The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator

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The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator

J.J. Duistermaat

Reprint of the 1996 Edition

J.J. Duistermaat (deceased)

Originally published as Volume 18 in the series Progress in Nonlinear Differential Equations and Their Applications

e-ISBN 978-0-8176-8247-7 ISBN 978-0-8176-8246-0 DOI 10.1007/978-0-8176-8247-7 Springer New York Dordrecht Heidelberg London © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper www.birkhauser-science.com

J. J. Duistermaat The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator

Birkhauser Boston • Basel • Berlin

J. J. Duistermaat Mathematisch Instituut Universiteit Utrecht 3508 TA Utrecht The Netherlands

Library of Congress Cataloging-in-Publication Data Duistermaat, J. J. (Johannes Jisse), 1942The heat kernel Lefschetz fixed point formula for the spin-c dirac operator / J. J. Duistermaat p. cm. -- (Progress in nonlinear differential equations and their applications ; v. 18) Includes bibliographical references and index. ISBN 0-8176-3865-2 1. Almost complex manifolds. 2. Operator theory. 3. Dirac equation. 4. Differential topology. 5. Mathematical physics. I. Title. II. Series. QC20.7.M24D85 1995 515'.7242--dc20

Printed on acid-free paper

© Birkhauser Boston 1996

95-25828 CIP

W)®

Birkhiiuser LLWJ

Copyright is not claimed for works of u.s. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN 0-8176-3865-2 ISBN 3-7643-3865-2 Typeset from author's disk by TeXniques, Boston, MA Printed and bound by Quinn-Woodbine, Woodbine, NJ Printed in the U.S.A. 987654321

Contents 1

2

3

4

5

Introduction 1.1

The Holomorphic Lefschetz Fixed Point Formula

1.2

The Heat Kernel

1.3

The Results

. . . . . .

The Dolbeault-Dirac Operator 2.1

The Dolbeault Complex . . .

2.2

The Dolbeault-Dirac Operator .

1 1 2 3

7 7 12

Clifford Modules

19

3.1

The Non-Kahler Case

19

3.2

The Clifford Algebra ..

22

3.3

The Supertrace . . .

27

3.4

The Clifford Bundle . .

29

The Spin Group and the Spin-c Group

3S

4.1

The Spin Group . . . . . . . . . . .

35

4.2

The Spin-c Group . . . . . . . . . .

37

4.3

Proof of a Formula for the Supertrace

39

The Spin-c Dirac Operator

41

5.1

The Spin-c Frame Bundle and Connections

41

5.2

Definition of the Spin-c Dirac Operator

47

v

vii

viii vi

6

7

8

9

Contents

Its Square

53

6.1

Its Square .. . . . . . . . . . . . .

53

6.2

Dirac Operators on Spinor Bundles

61

6.3

The Kahler Case . . . . . . . . . .

63

The Heat Kernel Method

69

7.1

Traces

.

69

7.2

The Heat Diffusion Operator. .

72

The Heat Kernel Expansion

77

8.1

The Laplace Operator .

77

8.2

Construction of the Heat Kernel

79

8.3

The Square of the Geodesic Distance

81

8.4

The Expansion . . . . . . . . . . . .

92

The Heat Kernel on a Principal Bundle

99

9.1

Introduction . . . . . . . .

99

9.2

The Laplace Operator on P

100

9.3

The Zero Order Term

105

9.4

The Heat Kernel

108

9.5

The Expansion.

110

10 The Automorphism

117

10.1 Assumptions . . . . . . . . . .

117

10.2 An Estimate for Geodesics in P

121

10.3 The Expansion

.

11 The Hirzebruch-Riemann-Roch Integrand

125

131

11.1 Introduction . . . . . . . . . . . . .

131

11.2 Computations in the Exterior Algebra

133

11.3 The Short Time Limit of the Supertrace

143

Contents

ix

12 The Local Lefschetz Fixed Point Formula 12.1 The Element go of the Structure Group 12.2 The Short Time Limit 12.3 The Kahler Case .

147 147 151 155

13 Characteristic Classes 13.1 Weirs Homomorphism 13.2 The Chern Matrix and the Riemann-Roch Formula 13.3 The Lefschetz Formula. . 13.4 A Simple Example.

157 157 159 164 169

14 The Orbifold Version 14.1 Orbifolds . 14.2 The Virtual Character . . . 14.3 The Heat Kernel Method . 14.4 The Fixed Point Orbifolds 14.5 The Normal Eigenbundles 14.6 The Lefschetz Formula. .

171 171 176 177 179 181 183

· ....

15 Application to Symplectic Geometry 15.1 Symplectic Manifolds . 15.2 Hamiltonian Group Actions and Reduction 15.3 The Complex Line Bundle. 15.4 Lifting the Action . . . . . . 15.5 The Spin-c Dirac Operator . 16 Appendix: Equivariant Forms 16.1 Equivariant Cohomology. . 16.2 Existence of a Connection Form 16.3 Henri Cartan's Theorem 16.4 Proof of Weil's Theorem . 16.5 General Actions . . . . . . .

· .... · ....

· .... .

187 188 192 201 205 213 221 221 225 227 234 234

Preface When visiting M.I.T. for two weeks in October 1994, Victor Guillemin made me enthusiastic about a problem in symplectic geometry which involved the use of the so-called spin-c Dirac operator. Back in Berkeley, where I had spent a sabbatical semester!, I tried to understand the basic facts about this operator: its definition, the main theorems about it, and their proofs. This book is an outgrowth of the notes in which I worked this out. For me this was a great learning experience because of the many beautiful mathematical structures which are involved. I thank the Editorial Board of Birkhauser, especially Haim Brezis, for suggesting the publication of these notes as a book. I am also very grateful for the suggestions by the referees, which have led to substantial improvements in the presentation. Finally I would like to express special thanks to Ann Kostant for her help and her prodding me, in her charming way, into the right direction. J.J. Duistermaat Utrecht, October 16,1995.

1 Partially

supported by AFOSR Contract AFO F 49629-92

Chapter 1 Introduction 1.1

The Holomorphic Lefschetz Fixed Point Formula

Let M be an almost complex manifold of real dimension 2n, provided with a Hermitian structure. Furthermore, let L be a complex vector bundle over

M, provided with a Hermitian connection. We also assume that ]{*, the dual bundle of the so-called canonical line bundle ]{ of M, is provided with a Hermitian connection. We write E for the direct sum over q of the bundles of (0, q)-forms; in it we have the subbundle E+ and E-, where the sum is over the even q and odd q, respectively. Write rand r± for the space of smooth sections of E ® Land E± ® L, respectively. From these data, one can construct a first order partial differential operator D, the spin-c Dirac operator mentioned in the title of this book, which acts on r. The restriction

D+ of D to r+ maps into r- , and the restriction D- of D to r- maps into r+. If M is compact, then the fact that D is elliptic implies that the kernel N± of D± is finite-dimensional, and the difference dim N+ - dim N- is

equal to the index of D+. The Atiyah-Singer index theorem applied to this case [7, Theorem (4.3)]

1 J.J. Duistermaat, The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8247-7_1, © Springer Science+Business Media, LLC 2011

Chapter 1. Introduction

2

expresses this index as the integral over M of a characteristic class in the De Rham cohomology of M, equal to the product of the Todd class of the tangent bundle of M, viewed as a complex vector bundle over M, and the Chern character of L. These characteristic classes are given by polynomial expressions in the curvature forms of the given bundles. If M is a complex analytic manifold, then the index of D+ is equal to the Riemann-Roch number of M, and the integral formula generalizes the one which Hirzebruch [39] obtained for complex projective algebraic varieties. If , is a bundle automorphism of L which leaves all the given structures invariant, then it induces an operator in r which commutes with D, and one can form the virtual character

(1.1) Under the assumption that the fixed point set M' of , in M locally is a smooth almost complex submanifold and that the action of , in the normal bundle is nondegenerate, the equivariant index theorem of Atiyah-Segal and Atiyah-Singer expresses the virtual character as the sum over the connected components F of lVI', of similar characteristic classes of the F's. In the complex analytic case, this is called the holonl0rphic Lefschetz fixed point

formula, cf. Atiyah and Singer [7, Theorem (4.6)]. In the case of isolated fixed points, it is due to Atiyah and Bott [5, Theorem 4.12].

1.2

The Heat Kernel

The operator Q+ == D-

r- to r-.

0

D+ maps

r+

to

r+,

and Q- == D+

0

D- maps

Each of the operators Q+ and Q- is equal to a Laplace operator,

plus a zero order part which involves curvature terms. The corresponding heat diffusion operators e -t Q± are integral operators with a smooth integral kernel K±(t, x, y), t

> 0, x, Y

E M. Along the diagonal x ==

y, and for

3

1.3. The Results

t 1 0, these kernels have an asymptotic expansion of the form

K±(t,

X,

x)

rv

t- n

Lt 00

k

Kt(x).

(1.2)

k=O

In this asymptotic expansion, each of the coefficients Kk(x)± is given by a universal polynomial expression in a finite part of the Taylor expansion of the geometric data at the point x. It was observed by McKean and Singer [57, p. 61] that

indexD+ =

JM tracec K;(x) -

tracec K;;(x) dx,

(1.3)

and they asked the question if not, by some fantastic cancellation, the higher order derivatives in the expression for K~ (x) cancel, to give that the integrand in (1.3) is equal to a characteristic differential form whose cohomology class is equal to the one of the index theorem. This would give a direct analytic proof of the index theorem, with the advantage of having a local interpretation of the integrand. Actually, in [57] the question is asked for the Euler characteristic of M, but it obviously can be generalized to arbitrary index problems.

1.3

The Results

It turned out that, also in the presence of an automorphism " the fantastic cancellation indeed takes place. See Theorem 11.1 and Theorem 12.1. In the complex analytic case, the result is referred to as a local holomorphic

Lefschetz fixed point forn1ula. It is the purpose of this book, to explain both all the ingredients in the formula, and how the answer comes about. In it, we will apply the methods of Berline, Getzler and Vergne [9, Ch. 1-6], and show how these work in the case of the spin-c Dirac operator. (For the comparison: our L is their W, the letter W is the classical notation of Hirzebruch [39]. We have chosen the letter L, because of the connotation of a "linear system".)

Chapter 1. Introduction

4

For the index, the result is due to Patodi [64] in the Kahler case, with another proof by Gilkey [28], who in [31] extended the result to almost complex manifolds. In the presence of an automorphism" the local formula had been obtained by Patodi [65] under the assumption that the connected components of the fixed point set M' of , in M are Kahler manifolds. A proof for general almost complex manifolds has been indicated by Kawasaki [45, pp. 156-158]. One can also obtain the result in this general setting as a consequence of the local Lefschetz formula for the spinor Dirac operator of Berline and Vergne [11], cf. [9, Theorem 6.11]. That is, by using the comparison (6.20) between the bundle E of (0, q)-forms and the spinor bundle S, and observing that it suffices to work locally, where spin structures always exist. The local formula is particularly suited for the generalization of the Lefschetz formula to compact orbifolds, which we will explain in Chapter 14. I learned this from the proof of Kawasaki [45] for the Riemann-Roch number. For arbitrary elliptic operators on compact orbifolds, the Lefschetz formula has been obtained by Vergne [73]. She used the theory of transversally elliptic operators of Atiyah [2], as Kawasaki [46] did in his proof of the index formula for orbifolds. The use of the local formula avoids the use of the commutative algebra of [2], which may make it more accessible to analysts. Strictly speaking, this work contains no new results. However, the spin-c Dirac operator is a very important special case among the general Dirac-type operators. As described above, it came originally from the study of complex analytic manifolds. On the other hand, every symplectic manifold (phase space in classical mechanics) also carries an almost complex structure and hence a corresponding spin-c Dirac operator. We will discuss the application of the theory to this case in Chapter 15. As a third application, we mention that recently the Seiberg-Witten theory, an Sl gauge theory which uses the spin-c Dirac operator, has led to striking progress in the differential topology of four-dimensional compact oriented manifolds. Here one works with spine Dirac operators which are defined in terms of spin-c structures which do not

1.3. The Results

5

necessarily come from an almost complex structure. See the Remark in front of Lemma 5.5. For an exposition of Seiberg-Witten theory see for instance Eichhorn and Friedrich [26] or Morgan [61]. The importance of the spin-c Dirac operator makes it worthwhile to work out the beautiful constructions of [9] for this special kind of Dirac operator. A large part of the exposition has a wider scope than just the spin-c Dirac operator. For instance, Chapter 8 is an exposition of the asymptotic expansion of heat kernels for generalized Laplace operators, following [9, Ch. 2]. Chapters 9 and 10, on the Berline-Vergne theory of heat kernels on principal bundles, are also written for more general operators than only the spin-c Dirac operator. The point of this theory is, that it gives an explanation for the similarity between the factor det l-~-R , which appears in the index formula, and the Jacobian of the exponential mapping from a Lie algebra to the Lie group. (Here R denotes curvature.) Lemma 9.5 and Lemma 9.6 form the starting point of this explanation. Although in general we tried to keep our notations close to our main reference [9], we apologize that at some points we ended up with a different choice. Finally, in Chapter 13 the formulas of Theorem 11.1 and Theorem 12.1 are translated into the language of characteristic classes, in which the formulas of Hirzebruch and Atiyah-Singer originally were phrased. We use the occasion to explain, in Chapter 16, the Weil homomorphism in its natural setting of equivariant differential forms in the presence of an action of a Lie group, and under the assumption that the action admits a connection form. I am very grateful to Victor Guillemin for arousing my interest in the subject, in connection with the question how the Riemann-Roch number of a reduced phase space for a torus action is related to multiplicities of intermediate phase spaces. And I apologize for spending so much time on writing up this text, instead of "adorning the dendrites". Finally I would like to thank the Department of Mathematics of DC Berkeley, for providing me with an ideal environment to work on this.

Chapter 2 The Dolbeault-Dirac Operator In this chapter we set the stage, by introducing complex and almost structures, the Dolbeault complex and Hermitian structures. The holomorphic Lefschetz number, defined as the alternating sum of the trace of the automorphism acting on the cohomology of the sheaf of holomorphic sections, will be expressed in terms of a selfadjoint operator, which is built out of the Dolbeault operator and its adjoint; the Dolbeault-Dirac operator in the title of this chapter. This material is very well-known but, also in order to fix the notations, we have taken our time for the description of these structures. Just for convenience, we will assume that all objects are smooth (infinitely differentiable).

2.1

The Dolbeault Complex

Let M be a manifold of even dimension 2n, provided with an almost complex structure J. That is, for each x E M, J x is a real linear transformation in T x M such that J x 2 == -1. A real linear mapping A from T x M to a complex vector space V is called complex linear and complex antilinear with respect 7 J.J. Duistermaat, The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8247-7_2, © Springer Science+Business Media, LLC 2011

8

Chapter 2. The Dolbeault-Dirac Operator

to the complex structure J x in T x M, if == i

A(v), v

E

Tx M

(2.1)

== -i

A(v), v

E

T x M,

(2.2)

A (Jx(v)) and

A (Jx(v)) respectively.

The space of complex linear and complex antilinear forms (V == C) on T x

M

is denoted by T~ M(l, 0) and T~

M(O' 1),

respectively. With this

notation, the space of complex linear and antilinear mappings from T x M to

V becomes equal to T~

0 V and T~

M(l,O)

M(O' 1)

® V, respectively. One

has the complementary projections 7r(1,0) :

~

r-7

~(1,0) :== ~ (~- i~

0

Jx ),

(2.3)

and

(2.4) from T*x

M

® C onto T*x

M(l,O)

along T*x M(O' 1) and from T* ' x

M

® C onto

T~ M(O' 1) along T~ M(l, 0), respectively.

A complex-valued function

f

on M is called complex-differentiable or

complex-analytic, or holomorphic, if, for every x E M, dfx is complex linear. If [) ==

7r(0,1) 0

d denotes the operator d followed by the projection

(2.4), then this condition is equivalent to the differential equation One also writes f)

==

7r(1,0)

0

d, so that d

== f) + [) on functions,

af

and f) f

== O.

== df

if and only if f is holomorphic. Let p, q, r E Z?O, with P + q == r. A complex-valued antisymmetric r-linear form on T x M is called of type (p, q), if it is equal to a finite sum of forms a 1\ {3, where a E AP T;

M(l,O)

forms of type (p, q) is denoted by T~ ArT* M x

0 C

==

and {3 E Aq T;

M(p, q).

~

Q7

p, q,p+q==r

M(O' 1).

The space of

The point is that T*x

M(p,q)

,

(2.5)

2.1. The Dolbeaul! Complex

9

so we have the projection 'lfp,q from AT T; M

@

C onto T;

M(p,q)

along the

sum of the other components. An L-valued version is obtained by tensoring T;

M(p,q)

with Lx. A (p, q)-form w x on T x M, which depends smoothly on

x E M, is called a (p, q)-form on M. The space of (p, q)-forms on M is

denoted by O(p,q)(M). In particular we will be interested in the case p == 0, for which we will use the following abbreviation throughout: E~ :==

Note that

E'l:

= 0 if q

T; M(O, q)

==

Aq T;

E~ ==

M(O' 1),

> n, and dime E'l:

C.

= ( ; ) if 0 ::; q ::;

(2.6) n. We will

write n

(2.7)

Ex :== EBE~, q=O

E+ == E xeven == x E; == E~dd

==

~ 'l7 even q

EB

Eqx'

E~.

(2.8) (2.9)

odd q

With the exterior product of forms and the splitting in E: and E;, Ex is a supercommutative superalgebra over C. (See [9, Section 1.3] for the definition of such algebras.) The Ex, x EM, form a complex vector bundle E over M with subbundles

E+ == UxEM E; and E- == UXEM E;. The space of sections of E, E+ and E- is equal to the direct sum of the spaces [2(0, q), where q runs over all the even and the odd integers 0

:s; q :s; n, respectively.

In Chapter 5 we will introduce the spin-c Dirac operator, which will be used in the general case of an almost complex structure. In order to motivate its definition and to understand its relation to complex analysis, we assume in the remainder of this chapter that M is a complex analytic manifold. This means that around every x E M there is a system of local coordinates in

10

Chapter 2. The Dolbeault-Dirac Operator

which J is equal to the standard complex structure of C n . This is equivalent to the condition that, at every x EM, there exist n holomorphic functions Zj in a neighborhood of x in M, such that the dZj at x are linearly independent over C. In such coordinates

Zj,

each (p, q)-form is of the form

W

==

L

WJ,K

dZ J

/\

dzK

(2.10)

,

J,K

where J and K runs over the set of strictly increasing sequences J == (ji)f==l and K == (k i );==l' respectively, each WJ,K is a complex-valued function, and dz J

== dZj1

dZK

== dZk1

/\ /\

dZ j2 dZk2

/\

/\

/\

dz jp ,

/\

dzkq •

(2.11) (2.12)

From this we see that dw is the sum of a (p + 1, q)-form and a (p, q+ 1)-form. Or, one again has d ==

a+ 8, if one writes a == 7f(p+l, q)

d

(2.13)

[) == 7f(p, q+l) 0 d

(2.14)

0

and

on (p, q)-forms. This implies that for each (0, I)-form w,

7f(2,0)

dw == O. For a general

almost complex structure J, it need no longer be true that d == 8

+ 8.

For each x E M, one has the antisymmetric bilinear mapping [J, J]x from T x M x T x M to T x M, which is defined by

[J, J](v, w) == [Jv, Jw] - J [Jv, w] - J [v, Jw] - [v, w],

(2.15)

for any vector fields v and w in M. Using the formula

(dw)(v, w) == vw(w) - ww(v) -

W

([v, w]),

(2.16)

11

2.1. The Dolbeault Complex one gets for each (0, I)-form w that

dw(v - i Jv, So the condition that

7r(2,O)

W -

i Jw) == w ([J, J](v, w)).

dw == 0 for every (0, I)-form w is equivalent

to the condition that [J, J] == O. The theorem of Newlander and Nirenberg now says that an almost complex manifold (M, J) is complex analytic if and only if [J, J] == O. This theorem is already valid if the first order derivatives of J are Holder-continuous. Cf. Newlander and Nirenberg [62], Hormander [41], Malgrange [55]. We continue the discussion of complex analytic manifolds. Identifying the types in 0 == d 2 == f)2 + f)[) + [)f) + [)2 on Q(p,q)-forms, one sees that f)2

== 0,

f)

a+ af) == 0, and [)2 == O. In particular, the operator 8 defines a

complex

(2.17) called the Dolbeault complex. On the sheaves of locally defined forms, this sequence is exact, and one gets the theorem of Dolbeault that

(2.18) where the right hand side denotes the q-th cohomology group of the sheaf

(] (O(p,ol) of halamorphic (p, O)-forms over M. See for instance Griffiths and Harris [33, p. 45]. If M is compact, then the ellipticity of the complex yields that the spaces in the left hand side are finite-dimensional. We will mainly be interested in the case that p == O. A holomorphic vector bundle Lover M is defined as a complex vector bundle over M for which the retrivializations are given by elements of GL(l, C) which depend holomorphically on the base point. (Here l

==

dime Lx.) All the above remains valid for L-valued (p, q)-forms, that is, the sections of the vector bundle

12

Chapter 2. The Dolbeault-Dirac Operator

In particular, we have the "twisted Dolbeault complex" defined by

a(O,q) : Eq

@

L

~

Eq+l

@

L,

and the corresponding cohomology groups

ker8(O,q)/range8(O,q-l)

~Hq(M,

O(L)) ,

(2.19)

the q-th cohomology group of the sheaf O(L) of holomorphic sections of L over M. If M is a compact complex analytic manifold, then an important quantity is the Riemann-Roch nunlber

RR(M, L) :==

n

L( -l)q dime Hq (M, O(L)).

q=O

(2.20)

More generally, if 1 is a complex analytic automorphism of L, then 1 acts on 0 (L), and one can define its holomorphic Lefschetz number n

X(1) == XM ,L(1) :== L(-1)qtracee1IHQ(M,O(L»·

q=O

(2.21)

This is a generalization because XM,L (l) == RR(M, L). If L is a holomorphic complex line bundle over M for which K*

@

L is

positive, then Kodaira's vanishing theorem says that Hq (M, O(L)) == 0 for every q > 0, cf. (6.34). If the latter is the case, the holomomorphic Lefschetz number is equal to the trace of the action of 1 on the space HO (M, O(L)) of all holomorphic sections of Lover M, and the Riemann-Roch number is equal to the dimension of that space.

2.2

The Dolbeault-Dirac Operator

In order to define adjoints, we now introduce Hermitian structures hand h L in the tangent bundle T M of M and the fibers of L, respectively. For each x E M, h x is a complex-valued bilinear form on T x M, such that

h x ( v, v) > 0 if vETx M, v

i= 0,

(2.22)

13

2.2. The Dolbeault-Dirac Operator and

h x (Jx(v), w)

== i

hx(v, w)

==

-h x (v, Jx(w)) , v, w

E

T x M.

(2.23)

Similarly with h x and T x M replaced by h~ and Lx, respectively. It follows that the real part (3 == Re h of h is a Riemannian structure in M, and J x is antisymmetric with respect to (3x. Furthermore, the imaginary part (J

== 1m h is a nowhere degenerate two-form in

symplectic for

(Jx.

M, and J x is infinitesimally

Finally,

(3(v, w) ==

(J

(lv, w)

shows that choosing two of the three structures J, (3,

(2.24) (J

determines the third.

In order to get a Hermitian structure in E, we begin by observing that (2.25)

is a complex linear isomorphism from T x

M

onto T; M(O,l). Using this

isomorphism, we transplant the Hermitian structure of T M to a Hermitian structure h(O,l) in T* M(O' 1). That is, h(O,l) is determined by the condition that, if ej, 1 S j S n, is a unitary local frame in T M for h, then Ej :== h ej forms a unitary local frame in T* M(O' 1) for h(O, 1). It is dual to the frame ej, in the sense that (ej, Ek) == bjk. The Hermitian structure h(O,q) on Eq == T*

M(O,q)

the conditon that the E K form a unitary local frame in

can now be defined by Eq,

if for each strictly

increasing sequence K == (k i )i=l we write (2.26)

The Hermitian structure

hE

on the direct sum E of the

Eq

is defined by

requiring the summands to be mutually orthogonal. And the Hermitian structure hE@L on E 0 L by the condition that if ej and lk are unitary local frames in E and L, respectively, then the ej 0 lk form a unitary local frame in E 0 L.

14

Chapter 2. The Dolbeault-Dirac Operator

The Hermitian L 2 -inner product of two sections u and v of E 0 L is now defined as

(u, v) =

1M h

E0 du,

v) dx,

(2.27)

where dx denotes the standard volume form in M, such that (2.28)

(Declaring dx to be positive determines the orientation of M.) If D : r (M, E 0 L) ~ r (M, E ® L) is a differential operator, then the (formal) adjoint D* of D is defined by

(D(u), v)

==

(u, D*(v)) ,

u, v

E

r

(M, E ® L),

(2.29)

where one of the u, v is compactly supported. The adjective "formal" is added, if one wants to stress that no attention is being paid to questions of L 2 -closure. In complex local coordinates, and a local holomorphic trivialization of L, we get for a constant Cl-valued (0, q)-form wand real-valued linear form ~ on en ~ R 2n :

Because of its frequent occurrence, we will use the abbreviation (2.30)

for the linear mapping of taking the exterior product with a E T; M(O' 1) from the left. Using the invariance properties of principal symbols, we get

that the principal symbol of the operator [) at ~ E T; M is equal to (2.31)

The principal symbol at ~ E T; M of the adjoint operator (2.32)

2.2. The Dolbeault-Dirac Operator

15

is equal to the adjoint of (2.31) with respect to the given Hermitian structures. In order to compute this, we observe that for every r and every pair of strictly increasing sequences J == (ji)i=l' K == (ki)i:i, we have h x (E r /\ EJ' EK ) == 0, unless IKI == {r} U IJI, and in this case the inner product is equal to

(-1)8, where s is equal to the number of i such that ji < r. Since the same answer is obtained for hx (E J , i (e r ) EK

),

we conclude that the adjoint

of e (E r ), the operator of taking exterior product with Er from the left, is equal to i (e r ), the operator of taking inner product with ere Writing ~r

== (e r , ~),

TJr == (Je r , ~), we get ~(O, 1)

==

n

L: ~ (~r + i TJr)

r=l

Er

(2.33)

.

Moving over every Er-term, and using that hx is complex linear in the first variable and complex antilinear in the second variable, we get n

(J

a* (~) w == - ~

L: (~r -

r=l

i TJr) i ( er ) w.

Since the antilinearity of w yields that

the result is

1

- (C) ~ --

(J 8*

-

2i ·I

(4- 1~C) fJ

.. Eq+l x 0 Lx

We now consider the operator D = 2 (8 + with principal symbol equal to 2 ((Ja

+ (Ja*).

8*),

---*

Eqx 0 L x .

acting on

(2.34)

r (M, E ® L),

Here the factor two has been

inserted in order to cancel the halves in (2.31), cf. (2.33), and in (2.34). 1 In [9, p. 138], [32, p. 184] and [33, p. 80] the more customary convention is used of taking the Hermitian inner product in E q, such that h x (tK' tJ() == 2 q for tK E Eq. As a consequence, (2.34) is equal to 1/2 times the formula for the symbol of 8*, obtained in [32, p. 184] and [9, p. 138].

Chapter 2. The Dolbeault-Dirac Operator

16

For anyone-form a and vector v we have e( a)2 == 0, i( V)2 == 0, and

i(v) (al\w) == (v, a)w-al\(i(v)w), and therefore

(e(a) + i(v))2 == (v, a). It follows now from (2.31) and (2.34), that (2.35)

It is quite remarkable that the square of the principal symbol is a scalar, because the principal symbol itself is a linear transformation in Ex

@

Lx,

which is far from diagonal. It actually has no diagonal terms at all because it maps E~ 0 Lx to the sum of E~+l 0 Lx and E~-l @ Lx. In [9], an operator

D which satisfies (2.35) is called a generalized Dirac operator. Since the generalized Dirac operator 2

(8 + 8*) is defined in terms of the operator a,

which appears in the Dolbeault complex (2.17), I got into the habit of calling it the Dolbeault-Dirac operator. From (2.35) we see that if ~ E T x M (real) and ~

=I

0, then (JD(~)2 is

invertible, and hence (JD(~) is invertible as well. That is, the operator D is elliptic. If M is compact, then this implies that the kernel of D is finitedimensional and the range of D is equal to the orthogonal complement of the kernel of D*

= D.

However

or So D u ==

°

~ (D u, D u)

=

82 = 0 implies that (8*) 2 = 0 and hence

(8 u, 8u) + (8* u, 8* u) .

is equivalent to [) u == 0 and [)* u == O. Since the latter equation

means that u is in the orthogonal complement of the range of 8, we get from the Dolbeault theorem (2.18) that (2.36)

2.2. The Dolbeault-Dirac Operator

17

This allows us to translate (2.20) and (2.21) in terms of the kernel of

D = 2 (8 + 8*). Note that D maps sections of Eq ® L to sections of (Eq+l EB Eq-l) ® L, so we have the operators (2.37) and

D- = Dlrc M , E-0L) : r (M, E- ® L)

---t

r

(M, E+ ® L) .

(2.38)

Collecting the terms with the same sign in (2.20), we get

RR(M, L)

dime ker D+ - dime ker Ddime ker D+ - dime ker ( D+) * = index D+ , (2.39)

the index of the operator D+. And doing the same in (2.21):

x(t) == tracec 'Y Ikef D+

-

tracee '"Y Ikef D -

.

(2.40)

Chapter 3 Clifford Modules 3.1

The Non-Kahler Case

In general, if the manifold is not Kahler, then the Dolbeault-Dirac operator D

=

2 ( 8 + 8*) is not the most suitable one for getting explicit formulas

for (2.39) and (2.40). For instance, if M is a complex analytic manifold and

n == 2, then Gilkey [29, Thm. 3.7] proved that the difference tracec Kt(x) - tracec K 1 (x) of the traces of the coefficients of t- I in the asymptotic expansion (1.2) is equal to a universal constant times d 8a

== 88a == 8 d a.

(3.1)

Here the Kahler form a, cf. (2.24), is viewed as a (1, I)-form. Moreover, the integrand in (1.3) is equal to a universal constant times the Laplacian applied to (3.1), plus terms involving at most second order derivatives of the metric. This follows from the formulas for E 2 and E 4 of Gilkey [30, p. 610], using that, in the computation of the supertrace, the linear contributions of the scalar curvature drop out. So the integrand in (1.3) involves fourth order derivatives of

(J.

I am grateful to Peter Gilkey for helping me with 19

J.J. Duistermaat, The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8247-7_3, © Springer Science+Business Media, LLC 2011

Chapter 3. Clifford Modules

20

these references. Instead of the Dolbeault-Dirac operator, we shall therefore use the so-called spin-c Dirac operator. This is an operator with the same principal symbol as the Dolbeault-Dirac operator, and defined in terms of a carefully chosen connection in the bundle E ® L, see (5.14). The definition involves the bundle C (T* M) of Clifford algebras of the duals T~ M of the tangent spaces, which comes naturally with the principal symbol (2.35). Sections of C (T* M) act on sections of E

L by means of the so-called Clifford multiplication, and the crucial property of the connection which is @

used in the definition of the spin-c Dirac operator is that it yields a Leibniz rule of the form (3.39). Such a connection is called a Clifford connection. In this chapter we will discuss the Clifford algebras. The Clifford algebra

C(2n) contains as a natural subgroup the spin group Spin(2n), which is related to the rotation group. And the complexification of C(2n) contains the slightly larger spin-c group Spine (2n), which contains the unitary group U (n) in a natural way. These groups will be introduced the next chapter, and will be used in the definition of the connection in E ® L and the spin-c Dirac operator in Chapter 5. In the condition (3.39) for a Clifford connection, there appears a connection in the bundle C (T* M) of Clifford algebras, which is defined in terms of the so-called Levi-Civita connection of the Riemannian structure (3. This is the unique connection in T M, which leaves (3 invariant and is torsionjree. Recall that for a general linear connection in the tangent bundle, with the corresponding covariant differentiation \7 of vector fields in M, the torsionTx at x E M is the bilinear mapping from T x M x T x M to T x M, defined by

T(v, w) == \7 v w - \7 w v - [v, w],

(3.2)

for any vector fields v, w in M. So the connection is torsion-free, if and only if

[v, w] == \7 vw - \7 wv,

(3.3)

expressing the Lie bracket of vector fields in terms of the covariant derivatives. Before we continue, we warn that, in general, the Levi-Civita connec-

3.1. The Non-Kahler Case

21

tion does not leave the almost complex structure invariant. If it would, then for any pair of vector fields v, w in M, each of the expressions

\lv (Jw) - J\Jvw, -\J w (Jv)

+ J\Jwv,

-\lJwv - J\lJw(Jv) , \lJvw+J\lJv (Jw) , is equal to zero. Summing, one gets that

[v, Jw] + [Jv, w] + J [Jv, Jw] - J [v, w] == 0, which is equivalent to [J, J]

== 0, or M is complex analytic. See the com-

ments after the definition (2.15) of [J, J]. A similar computation shows that dO"

== 0, so (M, J, h) is a Kahler manifold, and this conversely implies that

the Levi-Civita connection leaves J invariant, cf. [9, Proposition 3.66]. So the warning is that in the non-Kiihler case, a connection in T M, which leaves

both (3 and J invariant, and therefore also hand 0", necessarily has nonzero torsion. The fact that in the non-Kahler case the Levi-Civita connection does not leave the almost complex structure invariant is the reason for replacing the Dolbeault-Dirac operator by the spin-c Dirac operator. For the spin-c Dirac operator, we will get that the difference of the traces of K+(t, x, x) and

K-(t, x, x) converges as t 1 0, and the limit can be determined explicitly (Theorem 11.1). A similar result holds in the presence of an automorphism , (Theorem 12.1). Since the spin-c Dirac operator has the same principal symbol as the Dolbeault-Dirac operator, this leads, if M is compact, to a determination of the Riemann-Roch number and the virtual character as integrals over M and M' of these respective limits. In comparison with the Dolbeault-Dirac operator, the drawback of the spin-c Dirac operator is that in the case of a complex analytic manifold which is not Kahler, its kernel does not consist of holomorphic sections. Also, in general the square of the spin-c Dirac operator does not preserve the degrees of the differential

Chapter 3. Clifford Modules

22

forms (it only preserves the degree modulo two), so it cannot be used to get information about sections in the kernel of D which have a given degree as a differential form.

3.2

The Clifford Algebra

The point of departure for the definition of the spin-c Dirac operator is the formula (2.35) for the principal symbol, which leads naturally to a Clifford module structure on the fibers Ex of the bundle E. We shortly repeat the definitions and some of the properties which we will need. We refer to [9, Chapter 3] and Lawson and Michelsohn [51, Ch. 1] for some more details. If V is a real vector space, provided with an inner product Q, then the Clifford algebra C(V) == C (V, Q) is the algebra over R, generated by V, and with the relation v . v == -Q (v, v) ,

for each v E V.

(3.4)

More precisely, C(V) is defined as the quotient of the tensor algebra Q9 V of

V (with unit) by the two-sided ideal I, generated by the set of the elements v 0 v + Q (v, v), with v E V. Note that (3.4) is equivalent to v·w+w·v==-2Q(v,w),

v,wEV.

(3.5)

If one works over the complex numbers, then the rank of Q determines the Clifford algebras up to isomorphism, whereas in the real case the signature also matters. We will follow the original convention of Clifford [17] (whose point is that V can have any dimension) and take Q to be positive definite. In Dirac's papers [18], V is four-dimensional and Q has the Lorentz signature, whereas in Brauer and Weyl [15], Q is chosen to be negative definite (and

v + I : V ~ C(V) is an injective linear mapping from V to C(V), which will be used in order to identify V with a linear subspace of C(V). One also has 1 ~ V. The grading of the tensor dim V arbitrary). The map v

r---t

3.2. The Clifford Algebra

23

algebra 0 V passes to a filtration of C(V): C::;q (V) is the linear subspace of

C(V) generated by the

(3.6) with Vj E V and k :::; q. Note that C::S 1 (V) == REB V, and that

(3.7) The grading of 0 V also leads to a grading modulo two in C(V); the vector space in C(V) generated by the (3.6) with k even and with k odd will be denoted by C+ (V) and C- (V), respectively. We have the following properties:

C(V)

C+(V) E9 C-(V),

(3.8)

C+(V) . C+(V) C C+(V),

(3.9)

C+(V) . C-(V) c C-(V),

(3.10)

c C-(V),

(3.11)

C-(V) . C-(V) c C+(V).

(3.12)

==

C-(V) . C+(V)

This means that C(V) is a superalgebra; not supercommutative as soon as

Q =I O. We also have C::;l(V) n C-(V)

V and C::S 1 (V) n C+(V) == R. In particular, C+ (V) is a subalgebra of C(V), and the invertible elements in it form a Lie group, which will be denoted by C+ (V) x. Clifford [17] remarks that if V

==

== R 3 and Q is the standard inner product in R 3 , then C+ (V) is the

algebra of the quatemions.The basic property of the algebra C(V), which actually determines it up to isomorphism, is the following. Let A be any real algebra with unit, and let ¢ be a real linear map from V to A such that

¢(V)2 == -Q (v, v),

for each v E V.

(3.13)

Then there is a unique extension of ¢ to a homomorphism from C(V) to A,

which also will be denoted by cP. If A happens to be a complex algebra, then

Chapter 3. Clifford Modules

24

the map ¢ : C(V) ~ A has a unique extension to a complex linear map from C(V) 0 C to A, which automatically is a homomorphism of complex algebras and again will be denoted by ¢. As a first application of this principle, consider the linear mapping CA, which assigns to each ~ E V the endomorphism CA(~) in the exterior algebra

AV of V, defined by CA(~)(W) == ~ /\

W -

i (Q~) w.

(3.14)

This induces an algebra homomorphism

CA : C(V)

~

End (AV).

Then the map (JA :

a ..-..+ cA(a)(l) : C(V)

has the property that for any VI, (J A

(VI· V2 . . . . . Vk)

== VI /\

V2, ... , Vk

~

AV

(3.15)

E V:

V2 /\ ... /\ Vk

modulo

EB A

k-

2l

V.

(3.16)

l~I

It follows that

(J A

is a linear isomorphism, and preserves the filtration and

the grading modulo 2. The formula

defines a Q-dependent product

(a, f3)..-..+ a .Q f3 in AV, which can be viewed as a "lower order perturbation" of the wedge

product, because if a E Aav and f3 E AbV, then

3.2. The Clifford Algebra

25

If a±, b± E C±(V), then one defines the supercommutator of a == a+ and b == b+

+ b- by

+ a-

(3.17) If v E V, then the operator ads(v) : a ~

[v, a]s

in C(V) is a superderivation in the sense that ads (v ) (a . b) == ads (v ) (a) . b ± a . ads (v ) ( b),

(3.18)

if a E C±(V). Since i(Qv) is a superderivation in AV, it follows that the formula (3.19) which is valid for a E V by the definition of C(V), extends its validity to arbitrary a E C(V). In particular, ads(v) maps C::;k(V) into C::;(k-1) (V). This in turn can be used to prove that the algebra C(V) is simple, which means that if I is an nonzero two-sided ideal in C(V), then I == C(V). Indeed, let a E I, a

a

tJ. C::;(k-1)(V).

=1=

0, and let k be the order of a, that is, a E C::;k (V) but

Then we see from (3.19) that we can find

VI, ... ,

Vk E V,

such that the element

is nonzero. But t E C::;O(V) == R, so the fact that I is an ideal and tEl now implies that I == C(V). We now assume that V == W*, the dual of a real vector space W of dimension 2n, and that W is provided with a complex structure J and a Hermitian bilinear form h. Then (3 == Re h is an inner product in Wand we take Q == (3-1 equal to the corresponding inner product in V == W*. Finally, the natural complex structure in V is given by

J :== -J' : V ~ V,

(3.20)

Chapter 3. Clifford Modules

26

if J' denotes the real transposed of J. In this notation, (2.4) reads: ~(O,l) == ~ (~_ i J~).

(3.21)

In this case, the complexification V ® C is equal to the direct sum of the complex linear subspace V(l,O):== {~+iJ~

I ~ E V}

(3.22)

of forms on W which are complex-linear with respect to J, and the space V(O,l) :== {~ -

i J~

I ~ E V}

(3.23)

of forms on W which are complex-antilinear with respect to J. Write E ==

AV(O, 1) and define, for each ~ E

V,

the complex linear transformation c(~)

inEby c(~)(w) :== (~- i J~)

/\ w - i

(Q~)

w.

(3.24)

A computation as in (2.35) yields that C(~)2 == -Q (~, ~) . 1,

so c : V

-t

(3.25)

End( E) has a unique extension to a homomorphism of complex

algebras c : C(V) 0 C

-t

End(E).

(3.26)

The symbol c(a) stands for multiplication in E by the element a of the

Clifford algebra C(V). In this way the homomorphism (3.26) turns E into a C(V)-module, a Clifford module.

Lemma 3.1 Let E be the exterior algebra of V(O, 1). Then the homomorphism (3.26), which extends (3.24), is an isomorphism ofthe complex algebra C(V) 0 C onto End(E). Proof The argument that the algebra C(V) is simple works the same for the complexified algebra C(V) 0 C. Since the homomorphism c is clearly

3.3. The Supertrace

27

nonzero, its kernel is a two-sided ideal I in C(V) ® C, so I

== 0, or c is

injective. On the other hand, C(V) @C and End(E) have the same dimension over C, namely 22n

== (2 n )2. The conclusion is that the linear mapping c is

surjective as well. D One can view Lemma 3.1 as a structure theorem for the Clifford algebra

C(V), but we will use it in order to transfer Clifford algebra structures, such as its filtration, to End( E). With the notation of Chapter 2, we will apply this

== T; M, W == T x M, J == Jx , Q == f3x -1, and E == Ex. Comparing the sum of (2.31) and (2.34) with (3.24), we see that aD == i c. This is the to V

reason for our choice of c in (3.24)1. If E+ and E- denote the sum in E of the spaces AqV(O, 1) with q even and with q odd, respectively, then we have, analogously to (3.8) -

(3.12): E

== E+ E9 E-,

®C) 'c E+ C E+, (C+(V) ® C) 'c E- c E-, (C-(V) ® C) 'c E+ C E-, (C-(V) ® C) 'c E- c E+. (C+(V)

(3.27) (3.28) (3.29) (3.30) (3.31)

In particular, E+ and E- are C+ (V) 0 C-modules.

3.3

The Supertrace

We interrupt the definition of the operators D±, for a preview on the proof of the local formula for (2.40). In the computations, we will meet the supertrace strc A :== tracec A++ - tracec A-lIn [9, p. 110], the choice for c(~) is which has the same square.

J2 times the difference of e (~(O, 1))

(3.32) and i (Q~),

Chapter 3. Clifford Modules

28

of endomorphisms A of E. Here we denoted by Ajk the restriction of A to E k , followed by the projection from E to Ej along the complementary subspace. If A == c(a), then c(a+) and c(a-) corresponds with the diagonal and the antidiagonal part of the block decomposition of A, respectively. If

A == c(a) and B

==

c(b), then with the abbreviation t == tracec:

+ b- a-) = t (A+- B-+ + B+- A-+) strc c(a- b-

=

t (A-+ B+-

(t(A+- B-+) - t(B-+ A+-))

+ B-+ A+-)

+ (t(B+- A-+) -

t(A-+ B+-))

=0

Here the last identity can be seen by identifying E+ and E- by means of an isomorphism. So, with the supercommutator of (3.17), we have strc c ([a, b]s) == 0 for all a, b E C(V) (YA

@

C. Using (3.19), we see readily that

([C(V), C(V)]s) == A~(2n-l)V :==

2n-l

EB AjV. j=O

It follows that strc( A) == 0 if and only if (Y A (A) belongs to the codimension one linear subspace A~(2n-l)V of AV. In other words, the supertrace of A only depends on the component in A2nv, the volume part, of (YA (c-1(A)) in

AV. The inner product in V, together with the orientation of V, induces an isomorphism of A 2 nv with R. Let us write vol(w) E R for the real number, which in this way is assigned to the volume part of w E AV. In [9], the linear form vol on AV is called Berezin integration, see also [8, pp. 52, 53]. After tensoring with C, one now gets 2

strc A =

or

vol

(0"/\

c- 1 (A)) ,

A E End(E).

(3.33)

2This is the formula in [27, Th. 1.8] or [9, Prop. 3.21], with n replaced by 2n, because in our case the real dimension of V is equal to 2n.

3.4. The Clifford Bundle

29

A version of (3.33) has been used by Patodi [64] in his proof of Hirzebruch's Riemann-Roch theorem. Another proof of (3.33) will be given in Section 4.3. The identity (3.33) will be one of the keys in the proof of the local formula for (2.40). See Chapter 11. It can be viewed as a transfer of the computation of the supertrace in the endomorphism algebra of E, to a determination of the volume part in the exterior algebra of V. This leads in a natural way to an expression of the integrands, in (7.4) and (7.6), in terms of differential forms.

3.4

The Clifford Bundle

In our application of the Clifford multiplication, we will take, for each x E

M, the vector space V equal to T~ M, so W == T x M, J == Jx and Q == (3x -1. In this way we get the Clifford algebra C (T~ M) and the Clifford multiplication (3.34) The C (T~ M), x E M, form a smooth bundle C (T* M) over M, and for each section a and w of C (T* M) and E, respectively, we get the section

c(a) (w) of E, defined by (3.35) On the other hand, the Clifford algebra of R 2n , provided with the standard inner product, is denoted by C (2n). Each A E SO(2n) induces an automorphism Ac of C (2n).For each orthonormal frame

Ix in T xM, regarded as a

n

linear isomorphism from R onto T x M, the inverse of the transposed linear mapping lx' is a linear isomorphism from (R n)* onto T~ M. Identifying (R n )* with R n and regarding T~ M as a subset of the algebra C (T x M), we get

Chapter 3. Clifford Modules

30 SO

(Ix,)-I extends uniquely to an isomorphism Ix,c : C (2n) ~ C (T; M).

For every A E SO (2n), we have

Ix,c

0

Ac == (Ix

0

A)c ,

which justifies dropping the subscripts C from the notation.The bundle SOF M of oriented orthonormal frames in T M is a principal SO (2n)bundle, if we let A E SO (2n) act on it by Ix ~ Ix

0

A-I. We see that

the fibers of the mapping

(Ix, a)

~

Ix(a) : SOF M x C (2n)

~

C (T* M)

are equal to the orbits of the SO (2n )-action on the Cartesian product, if the action of A E SO (2n) is given by (3.36) In this way C (T* M) is identified with the associated bundle C (T* M)

== SOF M

xSO(2n)

C (2n) .

(3.37)

The identification (3.37) allows us to identify sections of C (T* M) with SO (2n)-equivariantmappings fromSOF M toC (2n). An SO (2n)-invariant connection in SOF M, which is the same as a linear connection in T M which leaves the Riemannian structure {3 invariant, then leads to the covariant differentiation \7 of sections of C (T* M), defined by (3.38) Here v denotes a vector field in M,

Vhor

is its horizontal lift in SOF M, and

in the right hand side of (3.38), a is viewed as a C (2n )-valued function on SOF M. The connection in T M used in the right hand side of (3.38) is

3.4. The Clifford Bundle

31

the Levi-Civita connection, the torsion-free one which leaves (3 invariant. The right hand side of (3.38) defines a section of C (T* M) because it is SO (2n )-equivariant. Our next task is to provide the bundle E with a Clifford

connection, a linear connection \7 == \7 E, such that we have the Leibniz rule

\7~ (c(a)(w))

=

+ c(a) (\7~ w) ,

c (\7 v a) (w)

(3.39)

for each vector field v in M and sections a and w of C (T* M) and E, respectively. It is sufficient to have this for sections a of T* M, that is, for one-forms in M. The complication however is that in (3.39) the product is the Clifford multiplication, defined by (3.24), with Q == (3-1. Another way of phrasing (3.39) is that the Clifford multiplication c is covariantly constant. A first attempt to do this is to introduce the bundle UF M of unitary frames with respect to the almost complex structure J. The unitary frame ex gives rise to the identification n

Z

~

L

Re Zr er

+ 1m Zr J x er : en ~ T x M,

r==1

which we also denote by ex. Note that ex multiplication by i in

en.

0

J

== Jx 0 ex, if J denotes the

The inverse of the pull-back e; by ex defines a

complex linear isomorphism of exterior algebras

if E( n) denotes the algebra of complex-antilinear forms on mapping A : V

---7

en. Each linear

V which commutes with J induces a complex-linear

transformation

A(A) Here A' : W

:==

(A')* : E

---7

E.

(3.40)

W denotes the transposed of A, a real linear mapping which commutes with J. So the pull-back (A')* by A' maps complex antilinear forms on W to complex antilinear forms on W. In this way, it defines a homomorphism from E to itself. In turn, the mapping A ~ A(A) is a ---7

32

Chapter 3. Clifford Modules

homomorphism of algebras

A : End (V, J)

~

End(E).

(3.41 )

Although it makes the notation a bit more cumbersome, we have chosen to distinguish A(A) from A in the notation. In the case that W

== en, we have

A(A) E End (E(n)) and

(ex

0

A')A

0

A(A) == ex,A.

So the mapping

leads to the identification

E == UF M

xU(n)

E(n)

(3.42)

of E with a bundle associated to the unitary frame bundle. Here the action of A E U(n) on UF M x E(n) is given by

(e, w) ~ (eoA- 1 , A(A)w).

(3.43)

The unitary frame bundle UF M is identified with a subbundle of the oriented orthogonal frame bundle SOF M by assigning to the unitary frame e (el' e2, ... , en) the oriented orthonormal frame (3.44) Let us assume for a moment that the horizontal spaces for the connection (the Levi-Civita one) in SOF M are tangent to UF M, so that these define a connection in UF M. Then we use (3.42) to identify the sections of E with U(n)-equivariant mappings from UF M to AC n and to define the covariant derivative with respect to the vector field v by (3.45)

33

3.4. The Clifford Bundle

The Leibniz rule (3.39) then follows from (3.38), (3.45) and the Leibniz rule Vhor

(c(a) (w)) == c (Vhor a) (w) + c(a)

(VhorW).

(3.46)

Note that in (3.46) the Clifford multiplication

c : C (2n)

~

End (E(n))

does not depend on the point x in M. Now the condition that the horizontal spaces are tangent to the unitary frame bundle just means that the Levi-Civita connection leaves the almost complex structure invariant, which, as we have seen before, is equivalent to the condition that (M, J, h) is a Kahler manifold. In the non-Kahler case, we will follow Kawasaki [45, p. 156], and exhibit

E as an associated vector bundle with a bigger structure groupSpinc (2n), which contains U(n) and projects to SO(2n). This associated vector bundle then can be provided with a Clifford connection, which in turn is used in the definition in Chapter 5 of the spin-c Dirac operator. In the Kahler case, the spin-c Dirac operator is equal to the Dolbeault-Dirac operator, but in the non-Kahler case it is different.

Chapter 4 The Spin Group and the Spin-c Group 4.1

The Spin Group

We begin the definition of the group SpinC(V), introduced by Atiyah-BottShapiro [4], with the definition of the slightly smaller spin group Spin(V). If~, Tj, ( EVe C(V),

then

So, if we write

then 7' ( a) (()

:== a . ( - ( . a == 2 Q (~, ()

Tj -

2 Q (Tj,

() ~.

( 4.1 )

This shows that 7' ( a) leaves V invariant, and in V acts as an antisymmetric operator, an element of the Lie algebra so(V) of the rotation group SO(V) in V with respect to the inner product Q. The linear subspace spin(V) of C~2(V) n C+(V), spanned by these elements a, is a Lie subalgebra of

35 J.J. Duistermaat, The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8247-7_4, © Springer Science+Business Media, LLC 2011

36

Chapter 4. The Spin Group and the Spin-c Group

C+(V), and 7' : spin(V) ~ so(V) is an isomorphism of Lie algebras. The spin group Spin(V) of (V, Q) is now defined as the connected Lie subgroup of C+ (V) x , with Lie algebra equal to spin(V). The tangent map at the identity of the homomorphism

(4.2) is equal to the isomorphism 7' from spin(V) onto 50(V) defined in (4.1), so 7 is a covering of Lie groups from Spin(V) onto SO(V). If~, 1] E V, Q(~, ~)

== Q(TJ, 7]) == 1 and Q(~, 7]) == 0, then

Since

we get the element 00

expc (t~·

1])

L

==

== cos t + (sin t) in the group Spin(V). Taking t

==

1f,

k==O

t (~.

~ . 1]

1])k

(4.3)

we see that -1 E Spin(V). Since

7( -1) == 1, the covering is not an isomorphism. Since the universal covering of SO(V) is two-fold if dim(V) > 2, we see that in this case Spin(V) is simply connected and isomorphic to the universal covering of SO(V). Themappingc : Spin(V)

~

End(E) defines arepresentationofSpin(V)

inE. SinceSpin(V) c C+(V), we see from (3.28) and (3.29)thatc (Spin(V)) leaves E+ and E- invariant. It is one of the points in Brauer and Weyl [15], that the representations of Spin(V) in E+ and E- are irreducible, and have highest weight equal to the fundamental ones, which are not equal to the highest weights of representations of SO(V).

37

4.2. The Spin-c Group

4.2

The Spin-c Group

The unitary group U (V) of V is defined as the group of the elements of

SO(V), which commute with J. If A E U(V), then its complex linear extension in V @ C, also denoted by A, leaves the subspace V(O, 1) of V @ C invariant. The restriction of A to V(O, 1) has a unique extension A(A) to an automorphism of the exterior algebra E == AV(O, 1) , which actually coincides with the mapping A(A) defined in (3.40). Using the isomorphism

(4.4)

c : C(V) ® C ~ End(E), we get an embedding c- 1 oA of U(V) as a subgroup U of C(V)

@

C.

In order to identify the Lie algebra of U, we investigate, for each ~ E V, cf. (3.20), (3.21): c(~)

c (J~)

+ i~) -

[e (~ -

iJ~)

- i

(Q~)]

0

[e (J~

-

iJ~)

- i

(Q~)]

0

[i e (~ - iJ~)

[e{~

i e (~ - iJ~)

0

2i e (~- i J~)

i (Q~) - i i (Q~) 0

0

i (QJ~)]

+i

i (Q~)]

e (~ - iJ~)

i (Q~) - i Q (~, ~).

Here we have used that i( Jw) == -i i( w) on antilinear forms in combination with QJ

== -QJ' == JQ (which follows from the antisymmetry of J with

respect to Q). Now for any a E

and w E W, the operator e(a)

V(O,1)

0

i(w) is a

derivation of the exterior algebra E ofV(O' 1), which on V(O, 1) acts by sending

w to (w, w) a. So, if a ==

~

. J~ + i Q(~,

~), then

c(a) is equal to A'(A),

the derivation of E, induced by some linear transformation A in W, which commutes with J. Since ~ . J~ == -J~ . ~

== ~ (~ · J~ - J~ . ~) ,

we see from (4.1) that

r'(a) : (

~ 2

[Q (~, ()

J~

- Q (J~, ()

~]

Chapter 4. The Spin Group and the Spin-c Group

38

belongs to the Lie algebra u(V) of U (V). Running through the various identifications, we see that c (a) == A' (T' (a)) .

(4.5)

We also get that T'(a), considered as a complex linear transformation in V, has complex trace equal to 2i Q(~, ~). Therefore

a - ~ tracec T'(a)

==

a ~ i Q (~, ~)

== ~ · J~ E spin(V).

(4.6)

It follows from (4.6), that the Lie algebra of

U is contained in spin(V)

:== c- 1 oA (U(V))

+ u(l), where u(l)

(4.7)

== iRis the Lie algebra of U(l),

the unit circle in C, viewed as a subset of C(V) 0 C. So if we define Spinc (V) :== Spin(V) . U (1), then we see that U

c

(4.8)

SpinC(V), and actually SpinC(V) is the group generated

by U and Spin(V). Note that U(l) commutes with everything, so Spin(V) n U(l) belongs to the center of Spin(V), which is equal to {±1}, because the center of SO(V) is trivial. It follows that the product map p : (s, z) ~ s . z induces an isomorphism Spin(V) x U(l)/{±l} ~ SpinC(V),

(4.9)

where -1 denotes the element (-1, -1) in Spin(V) x U(l). One sometimes sees the left hand side of (4.9) as the definition of SpinC(V), but we prefer to have Spine (V) defined as a subgroup of (C+ (V) 0 C) x, the group of invertible elements in C+ (V) 0 C. The injection ofU(l) into SpinC(V), followed by the projection 1r from SpinC(V) onto SpinC(V)/ Spin(V), has kernel equal to ±1. This means that there is a unique isomorphism i :

SpinC(V)/ Spin(V) ~ U(l),

(4.10)

4.3. Proof ofa Formulafor the Supertrace

39

such that ~ (z· Spin(V)) =

Z E

Z2,

U(l).

(4.11)

We will also write ~(8) = ~ (8 . Spin(V)), if 8 E SpinC(V). Finally, the homomorphism (4.2) extends to the homomorphism T :

a 1---+

(v

1---+

a · v· a-I) : SpinC(V)

which has kernel equal to U(l), the unit circle in C

----+

SO(V),

(4.12)

c C(V) ® C.

Integrating (4.5), we get

c(u) = A 0 r (u ), In turn, this implies that

rlu

u E U.

(4.13)

is an isomorphism from U onto U(V), with

1

inverse equal to c- oA. The equation (4.6) shows that

{z· u I u

E

U, z

E C, Z2 dete

r(u)

= I} C Spin(V)

(4.14)

is equal to the preimage in Spin(V) of U(V) under the double covering r : Spin(V)

4.3

-t

SO(V).

Proof of a Formula for the Supertrace

As an application of the previous section, we give a proof of the formula (3.33), which expreses the supertrace in terms of the volume part. Let

Ej

be

a unitary frame in V; consider the element n

a =

L

Ej ·

J Ej

+i n

E u.

j=l

Then (4.6) shows that c(a) is equal to the derivation of E, which acts as multiplication by 2i on E 1 . It follows that

strcc (expc(ta)) =

t(

q=O

-1)q (

n ) e2iqt = q

(1 _ e2it )n.

40

Chapter 4. The Spin Group and the Spin-cGroup

On the other hand, we read from (4.3) that

expc(ta) == eitn

n

II (cost+sintEj. JEj) , j=l

so

vol ((JA (expc(t a)))

= eitn (sin tt = (2i)-n (e 2it -1

Comparing the two expressions now yields (3.33).

r.

Chapter 5 The Spin-c Dirac Operator In this chapter we start by viewing E as a principal bundle for the group SpinC (2n), which contains the unitary group. The fact that SpinC (2n) also contains the spin group Spin(2n), which is a twofold cover of SO(2n), allows us to introduce a connection in this principal bundle which has the desired compatibility with the Levi-Civita connection. Using this connection in E, we will give the definition of the spin-c Dirac operator Din (5.14). In Lemma 5.5 it is established that D is selfadjoint and has the same principal symbol as the Dolbeault-Dirac operator.

5.1

The Spin-c Frame Bundle and Connections

The principal Spinc (2n )-bundle over M, which will replace the unitary frame bundle UF M in (3.42), is the spin-c frame bundle Spinc F M == UF M

xU(n)

SpinC (2n),

(5.1)

where the action ofU(n) on UF M x SpinC (2n) is given by

(e,r)l-+(eoA-1,c-1oA(A).r),

AEU(n).

41 J.J. Duistermaat, The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8247-7_5, © Springer Science+Business Media, LLC 2011

(5.2)

Chapter 5. The Spin-c Dirac Operator

42

Admittedly, the name is not ideal, because the elements in the fiber, the "spin-c frames", do not have a straightforward interpretation as bases in some vector space. Since the action of U(n) on UF M is free, the free action of Spinc (2n) on itself by right multiplication passes to a free action of SpinC (2n) on Spinc F M, exhibiting the latter as a principal SpinC (2n)bundle over M. Since the Levi-Civita connection of the oriented orthonormal frame bundle SOF M in general does not restrict to the subbundle UF M, we will not use the definition (5.1) in order to define the connection in Spinc F M. Instead, we consider the mapping a :

(e, r)

~

(f(e), T(r)) : UF M x SpinC (2n)

~ SOF M x

SO(2n) (5.3)

with f(e) defined as in (3.44), and the mapping (3 : (e, r) ~ (e, ~(r)) : UF M x Spin C (2n) ~ UF M x U(l).

(5.4)

Using (4.13), we see that a maps U(n)-orbits into SO(2n)-orbits for the action The quotient for the latter action is isomorphic to SOF M, so a induces a bundle mapping

a : Spinc F M

-f

SOF M.

On the other hand, combining (4.11) and (4.14), we see that (3 maps U(n)-orbits in UF M x SpinC (2n) into U(n)-orbits in UF M x U(l) for the action

A

1---+ ((

e, z)

1---+

(e

0

A-1, Z detc A)) .

(5.5)

The quotient ofUF M x U(l) by the U(n)-action is a principal U(l)-bundle. It can be identified with the unit circle bundle U (K*) of the dual K* of the canonical line bundle K

== T* M(n,O) of (M, J). Indeed, an element

WET; M(n,O) can be identified with the mapping

5.1. The Spin-c Frame Bundle and Connections

43

which satisfies

w (eoA-1)

= detc

(eoA-1 oe- 1 ) w(e)

=

(detCA)-l w(e).

The mappings a and 13· together define a bundle map 'Y: SpincFM ~ ~* (SOFM x U(K*)) ,

(5.6)

where ~ : x ~ (x, x) denotes the embedding of M onto the diagonal in

M x M. If r E SpinC (2n), T(r) == 1 and i(r) == 1, then r == z E U(l) and (4.11) shows that r

== ±1. Using this, one obtains that (5.6) is a double

covering. Now choose a U(l)-invariant connection in the principal U(l)-bundle

U(K*). (This is the same as a linear connection in the line bundle K* for which the Hermitian structure in K* is covariantly constant.) Together with the Levi-Civita connection in SOF M, we get a connection in the product bundle, which then is pulled back by

~

and 'Y to a connection in the spin-

c frame bundle. That is, the horizontal space in the tangent space at p of Spinc F M consists of the vectors v, such that Tp a(v) is horizontal in the tangent space at a (p) of SOF M, and T p 13(v) is horizontal in U (K*). The first condition determines v up to the addition of a scalar in the fiber direction, whose freedom then is eliminated by the second condition. Since a and

13

are right SpinC (2n)-equivariant, the connection is right

Spinc (2n )-invariant. Here the action on SOF M is the right action of SO(2n), via the homomorphism T, defined in (4.12). And the action on U (K*) is the right action ofU(l), via the homomorphism

L,

defined in (4.10).

The mapping e ~ (e, 1) defines an embedding of UF Minto Spinc F M. We note in passing that if (M, J, h) is a Kahler manifold, then the fact that the Levi-Civita connection in SOF M is tangent to the subbundle UF M implies that the connection in Spinc F M, which we just defined, is tangent to the subbundle UF M of Spinc F M.

Lemma 5.1 The embedding UFM x E(n) ~ SpincFM x E(n)

44

Chapter 5. The Spin-c Dirac Operator

leads to an isomorphism

E

= UFM xU(n) E(n) ~ SpincFM xS pin (2n) E(n), C

(5.7)

if, in UFM x E(n), the U(n)-action is given by (3.43) and in SpincFM x E(n) the action of s E SpinC (2n) is given by

(g, w)

~

(s· g, c(s)(w)).

Proof The SpinC (2n)-orbits in Spinc F M x E(n) can be identified with the U(n) x SpinC (2n)-orbits in UF M x SpinC (2n) x E(n), where the action of

(A, s) E U(n) x SpinC (2n) is given by

(e, r, w) Ifr

I---t

(e 0 A- 1 , c- 1 oA(A)· r· S-l, c(s)(w)).

= 1 and c- 1 oA(A)· 1· S-l = 1, then A(A) = c(s). This proves (5.7).

D This means that the embedding of UF Minto Spinc F M exhibits E as a vector bundle which is associated to the principal SpinC (2n)-bundle Spinc F M by means of the representation c : SpinC (2n) ~ End (E(n)). In the same way as in (3.45), the Spinc (2n )-invariant connection in Spinc F M leads to a covariant differentiation of sections of E, which we again denote by~.

If (M, J, h) is a Kahler manifold, then \7 is equal to the covariant

differentiation defined in (3.45).

45

5.1. The Spin-c Frame Bundle and Connections

Lemma 5.2 The Clifford algebra bundle C (T* M) is isomorphic to the associated SpinC (2n)-bundle

(5.8) where the action of 8· E Spin C (2n) on SpincFM x C(2n) is given by

(p, a)

f-t

(sp, s· a· s-1) = (sp, T(s)(a)).

(5.9)

Proof Using the definition (5.1) ofthe principal Spinc (2n )-bundle Spinc F M, we see that (5.8) is equal to the space of orbits in UF M x SpinC (2n) x C(2n) for the action of A E U(n),

If 8

== c- 1 oA(A), then S

E

8

E Spin C (2n), given by

U, so (4.13) yields that

A 0 7(S) == c(s) == A(A), which implies that 7(8) == A. So the embedding (e, a) ~ (e, 1, a) induces an isomorphism C (T* M) == UF M

xU(n)

C(2n) ~ Spinc F M

xS pin c(2n)

C(2n). (5.10)

This is the isomorphism meant in the lemma. 0 Again, as in (3.45), the Spinc (2n )-invariant connection in the spin-c frame bundle leads to a covariant differentiation \7 of sections of C (T* M). Fortunately, we have:

Chapter 5. The Spin-c Dirac Operator

46

Lemma 5.3 The covariant differentiation in (5.8) is equal to the covariant

differentiation in (3.37). In each case the covariant differentiation is defined by (3.38). The horizontal lifts of vector fields in M are with regard to the previously defined connection in SpincF M and the Levi-Civita connection in SOF M, respectively. Proof Sections of C (T* M), viewed as a principal SpinC (2n)-bundle, are mappings a from UF M x SpinC (2n) to C(2n) such that, for each e E UF M,

r, s E SpinC (2n), and A E U(n):

a (e 0 A-I,

c- I oA(A)· r· 8- 1 )

This implies in particular that a (e, -r)

=

T(8) a (e, r).

== a (e, r). Therefore, using (5.6), a

can be identified with a mapping from SOFx M x UFx M x U(l) to C(2n), such that, for each

f

E SOF x M, e E UFx M, z E U(l), and A E U(n),

C

s E Spin (2n):

a

(1

0

T(8)-1, e 0 A-I, /,(:) dete A) = T(8) a (j, e, z).

From the fact that T

(y . s) ==

T

(s)

for any y E C, we see that a (f, e, z) does not depend on z. It follows in turn that it does not depend on e either. So the covariant derivative of a is equal to the horizontal derivative of a, viewed as a mapping from SOF M to

C(2n), which satisfies, for each f E SOF M and A E SO(2n): a (J 0 A-I) = A (a(j)) .

(5.11)

That is, a is viewed as a section of (3.37). 0 Using the Leibniz rule (3.46), we arrive at the following conclusion.

5.2. Definition of the Spin-c Dirac Operator

47

Lemma 5.4 Choose any U(l)-invariant connection in U (K*), the unit cir-

cle bundle in the dual K* of the canonical line bundle K of (M, J). Then the covariant differentiation \7 on E, defined by the Levi-Civita connection in SOFM and the connection in U (K*), is a Clifford connection. If L is a complex vector bundle over M, then the L-valued (0, ·)-forms are the sections of E ® L. We assume that a connection in L is chosen which leaves the Hermitian structure h L in L invariant, and denote the corresponding covariant differentiation of sections of L by \7 L . Then the covariant differentiation \7 of sections of E 0 L (which depends on the choice of \7 L , although this will be suppressed in the notation) is defined by the Leibniz rule \7 v

(~Wjkej ® lk) == L ],k

(VWjk)

ej ® lk

+ WjkVhorej ® lk + Wjkej 0

\7L lk·

j,k

(5.12) Here v is a vector field in M, the ej and lk form an arbitrary local frame in E and L, respectively, and the complex-valued functions Wjk are the coefficients of W with respect to the local frame ej 0

lk

in E 0 L. A straightforward

calculation shows that the right hand side in (5.12) is independent of the choice of local frames, so the local definitons (5.12) piece together to a global differential operator on M. It is also clear that \7 is determined by the conditions that it is a covariant differentiation and that \7 v (w 0 A) ==

vhorw

0 A + W 0 \7 LA,

(5.13)

for every local section wand A of E and L, respectively.

5.2

Definition of the Spin-c Dirac Operator

At long last, we are ready for the definition of the spin-c Dirac operator, for which we want to have the local heat kernel formula.

Chapter 5. The Spin-c Dirac Operator

48

Definition Given connections in the dual K* of the canonical line bundle K == T* M(n, 0), and in the complex vector bundle L, the spin-c Dirac operator D, acting on sections of E 0 L, is given by 2n

Dw =

l:= c(cPj) (V'fjW).

(5.14)

j=l

Here

(h )~:1

is an arbitrary (not necessarily oriented or orthonormal) local

frame in T M and (cPj )~:1 is the corresponding dual frame. That is

Since (5.14) does not involve differentiations of the frames, it is independent of the choice of the local frames, so the local definitions (5.14) lead to a globally defined operator D in

r

(M, E 0 L).

Remark A straightforward generalization can be given as follows. A spin-c structure on M is a principal SpinC (2n)-bundle P over M, together with a complex line bundle

k

with Hermitian structure and connection over M, and a twofold

=

U(k)), which intertwines the action ofr E Spin (2n) on P with the action of (T(r), t(r)) on Q. All the covering from Ponto Q

,6.* (SOFM x

C

previous definitions can now be repeated, in which K* is replaced by K. This leads to a Dirac operator, acting on sections of E

@

L, in which now

E == P xSpinc E(n). The reader may verify that most of the results in the sequel carryover to this more general operator D, just by replacing K* by

k

at every occasion.

However, the relatioQ with the almost complex structure is lost. For a compaGt oriented manifold M of any dimension, a spin-c structure exists on lvl if and only if the Stiefel-Whitney class

W2

E H 2 (.l\1, Z/2Z)

is equal to the reduction modulo two of some c E H2 (M, Z). Hirzebruch

5.2. Definition of the Spin-c Dirac Operator

49

and Ropf [40, 4.1,iv)] proved that every four-dimensional compact oriented manifold M satisfies this condition; it therefore can be provided with a spin-c structure. On the other hand, it is a result of Wu Wen-Tsun [76, Th. 10, p.74] that a four-dimensional compact oriented manifold M can be provided with an almost complex structure, if and only if there exists a class 4 2 2 C E R (M, Z)suchthatw2 == cmod2andc == 2e+PI. Heree E H (M, Z) and PI E H 4 (M, Z) are the Euler class and the first Pontryagin class of M, respectively. See also [40, Th. 4.6 and 4.1,ii)]. For some more details on spin-c structures, cf. Lawson and Michelsohn [51, Appendix D]

Lemma 5.5 The spin-c Dirac operator D is selfadjoint. Its principal symbol is given by (JD(~)

== i c(~),

(5.15)

which in turn is equal to the principal symbol ofthe Dolbeault-Dirac operator

2 (a + 13*).

Proof Let wand 1/ be sections, one of which has compact support contained in an open subset of M in which we have a local frame Ij. Then we get, with the notation (5.14):

(W, D*v) = (Dw, v) = = -

~/M

~1 h(c(

o.

(6.34)

See also [75, Ch. IV, Thm. 2.4]. The idea is to apply Bochner's technique to conclude from Dw == 0 that, if we write R for the sum over j in the right hand side of (6.27),

which in view of the positivity of R then implies that the parts of degree q > 0 in w vanish. The conclusion (6.34) then follows from the isomorphism of Hq (M, 0(£)) with the kernel of D == 2(8 + 8*) in the space of sections of T* M(O,q) 0 L.

Chapter 7 The Heat Kernel Method 7.1

Traces

The goal of the heat kernel method is to express (2.40) as an integral over the fixed point set M'Y in M of the transformation 'Y. Here M'Y == M if 'Y is the identity. The method is based on the following observations about arbitrary elliptic differential operators D, acting on sections of a smooth vector bundle

F over a compact manifold M, which admits a a direct sum decomposition F == F+ E9 F-. In our case, F == E 0 L, with the splitting F± == E± 0 L, and D is the spin-c Dirac operator. For the required facts about trace class operators, see for instance Hormander [42, Sec. 19.1], or Duistermaat [19]. For the general vector bundle F, we write

r

==

r(M, F), and

r±

==

r(M, F±) for the space of smooth sections of F and F±, respectively. It is assumed that D+ : r+ ~ r- is an elliptic differential operator, so its adjoint, denoted by D-, maps r- to r+. If D == D+ EB D-, then D is a selfadjoint elliptic operator on

r±.

r,

with D± equal to the restriction of D to

Write N± == ker D± ==

r± n ker D

69 J.J. Duistermaat, The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8247-7_7, © Springer Science+Business Media, LLC 2011

70

Chapter 7. The Heat Kernel Method

for the kernel (null space) of D± in in r±, and let C± be the orthogonal complement of N± in r±. Then C+ is equal to the range of D- and D- is an isomorphism from C- onto C+. Similarly, C- == D+ (r+) and D+ is an isomorphism from C+ onto C- . Now let T be a trace class operator in r, which commutes with D and also preserves the spaces r±. We write T± for the trace class operator in r± which is obtained by restricting T to r±. The condition T D == D T then translates into the two conditions T- D+ == D+ T+ and T+ D- == D- T- . It follows that T± leaves N± and C± invariant, and that tracec T± == tracec T± IN±

+ tracec T± Ic± .

However, if we write D~ == D+lc+, then

T +Ic+ == (+) Dc

0

T _Ic-

0

(+)-1 Dc ,

and this shows that

Hence, in the computation of the difference of the trace of T+ and T-, the contributions coming from C+ and C- drop out, and we get the Lefschetz principle

tracec T+ IN+ - tracec T-/ N - == tracec T+ - tracec T-.

(7.1)

Now we assume that T± are integral operators with smooth integral kernels K±(x, y):

Here, for each x E M and y EM, K±(x, y) is a linear mapping from the fiber Pi over y to the fiber F:- over x. And dx denotes a smooth density in M.

71

7.1. Traces

The point is, that on a compact manifold an integral operator K op with smooth kernel K(x, y), acting on sections of a vector bundle F, is of trace class, and the trace is equal to the integral tracee K op =

1M tracee K(x, x) dx

(7.2)

of the trace of the integral kernel, over the diagonal in M x M. Note that

K(x, x) is a linear mapping from Fx to itself, so it makes sense to take its trace. In order to prove (7.2), one takes an L 2 -orthonormal basis ek(x) of sections of F, and writes

(7.3)

K(x, y) == E(Kopek' el)· el(x) ® ek(Y)*. k,l Here ek(Y)* denotes the linear form f ~ (f, ek(Y)) on

F:.

One obtains

(7.3) by checking that the integral operator, with kernel equal to the right hand side, gives the same result as K op when applied to a smooth section u of F. If we then let u approach the Dirac-delta function at Y, we get (7.3). Now putting x == y in (7.3), taking the trace of the resulting endomorphism of F x , and integrating over x, one obtains

Inserting (7.2) in (7.1), we get tracee T+IN+ - tracee T-IN- =

1M stre K(x, x) dx,

(7.4)

where the supertrace strc K(x, x) of the endomorphism K(x, x) of Fx ==

Fi EB F x- is defined by strc K(x, x) :== tracec K+(x, x) - tracec K-(x, x).

(7.5)

The operator I in (2.40), which commutes with D and leaves r± invariant, is not an integral operator with smooth kernel. Rather, its integral kernel-is

Chapter 7. The Heat Kernel Method

72

a Dirac-delta type of distribution along the graph x

== ry(y) of rye However,

if we have an integral operator T with smooth kernel as above, such that ry commutes with T, then ry

0

kernel

(-y 0 Tu)(x)

=

T == Tory is an integral operator with smooth

1M 'YIF1'-

I

X

K

('Y- 1X ,

y) u(y) dy.

If we moreover assume that T± acts as the identity on N±, then we get, by replacing T by ry T in (7.4), tracec 'YI N + - tracec 'YI N - =

1M strc ('YIFx

0

We have also simplified the notation by writing y

K(x, 'Y x )) dx.

(7.6)

== ryx; it is assumed that ry

preserves the density in M.

7.2

The Heat Diffusion Operator

In the heat kernel method, the above program is carried out with

(7.7) the heat diffusion operator generated by the operator D 2

== D- D+ (f)D+ D-.

For each t > 0, T(t) is an integral operator with integral kernel K(t, x, y), which depends smoothly on all the variables (t, x, y). T( t) can alternatively be defined as the solution of the following ordinary differential equation for t

> 0, with initial condition for t 1 0,

d~~t) + Q 0 T(t) =

0,

lim T (t) == 1.

(7.8)

tlO

Obviously e- tQ acts as the identity on the kernel of Q, which is equal to

the direct sum of the kernel of D+ and the kernel of D-. Also, e t Q leaves r± invariant. The fact that ry commutes with D, hence with Q

== D 2 , implies

that ry commutes with T( t). So all the conditions are satisfied and, with

T == T(t), (7.6) becomes tracec 'YI N + - tracec 'YI N - =

1M strc ('YIFx K(t,

X,

'Y x )) dx.

(7.9)

73

7.2. The Heat Diffusion Operator Note that the left hand side is independent of t

> 0, so apparently the right

hand side is independent of t as well. The next step is the observation that K(t, x, y) is rapidly decreasing as t

1 0 for x

-=I- y, and has an asymptotic expansion for t lOin powers of t

along the diagonal x

== y of the form

L

00

K(t, x, x) ~ t- d / 2

t k Kk(x),

t

1 o.

(7.10)

k=O

Here d denotes the dimension of M, in our case d == 2n. Each of the coefficients K k (x) is given by a universal rational expression in the derivatives of the coefficients of the operator P at the point x, where the order of differentiation and the complexity of the expressions increase with k. See Gilkey [32, Sec. 1.7] for a proof in this general context. If we insert this in the right hand side of (7.4), then we get the conclusion that

if k

i- d/2, whereas index D+ = dime N+ - dime N- =

r

JAl

stre K d/ 2(X) dx.

(7.11)

In particular, the index of D+ can only be nonzero if the dimension of M is even, say 2n, and then it is equal to the integral of the supertrace of the n-th coefficient in the asymptotic expansion for the heat kernel on the diagonal. If 1 is not equal to the identity, then the only contributions to the integral (7.9), which are not rapidly decreasing as t

1 0, come from a shrinking region

near the set. of fixed points of 1. In the case of nondegenerate isolated fixed points, the asymptotic expansion actually starts with a constant term (as a function of t), which can easily be determined, see for instance [32, Lemma 1.8.3]. If the fixed point set Mr of 1 in M locally is a smooth manifold and the action of 1 is nondegenerate in the normal bundle, then one gets a conclusion analogous to (7.11), but with the integration over M replaced by

74

Chapter 7. The Heat Kernel Method

an integral over the fixed point set, d replaced by the dimension of M'Y and with a suitably modified integrand. The last step of the heat kernel method is to express the integrand

strCKd/2(X) in (7.11) in terms of the geometric data, used in the definition of the operator D. For higher dimensions d == 2n, this seems at first sight to be an intractable quantity because K d/2 (x) is a complicated expression involving derivatives of high order of the geometric data at the point

x. We will follow here the method of Berline-Vergne, to explain why the supertrace of K n (x) is given by an explicit polynomial expression in curvatures (and no derivatives of curvatures). It consists of introducing a principal Gbundle P over M, a heat kernel KP(t, p, q) onR>o x P x P, and then writing the heat kernel K(t, x, y) as an integral over 9 E G of KP(t, p, 9 q) p(g), where p and q are in the fiber in P over x and y, respectively. See (9.24). For (7.11), we have x if 9

=I

== y and we may take p == q, but then still q =I 9 q

1. This makes that even for (7.11), we will need information about

the leading term of the heat kernel expansion away from the diagonal. The required information will be provided by Theorem 8.1, the proof of which is the main goal of Chapter 8. The factor 9

t---t

p(g) is a homomorphism from G to the ring of endomor-

phisms of E, in which one has the filtration coming from the Clifford algebra

C(E) via the isomorphism of Lemma 3.1. The supertrace depends only on the highest order part of the endomorphism with respect to the filtration. In Chapter 11 we will see how the factor p leads to the desired formula for the supertrace in terms of the curvatures. If ry

=/:

1, then one gets a local formula by multiplying the supertrace

integrand with a test function 7/J with support in a small neighborhood of a point of the fixed point set M'Y, where M'Y is a manifold. Integrating, in (7.6), over the fiber N x through x E M'Y of a ry-invariant normal fibration

N to M'Y, one is left with an integral over M'Y. The point is that in the latter integral the integrand converges for t 1 0 to 1/J times an expression in

7.2. The Heat Diffusion Operator

75

terms of curvatures and the tangent action of, on T x (Nx ). So in this case the supertrace integrand converges as t lOin distribution sense, with limit equal to a density in the fixed point manifold M'. Furthermore, this density has a natural geometric interpretation. For these local results, one does not need the manifold M to be compact, or the operator e -t Q to exist, the only ingredient is the heat kernel expansion, defined as a formal power series. In this sense the local formula is a stronger statement than the integral formula for the virtual character. Another remarkable feature is that even for the global integral formula, the proof uses only analysis and differential geometry, and no arguments from algebraic topology, like K -theory, have been used. Admittedly, the proof uses quite a lot of differential geometric constructions, and it depends on ones background, whether one would consider it simpler than the proofs which combine differential geometry with algebraic topology. For an exposition of the latter, see for instance Lawson and Michelsohn [51, Ch. 3].

Chapter 8 The Heat Kernel Expansion 8.1

The Laplace Operator

Our operator Q == D 2 , the square of the spin-c Dirac operator, has scalar principal symbol. So for the discussion of the asymptotic expansion of its heat kernel, we may restrict ourselves to the case that Q is a second order differential operator, acting on sections of a complex vector bundle F over a d-dimensional Riemannian manifold (M, {3), with principal symbol given by (jQ(~)

== {3-1 (~,

~) . 1,

~ E T* M.

(8.1)

The goal of this chapter is to prove the asymptotic expansion in Theorem 8.1, for the integral kernel of the operator e- t Q, as t ! O. In it, the covariant differentiation \7 of sections of F is used, which is introduced in Lemma 8.1. The appearance of the geodesic distance d(x, y) in the factor e- d (x,y)2/4t is motivated by the equation (8.20) and Lemma 8.3. The quantity j(x, y) in (8.48) is the Jacobian at x of the exponential mapping, centered at y, cf. (8.44) and (8.46). The factor j (x, y)-1/2 in (8.48) is responsible for the appearance of the Todd class in the formula for the Riemann-Roch number and the Lefschetz number, in Proposition 13.1 and Proposition 13.2, respectively. 77 J.J. Duistermaat, The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8247-7_8, © Springer Science+Business Media, LLC 2011

78

Chapter 8. The Heat Kernel Expansion If \7 is a covariant differentiation in F, with corresponding Laplace

operator

~

defined by (6.1), then (8.1) is equivalent to the condition that Q

and ~ have the same principal symbol, or that R == Q - ~ is of order at most one. In general R will actually be of order one. However, we have: Lemma 8.1 There is a unique covariant differentiation \7 in F, such that if ~

denotes the corresponding Laplace operator, the operator R == Q -

~

is

oforder zero.

Proof For convenience of notation, we take for fj a local orthonormal frame. Then we can write

Q - ~ ==

LR

j

\7 fj

+ zero order,

(8.2)

j

in which the coefficientsRj

== Rj , x are uniquely determined endomorphisms

of Fx , depending smoothly on x. Another covariant differentiation ~ in F is obtained by adding to \7, at each x E M, a linear mapping

That is, for each vector field v in M, we have ~ v == \7 v

+ r( v).

If Ii is the

Laplace operator defined by \7, then

Ii ==

~

+2 L

f(fj) \7 fj

+ zero order.

j

So, modulo zero order operators, Q -

Ii

is the first order operator with

coefficients equal to R j - 2 f (Ij). We get that these vanish, if and only if f is given by

f(v) == ~

L/3 (v, Ij) R j j

for each vET M. 0

(8.3)

8.2. Construction of the Heat Kernel

79

In classical quantum mechanics, the zero order part R of a Schrodinger operator Q ==

~

+R

is called the potential if the Laplace operator

~

is

accepted as the kinetic energy Hamiltonian. Since in our application we are quite far from this context, we will not use this terminology here. If Q is equal to the square of the spin-c Dirac operator D, then (6.10) shows that the connection is equal to the one in E ® L, which was used in the definition (5.14) of D. And R is equal to a sum of terms in which curvature operators are combined with Clifford multiplications.

8.2

Construction of the Heat Kernel

One says that the section u(t, x) ==

Ut(x)

of the vector bundle F over

R>o x M satisfies the heat equation for the operator Q if

~~

+ Qu == 0,

(8.4)

and one says that u has initial value Uo if lim Ut == tiO

(8.5)

Uo.

One can make various choices about the topologies in the space of sections with respect to which one requires (8.5) to hold. In our setting of smooth objects, it is natural to assume convergence with respect to the Coo -topology. Then (8.4) implies that also all derivatives of Ut with respect to t converge in the Coo -topology as t

1 0: · 8kUt 11m 8t k tiO

_ -

If M is compact, then a solution unique. It is given by

Ut

==

(.

U

-

Q)k Uo.

==

Ut

of (8.4) and (8.5) exists and is

T(t) Uo, with T(t) as in (7.7). In terms of the

heat kernel K(t, x, y), this means that

u(t, x) =

1M K(t, x, y) uo(y) dy.

(8.6)

80

Chapter 8. The Heat Kernel Expansion

We will choose the density dy to be the one for which the orthonormal frames in the tangent spaces with respect to (J have unit density. Using the existence of a heat kernel for the adjoint of P with respect to some Hermitian structure in F, one also has a strong uniqueness result, saying that if E ilt

are distributional solutions of (8.4) on ]0,

to 0 in distribution sense as t

1 0, then Ut

E [,

such that Ut

== ilt for all 0

> 0, and Ut and - ilt

0 and p, q E P is a linear map from E @Ln(q) to E 0L n (p) , denote the integral kernel of the heat diffusion operator e- tQ , where Cd is as in (9.22). If M is not compact, then we take for K

P

an asymptotic heat kernel, as discussed after Theorem 8.1.

y) of e- tQ with K P, we begin by observing that a linear mapping K (x, y) from E y 0 L y to Ex 0 Lx corresponds, if x == 1r(p) and y == 1r(q), to a linear mapping K(p, q) from E 0L y to E@L x , such that In order to be able to compare the integral kernel KM (t,

K(g p, h q) == p(g)

0

K(p, q)

0

p(h)-l,

g, h

E

X,

G.

(9.23)

The h-behavior is necessary and sufficient in order to get a uniquely defined element when applied to an element

1/

of

r == fG,

and the g-behavior in

order to get an element of r when K op has been applied.

With this identification, we can now formulate the following result.

Lemma 9.4 If KP(t, p, q) denotes the (asymptotic) integral kernel ofe-tQ, with Qas in (9.22), then the integral kernel K M (t, x, y) ofe- tQ is given by (9.24)

Proof We apply Lemma 9.3. The fact that Cas commutes with both operators leave

r invariant now implies that

Qand that

Writing this in terms of the integral kernels, and using that

(9.25)

9.4. The Heat Kernel we get, for any

1/

109

E f, PEP, x ==

7r(p),

1M K M(t, X, y) I/(Y) dy = (e- tQ 1/) (x) = e- t Cas (e- tQ 1/) (p) = e- Cas 1M (fc KP(t, p, 9 q) I/(g q) d9) ~Gq) = e- Cas 1M (fc KP(t, p, gq) p(g) I/(q) d9) ~Gq). t

t

0

Comparing the first and the last term, we get (9.24). That is, after we have verified that the right hand side in (9.24) has the required property (9.23) of being an integral kernel for an operator on NJ. In order to see this, we use that the G-action in f' commutes with Q. This makes that the G-action commutes with e- tQ , which in turn means for its integral kernel that

KP(t, gp, gq)

== p(g)

0

KP(t, p, q)

0

p(g)-l,

9 E G.

Indeed, using (9.2) and the G-invariance of the density in P, we get for an integral operator A with integral kernel K that

(g AI/)(p)

=

p(g)

t

K(g-1 p, q) I/(q) dq,

K(p, q)op(g) l/(g-1 q) dq

=

t

p(g) (AI/)(g-1 p)

=

whereas on the other hand

(A(g 1/)) (p)

=

t

K(p, gp)op(g) I/(q) dq.

We now replace p and q in (9.24) by gl P and g2 q, respectively. Then we make the substitution of variables -

9 == g1 9 g2

-1

.

Using the invariance of the density in Gunder left- and right-multiplications, and the fact that Cas commutes with p( G), we get that

K M (t, g1 p, g2 q)

=

P(g1)

0

K M (t, p, q) 0 P(g1)-1

0

P(g1)

0

p (92 -1)

This implies the property (9.23) for the right hand side in (9.24). 0

.

110

Chapter 9. The Heat Kernel on a Principal Bundle

9.5

The Expansion

The next step is to substitute the expansion (8.48), with Q replaced by the operator

Q of (9.22), in the right hand side of (9.24).

If we are aiming at

(7.4), then we may put q == p, and we have to determine the constant term in the asymptotic expansion, for t

1 0, of the supertrace of

e- t Cas (47Tt)-! dim?

.L t k Ak(p, 9 p)

fc e-

d(p,gp)2/4t j(p, 9 p)-1/2

00

0

p(g) dg.

(9.26)

k==O

Here j == jp is the determinant as defined in (8.44), but now for the Riemannian structure in P. The leading term A o(p, 9 p) is equal to the parallel transport in the bundle 1r* L along the geodesic from 9 p to p, defined by the connection (9.21). It is an important remark that, due to (9.22), all the terms

Ak(p, 9 p) are of the form 1 0 Tk(p, 9 p), where Tk(p, 9 p) E End (Lx), x == 1r(p). The action on E has been transferred to the factor p(g), which acts on the fixed vector space E. In the integral (9.26), one only gets contributions which are not rapidly decreasing, which come from 9 in a shrinking neighborhood of 9

== 1, so

it becomes natural to make the substitution of variables 9 == exp X, where X varies in a small neighborhood of 0 in g. The following lemma is a preparation for the computation in Lemma 9.6, of the tangent map of the exponential mapping at

XP,p E

T p P.

Lemma 9.5 Let P be a Rienlannian manifold and let v be a vector field in P, such that the solution curves of v are geodesics, and that the flow etv of v consists of isometries. For each PEP, let B (p) denote the linear endomorphism B(p) : u ~ -2 (\7 u v)(p). Then (9.27)

9.5. The Expansion

111

Proof Let v denote the vector field in T P, which has flow equal to t

~

T e t v.

Let w denote the geodesic vector field in T P, as in (8.23). The condition

that the v-flow consists of isometries means that v commutes with w. The condition that the v-solution curves t ~ p( t) are geodesics means that the curves t ~ (p(t), p'(t)) in T P, which by definition are solution curves of

V, are also solution curves of w. In other words, the vector field w -

v is

identically equal to zero on the submanifold

v(P) == {(p, v(p)) I pEP} of T P, the graph of the vector field v. If v(p) == 0, then the Tpet v form a one-parameter group of rotations in Tp P, so the solution curves of v near p can only be geodesics if (\7 v) (p) == O.

:I 0, and choose local coordinates a near

We therefore may assume that v(p)

p in which the vector field v is constant. In such local coordinates the vflow for t == 1 is a translation, so its tangent map from T p P == R d to T expp v(p) P = R d is equal to the identity. Here d = dim P. In the corresponding local coordinates (a, b) in T P, the linearization of w -

v is equal to the right hand side of the system of ordinary differential

equations

da == b dt '

db dt = A a + B b,

for some d x d-matrices A and B. And the tangent space to v(P) now corresponds to b == O. The condition, that w -

A

==

o.

v == 0 on v(P) implies that

So we get that b - B a == c is a constant and that

da dt = B a + c. The method of variations of constants of Lagrange yields the solution

a(t)

= e tB

a(O)

+ fat e(t-s)B (c) ds.

112

Chapter 9. The Heat Kernel on a Principal Bundle

For the computation of Tv(p) expp ' we take a(O) and we get e B -1 Tv(p) expp =

== 0, so c == b(O), and t == 1,

-----n-.

We shall now prove that, for each j and k (9.28) Here OJ == 0/OPj, viewed both as a differential operator, and as a (constant) vector field. This then completes the proof of (9.27). A computation in local coordinates with (8.23) shows that

On the other hand, the fact that v is an infinitesimal isometry, and that we had chosen the local coordinates such that the vector field v is constant, together imply that

Substituting

OJ (v, Ok)

=

(V 8j v, Ok) + (v, V8j Ok)

and the same with j and k interchanged, and using that

we get

Here we used in the second identity that v commutes with the OJ, so

This proves (9.28). D

9.5. The Expansion

113

Lemma 9.6 In the following statements, we identify the vertical spaces Vp with 9 and write X instead of X P, p. With these notations, we have a) The curves s ~ (exp s X) p, the orbits in P of the one-parameter subgroups of G, are geodesics in P. b) For each pEP and X E g, Txexpp maps H p T xM and ~ ~ 9 to Vexpx (p) ~ g, and

~

T x M to Hexpx(p)

~

(9.29) (9.30) c) The parallel transport along the geodesic s ~ exp(s X)(p), with s

running from 0 to 1, is equal to the linear transformation A o (expX (p, ) p) == e

_!X·Rl 2

(9.31)

in Lx, x == 1r(p). Here we used the abbreviation I

X· R ==

EXkRk,

(9.32)

k

in which X k == (X, e%) is the k-th coordinate of X E 9 with respect to the orthonormal basis ef, and the Rl are as in (9.18). d) The Jacobian of the substitution 9 == expX is equal to d (expX) = det (

1

Proof a) follows from (9.5) with Y == X.

e-adX) dX.

adX

(9.33)

114

Chapter 9. The Heat Kernel on a Principal Bundle

Since G also acts by isometries on P, we can apply Lemma 9.5. In view of (9.6) and (9.5), this leads to b)l. Note that (9.30) coincides with the familiar formula

d dE exp( -X)

exp(X

0

+ EY)ll=o =

1- e- adx adX Y

for the derivative of the exponential mapping from the Lie algebra to the Lie group; this in turn implies d). Finally we observe for c) that, for all s E R, 1f

(exp(s X) p)

so the differential equation ~ x A

== x,

== 0, cf. (9.21), is a linear differential

equation with constant coefficients. The operator which assigns to the initial values at s == 0 the solution at s == 1, is given by (9.31). 0 We have now made all the preparations for the main result of this chapter:

Theorem 9.1 With the assumptions as in Lemma 9.3, we get that the integral kernel ole-tQ along the diagonal x == y has the expansion

e- tCas

(47rt)-~dimp

1

e-IIXlI2j4t

II

Here pEP is such that 1f(p)

f

t k Bk(p, X)

0

eP/(X)

dX.

(9.34)

k=O

== x, and the coefficients B k satisfy (9.35)

and depend smoothly on p and X. Furthermore,

(9.36) 1 In

the formula in [9, Th. 5.4], corresponding to (9.30), the sign of a seems to be wrong,

because in [9, (5.2)] a left multiplication by exp( -a) has been used, whereas the action on Pin [9, Th. 5.4] is from the right. For the computation of jg(adX) in Theorem 9.1 below, this has no effect, because the adjoint of Jg(ad X) is equal to J g(- ad X), and these have the same determinants.

9.5. The Expansion

115

in which

(For the notations, see also (9.4) and (9.32).) The integration in (9.34) is only performed over a small neighborhood of the origin in g; another cutoffyields the same expansion. Proof With the substitution 9 == exp X, we get

d(p, 9 p)2 ==

IIXI1 2 ,

and

p(g)

==

eP'(X) .

Also, j (p, gp) = j (g-lp, p) = j (exp-Xpp, p).

For the parallel transport, we have:

Using Lemma 9.6, we see that (9.26) gets the form as described in the proposition. D In Chapter 11, we will determine the constant term, for t 1 0, of the supertrace of (9.34), in the case that Q == D 2 and D is the spin-c Dirac operator. This will yield the formula (11.17) for the integrand in the integral formula (7.11) for the index of D+. For this, one may skip Chapter 10, which is a preparation for the local Lefschetz formula in the presence of an automorphism,.

Chapter 10 The Automorphism The goal here is Theorem 10.1 below, which is the modification of Theorem 9.1 in the presence of an automorphism ,. The only contributions to the asymptotic expansion come from neighborhoods of fixed points in M of" which in the principal G-bundle P correspond to fixed points modulo the action of G. The extra ingredient needed here is the asymptotic estimate in Lemma 10.1 of the geodesic distance between two points in P which are close to the same G-orbit.

10.1

Assumptions

'M

Let, == be a transformation in M which preserves all the geometric data used to define the spin-c Dirac operator D. That is, for each x E M, we have (10.1 ) Furthermore, , is an isometry in M with respect to the Riemannian structure

'L

(3. And finally, we assume that there is a bundle automorphism of L, which preserves the Hermitian structure and the connection in L, and which covers'M' in the sense that 7r L from L onto M.

0 'L

==

'M 0

7r L'

if 7r L denotes the projection

117 J.J. Duistermaat, The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8247-7_10, © Springer Science+Business Media, LLC 2011

Chapter 10. The Automorphism

118

Remark According to a theorem of van Dantzig and van der Waerden [25], the action of the isometry group I of a connected Riemannian manifold M is proper. That is,

(¢, x)

~

(¢(x), x)

is a proper mapping from I x M to M x M. We may assume that

~

has a

fixed point x, so ~ E Ix if Ix denotes the set of ¢ E I such that ¢(x)

== x.

It follows now from the properness of the action of I that the group Ix is compact. In the applications, it may occur that ~ is an automorphism of the almost complex manifold (M, J), covered by an automorphism ~L of L, and that ~L belongs to a compact group K of transformations. By averaging an arbitrary Riemannian structure in M and Hermitian structure in Lover K, one arrives at a Riemannian structure in M and Hermitian structure in L, which are invariant under ~ and From a given

~ L'

respectively.

~-invariant

Riemannian structure {3 en

~-invariant nonde-

generate two-form a, one can also get a ~-invariant almost complex structure J by taking

Jx

== ax-1 0 {3x 0 Px -1 ,

in which Px is the positive square root of the positive definite symmetric transformation _(A x )2. (See (15.41) for more details.) In this way, the theory is applicable to transformations

~,

which belong to a compact group

of automorphisms of a symplectic manifold (M, a). The condition that ~ is an isometry of (M, f3), in combination with (10.1), yields that the mapping

(10.2) defines a bundle automorphism T ~ of UF M which

We have a

covers~.

similar bundle automorphism of SOF M, also denoted by

T~.

ding e

T~'s,

~

f(e) of UF Minto SOF M intertwines the two

not much danger of confusion.

The embedso there is

10.1. Assumptions

119

The transformation T'Y commutes with the action e ~ e

° A-I of A

E

U(n), so it induces a transformation Ip in the principal Spine-bundle P == SpincFM, which commutes with the action of G == Spin C (2n) in P. Also, because 'Y is an isometry of (M, (3), the transformation T'Y in SOF M leaves the Levi-Civita connection in SOF M invariant. And because T'Y commutes with J, it leaves the Hermitian structure and the connection in K* invariant. Since the connection in P was defined as the pullback of the connection in ~*

(SOF M x U(K*)) by means of the covering 'Y in (5.6) (sorry for the

clash of notations), it follows that 'Yp leaves the connection in P invariant. Finally, we observe that the induced action of I on E, which we could also denote by T 'Y, coincides with the one on (5.7), obtained by letting 'Y p act on the first factor. In the general setting of the previous chapter, we assume now that the Riemannian manifold (M, (3) is oriented, and that I is an orientation preserving isometry of (M, (3). Furthermore, it is covered by a bundle automorphism 'Yp of P which commutes with the G-action in P and preserves the connection in P. And finally"

is also covered by a bundle automorphism 'YL of L,

which preserves the Hermitian structure and connection of L. These data lead to the definition of the operator lOp, acting on sections w of

E

@

(/'op(w

L, by means of

@

A)) (p) = w (/,;1 (p))

@/'L,y

Ay ,

pEP, Y =

/,-1 07f(p) E M. (10.3)

It is clear from all the invariance properties of , mentioned above that lOp commutes with Q, Cas, and

(j.

So, in the same way as we arrived at (9.24),

we now get that the integral kernel of 'Yop

is equal to

e- t Cas

fa K

P

° e -tQ == e -tQ °lOp

(t, p, g/'p(q)) 0 p(g) 0

(1 @ /'L'1f(Q»)

(10.4)

dg.

(10.5)

Chapter 10. The Automorphism

120

For the study of (7.6), we may restrict to p == q. Substituting the asymptotic expansion (8.48), we see that we only get contributions which are not

!

p and 9 '7 p (p) is of order t 1/ 2 . Since '7 p covers '7, this implies that the distance between x == 1r(p) and '7(x) also is of order t 1/ 2 , or that x is at a distance of order t 1/ 2 to a fixed point Xo decreasing as t

0 , if the distance between

of '7 in M. In a neighborhood of a fixed point

Xo,

and in normal coordinates, the

isometry '7 == '7 lvf of (M, (3) has indeed a very simple description. It maps geodesics to geodesics, whereby the velocity vectors are transformed by means of T 'Y. (In other words, the transformation T'7 in T M commutes with the geodesic flow in T M.) In particular we have

(10.6) That is, the exponential mapping, centered at the fixed point xo, intertwines the linear transformation T XQ '7 in the tangent space T XQ M, with the transformation '7 in a neighborhood V of Xo in M. Note that T XQ '7 leaves the inner product

(3XQ

in T XQ M invariant. And that the assumption that '7 is an ori-

entation preserving isometry implies that T XQ '7 is an orientation preserving orthogonal linear transformation, a rotation, in T XQ M. The fixed point set ofTXQ '7 in T XQ M is a linear subspace ~Q M ofTXQ M. The orthogonal complement NXQ of T;Q M in T

XQ

M is invariant under T

XQ

'Y,

with 0 as the only fixed point. If we write

(10.7) this means that '7N

-

1 : N XQ

~

N XQ

is invertible.

(10.8)

In view of the decomposition of the rotation T XQ '7 into planar rotations, this implies that the dimension of N XQ is even, say equal to 2l. These facts imply that the fixed point set V, == M'

n V of '7 in V is

a smooth submanifold of M, of even codimension equal to 2l. Globally,

10.2. An Estimate/or Geodesics in P

121

this implies that each connected component F of MI is a closed, smooth submanifold of M. We also observe that the eigenvalues of T xo

,

do not

change, as Xo varies in F. The only warning is that in general, the normal action of , will be different along different connected components F of MI, and even the codimensions l == l F of the F can vary. Using a partition of unity, with smooth functions with support in normal coordinate neighborhoods as above, the study of the integral (7.6) is reduced to an integral over U, where the integrand has been multiplied by a test function ?jJ(x), which is a smooth real-valued function with compact support in U. We will perform the integration by first integrating over N xo and then over UI. Actually, we will not need to perform the integration over UI, because it will turn out that the integral over N xo already has an asymptotic expansion in powers of t, with an explicit expression for the leading term. Moreover, the remainder term in the asymptotic expansion is of order t k for large k, locally uniformly when Xo E F varies. Also, the asymptotic expansion can arbitrarily often be differentiated termwise with respect to both Xo and to t.

10.2

An Estimate for Geodesics in P

In order to prove the announced results on the integration over N xo , we first need a quite precise estimate for the distance between P and

g,

p

(p), cf.

Lemma 10.1 below. Recall that we need only to consider points p, which are close to a point Po E P, where Xo == 7f(Po) is a fixed point of, in M. But, in general, this

does not imply that Po is a fixed point for, p in P. Indeed, in our situation,

,p

is the transformation in P which is induced by the transformation T, in

T M, which leaves T Xo M invariant but has no nonzero fixed points in the linear subspace N xo • We see that unless, == 1, T, acts without fixed points on UF M, and hence also, p has no fixed points in P.

Chapter 10. The Automorphism

122

However, the fact that I(XO) == Xo is equivalent to the condition that

1 p(Po) E G P, or to the existence of a (uniquely determined) 90 E G, such that go Ip (Po) == Po· We will therefore write g O 'P==h o 8,

8==goo,p.

(10.9)

Then {) is a transformation in P, still covering " such that {)(Po)

== Po. We will

with

h==ggo-l,

subsequently perform the substitution of variables 9

== h go in the integral

(10.5). Since now we can restrict to h close to the identity element in G, we subsequently will substitute h

== exp X, with X in a small neighborhood of

oin g, in order to arrive (after the integration over N xo ) at an integral which is similar to (9.34).

In the integration over N xo ' we have an exponential factor, with in the exponent -1/ 4t times the square of the distance from P to (exp X p) (q), withp and (10.10) close to each other. The integration over N xo means that we take x where v varies in a small neighborhood of the origin in N xo '

==

v, and then choose expxo

pEP such that 1r (p) == x. A natural choice is to take p(v) ==

exp po (Vhor),

(10.11)

in which Vhor is the horizontal lift of v in T po P, the vector in H po which projects to v. Note that the choice (9.3) of the Riemannian structure in P gives that

Vhor

is orthogonal to X P, po. The curve t ~ p( tv) is a geodesic,

starting at Po, with horizontal tangent vector v. Since both 1 P and 90 are isometries which preserve the connection, t

~

8 (p( tv)) is also a geodesic,

starting at Po, this time with the horizontal tangent vector (10.12) if we identify all horizontal vectors over Xo with their projections in T xo M. The following lemma serves to get a sufficiently accurate estimate for the geodesic distance between p( tv) and (exp X)

° {) (p( tv) ).

123

10.2. An Estimatefor Geodesics in P

Lemma 10.1 For X E g, pEP and v, w E H p , we have the locally uniform estimate

= IIXI\2 (v, J-l w) + (w, J- 1w)) + 0 (t 3) ,

d (expp(tv), expX eXPp(tw))2

+t 2 ((v, J- 1 v) - 2 as t

~

(10.13)

O. Here J == J M (X . f1 p ), as in (9.29).

Proof 1 Write

p(t) == exp p (t v) .

(10.14)

It will be convenient to replace t v and t w by t l v and t 2 w, respectively, with indepedently moving t 1 and t 2 • Let u(t 1 , t 2 ) be the element in Tp(tl) P, close to X ==

Xp,p(tI),

such that (10.15)

This gives that the left hand side in (10.13) is equal to the square of the length of u(t l , t 2 ). Note that p(O) == p, p'(O) == v, and u(O, 0) == X. In the sequel, we will write ai == a/ati. Using (9.29), we get

v

+ J alU(O, 0) == 0,

J a2U(0, 0) == w.

(10.16)

Since J leaves H p invariant and u(O, 0) == X is orthogonal to H p , we get that

as t

(u(t), u(t)) = IIXI1 2 + 0

(II t ln .

1 o. Now consider the map

f.

from an open neighborhood T of {O} x {O} x

[0, 1] in R into P, defined by 3

(10.17) 1 We

follow the clever proof of [9, Prop. 6.17].

124 We will view

Chapter 10. The Automorphism

ai, i =

1, 2, 3, depending on the context, either as differential

operators, or as vector fields in T. We will also make a somewhat serious use of the pullback bundle E* T P ofT P. It is defined as the set of (t, w) E TxT P, such that E(t) = 7r(w) if 7r denotes the projection from T Ponto P. The projection to the first component defines a projection from

E*

T Ponto T, which turns

E*

T Pinto

a vector bundle overT; the fiber over t E T is identified with Tf(t) P. Sections of E* T P are mappings s : T

~

T P, with the property that 7r 0 s

are sometimes also called "vector fields in P, lying over

E".

=

E.

These

Examples of

such sections are the partial derivatives Oi E. For such a section s, the covariant derivative at t E T, in the direction of a vector T E T t T, is defined by

These definitions can actually be given for an arbitrary connection in an arbitrary bundle over P, but we will apply this here only for the case of the Levi-Civita connection in T P. The point of the whole construction is that all formulas for connections (such as concerning torsion and curvature) hold as if T were a submanifold of P and smooth sections of E* T P were restrictions to T of smooth vector fields in P. As a function of t 3 , the right hand side in (10.17) is a geodesic, so

This yields

03 2 (01 E, 03 E)

=

(\783 \78 1

= (\783 \783

03 E, 03 E)

==

01 E, 03 E)

(R (03 E, 01 E) 03 E, 03 E) =

o.

Here we used in the first equation that the Riemannian structure is covariantly constant, in the second equation that the connection is torsion-free, in the third equation that the vector fields 03 and 01 commute, and in the last equation

10.3. The Expansion

125

that the curvature is anti-symmetric with respect to the inner product. It follows that (01 t, 03t) is an affine function of t 3 • Since

°

we get 01 t == when s == 1. On the other hand, if t 3 and 03t == w(t 1 , t 2 ), so

== 0, then 01 t == p' (t 1)

From this, we obtain

01 (u, u) == 01 (03 t , 03 t ) == 2 (\7 81 03 t , 03 t )

== 2 (\7 8301t, 03t) == 203 (Olt, 03t) == -2 (p'(t 1 ), u(t 1 , t 2 )) . Using that \7 8 l P'

== 0, because

tl ~

p( t I ) is a geodesic, we get, in

combination with (10.16), that

812 (U, U)ltl=t2=O=2

(v,

82 81 (u, U)l tl=t2=O = -2

]-lV),

(v,

]-lW).

On the other hand, the geodesic distance between PI and exp X p (P2) is the same as the geodesic distance between exp( -X p ) (PI) and P2. Since the adjoint of - X . Op is equal to X · Op, we therefore get that

This completes the proof of (10.13). 0

10.3

The Expansion

We are now ready for the proof of the following result.

Chapter 10. The Automorphism

126

Theorem 10.1 Let F be a connected component of the fixed point set M' of"l in M; F is a smooth submanifold of M of codimension 2l F in M. Let x E F, and let

~

be a smooth real-valued function, with compact support,

contained in a small normal coordinate neighborhood U around x. Let N x denote the orthogonal complement ofTxF in TxM. Then the integral over the diagonal in N x x N x, of the integral kernel of (10.4), is asymptotically of the form e- tCas

(47ft)IF-~

dim?

1

e-IIXII2/4t

g

f. t k Ck(p, X)

p(go) dX.

0 eP/(X) 0

k=O

(10.18)

Each ofthe coefficients C k satisfies (10.19)

and depends smoothly on p and X and finitely many derivatives of ~ at x. Furthermore, if~(x) == 1,

Co(p, X)

=

jg(adX)1/2 j F (-X· f2 p )-1/2

(1 0 e~ X·R

1

'I'L,X)

.det (1 - 'l'N -1 e- x .np ) -1/2 det (1 - 'l'N )-1/2 . (10.20)

Here jg is as in (9.37). The space TxF is invariant under the mapping J M (-X· 0p), defined in (9.29), and jF( -X · f2 p) denotes the determinant

of the restriction J F (-X· f2 p ) of J A1 (-X· f2 p ) to TxF. Finally,

"IN

is

restriction ofTx"l to N x. The integration in (10.18) is performed only over a small neighborhood ofthe origin in g; another cutoffyields the same expansion.

Proof We resume the discussion in the paragraphs preceding Lemma 10.1. With p(v) as in (10.11), v E H x , we get from (10.12) and (10.13) that

¢(v)

:==

d(p(v), expX p 8(p(v)))2 -

/IX11 2

127

10.3. The Expansion satisfies ¢(v)

== q(v) + 0 (1IvIl 3 )

as v ~ 0, where q(v) is the quadratic form

We will use the fact that I Nand J commute, and that order to simplify q(v) to

q(v)

= 2

'N is an isometry, in

(v, J- 1 (1- 'YN) v).

Next, we will write it in the form q(v) == (Q v, v) for the symmetric operator (10.21) For X == 0 we have J

==

1, and because (10.22)

we see that

is positive definite in N x . It follows that, for all X in a neighborhood of 0 in g, Q(X) is still positive definite. By the Morse lemma with parameters, we can therefore find a regular substitution of variables v == v(w), smoothly depending on p and X, such that v(O) == 0 and ¢(v(w)) ==

Ilw11 2 • Ifwe write A ==

d~~) Iw=o, then we get

A* Q A == 1; so for the Jacobian at w == 0 we get det d~~) \w=o == det Q(X)-1/2.

(10.23)

The integration of K P in (9.24) over v therefore amounts to an integration of the form

I = (41Tt)-IF

r

JN

e-lIwll2j4t x

a(w, x, X, t) dw,

Chapter 10. The Automorphism

128

where we borrowed the appropriate power of 47ft from the power in front of the asymptotic expansion for K

P

.

(Recall that dim(Nx )

== 2l F .) "The

amplitude a has an asymptotic expansion of the form

L t k ak(w, x, X) 00

a(w, x, X, t)

rv

k=O

as t

1

0, which is smooth and locally uniform in the parameters. The

substitution w

== t 1/ 2 Z now yields an expansion of the form 00

I

rv

L t k Ik(x, X),

t

10

k=O

for the integral, which is locally uniform and can be differentiated termwise with respect to all the parameters. Furthermore, Io(x, X)

== ao(O, x, X),

> 0 are expressions in corresponding higher order Taylor expansions at w == 0, of the al(W, x, X) with l :S k. whereas the I k for k

For the identification of (10.20), we observe that2 : (10.24) Using (10.22), and the facts that X . Op commutes with IN' and that IN *

==

IN -1, we can rewrite (10.21) as

Q(X) =

j-1

(1 - TN -1 e- x ,op ) (1 - TN)

on N.

(10.25)

The factor j-1 in (10.25) leads to a factor

in det Q(X) -1/2, which combined with the factor (det J (- X · Op)) -1/2 which we had in the expansion of K P leads to the factor jF (-X · Op)-1/2 in (10.20). 21t seems that in the formula for A * in [9, p. 205], a wrong sign of a has crept in.

10.3. The Expansion

129

We finally observe that since I points, the determinant of 1 -

is a rotation without nonzero fixed

'N is positive. In order to prove this, we use N

the decomposition into planar rotations. For a planar rotation R over the angle

Q,

the determinant of 1 - R is equal to (1 - cos Q)

2

+ sin 2 Q ==

2 (1 - cos Q) > 0

if Q is not equal to an integral multiple of 27r. It follows that, for small X, also the determinant of 1 - IN -1 e- x ,op is positive. And because the determinant of Q(X) was positive, the determinant of the restriction to N of J- 1 is positive as well. This allows us to take (positive) square roots of each of these determinants, completing the proof of (10.20). 0 In the special case of the spin-c Dirac operator, the linear transformation

T xo ' commutes with the almost complex structure Jxo in T xo M, cf. (10.1). (Sorry for the clash of notation with J == J M (X · Op).) That is, T xo I is a unitary transformation in T Xo M. It follows that both T Xo F and N xo are invariant under J xo • That is, J restricts to an almost complex structure in F, and IN is a complex linear transformation in N xo ' with all eigenvalues on the unit circle and not equal to one. In Chapter 12, we will see that, again for the spin-c Dirac operator, the integral over N xo of the supertrace converges for t

1 o.

The limit is an

expression in terms of the geometric data at xo, and the test function VJ enters only via multiplication by 'ljJ(xo). That is, no derivatives of VJ at Xo appear. This means that the function x ~ strc K(t, x, x)

converges, for t

1 0, in distribution sense, and the limit is a smooth density

in each connected component F of M'.

Chapter 11 The Hirzebruch-Riemann-Roch Integrand 11.1

Introduction

In this chapter, we will determine the constant term, for t

1 0, of the super-

trace of (9.34) in the case that Q == D 2 and D is the spin-c Dirac operator. This will yield the formula (11.17) in Theorem 11.1 for the integrand, in the integral formula (7.11) for the index of D+. That is, in (9.34) we now take E == E(n), the direct sum over q of the space of (0, q)-forms on en. The group G is equal to SpinC (2n), cf. Chapter 4, acting on E( n) by means of the Clifford multiplication c. Since both G and 9 appear as subsets of the complex Clifford algebra C(2n) ® C, and c is a linear mapping from C(2n) 0

e to End (E(n)) (actually, an isomorphism

of algebras), we can write p == c Ie and p' == c Ig.

The Clifford multiplication by elements of G and g, respectively, leaves the decomposition of E(n) into its even part E(n)+ and its odd part E(n)invariant, so the same is true for the integrand in (9.34). In the integral (7.4), the integrand is the supertrace of the integral kernel along the the diagonal, which is the trace of the restriction to E+ 0 L minus the trace of the restriction 131 J.J. Duistermaat, The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-8247-7_11, © Springer Science+Business Media, LLC 2011

132

Chapter 11. The Hirzebruch-Riemann-Roch Integrand

to E- 0 L. Therefore we have to take the supertrace of (9.34), which here is the trace of the endomorphism of E(n)+

@

Lx, minus the trace of the

endomorphism ofE(n)- 0 Lx. One of the keys is formula (3.33), which expresses the supertrace of an endomorphism of E in terms of the volume part of a corresponding element of the exterior algebra. Although the correspondence is not a homomorphism of algebras, it is close enough to get that, for the computation of the leading term in the asymptotic expansion, one may replace the algebra of endomorphisms by the exterior algebra. For the precise statements to this effect, see Lemma 11.2 through 11.6. The translation of the formulas in terms of characteristic classes will be discussed in Chapter 13, cf. Proposition 13.1. The modifications in the presence of an automorphism 'Yare treated in Chapter 12.

r,

In this chapter and the next, we will use the notation of Chapters 3 and 4, and write V, C(V), E, Spin(V), and SpinC(V) instead ofR2n ~

C(2n), E(n), Spin(2n), and Spin (2n), respectively. C

( R2n

Several of the factors in (9.34) will turn out to be quite innocuous. To begin with, the Casimir element Cas E End(E) actually acts as a scalar multiplication. In order to prove this, we observe that it is equal to the c-image of the element Case :==

L ek 2 E C(V) 0

C,

(11.1)

k

whereek == efdenotesanorthonormalbasising c C(V)0C,cf. (9.15). The statement is therefore equivalent to the fact that Case is a scalar, cf. Lemma 11.1 below. In the proof, we also introduce a special kind of orthonormal basis in g, which will be used throughout this chapter. Lemma 11.1 Case is a scalar. Proof The Lie algebra of 9 == spinc (V) is equal to the direct sum of the Lie algebraspin(V) ofSpin(V) and u(l) == i R, the Lie algebraofU(I) C C(V),

11.2. Computations in the Exterior Algebra

133

so we can take the orthonormal basis of 9 to be equal to an orthonormal basis of spin(V), together with an element of u( 1). Since the latter is a scalar, its square is a scalar too. We choose the inner product (3 in spin(V) to be a suitable negative multiple of

(a, b)

~ trace (7'(a)

0

7'(b)) ,

where 7 is the two-fold covering from Spin(V) onto SO(V) defined in (4.2), so

7'

is an isomorphism of Lie algebras. This choice ensures that {3 is

conjugacy invariant. If (cPi);~l is an orthonormal basis in V, then the

form a basis in spin(V). We read from (4.1) that cPi · cPj and cPk . cPl are orthogonal ifi < j, k < land {i, j} =I {k, l}. Indeed, if {i, j} and {k, l} are disjoint, then the composition of 7' (cPi . cPj) and

(cPk . cPl) is equal to zero. And if for instance j == k, then this composition maps ( to 4 ( cPl' () cPi; this mapping has trace equal to zero. On the other hand, the trace of 7' ( cPi· cPj )2 7'

is equal to -8, so after multiplying (3 by a suitable positive constant, we get that the cPi · cPj, with i < j, form an orthonormal basis in spin(V). Now, if i < j, we get

which is a scalar. D

11.2

Computations in the Exterior Algebra

Since taking the trace is a linear form, we can bring it under the integral sign. Since also tracec (A 0 B) == tracec A . tracec B,

(11.2)

Chapter 11. The Hirzebruch-Riemann-Roch Integrand

134

the traces factorize with respect to the action on E and on Lx. Using the formula (3.33) for the supertrace, we see that the only factor in the supertrace coming from End (E) is equal to strc (eC(X») =

or

vol (O"A (expc X)) .

(11.3)

Here 0" A is the linear mapping from the Clifford algebra C (V) ® C to A V, the space of all complex-valued exterior forms on W, which was defined in (3.15). The multiplication dot refers to the mUltiplication in the Clifford algebra, which is also used in the power series 00

expc X ==

L

k=O

in C(V) ®

~X

k

c.

The following lemma is a slight variation of [9, Prop. 3.13]. Lemma 11.2 For sufficiently small T E End(V), we define

c(T) = (det cosh~Tf/2 E R, and

H(T) Then,

if X O"A

=

Ca~~T)

1/2

(11.4)

(11.5)

E End(V).

E spin(V) is sufficiently small:

(expc X ) == C(T'(X)) AH(T'(X)) (exPAO"A(X)).

(11.6)

Here T'(X) is viewed as a linear mapping from V to itself. For any A E End(V), AA denotes the induced action (of pull-back by means of A') on

the exterior algebra AV ofV. Finally eXPA denotes the exponential power series in the algebra AV. Proof Using the decomposition of V into planes of infinitesimal rotations for T'(X), we get the existence of an orthonormal basis such that

cPj

in V and OJ E R

n

X ==

L OJ

j=1

cP2j-1 . cP2j.

(11.7)

11.2. Computations in the Exterior Algebra Since cP2j-1 • cP2j commutes with using (4.3), that

cP2k-1 • cP2k

n

expc X =

135 in C(V) if j =1= k, we get, also

n

II exPc ()j cP2j-1

0

cP2j

II (cos Bj + sin Bj cP2j-1 . cP2j)

=

j=l

0

j=l

Since i(QcPj) (cPk) = 0 if j =1= k, this in turn leads to

a A(expc X)

=

n

II (cos Bj + sin Bj cP2j-1 /\ cP2j) , j=1

see (3.15) for the definition of a A. On the other hand, n

aA(X)

=

LB

j cP2j-1 /\ cP2jo

j=l

Since the elements

cP2j-1/\ ¢2j

commute in the algebra AV, and have square

equal to zero, we get n

eXPA aA(X) =

II (1 + Bj cP2j-l /\ cP2j) .

j=1

If i

=1=

j, we will denote by

Ji,j

the linear transformation in V such that

(11.8) Then

n

T'(X)

=

L2Bj J2j -

1 ,2j,

j=1

so e~TI(X) is equal to the rotation in the

cP2j-1,

cP2j-plane over the angle Bj •

It follows that cosh ~T'(X) is equal to multiplication by co.s Bj in the cP2j-l, cP2j-plane, and hence

(detcosh~T'(X))

1/2

=

II cos OJ. n

j=1

136

Chapter JJ. The Hirzebruch-Riemann-Roch Integrand

Furthermore,

rrn

tanh~T'(X) _

'(X) 2T

sin OJ

e. j=l J

-

!

.

.

e. J2)-1,2), COS J

which acts on cP2j-l 1\ cP2j as multiplication by • Uj Ll SIn (

OJ

COS

OJ

)2 .

This proves (11.6). D If cPj is an orthonormal basis in V, then we write, for each strictly increasing sequence J == (ji)f=l' (11.9) Together, the cPJ form a basis of the vector space AV. A natural inner product in AV is the one for which the cPJ form an orthonormal basis. Note that (11.10) for any w E AV. Applying this to w == eXPA aA(X), using (11.6), and applying (11.10) again to the resulting elements AH (T' (X) )(cP J ), we get aA(expC X ) == C(T'(X)) LPJqJKcPI(, J,K

(11.11)

in which

and Note that AH (T' (X)) preserves the grading of AV, so we only get contributions when cPJ and cPI( have the same degree. The idea is now to collect all the scalar factors, depending on X, bring

these in front of eXPA (j A(X), and then perform the integration over X. After

11.2. Computations in the Exterior Algebra

137

summing over J and K and taking the volume part, we then end up with the supertrace of (9.34). The integration in (9.34) is over 9 == u(l) EB f), in which 9 == spinC(V) and f) == spin(V). If we first perform the integration over u( 1), taking along a factor (47ft) -1/2, then we are left with an integral over f) with the same properties. We now study the integral over f). In the sequel, we will make systematic use of the commutative algebra

A+ V :==

n

LA

2j V,

(11.12)

j=O

the even part of the exterior algebra AV of V.

Lemma 11.3 Let f be a smooth, compactly supported function on f) spin(V), in which we take the orthonormal basis er == e~, consisting of the

cPi . cPj, with i < j. We write

for the corresponding basis of A 2 V. Then we have an asymptotic expansion, for t

1 0, of the form

1

(41ft)-4dim~

f(X) eXPAaA(X) dX '"

e-IIXIl2/4t

in which Wk E 2:j~k A 2jV. Moreover, degree 2k is given by

(2

a

f

t k Wk,

(11.13)

k=O

ry

if 0

:::; k :::; n, then the part ofwk of

L ~7' ar) k f(O) r

mod

L

A2j V.

(11.14)

j(m). ii) preserves the Hermitian structure in L, which means that maps U (L) to itself. The restriction of to U (L) will also be denoted by . iii) *O == O. Note that the automorphism is equivalently given by a diffeomorphism of U(L) which commutes with the U(l)-action and leaves 0 invariant. The automorphisms of L form a group, which will be denoted by Aut(L ). In the same fashion an infinitesimal automorphism of L is defined as a

U(l )-invariant vector field v in U( L) such that £( v)O == O. The infinitesimal automorphisms of L form a Lie algebra of vector fields, which will be denoted

by End(L).

Chapter 15. Application to Symplectic Geometry

206

Now a lifting to Aut( L) of the action of G in M will be defined as a smooth homomorphism 9 ~ gL from G to Aut(L) such that, for each 9 E G,

1f 0

gL == gM

0 1f.

In the same fashion a lifting to End( L) of the infinitesimal action of 9

in M will be defined as a homomorphism of Lie algebras X to End(L) such that, for each X E g, the projection

1f

~

XL from 9

intertwines XL with

XM . Here the Lie algebra structure in End( L) is the group-theoretic one, which is opposite to the Lie brackets of vector fields, cf. the comments in front of Lemma 9.1. So the homomorphism property means that (15.35) where in the right hand side we have written the standard Lie brackets of vector fields. Finally, if v and ware vector fields in manifolds Land M, respectively, and 1f : L

~

M is a smooth map, then we say that 1f intertwines v with w if

W1f (l) == T l 7r (Vl) , l

E

(15.36)

L.

This is equivalent to the condition that, for every solution curve 'Y in L of ry'(t)

8'(t)

== v('Y(t)), 8 ==

1f 0

ry is a solution curve in M of the equation

w(8(t)). This can be used to prove that, if 1f intertwines Vj with Wj, j == 1, 2, then 1f intertwines [VI, V2] with [WI, W2]. ==

Note that, for each vector field W in M there is a unique vector field v in

U(L) such that 1f : U(L) the sense that i( v)8

==

o.

~

M intertwines v with wand v is horizontal in

We write v

Proposition 15.2 A mapping X

~

== w hor in this case. XL from 9 to the set of vector fields in

U(L) is a lifting to End(L) of the infinitesimal action X

X M of 9 in M if and only if there exists a momentum mapping J-l : M ~ g* for the Hamiltonian action in M, such that ~

(15.37)

15.4. Lifting the Action

207

Proof The condition that 1r intertwines XL with X M means that there exists

Ix on L such that XL == x2?r + Ix 21ri L, and in this case XL is U(l)-invariant if and only if Ix is constant on the fibers, a smooth real-valued function

which means that I x condition that X

r--7

==

1r* ¢ x

for some smooth function ¢ x on M. The

XL is a linear mapping now is equivalent the existence

of a smooth mapping ¢ : M ~ g* such that, for every X E g, ¢x

== (X, ¢).

So from now on we may restrict our attention to (15.38) The invariance of e under XL is equivalent to

£ (XL) e == i (XL) de + d (i (XL) e)

o

+ 271"i d 71"* (X, 271"i 71"* (- i (X M ) a + d(X, ¢)) 271"i 71"* (d(X, 11) + d(X, ¢)) , - i (XL) 21ri 71"* a

¢)

where we have used (15.34) and (15.29) to get to the second line. This means that d(X, J--l)

+ d(X, ¢) == 0, because 71" is a submersion, or -¢ ==

J--l

+ v for

a constant element v of g* . In order to compute the Lie brackets of the vector fields XL and YL for

X, Y E g, we write

for the horizontal and vertical part of XL, respectively. In order to compute

[Xfor, YEor], we start by observing that

1r

inter-

twines it with [X M , YlV1 ] == -[X, Y]M, so there is a smooth real-valued function I on U (L ), such that

[X2o r, YEor]

=

-[X, y]~r + f 21ri L .

Chapter 15. Application to Symplectic Geometry

208

Applying () to both sides and using (2.16), we get

f

hor yhor) == rr* a (X hor yhor) _~ dB (X L 27Tt ' L L , L rr* (a (X M, YM)) == -rr* ([X, Y], J-L),

where in the last identity we used (15.15). The conclusion therefore is that

[X2o r, YEo r] = -[X, y]~~r - 7r*([X, Y], f-L) 27ri L . In a similar fashion we getthat7r intertwines [X2or , YEo r] with [X M , 0] = 0, so there is a smooth real-valued function f on U (L) such that [XrOf , Yl ert ] =

f 27ri L .

Applying () and using (2.16), we get this time

f

7r*(], (X2o r, Yl ert )

+ Xror 7r*(Y, O. Since h commutes with dg, also

H commutes with dg, and we now get, because (l' is basic: (l'

This shows that

(l'

== H (l' == dg (H (3(O)) == d (H (3(O))

is equal to the exterior derivative of a basic form, which

means that its cohomology class in Hbas(P) is equal to zero. This completes the proof that ibas is injective. In order to determine the basic form A in (16.29) in terms of a, we observe that the fact that the ni are horizontal, implies that S maps Aj, k, l to Aj,k-l,l+2. On the other hand, C maps Aj,k,l to Aj+2,k-l,l+2. Recall also that E and V act in

Aj, k, l

as multiplication by k and j, respectively.

16.3. Henri Carlan's Theorem Write aj, k, l for the

Aj, k,

l-component of any a E A, that is,

L

a ==

233

aj,k,l,

aj,k,l

E Aj,k,l.

j,k,l

+ A, then reads, written out in components:

The equation (16.29), a == h v aj,k,l

if (j, k)

I:

==

-s v

C vj -

j ,k+1,l-2 -

2 ,k+1,l-2

+ (k + j) vj,k,l

(16.30)

(0, 0), and aO,O,l

==

_Sv O,1,l-2

+ AO,O,l.

(16.31 )

The equation (16.30) for j == 0 can be written as VO,k,l

==

1 S vO,k+l,l k

+ !k aO,k,l '

k >_ 1.

Since vj, k, l == 0 if k is greater than the polynomial degree of v, this leads to a unique determination of va, 1, l in terms of a: VO,l,l

==

L

~ Sk-1 aO,k,l-2(k-1).

k~l

If we now combine this with (16.31), we get AO,O,l

==

L

-b. Sk aO,k,l-2k.

k'20

But this just means that because

(3(0) ==

L

~

Sk

(3,

(3

E

A.

(16.32)

k~O

Since S commutes with V, and a

~ ahor

is equal to the linear projection

onto the kernel of V along the sum of the eigenspaces of V for the nonzero eigenvalues, we get that

Chapter 16. Appendix: Equivariant Forms

234

Combining this with (16.32), we get (16.20). D Note that (16.30) also leads to an explicit determination, in terms of w, of all

vj,k,l

with (j, k)

=1=

(0,0). This leaves the freedom in

v

of adding a

basic form D. Since k acts as zero on the space of basic forms, we get a full explicit determination of the equivariant form k v in (16.29).

16.4

Proof of Weil's Theorem

For the proof of Theorem 13.1, we apply Theorem 16.1 to a(X) == f(X), for some f E C[g]AdG. That is, a is an equivariantly closed form which is a constant, when regarded as a differential form in P. We get that f(O) is a closed, basic form, and equivariantly cohomologous to f(X). Since also

f(O) is equivariantly cohomologous to f(X), we get that f(O) - f(O) is equivariantly cohomologous to zero. The injectivity of ibas implies now that there exists a basic form (3 in P such that f (f2) -

f (0) is equal to the exterior

derivative of (3.

16.5

General Actions

We conclude this chapter with the remark that in general equivariant cohomology has "very big" contributions from G-invariant neighborhoods of points PEP, where the stabilizer subgroup Gp is not discrete. Suppose that the G-action in P is proper. Then the compactness of Gp implies that, by restricting S to a suitable neighborhood of p, there are local coordinates around p in S which the action of Gp is linear. The radial contraction in S in the local model (16.16) then commutes with the G-action, so we have a G-equivariant retraction of U to the orbit G . p ~ G / G p through p. This leads to the conclusion that (16.33)

16.5. General Actions

235

the ring of the Ad Gp-invariant polynomials on gpo As soon as dim gp > 0, this ring has nonzero contributions in arbitrarily high degree. Using MayerVietoris sequences as in Bott and Tu [14, Ch. II], one can conclude that, as soon as the action of G in P is not locally free, then

He (P) =1= 0 for arbitrarily

large m. This is in strong contrast with ordinary de Rham cohomology. Related to this is the infinite-dimensionality of the classifying spaces which occurs in the topological definition of equivariant cohomology, cf. Atiyah and Bott [6]. Also related to this is the localization formula at the fixed points of a torus action, of Berline-Vergne [10] and Atiyah-Bott [6], for integrals of equivariant forms. See also Berline, getzler and Vergne [9, Ch. 7].

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Index adjoint operator, 14

curvature operator, 54

A-roof genus, 162

curvature form, 55, 227

asymptotic heat kernel, 96

double grading, 222

automorphism of line bundle, 205

Dirac operator, generalized, 16

basic cohomology, 224

Dirac operator, spin-c, 48

basic form, 224

Dirac operator, spinor, 62

Berezin integration, 28

divergence, 49

Bianchi identity, 60

Dolbeault complex, 11

canonical line bundle, 42 Casimir operator, 103

Dolbeault-Dirac operator, 16. equivariant cohomology, 223

characteristic classes, 159

equivariant differentiation, 222

characteristic forms, 159

equivariant form, 221

Chern character, 162

equivariantly closed, 222

Chern classes, 161

equivariantly exact, 222

Chern form(s), 161,204

Euler vector field, 88

Chern matrix, 161 Clifford algebra, 22

exponential mapping, 87 exterior product, 14

Clifford algebra bundle, 45

fixed point orbifolds, 180

Clifford connection, 31

frame, oriented orthonormal, 30

Clifford module, 26

frame, unitary, 31

Clifford multiplication, 26

frame, spin-c, 42

coadjoint representation, 192 connection form, 55, 225

geodesics, 85

covering a transformation, 117

geodesic distance, 87 245

246 geometric quantization, 201 Hamiltonian action, 192

Index

Newlander-Nirenberg theorem, 11 normal coordinates, 88

Hamilton vector field, 189

orbifold, 171

heat diffusion operator, 72

orbifold charts, 171

heat equation, 79

orbifold spin-c Dirac operator, 176

heat kernel, asymptotic, 96

orbit type, 174

holomorphic Lefschetz formula, 2

orbifold vector bundle, 172

holomorphic Lefschetz number, 12

oriented orthonormal frame, 30

horizontal form 55, 224 horizontal part, 228 horizontal subspaces, 100

Poisson structure, 198 positive line bundle, 67 potential, 79

index, 1, 17

principal stratum, 175

infinitesimally free, 225

proper action, 118

initial value, 79

pullback bundle, 124

inner product, 15 integral cohomology class, 161 integral symplectic form, 201

reduced phase space, 196 reduction at an orbit, 199 resonance with orbifold, 182

Jacobian of exp, 93

Riemann-Roch number, 12

Kodaira's vanishing theorem, 68

scalar curvature, 55

Laplace operator, 54

simple algebra, 25

Lefschetz principle, 70

slice, 226

Levi-Civita connection, 20

spin-c frame bundle Spinc F M, 41

lifting of action, 206

spin-c group Spin C (2n), 38

locally free, 225

spin-c Dirac operator, 48

momentum mapping, 192 multiplicity of stratum, 175 multiplicity of orbifold, 175 neutral shift, 193

spin-c structure, 48 spin group, 36 spin structure, 61 spinor bundle, 62 stratification in orbit types, 174

Index

stratification of orbifold, 175 structure constants, 228 structure equations, 228 superalgebra, 23 supercommutator, 25 superderivation, 25 supertrace, 27, 71 symplectic structure, 188 taking exterior product, 14 taking inner product, 15 Todd class, 163 Todd form, 163 torsion, 20 torsion-free connection, 20 total degree, 222 unitary frame, 31, 159 unitary frame bundle, 160 unstable manifold, 91 verticality, 231 vertical subspaces, 100 virtual character, 2

V -manifold, 172 volume part, 28 weight, 165 Weil homomorphism, 159

247