=C,y-4K(at+b)

4

d +d

for any rOtecttant C.

Proof: It. ti,Ilows itiitne( lately from (3.6)that the equations are invariof fi by a constant. So without loss of generality, ant under flue computation, we find that t; we, cau set C. = 1. 1S, direct r2(K(1); i)

h' (t); t)

'a'z!

-

ro(K(i); t)

K (1) t) + 3ry7crz('y-4K(); t) + 6 Ysc2r2('Y-4K( ); t).

Applying (:3.ti). ?,l.?N 1

ztud (4.10), we get

(4.10)

c2(2(t); f) eo(z(i); t)

-

,

. (.i); t) = y-4r2(K(t); t) = r2('y-4K(t);t)

'' r ;,(

t f ); t) -

t) - 67-2C2C2(Z(); t) (4.11)

Differential Equations from Mirror Symmetry

518

It is interesting to see how the new solution given is constructed. Fix a solution (z(t), K(t)) to the system (3.6). And consider the same

system with z, K replaced by z, K. Set z(t) = ct+d z() where z(t) is assumed to be a given solution. We would like to solve for k(t) in terms of the given K(t). We shall see that the general solution is

precisely k(t) = Cy-4K(J). We must solve r2(K(t); t) ='y4r2(K(t); t)

ro(K(t); t) =ysro(K(t); t) +

t) + 6yoc2r2(K(t); t). (4.12)

Set y(t) = Log K(t). Then we get y22

r2(K(t);t) =lye (4.13)

y,. yi2

ro( K (t) t) = Y(4) 2

+4

2

16

Thus the first equation in (4.12)becomes d2y(t) dt2

1 dy() 2 = 4(

dt

)

4

y(

&y(t) _ 1 df (t) 2 dt2

4(

dt ) )'

(4.14)

Since y(t) is given, this is a Ricatti equation in the variable `iit) . It's easy to verify that its general solution is

J(t)

y(

b

4Log y. + d) -

(4.15)

By the standard classical method, we find that the general solution to (4.14) is 4Cexp(2 j(t))

yg-(t) = P(t) - f 1 +C f exp(z l(t))

(4.16)

where C is an arbitrary constant. Writing it in terms of K(t), we get

K(t) =

4K(a+d) y 4 (B + C f y-2K( +d)1/2)

Now this satisfy the second equation in (4.12) if C = 0.

(4.17)

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519

5. ODEs for the Yukawa Coupling - Existence Our problem now is to construct an element P[a, C3] of I(c2ico), which will give a nontrivial differential equation for K(t). The last section indicates that we should look for a covariant element P[a, /3] such that (5.1)

P[c2, Co] = 0

identically. We shall do this in three steps. Let a, /3 be formal vari-

able of weight 2,4 respectively. We shall call a differential polynomial P[a, C3] a simple covariant of weight k if

P[c2, co] = p(z)z

ti

(5.2)

for some rational function p(z). It is obvious that P[a, 3] is automatically a covariant. First we find two simple covariants Pl[a, 0], Qi[a, p] of lowest weights. Our original system (3.6) then becomes PI [r2[K], ro[K]] =P(Z)Z'IP' i

Qi[r2[K],ro[K]]

=q(z)zIIQI I

for some rational functions p(z), q(z). We then eliminate z from this pair of equations. Finally we prove that the resulting equation governing K is nontrivial.

Proposition 5.1. Let p[a, /3] =100/3 - 9a2 - 30a" X[rx, l3J = - 32tr[a, /3]2a - 45(dtp[a, 8])2 + 40p[a, p] d p[a, p] (5.4)

They are the unique simple covariants of weights 4,10 respectively. Moreover any other simple covariant not proportional to either of them has higher weight, Note that these two simple covariants are in fact universal: they are independent of the original 4th order differential operator. Thus they are independent of the initial data (a2, ao). Proof: We prove this by solving exactly the condition (5.2) in the differential graded algebra generated by a, /3 up to weight 10. Since a, /3 have weights 2, 4, the only graded pieces we have to look are of weights between 2,10. It is easy to work out a basis in terms

520

Differential Equations from Mirror Symmetry

of a, a', .., /3, ,0' for each graded pieces. Solving the above condition yields the desired result. It is interesting to note that the equations (5.4)defines an isomorphism c : (a, a) '* (P[a, a], X[a, /3]) (5.5) between the two fields of differential rational functions: one is generated by p, X, the other by a, 3. We also note that p[c2, co] =p[a2(z), ao(z)]z,4

X[c2, co] =X[a2(z), ao(z)]z"o

Here we have slightly abused the notation: p[a2(z), ao(z)] really means 100ao(z) - 9a2(z)2 - 30a'2 (z), and similarly for X[a2(z), ao(z)]. Thus

both are just rational functions in the variable z. If p[a2(z),ao(z)] = 0 (or X[a2(z),ao(z)] = 0), then it's easy to show that p[r2[K], ro[K]] = 0 (or X[r2[K], ro[K]] = 0) is a nontrivial ODE for K. Thus from now on, we assume that neither is zero. Set p(z) =P[a2(z), ao(z)] (5.7)

q(z) =X[a2 (z), ao(z)] Then we have

5X[C2, co]P[c2, co]' - 2X[c2, co]'P[c2, co] = (5gdp dz

- 2p`)z'15. dz

(5.8)

It follows that P[C2, C01'

X[c2, co]2

(5X[C2, c0]P[c2, co]' - 2X[c2, co]'P[c2, co])2 X[C2,co]3

Lemma 5.2.

P'

-q2 = (5qg - 2p

-

)2

(5.9)

q3

For any f (z), g(z) E C(z), there is a nonzero

polynomial Q(a, b) such that Q(f (z), g(z)) = 0 identically. If we

write f = fi/f2, g = gi/g2, then such a Q may be choosen so

Lian and Yau

521

that deg, Q and degb Q are both bounded from above by Nmaz max(max(deg fl, deg f2) + max(deg gl, deg 92) - 2,1).

Proof: We consider the system of homogeneous linear equations resulting from the coefficients of the powers of z in

N

E a12f (z)'g(z)j = 0.

(5.10)

It is easy to verify that for N = N,iaz, the linear system has more variables than equations. 0 In particular if we now take f, g to be the two right hand sides of (5.9), then it follows that we have a nonzero polynomial Q with the above vanishing property. Fix such a Q and define a new differential polynomial in the variables £, 77 of weights 4,10 respectively:

5

rl]

g5NQ( 2, 77

(5776i

3

It is obvious that R[e, q] is nonzero. Also by construction we have R[p[c2, co], X[e2, co]] = 0 identically.

Proposition 5.3. R[p[r2[K], ro[K]], x[c2[K], co[K]]] = 0 is a non-

522

Differential Equations from Mirror Symmetry

trivial ODE for K. Proof: By direct computation, we find p[r2[K], ro[K]] _ 2 K' K(3) - 10K 3 K(4) 175 K'4 - 280 K K12 K" + 49 K2 K"2 +70K K4 X[r2[K], ro[K]] =4 (1225000 Kj10 - 4900000 K K's K" + 6737500 K2 K16 K"2r - 3626000 K3 Kj4 Kj3

+ 689675 K4 K'2 K"4 - 360836 K5 Kj5 + 1225000 K2 K'7 K(3) - 2940000 K3 K'5 K" K(3) + 1690500 K4 K'3 Kj2 K(3)

+ 695800 K5 K' Kj3 K(3) + 367500 K4 Kj4

K(3)2

- 705600 K5 K'2 K" K(3)2

- 235200K6 K0 K(3)2 + 117600 K6 K'

K(3)3

- 175000 K3 K'6 K(4) + 420000 K4 Kj4 K" K(4) - 562800 K5 K'2 K0 K(4) + 203000 K6 Kj3 K(4) + 42000 K5 Kj3 K(3) K(4) + 142800 K6 K' K" K(3) K(4) K(4)2 K(4) - 12000 K6 Kj2 K(4)2 + 94500 K5 Kj3 K" K(5) - 26400 K7 K"

- 16800 K7

K(3)2

- 103950 K6 K' Kj2 K(5) - 31500 K6 K'2 K(3) K(5) + 37800 K7 K" K(3) K(5) + 9000 K7 K' K(4) K(5)

- 1125 K8

K(5)2

-17500 K5 K14 K(6)

+ 28000 K6 Kj2 K" K(6) - 4900 K7 K0 K(6)

- 7000 K7 K' K(3) K(6) + 1000 K8 K(4) K(6)) /K10 (5.12)

Observe that R[6, q] has either a nontrivial leading power in rl', or else it has a leading power in r/. In the first case, we see

Lian and Yau

523

R[p[r2[K], ro[K]], x[c2[K], co[K]]] has a nontrivial leading term containing the 7th derivative K(7). In the second case we have a nontrivial leading term containing K(6). In either case, the resulting ODE is nontrivial. This completes the proof of our main theorem.

6. Remarks and examples The proof above indicates also that we can put a bound on the minimal weight of our ODE for K. Since Q obviously has weight zero, the differential polynomial R above has weight 5N. Thus the minimal weight is no more than SN,,,,ax. We now return to the discussion of the Picard-Puchs equation of a family of Calabi-Yau 3-folds. In the case of the mirror quintic, we have the famous Picard-Fuchs equation

(84 - 5z(58 + 4)(58 + 3)(5e + 2)(58 + 1)) f (z)) = 0

(6.1)

where 8 := zd . In this case one finds that N,.,,. = 1400. This turns out to be far from optimum. Using the computer, we find an ODE for K of weight 360.

In practice, we can simplify the computation for an ODE significantly as follows. If we regard 5716' - 277'e as another variable, then the construction in the last section guarantees that there have a polynomial in , 77, 6 := 5t ' - 217' which gives a nontrivial ODE for

K. Thus it is enough to look in the polynomial ring in , rl, 5 where the weights of the variables are respectively 4,10,15. Our construction guarantees that any zero polynomial quasi-homogeneous polynomial P(g, il, 6) satisfying P(P[c2, c0], x[c2, co], 5P[c2, co]'x[c2, co] - 2P[c2, co]x[c2, COY) = 0

(*) (6.2)

identically gives a a nontrivial ODE for K. We want to find one with the minimum weight.

Lemma 6.1. A necessary and sufficient condition for a nonzero polynomial P(e, rl, 5) satisfying (*) to be a minimum of weight n is that rl, 5) satisfying (*) of weight n - 1, n - 2, n - 3, every polynomial

or n - 4 is zero. Proof: The necessity of the condition is clear. To show sufficiency, note that if T(e, rl, 5) is a polynomial of weight lower that n - 4 and

524

Differential Equations from Mirror Symmetry

satisfies (*), then we can multiply it by a suitable power - say k so that C'T (x,17, 5) has weight n - 1, n - 2, n - 3, or n - 4 because e has weight 4. Now it must be zero, implying that T itself is zero. When we try to find a polynomial P(e, 17, 6) satisfying (*) and of minimal weight, say n, the lemma saves us from checking the the lower weight polynomials - all we have to check is that there is none in weights n - 4 to n -1. For example, in the case of the mirror quintic. We verify by computer that there are no nonzero polynomial T(C, rq, 5)

satisfying (*) and has weight between 359 and 356. Moreover there are exactly 127 monomials of weight 360 in the three variables. Up to multiple, exactly one linear combination of them satisfies (*). Here we shall give the simplest example known to us: here we consider the family of Calabi-Yau 3-folds X mirror to the complete intersection of 4 quadrics in p7. In this case, the minimal polynomial satisfying (*) has weight 180:

P(5,X,p) _

3783403212890625 X is + 52967644980468750 x15 52 + 292835408677734375 X12 54 + 833559395864062500 X9 56 + 1301823644717109375 X6 5$ + 1064406315612768750 X3 510 + 357449882108765625 512 + 9097175898878906250 X16

+ 75543680906950781250 X13 52 p5

- 55168781762820937500 X1o 54 p5 - 1235933279927738437500 X7 56 p5 - 2628328829388247068750 x4 8 p5 - 1669442421173622243750 x 510 p5 X14 p'° - 316395922222462973709375 - 1041303693581386404075000 X1152 pl0 Xs 54 pl0

+ 1397061241390545045311250 - 978071752628929206000 x5 56 plo

8 p1° ply + 726375263921582813504122500 X12 + 3088961515882945520173125 X2

- 2401967567306257982918892000 X9 52 p15 54 p15 + 2931906039367569842399977800 X6 - 2592007729730548310729752000 X3 56 p15 15

+ 2647721143185632529213250068P Xlo p20 + 731773527868504699561324929024 X7 52 p20 - 1384453886791545382987331665920

p5

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525

+ 1003786188392583028918031769600 x454 p2o - 92650299984331138894225408000 x Ss p20

+ 264379950716374035480557566033920 x8°25 - 323653884996678359415539902709760 x582°25 + 105122101152057682020817226956800 x284 p25 + 48853700167414249640038923438653440 x6 p3°

- 33153423760664989683513831467253760 x382p30 + 528120679253988321156369324441600 S4 p3°

+ 4965538896010513223822010617996247040 x4°35 - 1238934080748073699029086124292177920 x 82°35 + 265021771162266355900761945816768184320 x2 p4° + 5822406825670998196401392296588763725824 p45

Note that since S, x, p are of weights 15,10,4 respectively, P is a quasihomogeneous polynomial of weight 180. Each of the 37 terms in this polynomial corresponds to a partition of 180 by 15,10,4. In the second example, we consider the complete intersection of

two cubic in P5. In this case, we find that the ODE for K is given by a polynomial of weight 330. There are 108 partitions of 330 by 15,10,4. Thus this polynomial has a maximum length of 108. We have verified that in all cases above, our differential equation determines the Yukawa coupling K(t) up to the first two Fourier coefficients.

526

Differential Equations from Mirror Symmetry References

[1]

[2]

[3]

[4]

P. Candelas, X. de la Ossa, P. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B359 (1991) 21-74. S. Ferrara, C. Kounnas, D. List, and F. Swirner, Automorphic Functions and Special Kdhler Geometry, In Essays on Mirror Manifolds, Ed. S.T. Yau, International Press 1992. A. Klemm, B.H. Lian, S.S. Roan, and S.T. Yau, A Note on ODEs from Mirror Symmetry, Conference Proceeding in honor of I. Gel'fand (1995). A. Strominger, Commun. Math. Phys. 133 (1990) 163.

Surveys in Differential Geometry, vol. 5

Heat Kernels, Symplectic Geometry, Moduli Spaces and Finite Groups Kefeng Liu

1. Introduction In this note we want to discuss some applications of heat kernels in symplectic geometry, moduli spaces and finite groups. More precisely we will prove the nonabelian localization formula in symplectic geometry, derive formulas for the symplectic volume and intersection numbers of the moduli space of flat connections on a Riemann surface, and obtain several quite general formulas for the numbers of solutions of equations in finite groups. Several new formulas for the push-forward measures by various maps between Lie groups are also obtained. In solving these problems, we will use the corresponding heat kernels on Euclidean spaces, on Lie groups and on finite groups. The discussions of some aspects of the first two applications have appeared in [Liu], [Liulj and [Liu2j, here we will only sketch the main ideas. The purpose to include them here is to unify the discussions by using heat kernels. The third and the fourth applications were worked out through many discussions with P. Diaconis. Several results for compact Lie groups in Section 4 were also motivated by his conjectures and his results for finite groups The main idea we use is very simple and goes as follows. We consider a map between two spaces f : M -a N. In many problems we are interested in understanding the inverse image f (xo) for some point xo E N. Assume that there is a heat kernel H(t, x, xo) on N. We consider the integral of the pull-back of H(t, x, xo)

by f: 1(t) =

H(t, f(y), xo)d y Jmr

with respect to certain measure dy on M. We then perform the computations of this integral in two different ways, similar to the heat kernel proofs of the Atiyah-Singer index formula and the Atiyah-Bott fixed point formula. On one hand, when t goes to 0, I(t) will localize to an integral over a neighborhood of f-1(xo) C M, on the other hand, by using the special properties of symplectic manifolds or representation theory of the corresponding groups, we can explicitly compute I(t) globally in our cases. In this way, we obtain formulas about the geometry and topology of f -1(xo ) 01990 International Press

527

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KEFENO LIU

in terms of certain global information on M and N. We will give several examples here, to all of which we can apply this simple idea. (1). The moment map p, : M - g", where M is a symplectic manifold, and g" is the dual of the Lie algebra of the group G acting on M. In this case we are interested in the symplectic reduction MG = IA-I(O)/G. (2). The holonomy map f : G20 -a G where G is a compact Lie group and f is the product of the commutators 8 .f(z1,Ul,...yg,yg)

_ II [zi,U.i] j=1

In this case we are interested in the space f-I(c)/ZZ, which is the moduli space of flat connections on a flat G-bundle on a Riemann surface. Here c E G and Zc denotes the stabilizer of c. More general examples were studied in [Liu2]. (3). Equivalently we can consider the map

f:

n G20xfOy-+G

j=1

where O.f is the conjugacy class in G passing through cf E G. Another interesting map is the n-commutator map f(n) : Gn -+ G with .f(n(zl,z2,...,zn)=[zl,[z2,...[zn-1,zn1]]

In this case, an inductive formula for the push-forward by f (n) of the Riemannian measure on GI can be obtained. (4). For a finite group G, we can consider the maps similar to those in (2) and (3) or more generally we can consider n

f: Gnxflff5 +G j=1

where HI,

, Hn are subgroups of G and n f(z1i...

,znizl,...

z1

zn)

yjzj1

,

j=1

In this case the integral to define I (t) is replaced by the sum over Gn x n! H and the limit as t goes to 0 gives the number of solutions to the equation in Gn x rj? 1 H5 : TnT

11

z jz,z 1 = e.

j=1

More general examples than the above cases will also be considered in this note.

The interesting point here is that, the heat kernel method supplies a unified way to deal with several seemingly unrelated problems in geometry, topology and finite group theory. In Section 2 we derive a nonabelian localization formula in

symplectic geometry from the heat kernel point of view. Here we use the heat kernel of Euclidean space. Note that, different treatment of the result has been

HEAT KERNELS, SYMPLECTIC GEOMETRY, MODULI SPACES...

529

discussed first by Witten [W], then in [Wu], [JK] and [Liu]. In Section 3 we obtain the formulas for the symplectic volume and some intersection numbers of the moduli space by using the heat kernel of the Lie group G. Some detailed discussion in this section has appeared in [Liul], [Liu2], and in [BL]. In Section 4 we derive various formulas for the push-forward by those maps in (2) and (3) of the R.iemaniann measures. In Section 5 we derive several formulas for counting the numbers of solutions of certain polynomial equations in finite groups. All of the results in this section grew out of discussions with P. Diaconis who has quite different proofs for these formulas by using combinatorics methods. One of the formulas in Section 5 has been proved and used as the main tools in [St] to solve the long-standing Brauer p-block conjecture. I want to dedicate this paper to the memory of Prof. Qi-Ming Wang who passed away ten years ago. In 1988, after reading my master thesis, Prof. Wang encouraged me to follow Prof. S: T. Yau to study geometry. This is the turning point of my career. Finally I would like to thank P. Diaconis for the many stimulating and enlightening discussions.

2. Symplectic reduction Let M be a compact symplectic manifold with a symplectic form w. Assume that the compact Lie group G acts on M. Let g denote the Lie algebra of G and g* be its dual. Assume that the G-action has a moment map JA : M -3 g*. With the metric induced from the Killing form on 0, we can identify g* to the

Euclidean space R. Let us denote the metric on g* by < , >. Let 1

H(t,x,xo) = (4irt)I1(-II-

_

2 911

4t

be the heat kernel on g*. Assume 0 E g* is a regular value of it. We are interested in studying the symplectic reduction MG =,u 1(0)/G. We consider the integral of the pull-back of the heat kernel against the symplectic volume on M :

I(t) = f H(t, IA(U), 0)e . As in the derivation of the index formula, we will compute this integral in two different ways: local and global. (i) Local computation. Let t go to 0, then the integral I(t) localizes to a neighborhood of µ I (0), which can be identified as p-2 (0) x B6 where B6 denotes a small ball of radius d in g*. More precisely we can write the above integral as

1(t) =

H(t, p(y), 0)e" + O(e a'/at). µ-s(o)xBa

When restricted to µ-l(0) x B6, the symplectic structure w has a canonical expression in terms of the induced symplectic form wo on the symplectic reduction

MG = p-'(0)/G:

KEFENG LIU

530

w = 1r*wo + d(a, 8)

where a : p-1(0) -+ Ma denotes the quotient map which gives a principal 0bundle, and B is a connection form of this bundle. Also a denotes the coordinate on g* and (a, 8) denotes the obvious paring. And in this local model p is just the simple projection map: µ : p-1 (0) x g* --> g* defined by µ(x, a) = a. By substituting the local expression of w into the integrand we obtain

I(t) =

µ_,(0)xg.

H(t, a, 0)e"'"o+d(a,s) +

O(e-62/4t).

We now use the equalities:

d(a, B) _ (da, B) + (a, dB), and F = d8 - 8 A 8 where F denotes the curvature of 0. From these we derive

I (t) =

µ '(0)

e"`W0 A B

Jg` H(t, a, 0)e(°,F)Da +

0('_62 /4t)

where Da denotes the volume form of g* and AO = 81 A . A 8,,. Here we have used the fact that, by a degree count, in e(d°,g) only the term Do A 0 will contribute to the integral. Note that A8 is actually a volume form of G. By performing a simple Gaussian integral on g* = R", we get

I(t) = IGI

Je"'0-t +O(e-a'/4t) Ma

where < , > also denotes the inner product on g induced from the Killing form, and IGI denotes the corresponding volume of G.

(ii) Global computation. On the other hand we can rewrite r(t) in terms of equivariant cohomology class.

a t<w,w>ew+:(v,w)d

I(t) g

9

fm JM

Here < denotes the inner product induced on g by the Killing form. Note that the first identity actually corresponds to the Fburier transform of H(t, x, 0). By comparing the computations in (i) and (ii), we have the following equality of Witten which we summarize as a proposition.

Proposition 1: We have the following formula:

IGI r Mc

6W0-t = limt-,o

f"gJM et<w,w>eW+:(4,w)dP

This formula expresses the geometric and topological information on Ma in terms of the information on M. This is the spirit of the nonabelian localization of Witten.

HEAT KERNELS, SYMPLECTIC GEOMETRY, MODULI SPACES...

531

By using an idea of Wu [Wu], we can actually go a little bit further. Let T denote the maximal torus of G and {P} be its fixed point components. By using Atiyah-Bott localization formula, we can express the integral /m

eW+i(µ,w)

fm as the sum of the integrals over the fixed point components {P}. Here note that w + i(p, (p) is a G-equivariant cohomology class. The interested reader can also interprete the derivation of the general formula in [Liu) by using the idea in this section. 3. Moduli spaces of flat connections

In this section we will derive the symplectic volume formula and some results about the intersection numbers of the moduli space of flat connections on a principal flat G-bundle on a Riemann surface. Some of the discussions about moduli spaces in this section are basically contained in [Liul] and [Liu2] where we considered maps like

f: G29xGn-+G with n

9 ,xg,y9;Z1,...Zn)

f(xl,y1,...

_ II[xj,yjI Hzicizc 1 7

j=1

i=1

for certain fixed generic points c1, , C. E G. Here G is a semisimple simply connected compact Lie group. We refer the reader to [Liul] and [Liu2] for the details of the study of these maps by using heat kernels. Here let us consider an equivalent map given by n

f : G29 x f O0, -+ G i=1

with 9

n

f(xl,... ,y9i Z1,... ,Zn) = II[xj,yjl ITzi j=1

i=1

which is more commonly used to describe the moduli spaces. Here x3, yj E G and zi E Oc, . Recall that On is the conjugacy class of ci in G. Now let us start to derive the formulas. We equip G with the bi-invariant metric induced by the Killing form, then the explicit expression of the heat kernel on G is given by

1 E dA

JGJ

XA(xy-1)etp.(A)

,P,

where ]Gi denotes the volume of G and P+ denotes all irreducible representations

of G, which can be identified as a lattice in t', the dual of the Lie algebra of the maximal torus T. Also dA and xA denote the dimension and respectively the character of the representation A, and pc(A) = IA + pl2 -1p12 with

532

KEPENO LIU

P=12 EO+ the half sum of the positive roots. Recall that the moduli space we are interested in is just

Mc = M01,...,, = f-I(e)/G where G acts on G29 x rj 1 Oc, by the conjugation ry:

ry: G-3G2gxfO' i with 'y(w)(zl,... ,bg,zl ... ,z*+) = (wzlw 1,... )wygw 1;wz1w 1,... ,Wznw-1).

Similarly we consider the integral

I(t) =

JhEGao xJ1 Oaf

H(t, f (h), e)dh

where dh denotes the volume form of the bi-invariant metric on G2g x f Ij O., induced from the Killing form. We will again perform the computation of 1(t) in two different ways: local and global. (i) Local computation. Let Z(G) denote the center of G and IZ(G) I denote the number of elements in Z(G). As t goes to 0, a computation of the Gaussian integral as in [Liul] gives us I (t) IZ(G)I

L.

dve +O(e-62/4t)

where dv, is the Reidemeister torsion r(C,) of the complex

C:

0-ig

g29x11 Z-" I

g->0

where Z denotes the tangent space to O.f at cf. It is clear that Zf can be viewed as the image of the map (I - Ad(c3)) By using the Poincare duality of the cochain complexes of the Riemann surface,

one can show that the torsion r(C.) is related to the L2-volume of Mc in the following way:

r(CC)2 = j(c)2lldet TMcIIi2 where

j(c)2 = III det(I - Ad(cj))I i can be considered as the torsion of the boundary of the Riemann surface.

HEAT KERNELS, SYMPLECTIC GEOMETRY, MODULI SPACES...

533

But from the definition we know that, up to a normalization by a factor of 21r, the L2-volume is exactly the symplectic volume: (2x)2%

N,!

= I1detTMCIIL2

for the moduli space M. Here No denotes the complex dimension of Mc, and we denotes the canonical symplectic form on MM induced from the Poincare duality. So we have

N T(Cic) = dvc = (27r)21Ij(c)I

Nc-

where

I7(c)I=HI det(I-Ad(ci))I

.

We refer the reader to [W], [BL], [Liul], and [Liu2] for a proof of this relation. (ii) Global computation. The global computation is achieved by using the character relations: (wyzy-lz-1)dz = Ia

ix

IXA(wy)Xa(y-1),

f xa(wy)x,\(y-1)dy = and the formula Ioeh(g)dv9 = I

IZi 11G h(gcig-')dg

e

for any continuous function h on Here Z,, denotes the stabilizer of c,,, IZ, I its induced Riemannian volume, and dv9 is the induced volume form on 0,,. And note that we have used the notation:

J(cj) = det(I - Ad(ci)). By applying these formulas inductively we get

1XA(IJ[xi,U.i]11z{)11dx dy211dx; (TiO.r =IGI2s+..9(c)2

77

n,2:Xa(ct)

11i I2c:I

By putting the above computations together, we obtain the following formula which we also summarize as a proposition.

Proposition 2: We have the following formula for the 3Vmplectic volume of Mc:

534

KEFENG LIU

IMe`'a _ I Z(G

)I

IGI29+n-2fj(c)I tt (21021. llj IZciI

!!j XA(cJ) aEP+

d2

n-2 a

tPe(a)

+ O(e

62/4t) .

As in [Liul] or [Liu2], we can then take derivatives with respect to the cj's on both sides of the above identity to get intersection numbers on the moduli space Mc or Mu for u an element in the center Z(G). We refer the reader to [Liul] and [Liu2]

for the details of the derivation. Here we only mention an interesting vanishing theorem that we can easily obtain from the above formula. For simplicity let us take n = 1. The general case is the same. In fact let us introduce a function on the dual Lie algebra t of the maximal torus T, tr(X) = naen+ < a, A >, from which we construct a differential operator ir(8) such that, for C E t with u exp C = c, a(8)eGX, c) = tr(,\)e(a, c)

where (A, C) denotes the natural pairing. By applying 7r(8)29 to 1(t), we get, up to a constant,

y I(t) = IZ(G)I IGI"-IIj(c)I tr(8)2irN Z.I

daXa(c)e tP-(a) +O(e 62/4t ) AEP+

Note that, if c # e where e denotes the identity element of G, then the sum on the right hand side, which is just the beat kernel H(t, c, e), has limit 0 as t goes to 0. This in particular implies that limtyol(t) is a piecewise polynomial in C of degree at most 2gI0+I. From this one easily deduces certain new vanishing theorems for the intersection numbers of the moduli spaces. See [Liul] and [Liu2] for the details on the explicit formula for certain intersection numbers. We remark that, to get more complete information about integrals on the moduli space M., we can consider the integral I(t) = 1029

xn;O,,

F(h)H(t, f(h), e)dh

where F(h) is some G-invariant function on G29 x 11j O,f . On one hand, as t goes to 0, I(t) has limit given by IGI

L. Pew` IZ(G)I JMQ where P denotes the function on Mc induced by F(h). This is the result of the local computation. Note that in principle the integral of any cohomology class on M,: can be written in the above form. On the other hand, by Peter-Weyl theorem, we know that F(h) can be expressed

as the combinations of the characters of G. By using the product formula for characters XL V

HEAT KERNELS, SYMPLECTIC GEOMETRY, MODULI SPACES...

535

where C',\ is the Clebsch-Gordon coefficients, we can perform the global computation to get an explicit infinite sum over P+. As an example the interested reader

may try to derive the formula by taking F(h) = f ,=I I Xµ; (x.,-.I) for k < g - I.

4. The push-forward of measures Now motivated by a conjecture of Diaconis, we consider the push-forward measure by the following map:

f : G29 --* G, f(xi, ... , y9) _ JJ(xi,YA.

i

Let dh denote the Riemannian volume of Gag, then Diaconis conjectures that the push-forward measure is given by the following formula:

XA( )dx

IGP9_I

f.dh(x) =

dX

XEP+

if the right hand side converges. Otherwise we should use the normalized limit

f.dh(x) = IGIas-Ilhnt o

XA(g1)E

AEP+

2etP-(A)dx. A

This conjecture can be proved by using the above heat kernel idea. Assume f.dh(x) = F(x)dx as forms on G, then we only need to show that

F(x) =

XX(X-1

IGIas_I

)

dag_I

AEP+

X

It is clear that we have the identity

I (t) _ f H(t, f (h), x)dh = j H(t, y, x)F(Y)dy. GZa

(^,

Note that, as t goes to 0, the right hand side has limit given by F(x), and the right hand side can be calculated by using the character relations. This immediately gives the above conjectured equality. In fact we can prove similar formula for the push-forward of the Riemannian measure by the more general map

f: G ay x i

i from which we get the following formula.

i

Proposition 3: Let dh denotes the Riemannian measure on Gag x fl . 0c; . Then the following formula holds on G ni XA(ci)

f.dh(z) = IGlae+n-a 9(c)2

i IZ., I

6X9+n-I

AEP+

XX(

a I)dz.

KEPENQ LIU

536

Here we assume the right hand side converges, otherwise we should consider the normalized limit as above. Now we want to consider a slightly different situation. Let Hj, j = 1, , n, be subgroups of G, we consider the following map

f : G"x HHj-+G,

j

j

The same argument of using heat kernel will give us an interesting formula for the push-forward measure. Indeed we consider the integral

I(t) =

JhEQ" x flt Hf

H(t, f(h),x)dh = r G

As t goes to 0, the right hand side is F(x), and the left hand side can be calculated by using the character relations, from which we get

Proposition 4: Let dh denote the Riemannian measure on G" x 11, Hj. Then the following formula holds:

Jj'

f.dh(x) = IGI°-I AEP+

XA(x( -1)dx. a

One may also consider the map

f: G29x0- G with

f (xl, ... , y9; z) = jj[xj, yj]z2,

j

as well as the map

f: (:i2'xGxG-.G with

f(xl,... ,y9;w;z) _ ll[xjyj)wzw 'z.

j

We leave these as exercises to the interested reader. In these cases one needs

the formula Ja

xa(x)dx = IGIfa

where fa = 1 if A has real structure, -1 if A has a quartenionic structure, 0 otherwise.

More generally, we can consider the problem of solving equations fj(xl,...,xm)=cj, j=1,...,n in Lie groups. To understand the measure or number of solutions of these equations, we may apply the heat kernel on the corresponding group G to the map

HEAT KERNELS, SYMPLECTIC GEOMETRY, MODULI SPACES...

537

f : Gl -+ Gn

with f = (fi,

, fn), and consider the integral

1(t) = fG.,

H(t, fi (h), cj)dh.

In many cases we can derive various interesting formulas. As an example, let us consider the map

f(n): Gn -1 G, f(n)(x...... xn) _ [XI, [x2, [... , xn]]] This is the n-commutator map. We will derive an inductive formula for the pushforward measure, Let us define Qn(x) by the identity : f1n)dh(x) = Qn(x)dx.

Then we will prove

Proposition 5: The following formsla holds:

Qn(w) = lime-,o E etna(A) AEP+

xA(g)xA(g-1w-1)Qn-1(g)dg

JG

To prove this formula, we consider the integral

J ' H(t, f(n)(h), x)dh = JG H(t,v,x)Qn(v)dy.

1(t) =

G

As t goes to 0, the right hand side is nothing but Qn(x), while the left hand side is reduced to the computation of the integrals like

f

G*

XA

([xl, [... , xn]]x 1) H dzj j=1

which, after integration with respect to x1 by using the character relation, is equal to 101

TA 1G°-t

XA([x2,

[...

, xn]])XA([x2, [... , xn]]-1x-1) 11 dxj j=2

Jd J

j=2

1y 1)Qn-1(w)dw. JdJ I XA(w)XA(w. Here the second identity is from the change of variable w = [r2, [... , X.11-

By putting all of the above formulas together, we get the wanted formula.

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KEFENG LIU

5. Counting solutions in finite groups It turns out that all of the results for compact Lie groups in the above section have analogues for finite groups. Even the proofs are basically the same, if we replace the integrals in the last section by sums. As pointed to me by P. Diacoais, the heat kernel for a finite group G is given by a similar expression:

H(t,x,y) =

G

E da Xx(xy-1)e ts.(X) aEp+

where IGI denotes the number of elements in G, P+ still denotes all of the irreducible representations which is a finite set, and pc(A) is a function on P+. In fact the only

property we need for this function is that its limit, as t goes to 0, is the delta function:

E(xy 1) = 1 E dA XA(xy-1) IGI

ep+

where 6(xy-1) = 1 if x = y and 0 otherwise. Let Ocf with cl, , cn E G denote the conjugacy class of cj in G. Let S9,,, denote the number of solutions in G29 x 11j Oc, of the equation 77 n-

JJ[zj,yu]llz,=e j=1

j=1

where zj,yj E G and zj E Ocf. This is related to the Chern-Simons theory with finite gauge group. Let S be a compact Riemann surface. For n = 0, S9,,, counts the number of elements in the set Hom (r1(S), G). We will derive a general formula for S. The special case when n = 0 was proved in [FQ] by using topological field theory. As pointed out in [FQ], such formula was actually known to Serre. It is interesting to consider some v a r i a t i o n s of the above equation. Given subgroups H I ,--- , H. in G, we may consider the equation in Gn x 11j Hj: n

rl xjwjxj' = e 1=1

where xj E G and wj E Hj. We can also consider the n-commutator equation [xl, [x2, ... , xn]] = e. In all of these cases we will give general formulas. Certainly one may try to find more general equations by combining the above equations together, or try to figure out equations of other types. As in the compact Lie group cases, all of the formulas will follow naturally from

the heat kernel trick. For simplicity and compatibility with last section, let us introduce a notation: for a finite set G and any function f on it, let us write

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539

f f (9)dg = , f (g) gEG

We start from the first problem. We consider the map

f: Ga9 x11O,,-+G with

f(T1,y1,...

n

,xg, ygi

n

xn) _ JI[xj,yj]11 zj,

zl,...

j=1

j=1

and introduce the integral, more precisely the sum,

1(t) = fagxfl

Gut

OH(t, f (h), e)dh.

(i) Local computation. As t goes to 0, the delta function property of the heat kernel tells us that the limit is exactly the number S9,n. (ii) Global computation. On the other hand we can explicitly calculate the integral by using the formulas fG Xa(wyzvz

fG

1)dz = Id G1 x.\(wy)x.\ (y-'),

XA(wy)XA(y-1)dy = G,\Xa(w),

and if Z., denotes the stabilizer of cj, foe h(9)dvg =

h(9cj9 1)dg.

1141 II

s

Here as our convention, the integral means taking sums over G, and IGI, IZ,,j I denote the number of elements in G and Z,, respectively. By putting the above two computations together, we get the following:

Proposition 6: We have the following formula: IGlae+n-1

Sg,n

n 1 XA(cj)

lilj=llZQ' I ,EP+

d2g+n-a

x

For the second equation, let us define Qn(w) to be the number of solutions in Gn for the n-commutator equation, then we will derive an induction formula which was first derived by Diaconis by using combinatorics:

Proposition 7: The following formula holds:

Qn(w) = E E X (g)XA(g lw 1)Qn-1(9) AEP+ gEG

KEFENG LIU

540

To prove this we consider the map f(x1,...,xn)_(x1,[x2,...,xn]1

f(n): Gn-+G, and the function:

I (t) = fG' H(t, f (n) (h), w)dh = fG H(t, v, w)Qn(&)d9 (i) Local computation. As t goes to 0, the limit is dearly the function Qn(w). (ii) Global computation. By using the character formulas, similar to the compact Lie group case, we get

f XA(f(h)w-1)dh= Idal fGXa(9)Xa(9 1w1)Qn-1(9)d9 GA

We then obtain the wanted formula by identifying the above two computations. For the third equation, let Rn be the number of solutions in Gn x H 1 Hj such

that, for zj E G and zj E Hj, n

II xjzjzf 1 = e. j=1

Let us consider the map n

f: Gnx[fHj-rG j=1

with

f(z1i...

,xn;zt,...

zn )

_

f n

zjzjzj 1 ,

J=1

Again we consider the function

1(t) = f * G x fl7

H(t, f (h), e) dh, H1

and perform the local and global computations. (i) Local computation. As t goes to 0, the limit is given by the Rn. (ii) Global computation. Still by using the character formulas we get the expression: n n Xa(f zjzjxj ) Hdxjdzj = IGIE 1=1 AEP.1. j 1

Q^xj[iHt

So we have obtained our next proposition:

rIj fHf XA(zj)dzj

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541

Proposition 8: We have the following expression for Rn:

Rn =

11f E: JEH; XA(zi)

IGIn_1

2

AEP+

A

More generally, we can find the number of solutions in finite groups of some general equations like f1(x1,...,xn)=e,, C, xf EG,

7=1,...,m

by considering the map

f : Gn - Gm, f (h) _ (f1(h), ... , fm(h)) and the function from the heat kernel:

f

I(t) = /

11 H(t, f1(h),c I)dh.

G1^ j=1

Then similarly we can perform the global and local computations as above to get explicit formulas. We leave to the reader as an exercise to find the numbers of solutions in the equations like 9

fl[xa,ya]z2 = e, in G29 x G j=1

and 9

fl[xi,yi]wzw'1z = e, in G29 x G x G. j=1

Another interesting problem is to consider the integrals, or sums, like

I(t) =

F(h)H(t, f (h), e)dh JGae

for some G-invariant function F on G20. As t -+ 0, the limit of 1(t) compute

E P(h)

hem

where M denotes the set of solutions of one of the equations in this section. By

expressing F(h) in terms of the characters of G, we can obtain certain explicit expression of this sum in terms of the representation data of G.

Note that the numbers of solutions of some of the above equations correspond to the Hurwitz numbers, the numbers of coverings of a given Riemann surface. In [St], the formula in Proposition 6 was used to derive the general case of the Brauer p-block conjecture.

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KEFENG LIU

6. Concluding remarks There are some other interesting cases to which one may consider to apply the above heat kernel method. In [Liu2] we tried to find invariants for knots and 3manifolds. It may be interesting to combine this method with the work in [Hu]. As one can see that, symplectic structure has played the important role in our computations, such as in the derivation of the nonabelian localization formula and the intersection number formulas for moduli spaces. Recently P. Xu has explained to me some very general constructions of moment maps to Poisson manifolds. It should be interesting to apply our method to such general situation. An interesting example is to consider the group-valued moment map as introduced in [AMM], one may consider the integral of the pull-back of the heat kernel on the Lie group G by this map against the measure introduced in [AMM]. This should give us some quite interesting formulas relating the symplectic reduction and the representations of the Lie group. More precisely this should give us formulas similar to Witten's nonabelian localization formula. I thank Prof. S. Wu for the discussions concerning this point.

References [AMM] [BL]

IN [Hu] [J1

A. Alekseev, A. Malkin, E. Meinrenken, Lie group valued moment maps,

preprint 1997. J: M. Bismut and F.Labourie, Symplectic geometry and the Verlinde formula, prerint 1999. D. Freed, F. Quinn, Chern-Simons theory for finite group, Common. Math. Phys. 156 (1993), 435-472. J. Huebschmann, Extended moduli spaces, the Kan construction , and lattice gauge theory, Topology 35 (1999), 555-596. L. Jeffrey, F. Kirwan, Localisation for nonabelian group actions, Topology 34

[Liu]

(1995), 291-328. K. Liu, Remarks on nonabelian localization, IMRN Int. Math. Res. Notices, 13

[Liul]

K. Liu, Heat kernel and moduli space, MRL Math. Res. Letter, 3 (1996), 743-

(1995),683-691. [Liu2] [St]

M [Wu]

762.

K. Liu, Heat kernel and moduli space II, MRL Math. Res. Letter, 4 (1997), 569-588.

S. P. Strunkov, On equations in finite groups and invariants of representations for their subgroups, J. Pure and App. Alg. 107 (1996) 303-307. E. Witten, On quantum gauge theory in two dimensions, Comm. Math. Phys. 141(1991), 153-209.

S. Wu : An integral formula for the square of moment maps of circle actions, Lett. Math. Phys. 29 (1993), 311-328.

Survey. in Differential Geometry, vol. 5

A Brief Tour of GW Invariants Jun Li Department of Mathematics Stanford University Stanford, CA 94305 and Gang Tian Department of Mathematics Massachusetts Institute of Technology

Cambridge, MA 02139

The machinery of virtual moduli cycle was initiated by authors in 1995-96 to attack a class of topological invariants, especially Gromov-Witten invariant. The development of this machinery went through several stages during which several people made key contribution to its development. This machinery is essential to the study of mathematical problems in super-string theory. It also played key role in symplectic topology and enumerative problems in algebraic geometry. In this article, we will lead a brief tour though the development of this machinery. We will explain the basic idea to its construction and the means to its application. We hope that this will provide a manageable account on how to apply this machinery for non-specialists. The era of using non-linear analysis to define topological invariants began with Donaldson's celebrated work of his invariants of smooth 4-manifolds. This invariant is a counting invariant that counts ASD-connections of principle G-bundles over 4-manifolds. The prototype of a similar invariant on counting rational curves in a symplectic manifold was proposed by Ruan in early 90's [Rul], inspired by Gromov's work on pseudo-holomorphic maps [Gr] and Witten's work on non-linear a-models in string theory [Wil, Wi2]. This counting began to attract attention in part because of the notion of the Quantum cohomology and the Mirror symmetry conjecture. However, its mathematical foundation was not developed until the end of 1993 when Ruan and the second author constructed the symplectic invariants and proved their basic property for the class of semi-positive symplectic manifolds. This class of invariants are now referred to as Gromov-Witten invariants. Overtime, people began to ponder if similar invariants by counting curves can be defined for any symplectic manifolds and algebraic varieties. The notion of stable morphisms was introduced by Kontsevich [Ko] to construct a compactifled moduli spaces for this purpose. In 1995, the authors succeeded in constructing such invariants of ®1999 International Pram 543

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JUN LI AND GANG TIAN

smooth projective varieties. They reported their construction in the Santa Cruz conference in algebraic geometry that year [LT3]. Their method is to construct the virtual moduli cycle of the moduli space of stable morphisms and define GromovWitten invariants as the integration of the tautological topological classes against this cycle [LT1]. They constructed such cycles by constructing virtual normal cones using Kuranishi maps of the moduli space. In early 1996, Behrend-Fantechi [BF] gave an alternative construction of such cycles. They constructed similar cones as

Artin stacks over the moduli stack. In 1996, Fukaya-Ono [FO] and the authors [LT2] independently constructed the Gromov-Witten invariants for general symplectic manifolds, after constructing the virtual moduli cycle in analytic category. Different constructions were achieved later by Siebert [Sil] and Ruan [Ru2]. Finally, the authors in 1997 [LT4] and Siebert in 1998 (Si2] proved that the algebraic construction and the analytic construction of Gromov-Witten invariants coincide in case the underlining manifold is a smooth projective variety. All these work concluded the task of constructing Gromov-Witten invariants. Research on Gromov-Witten invariants over the years has contributed, directly

or indirectly, to the progress in other mathematical research. Here we can only single out a few such areas. Among them is the Mirror Symmetry Conjecture for quintic Calabi-Yau manifold, which relates the virtual counting of rational curves to variations of Hodge structure of its mirror. After work of many, including Candelas et al [Ca], Kontsevich [Ko] and Givental [Gi], this conjecture was finally proved rigorously by Lian-Liu-Yau [LLY] in 1997. Because of the limitation of time and

space, we will not be able to cover this and others in any details, suffices to say that they include the development of Floer-homology for symplectic manifolds and the proof of the Arnold conjecture, the localization formula for equivalent virtual moduli cycles, the research for relative GW invariants and the research for Frobenius manifold structure.

We now describe the composition of this paper. The main body of this survey paper consists of three parts, in order they discuss the algebraic construction of the the GW invariants, the symplectic construction of the GW invariants and the equivalence of these two constructions. In the first part, we will describe the geometric motivation of our original construction of virtual moduli cycles. We will explain why the obstruction theory of the moduli space is an essential ingredient for this construction and how Kuranishi maps and normal cones appear naturally in this construction. We will describe briefly the notions that appear in this construction and state the main theorem. We will refer the details of this construction and other related issues to our original paper (LT1]. For the construction of symplectic GW invariants, we will present a slightly different approach as to the one presented in [LT2, LT4]. The purpose for doing this is to present a construction that requires as little smoothness as possible. We will introduce a class of weakly Fredholm V-bundle and show that they admit Euler classes, just like the usual case. The new ingredient is that to a weakly Fredholm V-bundle E -+ X, among other things, the base space X is only assumed to have a locally closed partition (called a stratification) into smooth Banach orbifolds, and the bundle E is only assumed to be smooth along each stratum. The benefit of this, as is clear for the GW invariants, is that many moduli spaces admit geometric stratifications of which their strata are easily seen to be smooth. It is to study the normal structures of these strata in the total spaces where the technical difficulty arises. Our approach thus provides a more topological construction that bypass

A BRIEF TOUR OF OW INVARIANTS

645

many of the tedious, if not impossible, technical study of singularities. We hope this will be proved to be useful in the study of similar problems in the future. In the last part, we will briefly mention the equivalence of the above two constructions.

1. GW invariant as counting invariant We begin with a brief account of GW invariants and its first construction by Ruan and the second author for semi-positive manifolds. Let (X, w) be a smooth symplectic manifold and J an almost complex structure tame to w. For nonnegative integers g, n and homology class A E H.(X, Z) we form the moduli space fitg,k (X, A) of the equivalence classes of pseudo-holomorphic maps f : C -3 X such that C is an n-pointed genus g smooth Riemann surface', f is pseudo-holomorphic and f.([E]) = A. In this article, we will use f :C -+ X to denote such a map with marked points on C implicitly understood. In case we want to stress the marked points, we will use f : D C C -+ X with D C C the divisor of the marked points.

Here we say that f : C -i X is equivalent to f: C' -a X if there is an isomorphism p : C -> C' as pointed Riemann surfaces so that f' o p = f. By choosing w and J generic, we can make 91ig,k (X, A) smooth (indeed with at most orbifold singularities) of complex dimension (1.1)

It has an obvious evaluation map ev gllg,k (X, A) -+ X h

that sends any map (f, C, zi,

, xk) to (f (xi), ... , &k)) - Then for any class [a] E H. (X), we choose its cycle representative a C X and investigate the set (1.2)

ev '(a) C `mg.k(X,A).

If we choose [a] to be a degree 2r class, the above set will be a finite set if various transversality conditions are satisfied. By incorporating the almost complex structure of X, it is standard to show that it is an oriented set. Thus we can algebraically count this set. This counting is the GW invariant of (X, w). For the moment, this counting still depends on the choice of the symplectic

form w, the almost complex structure J and the cycle representative a. These combined form the defining data of the set ev'(a). To simplify the notation, we will use A to denote this set of defining data and use ' (A) C ` g,k (X, A)A to denote the pair in (1.2) defined based A. We say A is generic if various transversality conditions hold in constructing the set 4(A). We now explain why the algebraic count #P(A) is a topological invariant. Let A be any generic defining data. Then #I(A) is an invariant if it remains constant after perturbation of A. In [RT1], Ruan and the second author showed that in case g = 0 and (X, w) is semi-positive2, then any two generic defining data Ao and Al l We defer the definition to Chapter 3. aA symplectic manifold (X,w) is semi-positive if for any pseudo-holomorphic map f :C -a X we have f. ([C]) . cl(X) > 0.

JUN LI AND GANG TIAN

546

in the same homotopy class can be connected by a generic path At such that

di(A[o,1]) = U I(At) tE[0,1]

form an oriented cobordism between.(A0) and 1)(A1). To accomplish this, they first showed that §(Alo,1]) is a smooth manifold with boundary fi(A1) - I(Ao). The critical step is to show that it is also closed. For this, they compactified the moduli space 91to,k(X, A)A, and studied the closure of I(A[o,1]) in the union of these compactifications. Relying on semi-positivity condition, they showed that (I (A(0,1)) is indeed closed, thus proving the counting #1'(A) does not depend on the variation of A. This defines the GW invariant

that counts rational curves in X. We note that the class of semi-positive manifolds includes all Calabi-Yau manifolds. Later they generalized their construction to cover the GW invariants of all genus for semi-positive manifolds [RT2]. However, their approach relied on semi-positivity condition in part because in general the compactification of g,k(X,A) is too large to prove the independence of the counting #'(A).

2. GW-invariants of algebraic varieties 2.1. Why virtual cycle and what is a virtual cycle. In this section, we will discuss the algebraic construction of virtual moduli cycle and its application to defining GW invariants of smooth projective varieties. In the following, we will use morphisms, schemes and other commonly used terminology in algebraic geometry. For those unfamiliar to such terminologies, they can simply replace morphisms by holomorphic maps and schemes by complex varieties, possibly singular.

The first step to the construction of GW invariants for general varieties (or symplectic manifolds) is to think of it as the outcome of integrating certain tautological topological classes on the moduli space over its fundamental class. For this, we need to work with the compactified moduli space. Let X be a smooth projective variety and 9W9,k (X, A) the moduli space of stable morphismss f : D C C -i X from n-pointed (arithmetic) genus g nodal curve D C C to X of degree f.([C]) = A. The notion of stable morphism was introduced by Kontsevich [Ko]. The moduli space s.1Tt9,k (X, A) admits a natural evaluation map ev. When the moduli space 9Dt9,k(X,A) has pure dimension r (in (1.1)) and the subset of maps with smooth domains is dense in fi19ik(X,A), then the GW invariant is (2.1)

,9,k(a) =

J larva.. (X,A)]

ev'(a),

a e H*(Xt,Q).

The equivalence of this with the one based on counting is apparent from Poincare duality. This approach has the advantage that even in case the moduli at9,k (X, A) has bigger than expected dimension, we can still define the integral (1.1) if we can replace [%t9,k (X, A)] with an cycle that "imitate" the fundamental class of the 3A morphism f : D C C - X is stable if D C C Is a pre-stable n-pointed nodal curve and if there is no non-trivial vector field of C vanishing at the marked points and the nodal points of C so that f. (v) = 0. An equivalent definition can be found in section 3.5.

A BRIEF TOUR OF GW INVARIANTS

547

moduli space. One way to visualize this cycle is this: We deform X to a family of varieties, say Xt with Xo = X. Then A E H2(X,Z) induces At E H2(Xt,Z), at least for small t (in analytic topology). Then'.JDte,,,(Xt, At) form a family of moduli spaces not necessarily flat. Now imagine for t 96 0 we can find Xt so that for all t 0 0 the spaces Wt,,.(Xt, At) have expected dimensions. Then the limit l m[Ni9,n(Xt, At)] when t -10 will be a cycle in H2,.(91t9,k(X,A)). This limit cycle will be the virtual moduli cycle. It is to construct this cycle the machinery of virtual cycles was developed.

2.2. Algebraic construction of virtual cycle. We now discuss the algebraic construction of virtual cycle. We begin with an example that will illustrate this construction. Let

E (2.2)

W be a section of a vector bundle over a smooth projective variety W. Let X = s I(0) be the subscheme of W. We think of W as the ambient space and s the defining equation of X. Let n = dim W and m = rank E. Since X C W was cut out by m equations, the expected dimension

exp.dimX=n-m. In case X has pure dimension n - m, we will take its fundamental class [X] its virtual cycle [X]vir. Note that this is the Euler class of E, viewed as a class in An_mW.4 In general, we demand [X]4r E A.X to be the class so that it will remain constant if we vary X by varying the section a. This leaves us the only option that [X]vir is the Euler class e(E). To make it a class in A.X, we will take the localized Euler class

eloc(E, s) E An-ms '(0) = An-mX thanks to the section 8. The localized Euler class is constructed as follows [Fu]: Let Nxxw be the normal cone to X in W. It is canonically a subcone of EIx. Let nB:X - EIx be the zero section and nj:A.EJx -+ A.X the Gysin map. Then eloc(E, a) = ii [Nxlw] E A.X. Analytically, [Nxgw] can be viewed as the limit current (cycle) of rt, when t -> 0, where r, is the graph of s in E and i [Nx1w] can be viewed as a class in H.(X) resulting from intersecting Nxxw with a generic section of Eix. This interpretation shows that i ,[NxiW] is the usual Euler class when viewed as an element in H. (W). It is this localized Euler class which we define to be the virtual cycle [X]vir To push this construction to a general moduli space, say M, we face the improbable task of embedding it naturally into some ambient space W as the vanishing locus of a section s of a vector bundle E over W. Here we stress the naturality of such choice since otherwise the virtual moduli cycle will not be a naturally defined object. For instance, if M C W is the zero locus of a section s of E, a section of E ® Cw, then M will be the zero locus of a ® 0, where 0 is the zero section of the trivial line bundle Cw. However, e(E) 96 e(E 9 Cw). To satisfies this "naturality" condition, we ask ourselves what is the canonical defining equation(s) of a moduli 4Z.W is the group of algebraic cycles and A.X is Z.X modulo the rational equivalence relation.

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JUN LI AND GANG TIAN

space. The key to this question rests on the notion of the obstruction theory and its Kuranishi map. We recall the notion of an obstruction theory to deformation of a point p in a scheme X. DEFINITION 2.1. (X, p) is said to have an obstruction theory with coefficients in V, where V is a finite dimensional C-linear space, if the following holds: For any triple (A, I, cp), where A is an Artin ring, I C A is an ideal annihilated by the maximal ideal in C A and cp : Spec All --> X is a morphism such that go(0) = p, there is a function Ob(A, I, W) E V ®C I such that (1) Ob(A, I, gyp) = 0 if and only if there is an extension gyp: Spec A -+ X of

We skip the exact wording of the base change property since it does not affect our discussion later. It can be found in [LT1]. The following example illuminates the notion of obstruction theory. EXAMPLE 2.2. Let W be a smooth scheme and X C W be a subscheme defined by the vanishing of a section s of a vector bundle E. Let p E X and (2.3)

V = Coker{ds(p) : TpW --i Elp).

Then $ induces an obstruction theory of (X,p) with coefficients in V.

There is an obvious way to construct this obstruction theory. Since W is smooth, 9: Spec All -+ X extends to -i: Spec A -+ W. Since s o pb(Spec A/I) = 0 and m I = 0, sob induces an element in EIp ®C I. We denote by Ob(A, I, (p) E V ®c I

its image in V ®cI under the quotient map EIp ®c I -+ V ®cI. It is straightforward to show that this assignment satisfies all requirements of an obstruction theory, hence is an obstruction theory to deformation of p in X. Now let (X, p) be any scheme endowed with an obstruction theory with coefficients in V. Let X be the formal completion of X along p. (X sometimes is called

the germ of X at p.) Assume n = dimTX and m = dim V. We pick a dual basis (z) = (zt, , of TX. Following Kuranishi's construction of his map, there is a pair (t, $), where (2.4)

t:X y Spec C[[z]]

and 4' E V ®C[[z]]

so that X is the vanishing locus of 0 and the obstruction theory induced by 4' coincide with the obstruction theory given. We call (t, 1) a Kuranishi map of the obstruction theory.

It is instructive to work out an explicit example. Let s(z) = 2zs + z4. It defines the subscheme X = {z3 = 0} in C. For triple A = C[t]/(t3), I = (t2) and 92 (t) = z : Spec All to X, the obstruction class Ob(A, I, ep) = 0. Obviously 92

extends to fi(t) = z : SpecA -+ X. Now let A' = C[t]/(t4) and P = (ta). We compute Ob(A', I', 'ps) = 2, the coefficient of the leading non-zero term in s. This reveals the fact that the leading non-trivial terms in a Kuranishi map is determined by the obstruction theory, and the non-uniqueness of Kuranishi maps comes from the freedom of higher order terms. In this case, any function 4'(z) = 2zs + O(z4) is a Kuranishi map.

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So far we have settled the question of finding natural defining equation of a moduli space: The moduli space should come with an obstruction theory and the associated Kuranishi maps will serve as its defining equation. The strategy to construct the virtual cycle is dear now: We will use the Kuranishi maps to construct a virtual Done in a vector bundle E and use the Gysin map of its zero section to define the virtual cycle. There remain two technical difficulties: One is that the Kuranishi maps are not unique; The second is that they are local. They only exist as formal functions. We are still in the set up of (2.2). According to the plan, we intend to reconstruct the normal cone Nxxw near p by the Kuranishi maps of the obstruction theory. Let (c, ) be a Kuranishi map of the obstruction theory of (X, p), as in (2.2). We consider the normal cone (2.5)

N, := Nx/specc[[sjj C V X X

which is canonically a subcone of V x X. The cone N, is closely related to the normal cone NX1w at p.

LEMMA 2.3. Let the notation be as before and let 1I, 42 E V 0 C[[z]] be any two Kuranishi maps of (X,p). Then there is a vector bundle automorphism ¢ of V x X (as bundle over X) that restricts to the identity homomorphism at the dosed fiber V x {p} such that .-I (N,,) = N,2.

The upshot of this Lemma is that though Kuranishi maps are not unique, the normal cones they defined are almost unique. The only ambiguity comes from those automorphisms whose restrictions to the closed fiber are the identity automorphisms.

We now investigate the choice of the vector bundle E. It is understood now that the key that makes E the vector bundle of our choice is that the sheaf

T= Coker{fl ,Ix -+Ox(E)), which has the property that at each point p E X the space T ®ox Cp is the obstruction space to deformation of p in X, is a quotient sheaf of Ox(E). We (2.6)

call T the obstruction sheaf. We remark that not all obstruction theories has this property. Those that do will be called perfect obstruction theories. DEFINITION 2.4. A Scheme Y is said to have a perfect obstruction theory if there is a perfect complex T' = [T -> T2] in the derived category 1)6(X) of which the following holds: For any triple (A, I, gyp), where A is a C-algebra, I C A an

ideal so that 12 = 0 and W : Spec A/I -f X is a morphism, there is a function E 42 (7'9 ®ox I) such that (1) Ob(A, I, cp) = 0 if and only if there is an extension 0: Spec A -I X of P. (2) If Ob(A, I, io) = 0 then the space of all extensions 'P are parameterized by

I)I(T' (gox I). (3) the function

satisfies base change property.

This definition is almost the same as Definition (2.1) except that now A can Thus this covers deformation of all points in the sense of Grothendieck. For those who are unfamiliar with the derived category, they can replace [Ti -* T2] with a complex of locally free sheaves of Ox modules E = [El -+ E2], and 4' (T ®ox I) by the kernel and the cokernel of Ei ®o,, I -+ E2 Oox I for i = 1 and 2 respectively.

be any

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Let Y be any projective scheme endowed with a perfect obstruction theory with

associated complex T'. Let E be a vector bundle on Y so that the 42(71) is a quotient sheaf of Ox (E). For any p E Y, we denote by k the formal completion of Y along p and VP the vector space h2 (T' ®r.p ), the obstruction space to deformation

of pin Y. BASIC CONSTRUCTION 2.5. Let the notation be as before. There is a cone scheme N so that for each p E Y there is a vector bundle homomorphism

0:Elf, -4VP xY that extends the given homomorphism El, - Vp so that (2.7)

NJ f

= 0-'(No).

In example (2.2), the cone so constructed is exactly the normal cone Nxiw when using the obstruction theory induced by the section s with the associated complex S11", 4X -+ Ox(E) and the vector bundle E. DEFINITION-THEOREM 2.6. Let X be any projective scheme endowed with a perfect obstruction theory with associated complex T' = [Ti -3 T2]. Then X has a canonical virtual cycle [X]°I' which is constructed according to the following recipe:

(1) We pick a vector bundle E over X so that the sheaf cohomology 42(T') is a quotient sheaf of Ox (E). (2) We form a cone N C E according to the basic construction (2.5). (3) We define the virtual cycle [X]vir = 171[N].

Here is a few remarks. First, since T' is a perfect complex with two terms, its cokernel is a well-defined sheaf of Ox-modules. Because X is projective, the vector bundle E always exists; Second, the existence of the cone N satisfying rule (2.7) does not follow from the existence of the Kuranish maps. The existence is proved by constructing a relative Kuranishi family. Once such cone exists, then it will by unique, following (2.7); Thirdly, the cycle [X]°Ir so defined is independent of the

choice of E. One caution, the virtual cycle only exists as a class in A.X. It does not have a canonical cycle representative, just like the Euler class has not canonical submanifold representative.

2.3. GW Invariants. we now show that this machinery can be applied to construct the GW invariants of all smooth projective varieties. Following the blueprint of the construction, we need to describe the canonical obstruction theory of the moduli space OJlg,k (X, A) of stable morphisms, to workout a global vector bundle that makes the obstruction sheaf its quotient sheaf and then use the Gysin map to define the virtual cycle.

Let f : D C C -+ X be a stable morphism. It associates a complex Af = [f*f x - flc(D)] (in the derived category) of degree -1 and 0. The obstruction theory off is given by the following Lemma. LEMMA 2.7. The space of first order deformation off is naturally parameterized by the extension group Extcl(Af, Oc). The deformation functor D2orf admits a canonical obstruction theory with coefficients in the extension group Ext2 (Af, Oc).

The proof of this Lemma is a standard deformation-obstruction computation, and can be found in [LT1, Rn]. Now we let (z) be a multi-variable dual to the first extension group Ext (Af, Oc) and let J be the ideal generated by the components

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of the Kuranishi map of its obstruction theory. Then over SpecC[[z)]/J there is a universal family to deformation of f . Sometimes the scheme Spec C[[z]]/J is called the universal family to deformation of f. The obstruction space fits into the exact sequence of cohomology groups

--> Ext1(llc(D),Cc) -4 H'(f`fz) -+Exta(Af,Oc) -4 0. An easy consequence of this is the following smoothness result for convex manifolds5.

COROLLARY 2.8. Let X be a smooth convex variety and f : D C C -> X be a stable morphism of which C has genus 0. Then there is no obstruction to deformation of f. It is known that the germ of T19,k (X, A) at f is a quotient of the universal family to deformation of f . LEMMA 2.9. Let Aut(f) be the automorphism group of f. Then G acts naturally on Ext' (Af, Cc) and on the universal family Spec C[[z]]/J to the deformations off . Further, the germ of 9Jt9,k (X, A) at f is canonically isomorphic to the quotient (Spec C[[z]]I J)I G.

Note that it is possible that the local universal family to deformation off is smooth while the the moduli space is singular at f. In general there are two sources for the singularities of a moduli space: One is from the non-trivial automorphism group of an object and the other is the non-trivial obstruction to deformations. One essential precondition for constructing virtual cycle is the existence of a perfect obstruction theory. When S C 9Dtg,k (X, A) is an open subset over which a universal family exists, say f :D C C -+ X, then S has a perfect obstruction theory with the associated complex (2.8)

Extc/s(Af, Cc) __!L+ Ext01s(Af, Oc).

Because there are maps f E 9Jtg,k (X, A) that has non-trivial automorphism groups, the moduli space does not admit universally families. Nevertheless, since all such automorphism groups are finite, we have the following

LEMMA 2.10. For any points f E Ttg,k(X,A) we can find a scheme S acted on by G = Aut(f) and a G-equivariant family of stable morphisms over S so that the tautological morphism SIG -3 9Jt9,k(X,A) induced by this family is an open neighborhood (i.e. open embedding) of g E Mg,k(X, A). Let f :D C C -a X be such family over S. Then S admits a canonical perfect obstruction theory with associated complex (2.8).

The Lemma suggests that we should work with pairs (S, G), called charts of 9Jt9,k (X, A), and their tautological families. When all charts S are smooth, this is the classical notion of orbifold. For us, we need to deal with singular charts as well. This leads us to the notion of Mumford-Deligne stacks or Q.schemes. (We will call it stack since we will only work with Delign-Mumford stacks.) The notion of stack is widely accepted nowadays among algebraic geometers working with moduli problems. It is less so for people working in other areas. To completely develops the notion of stacks from scratch is technically demanding. 5A complex manifold is said to be convex if for any morphism f : P1 -+ X we have A1(f'1,c) = 0.

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However, the underlining idea behind it is quite simple. It was developed exactly to study a moduli space like 9R9,k (X, A): it has a collection (Sc,, Ga), where G. are finite group acts on Sc,, so that each Sa/Ga is an open subset of 9)19,5(X, A), and they together form an open covering of 9R9,k (X, A). Each (Sc,, Ga) in this collection

is called a chart of the stack. The informed readers will now suspect that there should be compatibility condition of (Sa,Ga) and (Sp,Gp) in case Sc, fl Sp 54 0,6 etc.. This is certainly the case. For brevity we will not state them here but instead say they satisfy the obvious compatibility condition. The upshot of using charts (Sc,, Ga) instead of open subsets Sa/Ga is that the former carry certain important data of which the later does not. For instance, an ordinary orbifold has smooth charts but not smooth neighborhood. Our moduli space 9)19,k (X, A) has charts carrying tautological families. We are now ready to construct the virtual moduli cycle of 9)19,k (X, A). We pick

a collection of charts of 9R9,k(X,A), say {(Sc,Gc)}, with fa:Da C Ca -+ X the tautological family over Sc,. The associated complex of the obstruction theory of fa is Ext%/s. (Aa, Os.), where A. = [f .*X - f1c,. (D.)]. Following the recipe, it remains to find a global vector bundle over 9R9,k (X, A) that makes the obstruction sheaves its quotient sheaves. Working with stack, it amounts to find a collection of vector bundles Ea over Sc,, each is Ga-equivariant, and Ga-equivariant quotient homomorphisms Osa (Ea) --+ Ext2C.15, (Aa, Oc ),

that are compatible over the intersections Sc, fl Sp. In our case, such a vector bundle can be constructed explicitly [LT1]. We still denote it by {Ea}. Then from the Basic Construction 2.5, we can construct (virtual normal) cone cycle Na C Ea. Since such construction is unique, Na C E. are Ga-equivariant and are compatible. In the end, we need to intersect this collection of cone-cycles [Na] E Z.Ec

with the collection of zero sections of Ea. One way to do this is to descend E. to a V-vector bundle on 9R9,k(X,A) which contains the descendents of [Na] as a subcone cycle. The other is to view the collection E. as a vector bundle over the stack 99l9,k(X,A). We denote E the collection {Ea} either as a V-vector bundle or a vector bundle over a stack. We denote by [N] C E the corresponding virtual normal cone cycle. As before, if we let rj be the zero section of E, then the virtual moduli cycle should be ['WZs,k(X,A)]"' = i& ([NJ) E A.9)la,k(X,A),

where 0% is an appropriately defined Gysin homomorphism of the zero section iM. Such Gysin homomorphism is known to exists [Vi]. The result of this Gysin homomorphism takes values in the group of cycles with rational coefficient. Once we have the virtual moduli cycle, we define the GW invariant to be

a E A`Xk. lA,9,k(a) =< [9R9,k(X,A)]"Ir,ev*(a) >E Q, This completes the task of constructing the GW invariants of smooth projective varieties.

3. Symplectic GW invariants The main purpose of this chapter is to construct the virtual moduli cycles for general symplectic manifolds. Accordingly, in this chapter we will work with 6The intersection is So x! is ,,(X,A) Sp in Grothendieck's topology.

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553

analytic category. For instance, we will view the moduli space of k-pointed genus g stable curves 9Ji,k, 2g + k > 3, as a compact orbifold. Let X be a smooth symplectic manifold with a given symplectic form w of complex dimension n, and let A E H3(X, Z). We state the main theorem of this chapter. THEOREM 3.1. Let (X,w) be a compact symplectic manifold of complex dimen-

sion n. Then for each g, k and A, there is a virtual fundamental class eA,9,k(X) E Hr(ll9,k X Xk, K), where r = 2c3(X)(A)+2(n-3)(1-g)+2k. Moreover, this eA,9,k(X) is a symplectic invariant.

As an application, let us define the GW-invariants now. Let 2g + k > 3. We define (3.1)

*A,9,k : H*(99,k,Q) x H'(X', Q t-4 Q,

to be the integrals (3.2)

oA g,k (Q, a) =

JA.o.e (X )

pr* /3 A pre a

where /3 E H* (9X9,k, Q), a E H* (X', Q) and pr; is the projection of 9ng,k x Xk to its component. All ryA g,k are symplectic invariants of (X, w). To construct such cycles, we will introduce the notion of smoothly stratified orbispaces, weakly pseudocycles and weakly Fredhohn V-bundles over any smoothly stratified orbispaces. This is what we will do in the first part of this chapter. After this we will sketch the construction of the virtual moduli cycles for weakly Fredholm V-bundles. Finally, we will apply this construction to stable maps and consequently prove the above theorem.

3.1. Smoothly stratified orbispaces. In this section, we introduce a class of topological spaces that admit a class of stratified structures. In this note, all topological spaces are Hausdorff. We first introduce the notion of smoothly stratified spaces. DEFINITION 3.2. By a smoothly stratified space we mean a topological space X with a locally finite? partition X = U.E1 X, , called a stratification, by locally closed subsets Xa, called strata, such that each Xa is a smooth Banach manifold.

Note that our definition differs from the usual one in that we do not require knowledge of the normal cone structure along each stratum X0. Let X and Y be smoothly stratified spaces. We will denote by Xt*P the topological space X without

the stratification. A map f : X -+ Y between two smoothly stratified spaces is a map f : PIP .. lrtop such that it maps strata of X into strata of Y. f is smooth if f Ix are smooth. A smooth manifold is viewed as a smoothly stratified space whose stratification consists of a single stratum X. In the following we will call smoothly stratified space simply stratified space if no confusion will arise. Next we introduce the notion of smoothly stratified orbispaces and their local uniformization charts. We let X be a topological space with a locally finite, locally closed stratification X = VaE1Xa'By this we mean for each a E X the set to E 1 I a E Xa } is finite.

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DEFINITION 3.3. Let X be as before. A local uniformization chart of X consists of a smoothly stratified space U = UaerU,, a finite group Gu acting effectively on

U by smooth maps and a GU-invariant continuous map iru : U`P - Xt0P such that (1) U = nu(U) is an open neighborhood of X; (2) each stratum Ua of U is invariant under Gv; (3) iru induces a covering map from U/G{I onto U as stratified spaces that respect

the induced stratum U/G{I of U/GU and U n Xa of U. Again, if there is no confusion we will call (t, GO) a chart and call U the quotient of the chart (U,GU). Let V C U be an open subset, then (irj1(V),Gij) defines a new chart of X, called the restriction of (U, GO) to V. For simplicity, we will often denote it by (U, GU) Iv. When p E U, we sometimes call (U, GU) a chart of p. Now let X be as before and let (U, Gu) and (Cl, Gf,) be two charts of X. We say the later is finer than the former if there is a homomorphism GV -+ Gtr and an equivariant smooth covering map Wvv belongs to the square

TV' (U n v) (3.3) RV l

UnV

f22+ irj' (U n v) 9ru l

UnV

Note that if U n V = 0, then U is automatically finer than V. We now define the notion of smoothly stratified orbispace. DEFINITION 3.4. We say that X is a smoothly stratified orbispace if it can be covered by a collection of charts as in the Definition 3.3 such that for any two charts

(Ui, GU,) and (Of, Go,) and x E UinU there is a chart (V,Gc,) of x E X such that (V, GV) is finer than both (Ui, GU.) and (Uj, GUa ). Later, we will abbreviate smoothly stratified orbispaces to SS-orbispaces. Clearly,

each smoothly stratified space is a SS-orbispace. A smoothly stratified space is smooth if all its charts, namely the Ui in the Definition 3.4, are smooth. In particular, a finite dimensional smooth smoothly stratified space is an orbifold. Note that according to this definition, each stratum X. of X is a Banach orbifold. If X and Y are two SS-orbispaces, a map 1: X -+ Y is a map f tOP : X EOP - YtOP

such that for each x E X there is a chart (U, GU) of x E X, a chart (V, GV) of f (z) E Y, a homomorphism Gv - Gc and a Gv-equivariant map f : U - V such that it descends to ft°P1u:U -+ V C X. We call such % a local representative of f. We say f is smooth if all its local representatives are smooth. A smooth map f : X -4 Y with smooth inverse f -I is called an isomorphism.

3.2. Weakly pseudocycles. In this subsection, we will work with a continuous map f : X - Y between two topological spaces X and Y. All homology theories are with rational coefficients. We begin with pseudomanifold. An oriented pseudomanifold S of dimension

d in X is a pair (p, M), where M is an oriented, smooth manifold of dimension d with boundary OM with corners and p : M -+ X is a continuous map. For a sA manifold of dimension d with boundary with corner is is a manifold with boundary except that the neighborhoods of a boundary point are isomorphic to the neighborhoods of a boundary

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pseudomanifold S = (f M) we define OS to be (f, 8M). We define -S to be (f, M), where M is M with the reversed orientation. We define the sum of two pseudomanifolds (f, M) and (g, N) to be (f JJ g, M U N). We then can define the sum E niai, where ni E Z, of pseudomanifolds inductively. For any (f, M), we say it is 0 if there are two disjoint open subsets UI and U2 of M so that the complement of their union in M is an at most d - 1 dimensional set°, such that there is an orientation reversing isomorphism p : UI -+ U2 so that flu, = f o p. We can talk about rational pseudomanifolds, which are formal sum of oriented pseudomanifolds with rational coefficients. A rational pseudomanifold is zero if an integer multiple of it is integral and is zero. Let f :X - Y be a continuous map between topological spaces. We now define weakly f-pseudocycles. DEFINITION 3.5. A rational weakly d-dim'l f -pseudocycle is a triple (p, M, K),

where (p, M) is a rational pseudomanifold in X, K is a closed subset in Y of homological dimension1° no more than d - 2 such that the closure of M - (f o p)-1(K) is compact in M and the rational (d - 1)-dimensional pseudomanifold (p, OM - (fop)-1(K)) is 0. Two such cycles (pi, M1, KI) and (p2, M2, K2) are quasi-cobordant if there is a (d + 1)-dimen8ion4l pseudomanifold (p, M) in X, a closed subset K C YtOP of homological dimension no more than d - 1 such that M - (f o p) -1(K) is compact in M, KI and K2 are contained in K and (PI, MI) - (p2, M2) = (p, OM). Two such cycles S and S' are equivalent if there is a chain of weakly f -pseudocycle So,

, SI

so that S = So, Si is quasi-cobordant to Si+i and St = S. The usefulness of this definition is illustrated by the following Proposition. PROPOSITION 3.6. Let (p, M, K) be a weakly d-dim 'l f -pseudocycle. Then f o

p : M -+ Y defines a unique element in Hd(YtO9, K) = Hd(YtOP). Further, two equivalent such cycles define identical elements in Hd(Y).

PROOF. First, the two homology groups are canonically isomorphic since K has homological dimension < d - 2. Now we show that f o p defines a cycle in Hd(YtOP, K). Since M is smooth and M - (f o p)-1(K) is pre-compact in M, we can give M a triangulation so that the boundary of f o p(M) is contained in K, thus it defines an element in Hd(Y,K). For the same reason, two quasi-cobordant and thus two equivalent such cycles define identical elements in Hd(Y).

0

3.3. Weakly Fredholm V-bundles. In this section, we introduce the notion of weakly Ftedholm V-bundles over a SS-orbispace X = UacrXa. We assume throughout this chapter that 0 E I and the stratum Xo is dense in X. Also, if U is a chart of X, then U has a stratification UOEJUO so indexed that Ua corresponds to the stratum U fl Xa. DEFINITION 3.7. Let X be an SS-orbispace. A V-bundle over X is a pair (E,P), where E is an SS-orbispace andP:E - X is a projection (of SS-orbispaces) such that X ( re a p . E) is covered by a set o f charts (ti, G t , , ) ( re a p . (E D,, Gtr,)) point of L = {xj >- 0, , xk > 0}. The boundary of such manifold is also a manifold with boundary with corner, following the convention that the boundary of L is the disjoint union of k manifolds L n {xi = 0). 98y this we mean it is the image of d -1 dimensional manifold. 10By this we mean max{k; Hk(K,Q) 0 0}.

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of which the following hold:

(1) If Ui is a chart over Ui C X then Efj, is a chart over P-1(Ui) and there is a projection PU, :EU. -+ ti representing P over Ui. (2) Let Ua be any stratum of Ui, then restricting to Ua the bundle E. IGa is a smooth Banach vector bundle over Ua with a Gut -linear action. Further the stratum EU, are pull-back of the stratum of U. (3) For any two charts (ti, Gut) and (%, GU,) of X (over Ui and Uj respectively) and x e Ui fl Uj, there is a chart Mk, GGa) of x finer than both (ti, GG, ) and (U,, Ge, )l such that there are Gu-equivariant isomorphisms

EUblunu -`'Pi7uUt(En,) and EU,,lu,nUb -`pir,u,(Euf) preserving the Banach bundle structures along each stratum, where lPurtlc is the map given in (3.3). (4) Let OEO, be the zero sections of EU., then they define a SS-orbi-subspace OE of E. We require that it is isomorphic to X under the projection P.

Now let E be a V-fiber bundle over X. A section of E is a map $ : X - E as SS-orbispaces so that P o 1: X -+ X is an isomorphism. We say $ has compact support if '-1(OE) as a subspace in Xt°P is compact. We fix a V-bundle over X with a section 1) with compact support. We now describe the weakly F4edholm structure of this bundle with section in terms of its local finite approximations.

DEFINITION 3.8. A local finite approximation of (E,P,'F) consists of a chart

(U,Gu) of X, a chart (EG,GG) of E, a representative 'Fu:U --Eu of (I and a finite rank equi-rank G0-linearized vector bundle F over U such that: (1) F is a Gu-equivariant SS-subspace of EU. Further restricting to each stratum Ua C U the inclusion Flua C EUlU, is a smooth sub-vector bundle. (3) U := 4):'(F) C U is a finite dimension equi-dimension SS-orbispace (with

strata U = UaEIUa). Further we require Uo = U fl to is dense in U and its complement has codimension at least 2 in U; (4) F := Flu is a continuous vector bundle. Further its restriction to each stratum

U. it is a smooth vector bundle and the map 4Flu,, where OF ='Fulu:U.- F, is a smooth section;

(5) At each to E 'F'1(0) n to, the differential d'F(w) : TTUo -+ EUOIw / FIw is surjective.

Such local finite approximation will be denoted by (U, GU, Efj, F). An orientation of (U, GU, Efj, F) is a Gu-invariant orientation of the real line bundle AtoP(TU)®ASOP(F)-1 over Uo. Such orientations are given by Gu-invariant non-vanishing sections of A'OP(TU) ® ALOP(V)-1 over Uo.

We call rank F - dim U the index of (U, GU, EU, F). Now assume that (U', GU EU,, F') is another local finite approximation of identical index over U' C X.

DEFINITION 3.9. We say that (U', GU EU F') is finer than (U, GU, EU, F) if we have the following:

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557

(1) (U', Gu,) is finer than (U, Gtj). (2) Let tp : irui (U fl U') -4 irjl (U n U') (which is Wuu, in (3.3)) be the covering map, then tp*F C ip*EU =_ Efj,Iu'nu is a smoothly stratified subbundle of F. (3) Let (U, F) and (U', F') be those defined in the Definition 3.8 for U and U' respectively, then for any w in Uo n tp'1(U), the natural homomorphism TWU'ITW(.)U --> (F'Irp*F)I. is an isomorphism.

(4) In case both (U, Gu, Efj, F) and (U', Gu Bu,, F') are oriented, then we require that the orientations of (U, Gu, Efj, F) and (U', G0,, Ejj,, F') are isomorphic through the isomorphism AtOP(TwU')

0 AS01'(F/I w)-1 2 AtOP(Tp(w)U) 0 At01(FIp(w))_1>

induced by the isomorphism in (3), where to is any point in Uo n tp-1(Uo). Note that from (5) of the Definition 3.8, Uo nw-1 (U) is a locally closed smooth submanifold of codimension rank F' - rankF in U'. Hence the homomorphism (3) in the above Definition is well-defined. Now let 21= {(Ui,Ge., Eu.,F;)};E, be a collection of oriented finite approximations of (X, E, f). In the following, we will denote by U; the open subsets of X such that A; = (Ui, Gu.) is a uniformization chart over Ui. We say 21 covers 4-1(0) if 4-1(0) is contained in the union of Uj in X.

DEFINITION 3.10. An index r oriented pre-weakly Fredhoim structure of the triple (B, E, f) is a collection 21= {(Ui, Gu,, Eut, Fi)}iEx of index r oriented local finite approximations such that 2 covers t-1 (0) and that for any (Ui, Gu,, Eu,, Fi ) and (U1, G%, B01, F1) in 21 with p E Uj n U1, there is a local finite approximation (U, Gjj, ED, F) E 21 such that p E U and (U, Gjj, Efj, F) is finer than both

(Ui,G(jt,EC,,,Fi) and (U.i,Gu;,Eu;,F1) It follows easily from this definition that for any number of local finite approxi-

mations {(Ui,G,,E,, Fi)}1
V-bundle if it admits an oriented pre-weakly Fredholm structure such that is compact and is contained in finitely many strata of X.

-1(0)

We will abbreviate a weakly Fredholm V-bundle to WFV-bundle. DEFINITION 3.12. Two WFV-bundles (X, E, 4)) and (X, E', V) are homotopic

if there is a WFV-bundle (X x [0,11,1,12) such that it restricts to (X, E, f) over X x {0} and to (X, W, V) over X x {1}.

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3.4. Construction of virtual moduli cycles. The following is the main result on constructing virtual moduli cycles. Let X be any SS-orbispace, let Y be a smooth orbifold and f : X' - Y°Op be a smooth map. We index the stratification X = UaE/Xa and Y = UaEjYa so that

f(Xa) CYa. THEOREM 3.13. Let (X, E, f) be a WFV-bundle of relative index r. Let 21 be the collection in the definition of WFV-bundle. We assume further that for any (U, Gu, Efj, Fij) in 21 and U = UaEIU,, the stratification, the representative fu,. : U. -a Ya is a submersion. Then we can assign to (X, E, 4) an equivalence class of weakly f-pseudocycle e(X, E, i), called the Euler class of (X, E, f). It depends only on the homotopy class of (X, E, il). Further, the image of this class under f. is a well-defined homology class in H,(Y,Q).

The class e(X, E, 1) or its image under f. is the virtual cycle we set to construct. Sometimes it is desirable to include the case where Y is a more general topological space. For instance if we want the Eider class to be a class in H,(X°op), then we can take f = idX : X -a X and hence Y = X will be an infinite dimensional space. The proof of Theorem 3.13 can be generalized to such case if f : X - Y satisfies certain topological condition. However, when Y is infinite dimensional, we have not found a satisfactory condition that is general enough to include many interesting examples. We will leave this to our future investigation. In the remainder of this subsection, we will briefly sketch how to use the WFV structure of (X, E, 1)) to construct the required Eider class. The complete account of this approach is a modification of the construction in [LT2, LT4] and will appear elsewhere [LT5].

The idea of the construction is as follows: Given a local finite approximation (U, G0, EU, F) over U C X, we can associate to it a finite dimensional model consisting of a finite dimensional SS-orbispace U, a smoothly stratified finite rank

V-bundle F over U and a smooth section 0: U - F. Following the topological construction of Eider class, we like to perturb 0 to a new section ¢: U -r F so that ¢ is as transversal to the zero section of F as possible. Here the difficulty with this is that F is only smooth along, not near, strata of U. Thus the notion of transversality is ill-defined near points where F is not smooth. Here is our solution: We find a homological dimension r - 2 set K C U so that we can perturb 0 to qi so that ¢ is transversal to the zero section of F away from K. Since the dimension of the zero locus of ¢ is r, the set K will not affect the construction of an r dimensional cycle. In the case we work with many local finite approximations, we will choose these perturbations so that their vanishing locus patch together to form a well-defined weakly f-pseudocycle in X. The Eider class of (X, E, is the equivalent class it represents. As is clear from this description, the main difficulty of this construction is to make sure that these perturbed vanishing locus patch together. We will do perturbation one chart at a time, and in the mean time make sure that the perturbation on the current chart agrees with the perturbations chosen in the preceding charts. One difficulty arises in that the intersection of charts are not open. However if the current chart is finer than all the preceding charts, then the intersection will be an open subset in the preceding charts and a locally closed subset in the current chart. Hence our first order is to pick a good set of charts to begin with. Let

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Gi, Ei, Fi)}ieK be the weakly smooth structure of (X, E, 4 ). For simplicity, we will denote the projection 1ru, : ti -4 Ui by iri, denote §-1 (Pi) by Ui, denote Fi1U; by Fi and denote the restriction §iJu, : U1 -+ EiJU, by Oi: Ui --3 Fi. Without loss of generality, we can assume that for any approximation (Ui, Gi, Ei, Fi) E 21 over Ui and any U' C Ui, the restriction (Ui, Gi, Ei, Ft)Ju, is also a member in 21. The following lemma enables us to pick a good set of charts.

LEMMA 3.14. There is a finite collection Z C K and a total ordering of .C of

which the following holds:

(1) The set to-1(0) is contained in the union U{Ui I i EC};

(2) For any pair i < j E C, the approximation (U3, G E F3) is finer than the approximation (U1, Gi, Ei, F1).

PROOF. We now outline the proof, which is elementary.

We will use the stratified structure of (X, E, 4;) as we did in [LT2]. Let X

a E lo be all the strata of X that intersect x-1(0). We give an order to to so that for each a E Io, P'1(0) fl Up>c, Xfl is compact. For convenience, we assume

Io = {1,... , k}.

To prove the lemma, we suffice to construct subsets C' C K, where a E lo and

0 < l < Ia, such that Al: For any distinct i, j E C,, Ui fl U3 = 9;

A2: For each aEIoandl
ZQ = 4-1(0) fl X. - u3>, U9ec; Up is contained in the image of finitely many manifolds of dimension less than I in X.,; A3: For any pair of distinct (i, j) E Ca x Cx., with (a, l) < (a', l'), i.e. a < a' or

a = a' and I < I', the chart (Ui, Gi, Ei, Fi) is finer than (U3, G3, E3, F3). In particular, all Ui with i E Uo<xa, C' cover $'i(0) ft X.. We will construct Ca inductively, starting from the largest a E Io. Note that C, may be empty. We now assume that for some a E lo we have constructed C«+I, C*' for all 1. We now construct C',. We first pick a finite C' C K such that {Uili E C'} covers Z«+I. This is possible since it is compact. Let 1. be the maximum of {dim(Ui f1 X,z)Ji E C'}. By assumption, l > 0. Since Ui is a SS-space +'-

with strata ir,'1(Ui fl X.), we can find open subsets V. C Ui such that (1) let fill = r,-' (Ui) then U{ fl 7r{ '(Xa,) is the same as the closure of t, n ir, 1(X.) in 7r{ 1(X ); (2) Let R = U; fl Ui fl ir; then R- - R (in Ui fl a; 1(Xc,)) is a smooth submanifold in Ui fl 7r{ ' (Xc,) of dimension less than la. After fixing an ordering on C', we define C-, to be the collection of charts (3.4)

(Ui, Gi, Ei, Fi) Iu ui
i E V.

Clearly, C- satisfies AI - A3. Now we assume that C; ,

, 4, have been con-

structed. We want to construct Cl I. For each x E Z',,, we can find a finite approximation (U1, Gi, Ei, Fi) such that it contains x and it is finer than all approximations in Up>iCa. We denote this collection by C` again. Since Z' is compact, we can pick a finite subcollection, still denote by C', such that {U1Ji E C'} covers Z. By our choice of the charts, Z. is contained in the image of finitely many submanifolds in X. of dimension less than 1. Next we choose U. C Ui such that ir{ 1(U;), i E C',

still cover Z;, and that if we let R = r; 1(U,) fl ZI., then R- - R is a finite union

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of smooth submanifolds of dimensions < 1- 1. Now we fix an ordering on C', as before, and define C` 1 to be the collection (3.4). By continuing this procedure for , 0, we obtain the desired C1 for 0 < I < I. This completes the proof of 1 - 2, the Lemma.

We now briefly sketch the strategy of the proof of the Theorem 3.13. Let (U;, G;, E;, Ft)0Ez be the collection of local finite approximations given by Lemma

3.14. We denote by O, : U; -+ F; the corresponding finite models. As stratified set U1 = UaEJ,UU,., where Io is the ordered index set in the proof of Lemma 3.14.

Since x'1(0) is compact, we can find open *, C U; with ir, 1(iri(*5)) = *5 such

that U{iri(*) I i E C} covers f-1 (0) and %n U; is precompact in U;.11 Note that by assumption, we can assume that for each a E Io, flu{,, : U, ,, -4 Y,, is a submersion.

Before we perturb the sections ¢i, we need to clarify by which we mean a perturbation is generic. We begin with the general situation. Let p: U -> V be a smooth map between two open smooth manifolds. Let K C U be a pre-compact subset and s E ru(E) is a smooth section of a smooth vector bundle over U. We say t is a p-generic perturbation of s relative to K with compact support if the following

holds. (1) The support of s - t is pre-compact in U and (2) there is a precompact open neighborhood K,bfa of K- C U so that the graph of t is transversal to the zero section in K'bha and p: Knbha n t-1(0) a V is a strong immersion. Here by a strong immersion p: A -a B between open manifolds we mean p is an immersion and further for any x, y E A so that p(x) = p(y) then p*T1A is transversal to p*TTA.

LEMMA 3.15. Let the situation be as before. Suppose dim U- rank E < dim V and f : U -4 V is a submersion. Then for any pre-compact K C U and s E ru(E), the space off -generic perturbations of s relative to K with compact support, denoted

Bo, is dense and open in the space B = {h E Fu(E) I h - s is compact} with Whitney COO topology.

PROOF. The proof is an easy application of the ordinary transversality theorem. It is clear that Bo C B is open. It remains to show it is dense. First, by the usual transversality theorem we can find a small perturbation t of s with compact support t - s such that the graph oft is transversal to the zero section in a neighborhood of K-. We then can find a diffeomorphism co : U -+ U sufficiently close to Idu with {x E U I v(x) # x} precompact in U so that p: W(t'1(0)) -+ V is a strong submersion in a neighborhood of K-. We pick an isomorphism ¢: E cp*E so that it is close to E E and away from a compact set L D K of U the isomorphism ¢ reduces to E E. Here of course we assume (PI U_L = Idu_L. Then the corresponding section (t) is a small perturbation oft and is in Bo. This proves the Lemma. We now show how to construct the perturbations of O; one at a time. Without

loss of generality, we assume r < dimY, since otherwise H,(Y) = 0. We begin with the first member in C, which is 1. To make the notation easy to follow, we denote the finite model q51 : U, -4 F, by O: U -+ F, W1 C U, by W and U,,, by Ua. We first look at the largest a E Z so that Ua # 0. Assume it is not dense in U. Recall f : X -+ Y is the map given and f : U. --4 Ya is a submersion. We then pick a small f-generic perturbation of ¢1,, relative to Ua n W- of compact 11 By this we mean its closure is compact.

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bbl

support, denoted by ¢, : U. -+ Ya. Since U, is not dense in U, by assumption, it has codimension at least 2. Thus dim f(o.-I(0) n W-) < r - 2.

Since Y is a smooth orbifold, we can find a closed neighborhood La C Y of f (¢a I(0) n W-) such that it has homological dimension < r - 2 and its boundary is a smooth orbifold in Y. Let Ma = f-I(La). Let B E G is the next largest element so that Up # 0. We next extend 4,a to Up. Since the vanishing locus of ¢a in W' n Up is contained in the interior point of m. nUp, we can extend ¢a to a neighborhood of Ua C U. UUp so that its zero locus in W is still in M. n (Ua U Up). Hence, we can find a small perturbation of OIu,uu,,

so that it is an extension of 0 it is an f-generic perturbation of ¢juuv, relative

to * - M. with compact support. Let the new section be 0,6 E rt,uu,,(F) Hence f : op I(0) n Wa -+ 1' is a strong immersion away from La. Without loss of generality, we can assume that f (O I (0)) intersects the boundary of La transversely, in the sense of orbifold. Since it has dimension < r - 2, assuming Up is not dense either, we can find a closed strong retraction neighborhood L of f (0i I (0) n W-) - La in Y. Hence La U L.'8 will have homological dimension < r-2

as well. We let Lp = L. U .V# and Mp = f-' (Lo). Continue this procedure, we can find a dosed neighborhood L C Y of homological dimension < r - 2 and all f-generic perturbation of 0 relative to W n U - f-I(L) of compact support. Let the perturbed section be Bbl. Let Ul C U = UI be a pre-compact neighborhood containing U n W- so that is transversal to the zero section in U1 - L. Let nI = the product of the number of the sheets of the covering UI/G2 -i UI with

the order of GI. We define Al = (1/nI)[irI( i I(0))j, considered as a rational pseudomanifold.

Next, we continue this procedure to U2. Let U12 be jpT, (UI), where F12 is defined in (3.3). Then 1p12 (UI n Ul - LI) is a smooth submanifold of U2. Hence, we can extend sv12 (0) to a neighborhood of R of U12 C U2, say 4', so that it is a small

f -generic perturbation of 02 relative to R- f- I (L) with compact support. We then extend this small perturbation to whole U2, say 2, so that there is a subset L2 C Y of homological dimension < r - 2 so that ¢2 is an extension of 912(5 I), it is an f-generic perturbation of 02 relative to WZ I(L2) with compact support. Let A= I (0))]. Clearly, AI = A2 when restricted to Ul n U2 - f-I(L2) Continue this procedure until we reach the largest ,n e G. We let L. C Y be the corresponding closed neighborhood of homological dimension < r - 2. Then by our construction, for each i, j E L,

Aiju,nui-f-'(L,,.) = Ajju,nuf-f-'(L.n) as pseudomanifolds. We let A be the patch together of these Ai. Since {*i I i E G} covers 4-I(0), if we choose the perturbations 4i sufficiently close to 4i for all i, then the pair (A, L,,,) will be a weakly f-pseudocyde. This cycle is the Euler class

of (X,E,.). 3.5. Stable maps. In this section, we review the definition of stable maps. Let X be a smooth symplectic manifold with a given symplectic form w, and let A E H2(X,Z) and let g, n E Z be fixed once and for all. In the following, by

562

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a k-pointed prestable curve, we mean a connected complex curve with k marked points and only normal crossing singularity. We recall the notion of stable C2-maps [LT2, Definition 2.11. DEFINITION 3.16. A k -pointed stable map is a collection (f, E, xi... , xk), where

(£, xi,... , xk) is a k-pointed connected prestable curve and f : E -+ X is a continuous map such that (1) the composite f o it is C2 -smooth, where 7r: E -1 E is the normalization of E and

(2) any component R C E satisfying (f o w)*([R]) = 0 E H2(X,Z) must have 3 > 2g(R)+ the number of distinguished points on R, where the set of distinguished points on E consists of all preimages of the marked points and the nodal points of E.

For convenience, we will abbreviate (f, E, xi,... , xk) to (f, E, (xi)) or to (f, £) or simply to f, if no confusion will arise. We call E the domain of f, with marked

points understood. Two stable maps (f, E, (xi)) and (f', E', (x,)) are said to be equivalent if there is an isomorphism p: E -+ E' such that f' o p = f and x, = p(xi).

When (f, E) _ (f', E'), such a p is called an automorphism of (f, E). We will denote by Aut f the group of automorphisms of (f, E). We let BA s k be the space of equivalence classes [f, E] of C2-stable maps (f, E) such that the arithmetic genus of E is g and f*([E]) = A E H2(X; Z). With X, A, g and k understood, we will simply write BAs L. as B. There is a natural topology on B, which we will define at Section 3.7. However, B is not a smooth Banach manifold as one hopes. It does admit a stratification with smooth strata, which we now describe. Given any almost complex structure J compatible with w, one can define a generalized bundle E over B as follows. Let (f, E) be any stable map and let

f : E -+ X be the composite off with it : E -+ E. We define A1 to be the space of all Ci-smooth sections of (0,1)-forms of t with values in f *TX . Here, by a P TX-valued (0,1)-form we mean a section v of T t ® P TX over t such

that J J. v = -v j, where j denotes the complex structure of E. Assume that (f, £) and (f', £') are two equivalent stable maps with the associated isomorphism p : E -+ E', then there is a canonical isomorphism A f i A f'i. It follows that the automorphism group Aut f acts on Ao'i and the quotient A o" / Aut f is independent of the choice of the representative in the equivalence class of (f, E). So we can define A0f1 to be Af,i/Autf. Put

E= U A. (flea It has an obvious projection P : E -+ B whose fibers are finite quotients of infinite dimensional linear spaces. There is a natural map

f r: B -* E satisfying P o f j= IdB, defined as follows: For any stable map f, we define 4'j(f) to be the image of df + J J. df j E A0 j' in A0f A. Obviously, we have oli(f) = 4J(fl) if f and f' are equivalent. Thus ' j descends to a map B -+ E, which we still denote by ' j.

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We will denote by MX,, ,k the moduli space of J-holomorphic stable maps with homology class A, having k-marked points and whose domains have arithmetic genus g. Clearly, MIA g k

=,P-'(0).

3.6. Stratifying the space of stable maps. In this section, we give a natural stratification of B = BA 9,n and study its basic properties. Given a stable map (f, E), we can associate to it a dual graph r f as follows: Each irreducible component Ea of E corresponds to a vertex va in r f together with a marking (ga, aa), where ga is the geometric genus of E. and as is the homology class fa([Ea]) in X; For each marked point xi of E. we attach a leg to va; For each intersection point of distinct components E. and E,6 we attach an edge joining va and vv and for each self-intersection point of E. we attach a loop to the vertex va. In the following, we will denote by Ver(r) the set of all vertices of r and Ed(r) the set of all edges of F. Clearly, the dual graph rf is independent of the representatives in [f], so we

can simply denote the dual graph of [f] by rlfl = rf. Moreover, the genus g of [1] is the same as the genus of rl fl, which is defined to be the sum of ga for all a E Ver(r) and the number of holes in the graph rl fl. Also, the homology class a of f.([E]) is the sum of all aa, which is defined to be the homology class of the graph r[fl Given any graph r with genus g and homology class a, we let B(r) be the space of all equivalence classes [f] of stable maps in B with rl fl = r. Clearly, B is a disjoint union of all B(r). The main result of this section is the following. PROPOSITION 3.17. For each dual graph r the space B(r) is a smooth Banach orbifold and the restriction of E to B(r) is a smooth orbifold bundle. Fhtrther the restriction of §i to B(r) is a smooth orbifold section.

PROOF. We first prove that B(r) is a Banach orbifold. It suffices to construct the neighborhoods of [f] in B(r). Let f = (f, E, (x;)) and let E be the normalization of E. Then (3.5)

t=UIt,i.EVer(r)

where to is the normalization of the component Ea corresponding to a. Recall that the distinguished points of to are preimages of marked points and the nodal points of E. Note that each edge e E Ed(r) corresponds to a node in E, thus has two distinguished points in t. We denote them by zei and zee. To avoid any confusion, we denote by EEOP the underlining real 2-dimensional manifold of t. A complex structure on Et0P is given by an almost complex structure which is a homomorphism j : TEt°P y TEt°P with ja = -Id. Two almost complex

structures j and j' give rise to the same complex structure if and only if there is a diffeomorphism 0 of top such that j' = d4' j (d4')-'. Let it be a family of almost complex structures such that jo is the given complex structure of E. Then the derivative v(jo) (d/dtjt)t=o is a ft-valued (0,1)-form. If it is another family of almost complex structures such that each ji induces the same complex structure as it does, then the corresponding (0,1)-form v(jo) can be written as v(jo) + & for some section it of T. This shows that local complex deformations of t are

parameterized by an open subset of H (Tt). Thus a local universal family 0

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of marked curve (2, (ii), (Z.1 -Z.2)), where ii E E is the preimage of xi E E and e E Ed(r), is of the form (3.6)

QXHPiXIIPe1XPe2, e

i

where Q, Pi, Pei and Pee are small neighborhoods of 0 in Hl(ft) and small neighborhoods of 2i, z1 and ze2 in t, respectively. Here as usual (ii) is the L, marked points and (zi,xe2) denotes 2 #Ed(r)-marked points, after fixing an ordering of Ed(r). By the Serre duality, we have HA(TE) = H20 (T`E®2). The latter is the space of holomorphic quadratic differential forms. Let V be a small neighborhood of 0 in the space of f *TX-valued vector fields along t in the C2-topology. Here f : E i X. For each v E V, we can associate to it a unique C2-map exp j(v) from E into X, where exp is the exponential map of a fixed metric on X. We denote by G f the automorphism group of f . It is a finite group and

acts naturally on H£(TE). This is because each a E Gf lifts naturally to an automorphism of E, so it induces an action on HA(TE). Without loss of generality, we can assume that both Q and fli Pi x f Ie Pa X Pee are G f-invariant. Now we define a natural Gf-action on U x V: Any point in U x V is of the form (q, (xi), (ze11 ze2), v), q E Q, xi E Pi, (ze1,' e2) E Pel X Pe2 and v E V.

Given any o, E Gf, we define o(q, (Xi), (zol, ze2), v) = (o (4), (o(xi)), (O(z'e1), Q(Z 2)), V o 1).

Next we construct a product manifold Xr as follows: for each edge or loop e of r, we associate to it two copies of X, say Xel, Xe2. Define Xr = 11. Xe1 X Xe2 We also define Ar = Ije De, where A. is the diagonal of Xe1 X Xe2. We can define a smooth evaluation map ir:U x V -+ Xr by ir(q, (XI), (zel, z2 ),V) = ]J (exp j (v) (zel ), exp j (v) (ze2 )) e

One can easily show that it is transversal to Ar. So * = 9''I (Or) is smooth. The finite group GI acts on W smoothly, thus W/Gf is a neighborhood of f in B(r). This proves the first part of the Proposition. We now prove the remainder part of the Proposition. Note that each point in W is represented by a stable map (f, E, (xi)) and (E, (Xi), (Z1, ze2)) E U. Let E f be the set of all CI-smooth, P TX-valued (0,1)-forms over E. Put

Ew=UEf fEW

Clearly, E* is a smooth Banach bundle, and G1, which acts on *, lifts to a linear action on E* such that the natural projection E* -+ * is G f-equivariant. Moreover, EIw = EW/G f, so E is an orbifold bundle over B. Finally, the section 411 lifts to a section (iw over * defined by

4w(f) = Obviously, it is smooth. This proves the Proposition.

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3.7. Topology of the space of stable maps. Now it is time to describe the topology of B. Let [f] be any stable map in B represented by a stable map (f, E, (xj)). We want to construct the neighborhoods of [f] in B. Let r be the dual graph of f. For each a E Ver(F), we define k,, to be the number of edges and legs attached to it. Here we count a loop to a as two edges. As before we still denote by E. the corresponding component and t. its normalization. We define Ver (1') C Ver(1') to be {a E Ver(r) ka < 3 and ga = 0} and define Ver,(F) the complement of Ver (1'). When of E Ver.(F), then EQ contains one or two distinguished points. We add two or one marked point(s) to Z. according to whether EQ contains one or two distinguished point. We also require that at these added points the curve E is smooth and the differential df is injective. Note that this is always possible since f.([EQ]) 0 0, by the stability of f. We denote by (yj)1
parameterized by admissible quadratic differentials on E. An admissible quadratic differential q is a meromorphic quadratic differential with at most simple poles at xi or y j and double poles at nodes satisfying: if wl, w2 are local coordinates of E near a node, i.e., E is defined by wjw2 = 0 in 0 near such a node, then lim

w, +O ,W2=0W,

q w, oOw,=owadw2

Neighborhoods of (E, (xi, y,)) in fifty,,+I can be constructed as follows: let Of be the automorphism group of (E, (xi, y,)), then a neighborhood U of (E, (xi, y;)) in 9728,,+I is of the form a/Gf, where a is a small neighborhood of the origin in the space of admissible quadratic differentials. For each y{, we choose a codimension two submanifold H1 C X such that H,

intersects f(E) uniquely and transversely at f(y,). We orient H; so that it has positive intersection with f (E).

We fix a compact set K C E\Sing(E) containing all marked points (xi, y,). We may assume that K is Gf-invariant. Let CU be the universal curve over U. We then choose a diffeomorphism ¢ from a neighborhood of K x 12 into CU such that F preserves fibers over U and restricts to the identity map on K x {(E, (x{, yj))}.

We also fix aE>0. To each collection of U, H,, K, 6 and 0 given as above, we can associate to it a neighborhood U = U(U, H,, K, 5, ¢) as follows: define U = U(U, H,,, K, b, ¢) to be

the set of all tuples (f', E', (xi, y,)) satisfying: (1) (E', (x;, y;)) is in 0; (2) f' is a continuous map from E' into X with f'(yj) E Hj; (3) f' lifts to a C2-map E' -+ X; (4) ]] f' 0 - f 11C2(K) < 5; (5) dx(f (E), f'(E')) < 5, where dx denotes the distance function of a fixed Riemannian metric on V. Note that the topology of £' may be different from that of E. We define -U(U, H,, K, b, ¢) to be a local uniformization chart of B. Given any tuple (f', E', (x{,yy)) in U, we call f' _ (f', E', (x;)) its descendant. One can show that f' is a stable map and gives rise to a point [f'] in B. Let U be the set of all equivalence classes of stable maps descended from tuples in U. Then U is a neighborhood of [f]. The topology of B is generated by all such neighborhoods U(U, H,, K, 5, ¢).

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Now assume that 6 and u are sufficiently small. Then there is a natural action of G f on t: let o E G f and f' = (f', E`, (x;, yf )) E U, then o acts on 12 by sending E' to o(E') and x; to o(x;). We define

o(f ) = (f' v-', o(E'), (a(x',), where for each j, y'' is the unique point in o(E') near yj such that f'(o-'(y'7 )) E H1. The existence and uniqueness of such y7 is assured by the assumption that HI are transversal to f (E). Note that y'f' may not be yf. Clearly, f' and o(f) descend to the identical [f'] in B. Conversely, if f' = (f', E', (xi', yf)) and f " _ (f", E", (x", y' )) descend to the same [f'] in U, then there is a biholomorphic x " . When 6 and a are sufficiently small, we may assume that r induces a biholomorphic map, denoted by

o, of (E, (xi)). This or acts on U as defined above. Thus one can easily see that o(f') = f". It follows that U is of the form U/Gf. Therefore, the neighborhoods U(U, H,i, K, 6, l) are the quotients of local uniformization charts U(U, Hf, K, 6, ¢). This proves THEOREM 3.18. Equiped with the topology described above, B is a SS-orbispace.

Moreover, we have

THEOREM 3.19. Assume that (X, w) is a compact manifold with a compatible almost complex structure J. Then the moduli space )Jtxq s,k is compact in B in the above topology.

The compactness theorem of this sort first appeared in the work of Gromov [Gr), further studied and extended by Parker and Wolfson, Pansu and Ye. A proof was also given in [RT1l for holomorphic maps of any genus, following Sacks and Uhlenbeck on harmonic maps in early 80's. Given a local uniformization chart t, we define

Eir =

U (f',E',(si,yj,))EU

where E(f, E, (s{ y;)) consists of all CI-smooth, f"TX-valued (0,1)-forms over the

normalization of V. All such Efj's form charts of E. One can show that the conditions in Definition 3.7 are all satisfied. Therefore, we have proved THEOREM 3.20. (B, E, §j) is a V-vector bundle.

4. Proof of the main theorem By Theorem 3.13, in order to prove our main theorem, we need to show that (B, E, 1a) is a WFV-bundle. By the results in the last section, we suffice to construct local finite approximations of (B, E, 4'J). We continue to use the notations developed so far. Let U be a chart of B with the corresponding group G, and let EU be the corresponding chart of E over U. We know that U is of the form U(U, H1, K, 6, 0). Let Ll be the local uniformization of U and CU be the universal curve over U. We may assume that U is sufficiently small.

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$67

Recall that a TX-valued (0,1)-form on CU x X is an endomorphism v : TCU H

TX such that J v = -v jcu, where jcu is the complex structure on CU. Let

A°"1(CU,TX)o be the space of all C°°-smooth TX-valued (0,1)-forms on CU x X which vanish near Sing(CU). Here Sing(CU) denotes the set of the singularities in the fibers of CU over U. Given each v E A°"i(CU,TX)o, we can associate a section of E(J as follows: For each f = (f, E, (xi, yf)) E U, we define vlf by vIf (x) = v(x,.f (x)), x E E. Clearly, vhf is a section on the fiber of EU over f. This way we obtain a section f I--} vlf over U. To avoid introducing new notations, we still denote this section by v. Now assume G' is non-trivial. Then for a E Gf the pull-back a*(vi) is a

section over U. Without loss of generality, we can assume the 1 I. JG1 I sections

of Eu is linearly independent everywhere. We define F = F(vi, , vt) to be the aubbundle in EU generated by the above 1 IG1I sections. F is a trivial vector bundle and is a G'-equivariant subbundle of ED. LEMMA 4.1. Suppose f = (f, E, (xi, y,)) E U and Sp(vl, , vi) If are transverse to L', where L f is the linearization of the Cauchy-Riemann equation at f. We further assume that in the definition of U, d is sufficiently small and K is sufficiently big. Then '- I (F) is a smooth manifold of dimension r + 1, where r is the index of L f which can be computed in terms of cI (X), the homology class of f (E), the genus of E and the number of marked points. Using this lemma, one can easily show that (U, G f, E0, F) is a local finite approximation. Furthermore, one can assign it a natural orientation by the canonical orientation on the determinant line det(L f) of L f. Note that L f is different from a J-invariant 8-operator Of by an zero-order operator, so det(L') = det(8f), hence, the canonical orientation is given by the canonical one on det(8f). Lastly, it is easy to see that strata of the finite model U of (U, F) mapped submersively to strata of 9 R s , k X k, if w e choose enough sections v i ,--- , vi. This completes the proof of the main theorem.

5. Final remark We now make a few remark to tie up the lose end of our presentation. The first is the equivalence of the algebraic and the symplectic definition of GW invariants. This was generaly believed to be true and was proved in [LT4, Si2]. THEOREM 5.1. Let X be any smooth projective variety with a Kohler form w. Then the algebraically constructed GW-invariants of X coincide with the analytically constructed GW-invariants of the symplectic manifold (XtO9, W).

The proof relied on the fact that we can find holomorphic finite approximation 0: U -+ F, in the symplectic construction of GW invariants, so that the obstruction theory induced by ¢ is identical to the canonical obstruction of 9Rg,k(X,A). We refer the details to our paper above. The second is some basic properties of the GW invariants. One property of virtual moduli cycle we stressed earlier is that if XT is a smooth family of peojective

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varieties with smooth base and if At extends to a global constant family in UtA.Xt, it exists after a base change of T if necessary, then the GW invariant yA, 9 k should be a local constant. This is indeed the case. It is proved by showing that the virtual cycle [971g,k(Xt,At)]vir is the image of the Gysin map i7e (9Jtg k(XT,AT)]°l°, where tl is defined by the square

09,k(Xt,At) -- `W19,k(XT,At)

{t}

-+

T

The second is that the GW invariants are expected to satisfy several important relations, among them the associativity (or composition) law. This again can be proved by studying some geometric (co)cycle A in the moduli of curves 9Jlg,k and the virtual cycle of W1g,k(X,A) xarts,,, A.

We should also mention that the alternative construction of Behrend-Fantechi [BF] used the notion of cotangent complex to represent the obstruction theory of the moduli stack 9Rg,k, and then constructed a cone, called virtual normal cone, as an Artin stack over the D-M stack 9.ltg,k(X,A). In the end, they use a vector bundle E over 911g,k (X, A), similar to the one mentioned earlier, to obtain a cone in E and hence obtain the virtual cycle using the Gysin map of the zero section of E. Recently this last step of using vector bundle E was relaxed by the work of Kresch [Kr] after he developed the intersection theory on Artin stack. It is interesting to note our symplectic construction of GW invariants did not use the existence of

such global vector bundle, thus it can serve as a hint that the global bundle E is not essential to the construction of virtual cycle. We note that the symplectic constructions by Siebert [Sil) and Ruan [Ru2] both relied on such global vector bundles.

References [Be] K. Behrend, Gromov-Witten invariants in alebraic geometry, Invent. Math. 127(1997), no. 3, 601-617.

[BF] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128(1997), no. 1, 45-88.

(Ca] P. Candelas, X. Ossa, P. Green and L. parks, A pair of Calabi-yau manifolds as an exactly solvable superconformal theory, Nucl. Phys. B359 (1991), 21-74. [DM] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publ. I.H.B.S. 45(1969), 101-145. [FO] K. Pukaya and K. Ono, Arnold conjecture and Gromov-Witten invariants, preprint, 1996. [Pu] W. Fulton Intersection theory, Ergebniase der Math. and ihrer Grenzgebiete 3. Folge Band 2, Springer 1984. [Gi] A. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices (1996), no. 13, 613-663. [Gr] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82(1985), no. 2, 307-347. [KM] M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164(1994), no. 3, 525-562.

[Ko] M. Kontsevich, Enumeration of rational curves via torus actions, The Moduli Space of Curves, edited by R.Dijkgraaf, C.Faber, G. van der Geer, Progress in Mathematics vol. 129, Birkhauser, 1995. [Kr] A. Kreach, Cycle groups for Artin stacks, preprint math.AG/9819166.

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[LTI] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. 11(1998), no. 1, 119-174. [LT2] J. Li and G. Tian, Virtual moduli cycles and Gromov- Witten invariants of general symplectic manifolds, Topics in symplectic 4-manifolds (Irvine, CA, 1996), 47-83, First Int. Press Lect. Ser., I, Internat. Press, Cambridge, MA, 199. [LT3] J. Li and G. Tian, Algebraic and symplectic geometry of virtual moduli cycles, Algebraic geometry-Santa Cruz 1995, 143-170, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, 1997. [LT4] J. Li and G. Tian, Comparison of algebraic and symplectic Gromov-Witten invariants, to appear in Asian J. Math. [LT5] J. Li and G. Tian, in preparation. [LLY] K-F. Liu, B. Lian and S-T. Yau, Mirror principle I, Asian J. math. 1(1997), no. 4, 729-763. [PW] T. Parker and J. Wolfson, Pseudo-holomorphic maps and bubnle trees, J. Geom. Anal. 3(1993), no. 1, 63-98.

[Rn] Z. Ran, Deformations of maps, Algebraic curves and projective geometry, E. Ballico, C. Ciliberto, eds, Lecture notes in Math. 1389. [Rul] Y. Ruan, Topological Sigma model and Donaldson type invariants in Gromov theory, Duke Math. J. 83(1996), no. 2, 461-500. [Ru2] Y. Ruan, Virtual neighborhoods and pseudo-holomorphic curves, preprint, 1996. [RTi] Y.B. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Diff. Geom. 42(1995), no. 2, 259-367. [R.'r2] Y.B. Ruan and G. Man, Higher genus symplectic invariants and sigma model coupled with gravity, Invent. Math. 130(1997), no. 3, 455-516.

[Sil] B. Siebert, Gromov-Witten invariants for general symplectic manifolds, preprint, 1996, (AG/9608005).

[Si2] B. Siebert, Algebraic and symplectic Gromov-Witten invariants coincide, preprint, 1998, (AG/9804108). [Vi] A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97(1989), no. 3, 613-670. [Wil] E. Witten, Topological sigma models, Comm. Math. Phys. 118(1988), no. 3, 411-449. [Wi2] E. Witten, Two dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990), 243-310, Lehigh Univ., Bethlehem, PA, 1991.

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