Dirac Operators in Riemannian Geometry (Graduate Studies in Mathematics)

Dirac Operators in Riemannian

Geometry Thomas Friedrich

Graduate Studies in Mathematics Volume 25

American Mathematical Society

Dirac Operators in Riemannian

Geometry

Dirac Operators in Riemannian

Geometry Thomas Friedrich Translated by

Andreas Nestke

Graduate Studies in Mathematics Volume 25

American Mathematical Society Providence, Rhode Island

Editorial Board Humphreys (Chair) David Saltman David Sattinger Ronald Stern 2000 Mathematics Subject Classification. Primary 58Jxx; Secondary 53C27, 53C28, 57R57, 58J05, 58J20, 58J50, 81R25.

Originally published in the German language by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, D-65189 Wiesbaden, Germany, as "Thomas Friedrich: Dirac-Operatoren in der Riemannschen Geometrie. 1. Auflage (1st edition)" © by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1997 Translated from the German by Andreas Nestke ABSTRACT. This text examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The presentation includes a review of preliminary material, including spin and spin0 structures.

An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature. Considerations of Killing spinors and solutions of the twistor equation on M lead to results about whether M is an Einstein manifold or conformally equivalent to one. An appendix contains a concise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study of four-manifolds.

This book is suitable as a text for courses in advanced differential geometry and global analysis, and can serve as an introduction for further study in these areas.

Library of Congress Cataloging-in-Publication Data Friedrich, Thomas, 1949[Dirac-Operatoren in der Riemannschen Geometrie. English] Dirac operators in Riemannian geometry / Thomas Friedrich ; translated by Andreas Nestke. p. cm. - (Graduate studies in mathematics, ISSN 1065-7339; v. 25) Includes bibliographical references and index. ISBN 0-8218-2055-9 (alk. paper) 1. Geometry, Riemannian. 2. Dirac equation. I. Title. II. Series. QA649.F68513 516.3'73-dc2l

2000 00-038614

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. © 2000 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. Q The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http: //www. ams. org/

10987654321

05 0403020100

Contents

Introduction

xi

Chapter 1. Clifford Algebras and Spin Representation 1.1.

Linear algebra of quadratic forms

1.2.

The Clifford algebra of a quadratic form

1.3.

Clifford algebras of real negative definite quadratic forms

1.4.

The pin and the spin group

1.5.

The spin representation

1.6.

The group Spin

1.7.

Real and quaternionic structures in the space of n-spinors

1.8.

References and exercises

Chapter 2.

Spin Structures

2.1.

Spin structures on SO(n)-principal bundles

2.2.

Spin structures in covering spaces

2.3.

Spin structures on G-principal bundles

2.4.

Existence of spin structures

2.5.

Associated spinor bundles

2.6.

References and exercises

vii

Contents

viii

Chapter 3.

Dirac Operators

57

3.1.

Connections in spinor bundles

57

3.2.

The Dirac and the Laplace operator in the spinor bundle

67

3.3.

The Schrodinger-Lichnerowicz formula

71

3.4.

Hermitian manifolds and spinors

73

3.5.

The Dirac operator of a Riemannian symmetric space

82

3.6.

References and Exercises

88

Chapter 4.

Analytical Properties of Dirac Operators

91

4.1.

The essential self-adjointness of the Dirac operator in L2

91

4.2.

The spectrum of Dirac operators over compact manifolds

98

4.3.

Dirac operators are Fredholm operators

107

4.4.

References and Exercises

111

Chapter 5.

Eigenvalue Estimates for the Dirac Operator and Twistor Spinors

113

5.1.

Lower estimates for the eigenvalues of the Dirac operator

113

5.2.

Riemannian manifolds with Killing spinors

116

5.3.

The twistor equation

121

5.4.

Upper estimates for the eigenvalues of the Dirac operator

125

5.5.

References and Exercises

127

Appendix A.

Seiberg-Witten Invariants

129

A.1.

On the topology of 4-dimensional manifolds

129

A.2.

The Seiberg-Witten equation

134

A.3.

The Seiberg-Witten invariant

138

A.4.

Vanishing theorems

144

A.S.

The case dim ML (g) = 0

146

A.6.

The Kahler case

147

A.7.

References

153

Appendix B. B.1.

Principal Bundles and Connections

Principal fibre bundles

155 155

Contents

ix

B.2.

The classification of principal bundles

162

B.3.

Connections in principal bundles

163

B.4.

Absolute differential and curvature

166

B.5.

Connections in U(1)-principal bundles and the Weyl theorem 169

B.6.

Reductions of connections

173

B.7.

Frobenius' theorem

174

B.8.

The Freudenthal-Yamabe theorem

177

B.9.

Holonomy theory

177

B.10.

References

178

Bibliography

179

Index

193

Introduction It is well-known that a smooth complex-valued function f : 0 ---* C defined

on an open subset 0 C R2 is holomorphic if and only if it satisfies the Cauchy-Riemann equation

8z

-0

with

a

2

(ax

+Z

-ay

Geometrically, we consider R2 here as flat Euclidean space with fixed orien-

tation. Changing this orientation results in replacing the operator j by the = (T - i3 Taking both operators together we differential operator 2 F obtain a differential operator P : C'(] 2; C2) --> C°°(R2; C2) acting via

P(f

=2i

x

Of 8x-

on pairs of complex-valued functions. An easy calculation leads to the following alternative formula for P:

P=(0i

i

0)8x01

01

)ay69

.

Denoting the matrices occurring in this formula by -/_- and -y., Yx=

i 0) 0

>

'Yy= ( 01

1

0),

yields

P = yx 8x

+ yy ey xi

Introduction

xii

as well as 'Yx 2- -E _ 'Yy,

'Yx'Yy + 'Yy'Yx = O.

The square of the operator P coincides with the Laplacian A on If82: P2

=

a2 axe -aye = A. 192

of the Laplacian within the class Thus we have found a square root P = of first order differential operators, and its kernel is, moreover, the space of holomorphic (anti-holomorphic) functions.

In higher-dimensional Euclidean spaces the question whether there exists a of the Laplacian was raised in the following discussion by square root P.A.M. Dirac (1928). Let T be a free classical particle in II83 with spin 2 whose motion is to be studied in special relativity. Denoting its mass by m, we have v"'' its energy by E and its momentum by p = 1-2/c2

E=

c2p2 -m2 c4.

In quantum mechanics T is described by a state function %(t, x) defined on RI x 1R3, and energy as well as momentum are to be replaced by the differential operators

E --+ ihat and p i--+ -ih grad, respectively. The state function 0 then becomes a solution of the equation

ih at =

c2h2A + m2c4

- jr. Mathematia cally speaking we now move to an n-dimensional Euclidean space and look involving the 3-dimensional Laplacian A = - a for a square root P =

of the Laplacian A = -

i-1

axti

The obvious as-

sumption that P should be a first order differential operator with constant coefficients leads to the ansatz

Pyi n

i=1

Now the equation p2 = A

axi

a holds if and only if the coefficients ryi ti

of P satisfy the conditions

y2=-E, i=1,...,n;

'Yi7j + 7j'Yi = 0,

i 0j.

Introduction

xiii

For n = 3, there is an obvious solution to these equations. The vector space C2

can be identified with the set of quaternions via C2

I

\

zl I = z1 + jz2, 2

= H -* )F3[ = C2 then correspond to multiplication by the quaternions i, j, k E H , respectively. Writing these as complex (2 x 2)matrices, we obtain and yl, y2, y3

y1=

(o 0i),

y2=

(0 -1),

73=

(0 o).

The algebra multiplicatively generated by n elements y1, ... , yn satisfying the relations

y2 = -E,

yiyj + y;yi = 0 (i 0 j),

is called the Clifford algebra Cn (W.K. Clifford, 1845-1879) of the negative definite quadratic form (Rn, -xi - ... - x,2n). Thus, the question whether there is a square root of the Laplacian leads to the study of complex representations K : Cn -* End (V) of the Clifford algebra. It turns out that Cn has a smallest representation of dimension dime V = 2[2]. The corresponding vector space is denoted by On and its elements are the Dirac spinors. Moreover, v"A is a constant coefficient first order differential operator acting on the space C°O(IIBn; An) of smooth An-valued functions on I[81.

Spinors can be multiplied by vectors from Euclidean space. In order to define this product we represent a vector x E an as a linear combination with respect to an orthonormal basis el,... , en, n

x=

xzei, i=1

and then define its product x V by a spinor 0 E An as n

x

_ E xzr, (yi) (0) i=1

From the defining relations of the Clifford algebra one immediately deduces the formula

x (x . 0) = -I Ixl l2V) In particular, the product vanishes if and only if either the vector x E W or the spinor b E On is equal to zero. There is no non-trivial representation e of the linear or the orthogonal group in the space An of spinors that is compatible with Clifford multiplication, i.e. which satisfies the relation A(x) . e(A) (0) = e(A) (x . V))

Introduction

xiv

for every A E SO (n; R), x E RI and E Z. Hence spinors on Riemannian manifolds cannot be defined as sections of a vector bundle that is associated

with the frame bundle of the manifold. It is for this reason that in differential geometry the question to what extent the concept of spinors could be transferred from flat space to general Riemannian manifolds remained open for decades. In 1938 Elie Cartan expressed this difficulty in his book "Lecons sur la theorie des spineurs" with the following words:

"With the geometric sense we have given to the word `spinor' it is impossible to introduce fields of spinors into the classical Riemannian technique." Only the development of the framework of principal fibre bundles and their associated bundles as well as the general theory of connections within differential geometry at the end of the forties made it possible to overcome

this difficulty. The group SO(n; R) is not simply connected. For n > 3 its universal covering, the group denoted by Spin(n), is compact and covers SO(n; R) twice. On the other hand, there exists a representation e Spin(n) -> GL(On) of the spin group which is compatible with Clifford :

multiplication. Considering now those special Riemannian manifolds Mn, today called spin manifolds, the frame bundle of which allows a reduction to the double cover Spin(n) of the structure group SO(n; R), we can define the vector bundle S associated with this reduction via the representation

e : Spin(n) -> GL(An), the so-called spinor bundle of Mn. Then spinor fields over Mn are sections of the bundle S and, as in the Euclidean case, the Dirac operator D can be introduced by the formula n

Do_ ti=1

Here 17 denotes the covariant derivative corresponding to the Levi-Civita connection of the Riemannian manifold. Therefore, spinor fields and Dirac operators cannot be introduced on every Riemannian space; but, nevertheless, they can be introduced for a large class. The existence of a Spin(n)-reduction of the frame bundle of Mn

translates into a topological condition on the manifold, i.e. the first two Stiefel-Whitney classes have to vanish:

wi(M') = 0 = w2(Mn). In dimension n = 4, for a compact simply connected manifold M4, this topological condition is equivalent to the condition that the intersection form on H2(M4;Z), considered as a quadratic form over the ring Z, is even and unimodular. The algebraic theory of quadratic Z-forms then implies that the signature v is divisible by 8. Surprisingly, in 1952 Rokhlin proved

Introduction

xv

a further divisibility by 2: the signature o -(M') of a smooth compact 4dimensional spin manifold M4 is divisible by 16: a(M4)/16 E Z.

This additional divisibility of the signature of a 4-dimensional spin manifol which does not result from purely algebraic considerations, was an essential aspect for the introduction of spinor fields and Dirac operators into mathematics. The consideration behind that may be outlined as follows. Could it be possible that there exists an elliptic operator P on every compact smooth 4-dimensional manifold with even intersection form on H2 (M4; Z), the index of which coincides with v/16? Today we know the answer to that question: it is essentially given by the Dirac operator on a spin manifold, eventually introduced for Riemannian manifolds by M.F. Atiyah in 1962 in connection with his elaboration of the index theory for elliptic operators. Since then it

has occured in many branches of mathematics and has become one of the basic elliptic differential operators in analysis and geometry. This book was written after a one-semester course held at Humboldt-University in Berlin during 1996/97. It contains an introduction into the theory of spinors and Dirac operators on Riemannian manifolds. The reader is assumed to have only basic knowledge of algebra and geometry, such as a two or three year study in mathematics or physics should provide. The presentation starts with an algebraic part comprising Clifford algebras, spin groups and the spin representation. The topological aspects concerning the existence and classification of spin reductions of principal SO(n)-bundles are discussed in Chapter 2. Here the approach essentially requires only elementary covering theory of topological spaces. At the same time, each result will also be translated into the cohomological language of characteristic classes. The subsequent Chapter 3 deals with analysis in the spinor bundle, the twistor operator and the Dirac operator in detail. Here the general techniques of principal bundles and the theory of connections are applied systematically. To make the book more self-contained, these results of modern differential geometry are presented without proof in Appendix B. Chapter 4 contains special proofs for the analytic properties of Dirac operators (essential self-adjointness, Fredholm property) avoiding the general theory for elliptic pseudo-differential operators. Eigenvalue estimates and solution spaces of special spinorial field equations (Killing spinors, twistor spinors) are the topic of Chapter 5. We mainly discuss the general approach, referring to the literature for detailed investigations of these problems. The book is concluded in Appendix A with an extended version of a talk on

xvi

Introduction

Seiberg-Witten theory given by the author in the seminar of the Sonderforschungsbereich 288 "Differentialgeometrie and Quantenphysik" in Berlin on February 9, 1995. Since the eighties a group of younger mathematicians at Humboldt-Univer-

sity in Berlin has been working on spectral properties of Dirac operators and solution spaces of spinorial field equations. Many of the results from this period are collected in the references. On the other hand, the present book may serve as an introduction for a closer study. I would like to thank all those students and colleagues whose remarks and hints had an impact on the contents of this text in various ways. I am particularly grateful to Dr. Ines Kath for her careful and detailed corrections of the text, and to Heike Pahlisch, whose typing of the manuscript took into account every single wish. Thomas Friedrich Berlin, March 1997

The English translation of this book has been prepared in the beginning of the year 2000. It does not differ essentially from the original text, although I made many changes in details which are not worth listing. During the last three years many new results have been published in this still dynamic area of mathematics. I included the corresponding references in the bibliography of the translation. During the academic year 1996/97 Dr. Andreas Nestke provided the exercises for students of my lectures at Humboldt University which furnished the starting point for this book. Two years later he had to leave the University. I thank him, as well as Dr. Ilka Agricola and Heike Pahlisch, for all the work and help related with the preparation of the English edition of this book. Thomas Friedrich Berlin, March 2000

Chapter 1

Clifford Algebras and Spin Representation

1.1. Linear algebra of quadratic forms We will start by recalling some facts from linear algebra. Let K be a field of characteristic 0 2. A bilinear form is a pair (V, B) consisting of a K-vector space V and a symmetric bilinear map

B:VxV--*K. The function Q : V --> K defined by Q(v) = B(v, v) is the corresponding quadratic form. Thus,

Q(Av)=A2Q(v) vEV, AEK. On the other hand, B can also be expressed by Q: B(vi, V2)

=

2

[Q(vl + v2) - Q(vl) - Q(v2)]

hence we will often identify B and Q. A bilinear form (V, B) determines a linear map from the vector space V to its dual V*, VE)

(V, B) is called a non-degenerate bilinear form, if this map is injective. So, (V, B) is a non-degenerate bilinear form if and only if

b'0; vEV 2wEV:B(v,w); 0. Let V be a finite-dimensional K-vector space and vl,... , v, (n = dimK V) a basis. Then the matrix M(V, B) _ (B (vi, vj))Z j.1 1

1. Clifford Algebras and Spin Representation

2

is symmetric. It is called the matrix of the form.

Definition. The rank of the bilinear form (V, B) is defined to be the rank of the matrix M(V, B), rank (V, B) := rank (M(V, B)).

Theorem (Lagrange). Let (V, B) be a finite-dimensional bilinear form. Then there exists a basis v1, ... , v,z for the vector space V with the property that the matrix M(V, B) is diagonal (bases with this property are called canonical):

/ Ai

0

Ar

M(V, B) =

0

0

r = rank(V, B).

,

0

)

If the field is K = R or C, then even more holds: Theorem (Sylvester). For every quadratic form over R there exists a canonical basis with respect to which the form has the following matrix:

/1

0

1

M(V,B) = 0

The number p of (+1)-entries and the number q of (-1)-entries in this diagonal matrix do not depend on the choice of this basis. The pair (p, q) is called the signature of the form, the number q its index.

1.1. Linear algebra of quadratic forms

3

Theorem. For every quadratic C-form there exists a canonical basis with respect to which the form has the following matrix:

/1

0

1

M(V, B) _

0

0

0)

We now consider a finite-dimensional non-degenerate bilinear form (V, B). If W C V is a subspace, we define the B-orthogonal complement W1 as

W1={vEV:B(v,w)=0 bwEW}. W is called an isotropic subspace if W f1 W1 $ {0}. W is called a nullsubspace, if W C W1 holds. Obviously, W is a null-subspace if and only if Qjw = 0 (or Blwxw = 0), i.e. Q (B) vanishes identically on W. Every null-subspace is isotropic but not vice versa.

Theorem (Witt Decomposition). Let (V, B) be a finite-dimensional nondegenerate quadratic K-form and W C V a maximal null-subspace. Then there exists a maximal null-subspace U C V such that

a) dimU=dimW, UnW = {0}. b) V=WeU®(W®U)1. c) For every vector 0 $ v E (W ®U)1 Q(v) $ 0. Moreover, for every basis w1 ... Wk in W there exists a basis u1 ... uk in U satisfying

B(uz7wj) = 8zj,

1 < i, j < k

(Witt basis).

Corollary. Let K be an algebraically closed field and (V, B) a finite-dimensional non-degenerate form. Then for every maximal null-subspace W

dimW <

[dirnV]

Proof. Consider the Witt decomposition V=W®UED (WED U)1.

It is sufficient to prove dim(W ® U)1 < 1. Let v, v' E (W ® U)1 be two non-trivial vectors. According to the preceding proposition, Part c), we have

for .,p E K

Av+µv'=04=#- Q(Av+µv') =0.

1. Clifford Algebras and Spin Representation

4

For fixed 0

p E K the quadratic equation in \ Q(Av + E.tv') _ A2 +

2tB(v, Q(v)

Q(v)

v') I\ + A2Q(v') = 0 Q(v)

has a solution. Hence, Q(Av + av') = 0, i.e. AV + µv' = 0, so v and v' are proportional.

1.2. The Clifford algebra of a quadratic form Let (V, Q) be a quadratic form over the field K K.

Definition. A pair (C(Q), j) is called a Clifford algebra for (V, Q) if 1) C(Q) is an associative lid-algebra with 1; 2) j : V --> C(Q) is a linear map and

j(v)2 = Q(v)

1

for all v E V;

3) if A is another K-algebra with 1 and u : V -> A a linear map satisfying u(v)2 = Q(v) 1, then there exists one and only one algebra homomorphism U : C(Q) -> A such that u = U o j.

-A

V

Proposition. a) For every quadratic form (V, Q) there exists a Clifford algebra (C(Q),j) b) If (C(Q),j) and (C'(Q), j') are both Clifford algebras for the same quadratic form (V, Q), then there exists an isomorphism f : C(Q) --> C'(Q) of the algebras satisfying f o j = j' C(Q)

\j

C'(Q)

V

/i

I

1.2. The Clifford algebra of a quadratic form

5

Proof. To show existence, consider the tensor algebra T(V) = K ® V

...

of the vector space V, and denote by I (Q) the two-sided ideal generated by all elements of the form (V (9 V) EB

{v ®v - Q(v) : v E V}.

Set C(Q) = T(V)/I(Q). If 7r : T(V) - C(Q) denotes the projection and i : V --> T(V) the natural embedding of the vector space into its tensor algebra, then

j =7roi determines a linear map j : V -> C(Q) for which, by construction, j (V)2 = Q(v) 1. Moreover, each linear map u : V -+ A into any algebra extends via U(vl (9 ... ® vk) = u(V1)

...

u(vk)

to an algebra homomorphism U : T(V) -+ A. If, in addition, u(v)2 = Q(V) 1, then we have _T(Q) C ker(U), and thus U induces a homomorphism

U : C(Q) -> A satisfying the relation we were looking for. Given another homomorphism U1 : C(Q) -+ A satisfying C(Q)

V

X

A

then u = U1 o j = Uo j. Hence U and U1 coincide on V C C(Q). On the other hand, the vectors from V generate the tensor algebra T(V) multiplicatively and hence the algebra C(Q) as well. Thus we immediately conclude U = U1. Uniqueness is a straightforward consequence of the third defining property of a Clifford algebra.

Corollary. The linear map j : V -> C(Q) from the vector space V of the quadratic form into its Clifford algebra is injective. The set j(V) C C(Q) generates the algebra multiplicatively.

For this reason we will often view the vector space V as a linear subspace of C(Q).

Proposition. The Clifford algebra C(Q) of a quadratic form is equipped with an involution /3 : C(Q) - C(Q) such that a) /3 is an algebra homomorphism and an involution, i.e. /32 = Id. b) Setting C°(Q) = {x E C(Q) : /3(x) = x}, C'(Q) = {x E C(Q) /3(x) = -x} we have the splitting C(Q) = C°(Q) ® C1(Q)

1. Clifford Algebras and Spin Representation

6

and the relations C°(Q) C°(Q) C C°(Q) , C°(Q) C'(Q) C C'(Q) as well as C'(Q) C1(Q) C C°(Q). In particular, C°(Q) C C(Q) is a subalgebra.

Proof. Consider the linear map u : V -3 C(Q), u2(v)

u(v) = -j(v). Since

= (-j(v))2 = j(v)2 = Q(v)

. 1,

there exists an algebra homomorphism 0: C(Q) -3 C(Q) satisfying

poj(v)=-g(v) for all v E V. Since, by construction,

(3o0oj)(v) =3H(v)) = -Oi(v) =j(v), ,

2 is the identity on the set j (V) C C(Q). Thus 02 = Id holds in general.

In addition to the involution : C(Q) -+ C(Q) each Clifford algebra also carries an anti-involution y : C(Q) -> C(Q). To construct it, we begin with a preliminary remark: If A is a Ilk-algebra, then one can define a new algebra

A on the set A by introducing the product

Now let (V, Q) be a quadratic form and C(Q) its Clifford algebra. Consider the algebra A = C(Q). For the linear map

V ')A=C(Q) the relation j (v) * j (v) = j (v) j (v) = Q (v) 1 holds in the algebra A. Thus there exists a unique algebra homomorphism 'y

: C(Q) -' C(Q)

such that

j(v) = yj(v),

v E V.

The properties of y as a map from C(Q) to itself are stated in the following

Proposition. For every

Clifford algebra there

C(Q) --> C(Q) with the following properties:

1) y is linear. 2) y o y = Id (y is an involution). 3) y(v) = v for all v E V C C(Q). 4) -y (x - y) = -y(y)'y(x),

x, y E C(Q)

exists a linear map

1.2. The Clifford algebra of a quadratic form

7

Given two bilinear forms (V,, B,) and (V2, B2), their direct sum is defined as the bilinear form (V, B) with V = V1 ®V2,

B(Vi,V2) = 0,

B1v1xvl = B1,

Blv2xv2 = B2.

Correspondingly, we will consider the direct sum Q1 ® Q2 of two quadratic forms.

Proposition. The Clifford algebra C(Q1(9 Q2) is isomorphic to the tensor product C(Q,)®C(Q2),

c(Q, ® Q2) = C(Q,)®c(Q2)

Remark. The tensor product C(Q1)®C(Q2) is to be understood as a tensor product of 7Z2-graded algebras. For two Z2-graded algebras A = A° A', B = B°6 B1 the tensor product A®B is the following Z2-graded algebra:

(A ®B)° = (A° ®B°) ®(A' (&B1),

(A ®B)1 = (A1 (&B°) ®(A° ®B1)

with product (a ® b?) (a' (a b)

=

(bib)

foranyaEA, bEB, aiEA'and& EBB. Proof of the proposition. Consider the linear map u : V1 ®V2 --j C(Q1)®C(Q2) defined by the formula u(v, + v2) = j, (v1) ®1 + 10 j2 (V2)

.

The multiplication rule in the algebra C(Q1)®C(Q2) implies

u2(v1+V2)=(v,®1+1(9v2)2=v1®1+v1®v2-v1®v2+1®v2, since 1 E C°(Q,), v1 E C'(Q,), and V2 E C1(Q2) belong to the corresponding sets of the Z2-grading. Thus, u2(v1 + V2) = (Q,(vl) +Q1(v2))1 ® 1 = Q(v, + v2)

1.

Therefore, u induces an algebra homomorphism U : C(Q, ® Q2) ---f C(Q,)W(Q2)

furnishing the isomorphism to be constructed.

Proposition. Let V be an n-dimensional vector space. Then the vector space C(Q) has dimension 2n, dimK C(Q) = 2n.

1. Clifford Algebras and Spin Representation

8

Proof. By the Lagrange theorem the quadratic form is the sum of n onedimensional quadratic forms:

Q=Q1®...®Qn. But the Clifford algebra of a one-dimensional quadratic form B : K x K --> K is easily computed,

and e2=Q(1).

C1(Q) = K e,

CO(Q)=K,

Hence, dimK C(Q) = 2 for a one-dimensional quadratic form. From the last proposition we conclude C(Q) = C(Q1)®... ®C(Qn)

and hence dimx C(Q) = 21.

Proposition. Let (V, B) be a quadratic form and v1, ... , vn a basis of V such that

B(vi, vj) = 0,

i

j.

Then the Clifford algebra C(Q) is multiplicatively generated by the elements , vn E V C C(Q) which satisfy the relations

vi, ...

v2 = Q(vi),

vivj + vjvi = 0,

i 0 j.

A particular basis of the vector space C(Q) is formed by the elements 1 and vil .... via, where 1 < it < i2 < < is < n with 1 < s < n.

Proof. V C C(Q) generates C(Q) multiplicatively, and vl,... , vn are a basis of the vector space V. Hence the vectors vi, ... , vn generate the algebra C(Q), too. Moreover, v? = Q(vi) for trivial reasons, and (vi + vj)2 = Q(vi + vj) = Q(vi) + Q(vj) = v? + v immediately implies

vivj + vjvi = 0,

i # j.

These 2n elements generate C(Q) linearly. Finally, since dim C(Q) = 2n, this system has to be a basis.

Example. Let us consider the trivial quadratic form Q - 0 on the vector space V. Then C(Q) = A * (V) is nothing but the exterior algebra of V.

Example. Let IK = IE8 and take V = R2 with the quadratic form Q = -x2 y2. Then C(Q) = } is the algebra of quaternions.

-

1.2. The Clifford algebra of a quadratic form

9

Proof. C(I[82, Q) is generated by el, e2 E R2. Thus 1, ei, e2, el e2 are a basis of the vector space C(Q). With the notation

is=e1i

j:=e2, k:=el e2,

the fundamental relations take the form

k.i=j, j2=j2=k2=-1,

i j=k, since e2 = -1 = e2,

e1e2 = -e2e1.

El

Example. Consider K = R, V = II81 and Q = -x2. In this case, C(Q) coincides with the algebra of complex numbers.

Proof. 1 and el E R1 form a special linear basis of C(Q) subject to the single relation ei = -1. Next we want to determine the centers of C(Q) and C°(Q), respectively, in the case of a non-degenerate form. In order to achieve this, consider a basis V1.... , vn of the vector space V such that

B(vi, vi) = Ai 00 and B(vi, vj) =0 for i

j,

i = 1,...

, n.

Let P be the set of all strictly ordered subsets of {1, ... n}: an element of ,

P is a subset P = {pl,... pk} where 1 -
...

VP1

VPk,

P = {Pi < p2 < ... < Pk}

P = 0

1,

Lemma. For P, R E P the equality VP vRVPi = (_1)IPIIRI-HPnRJvR

holds in C(Q), where I

I

denotes the cardinality of the corresponding set.

Proof. Note that the element vp is invertible in C(Q). Its inverse is given by -1 _ 1

VP

...

1

5Pk vPk

...

VP1.

P1 The formula to be proved then immediately follows from the relation vi vj _

-vj vi for i ; j. Proposition. 1) The center of the algebra C(Q) is equal to

Z(C(Q)) _

K

K®K[vl ...

vn]

for dim V even, for dim V odd.

1. Clifford Algebras and Spin Representation

10

2) The center of the algebra C°(Q) is equal to

(C0(Q)) _

K ®K[vl ...

for dim V even, for dim V odd.

vn]

K

Proof. Every element of the algebra C(Q) is of the form

a = E apvp. PEP

If a belongs to 2(C(Q)) or to Z(C°(Q)), then, in particular, vRa = avR necessarily holds for each set R E P with RI = 2. This implies apvp.

1: apvRVpvR-1 = PEP

PEP

Since RI = 2, we conclude E (- 1) (PnR) apvp = PEP

apvp, PEP

so that (-1)IPnRlap = ap for each set R C P with SRI = 2. This relation is a necessary condition to be satisfied by any central element. From this it follows straightaway that ap = 0,

if P 54 0, {1,

... n}, ,

i.e.

Z(C(Q)), Z(C°(Q)) C K ® K[vl ... vn]. If n is odd, then vi ... vn does not belong to C°(Q), hence Z(C°(Q)) = K. On the other hand, for n odd and P = 1,... , n} the lemma implies VPVRVp1 = (-1)IRI-IRIVR,

yielding VpVR = VRVP. In this case, the element vl all elements of the algebra C(Q), and we arrive at

...

vn commutes with

Z(C(Q)) = K ® K[vl ... vn]. The case dim V = 0 mod 2 is treated analogously.

1.3. Clifford algebras of real negative definite quadratic forms Let us first fix notations for some special Clifford algebras which will play an eminent role through the rest of the book: Cn = C(>18n, -x1 - ... - xn) - Clifford algebra of the n-dimensional real

Cn = C(1

, xi +... + xn)

negative definite form, - Clifford algebra of the n-dimensional real positive definite quadratic form,

1.3. Clifford algebras of real negative definite quadratic forms

Cn = C(Cn, zi + ... + zn)

11

- Clifford algebra of the n-dimensional complex quadratic form.

Remark. If Q is a non-degenerate complex quadratic form of dimension n (e.g. Q = -zi - ... -zn), then Q is equivalent to the form (Cn, z1 +...+zn) over the complex numbers and hence both their Clifford algebras coincide, C(Q) = Cn. Let us begin with the following general consideration concerning complexification. For a real quadratic form (V, B) the complexification (V OR C, BC) is defined as

Bc(vl 0 z, v2 ®w) = B(vl, v2)zw. On the other hand, if A is any real algebra, then its complexification A OR C

carries the product structure

(al 0 z) (a2 0 w) = (ala2) 0 (zw) turning A ® C into a complex algebra.

Proposition. Let (V, B) be a real quadratic form and (Vt, BC) its complexification. Then, in the sense of isomorphic C-algebras, C(VV, Bc) = C(V, B) OR C.

Proof. Define u : V OR C --> C(V, B) 0 C by u(v 0 z) = j(v) 0 z, where j : V -+ C(V, B) is the embedding of the real vector space into the Clifford algebra. Then, u(v ®z)2 = (j(v) (&z)2

=

j(v)2 ®z2

= B(v, v)z2 101 = Bc(v ®z, v ®z)

1.

Hence u extends to a homomorphism u : C(VV, BC) --> C(V, B) OR C of the

complex algebras. It is easy to check that u is an isomorphism. Corollary. One has the following identity of complexifications:

Cn = CnOR C = Cn,®RC.

Proof. The complexifications of the real forms xi+...+xn and -xi-...-xn are equivalent.

Proposition. The following graded real algebras are isomorphic: Cn+2 = Cn, OR C2

C1

n+2 = Cn ®C2.

The tensor product is to be understood as the usual one for algebras.

1. Clifford Algebras and Spin Representation

12

Proof. Choose an orthonormal basis of the vector space R'n+2 consisting of the vectors el, ... , en+2. The first n vectors then generate the algebras C, and Cn, where e', ... , e;n denote these same vectors considered this time as generators of C. Now define u : Rn+2 -- Cn, OR C2 by u(el) = 1 ®el, u(e2) = 1 ®e2i u(ei) = ei-2 ®ele2, 3 < i < n + 2. Then,

in

OR C2 the following equations hold:

u(el)2 = (1(9 ei)(1 ®ei) = 1®e2 = -1,

u(e2)2

= (1(9 e2)(10 e2) _ -1,

u(ei)2 = (e'i-2 (3 ele2)(4-2 (D ele2) = ei-2 0 ele2ele2 = 1 ® ete2eie2 = - 1, and of course for mixed terms

u(ei)u(ej) + u(ej)u(ei) = 0. Consequently, there exists an algebra homomorphism u:Cn+2--pC'n®II8 C2

fitting into the commutative diagram Cn+2 U

Rn+2

- Cn OR C2

Now u preserves the Z2-grading and is the desired isomorphism.

Example. The algebra C2 = C2 OR C is the complexification of C2. The latter algebra is generated by el, e2 satisfying the relations e2 = - 1 = e2,

ele2 + e2e1 = 0.

On the other hand, the underlying vector space of the complex algebra M2(C) has the basis E

0

_

(10

1)

,

\i

91 =

0

-iJ

92 =

(i

0T

(i

0

\

with the relations 91

=

_I = 92,

9192 + 9291 = 0.

This implies C2 = C2 OR C = M2 (C) .

Making repeated use of this example, we arrive at the following isomorphism.

Proposition. There is an isomorphism Cn+2 = Cn ®C M2(C). Proof. Cn+2 = Cn+2 OR C = (C;, OR C2) OR C = (Cn OR C) ®r- (C2 0 C) _ C.' ®CC2=C; ®CM2(C)

1.3. Clifford algebras of real negative definite quadratic forms

13

This isomorphism explicitly involves the one described in the proof of the previous proposition, i.e. Cn+2 = C,,, OR C2. Thus we obtain an explicit isomorphism C,' +2 = C° 0 M2 (C) which we spell out once again.

Corollary. Let e1, ...

, en+2 be the generating elements of the algebra Cn+2

... en those for the algebra C. Moreover, denote

and, correspondingly, ei,

,

by

i

91=

0i

),

92

=

\

O

i/

the generating elements of the algebra M2 (C) The isomorphism .

Cn+2 ' Cn OC M2 (C)

is then given by

e1*1®g1,

e2r--+1®g2,

3<jCn+2.

ei'-'(Set-2)®9192,

We will now apply these isomorphisms repeatedly and take into account the descriptions of the Clifford algebras involved, i.e.

C1'=C1®RC=C®R(C=C®C,

C2c=M2(C).

In summary, we immediately obtain the following proposition:

Proposition. a) If n = 2k is even, then Cn = M2 (C) ®... ® M2 (C) = .End (C2 (& ... ® C2) = End(C2"). k times

k times

b) If n = 2k + 1 is odd, then Cn = {M2(C)®...®M2(C)}®{M2(C)®...®M2(C)} = End(C2k)®End(C2k). These isomorphisms are explicitly described by the following formulas: The case n = 2k even:

ei -->E®...®E®9«(i)®7

. (9

T

times

where

if j is odd, 2 if j is even. The case n = 2k + 1 odd: If 1 < j < 2k, ej is mapped to ej i--> (E®...®E®9a(9)®T T, E®...®E(3 9a(j)®T 0 1

a (7)

[ 2 ] times

®T). times

The endomorphism e2k+1 is realized by e2k+1 H (iT ®... ®T, -iT ®... (S T).

1. Clifford Algebras and Spin Representation

14

Definition (complex n-spinors, Dirac spinors). The vector space of complex n-spinors is

forn=2k,2k+1.

pn:=cC2k =C2®...®C2 k times

The elements of An are called complex spinors.

Using this notation we obtain

if n = 2k is even,

Cn = End(An),

Cn = End(On) ® End(An), if n = 2k + 1 is odd. Moreover, the following diagram is commutative: cc

cc

2k+1

2k

End(A2k) `P

End(A2k+1) ® End(A2k+1)

Here cp is the diagonal map cp(A) _ (A, A) and the vector spaces A2k = A2k+1 coincide.

Denote by Kn the so-called spin representation of the Clifford algebra C. In the case of an even dimension, n = 2k, this is nothing but the isomorphism explained before, Kn : Cn

- End(An).

If n = 2k + 1 is odd, then kn consists of the isomorphism Cn = End(An) End(An) followed by the projection onto the first component, Kn : Cn

-) End(An) ® End(An) p-4 End(An).

The vector space of complex n-spinors is thus turned into a module over the Clifford algebra C,',.

1.4. The pin and the spin group Consider the real vector space Rn and the Clifford algebra Cn of the quadratic form -xi - ... - x,2n. W' itself is a linear subspace of Cn, R C Cn. For every vector x E Rn, the equality

x'x=-IIxIl2 holds in Cn and hence the inverse element x-1 is given by X

-1

-x

T1x,2.

1.4. The pin and the spin group

15

Definition. Pin(n) C Cn is the group which is multiplicatively generated by all vectors x E Sn-1. Therefore, the elements of Pin(n) are the products x1 ... x,,,, with xi E IISn, I xi 1. The spin group, Spin(n), is defined as I

Spin(n) = Pin(n) n CO,. Now we will define a homomorphism

A : Pin(n) -- O(n) from the group Pin(n) onto the group O(n) of orthogonal transformations of the Euclidean space Rn. To this end, recall the anti-involution

'Y:Cn-*Cn existing in every Clifford algebra. It has the particular property that y(x) _ x holds for all vectors x E ]l C Cn.

Lemma. If y E IEBn C Cn and x E Pin(n) C Cn, then the element x y y(x) belongs to the subspace R C Cn.

Proof. x is, by definition, the product x = xl ... x,,,, of vectors from the sphere Sn-1. Since

x ' y y(x) = xl ... xm Y ''Y(X

)

' ... y(xl),

we can suppose without loss of generality that m = 1, i.e. x E Sn-1. Next we choose an orthonormal basis in R' with x = el as the first vector. Then, writing

n

y=

yieie i=1

we get

n

n

E yiei

x . y . y(x) = e1

e1 = -y1e1 + E yiei,

(i=1

i=2

and hence x y y(x) is the vector in R which is the image of y under the reflection in the plane perpendicular to x. For x E Pin(n) we now define A(x) : Rn

Rn,

.\(x)y = x . y .'Y(x)

Then, obviously,

A(xlx2)y = x1x2YY(x1x2) = xlx2YY(x2)-y(xl) = A(xi)(A(x2)y)

and hence A : Pin(n) -> GL(n) is a group homomorphism. Every element

x of Pin(n) is the product x = xl ... x,n of vectors xi E S"-1, and the calculation above shows that A(xi) is a reflection. Therefore, A(x) itself is the superposition of reflections, so, in particular, an orthogonal transformation.

1. Clifford Algebras and Spin Representation

16

Proposition. a) A : Pin(n) --> 0(n) is a continuous surjective group homomorphism.

b) A-1(SO(n)) = Spin(n). c) ker(A) = {l, -1} -- Z2. d) For n > 2 , Spin(n) is a connected group. e) For n > 3 , Spin(n) is simply connected and A : Spin(n) -> SO(n) is the universal covering of the group SO(n).

Proof. a) directly follows from the well-known fact that every orthogonal linear map A : I[8n -* Rn is the superposition of reflections. Let 0 : C, -+ Cn be the involution of the Clifford algebra which defines the decomposition Cn = C° ® C. For every vector x E IlBn we have 0(x) -x. Now apply /3 to any element x = xi ... x,,,,: O(x) = )(XI) ...O(x-m,) _ (-1)mxl ... x,n.

From this we see that x = xi ... x,,,, belongs to Spin(n) if and only if m is even. On the other hand, A(x) = A(xi) ... A(x,,,,,) is the superposition of m reflections and, hence, A(x) lies in SO(n) if and only if m is even. This

proves b). Now suppose that A(x) = 1 holds in 0(n). Then xy-y(x) = y for all y E IlBn. Moreover, since \(x) E SO(n), x is the product of an even number of vectors, x = x1... x,n and m - 0 mod 2. Thus, x1...xmyxm...x1 = Y.

Multiplying this equation from the right by x1 ... x,,,, and taking into account that xi = ... = xm = -1 as well as m 0 mod 2, we obtain

x1...xmy = yx1...xm. The element x = x1 xm (m - 0 mod 2) of the Clifford algebra Cn thus commutes with every vector from IRE. Therefore, x belongs to the center of Cn and, at the same time, to the center of CO,. But, for every Clifford algebra,

Z(C(Q)) n Z(C°(Q)) _ K.

Hence, x E R', and from xj = 1 we see that x = ±1. It remains to be shown that Spin(n) is a connected group. As

A : Spin(n) --; SO(n) is surjective with ker A _ {1, -1}, it suffices to find a path in Spin(n) connecting the element (-1) E Spin(n) with the neutral element 1 E Spin(n). In the case n > 2, one such path is given by (0 < t < 1) y(t) = - cos(7rt) - sin(7rt)e1e2.

1.4. The pin and the spin group

17

y(t) indeed lies in Spin(n), since

tlel-sin y(t)= (cos(it) el+sin(it) e2I/ Icos1 \ \ /

it/ e2/ I

\(

in Cn.

Now we will describe the Lie algebra of the group Spin(n). For this we need the following general remark: Let A be a finite-dimensional associative real

algebra and A* C A the group of its invertible elements. A* is an open subset of A and, furthermore, a Lie group. Its Lie algebra a*, which can be identified with Ti(A*) A, thus coincides with A, a* = A.

The commutator of two elements al, a2 E A = a* is given by [al, a21 ala2 - a2al, and the exponential map

exp:a*=A-->A* is defined by the power series of the exponential function: °O

exp(a)

a an

- 1:

mt'

n=O

We are going to apply this observation in the case of the real Clifford algebra Cn. The spin group is a subgroup of Cn, Spin(n) C Cn,

and hence, to describe the Lie algebra spin(n), it suffices to determine the tangent space T1(Spin(n)) C Cn. We will proceed as follows: Let y(t) = xi(t) ... X2, (t) be a path in Spin(n) with xi(t) E Sn-1 and y(0) = 1. The tangent to -y at t = 0 is Tt (0)

=

dtl (0) ' x2 (0)

...

x2n, (0) + xi (0)x2 (0)

dtm (o).

Let M2 C Cn be the subspace of Cn spanned by the Clifford products ei ej, 1 < i < j < n. We will prove that each summand of dt (0) belongs to m2. Since -y(O) = 1, the first summand is equal to d

dt1(0) x11(0) But xi (t) xi (t) - 1 implies

dtl (0) x1(0)

dt1(0)xi(0) + xi(0)

dti(0) = 0

and, because of the relations in the Clifford algebra, ddt (0) and xi (0) are perpendicular as vectors in R1. Therefore, the first summand belongs to m2.

1. Clifford Algebras and Spin Representation

18

Analogously, the second summand coincides with x1(0) dt (O)x2 1(0)x1 1(0) _-x1(0) d2 (O)x2 (O)x1 1(0)

_ -{x1(0) dt (0)x1 1(0)} {x1(0)x2(0)XI 1(0)}. '

As dt (0) and x2(0) are perpendicular, the vectors x1(0) dt (0)xi 1(0) and xl(0)x2(0)xi 1(0) are perpendicular, too, and the second summand lies in m2. Altogether we arrive at the

Proposition. a) The linear subspace m2 C Cn

m2 = Lin(eiej : 1 < i < j < n) equipped with the commutator

[x, y] = xy - yx

is a Lie algebra which coincides with the Lie algebra of the group Spin(n) C C. b) The exponential map exp : m2 -+ Spin(n) is given by °°

exp(x)

i

_Y, x

-v.

i=o

c) If a : Cn -+ End(W) is a (real or complex) representation of the Clifford algebra and 0-ISpin(n)

: Spin(n) --> Aut(W),

the group homomorphism defined by restriction, then its differential (UjSpin(n))* : spin(n) = m2 -> End(W) is given by the formula (aaISpin(n))* = QJm2

Proof. The above calculation, first of all, shows that the Lie algebra spin(n) is contained in the subspace m2. Since dimR(m2) =

n(n - 1) 2

and dim(Spin(n)) = dim(SO(n)) = n(n - 1)/2, the two spaces have to coincide. This proves a) and b). c) follows from general facts concerning the differential of a homomorphism f : H --; G between two Lie groups.

1.4. The pin and the spin group Namely, f* :

19

h-B is uniquely determined by the commutative diagram exp

H--G as a homomorphism of Lie algebras. But in our situation the diagram

End(W)

m2 exp

exp

Aut(W)

Spin(n)

commutes, because u : Cn ---> End (W) is a homomorphism of algebras. This completes the proof.

Consider now the universal covering

A : Spin(n) - SO(n) and let us compute its differential

A. : spin(n) -> so(n). As before, we identify spin(n) with the vector space

m2 = Lin(ejej : 1 < i < j < n) and so (n) with the set of all skew-symmetric matrices. A particular basis of the vector space so(n) is formed by the matrices Ezj (i < j) defined by

0

j

i

...

...

0

0

...

-1

0

0 0

EZj _

j

1

0

...

...

0

...

0 0

Now we want to prove the formula '\*(ezej) = 2E,;j

for X, : spin(n) = m2 --3 so(n). The path

-y(t) = cos(t)+sin(t)eiej = -(cos(t/2)e;,+sin(t/2)ej)(cos(t/2)ei-sin(t/2)ei) is a subgroup y C Spin(n) with at (0) = eiej. Then, for k A('Y(t))ek = 616

i, j,

1. Clifford Algebras and Spin Representation

20

and, e.g. in the case k = i, we have

A(y(t))ei = (cos(t) + sin(t)eiej)ei(cos(t) + sin(t)ejei) = cos2(t)ei + 2sin(t) cos(t)ej - sin2(t)ei = cos(2t)ei + sin(2t)ej. g(A(y(t))ei) = lei. This computation proves the formula A (eiei) _ Thus 2Eij . The result can also be written invariantly.

Proposition. If z E spin(n) is an element of the Lie algebra, then for the differential A,, : spin(n) -+ so(n) the relation

A*(z)x = zx - xz holds for every x E Rn. In particular, the Clifford product zx - xz belongs to Rn(z E m2i x E Rn).

1.5. The spin representation Consider the Cn-module of n-spinors On. Via

Spin(n) C Cn C Cn - End(An), we obtain a representation rc of the group Spin(n) by restriction: K = KnjSpin(n) : Spin(n) -* Aut(On),

the spin representation of the group Spin(n).

Proposition. The spin representation is a faithful representation of the group Spin(n).

Proof. If n = 2k is an even number, then Cn = End(On) and the statement is trivial. This leaves the case n = 2k + 1. As vector spaces, O2k+1 = A2k, and the diagram Spin(2k)

Spin(2k + 1)

k2k

11k+1

GL(O2k) 1

GL(A2k+1)

commutes. This implies that the normal subgroup H := ker(K2k+1) has only the neutral element in common with Spin(2k),

H n Spin(2k) = {1}.

The subgroup A(H) C SO(2k + 1) is normal, since A : Spin(2k + 1) SO(2k + 1) is surjective. Moreover,

A(H) n SO(2k) = {E}.

Let A E \(H) C SO(2k + 1) be an element from this normal subgroup. The characteristic polynomial of A has odd degree, hence there exists a

1.5. The spin representation

21

unit vector vo such that A(vo) = vo. This means there is an element B E SO(2k + 1) for which BAB-1 E SO(2k). As A(H) is normal, this implies BAB-1 E A(H) f1 SO(2k), i.e. BAB-1 = E. Thus we proved that A(H) is the trivial subgroup of SO(2k+1). For H itself this leaves two possibilities, either H = {1} or H = {1, -1}. But (-1) E Spin(2k + 1) does not belong to the kernel of the spin representation. End(On,) can be considered as an endomorphism of On. This leads to the so-called Clifford multiplication of vectors and spinors, which is a linear map A vector x E Il8n C Cn C Cn

p: IR OR On -) On. Here p(x ®z)) is defined for x E JR and 0 E On by

µ(x (9 0) = K.(x)(0)Instead of p(x 0 v'), we will often simply write x cation extends to a homomorphism

This Clifford multipli-

p : A(II8n) OR An -- On

as follows: Using the orthonormal basis ei, ...

, en

of Rn, each element of

the exterior algebra A(IEB') can be written as

wk =

U

wili2 ...ikeil A ... A eik.

it<...
Y(wk ®4,) = l

wk

. Y = E wil...ik

..

eik

`Y7

il<...
where, as before, ei,, cp denotes Clifford multiplication of the vector eic by the spinor cp. A straightforward calculation leads to the formula

(xAwk) 0 =x (wk

0)+(x_jwk)

.0

for a vector x E IR and a multi-vector wk E A(1Rn). Now we will prove the

Proposition. Clifford multiplication p : A(W) OR On --+ On is a homomorphism of Spin(n) -representations.

Proof. For every element g E Spin(n) we have to check the formula (,\(g)wk) (r ' )_ We will do this by induction on the degree k of the multi-vector wk. For k = 1, wk is an ordinary vector x E II8n and the equation results from the K(g)

(W1.

1. Clifford Algebras and Spin Representation

22

following computation:

K(9)(x ')

K(9)Kn(x)(`f') =

Kn(9x9-1)K(9) ' = Kn(A(9)x)(K(9))

(,\(9)x) ' (K(9M Now we assume that the claim holds for multi-vectors of degree < k and consider wk+1 = x A wk. Then,

K(9)((xAwk)

)

K(9) (x ' (wk '

_

)) +

((x-j wk) .

(g)

(9)x ' (r, (g) (wk ' )) +

A

)

(9) (x wk) ' K(9)'b (,\(9)x_j A(9)wk) .

,(A(9)x) . ((A(9)wk) ' K(9) ) + ((A(9)x) A (A(9)wk)) ' r, (g) 'b = (A (g) w'+') r. (g)

This proves the proposition. Summarizing, we know now that vectors and multi-vectors can be multiplied by spinors. The product is always a spinor, and this multiplication is equivariant for the actions of the group Spin(n) on corresponding spaces. Next we will consider the case n = 2k in greater detail. In this case, the element el ... e2k belongs to the center of the algebra CO. As Spin(n) C CO, this element commutes with all elements from Spin(n). Hence the endomorphism

f = ikjC(el... e2k) : A2k

02k

is an automorphism of the Spin(n)-representation, i.e. f (n(g)0) = lc(g) f (0) for all g E Spin(n) and 0 E A2k. Since (el ... e2k)2 = (-1)k, f is an involution, f 2 = Idon . Thus the spin representation A2k decomposes into the eigensubspaces of f :

A2k = A2k ®02k,

A2k = {2O J E 02k : f (

)_

Definition (Weyl spinors). The spinors belonging to the subspaces 02k are called (positive or negative, respectively) Weyl spinors.

Proposition. a) dime 02k = dims 02k = 2k-1 b) If x E R2k is a vector and 0: E 02k, then the spinor x belongs to L. Thus Clifford multiplication induces homomorphisms m : R 2k OR Ak

A2k'

Proof. If x E R' is a vector (e.g. x = el), then, in the algebra Cn, we have the relation x (el... e2k) = -(el ... e2k) X.

1.5. The spin representation

23

Hence Clifford multiplication by the vector x anti-commutes with the involution f. Consequently, Clifford multiplication by a vector 0 0 x E R' maps the space L bijectively onto the space 02k

We will now prove that the spin representations A2k, A2k and A2k+1 all are irreducible representations of the spin group. To do this, we need the following preparation from linear algebra: Let V be a complex vector space and A = End(V) the algebra of all endomorphisms. Then A is a simple algebra, i.e. A does not contain any proper ideal. From this we immediately conclude the

Lemma. Let V and W be complex vector spaces with dim W < dim V

.

Then each homomorphism of algebras,

f : End(V) -* End(W), is trivial, f - 0. Proposition. The Spin (2k) -representation AZk is irreducible.

Proof. Consider the inclusions Spin(2k) C (C2k)° C C2k = End(A+ 2k ®A2k and assume that {0} # W g 02k is a Spin(2k)-invariant subspace. The products ei ej (i < j) belong to the group Spin(2k), and hence W is left invariant by them. On the other hand, the products eiej (i < j) multiplicatively generate the algebra (C20°. Thus we obtain a representation of this algebra in the vector space W: (C2k)° --> End(W).

C2k-1 = End(A2k_1) ® End(A2k_l), dim 02k_1 = 2k-1 and dim W < dim A+ = 2k-1 this representation f has to be trivial, an obvious As (C2 k)°

contradiction.

The same argument shows

Proposition. The Spin(2k + 1) -representation O2k+1 is irreducible.

Proof. In this case, Spin(2k + 1) C (C2k+1)° CC2k+1 = End(O2k+1) ® End(O2k+1) Again, let {0} W 0 O2k+1 be a Spin(2k + 1)-invariant subspace. Then, as above, W is left invariant by the action of the algebra (C2k+1)° and gives rise to a representation

f : (C2k+l)° -' End(W). Since (C2k+1)° = C2k = End(O2k) and dim W < dim 02k+1 = dim A2k, the

representation f is trivial, a contradiction.

1. Clifford Algebras and Spin Representation

24

The spin representation n : Spin(n) -> GL(An) is the representation of a compact group in a complex vector space. Thus there exists a Spin(n)invariant Hermitian scalar product in An. We will construct one such product satisfying an even stronger invariance property.

Proposition. In the vector space of n-spinors, On, there exists a positive definite Hermitian scalar product ( , ) with the invariance property

(x-0, x E I[8n, co, 0 E On. The spin representation rc : Spin(n) - GL(On) is a unitary representation with respect to this scalar product.

Proof. Let M2 = Lin(ezej i < j) C Cn be the Lie algebra of the group Spin(n). Consider g = an ® m2 C Cn. A simple calculation shows that g is a Lie algebra with the commutator z,WEg. :

The map cp : g - Cn+1 given by

OIm2 = Id,

lo(ei) = eien+1 for 1 < i < n

is the restriction of an algebra homomorphism 1 : Cn -} Cn+l.

is induced

by the map 1) : an -> Cn+l, l(ei) = eien+l, as can be seen from the equations

1 < i < n,

-D(ei)2 = -1,

l(ei)-D(ej) +

0,

1 < i < j < n.

Thus co : g --p Cn+1 maps the Lie algebra g bijectively onto the Lie algebra

of the group Spin(n + 1) and is, moreover, an isomorphism of these Lie algebras. Hence g is a compact Lie algebra. In view of the following remark this proves the assertion.

Remark. In the previous proof we made use of the following general result: Let g be a compact real Lie algebra and rc : g -* End(W) a representation in a complex vector space. Then in W there is a positive definite Hermitian

scalar product (,) with the invariance property (I(x)wl,W2) + (Wi,,(x)w2) = 0

for x E 9,Wi,W2 E W. To prove this we consider the compact group G corresponding to the Lie algebra g, an arbitrary positive definite product (, ) * in W, and we set (wl, W2) =

f(gwl,gw2)*dg, G

where dg is the Haar measure of the group G.

1.6. The group Spin

25

Proposition. If ic : Spin(n) - U(On) is the spin representation, then det (,(g)) = 1

for every group element g E Spin(n). In other words, the spin representation is a representation into the special unitary group SU(On) of the space of nspinors.

Proof. Basically, this is not a property special to the spin representation, but a consequence of the following observation: Consider the group homomorphism

f : Spin(n) -* S',

f (g) = det (s (g)).

As Spin(n) is simply connected, there exists a lift F : Spin(n) --+ R to the universal covering of S', f (g) = elriF(g) ,

which is a group homomorphism as well. Spin(n) is compact. Hence F(Spin(n)) C II8 is a subgroup which is contained in a bounded interval. This implies that F - 0 and thus f (g) = det (n(g)) - 1.

1.6. The group Spine The complex Clifford algebra Cam, comprises the group Spin(n) as well as the group Si of all complex numbers of modulus 1. Together they generate a

group which we want to denote by Spinc (n). Since Spin(n) fl Sl = {1, -1}, the group Spine (n) is apparently given by

Spine(n) = (Spin (n) x Sl)/{fl} = Spin(n)

xZ2

Sl.

The elements of Spine (n) are thus classes [g, z] of pairs (g, z) E Spin(n) x Sl

under the equivalence relation (g, z) - (-g, -z). We define several homomorphisms:

a) Let A : Spine(n) -+ SO(n) be given by A[g, z] _ .fi(g). b) i : Spin(n) -> Spine(n) is the natural inclusion, i(g) = [g, 1]. c) j : Sl -+ Spine(n) is the natural inclusion, j (z) = [1, z]. d) Let l : Spine (n) -+ S' be given by 1 [g, z] = z2. e) p : Spine (n) -+ SO(n) x Sl is given by p([g, z]) = (A(g), z2). Hence,

p=Axl.

1. Clifford Algebras and Spin Representation

26

Then the following diagram commutes:

Ii Spin(n)

1

Z

I

Spinc(n)

1

A

SO(n)

1

and all its rows and columns are exact. Moreover, p is a 2-fold covering of the group SpinC(n) over SO(n) x Sl. We will use this diagram to compute the fundamental group.

Proposition. Let n > 3. Then, a) The fundamental group ir1(SpinC(n)) is isomorphic to Z, and 1 : 7rl (SpinC (n)) --> 7ri (Sl) = Z is an isomorphism. b) Choose generators for the following groups: a E 7r1(SpinC(n)),

,(3 E -7rl(SO(n)),

y E 7rl(Sl)

with 1p(a) = y. Then for the homomorphism induced by the 2-fold covering pp : ir1(SpinC(n)) -+ rr1(SO(n)) x 7r1(Sl) the following formula holds: pp (a) = /3 + y.

Proof. a) follows directly from the exact sequence

1 --; Spin(n) -> Spinc (n) --> S1 -> 1 combined with 7rl (Spin(n)) = 1 for n > 3. Asp = A x 1, we still have to prove

A (a) = . 7r1(SO(n)) is isomorphic to Z2, and hence this is equivalent to A#(a) 0. Suppose that An(a) = 0. From the corresponding exact column sequence we see that there exists an element S E 7r1(Sl) with j (S) = a. This implies y = 1q(a) = ldjd(S) = 25

and y (as well as a) is not the generating element of the group 7r1(Sl).

1.6. The group Spin

27

Let n = 2k be an even number. The unitary group U(k) is a subgroup of SO(2k). Consider the homomorphism

f : U(k) - -> SO(2k) x S',

f (A) _ (A, det A).

Proposition. There exists a homomorphism F : U(k) -* Spin-(2k) such that the diagram

Spin (2k) p

U(k) ' SO(2k) x S1 commutes.

Proof. We have to prove that the group fp(7r1(U(k)) is contained in the If we choose a generating element d E irl(U(k)) = Z in set addition to the generating elements a, 3, y, then we have f f (6) _ l3 + y, and ,

the assertion follows by covering theory.

The same argument yields a lift of the homomorphism

f1 : U(k) --; SO(2k) x Sl,

f1(A) = I\A, det1 A /I

Proposition. There exists a homomorphism F1 : U(k) -+ Spin(2k) such that the diagram

Spin (2k) IP

U(k) fl

SO(2k) X Sl

commutes.

Remark. The homomorphism F : U(k) --> Spin(2k) can be explicitly described. Let A E U(k). Then there is a unitary basis fl, ... , fk in Ck with respect to which A has diagonal form eiO1

0

0

ei0k

A= If J : Ck -> Ck is the complex structure of Ck, then fj and J(fj),1 < j < k, are elements of the complex Clifford algebra C. We then define a

1. Clifford Algebras and Spin Representation

28

homomorphism F : U(k) -> Spin(2k) = Spin(2k) XZ2 S' by the formula k

F(A) = fl( cos C

Bl 2

l+ sin

g1

(2

fjJ(fj))

I

x e2

E a,

J

Since Spin(n) is contained in the Clifford algebra Cn, the spin representation of the group Spin(n) extends to a Spin- (n) -representation. For an element [g, z] from Spin (n) and any spinor 0 E On we then have i[g, z]ib = z . ,c(g)(,0).

Hence the space On of n-spinors becomes a Spin (n)-representation. The determinant of the endomorphism /c [g, z] : On - + On is given by the formula det ,c[g, z] = zdim(On)

In the case of an even dimension, n = 2k, the splitting 02k = 02k ®A2k is Spin(2k)-invariant and for the corresponding representations we have det rc±[g, z] = zdim(Ok)

This implies that the Spin (n)-representations det(A ) = and

(l)dimon±/2=l®

...®l

Adim(On) (A±)

(dim On /2 times)

are equivalent. For the particular case n = 4 we obtain the equation

A2(A)

A2(A) = l 4

= in the sense of Spin (4)-representations. 4

Proposition. There exists an injective homomorphism f : Spin (n) Spin(n + 2) such that the diagram

Spin (n) p

f

Spin(n + 2) A

SO(n) x Sl = SO(n) x SO(2) - SO(n + 2) commutes.

Proof. Spin(n) is a subgroup of Spin(n + 2), and S1 can be realized as a subgroup of Spin(n + 2) by the elements cost + sinters+len+2

= (cos (t/2) en+1 + sin (t/2) en+2) (cos (t/2) en+1 - sin (t/2) en+2)

The intersection of these subsets of Spin(n + 2) is {±1}, and so we obtain a subgroup of Spin(n + 2) isomorphic to Spin' (n).

1.7. Real and quaternionic structures in the space of n-spinors

29

The Lie algebra of the group Spinc (n) C CI, is the direct sum spine (n) = m2 ® iIR

and the differential p* : spine (n) ---+ so (n) x iR of the covering p is described by

p* (e«ep, it) = (2Eaa, 2it),

1 < a < R < n.

1.7. Real and quaternionic structures in the space of n-spinors Recall the following definitions of a real and a quaternionic structure in a complex vector space.

Definition. Let V be a complex vector space. A real structure is an lib-linear map a : V --+ V with the properties

a2 = Id,

a(iv) = -ia(v).

Definition. Let V be a complex vector space. A quaternionic structure is an JR-linear map a : V -+ V with the properties

a2 = -Id,

a(iv) = -ia(v).

First, we want to study real and quaternionic structures in the 3-dimensional spin representation A3 = C2 and their invariance properties under Clifford multiplication.

Proposition. a) In A3 there exists a quaternionic structure a : 03

A3 which commutes with /Clifford multiplication by each vector x E 1R3:

a p A3 --3 03 which anti-commutes In with Clifford multiplication by each vector from a 2-dimensional subspace x E R2 C R3, ,8(x

b) = -x/(c),

X E J2 C R3,

and commutes with e3: /3(e3 . ') = e3 . /3('0)-

,3 is not Spin(3)-equivariant.

V)E A3,

1. Clifford Algebras and Spin Representation

30

Proof. We realize 03 as the vector space A3 = C2 and make use of the realization of the spin representation of the Clifford algebra C3 described in Section 1.3:

/O i

i 0 e1-91-(0 -i),

e2=92=1 i

0),

e3=iT=

0

1

-1

0

A3 by the formulas

Define a, /3 : A3

a(z2)=( z12), )3(z2)=(z2 A straightforward calculation then leads to the result, e.g. /3 ( e 2

N(e3 ' b)

) = /3

)

(

=

(

)

( Z2 ) = ( zzl ) ,

e2 ( ) = e2 (z2 ) e3 ' 0(0) = e3 (z2

=

(izl )

2 ) _ ( zzl ).

Hence, /3(e2 - 0) = -e2/3(?) as well as /3(e3 - 0) = e3/(L).

In order to continue the construction, we will need the following algebraic preparation. If a : V -> V and /3 : W -* W are real or quaternionic structures, then their tensor product

a®/3:V®CW-+V®cW can be defined by (a (3 /3) (v (9 w) = a(v) ® /3(w).

a ®/3 is Ii -linear and anti-commutes with multiplication by i, since

(a® 3)(i(v (3w)) _ (a® /)(iv (gw) = a(iv) ®/3(w) = -ia(v) ®/3(w)

_ -i(a®,3)(v®w). Note that a ® /3 is correctly defined. In V ®cc W, the identity v ® w = -(iv) ® (iw) holds, and we have

(a ®/3)(-(iv) ®(iw)) _ -a(iv) ®/3(iw) _ -(-ia(v)) ®(-i/3(w)) Q(V) ®/3(w).

Thus a ® /3 is well-defined. a ®,Q again is a real (quaternionic) structure, since (a ® /3)2 = a2 ® /32 = ±Id .

Recall the real and quaternionic structures a, /3 : C2 --> C2 defined above and their commutation relations (note that e3 = iT is replaced by T) a91 = 91U7

a92 = 92a,

aT = -T a,

/3T = -T/3. Now we will construct one of these structures in each vector space Z n-spinors a,, : A, --k A,,. 091 = -910,

092 = -920,

of

1.7. Real and quaternionic structures in the space of n-spinors

31

Proposition. 1) Let n = 8k, 8k + 1. In On there exists a real Spin (n)-equivariant structure an : On -* On which anti-commutes with Clifford multiplication:

xERn and ')EOn. 2) Let n = 8k + 2, 8k + 3. In On there exists a quaternionic Spin(n)equivariant structure an : On -* On which commutes with Clifford multiplication:

an(x 0) = x an(O), x E R

and V) E L.

3) Let n = 8k + 4, 8k + 5. In On there exists a quaternionic Spin(n)equivariant structure an : On - On which anti-commutes with Clifford multiplication:

an(x

XEll Bn

and 0EOn.

4) Let n = 8k + 6, 8k + 7. In On there exists a real Spin(n)-equivariant structure an : On -> On which commutes with Clifford multiplication:

xEl8n and 2b EOn. Proof. We define an case by case. First case. n = 8k, 8k + 1. We have On = C2 ® ... ® C2 (4k times), and we set

an (a(3,3)®...®(a(9,3) Second case. n = 8k + 2, 8k + 3.

(2k times).

We have On = C2 ® ... ® C2 (4k+1 times), and we set

an = a ®(Q ®a) ®... ®(0 (3 a)

(2k times).

Third case. n = 8k + 4,8k + 5. We have A. = C2 ® ... ® C2 (4k+2 times), and we set

an = (a (& ,6) ®... ®(a ®0)

(2k + 1 times).

Fourth case n = 8k + 6, 8k + 7.

We have On = C2 ® ... ® C2 (4k+3 times), and we set

an=a® (/3(9 a)® ...0(,c3(9 a)

(2k+1times).

Using the presentation of Clifford multiplication and the commutation relations between a, 0 and gi, 92, T, respectively, described in Section 1.3, the properties listed above are easily checked.

1. Clifford Algebras and Spin Representation

32

Summarizing, we arrive at the following table for the real or quaternionic structures in An: an

quaternionic structures

real structures

n commutes with Clifford multiplication

6,7 mod 8

n

2, 3 mod 8

n anti-commutes with Clifford multiplication

0, 1 mod 8

n

4, 5 mod 8

We now ask whether, in the case of an even dimension n, the structures just constructed are compatible with the decomposition of Dirac spinors On into the sum of Weyl spinors On ®On . Let n = 8k + 2e (e = 0, 1, 2, or 3). The decomposition is defined by the operator

f = i 4k+e el ... e8k+2e = ieel

e8k+2e

For e = 1, 3, Clifford multiplication commutes with an. Since an is complex anti-linear, in these cases, an anti-commutes with f = ±iel ... e8k+2e For

dimensions n = 8k + 2, 8k + 6 this implies that the real (or quaternionic) structure an : An - On interchanges the summands in the decomposition On=An ED An-:

an (On) C On For e = 0, 2, Clifford multiplication anti-commutes with an. Thus an com.

mutes with f = ±el ...

e8k+2e,

an f = f an, and an preserves the

decomposition On = An ®On In summary, we obtain the .

Proposition. 1) The representation A k admits a real Spin(8k) -equivari ant structure. 2) The representation A8k+4 admits a quaternionic Spin(8k + 4)-equivariant structure.

1.8. References and exercises E. Artin. Geometric Algebra, Princeton University Press 1957. H. Baum. Spin-Strukturen and Dirac-Operatoren fiber pseudo-Riemannschen Mannigfaltigkeiten, Teubner Verlag Leipzig 1981.

H. Baum, Th. Friedrich, R. Grunewald, I. Kath. Twistors and Killing Spinors on Riemannian Manifolds, Teubner Verlag 1991.

1.8. References and exercises

33

B. Budinich, A. Trautman. The Spinorial Chessboard, Springer-Verlag 1988.

D. Husemoller. Fibre Bundles, McGraw-Hill 1966. H.B. Lawson, M.-L. Michelsohn. Spin Geometry, Princeton University Press 1989.

A.L. Onishchik, R. Sulanke. Algebra and Geometrie, Teil II, Deutscher Verlag der Wissenschaften, Berlin 1988.

Exercise 1. Let (V, Q) be a non-degenerate quadratic form. Determine all elements in the Clifford algebra C(Q) which anti-commute with every element of V.

Exercise 2. Let (Vi, Ql) and (V2, Q2) be two quadratic forms and f : V1 -> V2 a linear map such that Q1 (VI) = Q2 (f (v1))

for all vectors vi E V1. Prove that there is a homomorphism C(f) : C(Q1) -> C(Q2) of Clifford algebras such that the diagram C(Q1)

C(Q2)

commutes.

Exercise 3. In the text, the equations C1 = C, C2 = IHI were proved. Prove the following additional isomorphisms: ED

C6 = M8 (I),

C4 = M2 (H),

C5 = M4 (C)

C7 = M8 (I) ®M8 (R),

C8 = Ml6 (R)

Exercise 4. Prove the isomorphisms

Ci=I®R, C2'=M2(I), Cs=M2(C), C4=M2(H), = M2(H) ®M2(H),

Cs

= M4(H), Exercise 5. Prove the equation Ck_1 = Ck. C5

C81=M8(C)-

Hint: If Ck_1 = Ck-1 6 C1k-1 is the decomposition of the algebra Ck_1 and ek E IRk the last vector, then setting

f (xo + xl) = x0 + ekxl defines a homomorphism f : Ck_1 -4 Ck.

1. Clifford Algebras and Spin Representation

34

Exercise 6. Prove that Spin(3) Spin(4) Spin(5) Spin(6)

SU(2) = {q E IHI : IIgII =1},

SU(2) x SU(2), Sp(2), SU(4).

Exercise 7. Prove that the group Spinc (4) is isomorphic to the following subgroup H of U(2) x U(2): H = {(A, B) E U(2) x U(2) : det (A) = det (B)}.

Exercise 8. Let el,...

, en be an orthonormal basis of Rn and A. : spin(n) -so(n) the differential of the 2-fold covering .1 : Spin(n) -+ SO(n). Prove that for every element z E spin(n) the following equation holds:

z = 2 (A*(z)ez, ej)ezej = 4 (A*(z)ei, ej)ei, ej. i<j

z,j

Chapter 2

Spin Structures 2.1. Spin structures on SO(n)-principal bundles Let X be a connected CW-complex and let (Q, ir, X; SO(n)) denote an SO(n)-principal bundle over X.

Definition. A spin structure on the principal bundle Q is a pair (P, A) where

a) P is a Spin(n)-principal bundle over X, b) A : P -> Q is a 2-fold covering for which the diagram

commutes. Here the rows contain the action of the respective group on the corresponding principal bundle.

Definition. Two spin structures (PI, Al) and (P2, A2) are called equivalent if there exists a Spin(n)-equivariant map f : Pl -> P2 compatible with the coverings Al and A2:

Pl f P2

/2

Al

Q 35

2. Spin Structures

36

Denote by F a fibre of the SO(n)-principal bundle Q. F is diffeomorphic to the group SO(n) and, since n > 3, the fundamental group irl(F) consists of two elements: 7r1(F) = 7L2.

Let a E 7rl (F) be the non-trivial element. Denote by i : F --; Q the embedding of F into the space Q. Then

aF := i#(a) is an element of the fundamental group 7r1(Q). Consider a spin structure A : P --; Q on Q. To this covering there corresponds the subgroup

H(P,A) := A#(ir1(P)) C ir1(Q) H(P, A) C 7r1 (Q) is a subgroup of index 2.

Proposition. The element aF does not belong to H(P, A), aF ¢ H(P, A).

Proof. Suppose that aF E H(P, A). Then the inclusion i : F --3 Q lifts to a continuous map I : F - P such that the diagram

P A

F zQ commutes. I(F) C P is contained in one fibre F' of the Spin(n)-principal bundle P, and hence we obtain a map

I : F = SO(n) --f F' = Spin(n)

with

A o I = Idso(,,,)

For the induced homomorphisms of fundamental groups this implies A#I# _ Id,.1(so(,,,)) Since 7r1(SO(n)) = Z2 and 7r1(Spin(n)) = 1, this is a contradiction.

Proposition. The equivalence classes of spin structures on an SO(n)-principal bundle Q over a connected CW-complex X are in bijective correspondence with those subgroups H C irl (Q) of index 2 which do not contain aF,

aF 0 H. Proof. Let a subgroup H C 7r1(Q) of index 2 with aF 0 H be given. This subgroup defines a 2-fold covering

2.1. Spin structures on SO(n)-principal bundles

37

with connected total space P. Fix a point po E P and denote by y Q x SO(n) -* Q the action of the group SO(n) on Q. The map

P

P x Spin(n)

A

AxA

QxSO(n) µ- Q induces the homomorphism p# o (A x A)#,

71(P) = 7rl(P x Spin(n))

(n

#

7ri(Q) ®7l(SO(n))

N-#

i1(Q),

the image in 7rl (Q) of which coincides with H. Hence there exists a unique

continuous map µ : P x Spin(n) -> P with µ(po, 1) = po fitting into the commutative diagram P x Spin(n) µ P IAxA

IA

QxSO(n) µ Q It is easy to show that µ is an action of Spin(n) on P. For example, the map f (p) = µ(p, 1) : P -* P is the lift of the map A : P -> Q,

P...f.. P QlA

IA Id

Q

with initial condition f (po) = po. Uniqueness of the lift then implies f = Idp, i.e.

µ(p,1)=p

for all p E P. It remains to be proved that Spin(n) acts simply transitively on each fibre

of the map 7r o A : P -+ X over X. To show this, it suffices to check that (-1) E Spin(n) does not have any fixed point. Suppose that

µ(po, (-1)) = Choose a path 'y(t) (0 < t < 1) from 1 to (-1) in Spin(n). Then -y*(t) _ µ(po, 7(t)) is a loop in P; hence it defines an element of the fundamental group irl (P). The equivalence class of the loop Ay* (t) thus belongs to the subgroup H C ,7rl (Q) of the covering A P --+ Q. On the other hand, Ay* (t) = AA(po, y(t)) = A(A(po), A o 'y(t))

2. Spin Structures

38

Now A o -y(t) is a loop in SO(n) representing the non-trivial element a E 7rl(SO(n)). Finally, for the fibre F = \(po) SO(n) C Q we obtain the inclusion aF = A-y* (t) E H C ir1(Q),

in contradiction with the assumption on the subgroup H. Let us look at the exact homotopy sequence of the SO(n)-fibration it : Q --> X: (*)

...

) 7r2(X) 0' 7r1(F) #' 7r1(Q) #' 7r1(X)

) 1.

A subgroup H C iri (Q) of index 2 defines a non-trivial homomorphism

fH : 7r1(Q) - iri(Q)/H = Z2 and vice versa. The condition aF

H is equivalent to requiring that the

homomorphism fH 0 i# : 7L2 = 7r1(F)

ir1(Q) -' 7r1(Q)/H = Z2

is the identity. Hence we obtain the

Corollary. The spin structures on an SO(n)-principal bundle Q over a connected CW-complex X are in one-to-one correspondence with the homomorphisms splitting the sequence (*):

f : 7r1(Q) -' 7r1 (F)

and

f o i# = Id,,(F).

Corollary. If the SO(n)-principal bundle has a spin structure, then the following groups are isomorphic: a) 7r1(Q) = ir1(F)

7r1(X),

b) 7r2(Q) = 7r2(X)

Corollary. Let X be a simply connected CW-complex. An SO (n) -principal bundle Q has a spin structure if and only if 71 (Q) = Z2.

In this case, the spin structure is uniquely determined.

The last corollary can be generalized using the extension theory of finite groups. The preceding considerations show that an SO(n)-principal bundle Q over a CW-complex X has a spin structure if a) i# :7r1(F) = Z2 --+ 7r1(Q) is injective;

b) the exact sequence splits.

1 -+ 7ri (F) -#-) 7r1(Q) -- 7ri (X) -- 1

2.1. Spin structures on SO(n)-principal bundles

39

We now suppose that irl (X) is a finite group and, moreover, that condition a) is satisfied. Then 7r1 (Q) is an extension of the group Z2 = 7r1 (F) by 7r1(X). Recall the notion of a 2-Sylow subgroup of a finite group G. If G1 denotes the order of the group G and 2k is the largest power of 2 dividing IGJ, then each subgroup of G of order 2k is called a 2-Sylow subgroup. It is well-known that there exists at least one such subgroup in every group G. If

1--iZ2--->G--*IF -->1 is an extension of Z2 by r, then we have the following criterion for the splitting of this extension.

Proposition (Schur-Zassenhaus, Gaschiitz). The extension G of the group Z2 by the finite group r splits if and only if for every 2-Sylow subgroup G2 C G the extension

1--> splits.

Proof. Compare for example R. Kochendorffer, Lehrbuch der Gruppentheorie unter besonderer Beriicksichtigung der endlichen Gruppen, TeubnerVerlag, Leipzig 1966, or the translation, Group theory, McGraw-Hill, 1970.

A first conclusion is that for 7r1(X) finite, the principal bundle Q admits a spin structure if and only if the following conditions are satisfied:

a') i# : -7rl (F) --> ir1(Q) is injective; b') for each 2-Sylow subgroup G2 C 7rl (Q), the corresponding extension

1 --p (G2nZ2) --) G2 -pG2/(G2nZ2)

1

splits.

If G2 C 71 (Q) is a 2-Sylow subgroup of 7r1 (Q), then G2 = p# (G2) C 71 (X)

is a 2-Sylow subgroup of 7r1 (X) and vice versa. Thus, if every extension of Z2 by G2 splits (or, equivalently, H2(G2; Z2) = 0), then condition b') is automatically satisfied. We obtain the Corollary. Let X be a CW-complex with finite fundamental group satisfying the condition H2 (G2; Z2) = 0

for every 2-Sylow subgroup G2 C 7r1(X). Then an SO(n)-principal bundle Q over X has a spin structure if

i# : 7rl (F) -* in1(Q) is injective.

2. Spin Structures

40

In particular, this corollary applies in the case where the fundamental group 7r1(X) is of odd order. The only condition for the existence of a spin structure then consists in requiring the injectivity of the homomorphism i# : 7r1 (F) _k 7ri (Q)

We will now reformulate the description of spin structures thus obtained in the language of cohomology. For every CW-complex Y we have H1(Y; Z2) = Hom (Hl (Y); Z2)

= Hom (7r1(Y)/[7r1(Y), 7r1 (Y)] ; Z2) = Hom (7r1(Y) ; Z2)

Hence the homomorphism f : 7r1 (Q) --> 7r1 (F) = Z2 defines an element in

the cohomology group, f E H'(Q; Z2). On the other hand, H'(F; Z2) = Hom (Z2, Z2) = Z2, and the condition f o i# = Id,, (F) is equivalent to the requirement that the element f E H1 (Q; Z2) remains non-trivial after restriction to the fibre i* : H1 (Q; Z2) --> H1 (F; Z2) = Z2 . From this we obtain the Proposition. The spin structures on an SO (n) -principal bundle over a connected CW-complex X are in one-to-one correspondence with those elements f E H1 (Q; Z2) for which i*(f) 0 holds in H1 (F; Z2) = Z2 -

The SO(n)-fibration 7r : Q -+ X induces the following exact sequence of cohomology groups:

1 ---) H' (X; Z2) -' H1(Q; Z2) - H' (F; Z2) = Z2

H2 (X; Z2)

-* .. .

If 1 E Z2 = H1(F;Z2) denotes the non-trivial element, then w2(Q) := 8(1) E H2(X; Z2)

is called the second Stiefel- Whitney class of the SO(n)-principal bundle. The above sequence, moreover, immediately implies the

Proposition. An SO(n)-principal bundle over a connected CW-complex X has a spin structure if and only if w2 (Q) vanishes, W2(Q) = 0.

In this case, the spin structures are classified by H' (X ; Z2) .

Example. Consider the complex projective space Cpn. The group SU(n + 1) acts transitively on CP :

cpn = SU(n + 1)/S(U(n) x U(1)). The isotropy representation a

: S(U(n) x U(1)) -> U(n) C SO(2n)

2.1. Spin structures on SO(n)-principal bundles

41

is given by the formula

(B

0

l

I

= det B B.

det B

Let R = SU(n + 1) x, SO(2n) be the frame bundle. As CIPh is simply connected, the fundamental group iri (R) has at most two elements; it is the surjective image of iri(SO(n)) = Z2. In order to decide whether R admits a spin structure, we first compute the homomorphism o-# between fundamental groups induced from the isotropy representation a. The generating element of the fundamental group iri(S(U(n) x U(1))) is represented by the loop

(eit 1

0

0
0

e-it j

Thus a(y(t)), as a path in U(n), is given by e2it

eit

I

0'('Y(t)) _

0

eit 0

eit

and hence is equal to the (n + 1)-fold power of the generating element of 7ri(U(n)). This implies that the homomorphism U# : 7rl(S(U(n) x U(1))) = Z -* iri(SO(2n)) = 7L2

is described by o,#(1) = (n + 1) mod 2. Consider first the case that n is odd. Then a# is the trivial homomorphism, and thus there exists a lift &: S(U(n) x U(1)) -+ Spin(2n) of the isotropy representation v to the spin group. Hence

P := SU(n + 1) x& Spin(2n) is a spin structure on the frame bundle R. Finally, we also discuss the case that n is even. Consider now the fibration of R = SU(n + 1) xQ SO(2n) over the space SO(2n)/a(S(U(n) x U(1))) = SO(2n)/U(n) defined by the formula

[A, B] - B mod a(S(U(n) x U(1))), where A E SU(n + 1) and B E SO(2n). The fibre of this fibration is SU(n + 1)/Z +i. Hence the resulting exact sequence has the form

... --> Zn+i -* 7ri(R) ---r iri(SO(2n)/U(n)) = 1.

2. Spin Structures

42

Since 7rl (R) contains at most two elements, and n is even, in this case, 7r1(R) = 1. Summarizing, we obtain

i (R)

7G2

1

if n is odd, if n is even.

Since CIPn is simply connected, we can apply the corresponding criterion for

the existence of a spin structure on R. The bundle R has a spin structure if and only if irl (R) = Z2. This leads to the Proposition. The frame bundle R of the complex projective space C?n admits a spin structure if and only if n is odd, n - 1 mod 2.

Remark. It is important to note that two spin structures on one and the same SO(n)-principal bundle may not be equivalent even if the corresponding Spin(n)-principal bundles over X are equivalent. To see this, consider e.g. X = Rp2 and the trivial bundle Q = RP2 x SO(n). Because H'(RIP2; Z2) = Z2, there are two spin structures on Q. On the other hand, let P --} X = RIP2 be any Spin(n)-principal bundle. Since dim(RIP2) = 2,

9r,(Spin (n)) = 0,

this bundle has a section and is thus trivial. Hence the spin bundles of both these spin structures are isomorphic as principal bundles over X = II ]?2. The same effect occurs if we replace RIP2 by an arbitrary 2-dimensional manifold.

2.2. Spin structures in covering spaces Consider a simply connected CW-complex X (7r,(X) = 1) and an SO(n)principal bundle Q which is assumed to have (exactly) one spin structure (P, A). Let the discrete group r act from the left on the space X and on Q as a group of SO(n)-bundle morphisms. Set

x* = r\x, Q* = r\Q and let X -+ X* = r\X be the covering map. Then Q* is an SO(n)principal bundle over the space X*. We want to study the question of whether Q* admits a spin structure.

Since 7r1 (P) = 7ri (X) = 1, for every group element y E r there exist two lifts -y4 of the transformation y : Q -* Q such that the following diagram commutes:

P-+P Q7-

Q

If e is a left action of r on P with e(y) =

then P* = r\P obviously

becomes a spin structure on Q*. Conversely, if (P*, A) is a spin structure

2.2. Spin structures in covering spaces

43

on Q* and q : X --+ X* denotes the projection, then Q is the bundle q*(Q*) induced from Q* by means of this projection. Therefore, q*(P*) is a spin

structure on Q on which F acts as a group of automorphisms. But, since irl(X) = 1, the spin structure q*(P*) is equivalent to (P,A). Thus, in summary, we conclude:

Proposition. The spin structures in the principal bundle Q* over X* are in one-to-one correspondence with all left actions e of r on P satisfying e(y) = ly'.

We can also study the existence question for a spin structure on Q* by looking at the fundamental group. From

Q - Q* I X

*

we obtain the following commutative diagram:

Z2 = 7ri (F)

7rl

(F*)

1

1

= 7. (Q*)

-' 7r1(Q)

1 = 7rl (X) -.

F

7r1(X*) = r

By assumption, Q has a spin structure and hence 9r1(F) - irl (Q) is injective. Thus irl(F*) -- 7rl(Q*) is also injective, and we arrive at the

Proposition. The SO(n)-principal bundle Q* over the space X* has a spin structure if and only if the exact sequence

1 _' Z2=iri(F*)

'ir1(Q*)

splits.

Example 1. Let X = Sn = SO (n + 1) /SO (n) be the n-dimensional sphere and Q = SO(n+1) its frame bundle. Moreover, let the group r C SO(n+1) act freely on the sphere from the left. Then Q* = F\SO(n + 1)

2. Spin Structures

44

is the frame bundle of the space X* = F\S'. Q itself, as a bundle over Sn, has the spin structure P = Spin(n + 1). Thus Q* admits a spin structure if and only if the sequence

ir1(F\SO(n + 1)) - r -

1 -> Z2 = 7r1(SO(n))

+1

splits. Considering the covering A : Spin(n + 1) -* SO(n + 1), we obtain

F\SO(n + 1) _ .\-1(F)\Spin(n + 1) as well as 7r1(A-1(r)\Spin(n+1)) _ A-' (r). Hence the sequence in question is 1

) Z2

) A-1(r)

)r

)

1,

and the manifold X* = r\Sn is a spin manifold if and only if this sequence splits. For a group F of odd order JFI this sequence always splits. Otherwise, the splitting of the above exact sequence is equivalent to the splitting of the sequence 1

, (A-1(F2) n Z2)

; A-1(F2)

> F2

) 1,

where F2 C F is a 2-Sylow subgroup. Summarizing, we obtain the

Proposition. Let F C SO(n + 1) be a finite subgroup acting freely on the sphere S. The manifold F\Sn has a spin structure if and only if for every 2-Sylow subgroup r2 C F the sequence

1 -f (.\-'(r2) n z2) _, \-1(F2) ---> F2 -> 1 splits.

Let us consider the case n = 4k + 1 = 1 mod 4 separately.

Proposition. Let n = 4k + 1, and let r C SO(n + 1) be a finite subgroup acting without fixed points on the sphere Sn. Then F\Sn has a spin structure if and only if r contains no elements of order 2. In this case, H1(F\Sn; 7G2) = 0,

and hence the spin structure of F\Sn is unique.

Proof. Suppose that r does not contain any elements of order 2. Then F has no proper 2-Sylow subgroups, too, i.e. on F\Sn there exists a spin structure. Let A E F be an element of order 2. Since (Ax, Ay) = (x, y) for x, y E Rn+1 and A2 = 1, the matrix A is symmetric: (Ax, y) = (x, Ay). Hence A is diagonalizable with eigenvalues Al = ... = An = ±1. If 1 is an eigenvalue, then A has a fixed point on the sphere. This implies A = -E.

2.3. Spin structures on G-principal bundles

45

Set ro = {E, -E}. This leads to the following commutative diagram of coverings:

84k+1 __ Fo\s4k+1 = Rp4k+1

r\S4k+1

If F\S4k+1 has a spin structure, then pulling it back we construct a spin structure on the real projective space 1E8p4k+1 Thus we arrive at a contradiction, since R? admits a spin structure only for n - 3 mod 4. Finally, we will prove that H1(F\S'; Z2) = Hom (F; 7L2) = 1,

if F contains no elements of order 2. Actually, in this case, the order IFl

of the group F is odd, i.e. F has an odd number of elements. A nontrivial homomorphism f : F ---3 7Z2 defines by means of Fo = ker(f) and any yo 0 ker(f) a partition of the set r into r = Fo U yo Fo, and hence IFI = 2IFo1 - 0 mod 2, in contradiction with the assumption.

2.3. Spin structures on G-principal bundles Let G C SO(n) be a connected compact subgroup for which, moreover, in this section, it will generally be assumed that the inclusion induces an epimorphism

i# : 7r1(G) -' 7rl(SO(n)). Then the homogeneous space SO(n)/G is simply connected, 7rl(SO(n)/G) _ {1}. Consider a G-principal bundle (Q, in, X; G) over a CW-complex X. Then the associated bundle Q* = Q xG SO(n) is an SO(n)-principal bundle over X. Definition. The G-principal bundle Q admits a spin structure if the SO(n)principal bundle Q* admits one. Now we will derive a condition for the existence of a spin structure in this sense. To do so, denote by F = G and F* = SO(n) the fibres in the bundles Q and Q*, respectively. Consider the commutative diagram

F=G

- SO(n)=F*

z

i

Q* I SO(n)/G

46

2. Spin Structures

where the maps j, 1, m are defined as follows for q E Q, A E SO(n):

j(q) = [q, l],

m(A) = A mod G.

1[q, A] = A mod G,

The row Q --+ Q* -i SO(n)/G as well as the columns are fibrations. Hence we obtain the following commutative diagram of homotopy groups:

ir2(X)

1r2(X)

71(F*)

71(F)

ir2(SO(n)/G)

E'

7r1(Q)

3

7r1 (Q*) #

4

7rl(SO(n)/G)

Suppose that Q* admits a spin structure, and let f * : 7rl(Q*) -, 7rl(F*) be a homomorphism for which 7r1(F*) 7r1 (Q*) * rrl(F*) is the identity. Then consider the homomorphism

f = f* o j# : 7r1 (Q) -' 7r1(F*) = rrl(SO(n)) and conclude that

f o6= f*oj#os= f*os*oi#

-

Hence the following diagram commutes: 1r1 (F)

irl (F* )

Conversely, let f with this commutative diagram be given. Define f * irl(Q*) -+ 7rl(F*) as follows: For each element x E 7rl(Q*) there exists an element y E 7r1 (Q) such that j# (y) = x. Set f*(x) = f(y). If yl is another element from 7r1(Q) with j# (yl) = x, then there exists an element z E rr2(SO(n)/G) such that yl = y a(z). But this implies :

f (yl) = f (y)f (a(z)) = f (y)f (sa(z)) = f (y)i#a(z) = f (y) 1 in 7r1 (F*). Thus f* : 7rl(Q*) --> 7rl(F*) is uniquely defined. We still have to check the condition f* o s* = Id on 7r1(F*). Choose a E 7r1(F) with

i#(a) = a*

1 in 7r1 (F*). Then,

a* = i# (a) = f s(a) = f *j#s(a) = f *s*i# (a) = f *e* (a*).

2.4. Existence of spine structures

47

Summarizing, we obtain the

Proposition. Let G C SO(n) be a connected compact subgroup with

7r,(50(n)/G) = 1. A G-principal bundle Q over a connected CW-complex X has a spin structure if and only if there exists a homomorphism f : 7rl(Q) , ir1(SO(n)) for which the diagram iri (F) = 7rl (G)

7r1(SO (n) )

I

i1(Q) commutes.

This condition can again be reformulated cohomologically. The homomorphism f defines an element f E H1(Q;Z2) = Hom (ir1(Q),Z2) whose restriction to the fibre F, i* (f) E H'(F; Z2) = Horn (7r1(G), Z2), has to coincide with i# : 7r, (G) -f 7rl(SO(n)) = Z2. Hence we have the

Proposition. Let G C SO(n) be a connected compact group with 7rl (SO(n)/G) = 1.

A G-principal bundle Q over a connected CW-complex has a spin structure if and only if there is an element f E Hl (Q; Z2) for which the restriction to the fibre F coincides with i#, i* (f) = i# .

2.4. Existence of spine structures Consider an SO(n)-principal bundle Q with base space X. In analogy with a spin structure, we define

Definition. A spine structure on Q is a pair (P, A) consisting of a Spincprincipal bundle P over the space X and a map A : P --- Q such that the diagram

P x Spin's(n) - P I

IAxA

A

Q x SO(n)

Q

commutes.

Example. Because of the inclusion i Spin(n) -> Spinc(n), each spin structure on Q induces a spine structure. :

2. Spin Structures

48

Example. Suppose that the SO(n)-bundle Q (n = 2k) has a U(k)-reduction, i.e. there exists a U(k)-bundle R with Q=R

XU(k) SO(2k).

In Section 1.6 we constructed a homomorphism F : U(k) -+ Spin(2k) for which the diagram Spin (2k)

F'

U(k)

SO(2k)

commutes. Hence

P := R XU(k) Spin (2k)

is a spin structure on Q. In other words: Every U(k)-reduction of the SO(n)-bundle Q induces a spin structure in Q. The groups Spin(n) and SI are subgroups of Spin (n) whose elements commute with each other. Hence, if (P, A) is a spin structure, then

1) PIS' is a Spin(n)/{f1} = SO(n)-bundle isomorphic to Q,

P/S,=Q 2) PI := P/Spin(n) is an S'/{f1} = Sl-bundle over X, and the combination of bundle morphisms over the base space P --+ QxPI is a 2-fold covering. Here QxPI denotes the fibre-product of the SO(n)principal bundle Q with the S1 = SO(2)-principal bundle PI over X. QxPI is an (SO(n) x SO(2))-principal bundle over X. Because of the diagram (compare Section 1.6)

Spin (n)

Spin (n + 2)

SO(n) x SO(2) -i SO(n + 2) the (SO(n) x SO(2))-principal bundle QxPI admits a spin structure in the sense of Section 2.3. All in all, we obtain the

Proposition. If the SO(n)-principal bundle Q admits a spin structure, then there exists an S'-principal bundle PI over X such that the fibre product QxPI has a spin structure. Conversely, if such a bundle Pi with the given

property exists, then Q has a spin structure. Remark. Making use of characteristic classes, we can formulate this equivalently as follows:

a) Q has a spin structure;

2.4. Existence of spin structures

49

b) there exists an S'-bundle Pl such that w2 (Q x Pi) = 0; c) there exists an S1-bundle Pl such that w2(Q) =_ cl (PI) mod 2; d) there exists a cohomology class z E H2 (X; Z) such that w2 (Q)

zmod2. Hence, Q has a spin structure if and only if the Stiefel-Whitney class w2(Q) E H2(X; Z) is the Z2-reduction of an integral class z E H2(X; Z).

Corollary. If H2(X; Z) -* H2(X; Z2) is surjective, then every SO(n)bundle Q over X admits a spin structure. We will now discuss the existence question for a spin structure on an SO(n)principal bundle over a special class of base spaces X, and show that, sometimes, one can obtain a complete answer. Concerning the base space X we assume that

and 7r2 (X) is a finite group. Let Q be an SO(n)-principal bundle and P1 an Sl-bundle over X. On the one hand, the fibre-product QxP1 is an (SO(n) x SO(2))-principal bundle over X, and, on the other, a fibration over Q with fibre Si. Together, this yields the diagram 7r1(X) = 1

7r2 (X)

la Z2ED z ce - (0,a)

Z = 7r1(Sl) -- 7r1(Q
7r1(Q)=1

I 1

7r1(X)=1 of homotopy groups in which the column and the row are exact. As 7r2 (X) is finite, the image of 8 is contained in the subgroup Z2 and, therefore, there are two cases. First case: 8 = 0. Then Z2®9L = 7ri(SO(n) xSO(2)) f--; 7r1(Q
the bundle Q PI has a spin structure, i.e. Q itself admits a spin structure. Moreover, the exact row immediately implies 7r1(Q) = Z2.

2. Spin Structures

50

Second case: im(8) = ?L2.

Then the generating element of the group Z2 = 7ri(SO(n)) belongs to the kernel of the homomorphism

7ri(SO(n) x SO(2)) -; 7ri(QxPi). Hence there cannot exist any homomorphism f of the groups 7ri(SO(n) x SO(2)) Z-. 7ri(SO(n+2))

7r1(QxP1)

such that the given diagram commutes (i is the embedding of SO(n) x SO(2)

into SO(n + 2)). Consequently, QxP1 admits no spin structure, i.e. Q itself has no spin structure. Finally, 7ri (Q x Pi) = Z, and, therefore, the homomorphism 71 (Si) -f 7ri (Q x Pi) in the row of the diagram is surjective. This implies 7ri (Q) = 1. Summarizing this argument and applying, in addition, the criterion for the

existence of a spin structure on an SO(n)-principal bundle over a simply connected base space, we conclude:

Proposition. Let X be a CW-complex with 7ri (X) = 1 and 7r2 (X) a finite group. If Q is an SO(n)-principal bundle over X, then the following conditions are equivalent:

1) Q has a spin structure.

2) Q has a spin structure. 3) 71 (Q) = Z2.

If Q has no spin (spin) structure, then 7ri(Q) = 1.

Example. Consider the homogeneous space X5 = SU(3)/SO(3). The isotropy representation cp : SO(3) -+ SO(5) of this space induces an isomorphism cp# : 7ri(SO(3)) -+ 7ri(SO(5)) of fundamental groups. Let Q = SU(3) xso(3)SO(5) be the frame bundle of X5. Q is an SO(5)-principal bundle over X5. We will show that Q has no spin structure. To do so, start by computing the first and the second homotopy groups of X5 = SU(3)/SO(3) from the exact sequence

-

7ri( SO ( 3 )) = 7L2 = 1 -- 7r2( X5 ) --> 7ri(SU(3)) = 1 --f 7ri(X5) -* 1. 7r2

(SU( 3 ))

We obtain 71 (X5) = 1,

72 (X') = Z2.

2.4. Existence of spin structures

51

The fundamental group of Q is analogously determined from the sequence 7r2(SU(3) Xso(3) SO(5))

7rl(SO(3))

irl(SU(3) x SO(5)) --> 7rl(Q) --f 1. Since pp# is an isomorphism, we obtain 7rl(Q) = 1, i.e. Q admits no spin structure.

Next we are going to discuss when two spin structures (P, A), (P*, A*) on an SO(n)-principal Q bundle will be considered as equivalent.

Definition. Two spin structures (P, A), (P*, A*) on an SO(n)-principal bundle Q are called equivalent if

1) There exists a bundle isomorphism

'Pi

:Pi

of the S'-principal bundles Pl = P/Spin(n) and Pl = P*/Spin(n). 2) There exists a spin-bundle isomorphism 1 : P -> P* fitting into the commutative diagram

P* J IdxV) j QxPi QXP1 P

*

Remark. Given a bundle isomorphism

: P ---p P* with the commutative

diagram

P

P*

AXQ 4 4D induces an isomorphism 0 : P1 = P/Spin(n) - P1 = P*/Spin(n) of SI-bundles, and the diagram

P*

4)

P

IdxO

Q;P1-''QXPI commutes. The equality of two spin structures can thus be formulated equivalently by requiring the existence of a bundle isomorphism (D : P P* with the commutative diagram

P

P* A

Q

/A

2. Spin Structures

52

Definition. If (P, A) is a spin structure on Q, then the line bundle L = P1 XU(I) C = P XSpinc(n,) C

is called the determinant bundle of the spin structure. L is a complex line bundle over X, and the above consideration of characteristic classes yields the condition

w2(Q) - cl(L) mod 2 in H2(X; Z2). Thus we obtain the map

Spin(Q) -) {a E H2(X; Z2) : a - w2(Q) mod 2} from the set of all spin structures on Q to H2 (X; Z). Recalling that the existence of a spin structure on Q is equivalent to the existence of an element a E H2(X; Z) with w2(Q) = a mod 2, we conclude that the map from Spin(X;Z) to {a E H2(X;7G) : a = w2(Q) mod 2} is surjective. For fixed a E H2(X; Z) we have, on the other hand, the SO(n) x S'-bundle Q x P1, and a spin structure on Q is then only a reduction of this (SO (n) x S')-bundle onto the group Spin (n) - ) SO(n) x S'. These reductions are labeled by the elements of H'(X; Z2). However, in this last step we still allow for a gauge transformation of the S1-bundle P1. But the ambiguity resulting thereby may be controlled. Let P1 be an S1-bundle and Pl a reduction onto the 2-fold covering z -+ z2 of S1. A gauge transformation F of P1 is determined by a function f : X -+ S' with F(p1) = pi f (7r(pl)) The reductions Pl and F* (PjK) are equivalent as reductions of P1 if and only if the map

f# :7r1(X) -' 7r1(S1) = Z

takes values in 2Z c Z (f has a square root). The gauge transformation itself corresponds to an element of H1(X ; 7L), and the exact sequence

H'(X;Z) -1H'(X;Z)

H1(X;Z2)

0) H2 (X; Z) - H2 (X; Z) -) H2 (X; Z2) then implies that, for a fixed isomorphism class of the S'-bundle P1, the spin structures are labeled by H1 (X; Z2)/IM(W) = H'(X;7L2)/ker(j3) = im(,3)

On the other hand,

{a E H2(X; Z) : a w2(Q) mod 2} = im(b) = H2 (X ; Z) / ker o = H2 (X; Z) / im(3), and thus we obtain the

2.5. Associated spinor bundles

53

Proposition. Let Q be an SO(n)-principal bundle with spine structure. Then all the spine structures on Q are classified by H2 (X; Z).

Example 2. Let Q be an SO(2k)-bundle admitting a reduction R onto the subgroup U(k) C SO(2k). Let (P, A) be the canonical spine structure. Then for the determinant bundle we have

L = P X Spin' (C = (R X U(k) Spinc (2k)) X Spinc'(2k) C.

The homomorphism U(k) - Spin(c (2k) ---). S' is given by A , det (A). This implies

L=RXdetC On the other hand, if E = R X U(k) Ck is the associated complex vector bundle, then Ak(E) = R Xdet C.

This implies for the determinant bundle L of the spine structure the formula

L=A k (E). 2.5. Associated spinor bundles Consider an SO(n)-principal bundle (Q, ir, X; SO(n)) and denote by T = Q X SO(n) Rn

the associated real n-dimensional vector bundle. Let (P, A) be a spin or spine structure. The spin representation

n : Spin(n), Spinc(n) - U(An) now allows us to consider the associated complex vector bundle

S=Px,, AnS is called the spinor bundle for a given spin or spine structure, respectively. S is a complex vector bundle over X with an Hermitian metric. In the case n = 2k, this vector bundle splits into the sum of two subbundles S+, S-:

S=S+®S-, S =Pxk On. Clifford multiplication,

µ:Rn(g.On-->On

or

µ:A(Rn)®II2An

On,

is a homomorphism of the spin or spine representation, respectively, and, since

T=Q XSO(n)Rn=Px)118n, p induces a bundle morphism of the associated bundles

µ:TOS>S or

u:A(T)OS

) S.

2. Spin Structures

54

The real or quaternionic structures in On, respectively, pass on, in the case of a spin structure (not for a spin structure !), to the corresponding bundles.

The spin representations det(An) = Adim(on)(A,) and ldim(°m)/2 are equivalent where 1 : Spin (n) -+ S' is the homomorphism constructed above. This implies for the determinant bundle L of the spin structure the formula Ldim(S)/2

= Adim(S) (S)

and, in the case of even dimension n = 2k, we analogously obtain £dim(S±)/2 = A dim (St) (S±).

In forming the associated spin bundle S starting from a spin or spin structure, it may well happen that different spin structures lead to isomorphic spin bundles. We will now study this question in the case of a spin structure in greater detail. Let (Q, 7r, X; SO(n)) be a principal bundle. We think of Q as being given by

a) a covering X = U Uj of X by open sets Uj C X, iEI

b) a system of transition functions gzj : U2 n Uj - SO (n) such that 9ijgjk = 9ik,

9ii = 1.

Then, a spin structure (P, A) on Q is completely determined by a system of transition functions gzj : Uj n Uj -* Spin(n) satisfying the conditions A 0 gij = 9ij,

gijgjk = 9jk,

g22 = 1.

Hence two spin structures (P, A) and (P*, A*) are described by maps gzj, gij*

UznUj -> Spin(n) with A o gij = gzj = \ o gzj*.

Set eij = gzj [gzj*]-1. Then ezj is a map from UznUj to ker (A) = Z2 C Spin(n),

ezj:UinUj->7L2, and eij ejk = eik. Thus the system of transition functions ezj defines a real 1-dimensional bundle E. Since

for the spinor bundle S of the spin structure (P, A) (S* for the spin structure (P*, A*)) there are isomorphisms

S = S* OR E = S® ®(c E°,

2.5. Associated spinor bundles

55

where E° = E OR C is the complexification of E. As (±1) belongs to the centre of the Clifford algebra, this isomorphism is compatible with Clifford multiplication, i.e. the following diagram commutes:

T®S µ -- S T®S .0

E

WgIdES *

®E

From these considerations we obtain, e.g., the following

Proposition. Let X be a CW-complex whose second integral cohomology has no 2-torsion. Then the spinor bundles corresponding to possibly different spin structures on an SO(n)-principal bundle are isomorphic.

Proof. The complex line bundle E° is the complexification of a real line bundle, and hence, for the first Chern class, 2c1 (E°) = 0 in H2 (X; Z). Since

H2 (X; Z) has no 2-torsion, this implies ci (E°) = 0, i.e. E° is a trivial bundle.

Remark. The spin structures (P, A) and (P*, A*) are described by elements f f * E H'(Q; Z2) whose restrictions to H' (fibre; Z2) = 7G2 are non-trivial. Thus, f - f * vanishes after restriction to the fibre. From the exact sequence of the SO(n)-principal bundle, ,

0 -> H1 (X; Z2)

) H' (Q; Z2)

H' (SO (n); Z2)

we thus conclude that f - f * is an element of H' (X ; Z2). This implies

wi(E) = f - f*, where wl is the first Stiefel-Whitney class of the real bundle E.

Example. Let X = II p5 be the real projective space of dimension five and Q = IE8p5 x SO(3) the trivial SO(3)-principal bundle. Since Hi(R?5; Z2) = Z2 on Q, there are two spin structures (P, A) and (P*, A*). The bundle E is uniquely determined by wi (E) 0 in Hi (II81P5; 7L2). The first spin structure is trivial, and hence so is the corresponding spin bundle S. This implies for S*

S* = 2E° = Ec ®Ec. But, in general, c2(2E°) = ci(E°)2.

However, the bundle Ec is the associated bundle E° = S5 X Z2 c.

2. Spin Structures

56

If E° is trivial, then there exists a mapping f : S5 -+ S' with f (-x) _ -f (x), in contradiction with the Borsuk-Ulam theorem. Thus E° is nontrivial, i.e.

cl(E°)

0

in

H2(RP5;7L).

Since the map Z = H2(If TP5; Z)

is injective, this implies c2 (2E°)

a --> a2 E H4(II8IP5; Z)

0. Hence S* is not the trivial bundle.

2.6. References and exercises Th. Friedrich. Zur Abhangigkeit des Dirac-Operators von der SpinStruktur, Colloq. Math. 48 (1984), 57-62. J. Milnor. Remarks concerning Spin-manifolds, Differential and combinatorial topology (in honour of Marsten Morse), Princeton Univ. Press, 1965, 55-62.

R.C. Kirby and L.R. Taylor. Pin structures on low-dimensional manifolds, in Geometry of Low-Dimensional Manifolds, Part 2 (Durham, England, 1989; ed. by S.K. Donaldson), London Math. Soc., Lect. Note Series 151, Cambridge University Press 1990, 177-242.

Exercise 1. Let RP' be the n-dimensional real projective space. Prove that a) IMP' is orientable n = 1 mod 2. b) RPn has a spin structure <-=* n = 3 mod 4. Exercise 2. Which of the Graf3mannian manifolds G'n+k,k = SO(n + k)/[SO(n) x SO(k)] have a spin structure? Exercise 3. Let M3 be a compact, closed, and orientable 3-manifold. Then every SO(n)-principal bundle Q over M3 has a spine structure. Hint: If M3 is orientable, then by Poincar6 duality we have H2(M3;7G) = H1(M3;9Z)

=

71

3

and

H2 (M3; 7Z2) = Hl (M3; Z2)

z 7L2 .

[71, 71]

Chapter 3

Dirac Operators 3.1. Connections in spinor bundles Let (Mn, g) be an oriented connected Riemannian manifold and Q -+ Mn the SO(n)-principal bundle of positively oriented orthonormal frames. The Riemannian manifold has a uniquely determined torsion-free metric connection. Considering it as a covariant derivative on vector fields, we will denote this Levi-Civita connection by V. Viewed as a connection in the SO(n)-principal bundle, however, it is the so(n)-valued 1-form Z : TQ -> .so(n).

Let, in addition, a spine structure (P, A) together with the corresponding U(1)-bundle P1 and 2-fold covering 7r be given,

7r:P-->QxP1. Finally, we also fix a connection A in the principal bundle P1,

A : T P1 --> iR,

where we identify the Lie algebra of the group Sl = U(1) with the purely imaginary numbers. The connections Z and A together define a connection

ZxA:T(Q.so(n)®iR in the fibre-product Q x P1. It is not difficult to see that this connection lifts to the 2-fold covering 7r : P --+ Q x PI as a connection Z x A in the spine 57

3. Dirac Operators

58

principal bundle. The following diagram commutes:

T(P) !

spinC(n) = m2 ®iR

T(QxPI) Z-"

so(n) ®iR

where p* : spinC(n) --i so(n) ®iR is the differential of the 2-fold covering p :

SpinC(n) -> SO(n) x S'. The spin representation K : Spinc(n) -* GL(An) induces the spinor bundle S = P xSpinC(n) On

The sections

E F(S) of the spinor bundle can be identified with the map-

pings 0 : P -- On obeying the transformation rule b(p g) _ r,(g-1) (p), g E SpinC(n). On the one hand, the absolute differential of a section 0 with respect to the connection Z x A is computed by

DAB=dzb+k*(ZxA) and determines, on the other hand, a covariant derivative

VA : F(S) --} I'(T*M ®S) in the spinor bundle. A vector field X on the manifold Mn can be considered

as a function X : P - an with X (p g) = A(g-1)X (p), since the tangent bundle is an associated vector bundle to the principal bundle P. Then the Clifford product X V) is given by the function X : P --> On, (X - ')(p) = X (P) . ON. Using this, we obtain DA(X

.

)

= d(X b) + rc*(Z x A)(X - b)

= dX. Now let us transform i* (Z x A) (X i) algebraically. To do this, we insert a vector t E T(P). Then Z x A(t) is) is an element of m2®iR = spinC(n) and

r*(Z x A(tl) _ (y + is) X

=yX

+ X (iso).

However, since y E M2 and X E an, the following formula holds in the Clifford algebra Cn:

=X -y+A*(y)(X) Here A, : spin(n) -* so(n) denotes the differential. In our situation, A*(y) _

Z(d7r(t)) and thus

/-tip

K*(ZxA)(X

N

X{k*(ZxA)zb}+(A*(Z)X).

3.1. Connections in spinor bundles

59

Inserting this implies

DA(X 0) = X DAV) + (dX +,\*(Z)X) V = X DAV) + (VX)

,O.

All in all, we have thus proved:

E P(S) a spinor

Proposition. Let X, Y be vector fields on M7z and

field. Then, for the spinor derivative with respect to any connection A in

the U(1)-bundle P1,

VA(X' )=X'(VA'0)+(VYX)'

.

The spinor derivative VA is metric with respect to the Hermitian product in S, i.e. (VX

01) ,

+ (,V1).

Proof. The last formula is a consequence of the fact that the spin representation n : Spin -* GL(An) is unitary. We also want to specify local formulas for the connections. Let e : U C Mn --> Q be a local section of the frame bundle Q. e consists of an orthonormal frame e = (el, ... , e,) of vector fields defined on the open set U C Mn. The local connection form Ze = e*(Z) : TU --> so(n) is given by the formula

Ze = E wijEij i<j where the 1-forms wij denote the forms defining the Levi-Civita connection,

wij = g(Vei, ej), and Eij E so(n) are the standard basis matrices of the Lie algebra so (n). Analogously, we fix a section s : U -* P1 of the U(1)-principal bundle and obtain the local connection form

AS=s*(A):TU-->ill8. AS is an imaginary-valued 1-form defined on the set U, and e x s : U -- Q x Pl

is a local section of the principal bundle Q x Pl. Let

xe

be a lift of this

section to the 2-fold covering ?r : P -+ Q x P1. Since e

sx

wjjEjj, A'

)*(p*(Z x A)) _ (e x s)*7r*(Z x A) = (Z', A') i<j

and

T (P) Z-.- spin (n) = m2 ®iR d(exs) _-1 dTr

T(U)

d ex^)

iii

Pw

T(Q xPi)-"Z ` so(n) ®iIR

3. Dirac Operators

60

the local connection form Z x A

exs)

. exs) =

1

ZxA

2

is given by the formula 1

E wijeiej, 2As i<j

With respect to the section e x s, the section 0 E r(US) of the spinor bundle over U is described by a function' : U -- On,, and its covariant derivative is computed according to the formula VA 2J=

do

1

1

S

i<j

Remark (Special case of a spin structure). If (P, A) is a spin structure on Q, then P x spin(,) Spin`C (n) is an induced spin0 structure. The U(1)-bundle P1, in this case, is trivial with a canonical global section s : Mn -+ Pl. If we choose the connection A0 in Pl for which Ao 0 holds, all the formulas simplify correspondingly. Given a spin structure, we will simply denote the covariant derivative in the spinor bundle S with respect to this canonical connection A0 by V.

Remark (Special case of a U(k)-reduction). Let n = 2k be even, and let a topological U(k)-reduction R of the SO(2k)-principal bundle Q be given, R C Q. This is equivalent to saying that there is an almost-complex structure J : T(M2n) -> T(M2n) which is compatible with the metric g: 9(J(tl), J(t2)) = 9(ti, t2). The lift F : U(k) - Spin0(2k) fitting into the commutative diagram Spin" (2k)

U(k) '- SO(2k) x S1 together with f (A) = (A, det (A)), A E U(k), induces a spin0 structure on Q via P = R xU(k) Spinc(2k). The corresponding U(1)-bundle Pl is

Pi=RxdetS' with the 1-dimensional complex vector bundle E = R X &t C. This vector bundle can also be described differently. T(M2k) becomes a k-dimensional complex vector bundle by means of the almost-complex structure J and, by construction,

E=Ak(T).

3.1. Connections in spinor bundles

61

Summarizing, we conclude that each connection A in the U(1)-bundle P1 = R X det S1 (or in the vector bundle Ak (T) ) induces a covariant derivative 7A

in the spinor bundle. Particularly important is the case that a connection can be distinguished in the principal bundle P1 in an "obvious" way. This happens, e.g., if the Levi-Civita connection Z reduces to the U(k)-reduction R to a connection Z*. Then (M2k, g, j) is a Kahler manifold. In this case, Z* in turn induces a special connection AO in the associated bundle Pi = R Xdet Si, and we again obtain a distinguished covariant derivative which only depends upon the geometry of the base space. Now we return to the general situation and consider two connections A and A' in P1. The difference A - A' is an iI -valued 1-form on the manifold M' which will be denoted by 11:

A-A'=ri. From the local formula for the covariant derivative VA we immediately conclude that 1

2ri(X). for all spinor fields E F(S) and all vectors X E T(MT). A gauge transformation f : P1 -> Pi of the U(1)-bundle P1 is described by a mapping p f : M7z --> S1 satisfying

f(pi) = Pi of (7r(pi)) , where 7r : Pi -j Mn is the projection in the bundle. The connection f *(A) is given by

f*(A) =A+7r*A*(O) with the Maurer-Cartan form O = zx of the group U(1) = S'. This implies the formula V f *(A) %

- Ox

1

=2

dpµ X

f

for the corresponding covariant derivatives.

We now turn to the description of the curvature form of the connection Z x A. Let SZZ : TQ x TQ -> so (n) be the curvature form of the Levi-Civita connection with the components

OZ=EQijEzj, Qij:TQxTQ-->IR. i<j

The curvature !QA as a 2-form on P1 is simply QA = dA. The commutative diagram defining the connection Z x A immediately implies the formula QZXA

= 2 1: 7r*(SZtij)eiej ®27r*(dA) i<j

3. Dirac Operators

62

The 2-form dA is a form on the base space MI. From the general equation DZDZQf = p*(fZ)q, for the 2-fold absolute differential with respect to the connection Z we obtain the formula VA(VAO)

= 2

EQijeiej 0+

2dA

i<j

Here e = (ei, ... , eI) is a local orthonormal frame and (S1 )e = e* (Qz) _ Ei<j 52ijEij defines the corresponding components of the curvature form of the Levi-Civita connection. Using the structure equations of the Riemannian space (or, more generally, the formula for the curvature form, Q Z = dZ + 2 [Z, Z]) we can express the 2-forms Iij in terms of the forms wij = g(Vei, ej) of the Levi-Civita connection as well as the components Rijkl = g(VezVejek - VejVezek - V[e,,,ej]ek, el).

To this end, we begin with the following general remarks. Let (P, 7r, M; G)

be a G-principal bundle, Z : T(P) -f g a connection and p : G -; GL(Vo) a representation. The curvature form IZ = dZ + .1 [Z, Z] defines a 2-form p. (AZ) on the manifold with values in the endomorphisms of the associated

vector bundle V = P x,, Vo. On the other hand, a section 0 of this bundle can be identified with a function 0 : P -* Vo obeying the transformation rule 0(p g) = p(g-')O(p). Then DZcb is a tensorial 1-form of type p and hence a 1-form on the manifold M with values in V. The induced covariant derivative in V is thus given by '7XO

= DZq(X*),

where X* is a (horizontal) lift of X. The equation

RZ(X, Y) =VXVZO-VZVZ0-VZ

y] 0

determines the curvature tensor Rz, which is a 2-form with values in End(V), too. The curvature form and the curvature tensor are related by the following well-known formula.

Lemma. One has the identity Rz = P. (Qz) Proof. To prove this, consider vector fields X, Y on the manifold M and denote by X*,Y* the corresponding Z-horizontal lifts. The section VZO is given by DZc(Y*) = do(Y*) +Z(Y*)o = do(Y*). An analogous calculation shows that V Z V ZO coincides with X *Y*O, and thus

X*Y*(0) -Y*X*(0) = [X*, Y*] (O) - [X = [X*, Y*]vertw.

-

[X,Y]hor(cb) *, Y*]hor (0)

3.1. Connections in spinor bundles

63

On the other hand, the structure equation of the connection immediately implies [X*,Y*]vert = Z[X*,Y*] _ -SZ(X*,Y*). Hence,

Finally, if W E g is an element of the Lie algebra and IV- the corresponding fundamental vector field, then W (0) (p)

cb(p.etW)-O(p) = lim t-*o t (P(e-t,) - 1)

O(p) =

to

This implies the formula we wanted to prove, RZ(X, Y) 0 = P* (Q (X, Y))

Applying this to the components of the Levi-Civita connection, we obtain SZij (X, Y)

I:Rklij0,k(X)a,(Y)

= (f2Z(X, Y)ei,, ej) = (R(X, Y)ei, ej) = k,l

9

E Rijkl (dk A a,l) (X, Y), k,l

where ad, ... , vn is the frame dual to el, ... , en. Thus we arrive at the local formula for the curvature form SZZxA of the connection Z x A, QZxA

= 4i<jE

RijklO' k A o.1

eiej + 2 dA,

k,l

and the 2-form VAVA with values in the spinor bundle is calculated as follows:

VAVA,p = Here DADA

1

4

Rijklak A o-'

i<j

k,l

eiej 0 + 12 dA

is the 2-fold absolute differential of the spinor field V) and hence

a section of I'(A2 (& S). Starting from this 2-form with values in the spinor bundle, we construct a spinor-valued 1-form HA by a suitable contraction:

Definition. For a vector X E T(Mn) the 1-form HA is defined by n

H,A(X)

Then the following holds.

(VAVAO)(X,ea)

3. Dirac Operators

64

Proposition. Let Ric : T(MT) -> T(Mn) be the Ricci tensor of the Riemannian space considered as a symmetric endomorphism of the tangent bundle. Then one has the relation 2 Ric

H,p (X)

(X) z + 2 (X idA)

Proof. We are going to use the formula for V AV AO stated above, and we have to prove the following two relations: n

1) E ea (dA(X, ea)) V = (XidA) 0, a=1

2) E > E Rijklo'k A Qi(X, ea)eaeiej a i<j k,l

= -2Ric(X)

V).

The first one is trivial since the sum Ea=1 dA(X, ea) ea represents the decomposition of the form X_jdA with respect to the basis e1, following calculation takes place in the Clifford algebra Cn: Rijklo-k Aol(X,

...

, en. The

ea)eaeiej

a,k,l

i<j

Rijalo-1(X)eaeiej a,k,i<j

a,l,i<j

= 2 E RijkaQk(X)eaeiej a,k,i<j

= -2 E Rijkick(X )ej + 2 k,i<j

2

k,i<j

-2 E Rijkiok (X)ej + 2 i,j,k

E RjkaUk(X)eaeiej k,i<j

a54i , j

Rjkao'k(X)eaeiej k,i<j

a0i,j

-2Ric(X) + 2

RjkaQk(X)eaeiej. k,i<j a9-`i,j

However, the second summand vanishes. Namely, for a fixed index k this sum contains the Clifford product epeger, p < q < r, precisely three times, for the triples (a, i, j) = (p, q, r), (q, p, r) and (r, p, q). Hence the coefficient at epeger is proportional to the sum Rqrkp - Rprkq + Rpgkr = Rqrkp + Rrpkq + Rpqkr = 0

(Bianchi identity for the curvature tensor).

Remark. Taking into account that (VAVAO) (X, ea) = RS(X, e,,)0

= oX o a - V a0X - [x ea]

,

3.1. Connections in spinor bundles

65

the formula

e« (VAVAV)(X, e«) _ -2Ric(X)O + 1(XidA)

V)

can also be written as n

RS(X, e«) _ -2Ric(X)o + (XidA) 2

where RS(X,Y) = oXoY in the spinor bundle.

-

VYVX

-

V

is the curvature tensor

A spinor field 0 E F(S) is called VA-parallel (or simply parallel) if VA 1 = 0. Since

x2 =

(vXO, ' ) +

) = 0,

(V), V

the length of a parallel spinor field is constant. In the following we suppose that 0 does not vanish identically. DAB = 0 implies V AV Ab = 0, and hence the 1-form H,, vanishes identically. Thus, for every vector X E T(Mn),

Ric(X) 0 = (X-jdA)

V).

Ric(X) is a real vector while X_jdA is purely imaginary. For the evaluation of the last equation we need the following

Lemma. Let 0 E On be a non-trivial spinor and Z1, Z2 E I(Sn two real vectors. If

(Z1+iZ2) '=0, then for the vectors Z1, Z2 1) IZ1I = IZ2I, 2) (ZI, Z2) = 0.

Proof. Multiply the equation (ZI +iZ2)0 = 0 once again by (Zi +iZ2). In the complexified Clifford algebra Cc,

(Z1 + iZ2)(Zl + iZ2) = Zl - Z2 + i(ZIZ2 + Z2Z1) = {-IZ1I2 + IZ2I2} + i{-2(ZI, Z2)}. From (Z1 +iZ2)(Zl + iZ2)' = 0 and O 0 we then obtain Z112 = IZ2I2 as well as (Z1, Z2) = 0.

Consequently, the condition Ric(X) ip = (X-jdA) 1)

Ric(X)I = I z (X_jdA)I for all X E T(MT)

,

2) (Ric(X), (X_jdA)) = 0 for all X E T(M').

b implies

3. Dirac Operators

66

For the sake of brevity we denote by S the symmetric endomorphism S(X) _ Ric(X), and by A the anti-symmetric endomorphism A(X) = 2 (X-jdA) of the tangent bundle. The second equation implies

(S (X), A(X)) = 0 for every vector X. Inserting X + Y now leads to

(S(X), A(Y)) + (S(Y), A(X)) = 0 and hence

(X, SA(Y)) - (X, AS(Y)) = 0.

Thus, SA = AS, i.e. A and S commute. Hence, at a fixed point m E Mn, A and S can be diagonalized simultaneously: A has the form o

-WI

W1

0

0 o

-W2

W2

0

A= 0

-Wk

Wk

0 0

0

0

and S is similar to Al

0

S= t 0

An l

The condition I S(X) I = A(X) 1, X E T, now leads to the equations 1\1 = A2 = ±W1,

...

,

A2k-1 = A2k = ±Wk,

A2k+1 =

... _ An = 0.

For the lengths of the endomorphisms, n IIS112

= tr(S o ST) = tr(S2) i=1 k

IIAII2

= tr(AAT) = -tr(A2) =

2Tw2, i=1

3.2. The Dirac and the Laplace operator in the spinor bundle

67

we then obtain IIRicHI = IIAII.

However, the square of the length of A as an anti-symmetric mapping is twice the square of the length of the 2-form dA, IIAI12 = 211dAII2.

To summarize:

Proposition. Let (Ma, g) be a Riemannian manifold with a spine structure and A a connection in the U(1)-principal bundle P1 induced from this spine structure. VA denotes the induced covariant derivative in the spinor bundle. If there exists a VA -parallel spinor,0, VAO = 0,

then the following necessary conditions are satisfied: 1) JIRic1I2

= 2IIcAII2, where S1A = dA is the curvature form of the

connection,

2) rank(QA) = rank(Ric), and 3) the endomorphisms Ric and QA of the tangent bundle commute.

Corollary. Let (Mn, g) be a connected Riemannian manifold with a fixed spin structure. V denotes the canonical covariant derivative in the spinor bundle s (A = 0). If there exists a non-trivial parallel spinor 0, then the Ricci tensor of Mn vanishes identically, Ric - 0.

Remark. The conditions for the existence of parallel spinor fields listed above are only necessary. These conditions are not even locally sufficient compare the exercises.

Corollary. Let (M', g) be a connected Riemannian manifold which admits a spine structure. If the Ricci tensor has odd rank at least at one point, then Mn has no VA-parallel spinor fields for any spine structure and for any connection A in the U(1)-bundle of the spine structure.

Remark. In 1997 A. Moroianu studied the classification of Riemannian manifolds with parallel spine spinors in greater detail.

3.2. The Dirac and the Laplace operator in the spinor bundle We start from a Riemannian manifold (Mn, g) with a fixed spine structure and a connection A in the U(1)-principal bundle P1. These induce a covariant derivative VA : r(s)

) r(T* 9 s) = r(T (9 s)

3. Dirac Operators

68

in the associated spinor bundle S. Here and in what follows the cotangent bundle T* of Mn and the tangent bundle T will be identified by means of the metric g. Clifford multiplication and the Hermitian metric (,) in S behave as follows with respect to the covariant derivative VA (X, Y E r(T), 01, 2 E

r(S)) : VA((X./1 )=(DYX)'/1'0+X.VA

,

1,VXY'2)=X(/1,02) As in every vector bundle with connection, we can define the Laplace operator A in the bundle S.

E r(S) is a spinor field,

Definition (Laplace operator on spinors). If then 0 (') Lis defined by n

AA( )_

ve

-

e

n

div(ei)Ve

Using Stokes' theorem one then derives - as for every Laplace operator in a Hermitian vector bundle - the formula f (AA (01), 02) =

/'

J

/ (VAW1, VAW2)

_

Mn

f

/

/'

(01, DAMn

(02))

Mn

for two spinor fields 01, 02 with compact support contained in the interior of the manifold Mn. Here, (VAV)1, VA 02) is the scalar product on 1-forms, i.e. n

(DA

E(V 1, DA02) = i=1

1,

VA

e 02)

The Dirac operator in turn results from the composition of the canonical derivative with Clifford multiplication.

Definition (Dirac operator). The composition

DA = µoVA:

r(s)

) r(T*(9 S) = r(T(9 S) -r (s)

is called the Dirac operator. With respect to a (local) orthonormal frame e = (el, ... , en) on the manifold Mn, n

ei De

DA's _ i=1

Obviously, DA is a first order differential operator. Moreover, DA is an elliptic operator. Its symbol o-(DA) (X) : S -+ S, for a vector X E T, is given by Clifford multiplication:

a(DA)(X)(') = X 0.

3.2. The Dirac and the Laplace operator in the spinor bundle

69

This immediately follows from the formula n

DA(f 0) =

ei De (f

n

)=

i=1

ei

{df (ei) 4' +

f7e}

i=1

= grad(f) V)+ fDA(') We compute (DA'O, 01) : n

n

(,7

/,

(DAb, 0l) _ (ei Ve O, 4'1) i=1

-

e Y' , ei

/'

'Y1)

i=1 n

/

/

/1

/

{ei(`b, ei Yb1) - (Y , (Ve,ei) 4'1) - (Y , ei De 4 1)} i=1 n

n

ei(4', ei

01) -

i=1

div(ei)(0, ei 01) + (0, DA01) i=1

Considering the 1-form MO,01(X) = (V), X b1), we see that the first two summands are just its divergence, 6M"""1. Thus, (DA0, 01) = (V, DA'b1) + JMO>01

This implies that the Dirac operator is a symmetric operator with respect to the L2-product.

Proposition. Let

and ?P1 be spinor fields with compact support (contained

in the interior of the manifold). Then,

f (DAO, 01) = f ( DAbl). ,

Mn

Mn

Remark. If the dimension n = 2k is even, then the spinor bundle splits into the sum S = S+ ® S- of Dirac spinors. Since Clifford multiplication by vectors interchanges these summands, the Dirac operator decomposes into the sum of two operators, DA : F(St) -* r(SF).

Now we want to discuss a third operator acting on spinor fields, the socalled twistor operator TA. To this end, we need several preparations: First, Clifford multiplication µ : T ® S --+ S is a surjective homomorphism. Let ker(a) C T ® S denote its kernel.

Lemma. The formula n 1

P(X(94b) =X®0+ n i=1 defines a projection from the bundle T ® S onto the bundle ker(a) C T ® S.

3. Dirac Operators

70

Proof. A straightforward calculation shows that the image of P is contained in ker(A):

-X

0.

i=1

Analogously, one shows that P acts as the identity on ker(µ).

Definition. The twistor operator TA = P o VA is defined as the superposition of the covariant derivative with the projection onto the kernel of Clifford multiplication,

TA : F(S) -> F(ker(/2)). n

From DAB _

ei ® V AO we obtain the following formula for TA:

i=1 TA(b)

ei ®De +

= i=1

1

n

ei ®ei DA(Y') i=1

n 1

DA()) ei ® oe + -ei n Z

Corollary. A spinor field 0 E F(S) belongs to the kernel of the twistor operator TA if and only if, for every vector X E T, 1

+ nX DA(s) = 0.

Vx

Example. Consider a2 with its Euclidean metric and coordinates x, y. is an orthonormal frame. For the forms wii of Then e1 = j, e2 = the Levi-Civita connection we have wi,7 = 0. A spinor field is simply a mapR2 - f A2 = C2. The covariant derivative V coincides with the ping differential do, since wi.7 = 0. Contrary to the convention valid up to now, :

this time we will employ an equivalent though different realization of the Clifford algebra described by the matrices e1 =

If

=

( 01 0)'

e2=

(0 o)

(f) : a2 --* C2 is a spinor field, then of

of 0

1

DO _ (-1 0)

ax a ax

+

0

i

ay

\i

0

a

af

ay

az

2

3.3. The Schrodinger-Lichnerowicz formula

71

where a

a

1

a

a

a

a

1

az-2 ax-Zayaz2Cax+zay The kernel of the Dirac operator (DV) = 0) thus consists of the pairs of complex-valued functions f, g : R2 --> C which satisfy the Cauchy-Riemann equations

ofag=0. az az (BA)

Fix a vector by

E C2 and consider the spinor field

(x'y) = x (01 0)

i

(BA)

0/ \B/

+y \ .

Then

0)

B1

A)

ax

: R2 __+ C2 defined

(iA J

ay

and hence

D( ) = ( O1 0)

ax +

(0i

i

0'

90

= -2 I B

I

We want to show that 0 is a solution to the twistor equation. To this end, we have to check that a'O

ax+2 ( 1 0) D(O) =0

and y

+2

I

\

0

i)

0.

0

But this immediately follows from

ax+2 (01 ao

,

0

) D(

)=

1 (0 i D(O) = i 0)

ay -7 -2

1

( A) - (01 0) (B) 0, iB 0 i ) (A)= 0. iA

i

0

B

3.3. The Schrodinger-Lichnerowicz formula The square DA of the Dirac operator as well as that of the Laplace operator DA are second order differential operators. We compare these operators computing their difference D29 - OA:

3. Dirac Operators

72

DA - AA'O

E ei De (ej VA O) + EVA VA e ij i AV) Eei {(Deyej) V + ej Ve

+ E div(ei) 7 o Ve.,0}

i, j

b+ Eeiej V eV

E g(Deiej, ek)eiek V

i,j, k

)VAI

+ EVe De O +

e

+E DeVe + 'div(ei)Ve z i

EE g(Vei ej, ek)eiek' Ve + >eiej . De Ve j ilk

ij

The latter of these equations is a consequence of the definition of the divergence, i.e.

j:g(Veiej,ek)eiekVej211 = -

j

div(ej)DejO.

i=k

Now rewrite the following endomorphism:

-57 g(ej, V e2ek)eiek

Eg(Veiej, ek)eiek

i0k

i56k

- E g(ej, V eiek - V ekei)eiek i
1:g(ej, [6k, ei])eiek i
This implies DA

- DAO g(ej, [ek, ei])eiekVej w +

j i
eiej (De Ve - De, De ) Y'

i<j

eiej(Deve - oeoe - veie,]) =

2

eiejR'(ei,ej) 2, j

In Section 3.2, the identity

ej ' RS(ei, ej)zb = - I Ric(ei)

+ 2 (ei_jdA)

was proved. Multiplying by ej and summing over the index i yields 1

eiej Rs(ei, ej) _ - 2 i,j

ej Ric(ei) i

+

1

2E i

ej

(ei_jdA)

'.

3.4. Hermitian manifolds and spinors

73

But, in the Clifford algebra, eiRic (ei)

_

ij

Rii = -R.

Rjezei = -

Moreover, it is easily checked that for every 2-form 972 the equality

1: ei

(ei_J772)

=

2772

holds in the Clifford algebra C, Thus, altogether we obtain 4IP

D2AO-DA =2 ('-

+2dA-0.

Summarizing this yields the following formula, first proved by E. Schrodinger in 1932.

Proposition (Schrodinger-Lichnerowicz formula). Denote by R the scalar curvature of the Riemannian manifold and let dA = Q be the imaginaryvalued curvature 2-form of the connection A in the U(1)-bundle associated with the spin structure. Then one has DA

AAO +

R0

+ 2 dA 0. F-I

3.4. Hermitian manifolds and spinors Consider an almost-complex manifold (M2k, j) with almost-complex structure J, j2 = -Id. J acts on a 1-form wl E T*(M2k) via

(Jw1)(X) = w1(JX), X E T(M2k), and the complexification T*(M2k) OR C splits into the (fi)-eigensubspaces of J: Al

=

T*(M2k)

® C = A1'0 ®

A°,1,

where

A"° _ {w1 E T*(M2k) ® C : J(wl) = iw1},

= {w1 E T* (M21) ® C : J(wl) -iw1}. Let AM be the linear span of all elements u A w with u E AP(A1"°) and A°'1

w E Aq(A°,1). Then,

Ar = E Ap,q p+q=r

Denote by S2' and S p,q the space of sections of the bundle Ar and AM, respectively. The exterior differential acting on r-forms, d : S1r

,

cr+l,

3. Dirac Operators

74

decomposes with respect to this splitting. In particular, define the operators C7

Qp,q - Qp+l,q ,

0 : Qp,q - Qp,q+l

8 = 7rAp+1,4 o d, a = 7rAp,9+1 o d.

In general, 8 + 8 does not coincide with d. However, by induction one can show the following.

Lemma. The exterior differential d maps P P,7 into the sum cp-1,q+2 ® 52p,q+1 ®Qp+l,q 6 S2p+2,q-1

Thus,

dinp,9 = a + 8 mod S p-1,q+2 ®Qp+2,q-1 The forms from Q1,1 are exterior products a A j3. On the other hand, J acts on a 2-form w2 by

(Jw2)(X,Y) = w2(JX, JY). Hence, J(a A,3) = a A 0, and this implies the Lemma. Q11 = {w2 E S22 : J(w2) = w2}, 522'0 ED QO,2 = {w2 E S 2 : J(w2) = -w2}. Moreover, fix an Hermitian metric g compatible with the almost-complex manifold (M2k, j), i.e. a Riemannian metric g with the property g(JX, JY) = g(X, Y). Then 52(X, Y) = g(JX, Y) is a 2-form and, since

1(JX, JY) = g(J2X, JY) = -g(X, JY) = -52(Y, X) = 52(X, Y), the 2-form SZ belongs to Q1,1. Choose a local orthonormal frame of vector fields

el, e2 = J(ei),... , e2k-1, e2k = J(e2k-1) By means of this, the 2-form f can be expressed as Q = el A e2 +... + e2k-1 ^ e2k.

Since J(el + ie2) = e2 - iel = -i(el + ie2), the forms (el + ie2), ...

,(e2k-1 + ie2k)

are a basis of the fibre A°"1 at every point, and any A°,'-form is a linear combination of exterior products of r forms of this type. Next we will compute some algebraic identities. First, e2,,-1J(e2o A (e2,3-1 + ie20)) + e2a-1 A (e2a-J(e20-1 + ie2Q)) 0

ifa

i(e20-1 + ie20)

if a = /t,

/3,

3.4. Hermitian manifolds and spinors

75

and

e2«- (e2a-1 A (e20-1 + ie20))

+

e2a A (e2a-1-j(e2,3-1 +ie2Q))

f0

if cti

'3'

if a = ,3.

ie2Q)

This implies the Lemma. For every (0, r) -form 7?0,r E AO,r k

k

E e2a-1- (e2a A,0'r) + E e2a-1 A (e2aJ770'r) = 2r770 a=1 a=1 k

k

E e2aJ(e2a-1 A 70,r) + a=1 e2a A (e2a-1J770'r) _ -ir770,r.

a=1

Now let (P, A) be a spinC structure on the bundle of SO(n)-frames of the Hermitian manifold (M2k, J, g), and denote by S the associated spinor bundle. The 2-form S2 acts as an endomorphism in the bundle S, )S.

52 : S

We compute the eigenvalues of this endomorphism.

Proposition. Q : S -+ S has the eigenvalues i(k(Ic), - 2r), 0 < r < k, and the corresponding eigensubspaces have dimension

r

respectively.

The

spinor bundle splits into S = SO ® S1 ® ... ® Sk, where

Sr={iES: 1'0 =i(k-2r)'/b}. Proof. We will use the spin representation explicitly described in Section 1.3. In the space of Dirac spinors O2k = CC2 ®... ®C2 (k times) the operator

e2.-1e2a, 1 < a < k, is given by the matrix

e2a-le2a=E0 ...0 E0 91920 E9 ...®E with 9192 =

0 1

0

1

).

ThuQ = el n e2 + ... + e2k_1 A e2k is represented,

as an endomorphism of O2k, by

Q=(9192)®E®...E+ ... +E®...®(g192) The matrix 9192 has eigenvalues ±i. Let v(+1) and v(-1) be a basis of C2 consisting of corresponding eigenvectors. Then v(e1) ®... ®v(6k) (ea = ±1) is a basis of O2k = C2 ®... ® C2, and S2 acts on these basis elements by

(e) k

1(v(61) ®... ®v(ek)) = i

a=1

From this we conclude the assertion.

v(sl) ®... ®v(sk).

3. Dirac Operators

76

Now we will present an explicit isomorphism from the (0, r)-forms with values in So onto the bundle Sr, A°'r®So Sr, 0
?7°'r (g V)0 i)

77O,r

222

00

induces an isomorphism between the bundles A°,r ® So and Sr preserving inner products.

Proof. We first have to prove that the product i 0,r '50 belongs to Sr. To do so, we will use the formula

(x A wk) = x (wk

V))

+ (xJwk) 0

for x E Rn, wk E Ak and % E On. Next we obtain e2a-le2a(770'r - Oo)

= e2.-1 _

. {(e2a A 77°'r)

(71°,r A e2a-1 A e2a)

- ,/010 - (e2aJ?7°'r)

- 001

`Yo - (e2a-1J(e2a A?7(l'r)) . 4'0 -(P2a-1 A (e2ai?7o'r)) ' 0 + 0, '

since e2a-iJe2&J?7°,r = 0 for every form 77°,r E AO,r. Now we similarly rewrite the following expression: (77o'r

_ _

A e2,-1 A e2a) 00 (77O,r

,O,r

A e2a-1) . e2a - 00 - (-1)r+1(e2a-J(?701r A e2a-1)) 00

e2a-1 e2a Lo - (-1)r(e2a-1J710'r) e2a00

+(e2.J(e2a-1 A ?70'r)) . V0

= 77O,r e2a-1e2a 0 + e2a A (e2a-1J7lo'r) ')o) + e2,-J(e2a-1 A r70'r) 00

This computation implies 00) k

=

77O,r

(QV)O)

k

- E e2,,-1J(e2a A

?70'r) .

4'0 - E e2a-1 A

n/1

a=1 k

bo

a=1 k

+ 1 e2aJ(e2a-1 A71°'r)00 + T e2a A a=1

(e2a_J770'r) /,

(e2a-1J?70,r)V)o

a=1

= 770'r(Q')0) - 2ir?7°'r . V)0 = i770,r(k - 2r)qpo = i(k - 2r)77°'r 00.

Hence the spinor ?7°,r O o belongs to Sr. The remaining assertions follow directly from algebraic calculations.

Remark. The factor 2

z is necessary for the following reason: If 71°,1 = el+ ie2 is a (0,1)-form, then 177 °,1I2 = 2. On the other hand, from ele20o = it0 we obtain I(e1 + ie2)Jo12 = 4I0oI2.

3.4. Hermitian manifolds and spinors

77

Remark. In the isomorphism specified above we have to use the complexconjugate bundle A°'r, since to apply Clifford multiplication we first have to change the r-form into the corresponding r-vector. However, an Hermitian metric induces a complex anti-linear identification of vectors with covectors. An analogous calculation leads to the following

Proposition. The mapping /3r 2

1r,0 21]

:

A°,r,r ® Sk

--+ Sk-r defined by rlr 0 ®

k-

'Ok is an isometry.

Corollary. The spinor bundle S of an Hermitian manifold (with respect to an arbitrary spin structure) is isomorphic to S = (A°,0 +... + A°'k) ® So = (A°,0 +... + Ak,°) ® Sk, where

So={0ES:S2b=ik }, Sk={'bES:52 _-ik'}. In particular, So = Ak,° ® Sk and Sk = A0,k ® So.

We will now consider the case of the canonical spin structure of the Hermitian manifold (M2k, j, g), i.e. the one defined by the lift of the group homomorphism constructed before: Spiel (2k)

U(k) - SO(2k) The subbundles So, Sl, ... precisely correspond to the irreducible U(k)components of the representation O2k. If A E U(k) has diagonal form,

then k

2 Z 9 i -k7 /

/ \

/ \

11 (cos 12 I+ sin I l l e2j_1e2i So corresponds to the basis vector v(1) ® ... 0 v(1) and Sk to the basis F(A) = e j=1

7

vector v(-1) ® ... ® v(-1). Hence the endomorphism e2j_1e2j acts on So as multiplication by (+i) and on Sk as multiplication by (-i). Thus the bundle So coincides with the highest power of the complex tangent bundle (TM2k, j), while Sk is trivial. On the other hand, the determinant bundle G of the spin structure is again Ak(TM2k, J); hence

So=G=Ak(T), Sk=®1

3. Dirac Operators

78

in the case of the canonical spin structure. Now we turn to the following calculation involving first Chern classes. In general, for every spin structure we have Ldim(S)/2

= Adim(S) (S,)

This implies

2k-lcl(L) = cl(S) = cl((A°'0 + ... + A°+k) ®So) = 2kc1(So) + cl(A°,° + ... + A°Mk). For the canonical spin structure, So = L = Ak(T) implies -2k-1c1(M2k)

= cl (A°'0 + ... + A°,k).

Inserting this, we obtain the

Proposition. Let (P, A) be an arbitrary spinC structure on an Hermitian manifold, L the determinant bundle of this spin structure and So the corresponding subbundle of the spinor bundle. Then for the Chern classes the following relations hold:

cl(L) +cl(M2k) = 2c1(So),

cl(L) - cl(M2k) = 2c1(Sk)

Remark. For an Hermitian manifold (M2k, j, g) we defined the canonical spin structure by the lift F of the homomorphism f (A) = (A, det A):

Spin(2k) U(k) f+ SO(2k) x U(1) In this case, L = So = Ak(T). However, there is a second lift from U(k) to Spin (2k) related to the homomorphism fl(A) = (A, det a) We will call the spin structure corresponding to this the anti-canonical spin structure. In this case, L = Ak(T*) and So = 81. Furthermore, we will discuss the case that the given spin structure on (M2k, j, g) originates from a spin structure .

Given this, one obtains a spin structure by P° = P x Spin(2k) Spin(2k). The determinant bundle L is then trivial, L = Ol, and the (P, A)

.

spinor bundle again splits into S = (A°,° +... + AO,k) ® So. However,

(So)2 = Ak(T).

Indeed, the spinor bundle in this case is a vector bundle associated with the group Spin(2k). On the other hand, A2k has a real (quaternionic) structure j : A2k -' A2k which is spin equivariant and commutes (or anti-commutes,

3.4. Hermitian manifolds and spinors

79

respectively) with Clifford multiplication (compare Section 1.7). j induces a morphism in the spinor bundle which is complex anti-linear and, since

j(i(k - 2r)0) = -(k - 2r)ij (o) = (k - 2(k - r))ij(O), clj(O) = it maps the subbundle Sr into Sk_T. Going to the conjugate bundle 8k_, j can be considered as a complex linear morphism,

j: ST) Sk_T. Composing j : So -* Sk with,3k : Ak,0®Sk -+ So now defines an isomorphism

SO ®SO jSo®SkAkO®Sk®Sk=Ak'0 Hence,

So = Ako = Ak(Ao'1) = Aok.

Altogether, we obtain the following table.

canonical spin structure

G

So

Sk

Ak(T)

Ak(T)

01

61

Ak (T* )

So = AO k

Sk = Ak,O

anti-canonical spin structure Ak (T*) spin structure

®1

Fix a connection A in the U(1)-principal bundle P1 of the spin structure (or, equivalently, in G) as well as a connection Ao in the Hermitian vector bundle So. Then, on the one hand, there is the Dirac operator, DA : r(S)

r(S), k

and, on the other, aAo as well as 5% are operators in the bundle (L A") (9 i=O

So. Here 8A0 and aAo are defined by 0,r+1

2k 19A.

ei A

(1"7O'T ®VO) = (5,70,r) (g 0C) +

770,T

® 0A °0o,

i=1

2k

5A" (,70,r

® 00) _

(6*7,O,T)

® 0 - (E i=1

T1

®ve °o0

3. Dirac Operators

80

Note that for every vector X E T (M2k) and each (0, r)-form r70'r, the form X-Jr70,r is a (0, r - 1)-form. Thus in the formula for aAo the projection onto the (0, r - 1)-component is no longer necessary. The isomorphisms already defined,

ar:AO,r(9 So

)Sr,

k_

k

combine to an isomorphism a = L Car between the bundle j AO"r ® So and r=0

r=O

S. Hence we can compare DA with aAo + aAo . Note that a-'DAa becomes k

a complex linear operator in the bundle L Au," ® S. r=O

Proposition. There exists an endomorphism k

k

E:

(A0?')

®SO -->

®SO

A0,r r=0

r=0

such that a-1DAa = v(&0 + aAo) + EE depends on the almost-complex structure J, on the Hermitian metric g and on the connections A, A0.

Proof. a-'DAa and v2(5A0 + aAo) are first order differential operators. Hence it suffices to show that their symbols coincide. Fix a vector (covector)

X. Then the symbol v(DA)(X) : S -> S is Clifford multiplication by the vector X. Without loss of generality we choose the Hermitian basis el, J(e1),... e2k-1, J(e2k_1) with X = el. Now compute k

k

v(a-1 DAa) (X) : E A0,r ®So - } E A0,r ®So. r=0

r=0

To this end, first decompose a given (0, r)-form 77O,r into 77O,r

el_Jr7*0,r

= (e1 + ie2) A r70,r-1 + 77*0,r with

= e2.J

*O,r

77*o'r

V)o

= 0.

Then, U(DA) (X)a(T1o'r

=

/2/2

22

®,00)

= 21 el .

70,r . 00

el (e1 + ie2) r7o r-1 'po +

2r/2 el

(-1 + iele2) 71°'r-1 'bo + 2 (e1 A 7,*O,r)

Clifford multiplication by e1e2 commutes with

770,r-', and

. 00

e1e200 = i00

Moreover, e1Jr7*O,r = 0. Hence, o-(DA) (X)a(770'r (& 00)

770,r-1 00

2222

+

2

{(el

+ ie2) A 77*O,r} 00.

3.4. Hermitian manifolds and spinors

81

Applying a-1, we conclude that a-1o-(DA)(X)a(?7"' ®00) I

_

V'2_77 O,r-I + v'2-

(_X1Or +

(el + ie2)

A 17*0'r) ®1P0

2

J

11

(X +2JX) A770,r/ ®,00

Thus the symbol of a-1DAa is computed. For the symbols of aAo and aAo we have

a(5A0)(X) _ o(a)(X) ®Ids0,

o,(5A0)(X) _ c(a*)(X) ®Ids0,

and the mappings o(a)(X) : A°,r -+ AO,r+l and v(a)*(X) : AO,r - AO,r-1 respectively, are given by 0.(5)(X)(771'r)

=

X +2 JX

A77O,r'

te(a)*(X)(rl°'r) _ -XJ77°,r.

Consider the special case of a Kahler manifold (M2k, j, g). If Q is the bun of SO(2k)-frames and R its U(k)-reduction, then the Levi-Civita connection

reduces to the U(k)-principal bundle R. Choosing, furthermore, the anti-

canonical spin structure, we know that So = 81, and L = Ak(T*) = R X (det)-1 C. Thus the Levi-Civita connection induces a connection A in L.

As the connection AO in So = 191 we take the trivial one. In this case, the corresponding Dirac operator D just agrees with ,l'2-(a + a*):

Proposition. Let (M2k, J, g) be a Kahler manifold with the anti-canonical

spin structure. Then, 1) S

AO,O +

... + A°,k, and

2) the Dirac operator defined by the Levi-Civita connection coincides with v (a + a*).

Corollary. The space of harmonic spinors, {, E r(S) : Do = 0}, of a compact Kahler manifold (with respect to the anti-canonical spin structure) is isomorphic to k

E Hr(M2k;

0).

r=0

Finally, we discuss the case of a Kahler manifold (M2k, j, g) with fixed spin

structure. Then,

L = Ol,

S0

= A°'k,

Sk2

= Ak'0.

3. Dirac Operators

82

Thus Sk is a square root of the canonical bundle K = Ak(T*) = Ak,o of the Kahler manifold, and the spinor bundle is isomorphic to

S= (A0,0+A0,1 +...+AOk) ®Sk. Choose the trivial connection A in L = O1. As Sk = Ak,0 the connection in Sk is induced from the Levi-Civita connection. Summarizing, we have for the corresponding Dirac operator D the Proposition. Let (M2k, J, g) be a Kahler manifold with fixed spin structure. Then the following assertions hold: 1) Sk is a square root of the canonical bundle, Sk = K = Ak,o k

2) The spinor bundle is isomorphic to S = > AO,r ® Sk. r=0 3) The Dirac operator coincides with v"2-(6 + 8*).

Proposition. The space of harmonic spinors, {0 E I'(S) : DV = 0}, on a compact Kdhler manifold with spin structure is isomorphic to k

Hr(M2k, O(Sk)), r=0

where Sk is a line bundle for which Sk = K = Ak,o (describing the spin structure).

3.5. The Dirac operator of a Riemannian symmetric space Consider a Riemannian symmetric space Mm with isotropy group G. If K is the isotropy group of a fixed point mo E Mm, then Mm can be identified with the homogeneous space G/K. The Lie algebra g of the group G splits into

g=t+m and the following commutation relations hold: [e, f] C e,

[t, m] C m,

[m, m] C f.

Moreover, m is an Ad(K)-invariant subspace of the Lie algebra g, Ad (k) (m) C m

for

k E K.

Let (,) be a scalar product in the vector space g which is positive definite on m and has the invariance property ([X, Y1, Z) + (Y, [X, Z]) = 0

3.5. The Dirac operator of a Riemannian symmetric space

83

for all X, Y, Z E g. This scalar product defines a Riemannian metric on Mn which we also want to denote by (,). Right and left translations in the group G will be denoted by Rg and Lg, respectively, R9(91) = gig,

L9(91) = 991

The projection 7r : G -> G/K = Mm is a K-principal bundle. This principal

bundle has a canonical connection Z. The group K acts on G from the right. For X E t, the fundamental vector field of the K-action at the point g E G is given by (9) = dt (9 etx)t=0.

But this is precisely the left invariant vector field X determined by the vector X E t. Thus the vertical tangent space of the K-principal bundle 7r : G -+ G/K at the point g E G coincides with the space dLg(t), TT(G) = dLg(t).

Define the connection in the K-principal bundle as the splitting Tg(G) = TT (G) +T9 (G) with TT(G) = dLg(m). We have to check that this distribution {T9h (G) = dLg (m) }

is right invariant under the K-action. However, T9 (G) = dLgdLk(m) = dRkdRk-idLgdLk(m).

Each right translation commutes with every left translation, and thus Ty (G) = dRkdLgdRk-idLk(m) = dRkdL9Ad(k)(m) = dRk(TT (G)).

The canonical connection in the principal bundle (G, 7r, G/K; K) as a 1-form

Z : TG -+ t is easily described. Let O be the Maurer-Cartan form of the Lie group G, O : T(G) ---p g,

O(t) = dLg(tg).

Then Z = pre o O. Indeed, for X E t, the fundamental vector field X of the K-action is described by the left invariant vector field X, and this implies

O(X) = X, i.e. Z(X) = X. On the other hand, the kernel of Z is exactly Th(G). We also compute the curvature form of this canonical connection:

cZ = dZ + 2 [Z, Z] = prt(d0) + 2 [prte, prt0] If, however, O = Ot + Gm is the decomposition of the Maurer-Cartan form, then [O, O] = [Ot, Ot] + [Ot, Om] + [Om, Ot] + [em, Om] and the commutator relations imply pre [O, O] = [O', Of] + [Om, 9m].

3. Dirac Operators

84

Thus, S2z

= pre

(de+ 2 [O, O]) -

2 [19m, am]

and the structure equation dO + 2 [O, O] = 0 of the Lie group G leads to S2

2

_ - 12 [prmE), prma]

Let Q be the bundle of orthonormal frames on M. Then there exists an inclusion i : G --> Q such that the diagram

G

Q

G/K +- Mm commutes. To this end, fix an orthonormal basis el,... , e,,,, at the point mo and define i(g) = (dly(e1), ... , dl9(e, )), where 19 : Mm --> Mm is the action of g E G on Mm. Now we want to see that the Levi-Civita connection reduces to the K-principal bundle (G, 7r, G/K; K) and coincides there with

the constructed connection Z. Note that the tangent bundle T(Mm) of Mm = G/K is T(Mm) = G XAd(K) m,

the bundle associated with the representation Ad : K -* 0(m). Hence a vector field T on the manifold Mm is a mapping T : G -p m with the invariance property T(gk) = Ad(k-1)T(g). Z induces a covariant derivative Vz in T(Mm) = G XAd(K) m, and VZT = dT + [prf@,T].

By the invariance property of (,) this implies (OZT,T1) + (T, VZT1)

_ (dT,T1) + (T, dTl) + ([prtO, T], Tl) + (T, [prtO, T1])

(dT,T1)+(T,dT1) =d(T,T1), i.e.

Vz preserves the Riemannian metric. Analogously, one shows that Vz

is torsion-free. However, these two conditions uniquely determine the LeviCivita connection of Mm.

3.5. The Dirac operator of a Riemannian symmetric space

85

Fix a homogeneous spin structure of the symmetric space G/K, i.e. a homomorphism Ad : K -- Spin(m) such that the diagram Spin(m) IA

A

K - So (m) commutes. Let n : Spin(m) --f GL(A) be the spin representation. Then a spinor field 7P is identified with a function : G -* A satisfying the invariance condition V (gk)

= r.Xd(k-1)O(.4')

Let X be a left invariant vector field on the group G with X E f. Then, Xzb(g)

dtrcAd(e-tx),(s) = - *Ad*(X) (g),

=

and hence

Xzb = -ic Ad(X)O = -Ad*(X) 0, where Ad(X) zb is Clifford multiplication of the spinor 0 E A = A(m) by the element Ad*(X) E spin(m) C Viff (m) of the Clifford algebra. Consider the 1-form DZc with values in A. Then, (DZ?G)(X) = DO(X) + r,*Ad*(Z(X))V).

For X E f, the last computation (Z(X) = X!) immediately shows that DZ'(X) = 0. For X E m we obtain Z(X) = 0, and thus DZO(X) = X (V)). Choosing an orthonormal basis Xl,... , X,,,, in m, we obtain m

DZb _

Xi ®Xi(0), i=1

and, for the Dirac operator, this implies the simple formula m

i=1

We will now compute D2: m "/' D20 =

i,j=1

m

m

i=1

i,j=1

/ /' X,.Xj . (XXj(')) _ -EXi (0) '+ 2 E Xi'Xj . ([X,,Xj]o)

However, the vector field [Xi, Xj] belongs to t, and hence the above calculation shows that [Xi, Xj] (0) = -Ad*([Xi, Xj]) . 0-

3. Dirac Operators

86

Inserting this yields M 2

X2(0)

D2

i,j=1

i=1

Choose an orthonormal basis in the Lie algebra f, Y1, ... , Y. Then, from Y. ('b) = -Ad* (Y«)

'

we immediately conclude that Y« (0) = Ad-.(Ya) ' Ad* (Y.)

m and, finally, using the Casimir operator S2c

k

.qb _

= QG(V))

- > Ya of the group a=1

i=1

G, we arrive at the formula D20

k

X2

1 2

a=1

To treat the remaining term, we compute in the Clifford algebra Jif f (m) as follows. Let Ad. (Y,,) E spin (m), and let X1, ... Xm be an orthonormal basis in in. Then, ,

m

Ad, (Y,) =

4

i,j=1 m

A

E (Ad. (Y.) (Xi), Xj) Xi ' Xj

i,j=1 m

d

1: ([Y., Xi], Xj)Xi Xi i,j=1

and, analogously, m

Ad.([Xi, Xj]) = 4

([[Xi, Xj], Xp], Xq) Xp ' Xq. p,q=1

The remainder term thus coincides with 1R

E E E ([YY, Xi], Xj) ([Ya, Xp], Xq)XiXjXpXq a=1 i,j=1 p,q=1 in

in

1: E i,j=1 p,q=1

Xi,X ,X ,X XiX X X

3.5. The Dirac operator of a Riemannian symmetric space

87

Because of the invariance property of the scalar product (,) in g we also have k

T, ([Y., XiJ, Xj) ([Y., Xp],Xq) a=1

(Y., [Xi, Xj]) (Y., [Xp, XqJ) _ ([Xi, Xj], [Xp, Xq]) a=1

Hence the term under consideration simplifies to the expression

-16 i,j,p,q ([Xi, Xj], [Xp, Xq])XiXjXpXq The Levi-Civita connection is induced from the connection Z in the Kprincipal bundle. Thus for the curvature tensor R of the Riemannian space the following formula holds:

R(Xi, Xj) = [QZ(Xi, Xj), i.e.

(R(Xi, Xj)Xp, Xq)

([1Z(Xi, Xj), Xp], Xq) = -([[Xi, Xj], Xp], Xq) -([Xi, Xj], [XP, Xq])

In the Clifford algebra Vif f (m) = Cm, the remaining term has CO- and Cmas well as C4L-parts. The Cm- and C4,,-parts vanish, and for the C° -part we have

- d E([X" Xj], [Xi, Xjl)X'XjXiXj i<j

=4EII[Xi,Xj]II2=

8

I[Xi,Xj]112 = 8R,

where R is the scalar curvature of the space G/K. Summarizing, we arrive at the

Proposition. Let Mm = G/K be a compact Riemannian symmetric space with a homogeneous spin structure. Let SZG denote the Casimir operator of the Lie group G. Then,

D2=SlG+8R. Remark. The importance of this formula lies in the fact that it allows us to compute the eigenvalues of D2 purely by means of representation theory. Let A : G -> GL(VA) be an irreducible complex representation and

3. Dirac Operators

88

A.

gt(V,\) its differential. Then, m

A*(QG) =

-

E '\.(X,)1 z=1

k

- a=1 E) *(Y«)2

is an operator in VA. It is easy to show that this operator commutes with all automorphisms .(g) : Va -i VA (g E G). This follows immediately from the formula A(g)A*(X)A(g-1) = a*(Ad(g)X), X E g, g E G, which itself is the result of a straightforward calculation using the fact that Ad(g) g -> g maps the orthonormal basis consisting of {X1i ... , Xm, Y1, ... ,Yk} again onto an orthonormal basis of g. Let u be an eigenvalue of )*(c)G) and W. C V, the corresponding eigensubspace. Then W. is invariant under all .(g), g E G. The irreducibility of VA implies WN0 = 0 or VA. But WN, = 0 is excluded, as ). (cla) indeed has eigenvalues over C. Hence we have WN, = Va, i.e. A*(1 ) is a multiple of the identity.

Finally, consider the Hilbert space L2 (S) = L2 (G/K; S) of all squareintegrable sections of the spinor bundle over the compact Riemannian sym-

metric space. Then G acts as a group of unitary transformations on L2. Decompose L2 into the direct sum L2(S)

_

VA

AEA

of finite-dimensional irreducible G-representations VA, and compute each time the number c(A) with \*(I1G) = The spectrum of D2 is then given by

Spec (D2) _ {c(A) + 8R : A E A } . These computations were, e.g.

carried out in the case of spheres, cer-

tain Graf3mannian manifolds, and for the complex projective spaces CP271+1 (compare the references).

3.6. References and Exercises M. Cahen, A. Franc et S. Gutt. Spectrum of the Dirac Operator on Complex Projective Space Cp2q-1, Letters in Math. Phys., 18, 1989, 165176.

Th. Friedrich. Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkriimmung, Math. Nachr. 97 (1980), 117-146.

Th. Friedrich. Zur Existenz paralleler Spinorfelder fiber Riemannschen Mannigfaltigkeiten, Coll. Math. XLIV (1981), 277-290.

3.6. References and Exercises

89

N. Hitchin. Harmonic spinors, Adv. in Math. 14 (1974), 1-55. K.-D. Kirchberg. Compact six-dimensional Kahler spin manifolds of positive scalar curvature with the smallest possible first eigenvalue of the Dirac operator, Math. Ann. 282 (1988), 157-176. A. Lichnerowicz. Spineurs harmoniques, C.R. Acad. Sci. Paris 257 (1963), 7-9.

J.-L. Milhorat. Spectre de 1'operateur de Dirac sur les espaces projectifs quaternioniens, C.R. Acad. Sci. Paris Ser. I 314 (1992), 69-72. A. Moroianu. Parallel and Killing spinors on Spinc-manifolds, Comm. Math Phys. 187 (1997), 417-428.

E. Schrodinger. Diracsches Elektron im Schwerfeld I, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Phys.-Math. Klasse 1932, Verlag der Akademie der Wissenschaften, Berlin 1932, 436-460.

S. Seifarth, U. Semmelmann. The spectrum of the Dirac operator on the odd-dimensional complex projective space CP2m-1, Preprint des SFB 288 "Differentialgeometrie and Quantenphysik" No. 95 (1993). H. Strese. Uber den Dirac-Operator auf Graf3mannschen Mannigfaltigkeiten, Math. Nachr. 98 (1980), 53-58. S. Sulanke. Berechnung des Spektrums des Quadrates des Dirac-Operators D2 auf der Sphare and Untersuchungen zum ersten Eigenwert von D auf 5-dimensionalen Raumen konstanter positiver Schnittkrummung, Dissertation, Humboldt-Universitat zu Berlin 1981. Exercise 1. Let (M4, g) be a 4-dimensional Riemannian spin manifold with non-trivial parallel spinors V+, L- in the bundles S+ and S-, respectively. Prove that (M4, g) is flat.

Exercise 2. Prove that every parallel spinor 0+ in the bundle S+ over a 4-dimensional Riemannian manifold (M4, g) induces a complex structure J : TM4 -* TM4 such that (M4, g, J) is a Kahler manifold. Exercise 3. Prove that, in 2-dimensional Euclidian space i 2, the general solution to the twistor equation T() = 0 is given by

tl>(x,y) _ (D) with arbitrary vectors

-

(-XI+iY x

(BA)

02y)

(A) (C) D E CZ = A2D B ,

Exercise 4. The metric on M4 C II84,

M4 = {(xl, ... x4) E L ,

4

:x1 > 0, 0 < x2 < 7r},

3. Dirac Operators

90

defined by ds2

=

xl

(dxl)2 + xI(dx2)2 + xlsin2(x2)(dx3)2 +

xl +

xl + c xl (c > 0) is Ricci-flat but does not admit any parallel spinors. Exercise 5. Let (M4, g) be a Riemannian spin manifold and

c(dx4)2

E F(S) a

spinor field. If

7XV) =w(X)'') with a real-valued 1-form w, then

a) Ric - 0, b) dw = 0. Prove by examples that this does not hold for general complex-valued forms.

Chapter 4

Analytical Properties of Dirac Operators 4.1. The essential self-adjointness of the Dirac operator in L2 Let us recall some notions from the spectral theory of linear operators in complex Hilbert spaces. Let A be an (in general unbounded) operator with dense domain of definition, D(A), in the complex Hilbert space H. Denote the range of A by R(A). The graph, r(A) C H x H, consists of all pairs (x, Ax), x E D(A). In the sequel we will assume that its closed hull F(A) C H x H again is the graph of an operator A which then is called the closure of A. Then A acts via the formula A(x) = rli n A(xn) and its domain of definition, D(A), consists of all vectors x E H for which there exists a sequence xn E D(A) withn-roo lim xn = x such that, moreover, the sequence A(xn) converges in H. Differential operators always have a closure in this sense. The spectrum of an operator consists of three parts. First, there are the eigenvalues of A which form the so-called point spectrum 0p (A):

Qp(A) = {A E C: ker(A - A)

{0}}.

Furthermore, there are the residual spectrum, vr(A), as well as the continuous spectrum, Q,(A):

ar(A) _{AEC: ker(A - A) = 0,

R(A - A) 0 H}, 91

4. Analytical Properties of Dirac Operators

92

o (A) = {A E C : ker(A - A) = 0, R(A - A) = H, (A - A)-1 is unbounded}.

The remaining complex numbers form the resolvent set p(A):

p(A) = {A E C : (A - \)-1 is a bounded operator defined on R(A - \) = H}. To each operator A there corresponds an adjoint operator A* with the domain of definition

D(A*)={xEH:EyEH `dzED(A):(Az,x)=(z,y)} and A*(x) = y. This implies the relation (Az,x) = (z, A* (x))

for all vectors z E D(A), x E D(A*). Let A be a symmetric operator, i.e. (Ax, y) _ (x, Ay),

x, y E D(A).

A. (We will Then D(A) is contained in D(A*) and, in addition, abbreviate this as A C A*.) The double adjoint operator A** coincides with the closure A (von Neumann theorem):

A=A

CA.

Definition. The operator A is called self-adjoint if A = A*. In particular, self-adjoint operators are closed, A = A.

Definition. The operator A is called essentially self-adjoint if its closure A is self-adjoint, i.e. A = A*. The spectrum of essentially self-adjoint operators is real, a(A) = a(A) = a(A*) C 1181.

Finally, recall the notion of spectral measure. If S C C is a set of complex numbers and B(S) the o-algebra of all its Borel subsets, then a spectral measure F is a mapping

F : 13(S) -+ Proj(H) from the o-algebra B(S) into the set Proj (H) of all projectors in the Hilbert space with the following properties:

a) F(S) = IdH, b) for every x E H the equality Mx(B) = (F(B)x, x), B E B(S), defines a measure on the o-algebra B(S).

4.1. The essential self-adjointness of the Dirac operator in L2

93

Given two vectors x, y E H, then M.,y(B) = (F(B)x, y)

is a complex-valued measure. Any measurable function f : S -- C can be integrated against a spectral measure:

f f (s)dF(s). S

The value of this integral is an operator in the Hilbert space H which is bounded for a bounded function. Moreover, we have the formula

((I f (s)dF(s)

ff(s)d/2xv(s).

x, y

s S The spectral theorem for self-adjoint operators can now be formulated as I

follows:

Theorem. Let A be a self-adjoint operator in H with spectrum o(A) C R. Then there exists exactly one spectral measure F on the Q-algebra 13(v(A)) with

f \dF(,\).

A=

v(A)

We now turn to the situation for the Dirac operator DA on a Riemannian manifold (Mn, g) with fixed spin structure and fixed connection A in the determinant bundle of the spin structure. The space 1T (S) of all sections of the spinor bundle with compact support carries the scalar product (01, 02) =

f(i(x),2(x))dMm. Mn

Let L2 (S) be the completion of this space. DA is a symmetric operator in L2(S) with domain D(DA) = Fe(S) (compare Section 3.2). In this section, we will prove that DA is essentially self-adjoint whenever (Mn, g) is a complete Riemannian manifold. To start with, we state the following formula for DA:

Proposition. If f is a smooth function defined on the manifold Mn, grad(f) its gradient field and V) a spinor field, then

DA(f . 0) = fDA(0) + grad(f) DA (0) .

4. Analytical Properties of Dirac Operators

94

Proof. This formula is obtained by a straightforward computation:

DA(f

)=

ei '

VA

(f Y) _

+f

Ve

4)

i=1

e

i=1

ei (ei (f) 4

grad(f) ,+fDA(ib) Now we start to prove the essential self-adjointness of the Dirac operator DA. The argument will proceed along the lines of the the article by J. Wolf (see the references at the end of this chapter). The proof is subdivided into several steps. Let DA be the adjoint to DA. In the domain D(DA) we introduce the norm

N(b) where

110112 + I IDAI12,

denotes the norm in the Hilbert space L2.

Lemma 1. Let I'e(S) C D(DA) be dense with respect to the N-norm. Then DA is essentially self-adjoint.

Proof. Under the assumptions above we have to prove that D(DA) C D(DA). Let 0 E D(DA). By assumption, there exists a sequence on E Fe(S)

N(0 - On) = 0. This implies that rli On = 0 in L2 (S) and, n moreover, that DAWn) converges to DA(s) in L2. However, n is a smooth with 1li

spinor field with compact support, and hence DAW ) = DA(2b ). Thus the sequence DA('On) converges in L2. But the latter just means that b E D(DA).

Next we introduce the linear subspace D,(DA)

E D(DA) : V) has compact support}.

Lemma 2. F,(S) is dense in D,(DA) with respect to the N-norm. Proof. Choose a locally finite covering of the manifold M1 by charts (Ui, hi)

indexed by the set i E I such that each Ui C Mn is a compact subset. Let {fi}iEI be a partition of unity subordinate to this covering with supp(fi) C Ui. For a given spinor such that

E D,(DA), there are only finitely many indices i E I

supp(fi) n supp('')

0.

Denote those by i1, ..., ii E I. Then, 0 = b1 + ... + 0e with 'bj = fj 0, where 1 < j < 1. The spinor field O j can be considered as a 2[,/2] -tuple of functions with compact support defined on the space R1. Approximate 0j

4.1. The essential self-adjointness of the Dirac operator in L2

95

by the convolutions with an approximation of the delta-distribution. Let h : IIBn -- R1 be given by

h(x)

0

forlxl > 1,

e 11-x1'

for IxI < 1,

and let hE : R' - R1 be defined by hE = E h (1). Then bj * h, is a smooth function with compact support approximating bj in L2. Since DA (0i) belongs to L2, the sequence DA( j * h,) thus converges to DA( j). Applying the chart mapping again, we obtain smooth spinor fields with compact support i,l, j,2i ... defined in the Riemannian space with

k N(Oj - 'i,k) = 0. 1,k + ... +'1,k, we now conclude that k belongs to Pe(S) and lim N(b - L1k) = 0. Forming 'Ok = k-*oo

Lemma 3. If (Mn, g) is a complete Riemannian manifold, then D°(DA) is dense in D(DA) with respect to the N-norm.

Proof. Denote the interior distance between two points ml and m2 in the Riemannian manifold by d(ml, m2). Let mo E Mn be a fixed point and p(m) = d(mo, m) the distance fromm to mo. The triangle inequality implies Ip(m1) - P(M2)1 < d(ml, m2), i.e. p is a Lipschitz continuous function. Hence p(m) is differentiable almost

everywhere and the gradient grad(p) exists a.e., too. Furthermore, at each of these points I

I grad(p) I << 1. I

Consider the geodesic ball

1C(r) = {m E Mn : p(m) < r}. Since (Mn, g) is a complete Riemannian manifold, the closure Kr is a compact subset of Mn. Choose a C°°-function, a : II81 --* [0, 1], with the following properties: (i)

a(t)-1

for

-oo
a(t) =_0 for 2 < t < oo. Let the constant M be the maximum of the derivative I a' (t) I: (ii)

M= sup a'(t)j. 1
The function b,.: M' -> [0, 1] is defined by

b, (m) = a (P(m)) r

J

4. Analytical Properties of Dirac Operators

96

Then, br - 1 on JC(r) and supp (br) C 1C(2r). Moreover, br is Lipschitz continuous and the following inequality holds a.e.:

r a'

2

1

IIgrad(br)112 =

Let

2

Ilgrad(p)II2 < r2

belongs to D(DA). Furthermore,

E D, (DA) be given. Then 0, =

D,*q(br) = grad(br) V) + brDA(b),

and this implies for the corresponding L2-norms IIDA(I - 4)r)112

= II (1 - br)DA) - grad(br)'bII2 2IIDA( )II2+

2r22

f ICII2. Mn

Mn\IC(r)

Thus we obtain the following estimate for the N-norm:

N2(b- b) = II0-wrUI2+IIDA(0 -br)112 <

f

IIV)I12

From f 110112 < oc and f

that

2M2 2 I IDA(

)I 12+

I

f

V) I

I I2.

Mn

Mn\IC(r)

Mn\IC(r)

Mn

f

+

I DA () I I2 < oo we then immediately conclude

Mn

lim N2(b - fir) = 0.

r-roo

In particular, we have proved the following:

Proposition. Let (M'n, g) be a complete Riemannian manifold with spin structure. Then the Dirac operator DA is essentially self-adjoint in L2(S). We will show, in addition, that the kernels of DA and DA in L2 coincide. This will result from a more general inequality we are going to prove first.

Proposition. Let (Mn, g) be a complete Riemannian manifold and 0 a spinor field of class C2. Then, for the L2-norm I IDA( )I12 < tIID2

(V)) 112

+ t

'

11

I

I and any number t > 0, 112.

Proof. We will use the same balls 1C(r) and functions br as in the proof of the previous proposition. The equality DA(br,0)

= 2br grad(br)

+ b2DA(' )

4.1. The essential self-adjointness of the Dirac operator in L2

97

implies that for every positive number e > 0

f

I

f (b2r DA (0), DA (0))

I brDA(O) I2 =

K(2r+E)

(DA (b2

K(2r+e)

1C(2r+E)

2

f

(br grad(br)

f (brDA(0),)

' DA(s),)

K(2r+E)

1C(2r+E)

Here we used the fact that the support of the spinor in )Q2r). For e -+ 0 this implies

f

f (DA (,0), br'p) - f (b, DA (0), 2grad (b

II brDA(O)II2 =

IC(2r)

is contained

IC(2r)

IC(2r)

Now we will apply Schwarz' inequality, I (x, y) I < IxI IYI <_

2

IxI2 + 2t IyI2

for all t > 0. This allows us to estimate the last term in the equation (with t = 1): f (brDA(O), 2grad (br)

V))

fHbrDA()II2+2 f I Igrad (br) I I2

< 2

IC(2r)

IC(2r)

< 2

f

IC(2r) 2

IIbrDA(

)II2+2M f

K(2r)

11

112.

K(2r)

From 0 < br < 1 and another application of Schwarz' inequality, this time to the first term, we obtain

f (DA(),br)

2

K(2r)

f JIDA()II2+ 2t f IIII2. K(2r)

IC(2r)

Combining both relations yields

fHDA()H2
K(r)

IlbrDA(V))II2

K(2r)

f{

<2

f

K(2r)

2M A r2 II0II2+2IID2(4)II2+2II0II2J

f IIbrDA('O)II2 K(2r)

f {JI2 +

K(2r)

f IIbrDA(V))II2

2

IC(2r)

-

-

+ 4M2 ) Ct

I

110112

}

4. Analytical Properties of Dirac Operators

98

In the case f 110112 = 00, the inequality to be proved is trivial. If, however, Mn

the integral is finite, then we obtain the inequality to be proved for r -> oo,

t

IDA()112 < tIID2 (0)II2 +

IIV) 112.

Corollary. For a complete Riemannian manifold (Mn, g), the kernels of the operators DA and DA in L2 (S) coincide,

ker(DA) = ker(DA).

Proof. Let 'b E L2 (S) satisfy DAB = 0. By the regularity theorem for solutions of elliptic differential equations we first conclude that '0 is smooth. Hence we can apply the inequality from the previous proposition and thus obtain IIDA(b)II2 _
111

112=

tII0112.

II2 < oc then implies for t - oo that DA(b) - 0-

4.2. The spectrum of Dirac operators over compact manifolds Going from an operator A to its closure A does not change the spectrum, o, (A) = o-p(A) U o (A) U o,, (A): o, (A) = o- (A).

For a Dirac operator this means, in particular, that Q(DA) = v(DA). In the case of a complete Riemannian manifold DA is a self-adjoint operator; hence it has no residual spectrum:

Qr(DA) = 0 Thus a(DA) only consists of the point spectrum, Up MA), and the continuous spectrum, a (DA): o-(DA) = o-(DA) = o'p(DA) U Qc(DA):

For a compact Riemannian manifold the domain of definition of the operator DA coincides with r(S). If \ E o-p(DA) is an eigenvalue of the closure, then there exists a spinor field 0 E L2(S) with

0 E L2. The regularity theorem for elliptic differential operators then implies that ' DA(V)) _ Aw,

is smooth. Hence we have 0 E F(S) = D(DA), i.e. the point spectrum of DA coincides with the point spectrum of the closure: o-p(DA) = op(DA).

4.2. The spectrum of Dirac operators over compact manifolds

99

Continuous spectrum does not occur in the case of a compact base space. However, we have the

Proposition. Let (M', g) be a compact Riemannian manifold with spin structure. For the spectrum of the Dirac operator the following relations hold: i)

up(DA) = o(DA),

ii)

ac(DA) _ 0 = o'r(DA)

Together, these imply

iii)

o- (DA) = or(DA) = ap(DA) = O-p(DA)

Proof. We give a sketch of the proof. First, it is easy to see that the residual spectrum of DA is empty. Indeed, for \ E Qr(DA), there exists a spinor field cp E L2(S) such that (DA(0)-AW,c0)L2=O

for all ' E F(S). Choosing '' with support in a chart and transferring this equation to Euclidean space, we obtain an elliptic differential operator P (= DA - A) as well as a function cp E L2 (R') such that (P (0), cp) L2 = 0

E C'°(R'). By the regularity theorem for elliptic operators, cp is smooth. In this case, we can write the equation (DA(O) - 4), cp)L2 = 0 as (b, (DA - .\)(p)L2 = 0. This in turn implies DAcO = acp and cp E F(S) _ for all

D(DA), i.e. A is an eigenvalue of DA. If now o ,,(DA) is empty, we obtain ap(DA) U o-c(DA) = o- (DA) _ 0- (DA) = O'p(DA) U oc(DA)

The approximation spectrum va (A) of an arbitrary operator A : D(A) R(A) in a Hilbert space is defined by {A E C :

a sequence xn E D(A) s. t.

Ixn I I

= 1, HA(x) - Axn I H -> 0}

.

In general, vp(A) U o-, (A) C o (A) C o-(A).

Applying this inclusion to the case of a Dirac operator over a compact manifold, we at once conclude from the facts proved so far that Ca(DA) = 0-a(DA) = o(DA) _ 0- (DA)

Finally, it remains to be shown that o- ,,(DA) = up(DA). Let \ E o- (DA). Then there is a sequence of spinor fields yin E F(S) with IIInIIL2 = 1,

IIDA(On) - 4nIIL2 - 0.

4. Analytical Properties of Dirac Operators

100

On is smooth and we can thus apply the Schrodinger-Lichnerowicz formula: 1

2

II DA (0n) - AV).II L2 < I IDA('On)II2

=

I0.I I2

f(dAn,n)+A2Hnl.

f

IIVAV)nHIL2+

+'\2I

Mn

Mn

Since Mn is compact and I Yin I L2 = 1 as well as I DA (bn) the L2-lengths I I VA On I I L2 of the 1-forms DAn, I

DAY nI IL2

I

=

I

f

VA 1:(oe'On,

e

Mn i=1

-

AV), 112

L2 -3 0,

0n)dMn,

are bounded. Hence On is a bounded sequence in the Sobolev space Hl (S). By the Rellich lemma, the embedding H'(S) -> L2 (S) is compact. Thus we can assume that on converges to a spinor field V)o in L2, and this immediately implies DA(' o) = A'0o So we have proved that A is an eigenvalue of DA, and hence a, (DA) = up (DA)11 We still suppose that (Mn, g) is a compact Riemannian manifold with fixed spin`s structure. The first Sobolev norm of a smooth spinor field '0 E F(S) is given by

H' =

112

IIL2+IIoA and the corresponding Sobolev space H'(S) is the completion of F(S) with respect to this norm. Since Mn is compact, different connections A induce equivalent norms in the determinant bundle of the spinC structure. Moreover, the embedding II

II

I

H'(S) -, L2(S) is a compact operator (Rellich lemma). The Dirac operator DA is a contin-

uous operator DA : H'(S) , L2(S). This follows, e.g., from the estimate below for the norm of DA(s):

IIDAOIIL2 = ,

f(eiV,ejV)<

i,j=1Mn

fivtvi

id =1Mn n/1

2

f(Ioe 012 + IDe Y'I2) = n 1VA

I IL2 < n I

IHI

2,7 Mn

Applying the Schrodinger-Lichnerowicz formula for D22 (,0), we can, on the other hand, rewrite IIDA(0)IIL2 as follows:

IIDA()IIL2 = (DA1 ),0)L2 = IIVA IIL2 + f 4II0II2+ Mn

f 1(dA'0,0). Mn

4.2. The spectrum of Dirac operators over compact manifolds

101

The endomorphism 1 dA : S -> S is bounded, i.e. there exists a constant c > 0 such that for every point of the manifold Mn and any spinor 0 we have the pointwise inequality -cIIV)II2 <

(

dA-

0,0)

cII0II2.

Inserting this, we obtain the inequality (*)

ICI IH1 +

(Rrnin

-C- ll

2

112

IL2 < II DAV)II L2

H(4 +

Rax

110 1I1

+C-1I

110112

L2,

where Rmax = max {R(m) : m E Mn} is the maximum of the scalar curvature and Rmin its minimum. We will use this inequality to prove the following:

Proposition. Let DA be a Dirac operator over a compact manifold. The closure DA = DA of the Dirac operator DA is defined on the subspace H'(S) C L2(S). Proof. If V E D(DA) belongs to the domain of DA, then there is a sequence 0 in L2(S) and DA(',) converges in L2. From On E IT(S) such that On inequality (*) one immediately sees that 0, is a Cauchy sequence in H'(S). This implies that 4bn converges to V)* in H' (S) The embedding H' (S) --+ is continuous, hence b* coincides with ib. Thus 0 belongs to H'(S). L The converse is trivial. .

Proposition. Let A ¢ o'(DA) be a number which is not in the spectrum of the operator DA. Then (DA - A)-1 : L2(S) _' L2(S) is a compact operator.

Proof. The inequality (*) can be written as I(DA-A)-1(DA-A*IIH1 < II(DA-A)V)IIL2'+(c+1 +A2- Rrin III

Setting cp = (DA - A)b E im(DA - A), we have II (DA -

IIL2.

/

IICI122 +CII (DA - A)-1;pIIL2'

However, the operator (DA - A) is continuous in L2. Hence, for a suitable constant C*, we obtain /the estimate II (DA - '\)_'(P1 1H1 < C*IIsoIIL2.

4. Analytical Properties of Dirac Operators

102

Thus the image of the operator (DA - A)-' (A ¢ v(f)A)) is contained in the Sobolev space H1(S). The assertion then follows from the compactness of the embedding HI(S) -+ L2(S).

Proposition. There exists a complete orthonormal basis zbj, V)2.... of the Hilbert space L2 (S) consisting of eigenspinors of the Dirac operator DA, DA()n) = AnOn-

Moreover, lim

n- oo I An I = 00-

in such a way that A is not in Proof. Choose the real number A 0 the spectrum of DA. The operator (DA - A)-' : L2(S) -* L2(S) is compact and self-adjoint. The spectral theory of these operators leads to a complete orthonormal basis '01 i V)2, ...

in L2 (S) with

(DA - A)-12Nn = AnOn

and lim An = 0. Thus (u n- oo

0),

1

DA(V)n) = (An +

A

On) /

i.e. the spinor field On is an eigenspinor of the Dirac operator for the eigenvalue An = +n

Corollary. There exists a constant C > 0 such that for all spinor fields co E Hl (S) orthogonal to the kernel ker'(DA) the inequality I (DA(co), (P) L21 >- C ICI IL2

holds.

Inequality (*) admits still another interpretation. I

I DA'' I I L2

n

,

Since II11 '0 1122

->

we have

{II0IIL2 + IIDA0IIL2} < II IIHI <_ IIDA0IIL2 +

(c+ 1-

R41ri)

IIV) 112

11011

Hz and 11011 L2 + I I DA2b I 1 L2 are equivalent norms. In other words, the Sobolev space HI(S) can be defined as the completion of the space I" (S)

with respect to the norm

IIII* =

110112

2

+ IDA0IIL2.

The k-th Sobolev space Hk(S) is then the completion of F(S) with respect to the norm k

11011k

- E IIDA( )IIL 2

2.

i=0

4.2. The spectrum of Dirac operators over compact manifolds

If

103

= j c*Anon its decomposition in

E r(S) is a smooth spinor field and

n=1

L2 with respect to the complete orthonormal system 01, 02, ... consisting of eigenspinors of DA, one easily computes 00

IIDAk

An2,\2k

O11L2 =

n

n=1

This implies the

Proposition. Let 01, 02.... be the complete basis of the Hilbert space L2(S) consisting of eigenspinors. If 00

An VYn

n=1

is the L2 _representation of the spinor field zl' E L2(S), then 0 belongs to the Sobolev space Hk(S) if and only if the sum 00

A,,2,\nk
is finite.

We will use the previous proposition to define the (- and the 77-function for the Dirac operator. First, we prove that the embedding Hk --> L2 is a Hilbert-Schmidt operator for k > 2 dim(Mn). Lemma. Let E be a complex vector bundle with Hermitian metric and metric connection over a compact Riemannian manifold (Mn, g), and let Hk(E) denote the k-th Sobolev space. If k > 2 dim(Mn), then the embedding

Hk (E) ' L2 (E) is a Hilbert-Schmidt operator.

Proof. The Sobolev embedding theorem states that Hk(E) is contained in C° (E) and that this embedding is continuous for k > in. Fix a point mo E M' and an orthonormal basis el(m°), ..., el(mo) in the2 fibre Emo. Let cp : Hk(E) -+ Emo be the linear mapping cp(s) = s(mo). co is continuous, and the norm of cp can be estimated as follows: I IcPI I = sup

SEHk

1s(MO)11 I1SIIHk

< sup

I IS =: C.

sEHk 1S1IHk

By Riesz' lemma there exist elements s1i..., sl E Hk with

s(mo) = p(s) = (s, S1)Hke1(mo) +---+ (s, SI)Hkei(mo)

4. Analytical Properties of Dirac Operators

104

The norm IIWII2 is then computed to be IIs(m2)II2

IkoHI2 = sup Hk

ISHk

=

I(S'si2Hk I2

sup

EHk i=1

SIHk

and hence, for each index 1 < i < 1, ((1181122 k 11

112 > Sup

I2

=

IIsjIIHk.

H

Now let f1, f2, ... be an orthonormal basis in Hk(E). Then, 00

00

E Ifj(mo)I2 = E IV(fj)I2 j=1

j=1 0C

l

l

= EEI(fj,si)Hk12=EII8,112k < j=1 i=1

III(PII2
i=1

Integrating this inequality over the compact manifold M' yields 1

I

Ifj

12

L2

< l C2 vol (Mn).

The spectrum a(DA) of a Dirac operator only consists of eigenvalues, and the

closure DA defined on H'(S) is self-adjoint. Let F : 5(o-) -> Proj(L2(S)) be the spectral measure. Then, r

DA =J dF(A). a

For fixed t > 0, e-tae is a bounded function. Hence St

= fe_tA2dF(A) C

is a bounded operator in L2(S).

Proposition. i) The operators St, t > 0, form a semigroup, St1+t2 = St1St2, of bounded operators with norm I St < 1.

ii) The generator of this semigroup is D. iii) For t > 0, St : L2 (S) -> L2 (S) is a Hilbert-Schmidt operator.

4.2. The spectrum of Dirac operators over compact manifolds

105

Proof. Only the last point needs to be proved. First, note that for all co E L2 (S) the H'-norm I St (cp) I lHk is finite. Indeed, since k I

=

I V) 11H2

2

I IDA(0)1IHk, i=0

it is sufficient to prove that DA(St(cp))112, is finite. The operator DA o St is given by the integral

i

,

AkdF(A)

f e-t,\2 dF(A)

_ fAke_tdF(A)

a

0"

and Ake-ta2 is a bounded function of A (t > 0). Thus D,k9 o St is continuous in L2. The argument above now proves that the image of each operator St is contained in F(S),

St(L2(S)) C n Hk(S) = F(S). k=0

Moreover, as an operator from L2 to Hk(S), St is continuous. Choosing k > dim M', we obtain 2

L2 (S) St

Hk (S) - L2(S),

where Hk(S) -> L2(S) is a Hilbert-Schmidt operator. But then St : L2(S) L2(S) is a Hilbert-Schmidt operator, too.

E

Corollary. The function

e-tae :_ CD22 (t) is finite for t > 0.

AEO(DA)

Proof. The Hilbert-Schmidt norm of the operator St : L2(S) -> L2(S) is IStI1H-s

ISt(

=

e-2t 2 = D2 (2t),

)11i2 = AEQ(DA)

AEQ(DA)

where V),\ is a complete orthonormal basis in L2 (S) consisting of eigenspinors of DA.

By the same method we can define other spectral functions of the operator DA. Particularly important here is the so-called n-function. To define it, suppose that the kernel of DA is trivial. Then zero has positive distance to the spectrum o-(DA). Let z E C be a complex number with positive real part, Re(z) > 0. The function sgn(.A)

1 I

4. Analytical Properties of Dirac Operators

106

is bounded on the set a(DA). The corresponding operator

T(z) =

Ja

sgn

1

dF(A)

lAlz

is bounded in L2(S). The superposition DA o T(z) is given by the function dim(M) sgn(A)IT x , which is bounded if k < Re(z). Now assume Re(z) > 2 dim(M Then the operators DA oT(z), ..., Dkk o Choose k with Re(z) > k > T(z) are bounded in L2. Hence T(z) maps the space L2(S) into the space

Hk(S). The embedding Hk(S) -3 L2(S) is a Hilbert-Schmidt operator. Eventually, we obtain the

I

Mn) and ker(DA) = 0, then the operator Proposition. If Re(z) > 2 dim(fsgn()dF(A)

AI

T(z) is a Hilbert-Schmidt operator in L2(S).

The Hilbert-Schmidt norm/ is now computed by '\I-2Re(z)

IIT(z)IH-S = AEa

Define the so-called 77-function of the Dirac operator DA as

77DA(z) _ E sgn (A)IA

.

00AEa

Then we obtain the following result.

Proposition. Suppose that ker(DA) = 0. Then the function r7DA(z) is analytic in the half-plane Re(z) > dim(Mn). Remark. 77DA (z) has a meromorphic continuation onto the complex plane and is, in particular, analytic at the point z = 0. The 77-invariant of DA is then defined to be 77DA(0).

We also want to discuss the boundary case z = dim(Mn). Since T. r n/2 DA/2 o = J I,\In/Z dF(A) = f(sgn())f12dF(A), a

a

the operator

T* _

f a

IAIn/ZdF(A)

Hn/2 _._, L2 maps the space L2(S) into H,/2 (S) and DA/2 o T* : L2 is invertible, (DA'2 o T*)2 = IdL2. The kernel of DA is trivial and hence .

4.3. Dirac operators are Fredholm operators DA/2

107

H,12(S) --+ L2(S) is bijective, since the index of DA is equal to zero (compare the next section). Thus T* : L2 -* Hn/2 is bijective and its Hilbert-Schmidt norm as an operator in L2 coincides with the HilbertSchmidt norm of the embedding Hn/2 -* L2. The latter is infinite, and we obtain the :

Proposition. The 71-function 77DA(z) of a Dirac operator has a singularity at the point z = dim(Mn).

4.3. Dirac operators are Fredholm operators Proposition. The Dirac operator DA : H'(S) -> L2(S) over a compact Riemannian manifold is a Fredholm operator of index zero.

Proof. We have to prove that ker(DA) and L2/im(DA) are finite-dimensional vector spaces of equal dimension. In the vector space ker(DA) consider the balls

K1 = {z E H'(S) : DA (V)) = 0,

II0IIH1 = IIIIL2 + IDAbIIL2 < 1},

K° = {b E H'(S) : DA(b) = 0, II0IIL2 < 1}. Obviously, K° = K1. On the other hand, H1(S) -> L2(S) is a compact operator and hence K° = K' C L2(S) is compact. Considering ker(DA) now as a subspace of L2, the balls are compact in the L2-norm. Hence ker(DA) is a finite-dimensional vector space. We determine the orthogonal

complement of DA(Hl) in L2. A spinor cp from this space satisfies the condition (DA (0), co) L2 = 0

for all 0 E H'(S). Similarly to the proof of the fact that the residual spectrum or (DA) of DA in L2 is empty, we conclude first the smoothness of cp and then DA(W) = 0. Thus,

(DA(H1))1 = ker(DA)

and, last, to complete the proof it remains to be shown that DA(H') C L2 is a closed subspace. Assume that the sequence DA(On) converges to E L2 (S) in L2. Without loss of generality we can suppose that 0, is orthogonal to the kernel ker(DA). But then, IIDA(0n)II L2 >_ C'IIjjJL2

(compare the corollary in Section 4.2) and V),, is a Cauchy sequence in L2. Inequality (*) implies that 0n is a Cauchy sequence in H1(S) as well. Thus the limitn-+oo lim 0n = exists in H1(S). The operator DA : H'(S) --+ L2(S) is continuous, and 2b

noc DA( n) = DA(n

4

n) = DA(w*)

4. Analytical Properties of Dirac Operators

108

for 0* E H'(S). This means that 0 belongs to the image DA(H'). In the case of a manifold M2k of even dimension, n = 2k, consider the Dirac operators

DA : r(s:':) -} r(S ). They are Fredholm operators DA : H1(S±) - L2(S:F).

The index of DA will be denoted by Index(DA). Since DA is a self-adjoint operator, Index (DA) = dim ker(DA) - dim ker(DA ). The index of DA depends on the characteristic classes of the manifold M2k as well as on the first Chern class of the determinant bundle L of the spinC structure. We quote the corresponding formula without proof. The power series of the even function

_

t/2

t et/2 - e-t/2

sinh (t/2) can be represented in the form t 1 + A2t2 + A4 t4 + .... et/2 - e-t/2 = An easy calculation shows, e.g., that 1

A2

24,

A4

10 4.24

5760

Denote the Pontrjagin classes of a 4k-dimensional compact manifold M4k by pl, p2, ..., A. The class pj, 1 < j < k, is an element of the 4j-th cohomology group Hi (M4k). Introduce k formal variables x1, ..., xk and represent P1, ..., Pk as the elementary symmetric functions in the squares of these variables:

xi+...+xk=p1 k

Then 2rj

=i 2

X -.i/2 is a symmetric power series in the variables xi, ...

,

x

and hence defines a polynomial in the Pontrjagin classes. Denote this cohomology class by A(M41,): k

,A(M4k) = 1-f

x2/2

sinh (xi/2)

For example, for k = 1, 2 we obtain the formulas

A(M4)=1-pi, 4

k

=1,

4.3. Dirac operators are Redholm operators

A(Mg) = 1

- 24pi + 5760pl

109

k = 2. A manifold of dimension 4k + 2 also has k Pontrjagin classes, and we define A(M4k+2) by the same formulas. The index theorem for Dirac operators now reads as follows: 1740p2,

Proposition. Let (M2k, g) be a compact oriented Riemannian manifold with spin structure, and let c = ci(L) denote the first Chern class of the determinant bundle of the spin structure. The index of the Dirac operator D+ associated with a connection A in the U(1) -bundle of the spin structure is equal to Index (DA) =

e2°A(M2k).

J

M2k

As a consequence of this one can prove that certain characteristic numbers are integers.

Corollary. Let M2k be an oriented compact smooth manifold and take c E H2(M2k; Z) to be a cohomology class whose Z2-reduction coincides with the second Stiefel- Whitney class of M2k, c = w2(M2k) mod 2. Then

f e2°A(M2k) M2k

is an integer. As an example, we discuss the case of a 4-dimensional manifold M4 in greater

detail. The intersection form of M4 is defined by the cup-product in H2: H2 (M4; Z) x H2 (M4; Z) --+ Z,

(a, Q) -* (a U'3) [M4] .

Considering this quadratic form over the ring Z of integers as a real quadratic form, the resulting form has a signature (p, q). By Poincare duality we have p + q = dim H2 (M4; R). The number 0-(M4) = p - q is called the signature

of the 4-dimensional manifold M4. This signature is closely related to the Pontrjagin numbers and, by the Hirzebruch signature theorem, o (M4) = 3

f

pi.

M4

The A(M4)-class can thus be written as A(M4) = 1 - $a, and the polynomial e 2 A = (1 + c + c2) (1 ! a) yields the formula 2

$

-

Index (DA) _ $ (c2 - Q).

As an application of the formula just stated we prove the following result, originally due to Rokhlin.

110

4. Analytical Properties of Dirac Operators

Proposition. Let M4 be a smooth compact orientable 4-dimensional manifold with spin structure, w2(M4) = 0. Then the signature o(M4) is divisible by 16.

Proof. Since W2 = 0, the manifold M4 has a Spin (4)-structure. Consider the corresponding Dirac operator in the spinor bundle S. Then, 810- (M4).

Index (D+)

But S is a vector bundle associated with the group Spin(4). From the considerations in Section 1.7 we know that the spin representations 04 have equivariant quaternionic structures. These in turn induce parallel quaternionic structures in the spinor bundles S±, and hence the complex vector spaces ker(D+), ker(D-) also carry quaternionic structures. Thus, dime ker(D::) 0 mod 2, which immediately implies Index (D+) = 0 mod 2. This means that a(M4) O mod 16.

Remark. The condition w2(M4) = 0 is equivalent to the integral intersection form in H2 (M4; Z) being an even Z-form, Va E H2(M4; Z). a2 = 0 mod 2 However, it is a well-known algebraic fact that the signature o- of even forms

over the ring Z is divisible by 8. The Rokhlin theorem thus states an additional divisibility of the signature by 2 in the case of smooth manifolds with even intersection form! The smoothness of M4 is indeed necessary. There exist simply connected topological manifolds Mtp with w2(Mt p) = 0 and o(Mtop = 8. Remark. The parallel quaternionic structures in the spinor bundle S± used in the proof of the Rokhlin theorem exist in all dimensions n = 8k + 4 4 mod 8, if St is associated with the group Spin (trivial spinC structure). Thus we have e.g.: Let M8k+4 be an orientable compact smooth spin manifold. Then, Msk+4

is an integer.

Apart from results concerning the integrality of special characteristic numbers, a further application of the index formula for Dirac operators consists in deriving topological obstructions for the existence of Riemannian metrics with positive scalar curvature. The Schrodinger-Lichnerowicz formula, 1

1

DA=DA+4 R+ 1dA,

4.4. References and Exercises

111

immediately implies ker(DA) = 0 and hence Index(DA) = 0, if all eigenvalues of the self-adjoint endomorphism R + dA : S -- S are positive. Thus a 4 we have the

Proposition. Let (M2k, g) be a compact oriented Riemannian manifold with spine structure, and denote by c = cl(L) the first Chern class of the determinant bundle L of the spine structure. If L has an Hermitian connection A such that all eigenvalues of the endomorphism

4R+1dA=4R+21lA:S-+S are positive, then

f e,

A(M2k) = 0.

M2k

Corollary. Let M4k be a compact oriented spin manifold (w2(M4k) = 0), and assume that M4k

Then M4k admits no Riemannian metric of positive scalar curvature.

Example. In the last corollary the assumption W2 (M4k) = 0 is necessary. The complex-projective plane M4 = C]P2 with the Fubini-Study metric has a Riemannian metric of positive scalar curvature, and

A(ci2)8o-(Cp2)_-8#0. However, C p2 is not a spin manifold.

4.4. References and Exercises M.F. Atiyah, V.K. Patodi, I.M. Singer. Spectral asymmetry and Riemannian geometry Part I, Math. Proc. Cambridge Phil. Soc. 77 (1975) 43-69; Part II, ibid. 78 (1975), 405-432; Part III, ibid. 79 (1976), 71-79.

M.F. Atiyah, I.M. Singer. The index of elliptic operators III, Ann. of Math. 87 (1968), 546-604. M.H. Freedman. The topology of four-dimensional manifolds, Journ. Diff. Geom. 17 (1982), 357-453.

P.B. Gilkey. Invariance theory, the heat equation and the Atiyah-Singer index theorem, Publish or Perish 1984. K. Maurin. Methods of Hilbert spaces, PWN, Warsaw, 1965. K.H. Mayer. Elliptische Differentialoperatoren and Ganzzahligkeitssatze fur charakteristische Klassen, Topology 4 (1965), 295-313.

4. Analytical Properties of Dirac Operators

112

J. Wolf. Essential self-adjointness for the Dirac operator and its square, Indiana Univ. Math. J. 22 (1972/73), 611-640.

Exercise 1. Let (M4, g) be a compact Riemannian manifold with spine structure and DA the Dirac operator. Consider the heat equation,

m) = -DA

t > 0,

m),

with the Cauchy initial condition 24'(0, m) = 00 (m). Prove:

1) This Cauchy problem has at most one solution. 2) If St : L2(S) --> L2(S) is defined by

St = fe_tA2dF(A), a

then 0(t, m) = St(0o(m)) is the unique solution to the Cauchy problem.

Exercise 2. Let 0a, A E v(DA)), be a complete basis of L2(S) consisting of eigenspinors of the Dirac operator. Prove that the section E(t, m1, m2) in the bundle S ® S over M' x M" defined by e_t.\20a(ml)

E(t, m1, m2) =

(9 b (m2)

AEa

for t > 0 is smooth. Show, in addition, the following properties: 1) Ft

(t, m1, m2) = -D22A (E(t, m1, m2)),

2) 0(m) = lira f E(t, m1i m2)04'(m2)dm2 for ?G E L2(S). t-}OMn

Exercise 3. The operator St : L2(S) --> L2(S) is an integral operator with kernel E(t, m1, m2), i.e. AV)) (M) _

f

'I'/

Mn

Exercise 4. Let 0 < A <

be the eigenvalues of D. Prove the

following asymptotic formula for k --> oc: N Ck2/n. k

00

Hint: E

converges for Re(z) > n/2 and has a singularity at the point

z = n/2. Exercise 5. Let D be the Dirac operator of a compact Riemannian spin manifold (Ma, g). Prove that in the cases n # 3,7 mod 8 the 77-function of D vanishes identically.

Chapter 5

Eigenvalue Estimates for the Dirac Operator and Twistor Spinors

5.1. Lower estimates for the eigenvalues of the Dirac operator In this chapter we will consider a compact Riemannian manifold (M', g) with fixed spin structure and its Dirac operator D which, in this case, is exclusively determined by the Levi-Civita connection. By integration, from the Schrodinger-Lichnerowicz formula,

D2=0+41R, 4R0 for every eigenvalue A of we immediately obtain the inequality A2 > the Dirac operator, where Ro = min{R(m) : m E M' j is the minimum of the scalar curvature. However, this estimate is not optimal. We have (Th.

Friedrich, 1980)

Proposition. Let (M n, g) be a compact Riemannian manifold with spin structure, and A an eigenvalue of the Dirac operator D. Then, A2

>

1 n Ro. 4n-1

113

114

5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors

Moreover, if A = ±a nn1 Ro is an eigenvalue of the Dirac operator and b is a solution of the field equation

a corresponding eigenspinor, then

=T_

VX

2

n(nRo-

1)X-0

and the scalar curvature R is constant.

Proof. The idea of the proof is based on not using the Levi-Civita connection but, instead, considering a suitably modified covariant derivative in the spinor bundle. To this end, fix a real-valued function f : Mn -+ R1 and introduce the covariant derivative Vf in the spinor bundle S by the formula The algebraic properties of Clifford multiplication imply that Vf is a metric covariant derivative in the spinor bundle S: ,/

X (V,, Y'1) = (V

Let Of i=1

,''b1) + ('0, Vf '01).

V Ve - > div(ei)V be the corresponding Laplace operi=1

ator, and denote by

n

of,012

n

I2IoejO+ fei .012

IV i=1

i=1

the length of the 1-form Vf 0. We will compute the operator (D - f )2. First,

(D- f)2 = (D - f)(D - f) = D 2 - 2fD - grad(f) + f2, and the Schrodinger-Lichnerowicz formula implies

(D-f)2 =0+

1

R - 2f D - grad(f) + f2.

On the other hand,

Of

n

n

i=1

i=1

= - E(oej+fei) (oe + fei) -

div(ei) (Vex + fei)

0-2fD-grad(f)+nf2. Summing up, this yields

(D- f)2=Of+4R+(1-n)f2, and, by integration over Mn, we obtain the formula

f

Mn

((D-f)2gp,V))= f {Iof

Mn

012+4RIOI2+(1-n)f2IV12}

5.1. Lower estimates for the eigenvalues of the Dirac operator

115

Suppose now that DV = AV). Then we can insert the function f = n into the last formula and obtain A2

(n_

11011L2

n 1)

= 11V nI

IL2 + \21

n

1

I ICI IL2 +

4

f

RIV)I2.

Mn

An algebraic transformation yields

.2n

n2n

IL2 = Ilon 12 +

I

4

f

RI

12 >_

Mn

\2 >

n

1

Discussing the boundary case in this estimate, we immediately obtain the remaining assertions of the proposition. i.e.

- 4n-1 Ro.

The method of proof applied here may be refined in various ways. Consider,

for example, for a fixed smooth real-valued function f

:

M'z -+ R1 the

(non-metric) covariant derivative

txv = Ox + AX n

V) +,aX grad(f)

z/) + vdf (X)V)

V= - n and perform a 1 n -1' n 1' calculation with the length I I eµf V bI 1L22 similar to the one in the proof above. Then one obtains the inequality (0. Hijazi, 1986) with the "optimal" parameters µ = -

-

Proposition. A2

>

grad f12}. - n-2l n min{1R+0(f) 4 n-1 n-1

In particular, in dimension 2 the summand (grad f 12 drops out. Then the formula simplifies to

,\2 > min{1R+20(f)}. The Gauf3 curvature K of the Riemann surface (M2, 9) is equal to K = 2R, and we can choose f as a solution to the differential equation

20(f) = -K + Thus

vol(M2 g) (M2))

2

R + 2A(f) = Vol(M ol(M

)

fK=

-K +

M2

is constant, and we obtain

A2 > 2irX (M2)

- vol(M2)

2vo(M2))

116

5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors

Of course, the last inequality is interesting only for 2-dimensional Riemannian manifolds which, topologically, are spheres. Summarizing, we obtain the following proposition, originally due to Lott, Hijazi, and Bar.

Proposition. If (S2, g) is a Riemannian metric on S2, then, for the first eigenvalue of the Dirac operator, we have 47r A2

vol(S2, g)

The method we have outlined for estimating the eigenvalues of the Dirac operator may be refined even further when the Riemannian manifold carries additional geometric structures. Let us consider e.g. the case of a Kahler manifold (M2k, J, g) with complex structure J : T(M2k) -+ T(M2k). In this situation, consider the covariant derivative depending on two parameters f and h which can be chosen freely. Elaborating on the Weitzenbock formulas for Riemannian manifolds with additional geometric structures, one will in general obtain better estimates than in the general case of a Riemannian manifold. For example, the following inequality, first proved by K.-D. Kirchberg, holds for Kahler manifolds:

Proposition. Let (M2k, J, g) be a compact Kahler spin manifold and A an eigenvalue of the Dirac operator. Then, if k = dime M is odd, 11 Ro A2 >

4

I k&1Ro

if k = dime M is even.

Remark. The quaternionic Kahler case has been investigated by Kramer, Semmelmann, and Weingart in 1997/98.

5.2. Riemannian manifolds with Killing spinors By the proposition proved in Section 5.1, a spinor field which is an eigenspinor for the eigenvalue 2 n-1 Ro solves the stronger field equation

Vxz/i==F2\/n(no This leads to the general notion of Killing spinors. Definition. A spinor field 0 defined on a Riemannian spin manifold (Mn, g) is called a Killing spinor, if there exists a complex number µ such that Ox0=µX.0

for all vectors X E T. µ itself is called the Killing number of '.

5.2. Riemannian manifolds with Killing spinors

117

We begin by listing a few elementary properties of Killing spinors.

Proposition. Let (Mn, g) be a connected Riemannian manifold. 1) A not identically vanishing Killing spinor has no zeroes. 2) Every Killing spinor ' belongs to the kernel of the twistor operator T. Moreover, b is an eigenspinor of the Dirac operator, D( ,O) = -nµ 3) If b is a Killing spinor corresponding to a real Killing number µ E R1, then the vector field n

V" =

E (ei

,

4')ei

i=1

is a Killing vector field of the Riemannian manifold (Mn, g).

Proof. A Killing spinor restricted to the curve ry(t), fi(t) _ 0(-y(t)), satisfies the following first order ordinary differential equation along this curve: dt (t) = µ Y(t) fi(t)

Now z/'(0) = 0 immediately implies 4(ry(t)) - 0, and this in turn yield property (1). Starting from Vx = µX 0, we compute n

=-nµo i=1

i=1

and thus obtain (O)

T

ei ® (VejO + 1 ei DO) _

ei ® (µeii - µei 0) = 0. n i=1 For a fixed point mo E Mn and a local orthonormal frame el,... , en with Vei(mo) = 0 we compute the covariant derivative VxV''P:

_

i=1

n

n

/ (ei

'7xV0

"/'

Vx4', w)ei + I1

i=1 n

'/1

Y', Vx )ei

i=1

(ei X

µ

(ei

-

µ(eiO, X - Y')ea

Y, 'Y)ei +

i=1

i=1

n

E((ei - X -Xei) 0,O)ei i=1

This implies g(VxVO,Y) = µ((YX-XY)

, v); hence g(VxVO,Y) is antisymmetric in X, Y. But this property characterizes Killing vector fields on a Riemannian manifold.

5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors

118

Not every Riemannian manifold allows Killing spinors 0 0 0, and not every

number µ E C occurs as a Killing number. We now derive a series of necessary conditions. To this end, recall the Weyl tensor of a Riemannian manifold. Let Rijkl = g(VeiVejek - VejVeiek - V[ei,ej]ek, el)

be the components of the curvature tensor and n

Rij = E Raija a=1

those of the Ricci tensor. Then define two new tensors K and W by 1 R

Ki. =

n-2 2(n-1) gij - Rij

,

Wa,Qya = Ra,OyS - gp5Kay - gayKQS + g,3-,Ka5 + ga5KKy.

W is called the Weyl tensor of the Riemannian manifold. Because of its symmetry properties the Weyl tensor can be considered as a bundle morphism defined on the 2-forms of (Mn, g): W : A2(Mn) -> A2(Mn),

Wijklek A el.

W(ei A ej) _ k
With these notations we have the following

Proposition. Let (Mn, g) be a connected Riemannian spin manifold with a non-trivial Killing spinor 0 for the Killing number µ. Then: 1)

µ2 =

1

1

4n(n- 1)

R at each point. In particular, the scalar curvature

of (Mn, g) is constant and µ is either real or purely imaginary. 2) (Mn, g) is an Einstein space. 3) W(w2) 0 = 0 for every 2-form w2 E A2(Mn).

Proof. VX0 = µX 0 implies VXVyO = µ(VXY) . 0 + µ2Y X. hence

(VXVy - VyVX - V [X,y])V = µ2(Y X - X. Y)b. n

Computing £ ea R(X, ea)

now yields

a=1

n

n

E ea R(X, e.),0 =µ2E ea(edX - Xea) a=1

z(i = 2(1 - n)µ2zb.

a=1

On the other hand, in Section 3.1 we proved the formula n

ea R(X, ea)O a=1

Ric(X)

V).

and

5.2. Riemannian manifolds with Killing spinors

Hence, Ric(X) 0 = 4(n - 1),a2X

,

119

and, since 0 does not vanish at any

point, this implies

Ric(X) = 4(n - 1)µ2X. Thus (Mn, g) is an Einstein space of scalar curvature R = 4n(n -1)p2. The curvature tensor R(X, Y) in the spinor bundle S is related to the curvature tensor R(X, Y) Z of the Riemannian manifold (Mn, g) via the formula n

E ea R(X, Y)ea

R(X, Y)O = 4

Hence, because 4µ2 =

n R

1

,

.

the equation

VXVy' - VyV b - V[X,y]b =,u2(YX - XY) 0 can also be written as

0=0

{Ee « R(X,Y)e«+n(R and, for an Einstein space, n

e,, AR(X,Y)ea+n( R

1)(XAY-YAX)

a=1

coincides with W (X A Y). This, eventually, implies W(w2).,0 =0.

From the proof of the preceding proposition we can also deduce the following geometrical property of manifolds with Killing spinors:

Proposition. A Riemannian spin manifold admitting a Killing spinor 'O 0 with Killing number µ 54 0 is locally irreducible.

Proof. If Mn is locally the Riemannian product Mn = Mi X M2 -k, then we may consider vectors X, Y tangent to Mi and M2 -k, respectively. This implies R(X,Y)Z = 0, and from n

Ee,,R(X,Y)ea+n(R

1)(XY-YX) 0=0

we obtain Since a 0, the scalar curvature is different from zero. Moreover, X and Y are orthogonal vectors. But this implies = 0, hence a contradiction.

120

5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors

The next-to-last proposition shows that Killing spinors are divided into two

types, depending on whether the Killing number p is real or imaginary O/t

0):

real Killing spinors

pER

M" is an Einstein space of pos. scalar curvature R > 0

imaginary Killing spinors ,u E i R M' is an Einstein space of neg. scalar curvature R < 0

Since R = 4n(n - 1)µ2, real Killing spinors precisely correspond to eigenspinors of the Dirac operator for the eigenvalue ±1 nn R. The field equation V xV) = pX 0 could be generalized by allowing µ : M' -> C to be a complex-valued function. However, due to a result by A. Lichnerowicz, this does not lead to an actual generalization: i1

Proposition. Let (M', g) be a connected spin manifold, µ : Mn -> C a smooth function and 0 a non-trivial solution of the equation

Oxb=µX - V). If the real part Re(,u) $ 0 is not identically zero, then µ is constant and real. Hence zb is a real Killing spinor.

In low dimensions n = dim (Mn), the geometrical conditions for the existence of real or imaginary Killing spinors, respectively, are rather restrictive, and for n < 4 only Riemannian spaces of constant sectional curvature admit

this kind of spinor fields. Consider e.g. the case n = 3. Then (M3, g) is necessarily a 3-dimensional Einstein space, hence a space form. We meet the same situation in dimension n = 4. Proposition. Let (M4, g) be a connected Riemannian spin manifold with a non-trivial Killing spinor 0 for the Killing number p 0. Then (M4, g) is a space of constant sectional curvature. Proof. Decompose the Killing spinor V) =,0++0- according to the splitting of the spinor bundle, S = S+ ® S-. The equation for the Killing spinor then takes the form

Vx'0+ = µX'0-,

Vxb- =

Ax'b+

Define the set

N = {m E M4 : V)+(m)

= 0 or-(m) = 0}.

5.3. The twistor equation

121

N C M4 is a closed subset without inner points. Indeed, if N has inner points, then there is an open subset U C N C Mh on which, e.g. + vanishes, 0U = 0. This implies V L 0 and, since p 0 0, from the Killing equation we obtain 0 0++r- vanishes identically on the subset U, a contradiction to the fact proved above that non-trivial Killing

spinors have no zeroes. Thus U := M4\N is a dense open subset of M4. The condition on the Weyl tensor W of M4 now takes the form W(w2)0+ = 0 and W(w2)0- = 0. However, a simple algebraic computation using the realization of the C4module A4 = A+ (D 04 explicitly given in Sections 1.3 and 1.5 proves the following fact.

If 772 E A2 (R4) is a 2-form and + E A+, V)- E 04 are two non-trivial spinors, then 172. 0+ = 0 = 772. 0- implies that the 2-form 772 is trivial, 772 = 0.

Applying this, we immediately conclude that the Weyl tensor W vanishes on the set M4\N. However, this set is dense. Hence (M4, g) is a 4-dimensional Einstein space with vanishing Weyl tensor, i.e. a space of constant sectional curvature.

In view of this proposition, the search for necessary and sufficient conditions on a Riemannian space to have (real or imaginary) Killing spinors is interesting only in dimensions n > 5. There are extensive series of examples and studies, for which we refer to the book [BFGK] and the supplementary article [Bar] using the holonomy theory. In the case of real Killing spinors (e.g. in the important dimension n = 7) this classification problem has not been finally settled yet (compare the paper [FKMS]).

5.3. The twistor equation A spinor field V) belongs to the kernel of the twistor operator T if and only if

vx'b + 1 X D(O) = 0 for all vectors X E T(M'n). Here D() denotes the application of the Dirac operator to 0 (compare Section 3.2). Solutions ' of this field equation are called twistor spinors. Killing spinors are special solutions of this twistor equation. The essential difference between the field equation for Killing spinors and the twistor equation consists in the fact that the latter is conformally invariant. By a straightforward and elementary calculation one easily proves the following

122

5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors

Proposition. Let g and g* be two conformally equivalent Riemannian metrics on a manifold Mn. Then there exists an (explicit) isomorphism ker(T) ker(T*) between the kernels of the twistor operators T and T*. Of course, the field equation for Killing spinors does not show this conformal

invariance, because the (not conformally invariant) Einstein condition is necessary for the existence of Killing spinors. On the other hand, if (Mn, g) is a Riemannian spin manifold with Killing spinors and g* a Riemannian metric conformally equivalent to g, then, in general, (Mn, g*) carries no Killing spinors although it admits twistor spinors. In the compact case, this is the only difference between these two equations. Proposition. Let (Mn, g) be a compact connected Riemannian spin manifold with ker(T) 0. Then there exists an Einstein metric g* conformally equivalent to g such that the space ker(T) .^s ker(T*)

coincides with the space of real Killing spinors on (Mn, g*).

Proof. To begin with, the solution to the Yamabe problem for compact Riemannian manifolds implies that the metric g can be replaced by a conformally equivalent metric g* of constant scalar curvature. Hence, without loss of generality, we can assume that g itself has constant scalar curvature and that ker(T) ; 0. We have to prove that, in this case, every solution of the twistor equation is representable as the sum of real Killing spinors. The equation

1XD(O)=0

vX,O +

implies

Ve.Vea0+ n E Vea(ea . D(O)) = -0(O) + nD2( )

0= a=1

a=1

Applying the Schrodinger-Lichnerowicz formula, D2 = A +

4R, we

obtain

n D ()=4n-1R 2

If the (constant) scalar curvature vanishes, R = 0, then D2(o) = 0 and, since

0=

f(D2(),) = f Iv

Mn

is a parallel section. In case R ; 0, decompose _

I2,

Mn

n

nR (P+ +

-)

into

5.3. The twistor equation

with cp f =

1

2

123

v/ nnR1 ± Do. The scalar curvature is positive for R 0 0,

since all eigenvalues of D2 are positive. Moreover,

nR1D(V)

f D2()

21/n R1(Pt

The spinor fields cp± are thus eigenspinors for the smallest possible eigenvalue V7 Ri From the proposition proved in Section 5.1 we conclude that 2 n cp f are Killing spinors.

Now we want to prove a local version of the preceding proposition. To this end, we need the

Lemma. Let V) be a twistor spinor. Then,

Vx(D( )) = 2(nn

2)

(2(nR 1)X - Ric(X))

.

Proof. Differentiating once more, we derive from Vx0 + nX D('b) = 0 the equations

DeaVx0 + 12(V e«X) D(V)) + nX . Ve«(D(b)) = 0. VxVea + n-(Vxe«) D( ) + -e« Vx(D( )) = 0. These imply

R(X, ea) + -e,,, Vx(D(o)) - nX - Vea(D(o)) = 0. Multiplying by e« and adding up yields n

ea R(X, e.) V) = Vx(D(10)) - nX D2(

) - Vx(D(

U=1

1 Rn Inserting D2( ) = 0 and E ea R(X, ea)

4n-1

a=1

_ -2Ric(X)

fib, we

immediately arrive at the asserted formula. A consequence of the formula is that the equality

Re(Vx(D ),'') = 0 holds for every twistor spinor z/). Hence we can define the following first integrals:

124

5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors

Proposition. Let 0 be a twistor spinor defined on a connected Riemannian manifold. Then the functions

C(b) = Re(z(', D ), n

Q() =10121D 12 - C2(0)

[Re (DO, ei 0)]2

Q=1

are constant.

Proof. Differentiating C(0) yields

X (C(')) = Re(Vx D'0) + Re(', V (D0)) ,

(_x

+0=0.

An analogous, if a little longer, computation using the formula of the previous lemma also shows that

X (QM) = 0. 71

Remark. The first integral C(b) was introduced by A. Lichnerowicz (1987), the invariant Q(O) by Th. Friedrich (1989).

These first integrals of a solution 0 to the twistor equation occur in the following local result which shows that, locally outside its zero set, a given twistor spinor V can always be transformed conformally into the sum of two real Killing spinors. The proof relies on using the conformal weight of the twistor operator and can be found in [F2] (compare also [BFGK]).

Proposition. Let (M', g) be a Riemannian spin manifold with non-trivial twistor spinor o, and denote by N = {m E Mn : z/(m) = 0} its zero set. Then N only consists of isolated points, and with the Riemannian metric ,,14g defined over Mn\N the latter becomes an Einstein space with scalar curvature R*

_ 4(n - 1) n

(C2(V))

+ Q(0)).

With respect to the metric g*, the spinor field

(1 )

gy(m) is the sum of two

real Killing spinors.

Finally, we note that many examples of Riemannian manifolds with solutions 0 of the twistor equation admitting zeroes are known (see Kiihnel and Rademacher). The classification of Riemannian manifolds with these solutions of the twistor equation is not yet fully understood.

5.4. Upper estimates for the eigenvalues of the Dirac operator

125

5.4. Upper estimates for the eigenvalues of the Dirac operator Estimates from above for the eigenvalues of the Dirac operator depending upon the geometry of the base space can be attained in various ways. By constructing suitable test spinors and considering the corresponding Rayleigh quotients one obtains bounds depending on the injectivity radius and curvature data of the Riemannian manifold (compare [Bal]). A further method, based on a comparison with the sphere, was suggested by Vafa and Witten in 1984 and geometrically elaborated by H. Baum (compare [Ba]). In this section, we will describe the results obtained along these lines, but refer to the original papers for the details. The starting point of this method is the observation that the spinor bundle S° of the sphere S2m C R2m+1 is trivial. Therefore, one can choose a global trivialization of this bundle over S2m by spinors V)1, ... , zp2- whose covariant derivatives V i (1 < i < 2m) are known (take, e.g. real Killing spinors on S2m). Then the space F(So) of all spinors can be identified with the space C°°(S2m; .2m) = C°O(S2m; (C2") of (C2m-valued functions. For the covariant derivative in the spinor bundle the formula

Vx(u) = X(u) + (-1),2X . (u+ - U-) 2 holds, where u = u+ + u- is the decomposition of u E C, (S2,; A2m) according to the splitting A2m =

'

®D2m.

Now let f : M2m -* S2m be a smooth mapping from the compact connected Riemannian spin manifold M2m, into the sphere S2m. If S is the spinor bundle on M2" and f *(So) its pull-back by the mapping f , then there are two operators of Dirac-type in the bundle S ® f*(So). The Levi-Civita connection of the sphere S2m defines a connection Vf in the induced bundle f*(So), and hence 2m

D1 =Eei®(Vei ®1+1(9Vei) i=1

becomes a Dirac operator in S ® f *(S°). On the other hand, So, and thus f*(So) as well, has a flat connection V° defined by requiring the constant functions u E C°° (S2m, A2m) to be V°-parallel. Then f * (So) is a flat bundle, and we can consider

Doei®(Vei 0 1+1®V0

ei).

i=1

Of course, Do is unitarily equivalent to 2m copies of the Dirac operator D of the Riemannian manifold M2m. The difference of the operators, L f = D f - Do, is a self-adjoint bundle morphism in the vector bundle S ® f * (So),

126

5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors

and its L2-norm as an operator in the Hilbert space L2(S ® f *(So)) can be controlled: I ILf I

Here II df I

I

. = max{I

I

<_

2m-1Vm__I

I

I df I

I

..

: x E M2m}. Since Do = D f -L f, a perturbation

argument shows that there are at least as many eigenvalues of Do in the interval [- I ILf 11, I Lf I as the dimension dim ker(D f) of the kernel of the operator Df prescribes. But this dimension in turn can be computed using I]

I

index theory. We have

Index(D f) = 2--IA(M2,) ± deg(f ), S2m. This immediwhere deg(f) is the degree of the mapping f : M2m, ately implies the inequality I2m-lA(M2m) - deg(f) dim ker(D f) > I2m-1A(M2m) + deg(f) I + deg(fl)I.

>

21

Assuming now that the mapping degree deg(f) is greater than

... + Mk-1) + 1, where 0 < '\o < '\i < ... denote the eigenvalues of D2 over 2m-1(mo +

M2m and

mo, ml,... the dimension of the corresponding eigensubspaces, we see that the operator Do= 2m D has at least 2m (mo + ... + mk_1) +2 eigenvalues in the interval [_2m_1/

V

j

I df I I

o,

2m-1 v1m_I

Idf 11.].

Thus the Dirac operator D of the Riemannian manifold M2m, has its k-th eigenvalue still in this interval. Summarizing this argument leads to the following result.

Proposition (compare [Bau]). Let f : M2m --> S2m be a smooth mapping from the compact connected Riemannian spin manifold M2m, into the sphere. Denote by mo, ml,. . . the dimension of the spaces of eigenspinors of D2 for

the eigenvalues A (0 <'\o < A2 < ... ). If deg(f) >_ 2m-i (mo + ... + Mk-1) +

1,

then the k-th eigenvalue A2 is bounded by I) kI <_ 2m-'/I I df I

"C.

The construction of special mappings f : M2m - S21 now allows us to derive geometrical bounds on the eigenvalues of Dirac operators from this proposition. For example, one obtains the

5.5. References and Exercises

127

Corollary. Let (M2m, g) be a compact connected Riemannian spin manifold of even dimension with positive sectional curvature K. Denote by Kmax the maximum of the sectional curvature and by A the first eigenvalue of the Dirac operator. Then AI < 2m-1

Kmax 71

If M2m is a submanifold of the Euclidean space R2m+1, then, in particular,

we can choose the GauB mapping as the mapping f : Mgr" -> S. For surfaces M2 C R3 in 3-dimensional Euclidean space this leads to the estimate I)I <_ C(M2)max{Iµ(m)I : m E M2} where 1

C(M2) =

3 2

if the genus of M2 is g = 0, if the genus of M2 is g = 2 or 3, if the genus of M2 is g > 4,

and µ(m) is the greater of the two principal curvatures at the point m E M2.

To conclude this chapter we want to quote an inequality relating the first eigenvalue of the Dirac operator to the first eigenvalue of a Schrodinger operator in the case of a surface M2 C R3. The detailed discussion can be found in the cited paper. Proposition (compare [Agr/Fri]). Let M2 C 183 be a closed oriented surface with mean curvature H, and denote by D the Dirac operator on M2 corresponding to the induced spin structure. Then, '\1(D) <_ /L(0 + H2)

where al is the first eigenvalue of the Dirac operator D and µl the first eigenvalue of the Schrodinger operator A + H2 on the surface M2.

5.5. References and Exercises I. Agricola, Th. Friedrich. Upper bounds for the first eigenvalue of the Dirac operator on surfaces, Journ. Geom. Phys. 30 (1999), 1-22.

H. Baum, Th. Friedrich, R. Grunewald, I. Kath. Twistors and Killing spinors on Riemannian manifolds, Teubner 1991.

H. Baum. An upper bound for the first eigenvalue of the Dirac operator on compact spin manifolds, Math. Zeitschrift 206 (1991), 409-422.

Chr. Bar. Upper eigenvalue estimations for Dirac operators, Ann. Glob. Anal. Geom. 10 (1992), 171-177.

128

5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors

Chr. Bar. Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), 509-521.

Th. Friedrich. Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrummung, Math. Nachr. 97 (1980), 117-146.

Th. Friedrich. On the conformal relation between twistors and Killing spinors, Supplemento de Rendiconti des Circole Mathematico de Palermo, Serie II, No. 22 (1989), 59-75.

Th. Friedrich, I. Kath, A. Moroianu, U. Semmelmann. On nearly parallel G2 structures, Journ. Geom. Phys. 23 (1997), 259-286.

Th. Friedrich, E. C. Kim Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors, to appear in Journ. Geom. Phys. 0. Hijazi. A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), 151-162.

K.-D. Kirchberg. An estimation for the first eigenvalue of the Dirac operator on closed Kahler manifolds with positive scalar curvature, Ann. Glob. Anal. Geom. 4 (1986), 291-326. K.-D. Kirchberg. Twistor spinors on Kahler manifolds and the first eigenvalue of the Dirac operator, Journ. Geom. Phys. 7 (1990), 449-468. A. Lichnerowicz. Spin manifolds, Killing spinors and the universality of the Hijazi inequality, Lett. Math. Phys. 13 (1987), 331-344. Exercise 1. Let (Mn, g) be a compact connected Riemannian spin manifold and denote by K± the space of Killing spinors:

Kt= OEP(S): Vxo=f2

R

If dim(K+) + dim(K_) > 2[ /2', then (Mn, g) is isometric to the sphere Sn.

Exercise 2. Compute the space K± of Killing spinors for the sphere Sn and for Euclidean space Rn. Determine all solutions of the twistor equation for hyperbolic space H n.

Exercise 3. 118?3 has two spin structures. Each of the numbers ±2 2 is an eigenvalue of the Dirac operator associated with exactly one of these spin structures.

Exercise 4. Let (Mn, g) be a Riemannian spin manifold and Nn-1 C Mn an umbilic submanifold. Prove that the restriction of a Killing spinor on Mn to the submanifold Nn-1 is a solution of the twistor equation on N'

Appendix A

Seiberg-Witten Invariants

A.1. On the topology of 4-dimensional manifolds The topology of manifolds looks back at a long history which began in the last century (Riemann). In the works of many mathematicians (Poincare, Brouwer, Hopf, Morse etc.) during a first period lasting until the middle of the thirties of this century homological properties of manifolds were studied, the calculus of variations was developed and, in particular, complete proofs were given for the classification of compact 2-dimensional manifolds. The characteristic features in the topology of manifolds between 1935 and 1960 were the theory of characteristic classes (Whitney, Pontryagin), the calculation of the bordism ring (Thom) and the discovery of exotic differential structures (Milnor). In particular, it became obvious that in dimensions n > 4 the category Top(n) of n-dimensional topological manifolds does not coincide with the category Diff (n) of smooth manifolds, i.e. the natural mapping Diff (n) --j Top(n)

forgetting the differential structure is neither injective nor surjective. In connection with the solution of the Poincare conjecture in dimensions n > 5 and the proof of the h-cobordism theorem (Smale), the so-called surgery techniques were developed in the sixties (Wall, Browder, Novikov). They led to a far-reaching classification theory for certain classes of smooth compact manifolds in dimensions n > 5.

129

A. Seiberg-Witten Invariants

130

The situation in low dimensions, n = 3, 4, is rather special. On the one hand, the possible variety of forms is considerably larger already in dimension n = 3 than for surfaces, and the classification question is much more difficult. On the other hand, even in a 4-dimensional manifold there is

not enough room to apply the surgery techniques which were so successful

in higher dimensions. Dimension n = 3 turned out to be the last one in which Top and Diff coincide: every compact 3-dimensional manifold is triangulizable, every two triangulizations are combinatorially equivalent and, moreover, every such manifold admits exactly one differential structure. The Poincare conjecture in this dimension is still unsettled. Apart from the fundamental group 7rl (M4), the integral intersection form H2 (M4; Z) is the most important invariant of an orientable compact 4-dimensional manifold. In 1982, M. Freedman proved that for simply connected compact topological 4-manifolds this intersection form almost determines the manifold itself in

Top(4). In particular, each unimodular quadratic form over the ring Z of integers can be realized as the intersection form of a compact simply connected topological manifold M4. On the other hand, it was already known for a long time (Rokhlin 1952) that if a unimodular form of even type can be realized by a closed smooth 4-manifold, then its signature is divisible by 16. Take, e.g., the positive definite unimodular Z-form E8 of dimension 8, 2

-1

0

0

0

0

0

-1

2

-1

0

0

0

0

2

-1

0

0

0

0

0

-1

2

-1

0

0

0 0 0 0

0

0

0

-1

2

-1

0

-1

0

0

0

0

-1

2

-1

0

0

0

0

0

0

-1

2

0

0

0

0

0

-1

0

0

2

0 -1

E8 =

E8 is of even type and has signature 8, o- (E8) = dim E8 = 8. Hence there exists a simply connected topological manifold Mo with intersection form E8, and Mo is by no means smooth, i.e. the mapping Diff(4) -> Top(4)

is not surjective (and not injective as well). Thus the smooth topology in dimension 4 is already a completely different topic than the continuous.

Likewise, at the beginning of the eighties, S.K. Donaldson introduced a method for the study of Diff(4) based on associating with every smooth 4-dimensional manifold the moduli space of solutions of the self-dual YangMills equation in a non-abelian gauge field theory as an invariant, and on deriving new invariants from it. In this way, he was able to exclude further unimodular quadratic forms over the ring Z as intersection forms of smooth

A.1. On the topology of 4-dimensional manifolds

131

simply connected and closed 4-manifolds M4. For example, if H2 (M4; Z) is positive definite, then this intersection form has to be trivial. In particular, E8 ® E8 cannot occur as the intersection form of a smooth manifold even though the Rokhlin condition o-116 E Z is satisfied. This obstruction eventually led to the proof that there exist exotic differential structures in R4. On the other hand, using known algebraic surfaces, many unimodular Z-forms can be realized as intersection forms. Computations for connected sums of K3-surfaces with (S2 X S2) resulted in the conjecture that the form

(2k(-E8) ®m

0

0 )J

does occur as the intersection form of a smooth 4-manifold if and only if m > 3k (the so-called 11/8 conjecture; this inequality is equivalent to b2(M4)/1o(M4)1 > 11). Within the framework of Donaldson theory, this

a

inequality could only be proved for small k. Another application of the invariants constructed by means of non-abelian gauge field theory concerns splitting questions. A compact simply connected complex surface S with b2 (S) > 3 cannot be smoothly represented as a connected sum S = Xi#X2 with b2 (Xi) > 0. This led to the construction of different differential structures on compact simply connected 4-manifolds.

In autumn 1994, E. Witten suggested that all these results, and others reaching even further, could be obtained by considering the moduli space of a system of equations for a pair consisting of a spinor and an abelian con-

nection. The spinor has to be harmonic with respect to the abelian gauge field and, on the other hand, algebraically related to the curvature form of the abelian connection (Seiberg-Witten equation). This system of equations is the 4-dimensional analogue to the 2-dimensional Ginzburg-Landau model (1950) of superconductivity. Witten's claim was elaborated by many mathematicians in the following months and turned out to be right. In contrast to non-abelian gauge field theories with their non-linear equations, one could now return to an abelian theory, and the analytical theory of smooth 4-dimensional topology gets drastically simplified !

The first observation forming the base of Seiberg-Witten theory is that each orientable compact 4-dimensional manifold M4 has a spinC structure (though, possibly, no spin structure!). Hence spinors may be defined on it. We briefly sketch a proof: The universal coefficient theorem implies the formula

H3(M4; Z) = {H3 (M4; Z)/Tor(H3(M4; Z))} ® Tor(H2(M4; 7L)),

A. Seiberg-Witten Invariants

132

and, from Poincare duality, H2 (M4; Z) = H2(M4; Z), we conclude the relations Tor (H3 (M4; Z)) = Tor (H2 (M4; Z)) = Tor (H2 (M4; Z)).

Let T C H2 (M4; Z) be the torsion subgroup. Consider the exact sequence H2 (M4, Z)

H2 (M4, 7G) - H2 (M4, Z2)

H3 (M4, Z) ,...

H3 (M4, Z)

.

Then, im(/3*)

= {a3 E Tor(H3(M4; Z)) : 2ca3 = O}

{y2 E T : 2y2 = 0}.

2

The sequence {y2 E T : 2ry2 = 0} --+ T -4 T -+ T/2T is an exact sequence of 7L2-vector spaces. Thus, dimz2 (T/2T) = dimz2 {y2 E T : 2-y2 = 0} and,

since r(T) = T/2T, we obtain dimz2 (H2(M4; Z2)) = dimz2 (im(r)) + dimz2 ('m(8 )) = dimz2 (im(r)) + dimz2 (r(T)). The following inclusion is obvious:

r(T) C im(r) c H2(M4; Z2).

Consider x E im(r) with x = r(a) and a E H2(M4; Z). If y E r(T), then there exists an element /3 E T C H2(M4; Z) with r(6) = y. 0 is torsion and hence a U (3 in H4 (M4; Z) Z vanishes. This in turn implies

xUy=O for xEim(r),yEr(T). The set IT = {y E H2(M4; Z) : V y E r(T) y U y = 0} thus contains im(r). On the other hand, dimz2 M= dimz2 (H2(M4; Z2)) - dimz2 (r(T)) = dimz2 (im(r))

by Z2-Poincare duality. Hence, im(r) = r, i.e.

im(r) ={-yEH2(M4;Z):V y E r(T) yUy=O}. So we arrive at a more precise description of the image of the Z2-reduction r : H2 (M4; Z) --f H2 (M4; Z2) . We are going to use this to prove that

M4 has a spinC structure. The necessary and sufficient condition to be considered is that the second Stiefel-Whitney class w2 belongs to the image

Im(r). We thus have to check whether w2 U y = 0 for all y E r(T). By the Wu formulas for an orientable 4-dimensional manifold, w2 is the only cohomology class w2 E H2(M4; Z2) satisfying the condition x2 = W2 U x for all x E H2(M4;Z2). Now for y E r(T) we have y2 = 0, since y is the Z2-reduction of a torsion element from H2(M4; Z). This implies

W2Uy=y2=0

A.1. On the topology of 4-dimensional manifolds

133

for all y E r(T), i.e. the second Stiefel-Whitney class w2(M4) is the Z2reduction of an integral cohomology class. Altogether we obtain the

Proposition (Wu 1950; Hirzebruch and Hopf 1958). Every compact orientable 4-dimensional manifold M4 has a spinc(4) structure. Next we will collect a few special formulas from 4-dimensional spin algebra. The Hodge operator * : A' (R4) - A2 (1184) acting on 2-forms splits A2 into the self-dual and the anti-self-dual 2-forms: A2(I[84)

= A2 (II84) ® A2 (184).

The 4-dimensional spin representation A4 also decomposes into A4 = Al A4 with 04 C2. The endomorphisms ej ej : 04 04 induced by Clifford multiplication have the following matrices: ele2 e2e3

_ (i -i0 0 _

e1e3

i

0

i

0

1

= (1

0

0

e2e4

0

)'

i

1

= -1 0)

'

i

0

0)

ele4 = (-i e3e4 = (0

0

i

In particular, as endomorphisms in 04 eie2 - e3e4 = 0,

ele4 - e2e3 = 0.

ele3 + e2e4 = 0,

For a spinor c E 04 we define a 2-form w" by the formula W,11

(X,

y) =

CD)

(X . Y

+ (X,

Y)14)12,

where X, Y E 1184. wD is a 2-form with imaginary values. This results from the following calculations: w111(X,Y)

=

(X'Y'.D'.j))+(X,Y)I(DI2

= ((-YX - 2(X,Y))-D, (D) + (X,Y)1j)12

= -(YXc,(D) (X, y)

= (X.Y.

,')+(X,Y)I(D I2=(W,X.Y.

)+(X,Y)Iq I2

(Y X', (D) + (X, y)1.1) 12 = w'' (Y, X) = -w' (X, Y).

Using the explicitly described spin representation, we easily obtain the proof of the next

Proposition. 1) Let

E O4 and w2 E A2 . Then, w2

2) (&D ' , (D) _ -21-D 14 and 1w'

2

=

(D

21'14.

= 0.

A. Seiberg-Witten Invariants

134

E C2

Proof. Setting

= 04 yields

2

^el, e2) =

i{I.,pl12

- I'1)2I2} = 0'(e3, 64),

^ei, e3) = --b2C + '1)1'52 = -W = -iDAi

w'D(ei, e4)

ID

(e2, e4),

wD(e2i e3),

and the formulas simply result from inserting these entities.

A.2. The Seiberg-Witten equation Let M4 be an oriented (compact) 4-dimensional manifold. Fix a Spine (4)structure and a connection A E C(P) in the U(1)-principal bundle associated with the spinC structure. For -D E F(S+), consider the equations (SeibergWitten equation): DA (D =0,

Q+y4

To understand the Seiberg-Witten equation we compute, for an arbitrary section E F(S+) and any connection A E C(P), the integral 2

+IDA-1)

J

I2.

M4

Since QA and w' are forms with purely imaginary values, calculating their length amounts to 2

2

[QA+(ei,

i<j

I

14T -21: QA(ei,ej)(eiej-D,4))

= Icrj2+ IQAI2

ej) + 4w"' (ei, ej)

iw

i<j

1IW D12

+

QA I2 + 8

4

-

- 2 (Q++

On the other hand, by the Schrodinger-Lichnerowicz formula we have

f M4

ID9.1)12

=

f

I(DI2

+ (Q1), P)]

M4

This implies

f

M4

f [Ic2

M4

+

+

I2+8I,DI4].

4

A.2. The Seiberg-Witten equation

135

Hence the functional E(4), A) = fM4 IIQ+12 +

1 I(DI4]

4 solutions of the Seibergis non-negative and its zeroes are precisely the Witten equation. Proposition. Let (1), A) be a solution of DA = 0, SZA = --41w4' over a compact Riemannian manifold (M4, g) with scalar curvature R. Then, at each point, where

14)(x)12 < -Rmin,

Rmin = min{R(m) : m E M4}.

Proof. At a point x where 14) (x) 12 attains its maximum we have 0 < L I (p 12. However,

o < AI4)I2 = 2((VA)*vA(D, 4)) -2 (VA(p,

} -RITZ -

2 f (- 4 (1), -1)) - 2 (QA -1), 4))

_RIcl;I2 2

If now I(D max >

0,

+

then 0 <

VA.D) < 2((VA)*VA(D, (D) (D)

=

(4(D) = -RI.1) I2 _ 2

1 I.,D I4.

2

-R - 1/2I m

and

-Rmin

Let P - M be a U(1)-principal bundle and A : TP -. E51 = iR1 a connection. A gauge transformation on P is given by a function f : Mn --> S1 acting via f (p) = p f (7r(p)). Let O = zx = zdz be the 1-form O : TS' i1181

Then the action of the gauge group C; (P) = Map (M', S1) on C(P) is described by

f*(A) =A+7r*f*(O) and the curvature forms of A and f *(A), respectively, are QA=dA, f2f*(A)=f2A.

(L) = Map (M4, SI) in greater detail. Consider f : M4 --> S1. Then f acts on C(P) by We will describe the action of the gauge group !; (P)

f*(A)=A+7r*f

.

For connections in the fibre product R x P of the frame bundle with the U(1)-bundle of the spinC structure we thus obtain

LC®A

(A+rr*).

Here LC denotes the Levi-Civita connection of R. Hence, LC ® (A+7r*

)

LC ® A is a 1-form on R x P with values in 81 vanishing on TR. Lifting

A. Seiberg-Witten Invariants

136

both connections to the SpinC(4)-structure, we obtain, for the covariant derivatives in S,

Vt *(A)4) - Ot = df

(1)

f

.

This implies that for the Dirac operators DA, D f*(A) : F(S) -* F(S) the following formula holds:

Df*(A)(D - DAB= fgrad(f).

As the group !;(P) _ { f : M4 -* S'} acts on F(S) x C(P) by f ((D, A) _ (.1

- 4), f* (A)), we have

Df*(A) ('Dlf) = DA(`I'/f) + f grad(f) . (elf) =

f2 grad(f) (D + f2 grad(f) (D = fDA("D). -

This implies that if (,D, A) is a solution of the Seiberg-Witten equation, so

is f

((D, A)

_

(1'lf,f*A).

Now we turn to the definition of the moduli space for Seiberg-Witten theory. If M4 is a compact oriented Riemannian spinC manifold, P the U(1)-bundle of the spinCrstructure and L the line bundle, then define

ML = { (4),A) E F(S+) x C (P) :

DAI) = 0,

ci =

1 JJJ

Proposition. J)tL is compact.

Proof. Let F(L) = F = {w2 E A2(M4) : and let

dw2

[w2]DR = cl(L)},

be the only harmonic form in F(L). Since the curvature form E C(P) is gauge invariant, we obtain a mapping

wharm

!QA of A

P:fitL-NF(L), First Step. on F(L)). Proof.

= 0,

P[A,(D]=SZA.

P(J)tL) C F(L) is a compact subset (relative to the L2-topology

Suppose that DA) = 0 and Q++

Write the curvature form

QA as

QA = (1lA)+ + (IA) = Warm + dij1.

A.2. The Seiberg-Witten equation

137

Denote by x1 the harmonic 1-forms and im(d°) = im(d : A° --+ A'). Then we can choose 771 to be orthogonal to im(d°) E )'HI r(Al). From oI,D I2

= 2((VA)*VA(D, (b)-2(VAI),VAS) =

VA(D)

and f A I I2 = 0 we obtain M4 2IIVA4)IIL2

f (_2_1/24)

=

M4

<

f (-R/2) I4)I2 < {R<0}

f

(-R/2. (-Rmin))

Cl (R).

R
Hence there is an L2-bound

< Cl. Moreover,

IIVA(DIIL2

SS2A = *d* (cA+*clA) = *d*cA, b,QA = *d * (QA - *QA = *d * QA, Now we compute 5w ':

i.e. 811A = 2SSZA =

28(wd))

= SfA(t =

)- t

2 ((', VA(D)

This implies I ISQAI

and, by the C°-bound for

I

12

I

< C2 I

p1IL2IIVA.D IIL2 I

,

IIscAIIL2 <- C2 2

2 < C2. Now Since Q2 = wharm + dr71, this implies Iladii1Ili2

771

was chosen orthogonal to im(d°), i.e. d771 = 0. As A771 = 8dr71, we obtain 11 'L 771IIL2 <

C2. Since 771 1 'H1 = ker(A) and, because the spectrum of L is discrete, we have II?7 HL2 < C3II o

111L2

Thus, II"?

IIL2
IIo7711IL2
So far we have shown that the set of 1-forms 771 in question is bounded in the Sobolev space H2(Al(M4)). Since the map d : H2 (A' (M4)) -; H'(A2(M4))

is continuous, the set of all d771, i.e. the set of all 52A in the image of P : 3tL -+ F(L), is bounded in H1(A2(M4)). /Lastly, by the Rellich lemma, H1 (A2 (M4)) -> L2 (A2 (M4))

is a compact embedding. Hence P(mL) C F(L) is a compact subset.

A. Seiberg-Witten Invariants

138

Consider Pi : 'mt -3 C(P)/g(P), A] _ [A]. Then PI(ML) C C(P)/G(P) is a compact subset. Proof. This is Weyl's theorem. Namely, the mapping C(P) -* F(P), A -; QA is a fibration with compact fibre Pic(M4) = H' (M4; R)/H1(M4; Z) Tbi(M4). Since the diagram Second Step.

fitL

P,

C(P)/9(P)

F(L) commutes and P(9RL) C F(L) is compact, the same also holds for P1(! L) C

C(P)/g(P). Third Step. 9RL is compact.

Proof. For a given A E C(P)/!;(P) the pre-image Pi 1([A]) C mi. consists of the solutions of DAI)= 0,

maxj
This is a bounded ball in a finite-dimensional vector space.

A.3. The Seiberg-Witten invariant In this section, we will study the linearization of the space 9RL. The SeibergWitten equations are

DA)=0, The linearization of these equations at the point ((D, A) is an operator

P4,A) : r(s+) e r(A') --> r(s-) ® r(A+). For given W E r(S+) and 771 E r(A'), consider the variation

At=A-l-tr71,

''t = ,+tqf.

Then Q At = dA + td771, and thus, dt (S2At) lt=o = (di71)+. Moreover, from w'D(X, Y) = (XY(j), (I)) + (X, Y) I
_ (XY( at ) 1t=o,' ) +

d t_o) + (x, Y) (T, l)) + (X, Y) (t', XF)

_ (XYT, (I)) + (XY-1);'P) + (X, Y) (T' (') + (X, Y) (-I), qf)

A.3. The Seiberg-Witten invariant

139

= (XYT, (D) + (I, YxW) + (X, Y) (W, (D) + (X, Y) (4),W) = (XYT,') + (YXqf, V) + (X, Y) (W, -15) + (X, Y) ((D, W)

= (XYW, (D) + ({-XY - 2(X, Y) }W, -D) + (X, Y) (W, (P) + (x, Y) (XYW,,D) - (XYW, (D) - 2(X, Y) (`I', ) + (X, Y) (W 1)) + (X, Y) (1),W)

_ (XYW, ) - (XY, ) + (X,Y){(, ) -

def

Hence Q++ _ -4w( is linearized by (dr7')+ _ 0 linearizes to 771(D+ DAY = 0.

w `y(X,Y).

Analogously, DAB

Altogether, P( A) : r(S+) ® r(A') --4 r(s-) ® r(A+) is given by P(1(,A)(W,r)1) = (77115 +DAY,dr7++

4w

We compute the tangent space to the orbit C9(P) (4), A) at the point A) as follows: If ft : M -> Si is a smooth family of gauge transformations,

fo - 1, then

d(T,(1)) - = h) _ -h-t

d

1

dt

1 ) = dt

where h := dftit=o and ft *A - A = t Thus, .

dt Hence P(0,11,A)

(ft *A - A) t=0 = dh.

: r(A°) -> r(s+) ® r(A1) is given by

P( A) (h) = (-h4, dh). Indeed, we have P('D ,A) o P(, A) = 0. Namely, if T _ -h' and 771 = dh, then

DA R) = dh -D - DA (hl)) = dhl) - dh - 4P + hDAR) = hDA(C = 0 and dr7l = 0 as well as (since T = h4t, h is purely imaginary) w't""'(X,Y) =

-1)) + (XY){h(,D,

= h{(XY41), -1)) + (-cD, XY-15) +

h(,D, -1))}

0.

Next we want to compute the index of the elliptic complex (over the real numbers) (*)

r(A°)

P A4

r(s+) ®r(A') P

r(S-) ®r(A+),

A. Seiberg-Witten Invariants

140

Since operators of lower order than that determining the principal symbol do not contribute, the index we are looking for is equal to that of the complex

r(A°)

(°

r(S+) ®r(Al)

(DA'Pr+

r(s-) ®r(A+).

However, the latter is the sum of two complexes and, moreover, the index is additive: p

r(A')

IndexR = IndexR(r(A°)

r(A+))

r(S-)) pr+d r(A+)) - 2 Indexc (D+). = IndexR(r(A°) -L r(A') + IndexR (r (s+)

By the index formula we have

Index (DA+) = -c2 8

1(7. 8

This implies

IndexR = IndexR(r(A°)

The index of the complex r(A°) Index

(

*

d-

r(A') -> r(A+))

- 4c2 + 4a.

r(Al) ---+ r(A+) equals

1

2

1

2X+ 1

2v. Thus,

2

)=2X+2a'-4c

This calculation yields dimR ker(P(OD A)) - dimp,7-11 + dimR 7-12 = 4 (2X + 3o-) - 4 c2,

where %1, ,H2 are the first and the second cohomology of the complex (*), respectively. In conclusion, we obtain the formula dims 7-' + dimR 7-12 - dimR ker(P

4c2

- 4 (2X + 3Q).

Definition. A solution of the Seiberg-Witten equation is called reducible if 0. Then SZ+A 0. If (4), A) is irreducible, then ker(P(% A)) = 0. This implies dimR 7-ll - dimR 7-12 = 4 c2 - 4 (2X + 3Q).

Hence we have computed the virtual dimension v - dimR ML of the moduli space.

Proposition. One has v - dimR L = 4c2 -

4(2X + 3Q).

Now we want to prove the following generic vanishing theorem for the second cohomology 7-12 of the complex (*).

A.3. The Seiberg-Witten invariant

141

Proposition. Let M4 be a compact oriented manifold and c E H2(M4; Z) a Spinc(4) structure. For a generic set of metrics, g E Met (M4), the second cohomology 7-12 of the complex (*) is trivial for every irreducible solution ((D 0 0, A) of the Seiberg- Witten equation.

Proof. Fix a metric g E Met (M4) and consider only local perturbations of the metric in a neighbourhood U, g E U C Met (M4), so that we may identify the spinor bundles corresponding to different metrics. Then regard the Seiberg-Witten equation DAB = 0, (c A)+(g) _ -owl) as depending also upon the metric g E U. If gt = g + t h is a variation of the metric, then the variation of the operator is given by P(,b,A,9) :

r(s+) ® r(A') ® r(S2(T)) -- r(s) ® F(A2), p = (PS-,

pA2)

with components P(45 ,A,9) (P, 2

P(-P,A,9)

E, h)

h)

E,D + DA (D +

(dE)+(9)

dt

(D+t) jt=o

- 1 wc,W + dtd (1 A)+(9+t

h)

2

Of course, we only have to take into account variations of the metric transver-

sal to the action of the diffeomorphism group of M4, i.e. we can assume 0(h) = 0. Hence the variation of the Dirac operator involves, among othand, on the other hand, the variation of ers, the Clifford product d(tr(h)) the decomposition A2 = A+ ® A? only depends on the traceless part of the symmetric tensor h, since the Hodge operator * of a 4-dimensional manifold acting on A2 is conformally invariant. If now (Q, rl) E r(S-) er(A2) belongs to the cokernel of the operator P(D,A,g), then we deduce the conditions

(d(tr(h)) -D, IF) = 0,

(p(h), rl) - 0,

where p(h) describes the change of the *-operator. Varying h E r(S2(T)), we immediately obtain 7 0 and T - 0, since, as a solution of the elliptic does not vanish on a dense set in differential equation D9A 45 = 0 ((D $ 0), M4.

Corollary. Let M4 be a spin manifold with its canonical SpinC (4) structure

(L-O'). Then,

v-dim9te _-1+bl-b2 71

Let L -* M4 be a line bundle over M4 and cl (L) E HJR(M4; R) its Chern class in de Rham cohomology. A Riemannian metric g on M4 is called "L-nice" if there is no 2-form w2 such that

1) /w2=0;

A. Seiberg-Witten Invariants

142

2) *w2 = -w2; 3) [W2]DR = ci(L).

We first discuss under which conditions there exists a "nice" metric. The A-product defines a bilinear form A : H2 (M4; I(8) X HDR(M4; R) --> R.

For a metric g, denote by H2(g) the space of all harmonic forms. Then there is an isomorphism H2(g) HDR(M4; IIS). The Hodge operator * : H2 (g) -* H2 (g) is an involution. Hence each metric g defines an involution on de Rham cohomology,

*9 : HDR(M4; R) - HDR(M4; R), and *9 preserves the A-product. Let E+ (g) and E- (g) be the eigensubspaces for +1 of *9 . Then, dim E+ (g) = b2 .

First case. If b2 = 0 and cl (L) 0, then there exists a g-harmonic form w2 in cl (L) such that Lw2 = 0. As b2 = 0, we have *w2 = -w2. Hence, in this case, there is no nice metric for cl (L) at all. Second case. b2 = 1. In this case, HDR is a pseudo-Euclidean space with index (1, b2 ). If b2 = 0, then every metric is nice. Now suppose that b2 > 0. Then we have (HDR, A) ^ (1!862, z2 - xi xy2_1) and, for transversality reasons, it is obvious that an L-nice metric g can be chosen. However, two

-

such metrics cannot necessarily be joined, since E- (gt) has to stay in the range z2 - X12_..._ x222_1 < 0! Altogether, this implies:

If b2 = 1, b2 > 0 and c1 (L) < 0, then the space of L-nice metrics is not connected. (If c2 (L) > 0, then every metric is L-nice.) Third case. b2 > 2. It is clearx22+that any two metrics in E- (g) can be deformed within x 2 + x2 + + - yi - . yb- < 0 in such a way that 2

2

ci (L) is not met. This implies

a) if b2 > 2 and ci (L) > 0, then every metric is L-nice; b) if b2 > 2 and ci (L) < 0, then the space of L-nice metrics is connected.

Proposition. Let (M4, g) be an oriented Riemannian manifold and c a SpinC(4) structure with line bundle L. Let g be an L-nice metric. Then there are no reducible solutions of the Seiberg-Witten equation.

A.3. The Seiberg-Witten invariant

143

Proof. DA(P = 0, i i = -4w and 1) - 0 together immediately imply Q+y = 0. Hence QA is an anti-self-dual 2-form, *QA = -QA. Thus QA is also harmonic, LQA = 0, since dSlA = 0 and [Z,,QAI DR = cl(L). However, by

the assumption on the metric, there is no such form QAFor the moduli space YtL = DDtL(g) the following picture results.

First case. b2 (M4) > 2. Then, for a generic metric g on M4, we have that g is an L-nice metric, 7-12 ker((PI(15 A))*) = 0, and the space of these metrics is pathwise connected. Consequently, for such a metric, ML (g) is empty or a compact smooth manifold of dimension 1 2

dimfitL(g) = 4c

-

1 4(2X+3ov)

and the bordism class in the unoriented bordism ring N* is independent of the metric. Define the Seiberg-Witten invariant as

S - W(M4, c) _ [L(g)] E N*. Second case. b2 (M4) = 1. Case 2.1. c2 (L) > 0, b2 (M4) = 1. In this case, each metric g is L-nice and, generically, 712 ker((P(4, A))*) = 0. As above, the bordism class 9)tL (g) is uniquely determined.

c2 (L) < 0, b2 (M4) = 1. A generic metric g is L-nice and we have H2 ker((P(P A))*) = 0. Hence 9JtL(g) is empty or a compact smooth manifold, but the space of the metrics in question is not necessarily connected. A definition of Case 2.2.

S - W(M4, c) = [9XL(g)] E N* is only possible for a chosen connected component in the corresponding space of metrics (or following a different line of argument). Third case. b2 (M4) = 0. In this case, we have 712 Pz ker((Pl Al)*) = 0 for a generic metric. However, now there exist reducible solutions. If 1 0, then QA+ 0 implies that the curvature QA is the only g-harmonic form in cl (L). By the Weyl theorem, the set {(45

-0, A) :

c1(L)I /9(P)

is diffeomorphic to the Picard manifold Pic(M4) = H2(M4; IR)/H2(M4; Z) of M4. The moduli space ML (g) is empty or a compact manifold with singular set Pic(M4) = Tb1(1''14). Its "bordism class" is again unique.

A. Seiberg-Witten Invariants

144

Remark. To define the Seiberg-Witten invariant within the unoriented bordism ring by S - W(M4, c) = [k(g)] E 9t* is the simplest possibility. More refined invariants are obtained by the observations to follow.

Observation 1. On 9X L(9) there exists a universal S'-principal bundle. If m° E M4 is a fixed point, then we consider the subgroup go C 9(P) of the gauge group given by

go ={f :M4-}S': f(mo)=1}. Then i)tz,(g) _ {(', A) : DA (D = 0, Q++ _ -4w' }/Go is a principal bundle with structure group c/cro = Sl over 9 RL(g). Hence 9RL(g) has a distinguished cohomology class e E H2 (9 L (g), Z). Thus the Seiberg-Witten invariant can be understood as a pair ([TZL(g)], e) consisting of a bordism class and a cohomology element from H2. Observation 2. The tangent space T(41,A)9)ZL(g) can, for a generic metric, roughly speaking be identified with the sum

{w E r(S+) : DAT = 0} ®{ql E IF(Al) : 6771 = 0, (d?7')+ = 01. The former of these spaces is a complex vector space and hence has a canonical orientation. The second space results from the complex r(A°) r(Al) r(Al) or from the operator 6®d+ : r(Al) - r(A°) ®A(A+). Thus the determinant of the second space is given by det(b ® d+) = det ker(S ® d+) ® det coker(o ® d+) = det Hl ® det H. Now if an orientation in Hl (M4; Ilk) ® H+ (M4; R) is fixed, then ML (g) can be oriented. Considering the triple (M4, c, .O) consisting of

1) a 4-dimensional compact and oriented manifold M4, 2) a Spinc(4) structure c E H2(M4, Z), 3) an orientation Din HI (M4 ; R) (D H+2 (M4; R),

then, in the cases b2 (M4) > 2 or b2 (M4) = 1 and c2 < 0, the Seiberg-Witten invariant is defined in the oriented bordism ring S2* 0.

A.4. Vanishing theorems Consider a 4-dimensional manifold M4 with b2 > 1 and assume that M4 has a metric with positive scalar curvature, R > 0. Perturbing the metric, we can arrange it to be an L-nice metric as well. Then, ML (9) is the space of all flat connections,

9XL(g) _ { ( = 0, A) : SZA = 0} /c _ {A E C(P) : QA = 0} /q.

A.4. Vanishing theorems

145

If cl (L) E H2 (M4) is not a torsion element, then ML (g) = 0. Thus we have the

Proposition. Let M4 admit a metric of positive scalar curvature and suppose that cl (L) E H2 (M4, Z) is not a torsion element. Then, S - W (M4, c) = 0.

This holds for b2 (M4) > 2 in general. In the case b2 (M4) = 1, consider

as S - W (M4, C) the Seiberg- Witten invariant computed with respect to the component containing the metric of positive scalar curvature. Let g be a fixed Riemannian metric and ((D, A) a solution of DAB = 0,

SZA =

We already know that at each point 1.1)(x)12 < -Rmin. By the SchrodingerLichnerowicz formula, oI.CpI2

-1

=

-2

and hence I4)I4

< f(-ii .1)2 < Rminvol(M4).

M4

M4

This implies

f p-+12 6 f W-DI2 =

8

n

vol (M4).

M4

M4

M4

f Iq)I4 <- 1-R 2

As QA is a curvature in L,

f(cI2 - InA2)

ci(L) =

M4

0, this implies ci (L) - (2X + 3o) > 0, and hence

If 9RL (g)

2X + 3Q < ci(L) =

f(c2 - InAI2), M4

i.e.

f ISZAI2 < f IS2AI2 M4

Thus

f

I cl

M4

implies the

-

M4

I2 as well as f I1 I2 is bounded, if ML is not empty. This M4

A. Seiberg-Witten Invariants

146

Proposition. Let (M4, g) be a compact oriented manifold with fixed Riemannian metric g. Then there exist at most finitely many Spinc(4) structures c E H2 (M4, Z) for which fL (g) is not empty.

A.5. The case dim 9RL (g) = 0 The Seiberg-Witten invariant as an element in the bordism ring becomes a numerical invariant for dimTZL(g) = 0. If M4 is a compact oriented 4-dimensional manifold and c E H2 (M4; Z) a Spinc (4) structure with c2

- (2X + 3v) = 0,

then define the Z2-invariant nL(g) = number of points in the moduli space 9J L(g) mod 2.

In the case b2 > 2, the invariant nL(g) E Z2 is uniquely determined for a generic metric; for b2 = 1 a "component" in the space of L-nice metrics has to be fixed additionally. If, moreover, an orientation 0 in Hl (M4, IR) ® H+ (M4; R) is given, then no (g) E Z is defined as an integral invariant to be the alternating sum of the points in 9RL (g) which now carry a sign. However, this numerical Seiberg-Witten invariant does not exist for every manifold M4. Namely, the existence of a cohomology element c E H2(M4; IR) with c2 = 2X + 3v and c =_ w2(M4) mod 2 is a necessary condition. The number 2X + 3cr _ (2e + pl)[M4] has to be representable by an element c E H2(M4; ]I8). Here e E H4(M4; I[8) denotes the Euler class and pl the first Pontryagin class. In other words, the resulting necessary condition on M4 requires that the cohomology class 2e + pi E H4(M4; I[8) has to belong to the image of the quadratic intersection form H2(M4; I[8) E c -+ C2 E H4(M4; II8). Suppose that M4 has an almost-complex structure J : TM4 --+ TM4 inducing the orientation. Then TM4 is a complex 2-dimensional vector bundle over M4 having Chern classes cl E H2 (M4; Z) and C2 E H2 (M4; Z). Since the complex structure J is compatible with the orientation, the real characteristic classes w2, e and pi of M4 are related to ci and c2 via

ec2, w2C1 mod 2, p1=c -2C2. This immediately implies 2e + pi = ci. The argument can also be reversed. The following proposition holds.

Proposition (Hirzebruch and Hopf, 1958). Let M4 be a compact oriented 4-dimensional manifold. Then there exists an almost-complex structure on M4 inducing the orientation if and only if there is an element c E H2 (M4; Z) with c2 = 2e +p1.

A.6. The Kahler case

147

A.6. The Kahler case Let (M4, g, J) be a Kahler manifold (or, for the moment, just an Hermitian manifold) and denote by Pj C P(M4, g) the corresponding U(2)-principal bundle. Then M4 has the canonical Spin' (4) structure

Q = Pi x l Spin' (4) with the associated line bundle L = A2 (TM4). Consequently, for the canon-

ical Spin' (4) structure, cl (L) = CI (M4). As is well-known, the signature and the Euler characteristic of the Kahler manifold M4 are given by

a=

(C2

X=c2.

This implies

v - dimJJtL(g) = 4c2(L) - 4(2X+ 3a) =

4{c2

- (2c2 + ci - 2c2)} = 0.

Hence the virtual dimension of the moduli space 9A L (g) vanishes for a Kahler

manifold with respect to the canonical Spin' (4) structure. Remark. This computation of the virtual dimension can also be performed in a more general situation. Consider an oriented manifold M4 and an almost-complex structure J : TM4 --+ TM4, j2 = -Id. Then, det (J) = 1. The set of almost-complex structures has two components. One component consists of those J for which {X, JX, Y, JY} defines the given orientation, and the other contains those J defining the opposite one. Let 7+(M4, D) and 3-(M4, 0) be the corresponding bundles. If J is a section in 3+(M4, 37), then (TM4, j) is a complex vector bundle and the second Chern class, c2, coincides with the Euler class of M4. For c = cl (TM4, J) E H2 (M4; Z), 1

2

1

1

2

1

V - dim ML (g) = 4 c - 4 (2X + 3Q) = 4 c1 - 4 (2c2 + p1)

= 4ci -

4(2c2+ci -2c2) = 0.

For the Pontryagin class pi of the oriented real bundle TM4 we made use of the formula pl (TM4) = ci (TM4, j) - 2c2 (TM4, j).

Let Q(X, Y) = g(X, JY) be the Kahler form of (M4, g, j). SZ acts as an endomorphism in the spinor bundle 1 : S+ --+ S+ and has the eigenvalues ±2i. Indeed, with respect to a basis from the principal bundle Pj, Q = el A e2 + e3 A e4,

the endomorphisms el A e2 = e3 A e4 : S+ --+ S+ coincide and are both given

by the matrix

A. Seiberg-Witten Invariants

148

Let S+(±2i) C S+ be the corresponding subbundles. Furthermore, let (P E S+ be a spinor from S+(2i), i.e. SZ(D = 2i D. Identifying by means of the basis S+

C2, the spinor

has the components - =

Com-

puting the 2-form w' yields w' = iI-DI212 for (D E S+(2i). Analogously, one computes wD = -ij 2l for spinors

Denote by cl° the spinor in S+(-2i) _- A°'0 corresponding to the function 1. Then, in the chosen coordinates, 4(D° = I

I

and w"0 = -in.

The bundle L = A2T of the canonical spinC structure has a distinguished connection A0, the Levi-Civita connection. Considering the corresponding Dirac operator DA0 : r(S+) - P(S ) and identifying as above A°'0 ®A°'2

S+

S- = A°'1,

we see that DA0 coincides with the a-operator of the Dolbeault complex: DAo =V2 A ED j2*)

Now suppose that the scalar curvature R of the Kahler manifold (M4, g, J) is negative and constant, R = const < 0. Then -D _ R(b0 is a section in S+, and

DAo 1 = w11,

0,

_ -I,1)I2ic _ -(-R)ic = RiQ.

On the other hand, the curvature 11A0 in the line bundle L = A2T is given by the Ricci form p, OAo = ip

with p(X, Y) = g (X, J Ric Y). Here Ric : T --> T denotes the Ricci tensor.

In a basis, J : T -* T and the Ricci tensor are respectively given by the matrices

J_

0

-1

1

0

0 0

0

0 0

0

0 0

-1

0

1

0

Ric =

R11 R12 R13

R12 R22 R13

R13 R23 R33

R14 R24 R34

R14

R24

R34

R44

A.6. The Kahler case

149

Since J and Ric commute, we obtain the following special shape for the skew-symmetric endomorphism:

J o Ric =

0 -A D -C -D -C 0 -B C -D B 0

,

A = R11 = R22,

B = R33 = R44,

and hence p takes the form p = Ael A e2 + Be3 A e4 + C(el A e4 - e2 A e3) - D(el A e3 + e2 A e4). Now e1 A e4 - e2 A e3 and el A e3+ e2 A e4 are 2-forms in A2 . For the projection

p+ of the form p onto the subbundle A+ this implies the formula B A +B p

(el A e2 + e3 A e4).

2

Since A + B = R11 + R33 = (R11 + R22 + R33 + R44), we thus obtain the following relation, holding in 2every 4-dimensional Kahler manifold:

p+= 4 As

QAo = ip = i4Q = 4 the pair ('k, Ao) = (V-R-cDo, Ao) is a solution of the Seiberg-Witten equation.

Theorem (LeBrun 1994). Let (M4, J, g) be a compact Kahler manifold of constant negative scalar curvature, R < 0, and choose the canonical Spinc (4) structure. Then, for the Seiberg-Witten invariant, na(g) = 1

in 7G2

(b2 > 2 for the Seiberg-Witten invariant to be defined independently of the metric).

Proof. Let (-D, A) be an arbitrary solution of the equation DAI) = 0,

ul = 4w

Then, Iq) (M) 12 < -Rmin

= -R

and

81

IQAI2 =

8

Since p+ = E Q and I S2I 2 = 2, this implies (*)

f IpAI2 < J

M4

M4

8 R2

16 = = f R2I6I2

M4

I

M4

(RIIII) Ip+I2.

4

2

M4

A. Seiberg-Witten Invariants

150

On the other hand, since p is the curvature form of the Levi-Civita connection in the bundle A2T, it follows that cl(A2T)

47x2

f (Ip+12 - Ip

12),

M4

and hence f Ip+I2 = 27r2ci(A2T) +

2

M4

f

IAI2.

M4

The scalar curvature of the Kahler manifold is constant. Therefore, the Ricci form p is a harmonic 2-form, a consequence of the Bianchi identity. The harmonic form realizes the minimum of the L2-norm in each cohomology class. A is a connection in A2T, and hence

f

f 1P12 M4

AI2.

M4

Thus, (**)

f Ip+I2 < 27r2ci(L) + M4

f IQAI2 = f IpAI2. M4

M4

Combining (*) and (**) yields S2A = ip and 1.1) 12 = -R. The proof of the estimate I4)I2 _< -Rmin then immediately implies VA p __ 0. Decompose D according to the splitting of the spinor bundel S+ = A°'° ® A°'2 into (P =0,0®4,0,2

°,2 is a VA-parallel section in A°'2. If it is non-trivial, then we conclude that cl (A°'2) = cl (A2T) = 0. However, the curvature form of the Levi-Civita connection A0 in A2T is

52+90=ip+=i4S2#0, since R < 0. Hence (D°,2 vanishes and 4) is proportional to the standard spinor, ' = f 4D°. The length If I2 = -R is constant, since I.DI2 is constant. The connection A differs from the Levi-Civita connection A0 by an imaginary-valued 1-form 171, A - A0 = 771. As QA = ip, we have dr71 = 0. But

0=DA

1P =grad (f)'-11°+771f-,D°.

This implies grad f + 771 f = 0. Now consider the gauge transformation 9

+ :M4

1.

A.6. The Kahler case

151

Locally we can write f as f = ,/-ReiF and dg = Rdf . Then, g = ezF = Rdf Since grad(f) + f 771 = 0, this implies 771 = - s . In addition, we have .

found a gauge transformation g : M4 -f T1 with

A=AO - gg, i.e. (-cD, A) is equivalent to ( /

=g.(

c1)o),

0,A0).

Corollary. Let M4 be a compact oriented manifold with b2 > 2. If M4 has a Kdhler structure with constant negative scalar curvature, then M4 admits no Riemannian metric of positive scalar curvature. Remark. The preceding corollary is the special case of a more general fact. Consider a 4-dimensional compact and oriented manifold M4 with a symplectic structure w and assume that w A w > 0 defines the fixed orientation.

Choose an almost-complex structure J : TM4 , TM4, J2 = -Id, for which g(X, Y) = w(X, JY) is a Riemannian metric. From w A w > 0 one easily concludes that J is a section in the bundle 'j+(M4, D). Let c = c(w) E H2 (M4; Z) be the first Chern class of the complex vector bundle (TM4; J). Then, by the observations above,

v - dim i)lL = 0,

i.e. for a generic metric the moduli space is discrete.

Theorem (Taubes, 1994). Let (M4, s7) be an oriented compact 4-dimensional manifold and w a symplectic form with w A w > 0. Moreover, assume that b2 (M4) > 2, i.e. the Seiberg-Witten invariant is defined. If c = c(w) E H2 (M4; Z) is the Chern class of an almost-complex structure associated with w, then nc(M4) = 1 mod 2.

Corollary. Let (M4, sD) be an oriented compact 4-dimensional manifold satisfying the following two conditions:

1) b2(M4)>2, 2) M4 has a symplectic structure w with w A w > 0.

Then M4 admits no Riemannian metric of positive scalar curvature.

We will now turn to the case of a Kahler manifold (M4, g, j) where the Spin' (4) structure c E H2 (M4; Z) is not necessarily the canonical one of M4. As before, denote the line bundle by L. The Kahler form SZ acts as an endomorphism on the spinor bundle and has the eigenvalues ±2i there. Hence the spinor bundle splits, S+ = S+ (2i) ® S+(-2i).

A. Seiberg-Witten Invariants

152

L is isomorphic to A2S+ for each Spincc(4) structure, and hence we obtain the isomorphism S+ (2i) ® S+(-2i) = L. For every connection A E C(L) the Kahler form SZ is parallel, and hence

the decomposition S+ = S+(i) ® S+(-i) is

VA-parallel,

too. For a spinor

4D = (D+ +'- the integral formula now takes the following form:

f IAA + 4wY + IDA J2 M4

f {I'qA12+IDAj

4

(11)+12+II2)+ I (I-CD +I2+Ip-I2)2}

M4

This implies that for a given solution

(D+ + (D-, A) the pair -cD+ - -cD_, A) also solves the Seiberg-Witten equation. This remark, in turn, allows us to reduce the Seiberg-Witten equation considerably. Start from a solution of the equation,

DAB=O,

S2+A=-4W

with ( _ (D+ + 4)_, and note that, moreover,

DAB= 0,

Q+A=-4w4)

with -1)+ - -1)_. In a local orthonormal frame Kahler form ci has the form

el,... e4 on M4 the

i = el A e2 + e3 A e4

and the spinor P = (45+, 4D_) is given by its components. By the definition of w w'D

=

i(I.1)+I2

- I(p-I2)(el n e2 + e3 A e4) -5+-D-)(el A e3 - e2 A e4) + (D+1)_)(el n e4 + e2 A e3).

Therefore,

w(D +w = 2i(I-D+I2 - I.1)_I2)n.

Since Q++ = -4w` _

w

we obtain

= i(I.D+I2 - j)_I2)SZ and 4'+ . '- = 0. Thus, either (D+ - 0 or (D_ = 0. Moreover, S2 is a All-form, and hence, -41l,+9

since A°'2 n A2 _ {0} = A2'° n A? , we at once arrive at QO, A

2=0=QA °.

A. 7. References

153

The curvature form of the connection A in L is thus a (1, 1)-form. Hence A defines a holomorphic structure in L, and, therefore, in S+(±2i) as well. In this case, the Dirac equation, DA1+ = 0 (or 0, respectively), means that 4)t is holomorphic. The Chern class of L is given by

C+(L) = -QA

ci(L)

and hence, from 12 A cl = S2 A c1 , we conclude that

fc2Aci(L) = 1 fi+i2 - I`)-12)12 A c2

J :def M4

M4

J is the cup product of S2 and cl(L), hence a topological invariant. If J < 0, then 41)+ - 0; in case J > 0 we thus have (D_ = 0. Taking into account, in additon, the action of the gauge group leads to the

Theorem (Witten, 1994). Every solution of the Seiberg-Witten equation over a Kahler manifold M4 for the spinc structure c E H2 (M4; Z) corresponds to a pair consisting of a holomorphic structure in the bundle S+ (±2i) and an element of

PH°(M4; S+(-2i)) if S2 A c < 0, PH°(M4;S+(2i)) if S2Ac> 0. For bi (M4) = 0, the holomorphic structure in S+(±2i) is unique and )tL(g) IPH°(M4; S+ (±2i)). Remark. In general, O7tL(g) cannot be used to determine the SeibergWitten invariant S - W (M4, c) E J V,,, since the Kahler metric g is not generic.

A.7. References J. Eichhorn, Th. Friedrich. Seiberg-Witten theory, in Symplectic Singularities and Geometry of Gauge Fields (Warsaw; 1995), Banach Center Publ., vol. 39, Polish Acad. Sci., Warsaw, 1997, pp. 231-267.

M. Furuta. An invariant of spin 4-manifolds and the 11/8-conjecture, Preprint 1995.

P. B. Kronheimer, T. S. Mrowka. The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), 797-808.

D. Kotschick, J.W. Morgan, C.H. Taubes. Four-manifolds without symplectic structures, but with non-trivial Seiberg-Witten invariants, Math. Res. Lett. 2 (1995), 119-124.

C. LeBrun. Einstein metrics and Mostow rigidity, Math. Res. Lett. 2 (1995), 1-8.

154

A. Seiberg-Witten Invariants

C. LeBrun. On the scalar curvature of complex surfaces, Geom. Funct. An. 5 (1995), 619-628.

C. LeBrun. Polarized 4-manifolds, extremal Kahler metrics and SeibergWitten theory, Math. Res. Lett. 2 (1995), 653-662.

C. LeBrun. 4-manifolds without Einstein metrics, Math. Res. Lett. 3 (1996), 133-147.

J.W. Morgan. The Seiberg-Witten equation and applications to the topology of smooth four-manifolds, Mathematical Notes, Princeton University Press 1996.

J.W. Morgan, Z. Szabo, C.H. Taubes. A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture, J. Diff. Geom. 44 (1996), 706-788.

C. H. Taubes. The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), 809-822. C. H. Taubes. More constraints on symplectic forms from Seiberg-Witten invariants, Math. Res. Lett. 2 (1995), 9-14. C.H. Taubes. The Seiberg-Witten and the Gromov invariants, Math. Res. Lett. 2 (1995), 221-238. C.H. Taubes. From the Seiberg-Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996), 845-918. E. Witten. Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), 769-796.

Appendix B

Principal Bundles and Connections B.1. Principal fibre bundles Definition. Let E, X and F be three topological spaces. A mapping 7r E -+ X is called a locally trivial fibration with fibre F if for each point xo E X there exist a neighborhood U(xo) and a homeomorphism -DU(xo)

p-1(U(xo)) , U(xo) x F such that the diagram

7r-1(U)"

UxF

commutes. Then E is called the total space of the fibration, X its base and F = Ex = 7r-1(x) the fibre over x E X.

Example 1. Consider E = X x F and the projection 7r : E -* X. Then (E, 7r, X; F) is a locally trivial fibration.

Example 2. Let 7r : E -* X be an unramified covering, and let F be a discrete space whose points can be mapped bijectively onto 7r-1(x). Then 7r : E -+ X is a locally trivial fibration with fibre F.

Example 3. Let X = M" be a smooth manifold, E = T(MT) and 7r E -> X the projection of the tangent bundle. Then (T (MT ), 7r, Mn; R') is a locally trivial fibration with fibre F = IlBn.

Example 4. Let G be a Lie group, H C G a closed subgroup and E = G, X = G/H. Then the projection 7r : G -* G/H is a smooth locally trivial fibration with fibre H. 155

B. Principal Bundles and Connections

156

Definition. Let E and E* be two fibrations over X with projections it and 7r*, respectively. These fibrations are called equivalent if there is a homeomorphism f : E -* E* fitting into the commutative diagram

E

f

E*

Definition. A locally trivial fibration (E, it, X; F) is called trivial if it is equivalent to (X x F, 7r, X; F).

Example 5. Not every fibration is trivial. Let E = T(S2) be the tangent bundle of the sphere X = S2. Then T(S2) -p S2 cannot be the trivial fibration, since its Euler characteristic is two, X (S2) = 2 $ 0, and hence, by the Hopf theorem, there are no vector fields without zeroes.

Definition. Let p : E -- p X be a fibration. A mapping s : X --* E is called a section if it o s = Idx. In case s is defined on an open set U C X only, it is called a local section.

Let (E, it, X; F) be a fibration over X and f : Y -4 X a (e.g. continuous or smooth) mapping. Then define a bundle f*E over Y by

f*E={(y,e)EYxE: f(y)=7r(e)} with projection 7r* (y, e) = y. The fibration 7r* fibration induced by f.

Proposition. p* : f *E

f *E

Y is called the

Y is a locally trivial fibration with fibre F.

Definition. A 4-tuple (P, it, X; G) is called a G-principal bundle if 1) P is a topological space and G a topological group acting freely from the right on P; 2) 7r : P -> X is continuous and surjective, and ir(pl) = 7r(p2) if and

only if there exists an element g E G such that plg = P2 holds; and 3) it : P --s X is a locally trivial fibration in the sense of principal bundles, i.e. for every x E X there exist U, x E U C X and -1)U 7r-1(U) --+ U x G such that '11 U(p) = Or (p), cou(p))

and cpu :7r-'(U) --3 G has the property cpu(p g) = cou(p) . 9.

B.1. Principal fibre bundles

157

Remark. a) Every G-principal bundle is a locally trivial fibration with fibre F = G.

b) If (P, it, X; G) is a G-principal bundle and f : Y ---> X is continuous, then (f *P, 7r*, Y; G) is again a G-principal bundle. c) If P, G, X and all mappings are smooth, then the principal bundle is also smooth.

Definition. Let (P, it, X; G) and (P1, 7r1, X; G) be two principal bundles over the same base X with the same structure group G. These two principal bundles are called isomorphic if there exists a homeomorphism f : P ---+ P1, satisfying the following conditions:

1) The diagram

P

f P1 X

commutes.

2) f (p g) = f (p) g, i.e. f is compatible with the action of G. Proposition. If the principal bundle (P, it, X; G) has a section, then this principal bundle is isomorphic to the trivial G-principal bundle (X x G, 7r, X; G).

Example 1. Let G be a Lie group and H a closed subgroup. Then, setting

P = G, X = G/H and G = H, we see that (G, it, G/H; H) defines an H-principal bundle over G/H. Remark. Non-isomorphic principal bundles may well be equivalent as fibrations.

Example 2. Take X = S2 = C]P1 and P = S3 = {(w1, w2) E C2 :

Iw1I2 +

IW212 = 1}. Consider the fibration

7r : S3 -* CP',

lr(wl, w2) = [w1 : w2],

and two principal bundles 1 = (S3, 7r, Ce'; Sl) and 6 = (S3, it, CPI; S1) differing only in the action of the group S1 on S3. In i;'1 let the group S1 act on S3 by (w1, w2) . z = (wlz, w2z)

and in 2 by (WI, W2) . z =

(w1z-i,w2z-1).

Then S1 as well as 6 are principal bundles. 61 and 62 are not isomorphic as Sl-principal bundles. The principal bundle S1 is called the Hopf fibration.

B. Principal Bundles and Connections

158

Example 3. Let Mn be a smooth manifold and Lx(MT) _ {(VI, ... , v"n) E TXMn I det(vl, ... ,,0n) 54 0} the set of all frames at the point x E Mn.

The union L(Mn) _ U Lx(Mn) is called the frame bundle of M. Let xEMn

GL(n;1i8) act on L(Mn) by All

A1n

_ Anl

n

n

(vl,... ,vn)'

Ann

iAin

vjAil,...

i-1

i=1

Then (L(Mn), 7r, Mn; GL(n, R)) is a GL(n,118)-principal bundle over Mn.

Example 4. Let (Mn, g) be a Riemannian manifold and O(Mn; g) _{(411, ... vn) E L(Mn) : g(vi, v9) = Then O(Mn; g) is an O(n; R)-principal bundle over Mn.

a.7}.

Example 5. Let (M2n, w) be a symplectic manifold and Sp(M2n; w) (v1

... vn,, 191 ... 2Un) E L(M2n) :

(vi, vj) = w(wi, wj) = 0 1 w(vi, w.9) = ail

Then Sp(M2n, w) is a Sp(2n;118)-principal bundle over

.

J

M2n.

Example 6. Let Mn be a manifold with a fixed orientation O. Set

P=

{(v"1,

...

,v"n

) E L(M') :

{v"1,

...

,

v"n}

= O}.

Then P is a GL+(n;1[8)-principal bundle.

Definition. Let p = (P, 7r, X; G) be a G-principal bundle and A : G1 -> G a continuous (smooth) group homomorphism. A A-reduction is a pair (µ, f) consisting of a G1-principal bundle µ = (Q, ir, X; G1) over X and a mapping f : Q -+ P such that 1) the diagram Q

f

P

-1 X X

commutes, and 2) f (9' ' g1) = f (4') ' A(g1)

Examples 4, 5, 6 describe different reductions of the frame bundle to the groups 0(n;118), Sp(2n; R) and GL+(n; R), respectively.

B.1. Principal fibre bundles

159

Definition. Two A-reductions (µ, f ), (µ, f) of the principal bundle p are called equivalent if there exists an isomorphism D : Q --f Q of the G1principal bundles such that the diagram

Q

f

Q

it

P commutes.

Proposition. Let A : O(n; R) -} GL(n; R) be the canonical embedding of groups and Mn a smooth manifold. The set of all A-reductions of the frame bundle L(M') is in bijective correspondence with the set of all Riemannian metrics on Mn. Proposition. Let A : GL+(n; IIi) --> GL(n; l[8) be the canonical embedding of groups and Mn a smooth manifold. The set of all A-reductions of the frame bundle L(Mn) is in bijective correspondence with the set of all orientations on Mn.

Remark. For a given Men, each symplectic structure w defines an Sp(2n; R)reduction of L(M2n); compare Example 5. Conversely, given an Sp(2n)-

reduction, one can define at each point x E Men a 2-form wx : TM n X TxMn -> R. The resulting 2-form w will be non-degenerate. However, in general, dw = 0 will not hold. Thus not every reduction of the frame bundle L(M2n) to the subgroup Sp(2n; I[8) can be identified with a symplectic structure on Men.

Consider now a G-principal bundle (P, 7r, X; G) and a topological space F

on which G acts from the left, G x F -> F. Let G act from the right on

PxFby

Set

E=PxF/G:=PXGF and take the projection 7r : E --> X defined by

7r(e)_7r[p,f] _ir(p) The 4-tuple (E, p, X; F) is a locally trivial fibration, i.e. we have the

Proposition. If (P, 7r, X; G) is a principal bundle and F a space on which G acts from the left, then (E, p, X; F) with E = P xG F is a locally trivial fibration. The fibration (E, 7r, X; F) is called the bundle with fibre F associated to the principal bundle. Proposition. Let (P, 7r, X; G) be a G-principal bundle and F a G-space defining the associated bundle E = P xG F. Then there exists a bijection between the sections s in the bundle (E, 7r, X; F) and the maps s* : P --+ F with s*(p g) = g-1s*(p)

B. Principal Bundles and Connections

160

Now let G be a group and H C G a closed subgroup. Consider the G-space

G/H = F and the bundle E = P xG (G/H) associated to the principal bundle (P, 7r, X; G).

Proposition. The bundle (E, 7r, X; G/H) has a section if and only if the G-principal bundle (P, 7r, X; G) has a reduction to the subgroup H --4 G.

Example 7. Let G be a group and H a closed subgroup. Consider the trivial G-principal bundle P = X x G. A section in P XG (G/H) ^ X x G/H is then simply a maps : X -3 G/H, and the H-principal bundle corresponding to this section is

Q={(x,g) EX with the projection (x, g) -* x. Now fix

G = S0(3), H = S0(2) =

/ C

l 0)

S0(3)

and X = S2 = SO(3)/SO(2), where the identification SO(3)/SO(2) ^ S2 is defined by A -- A(e3); e3 is the third basis vector of the Euclidean space 1[83. Since e - H <--> e3 under this identification, we conclude that, for every mapping f : X = S2 -> SO(3)/SO(2) S2, Q = {(x, g) E S2 x SO(3) : 9-If (x) = e3}

is an S1-principal bundle which is a reduction of the trivial bundle P = X x G = S2 x SO(3). Take, e.g., f : S2 --+ S2 to be the identity. Then the resulting Sl = SO(2)-principal bundle is Q = {(x,g) E S2 x SO(3) : g-lx = e3}. Though Q is the reduction of the trivial SO(3)-principal bundle over S2, Q itself is not a trivial Sl-principal bundle over S2. This example shows that reductions of trivial bundles may well be non-trivial principal bundles.

As the last topic in this section we want to discuss vector bundles. To this end, consider again a G-principal bundle (P, ir, X; G) and, in addition, a vector space F = V (complex or real). Let G act on V via a representation

p : G --> GL(V). Then we obtain the associated bundle E = P xG V = P X PV with projection 7r : E -- X. Now define ll8 x E -- E or C x E --> E, respectively, by

EE) e=[p,v]- A.e=[p,Av] EE and an addition of two elements el, e2 satisfying 7r(el) = 7r(e2) by el = [p, v1],

e2 = [p, v2] -> el + e2 = [p, V1 + v2].

As G acts linearly on V, these operations are uniquely defined. This results in the following structure: Each fibre of P x P V = E is a vector space over

B.1. Principal fibre bundles

161

II8 or C, respectively, and for every x E X, there exist a neighborhood U, x E U C X, and a mapping -cDu : p-1 (U) -; U x V which is linear in each fibre.

Definition. A (real, complex) vector bundle over the space X is a fibration (E, 7r, X; Vn) such that each fibre is a vector space and for every xo E X

there exist an open set U, xo E U C X, and a homeomorphism bu p-1(U) -* U X R, (U X (Cn) which is linear in each fibre.

Example 8. Let Mn be a manifold and (L(Mn), ir, Mn; GL(n; Ia)) its frame bundle. Moreover, let p : GL(n) --; GL(118n) be the usual representation. Then the associated vector bundle is isomorphic to the tangent bundle: L(M') XGL(n) ll -_ T(Mn). To show this, define a mapping f : L(Mn) XGL(n) R' --+ T(MT) by

f A, ...

> vn; Cl,

..

,

Cn) = 1:vici = v" - C.

For A E GL(n; I[8) we have (v, c) = (VA, A-lc) and, at the same time, vAA-lc = v c. Hence, f : L(Mn) XGL I[8n -i T(Mn) is uniquely defined and, in addition, an isomorphism of vector bundles. Example 9. T*Mn ^_ L(Mn) xp* (Rn)*, where the mapping p* : GL(n; II8) GL((I[8n)*) is the dual representation in the dual space (R)*.

Example 10. Let Pk : GL(n; Il) -* GL(Ak((?n)*) be the representation in the space of k-forms of RI. Then, Ak(Mn) = L(Mn) xp, Ak((I[8n)*).

Example 11. Let = (S3, 7r, ClP 1; S1) be the Hopf fibration and H = S3 x p C the associated bundle, where p : S1 -- GL(C) is given by p(z)w = zw. H consists of the equivalence classes [(wl, w2), w] of pairs of complex numbers under the identification {(wl, w2), w} - {(wlz, w2z), z-1w}.

The mapping f : H --> H = {(1, e) E C?1 X C2 f (wl, w2; w) _ ([WI : ECP1

:

E l} defined by

(wlwW2W)) EC2

obviously defines an isomorphism between H and A. Hence H ---). C?I, as a vector bundle, is equal to

H={(1,1;) EC?1 >C2El} with projection p : H -* C?1, p(l, l;) = 1. H (H) is the so-called Hopf bundle (tautological bundle over Cl?').

B. Principal Bundles and Connections

162

B.2. The classification of principal bundles In this section, we will study the following question:

Let a space X be given. How many isomorphism classes of G-principal bundles (P, 7r, X; G) with structure group G exist over X? Theorem (First homotopy classification theorem). Let = (P, 7r, Y; G) be a G-principal bundle over the topological space Y, X a paracompact space and fl, f2 : X -+ Y two homotopic mappings, fl - f2. Then the induced principal bundles fl , f2 over X are isomorphic.

Definition. If X is a paracompact space, then denote by HFBG(X) the set of all isomorphism classes of principal bundles over X with structure group G.

Definition. Let G be a topological group. A universal G-bundle is a Gprincipal bundle c = (EG, 7, BG; G) such that for every CW-complex X the assignment

[X; BG] E [f] - f *G E HFBG (X) is a bijection. In other words, the following two conditions have to be satisfied:

1) For every G-principal bundle l; over X there exists a mapping f X -f BG with f*

2) For fo,f1:X->BGsuch that f c fl* Gwehave fo- fl. BG is called the classifying space of the topological group G.

Obviously, the classifying space of a topological group is not uniquely determined. Namely, we have the

Proposition. If ec = (EG, 7r, BG; G) is a universal bundle and B a topological space which is homotopy equivalent to BG, then over B there is also a G-principal bundle which is universal.

Proposition. Let 6c = (EG, 7r, BG; G) andG = (EG, 7r, PG, G) be two universal G-principal bundles with the CW-complexes BG, BG. Then there exist homotopy equivalences 0 : BG --> PG and : PG -> BG with

2/) o0-IdgG, 0o0-Id and ec='* c,

c=cb*6c

In the class of CW-complexes, the homotopy type of the classifying space of a group, if one exists, is uniquely determined. Now we can formulate the second homotopy classification theorem for principal bundles:

B.3. Connections in principal bundles

163

Theorem (Second homotopy classification theorem). 1) For every topological group G there exists a universal G-principal bundle Z;G = (EG, 7r, BG; G). This bundle, moreover, has the property that for every paracompact space X the assignment

[X; BG] E) [.f] -+ f*G E HFBG(X) is bijective.

2) For every topological group G there exists a universal G-principal bundle

(EG, ir, BG; G) such that BG is a CW-complex.

Example (G = Z2). The set PBz2 (X) = [X; B(Z2)] = [X; I[8IP°°] (X a CW-complex) is in bijective correspondence with the elements of H1(X; Z2). If is a Z2-principal bundle, then the element in H1 (X; Z2) corresponding

to this principal bundle is called the first Stiefel-Whitney class wl

E

HI (X; Z2).

Example (G = S'). If X is a CW-complex, then PBS, (X) = [X; B(S')] _ [X; C?'] is in bijective correspondence with the elements of H2(X; Z). For an S'-principal bundle 6, the element in H2 (X; Z) corresponding to this fibration is called the first Chern class of and will be denoted by cl (l;) E H2(X;Z).

B.3. Connections in principal bundles Consider a smooth principal bundle (P, ir, M'n; G). The vertical space Tp (P) of the projection 7r is the space Tp (P) = {X E Tp(P) : d7r(X) = 0}.

Lemma 1. For X E g denote by X (p) = at (p exp(tX)) t=0 the fundamental vector field of the G-action on P. Then

gEX->X(p)ETT(P) is a linear isomorphism.

Definition. The assignment Th : p E P -> Tph(P) C TpP (geometric distribution, Pfaff system) is called a connection on (P, 7r, M; G) if 1) TpP = Th(P) G Tp (P), 2) dR9(Tph(P)) =Tp9(P), and 3) Th is smooth. The projection onto the vertical space X (p) ®Y E Tp (P) ®Tph (P) , X E g defines a g-valued 1-form Z on P, Z : TP - g. Then we have the

B. Principal Bundles and Connections

164

Proposition. a) Z(X) = X for every X E g, b) (Rg)*Z = Ad(g-1)Z. Conversely, given a g-valued 1-form Z : TP --* g satisfying a) and b), then Tp (P) = {X E TpP : Z(X) = 0} is a connection. Now we will describe the local characterization of a connection: Let s : U C M -> P be a section. ZS = Z o ds = s* (Z) : TU g is the local connection define form. For two sections si Ui -* P, s j : Uj --r P and Ui n Uj :

gij:UinU1-Gby si (x) = sj (x) - gij (x)

Denote the Maurer-Cartan form of the group G by O : TG --> g, O(tg) _ dLg-1 (tg). The corresponding 1-forms on Ui n Uj are Oij = gz-(O):

Oij =

dLg-idgij.

Proposition. 1) Zsi = Ad(gij1(x))ZS' +Oij. 2) Let a family {(Ui, si)} with U Ui = M be given and let Zi : TUi --> g be a family of 1-forms such that

Zi = Ad(gzj')Zj + Oij. Then there exists precisely one connection Z : TP -i g with Zsi = Zi.

Example. Let (Mn, g) be a Riemannian manifold, V : TM -* T*M ® TM the Levi-Civita connection, and O(M, g) the bundle of orthonormal frames. In the Lie algebra so(n) we choose the basis {Xij}i<j, where Xij =Eij -Eji and Eij denotes the matrix with 1 in the i-th row and j-th column (and zero otherwise). Let s : U -* O(M, g), s = (Si,... , sn) be a local section and set ZS = E g(Vsi, sj)Xij. Then {ZS, s} defines a connection Z in O(M, g) i<j

and the 1-forms wij = g(Vei, ej) are the connection forms of the Levi-Civita connection.

Example. Let Mn be a differentiable manifold. The set of connections on the frame bundle L(Mn) is in bijective correspondence with the set of affine connections V : TM -+ T*M ® TM. Example. X = G/H is called reductive if there exists a decomposition g = hem with Ad(H) (m) C m. Consider the principal bundle t; = (G; ir, G/H; H) and the distribution G E) g -> Ty G = dLg(m). This is a connection in with dL,,, (T9 G) = Tag G.

Let a principal bundle (P, 7r, M; G) and a representation p : G -+ GL(V) be given.

B.3. Connections in principal bundles

165

Definition. A q-form w E Al (P, V) with values in V is called tensorial of type p if

1) w2(tl, , tq) = 0 if one of the vectors tz E TT is vertical, and 2) R*w = p(g-1)w, g E G.

Example. If Z, Z : TP -> g are two connections, then Z - 2 is a tensorial 1-form of type Ad on P with values in g. Conversely, if Z is a connection and w : T P -> g a tensorial 1-form of type Ad, then Z + w is again a connection.

Proposition. The vector space of tensorial q-forms of type p on P with values in V is isomorphic to the vector space Aq(M; E) of q-forms on M with values in the associated vector bundle E = P x P V V.

Corollary. Let C(P) be the set of all connections on P. C(P) is an of ne space with vector space A'(M;g). Here g denotes the bundle g = P XAd 9 associated by means of the representation Ad.

Definition. Let (P,,7r, M; G) be a principal bundle with connection. If X E X (M) is a vector field on M and X* E X (P) a vector field on P, then X* is called a horizontal lift of X if a) X * (p) is horizontal at every point p E P, and b) d7r(X*(p)) = X (7r (p)).

Proposition. 1) Let X E X(M) be given. Then there exists a uniquely determined horizontal lift X* E X (P) of X. X * is right invariant. If, on the other hand, Y E X (P) is a right invariant horizontal vector field, then there exists a vector field X E X (M) with X* = Y.

2) X,Y E X(M)

X* +Y* = (X +Y)*,

(fX)* _ (f

07r)X*,

[X, Y]* = projhor. [X*, Y*]

3) If Y is a horizontal vector field on P, then [X, Y] is horizontal for all X E g. 4) In particular, [X, Z*] = 0 for X E g, Z E X (Mn). Let y : [a, b] -> M be a curve (continuous, piecewise C2).

Definition. A curve y* : [a, b] -> P is called a horizontal lift of y if 1) lry* = y, and 2) y* is horizontal.

Proposition. Let y : I -> M be a curve in M and u E Py(o) a fixed point. Then there exists precisely one horizontal lift yu of y with y,*,(0) = u.

B. Principal Bundles and Connections

166

Definition. Let Ty : Py(a) - Py(b) be defined by ry(u) = yo(b). -ry is called the parallel transport along y.

Proposition. 1) ry does not depend on the parametrization of y.

2) ryRg=R9Ty VgEG. 3) ry is bijective.

Proposition. Let (P, 7r, M; G) be a principal bundle with connection Z, and suppose that M is connected. If the parallel transport r does not depend on

the curve, then there exists a horizontal section in P and, moreover, the principal bundle is isomorphic to the trivial bundle (M x G, prl, M; G) with flat connection.

For E = P XG F (F an arbitrary space), the parallel transport in E, rE

Ey(a) -3 Ey(b), is defined by rE[p, v] = [ry(p), v]. A connection in

the principal bundle induces a parallel transport in every associated bundle.

B.4. Absolute differential and curvature Definition. Let (P, 7r, M; G) be a principal bundle with connection Z, V an arbitrary vector space, and let w E Aq(P, V) denote a q-form on P with values in V. Define Dw E Aq+l(P,V) by (Dw)p(to, ,tq) = dw(prhto, ,prhtq) Dw is called the absolute differential of w. It defines a linear mapping

D : Aq(p, V) - Aq+i(P, V).

Proposition. 1) If w is a q -form of type p, then Dw is a tensorial (q+1)-form of type P.

2) Suppose that w E Aq(P; V) is tensorial of type p. Then,

Dw =

dw + p. (Z) A w with p* : p -* gl(V) and q

(p*(Z) A w) (to, ... , tq) = L(-1)op*(Z(ta))w(to, ... , ta, ... , tq). a=o

Now let p : G --+ GL(V) be a representation and set E = P xP V. The tensorial forms of type p coincide with the forms AP(M, E) on M with values in the vector bundle E. Hence the absolute differential can be viewed as an operator D : Aq (M; E) -r Aq+1(M; E). Definition. Let (E, 7r, M) be a vector bundle over M and F(E) the space of smooth sections. A mapping V : r(E) -+ r(T*M ® E) is called a covariant derivative in E if the following conditions are satisfied:

B.4. Absolute differential and curvature

167

1) V is linear, V(ei + e2) = Vel + Del. 2) V (f e) = df ®e -I- f Ve for f E C°°(M), e E F(E). The 1-form V e is also written as (V e) (X) = Vie.

Proposition. Let (P, 7r, M; G) be a principal bundle with connection, p G -> GL(V) a representation and E = P x p V the associated vector bundle. Then the absolute differential D : r(E) -3 A1(M;E) = P(T*M (9 E) is a covariant derivative.

D is called the covariant derivative in E associated with Z. It is occasionally also denoted by Vz, VE

Proposition. Let s E F(E) be a smooth section. Then, (Ds) (X)

dt(TtO(s('Y(t)))t=o,

X E TpM,

where y(t) C M is a curve with -y(O) = p,,-y(0) = X, and T% : Ey(t) --> Ey(o) is the parallel transport.

Corollary. Let y(t) C M be a curve and s(t) C E a curve over y(t). Then s(t) is the parallel transport of s(O) along y(t) if and only if dt Ds('y(t)) = 0.

Definition. Let (P, 7r, M; G) be a principal bundle and Z : TP - g a connection. Then Z is a 1-form on P of type Ad (i.e. Rg**Z = Ad(g-1)Z). By the preceding theorem,

Q:=DZ is a 2-form on P with values in g which is tensorial and of type Ad. SZ is called the curvature form of the connection. By the general identification {tensorialq-form of type p}

Aq (M; P x p V)

,

S2 can also be considered as. a 2-form on M with values in g = P XAd g, S2 E A2(M; g). We introduce a few notations: Let w E Ai(P; g), ,r E Aj (P; g) be two forms on P with values in g (or else, w E AZ(M; g), ,T E Aj (M; g)

two forms on M with values in the bundle g). Then define a form [w, T] E Ai+j (P; g) (or [w, r] E Ai+j (M; g) }, respectively) by

[w,r](X1,... ,Xi+j)

= iiji E (-1)9[w(Xg(1), ...

, X9(i)), r(X9(i+l), ... , Xg(i+i))]

9ESi+1

If Al,

,

Ae is a basis in g and w = wiAj, r = ri Aj, then, obviously, [w,r] = wZ Arj [Ai,Aj].

B. Principal Bundles and Connections

168

This bracket has the following properties:

a) [w,,r] = (-1)zj"[7, w]. b) For w E Ai(P, g), T E Aj (P; g), cp E A' (P; g),

7]+(-1)ii[[T,coj, w] =0. c) d[w, 7] = [dw, T] + (-1)i[w, dr]. d) If w is a 1-form and w E A' (P, g) (or w E A' (M; g)), then 2

[w, w] (X, Y) = [w(X ), w(Y)]

Proposition. Let (P, 7r, M; G) be a principal bundle, Z : TP --> g a connection and Il = DZ its curvature form. Then we have: 1) The structure equation: Q = dZ + [Z, Z] 2 2) The Bianchi identity: D11 = 0. 3) If w E Aq(P; V) is tensorial of type p : G -- GL(V), then .

DDw = p* (Sl) A w.

4) If w E Aq(P, g) is tensorial of type Ad : G -* GL(g), then

Dw=dw+[Z,w]. Corollary. Let X, Y be horizontal vector fields. Then,

Z([X,Y]) = -f2(X,Y). Proof. SZ = dZ+ 2 [Z, Z]. Inserting horizontal fields we obtain Z(X) = 0 = Z(Y), hence Q (X' Y) = -Z[X, Y].

Proposition. Let (P, ir, M; G) be a principal bundle and Z, Z two connections with curvatures SZ = DZ, = DZ. Then, 77 = Z - Z is a tensorial 1-form of type Ad and 1 fl=St+Dr7+ 2[77,r1]-

Considering S2,1 as s-forms on M and 77 as a 1-form on M with values in g = P xAd g 0, n E A2 (M, g), r7 E S2' (M, g), we have the same formula, this time with the operator D : A' (M; g)

A2 (M;

Definition. A connection Z on (P, ir, M; G) is called (locally) flat if there

exists an open covering Ui of M such that (PU,, Z) is isomorphic to (Ui x G, prl, Ui; G) with the canonical connection.

B.S. Connections in U(1)-principal bundles and the Weyl theorem

Proposition. Z is a locally flat connection

)-0

169

the bundle

Th(P) C TP of horizontal vectors is involutive.

Proposition. Let 7rl (M) = 0 and let (P, 7r, M; G) be a principal bundle with a locally flat connection Z. Then (P; Z) is isomorphic to (M x G) with the canonical connection. Definition. Let (P, 7r, M; G) be a principal bundle. A gauge transformation is a diffeomorphism f : P -+ P with

f = 7r, and 1) oTr,

2) f(p.g)=f(p).g Denote by 9(P) the group of all gauge transformations.

Proposition. 1) If Z E C(P) is a connection and f E Q(P) a gauge transformation, then f * Z E C (P) is again a connection. In other words, the group of gauge transformations acts on the set of all connections. 2) Each gauge transformation f is given by a mapping A f : P -> G,

f (p) = p µf (p). Then, (f*Z)p = Ad(µf(p)-1)Zp+dLµf(p)-idl flp

=Ad(/-tf1(p))Zp+µp0.

Proposition. Let the gauge transformation f : P - P be given by ,a f P - G, and let Z be a connection. Then, for the curvature forms S2z and Sjf*z

Qf*z

= Ad(µ f 1)QZ.

B.S. Connections in U(1)-principal bundles and the Weyl theorem In this section, we deal with the group G = S1 = U(1) = {z E C : zj = 1}.

If y(t) is a curve in G with y(0) = 1, then y(0) E T1S1 = C71. On the other hand, 1y(t)12 - 1 implies ' (0) E i118. Hence we obtain an identification 61 D '(0) , y(0) E iIi. The Lie algebra E51 can be identified with iR in

such a way that the diagram

/e

exp\ S1

with e : i]R -p S1, e(ix) = eix, commutes. Consider now the canonical form (Maurer-Cartan form) O : TS' -* 61 iIIB of the group. We will show that dz 0=-=zdz. z

B. Principal Bundles and Connections

170

In fact, if z E S1, F E TZS1 and -y is a curve with 'y(0) = z, 'y(0) = t, then

8(1 _ (dLz-1(t)) =

d (-'t) t-o = z dt It=o = 1tz = dt z 1

dry

z

i.e. e = z Moreover, f S = f dz/z = 27ri. Hence, go := 27rZ19: TS1 -R S1

S1

is a real-valued 1-form on S' with f cp = 1. Si

Now let (P, 7r, M'; S1) be an Sl-principal bundle over M. If f : P -> P is a gauge transformation, then set f (p) = p Izf(p), j if : P -> S'. Since

f(p.z)=f(p).z,

=p.µf(p) z.

Hence, z7i f (p z) = u f (p) z and, since S' is abelian, we have 1.1f (p z) _ Ft f (p). Thus A f : P --i S' is constant on the fibres and induces a mapping

9f : M'1 -p S1. Conversely, if a mapping µ : Mn --> S' is given, then

f(p)=p.µ(7r(p)) defines a gauge transformation. The group of gauge transformations thus coincides with the group of all mappings p : M' --+ Sl: )S1}.

{A:M' Fix a connection Z : TP -> iR on P. If f : P -> P is the gauge transformation corresponding to T if : Mn --> S1, then

f*Z=Ad(pfl)Z+µf0=Z+Pfe=Z+7r*µf8=Z+27rµfcp. Consider SZ = Qz : TP x TP -> iR, the connection form of Z. Since

R*f2 = Ad(z-1)t2 = SZ,

SZ is a tensorial 2-form on TP invariant under the action of right translations. Thus SZ is simply a 2-form on M'z with values in iR,

t2Z:TM' xTM' --+ iR. As 0 = Dt2Z = dt2Z + Ad* (Z) A Qz = dt2Z, QZ is a closed 2-form, i.e. =0

dttZ=0. If 2 is another connection, then

QZ =QZ+Dr7+ with 77 =

1

2[77,77] =t2Z+Dr7

Z. Now, 77 again is a tensorial 1-form with RZ77 = Ad(Z-1)77 =

77, hence a 1-form on M' with values in iR. This implies:

1) The curvature form S1Z of an arbitrary connection in P is a closed 2-form on M' with values in iR.

B.5. Connections in U(1)-principal bundles and the Weyl theorem

171

2) If SZZ, StZ are the curvature forms of two connections, then there exists a 1-form on Mn with values in iR such that SIZ

- Qz=drl.

The de Rham cohomology of a compact manifold is defined by

H2 (Mn;R) = Z2(Mn)/B2(Mn), where

Z2(Mn) _ {w2 : w2 is a 2-form and dw2 = 01, B2(Mn) = {w2 : there exists a 1-form µl with w2 = dµ1}. 1) and 2) imply that the class [-27rC2SZZ] E H2nR(Mn; I[8) is a uniquely determined element of the de Rham cohomology of Mn not depending on the choice of Z, but only on the principal bundle. This class will be denoted by

cl(P) E HDR(M;R) It is called the real Chern class of the SI principal bundle P. Set .F(P) = {w2 : w2 is a 2-form with dw2 = 0, [w2] = cl(P)}. Then, C(P)

obviously defines a mapping

Z-

-Qz

27r.

E.F(P)

: C(P) -+.F(P). We state some of its prop-

erties:

1) If Z and 2 are gauge equivalent connections, then 2) 0 is surjective.

(Z)

From 1) and 2) we obtain a surjective mapping

0: C(P)/G(P)

.P(P).

The first de Rham cohomology is defined by HDR(Mn; I[8) = Z' (M) 1B1 (M),

where Z' and B1 are the following spaces:

Z' (M) = {w' : wlis a closed 1-form, dwl = 01, B1(M) = {w' : there exists a function f on M with wl = df }. Let, moreover,

ZI(M;Z) =

fw' E Z for all closed curves y

wl E ZI(M) : ly

Since f df = f f = 0, we obviously have ZI (M; Z) D BI(M). Let 1

09-Y R(Mn;

HD'

Z) = Z1(Mn; Z)/BI (Mn)

B. Principal Bundles and Connections

172

be the so-called integral de Rham cohomology. Again consider w2 E .P(P) and denote by C,,2 (P) the set

C.2(P) =,O-1(w2)

_ {Z

E C(P) :

27ripZ

= w2}

3) Cw2(P) is a g(P)-invariant affine space with vector space Z'(Mn). 4) The set C,,2(P)/9(P) is in bijective correspondence with

Pic(M") =

HDR(M'; R)I

HDR(M'z; Z).

Summarizing, we state the so-called Weyl theorem.

Theorem (Weyl theorem). Let (P, 7r, M; Sl) be an S' -principal bundle over the compact manifold Mn with first Chern class cl(P) E HDR(M;lR), and set

,r(P) = {w2 :dw2 = 0,

[w2]

= cl(P)}.

Define a surjective mapping b : C(P)/G(P) -3 .P(P) by the assignment

Z .-- -nZ =

27ri

QZ.

Then each fibre r-1(w2) of 0 is diffeomorphic to the Picard manifold

Pic(Mn) = HDR(Mn; R)IHDR(Mn; Z) of Mn. In particular, HDR(Mn; lib) = 0 (e.g. for simply connected Mn) implies that V is bijective.

Example. Consider the Hopf fibration it : S3 --> CP1 = S2 with S3 = {(wl, w2) E C2 : Iw112 + IW212 = 1}

and the S'-action S3 X Sl - S3, ((wl, w2); z) = (WI Z, w2z)

We will construct a connection in this S'-principal bundle. Set 1

Z = {wldwl - wldwl + w2dw2 - w2dw2}. 2

Since z - 2 E i]R for every z E C, it follows that Z is a 1-form on S3 with values in iR. It has the following properties:

1) Z is invariant under the Sl-action, i.e. (Rx)*Z = Z, z E S'. 2) For ix E ilR and the corresponding fundamental vector field (ix) on S3 we have Z(ix) = ix. Thus Z is a connection in the bundle 7r : S3 -+ S2. We are going to compute its curvature. As S' is abelian, S2 = dZ, and thus, 0 = -dwl A dwl - dw2 A d1702

B.6. Reductions of connections

173

as a 2-form on S3 with values in M. Since S2 is a curvature form, S2 = 7r*1 for a 2-form f2 on CP' = S2. Let -y : S2\{north pole} --> C denote stereographic projection, and let SZ be a 2-form on C with Then ry o -7r : S3\{(w1, w2) : w2 5L 0} -f C is given by ry o 7r(w1, w2) = w2

and S2 has the form

--

dz A dz (1 + IzI2)2'

Hence, for the curvature form Il of the connection Z, we have f SZ/2iri = 1. S2

This equation means that cl(Hopf fibration)= -1. Remark. For the Sl-principal bundle (S3, 7r, CP'; S1) with Sl-action ((w1, w2), z) _ (W1z-1, W2z-1),

a similar argument shows that Z* _ -Z : TS3 i]R is a connection in that bundle. This implies SZ* = dZ* = -12 = dw1 A dw1 + dw2 A dzu2i -SZ

dzAdz (1 + Iz12)2 '

1

27ri S2

Hence, c1(6) = +1, and we have once again proved that this S'-principal bundle is not isomorphic to the Hopf fibration.

B.6. Reductions of connections Let (P, 7r, M; G) be a G-principal bundle over M'n, and let (P', 7r, M'z; G') together with f : P' -> P be a A-reduction of this bundle, i.e.:

f

1) A : G' -* G is a group homomorphism,

2) f : P' -> P is smooth and the diagram

P' commutes,

3) f (p'9') = f (p')A(9').

" M fI

P

B. Principal Bundles and Connections

174

Proposition. Let Z' : TP' (P',ir,M;G').

be a connection in the G'-principal bundle

1) There exists one and only one connection Z : TP -j g such that df : TP' -> TP maps the horizontal spaces with respect to Z' onto the horizontal spaces with respect to Z. 2) A*Z' = f *Z and, for the curvature forms, A*SZ' = f *Q.

Remark. 1) The connection Z constructed starting from the connection Z' is called the induced connection or the A-extension of Z'.

2) If G' is a subgroup of G and A G' -3 G the embedding, then Z' is called a reduction of the connection Z onto the subbundle :

(P', 7r, M; G).

Consider now a principal bundle (P, 7r, M; G) with connection, as well as a subgroup H C G and a subbundle (Q; nr, M; H). We ask the following question: When does a connection Z' exist in (Q, 7r, M; H) such that Z' is a reduction of Z? A provisional answer to this question is contained in the following:

Proposition. In the above notation, with the additional assumption that there exists a decomposition of the Lie algebra

g= lj em with Ad(H)(m) C m, we have: If Z : TP -+ g is a connection in (P, 7r, M; G), then Z' = prb o ZJTQ : TQ -> tj is a connection in Q.

If, in particular, Z : TP -+ g takes only values in the subalgebra [ , then Z reduces to a connection Z' : TQ -+ .

B.7. Frobenius' theorem The local Frobenius theorem in Euclidean space can be formulated as follows:

Theorem (Local Frobenius theorem in R n). Let U C R1 be an open subset and w1, ... wr 1-forms on U, n = r + s. Moreover, suppose that wr are linearly independent at every point and a) wl, . b) there exist 1-forms E on U with ,

.

,

r

dwz=EG'Awi. j=1

B.7. Frobenius' theorem

175

For x E U define

Es(x) = {t E TTU : w'(i) =

= wr'(t = 0}.

Then for every point xO E U there exists a regular s-dimensional surface piece FS with

1) xo E Fs, 2) y E Fs TyFs = Es(y)by an n-dimensional manifold, this leads to the notion of a

Replacing Ift

distribution:

Definition. Let Mh be a manifold. A differential system or a distribution on Mn is a selection of k-dimensional subspaces E. C TM' in every tangent space such that Ex depends smoothly on the point x in the following sense: For every x E Mn there exist a neighborhood U(x) and vector for tk on U(x) with Ey = Lin(tl (y), , Fk (y)) fields tl, ,

allyEU(x). Then Ek = U Ex is a smooth subvector bundle of the tangent bundle. X

Definition. Let Ek C TMn be a k-dimensional distribution. Ek is called integrable if the following condition is satisfied: For two vector fields fl, F2 on Mn with values in Ek, the commutator [tl, t2] also has values in Ek.

Theorem (Local Frobenius theorem on manifolds). Let Ek c TMn be an integrable k-dimensional distribution. For every point x E M there exist a neighborhood Ux and a submanifold x E Fk C Ux with TyFk = Ey for all

yEFk. Before turning to the global version of the Frobenius theorem we have to explain or extend a few general notions. To this end, recall the following definitions:

Definition. A smooth manifold without boundary is a pair (M, D), where 1) M is a topological T2-space with countable basis, and 2) D is a differentiable structure on M, i.e. a family D = {(UU, hi)}iET where Ui C M is open, hi : Ui -+ Vi C Rn are homeomorphisms and the hihi 1 are smooth.

Definition. Let (M, D) be a smooth manifold without boundary. A subset A C M is called a k-dimensional submanifold if

VaEA2 (U,cp)ED,aEU,cp:U-->VCllgn: cp(A fl U) is an open subset of I8k x {0}.

B. Principal Bundles and Connections

176

One then shows that (M, D(M)) is a manifold in the induced topology and with the atlas D(A) = {(A fl U, cPjAnu)}. Moreover, the embedding i : A --> M is smooth and di : TA --> TM is injective.

Definition. Let (M, D(M)) be a smooth manifold. A subset A C M is called a weak submanifold if there exist a smooth manifold (N, D(N)) and a differentiable mapping f : N --; M with the following properties: 1) f is injective.

2) f (N) = A. 3) df : Tn,N -3 Tf(n)M is injective.

If a E A is a point in the weak submanifold, then there is one and only one

n E N with f (n) = a. The space dfn(TnN) =: TaA is called the tangent space of A at the point a E A.

Example. Take M = T2 and let cp(t) = (eiat eiat), a//3 irrational, A = cp(R'). Then A C T2 is a (dense) weak submanifold which is not a submanifold.

Proposition. Let A C M be a weak submanifold and f: N -j A a model. For every point n E N there exists a neighborhood u E U(n) C N with the following properties:

1) f (U(n)) is a submanifold of M. 2) f : U(n) - f(U(n)) is a diffeomorphism.

Corollary. Let A C M be a weak submanifold, and let f : N -+ A, fl Ni -> A be two models. Then fi' o f : N - Nl is a diffeomorphism. Definition. Let Ek C TM be a distribution. An integral manifold of Ek is a weak submanifold A C M with TA = Ey for all y E A. Theorem (Global Frobenius theorem on manifolds). Let Ek C TM be an integrable distribution on a manifold M. Then for every point x E M there exists a weak submanifold A(x) with the following properties:

1) A(x) is an integral manifold of Ek, i.e.

TTA(x) = Ey V y E A(x). 2) A(x) is connected.

3) A(x) is maximal, i.e., if B is a connected integral manifold of Ek with A(x) C B, then A(x) = B.

B.9. Holonomy theory

177

B.8. The Freudenthal-Yamabe theorem Definition. Let G be a Lie group. A subset H C G is called a (weak) Lie subgroup if the following conditions are satisfied.

1) H is a subgroup. 2) H is a weak submanifold respecting the group structure, i.e. there exist a Lie group H and a smooth mapping f : H --* G such that a) f is injective,

b) f (H) = H, c) df is injective,

d) f is a group homomorphism.

Theorem (Freudenthal-Yamabe). Let G be a Lie group and H C G a subgroup with the following property: Each element of H can be connected with the neutral element e E G by a piecewise smooth curve, and this curve lies in H. Then H is a weak Lie subgroup.

B.9. Holonomy theory Let (P,,7r, M; G) be a principal bundle and Z : TP -> g a connection. (In this section, we suppose that M is connected.) Take, moreover, p E P and x = ir(p). If y is a curve in M (piecewise smooth) starting and ending at x, then we can consider the parallel displacement -ryPx - *Px. Let r-y(p) = p gy. Since -ry(p h) = Ty(p) h = p gy h for all h, it P5 is completely described by gy E G. The {g E G : there exists a loopy at x with ry(p) = p g} is an set 0(p) (algebraic) subgroup of G, obviously follows that Ty : Px

O(p) CG, p E P. In fact, if yl, 72 are two loops at x, then 7-yl*y2 = Tyl o rye, and hence gy1 *72 = g-

' gy2

Definition. The group O(p) is called the holonomy group of the connection Z with respect to the base point p E P. Furthermore, define

0° (p) = {g E G : 3 a loop y at x which is null-homotopic to -ry (p) = p g}.

For trivial reasons, 0°(p) C O(p) C G is a subgroup.

B. Principal Bundles and Connections

178

Proposition. 1) 0°(p) is a weak Lie subgroup of G.

2) 0°(p) is normal in /(p), and 0(p)/0°(p) is countable. Theorem (Reduction theorem of holonomy theory). Let (P, 7r, M; G) be a principal bundle with connected base Mn and Z : TP -> g a connection. For a fixed point p0 E P, denote by c(po) the holonomy group and by P(po) the set

P(po) = {p E P : there exists a horizontal path from po to p }. Then (P(po), 7r, M; O(po)) is a reduction of the principal bundle (P, 7r, M; G), and the connection Z reduces to this bundle.

Theorem (Ambrose-Singer, 1953). Let (P, 7r, M; G) be a principal bundle, M connected, and let Z : TP -* g be a connection with curvature form SZ = DZ. Let p0 E P be a fixed point and (P(po), ir, M, c5(po)) the reduction. Then the Lie algebra of the holonomy group ¢(p0) is generated by the elements SZ(X, Y) with X, Y E (TP)p and p E P(po) .

B.10. References S. Kobayashi, K. Nomizu, Foundations of differential geometry, Volume 1, Wiley 1963.

R. Sulanke, P. Wintgen, Differentialgeometrie and Faserbiindel, Deutscher Verlag der Wissenschaften, 1972. D. Husemoller, Fibre bundles, McGraw-Hill, 1966.

Bibliography

[1]

I. Agricola, Th. Friedrich, The Gaussian measure on algebraic varieties,

[2]

Fund. Math. 159 (1999), 91-98. I. Agricola, Th. Friedrich, Upper bounds for the first eigenvalue of the Dirac operator on surfaces, Journ. Geom. Phys. 30 (1999), 1-22.

[3]

I. Agricola, B. Ammann, Th. Friedrich, A comparison of the eigenvalues of the Dirac and Laplace operator on the two-dimensional torus, Manusc. Math. 100 (1999), 231-258.

[4] B. Alexandrov, G. Grantcharov, S. Ivanov, An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifolds admitting parallel one-form, Journ. Geom. Phys. 28 (1998), 263-270.

[5] B. Alexandrov, G. Grantcharov, S. Ivanov, The Dolbeaut operator on Hermitian spin surfaces, math.DG/9902005.

[6] B. Alexandrov, G. Grantcharov; S. Ivanov, Curvature properties of twistor spaces of quaternionic Kahler manifolds. Journ. Geom. 62 (1998), 1-12. [7] B. Alexandrov, S. Ivanov, Dirac operators on Hermitian spin surfaces, Preprint Univerity of Sofia 1998.

[8] B. Ammann, Spin-Strukturen and das Spektrum des Dirac-Operators, Dissertation Freiburg 1998

[9] B. Ammann, The Dirac operator on collapsing Sl-bundles, Seminaire de thdorie spectrale et geometrie vol.16 (1997/98), Univ. Grenoble I, Saint-Martind'Heres, 1998, 33-42.

[10] B. Ammann, Chr. Blir, The Dirac operator on nilmanifolds and collapsing circle bundles, Ann. Glob. Anal. Geom. 16 (1998), 221-253.

[11] L. Andersson, M. Dahl, Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global. Anal. Geom. 16 (1998), 1-27.

[12] M. Anghel, Extrinsic upper bounds for eigenvalues of Dirac type operators, Proc. AMS 117 (1993), 501-509.

[13] E. Artin, Geometric algebra, Princeton University Press, 1957. [14] M.F. Atiyah, Riemann surfaces and spin structures, Ann. Scient. Ecole Norm. Sup. 4 (1971) 47-62. 179

Bibliography

180

[15] M.F. Atiyah, R. Bott, A. Shapiro, Clifford modules, Topology 3 (1964), 3-38.

[16] M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and Riemannian geometry, Part I : Math. Proc. Cambridge Phil. Soc. 77 (1975), 43-69., Part II : Math. Proc. Cambridge Phil. Soc. 78 (1975), 405-432., Part III : Math. Proc. Cambridge Phil. Soc. 79 (1976), 71-79.

[17] M.F. Atiyah, I.M. Singer, The index of elliptic operators III, Ann. of Math. 87 (1968), 546-604.

[18] C. Bar, Das Spektrum von Dirac-Operatoren, Dissertation Bonn 1990, Bonner Math. Schr. 217 (1991). [19] C. Bar, Upper eigenvalue estimations for Dirac operators, Ann. Glob. Anal. Geom. 10 (1992), 171-177.

[20] C. Bar, Lower eigenvalue estimates for Dirac operators, Math. Ann. 293 (1992), 39 - 46. [21] C. Bar, Real Killing spinors and holonomy, Commun. Math. Phys. 154 (1993), 509-521.

[22] C. Bar, The Dirac operator on homogeneous spaces and its spectrum on 3dimensional lens spaces, Math. Ann. 309 (1997), 225-246. [23] C. Bar, Harmonic spinors for twisted Dirac operators, Arch. Math. 59 (1992), 65-79.

[24] C. Bar, The Dirac operator on space forms of positive curvature, Math. Ann. 309 (1997), 225-246.

[25] C. Bar, Metric with harmonic spinors, GAFA 6 (1996), 899-942. [26] C. Bar, Heat operator and zeta-function estimates for surfaces, Arch. Math. 71 (1998), 63-70.

[27] C. Bar, Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Glob. Anal. Geom. 16 (1998), 573-596. [28] C. Bar, On nodal sets for Dirac and Laplace operators, Commun. Math. Phys. 188 (1997), 709-721.

[29] C. Bar, The Dirac operator on hyperbolic manifolds of finite volume, Preprint SFB 256 (Bonn), No. 566, 1998.

[30] C. Bar, D. Bleecker, The Dirac operator and the scalar curvature of continuously deformed algebraic varieties, in Geometric Aspects of Partial Differential Equations (Roskilde, 1998), Contemp. Math., vol. 242, Amer. Math. Soc., Providence, RI, 1999, 3-24.

[31] C. Bar, P. Schmutz, Harmonic spinors on Riemann surfaces, Ann. Glob. Anal. Geom. 10 (1992), 263-273. [32] H. Baum, Spin-Strukturen and Dirac-Operatoren uber pseudo-Riemannschen Mannigfaltigkeiten, Teubner-Verlag Leipzig 1981. [33] H. Baum, The index of the pseudo-Riemannian Dirac operator as a transversally elliptic operator, Ann. Glob. Anal. Geom. 1 (1983), 11-20. [34] H. Baum, 1-forms over the moduli space of irreducible connections defined by the spectrum of Dirac operators, Journ. Geom. Phys. 4 (1987), 503 -521 [35] H. Baum, Varietes riemanniennes admettant des spineurs de Killing imaginaires, C.R. Acad. Sci. Paris Serie I, 309 (1989), 47-49. [36] H. Baum, Odd-dimensional Riemannian manifolds with imaginary Killing spinors, Ann. Glob. Anal. Geom. 7 (1989), 141-154.

Bibliography

181

[37] H. Baum, Complete Riemannian manifolds with imaginary Killing spinors, Ann. Glob. Anal. Geom. 7 (1989), 205-226. [38] H. Baum, An upper bound for the first eigenvalue of the Dirac operator on compact spin manifolds, Math. Zeitschrift 206 (1991), 409-422. [39] H. Baum, The zeta-invariant of the Dirac operator coupled to instantons, Preprint des SFB 288 No. 90, Berlin 1993.

[40] H. Baum, A remark on the spectrum of the Dirac operator on pseudoRiemannian spin manifolds, Preprint SFB 288 No. 136, Berlin 1994. [41] H. Baum, Eigenvalues estimates for the Dirac operator coupled to instantons, Ann. Glob. Anal. Geom. 12 (1994), 193-209. [42] H. Baum, The Dirac operator on Lorentzian spin manifolds and the Huygens property, Journ. Geom. Phys. 23 (1997), 42-64. [43] H. Baum, Strictly pseudoconvex spin manifolds, Fefferman spaces and Lorentzian twistor spinors, Preprint No. 250 SFB 288, 1997.

[44] H. Baum, Th. Friedrich, Eigenvalues of the Dirac operator, twistors and Killing spinors on Riemannian manifolds, in "Clifford Algebras and Spinor Structures" (ed. by R.Ablamowicz and R.Lounesto), Kluwer Academic Publisher 1995, 243-256.

[45] H. Baum, I. Kath, Normally hyperbolic operators, the Huygens property and conformal geometry, Ann. Glob. Anal. Geom. 14 (1996), 315-371.

[46] H. Baum, Th. Friedrich, R. Grunewald, I. Kath, Twistors and Killing spinors on Riemannian manifolds, Teubner-Verlag Leipzig/Stuttgart 1991.

[47] N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operators, Springer Verlag 1992.

[48] W. Biedrzycki, Spinors over a cone, Dirac operator and representations of Spin(4,4), Journ. Funct. Anal. 113 (1993), 36-64. [49] E. Binz, R. Pferschy, The Dirac operator and the change of the metric, C.R. Math. Rep. Acad. Sci. Canada V (1983), 269-274. [50] E. Bonan, Sur les varietes riemanniennes a groupe d'holonomie G2 ou Spin(7), C.R. Acad. Sci. Paris 262 (1966), 127-129.

[51] B. Booss-Bavnbek, K.P. Wojciechowski, Elliptic boundary problems for Dirac operators, Birkhauser-Verlag 1993.

[52] B. Booss-Bavnbek, G. Morchio, F. Strocchi, K.P. Wojciechowski, Grassmannian and chiral anomaly, Journ. Geom. Phys. 22 (1997), 219-244. [53] M. Bordoni, Spectral estimates for Schrodinger and Dirac-type operators on Riemannian manifolds, Math. Ann. 298 (1994), 693-718. [54] M. Bordoni, Comparaison de spectres d'operateurs de type Schrodinger et Dirac, Seminaire de theorie spectrale et geometrie no. 14, Institut Fourier Gre. noble (1996), 69-81.

[55] J.P. Bourguignon, P. Gauduchon, Spineurs, operateurs de Dirac et variations de metriques, Commun. Math. Phys. 144 (1992), 581-599. [56] C.P. Boyer, K. Galicki, New Einstein metrics in dimension five, math.DG/0003174.

[57] C.P. Boyer, K. Galicki, B. Mann, Geometry and topology of 3-Sasakian manifolds, Journ. Reine and Angew. Mathematik 455 (1994), 183-220.

[58] C.P. Boyer, K. Galicki, B. Mann, Some new examples of inhomogeneous hypercomplex manifolds, Math, Res. Letters 1 (1994), 531-538.

Bibliography

182

[59] C.P. Boyer, K. Galicki, B. Mann, Hypercomplex structures on Stiefel manifolds, Ann. Glob. Anal. Geom. 4 (1996), 81-105.

[60] C.P. Boyer, K. Galicki, B. Mann, On strongly inhomogeneous Einstein manifolds, Bull. London Math. Soc. 28 (1996), 401-408.

[61] C.P. Boyer, K. Galicki, B. Mann, E. Rees, 3-Sasakian manifolds with an arbitrary second Betti number, Invent. Math. 131 (1998), 321-344 (1998). [62] D. Brill, J. Wheeler, Interaction of neutrinos and gravitational fields, Revs. Modern Phys. 29 (1957), 465-479. [63] R. Bryant, Metrics with exceptional holonomy, Ann. Math. 126 (1987), 525576.

[64] R. Bryant, S. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke. Math. Journ. 58 (1989), 829-850. [65] V. Buchholz, Die Dirac- and Twistorgleichung in der konformen Geometrie, Diplomarbeit, Humboldt-Universitat zu Berlin 1998. [66] V. Buchholz, Spinor equations in Weyl geometry, Preprint SFB 288 No. 363 (1999), to appear in Suppl. Rend. Circ. Math. di Palermo. [67] V. Buchholz, A note on real Killing spinors in Weyl geometry, to appear in Journ. Geom. Phys. [68] J. Budczies, Dirac-Operatoren auf Riemann-Cartan-Raumen beliebiger Dimension, Diplomarbeit, Institut fur Theoretische Physik, Universitat Kbin, 1996.

[69] B. Budinich, A. Trautman, The spinorial chessboard, Springer-Verlag 1988. [70] U. Bunke, Upper bounds of small eigenvalues of the Dirac operator and isometric immersions, Ann. Glob, Anal. Geom. 9 (1991), 211-243. [71] U. Bunke, The spectrum of the Dirac operator on the hyperbolic space, Math. Nachr. 153 (1991), 179-190.

[72] U. Bunke, On the spectral flow of families of Dirac operators with constant symbol, Math. Nachr. 165 (1994), 191-203. [73] U. Bunke, On the glueing problem for the 77-invariant, Journ. Diff. Geom. 41 (1995), 397-448.

[74] U. Bunke, A K-theoretic relative index theorem, Math. Ann. 303 (1995), 241-279.

[75] U. Bunke, T. Hirschmann, The index of the scattering operator on the positive spectral subspace, Commun. Math. Phys. 148 (1992), 487-502. [76] J. Bures", V. Soucek, Eigenvalues of conformally invariant operators on spheres, Proceed. 18th Winter School "Geometry and Physics" (Srni, 1998), Rend. Sem. Mat. Palermo (2) Suppl. No. 59 (1999), 109-122.

[77] F.M. Cabrera, M.D. Monar, A.F. Swann, Classification of G2-structures, Journ. London Math. Soc. II. Serie 53 (1996), 407-416.

[78] M. Cahen, A. France, S. Gutt, Spectrum of the Dirac operator on complex projective space Qpl2q-1l, Lett. Math. Phys. 18 (1989), 165-176.

[79] M. Cahen, A. France, S. Gutt, Erratum to "Spectrum of the Dirac operator on complex projective space CP(2q'-1)", Lett. Math. Phys. 32 (1994), 365-368.

[80] M. Cahen, S. Gutt, Spin structures on compact simply connected Riemannian symmetric spaces, Simon Stevin 62, No. 3/4 (1988), 209-242.

[81] M. Cahen, S. Gutt, L. Lemaire, P. Spindel, Killing Spinors, Bull. Soc. Math. Belg. 38 (1986), 75-102.

Bibliography

183

[82] M. Cahen, S. Gutt, A. Trautman, Spin structures on real projective quadrics, Journ. Geom. Phys. 10 (1993), 127-154.

[83] M. Cahen, S. Gutt, A. Trautman, Pin structures and the modified Dirac operator, Journ. Geom. Phys. 17 (1995), 283-297. [84] E. Cartan, La theorie des spineurs, Paris, Hermannn 1937 (2. edition 1966). [85] D.M.J. Calderbank, Clifford analysis for Dirac operators on manifolds with boundary, MPI Bonn, 1996, Preprint No. 96-131.

[86] D.M.J. Calderbank, Dirac operators and conformal geometry, lecture at Banach Center Warschau, 22.02.1997.

[87] R. Camporesi, A. Higuchi, On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces, Journ. Geom. Phys. 20 (1996), 1-18. [88] C. Chevalley, Algebraic theory of spinors, Columbia University Press, New York 1954.

[89] A.W. Chou, The Dirac operator on spaces with conical singularities and positive scalar curvature, Trans. Amer. Math. Soc. 289 (1985), 1-40. [90] R.L. Cohen, J.D.S. Jones, Monopols, braid groups and the Dirac operator, Commun. Math. Phys. 158 (1993), 241-266.

[91] L. Dabrowski, A. Trautman, Spinor structures on spheres and projective spaces, Journ. Math. Phys. 27 (1986), 2022-2028.

[92] M. Dahl, The positive mass theorem for ALE manifolds, in Mathematics of Gravitation, Part I (Warsaw 1996), Banach Center Publ., vol. 41, part I, Polish Acad. Sci., Warsaw, 1997, 133-142.

[93] H. Dlubek, Th. Friedrich, Spectral properties of the Dirac operator, Bulletin de L'Academie Polonaise de Sciences, Series des Sciences Mathematiques XXVII (1979), 621-624.

[94] H. Dlubek, Th. Friedrich, Spektraleigenschaften des Dirac-Operators - die Fundamentallosung seiner Warmeleitungsgleichung and die Asymptotenentwicklung der Zeta-Funktion, Journ. Diff. Geom. 15 (1980), 1-26.

[95] H. Dlubek, Th. Friedrich, Immersionen hoherer Ordnung kompakter Mannigfaltigkeiten in Euklidischen Raumen, Beitrage zur Algebra and Geometrie 9 (1980), 83-101.

[96] M.J. Duff, B.E.W. Nilson, C.N. Pope, Kaluza-Klein supergravity, Physics Reports 130 (1986), 1-142.

[97] J.J. Duistermaat, The heat kernel, Lefschetz fixed point formula for the Spnnc Dirac operator, Birkhauser Verlag 1996. [98] J. Eichhorn, Elliptic differential operators on noncompact manifolds, in Sem. Anal. Karl-Weierstrass-Inst. Math. (Berlin 1986/87), Teubner-Texte zur Mathematik 106, Teubner-Verlag Leipzig (1988), 4-169.

[99] J. Eichhorn, Th. Friedrich, Seiberg-Witten theory, in Symplectic Singularities and Geometry of Gauge Fields (Warsaw, 1995), Banach Center Publ., vol. 39, Polish Acad. Sci., Warsaw, 1997, 231-267. [100] P.M. Feehan, A Kato-Yau inequality for harmonic spinors and decay estimate for eigenspinors, math.DG/9903021. [101] A. Fino, Intrinsic torsion and weak holonomy, Math. Journ. of Toyama University 21 (1998), 1-22. [102] A. Franc, Spin structures and Killing spinors on lens spaces, Journ. Geom. Phys. 4 (1987), 277-287.

Bibliography

184

[103] M.H. Freedman, The topology of four-dimensional manifolds, Journ. Diff. Geom. 17 (1982), 357-453.

[104] Th. Friedrich, Vorlesungen fiber K-Theorie, Teubner-Verlag Leipzig 1978. [105] Th. Friedrich, Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrummung, Math. Nachr. 97 (1980), 117-146.

[106] Th. Friedrich, Zur Existenz paralleler Spinorfelder fiber Riemannschen Mannigfaltigkeiten, Coll. Math. XLIV (1981), 277-290. [107] Th. Friedrich, Die Abhangigkeit des Dirac-Operators von der Spin-Struktur, Coll. Math. XLVII (1984), 57-62. [108] Th. Friedrich, Self-duality of Riemannian manifolds and connections, in "Riemannian Geometry and Instantons", Teubner-Verlag Leipzig 1981, 56-104.

[109] Th. Friedrich, A remark on the first eigenvalue of the Dirac operator on 4-dimensional manifolds, Math. Nachr. 102 (1981), 53-56. [110] Th. Friedrich, On surfaces in four space, Ann. Global Anal. Geom. 2 (1984), 257-284.

[111] Th. Friedrich, Riemannian manifolds with small eigenvalues of the Dirac operator, Proceedings of the 27th Arbeitstagung, Bonn 12.-19.06.1987, Preprint MPI in Bonn.

[112] Th. Friedrich, The geometry of t-holomorphic surfaces in S4, Math. Nachr. 137 (1988), 49-62.

[113] Th. Friedrich, On the conformal relation between twistors and Killing spinors, Supplemento de Rendiconti des Circole Mathematico de Palermo, Serie II, No. 22 (1989), 59-75.

[114] Th. Friedrich, The classification of 4-dimensional Kahler manifolds with small eigenvalue of the Dirac operator, Math. Ann. 295 (1993), 565-574.

[115] Th. Friedrich, Harmonic spinors on conformally flat manifolds with S-symmetry, Math. Nachr. 174 (1995), 151-158. [116] Th. Friedrich, Neue Invarianten der 4-dimensionalen Mannigfaltigkeiten, Preprint SFB 288 No. 156 (1995).

[117] Th. Friedrich, Dirac-Operatoren in der Riemannschen Geometrie, Adv. Lectures in Mathematics, Vieweg-Verlag Braunschweig/Wiesbaden 1997.

[118] Th. Friedrich, On superminimal surfaces, Archivum Mathematicum 33 (1997), 41-56.

[119] Th. Friedrich, On the spinor representation of surfaces in Euclidean 3-spaces, Journ. Geom. Phys. 28 (1998), 143-157. [120] Th. Friedrich, Cartan spinor bundles on manifolds, SFB 288 Preprint Nr. 303 (1998), to appear in Reports on Mathematical Physics. [121] Th. Friedrich, A geometric estimate for the periodic Sturm-Liouville operator whose potential is the curvature of a spherical curve, Preprint SFB 288 No. 382 (1999), math.DG/9904123, to appear in Coll. Mathematicum. [122] Th. Friedrich, Weak Spin(9)-structures on 16-dimensional Riemannian mar nifolds, Preprint SFB 288, No. 437 (2000), math.DG/9912112. [123] Th. Friedrich, Solutions of the Einstein-Dirac equation on Riemannian 3manifolds with constant scalar curvature, math.DG/0002182. [124] Th. Friedrich, R. Grunewald, On Einstein metrics on the twistor space of a four-dimensional Riemannian manifold, Math. Nachr. 123 (1985), 55-60.

Bibliography

185

[125] Th. Friedrich, R. Grunewald, On the first eigenvalue of the Dirac operator on 6-dimensional manifolds, Ann. Glob. Anal. Geom. 3 (1985), 265-273.

[126] Th. Friedrich, I. Kath, Einstein maniolds of dimension five with small first eigenvalue of the Dirac operator, Journ. Diff. Geom. 29 (1989), 263-279. [127] Th. Friedrich, I. Kath, Compact 5-dimensional Riemannian manifolds with parallel spinors, Math. Nachr. 147 (1990), 161-165.

[128] Th. Friedrich, I. Kath, Varietes riemanniennes compactes de dimension 7 admettant des spineurs de Killing, C.R. Acad. Sci. Paris Serie I, 307 (1988), 967-969.

[129] Th. Friedrich, I. Kath, 7-dimensional compact Riemannian manifolds with Killing spinors, Commun. Math. Phys. 133 (1990), 543-561.

[130] Th. Friedrich, I. Kath, A. Moroianu, U. Semmelmann, On nearly parallel G2-structures, Journ. Geom. Phys. 23 (1997), 259-286

[131] Th. Friedrich, E.C. Kim, The Einstein-Dirac equation on Riemannian spin manifolds, Journ. Geom. Phys. 33 (2000), 128-172.

[132] Th. Friedrich, E.C. Kim, Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors, math.DG/9906168, to appear in Journ. Geom. Phys. [133] Th. Friedrich, H. Kurke, Compact four-dimensional self-dual Einstein mar nifolds with positive scalar curvature, Math. Nachr. 106 (1982), 271-299.

[134] Th. F7riedrich, 0. Pokorna, Twistor spinors and the solutions of the equation (E) on Riemannian manifolds, Supplemento de Rendiconti des Circole Mathematico de Palermo, Serie II, No. 26 (1991), 149-153.

[135] Th. Friedrich, S. Sulanke, Ein Kriterium fur die formale Selbstadjungiertheit des Dirac-Operators, Coll. Math. XL (1979), 239-247.

[136] Th. Friedrich, A. Trautman, Clifford structures and spinor bundles, Preprint SFB 288 No. 251 (1997). [137] Th. Friedrich, A. Trautman, Spin spaces, Lipschitz groups and spinor bundles, math.DG/9901137, SFB-Preprint No. 362 (1999), to appear in Ann. Glob. Anal. Geom. [138] K. Galicki, S. Salamon, Betti numbers of 3-Sasakian manifolds, Geom. Dedicata 63 (1996), 45-68.

[139] P. Gauduchon, Hermitian connections and Dirac operators, Boll. Un. Mat. Ital. B (7) 11 (1997), 257-289.

[140] J. Gilbert, A. Murray, Clifford algebras and Dirac operators in harmonic analysis, Cambridge University Press 1991. [141] P.B. Gilkey, Invariance theory, the heat equation and the Atiyah-Singer index theorem, Publish or Perish 1984.

[142] P.B. Gilkey, J.V. Leahy, J.H. Park, Eigenforms of the spin Laplacian and projectable spinors for principal bundles, Nuclear Physics B 514 (1998), 740-752.

[143] S. Goette, U. Semmelmann, The point spectrum of the Dirac operator on noncompact symmetric spaces, math.DG/9903177.

[144] S. Goette, U. Semmelmann, Spin° structures and scalar curvature estimates, Preprint 1999. [145] A. Gray, Vector cross products on manifolds, Trans. Amer. Math. Soc. 141 (1969), 465-504.

Bibliography

186

[146] A. Gray, Weak holonomy groups, Math. Zeitschrift 123 (1971), 290-300.

[147] R. Grunewald, Six-dimensional Riemannian manifolds with a real Killing Spinor, Ann. Glob. Anal. Geom. 8 (1990), 43-59.

[148] R. Grunewald, On the relation between real Killing spinors and almost Hermitian structures, Preprint No. 271, Fachbereich Mathematik der HumboldtUniversitat zu Berlin, 1991.

[149] S. Gutt, Killing spinors on spheres and projective spaces, In "Spinors in Physics and Geometry", World Sci. Publ. Co., Singapore 1988, 238-248.

[150] K. Habermann, The twistor equation on Riemannian manifolds, Journ. Geom. Phys. 7 (1990), 469-488.

[151] K. Habermann, Twistor spinors and their zeroes, Journ. Geom. Phys. 14 (1992), 1-24.

[152] K. Habermann, The graded algebra and the conformal Lie derivative of spinor fields related to the twistor equation, Journ. Geom. Phys. 18 (1996), 131-146. [153] M. Herzlich, Scalar curvature and rigidity of odd-dimensional complex hyperbolic spaces, Math. Ann. 312 (1998), 641-657.

[154] M. Herzlich, A. Moroianu, Generalized Killing spinors and conformal eigenvalue estimates for Spin` manifolds, Ann. Glob. Anal. Geom. 17 (1999), 341-370.

[155] G. Hess, On the existence of a spectral function decomposition for the Dirac operator on a semi-Riemannian manifold, Preprint Universitat Munchen 1996. [156] G. Hess, Canonically generalized spin structures and Dirac operators on semiRiemannian manifolds, Dissertation Ludwig-Maximilian-Universitat Munchen 1996.

[157] O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Commun. Math. Phys. 104 (1986), 151-162. [158] O. Hijazi, Caracterisation de la sphere par les premieres valeurs propres de l'operateur de Dirac en dimension 3,4,7 et 8, C.R. Acad. Sci. Paris Serie I Math. 303 (1986), 417-419.

[159] O. Hijazi, Eigenvalues of the Dirac operator on compact Kahler manifolds, Commun. Math. Phys. 160 (1994), 563-579. [160] O. Hijazi, Lower bounds for eigenvalues of the Dirac operator, Journ. Geom. Phys. 16 (1995), 27-38.

[161] O. Hijazi, Spectral properties of the Dirac operator and geometrical structures, Lecture Notes in Nantes, March 1998, Http://ieca.u-nancy.fr/ hijazi/socrates.ps.

[162] O. Hijazi, A. Lichnerowicz, Spineurs harmonique, spineurs-twisteurs et geometrie conforme, C.R. Acad. Sci. Paris Serie I Math. 307 (1988), 833-838.

[163] O. Hijazi, J: L. Milhorat, Minoration des valeurs propres de l'operateur de Dirac sur les varietes spin Kahler-quaternioniennes, Journ. Math. Pures Appl. 74 (1995), 387-414.

[164] O. Hijazi, J: L. Milhorat, Twistor operators and eigenvalues of the Dirac operator on compact quaternionic Kahler spin manifold, Ann. Glob. Anal. Geom. 15 (1997), 117-131.

[165] O. Hijazi, J.-L. Milhorat, Decomposition spectrale du fibre des spineurs d'une variete spin Kahler-quaternienne, 332.

Journ. Geom. Phys. 15 (1995), 320-

Bibliography

187

[166] F. Hirzebruch, H. Hopf, Felder von Flachenelementen in 4-dimensionalen Mannigfaltigkeiten, Math. Ann. 136 (1958), 156 -172. [167] N. Hitchin, Harmonic spinors, Adv. in Mathematics 14 (1974), 1-55.

[168] N. Hitchin, Kahlerian twistor spaces, Proc. Lond. Math. Soc. III Ser., 43 (1981), 133-150.

[169] D. Husemoller, Fibre bundles, McGraw-Hill, New York 1966. [170] A. Ikeda, Formally self adjointness for the Dirac operator on homogeneous spaces, Osaka Journ. Math. 12 (1975), 173-185. [171] D. Johnson, Spin structures and quadratic forms, Journ. London Math. Soc. 22 (1980), 365-373.

[172] D. Joyce, Compact Riemannian 7-manifolds with holonomy G2 (Parts I and II), Journ. Diff. Geom. 43 (1996), 291-328 and 329-375. [173] D. Joyce, Compact 8-manifolds with holonomy Spin(7), Invent. Math. 123 (1996), 507-552.

[174] M. Karoubi, Algebres de Clifford et K-Theorie, Ann. Scient. Ec. Norm. Sup. 4 ser. 1 (1968), 161-270. [175] G. Karrer, Einfuhrung von Spinoren auf Riemannschen Mannigfaltigkeiten, Annales Academiae Scientiarum Fennicae, Series A 336/5, Helsinki 1963. [176] I. Kath, Varietes riemanniennes de dimension 7 admettant un spineur de Killing reel, C.R. Acad. Sci. Paris Serie I Math. 311 (1990), 553-555. [177] I. Kath, G2-structures on pseudo-Riemannian manifolds, Journ. Geom. Phys. 27 (1998), 155-177.

[178] I. Kath, Pseudo-Riemannian T-duals of compact Riemannian reductive spaces, Preprint No. 253 SFB 288 (1997), to appera in Transf. Groups. [179] I. Kath, Parallel pure spinors on pseudo-Riemannian manifolds, Preprint No. 356 SFB 288 (1998). [180] E.C. Kim, Die Einstein-Dirac-Gleichung uber Riemannschen SpinMannigfaltigkeiten, Dissertation Humboldt-Universitat, Berlin, 1999.

[181] R.C. Kirby, L.R. Taylor, Pin structures on low-dimensional manifolds, in "Geometry of Low-dimensional Manifolds" Part 2 (ed. by S.K.Donaldson), Lon-

don Math. Soc. Lecture Notes Series 151, Cambridge University Press 1990, 177-242.

[182] K: D. Kirchberg, An estimation for the first eigenvalue of the Dirac operator on closed Kahler manifolds with positive scalar curvature, Ann. Glob. Anal. Geom. 4 (1986), 291-326.

[183] K.-D. Kirchberg, Compact six-dimensional Kahler spin manifolds of positive scalar curvature with the smallest possible first eigenvalue of the Dirac operator, Math. Ann. 282 (1988), 157-176. [184] K.-D. Kirchberg, Twistor spinors on Kahler manifolds and the first eigenvalue of the Dirac operator, Journ. Geom. Phys. 7 (1990), 449-468. [185] K.-D. Kirchberg, Properties of Kahlerian twistor spinors and vanishing theorems, Math. Ann. 293 (1992), 349-369. [186] K.-D. Kirchberg, Some further properties of Kahlerian twistor spinors, Math. Nachr. 163 (1993), 229-255.

[187] K.-D. Kirchberg, Killing spinors on Kahler manifolds, Ann. Glob. Anal. Geom. 11 (1993), 141-164.

Bibliography

188

[188] K.D. Kirchberg, Holomorphic spinors and the Dirac equation, Ann. Glob. Anal. Geom. 17 (1999), 97-111.

[189] K.-D. Kirchberg, U. Semmelmann, Complex structures and the first eigenvalue of the Dirac operator on Kahler manifolds, Geom. Funct. Analysis 5 (1995), 604-618.

[190] T. Kori, Index of the Dirac operator on S4 and the infinite dimensional Grassmannian on S3, Japanese Journal of Mathematics 22 (1996), 1-36. [191] T. Kori, Chiral anomaly and Grassmannian boundary conditions, Preprint, in Geometric Aspects of Partial Differential Equations (Roskilde, 1998), Contemp. Math., vol. 242, Amer. Math. Soc., Providence, RI, 1999, 35-42.

[192] T. Kori, Meromorphic zero-mode spinors on locally conformally flat 4manifolds, Preprint, Waseda University Tokyo 1998.

[193] Y. Kosmann, Derivees de Lie des spineurs, Ann. Mat. Pura ed Appl. 91 (1972), 317-395.

[194] D. Kotschick, Non-trivial harmonic spinors on certain algebraic surfaces, Proc. Amer. Math. Soc. 124 (1996), 2315-2318.

[195] W. Kramer, U. Semmelmann, G. Weingart, Eigenvalue estimates for the Dirac operator on quaternionic Kahler manifolds, Math. Zeitschrift 230 (1999), 727-751.

[196] W. Kramer, U. Semmelmann, G. Weingart, Quaternionic Killing spinors, Ann. Global Anal. Geom. 16 (1998), 63-87.

[197] W. Kramer, U. Semmelmann, G. Weingart, The first eigenvalue of the Dirac operator on quaternionic Kahler manifolds, Commun. Math. Phys. 199 (1998), 327-349.

[198] M. Kraus, Lower bounds for eigenvalues of the Dirac operator on surfaces of rotation, Journ. Geom. Physics 31 (1999), 209-216. [199] M. Kraus, Lower bounds for eigenvalues of the Dirac operator on n-spheres with SO(n)-symmetry, Journ. Geom. Phys. 32 (2000), 341-348. [200] M. Kraus, Eigenvalue estimates for the Dirac operator on the 2-dimensional torus with Sl-symmetry, to appear in Ann. Global Anal. Geom. [201] W. Kiihnel, H.-B. Rademacher, Twistor spinors with zeros and conformal flatness, C.R. Acad. Sci. Paris, Serie I, 318 (1994), 237-240.

[202] W. Kiihnel, H.-B. Rademacher, Twistor spinors with zeros, Inter. Journ. of Math. 5 (1994), 877-895.

[203] W. Kiihnel, H.-B. Rademacher, Twistor spinors and gravitational instantons,.Lett. Math. Phys 38 (1996), 411-419.

[204] W. Kilhnel, H: B. Rademacher, Conformal completion of U(n)-invariant Ricci-flat Kahler metrics at infinity, Zeitschr. Anal. Anwend. 16 (1997), 113117.

[205] W. Kiihnel, H: B. Rademacher, Twistor spinors on conformally flat manifolds, Illinois. Journ. Math. 41 (1997), 495-503.

[206] W. Ktlhnel, H: B. Rademacher, Asymptotically Euclidean manifolds and twistor spinors, Commun. Math. Phys. 196 (1998), 67-76.

[207] R. Kusner, N. Schmitt, The spinor representation of surfaces in space, Preprint 1996 (dg-ga 9610005)

[208] H.B. Lawson, M.-L. Michelsohn, Spin geometry, Princeton University Press 1989.

Bibliography

189

[209] M. Llarull, Sharp estimates and the Dirac operator, Math.Ann. 310 (1998), 55-71.

[210] J. Lewandowski, Twistor equations in a curved spacetime,

Class.

Quant.

Grav. 8 (1991), 11-17.

[211] A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris 257 (1963), 7-9.

[212] A. Lichnerowicz, Varietes spinorielles et universality de l'inegalite de Hijazi, C.R. Acad. Sci. Paris Serie I Math. 304 (1987), 227-231. [213] A. Lichnerowicz, Spin manifolds, Killing spinors and the universality of the Hijazi inequality, Lett. Math. Phys. 13 (1987), 331-344. [214] A. Lichnerowicz, Les spineurs-twisteurs sur une variete spinorielle compacte, C.R. Acad. Sci. Paris Serie I Math. 306 (1988), 381-385. [215] A. Lichnerowicz, Sur les resultats de H. Baum et Th. Friedrich concernant les spineurs de Killing a valeur propre imaginaire, C.R. Acad. Sci. Paris Serie I Math. 306 (1989), 41-45. [216] A. Lichnerowicz, On the twistor spinors, Lett. Math. Phys. 18 (1989), 333345.

[217] A. Lichnerowicz, Sur les zeros des spineurs-twisteurs, C.R. Acad. Sci. Paris Serie I Math. 310 (1990), 19-22. [218] A. Lichnerowicz, La premiere valeur propre de l'operateur de Dirac pour une variete Kahlerienne et son cas limite, C.R. Acad. Sci. Paris Serie I Math. 311 (1990), 717-722.

[219] A. Lichnerowicz, Spineurs harmoniques et spineurs-twisteurs en geometrie kahlerienne et conformement kahlerienne, C.R. Acad. Sci. Paris Serie I Math. 311 (1990), 883-887.

[220] A. Lichnerowicz, Spineurs-twisteurs hermitiens et geometrie conformement kahlerienne, C.R. Acad. Sci. Paris Serie I Math. 314 (1992), 841-846. [221] J. Lott, Eigenvalue bounds for the Dirac operator, Pac. Journ. Math. 125 (1986), 117-126.

The Dirac operator and conformal compactification, [222] J. Lott, math.DG/0003140. [223] St. Maier, Generic metrics and connections on Spin- and Spina-manifolds, Ann. Global Anal. Geom. 16 (1998), 63-87. [224] K. Maurin, Methods of Hilbert Spaces, PWN, Warsaw 1965. [225] K.-H. Mayer, Elliptische Differentialoperatoren and Ganzzahligkeitssatze fur charakteristische Klassen, Topology 4 (1965), 295-313. [226] B. McInnes, Existence of parallel spinors on nonsimply connected Riemannian manifolds, Journ. Math. Phys. 39 (1998), 2362-2366. [227] M.L. Michelsohn, Clifford and spinor cohomology of Kahler manifolds, Amer. Journ. of Math. 102 (1980), 1083-1146.

[228] J.-L. Milhorat, Spectre de l'operateur de Dirac sur les espaces projectifs quaternioniens, C.R. Acad. Sci. Paris, Serie I Math. 314 (1992), 69-72. [229] J. Milnor, Remarks concerning Spin-manifolds, in "Differential and Combinatorical Topology" (in honour of Marsten Morse), Princeton 1965, 55-62. [230] M. Min-Oo, Scalar curvature rigidity of asymptotically hyperbolic spin manifolds, Math. Ann. 285 (1989), 527-539.

Bibliography

190

[231] A. Moroianu, La premiere valeur propre de l'operateur de Dirac sur les varietes kahleriennes compactes, Commun. Math.Phys. 169 (1995), 373-384, annonce dans C. R. Acad. Sci. Paris Serie I, 319 (1994), 1057-1062. [232] A. Moroianu, Spineurs et varietes de Hodge, Rev. Roum. Maths. Pures Appl. 43 (1998), 615-626.

[233] A. Moroianu, Sur les valeurs propres de l'operateur de Dirac d'une variete spinorielle simplement connexe admettant une 3-structure de Sasaki, Studii si Cercetari Matematice 48 (1996), 85-88.

[234] A. Moroianu, Formes harmoniques en presence de spineur de Killing kahleriens, C.R. Acad. Sci. Paris Serie I, 322 (1996), 679-684.

[235] A. Moroianu, Structures de Weyl admettant des spineurs paralleles, Bull. Soc. Math. R. 124 (1996), 685-695. [236] A. Moroianu, On Kirchberg's inequality for compact Kahler manifolds of even complex dimension, Ann. Glob. Anal. Geom. 15 (1997), 235-242. [237] A. Moroianu, On the infinitesimal isometries of manifolds with Killing spinors, Preprint SFB 288 No. 240, Berlin 1996.

[238] A. Moroianu, Parallel and Killing spinors on SpinC manifolds, Commun. Math. Phys. 187 (1997), 417-428.

[239] A. Moroianu, SpinC manifolds and complex contact structures, Commun. Math. Phys. 193 (1998), 661-673.

[240] A. Moroianu, Kahler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz, Ann. Inst. Fourier (Grenoble) 49 (1999), 1637-1659.

[241] A. Moroianu, U. Semmelmann, Kahlerian Killing spinors, complex contact structures and twistor spaces, C,R. Acad. Sci. Paris Series I Math. 323 (1996), 57-61.

[242] A. Moroianu, U. Semmelmann, Parallel spinors and holonomy groups, math.DG/9903062.

[243] P. van Nieuwenhuizen, N.P. Warner, Integrability conditions for Killing spinors, Commun. Math. Phys. 93 (1984), 227-284.

[244] B.E. Nilson, C.N. Pope, Scalar and Dirac eigenfunctions on the squashed seven-sphere, Phys. Lett. B 133 (1983), 67-71. [245] A.L. Onishchik, R. Sulanke, Algebra and Geometrie Teil II, VEB Deutscher Verlag der Wissenschaften, Berlin 1988.

[246] C. Ohn, A.-J. Vanderwinden, The Dirac operator on compactified Minkowski space, Acad. Roy. Belg. Bull. Cl. Sci. (6) 4 (1993), 255-268.

[247] R. Parthasarathy, Dirac operator and the discrete series, Ann. Math. 96 (1972), 1-30.

[248] T. Parker, H. Taubes, On Witten's proof of the positive energy theorem, Commun. Math. Phys. 84 (1982), 223-238.

[249] F. Pfafe, The Dirac spectrum of Bieberbach manifolds, to appear in Journ. Geom. Phys. [250] H.-B. Rademacher, Generalized Killing spinors with imaginary Killing function and conformal Killing fields, in "Global Differential Geometry and Global Analysis", Proc. Berlin 1990, Springer Lecture Notes Math. 1481 (1991), 192198.

Bibliography

191

[251] W. Reichel, Uber die Trilinearen alternierenden Formen in 6 and 7 Veranderlichen, Dissertation, Greifswald 1907. [252] S. Salamon, Spinors and cohomology, Rend. Sem. Mat. Univ. Pol. Torino 50 (1992)

[253] E. Schrodinger, Diracsches Elektron im Schwerefeld I, Sitzungsbericht der Preussischen Akademie der Wissenschaften Phys.-Math. Klasse 1932, Verlag der Akademie der Wissenschaften, Berlin 1932, Seite 436-460.

[254] S. Seifarth, U. Semmelmann, The spectrum of the Dirac operator on the complex projective space p(2q-1), Preprint SFB 288 No. 95, Berlin 1993. [255] S. Slebarski, The Dirac operator on homogeneous spaces and representations of reductive Lie groups, Part I, Amer. Journ. Math. 109 (1987), 283-302. [256] S. Slebarski, The Dirac operator on homogeneous spaces and representations of reductive Lie groups, Part II, Amer. Journ. Math. 109 (1987), 499-520. [257] M. Slupinski, A Hodge type decomposition for spinor valued forms, Ann. Scient. Ec. Norm. Sup. 29 (1996), 23-48. [258] S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. 136 (1992), 511-540. [259] H. Strese, Uber den Dirac-Operator auf Gralimannschen Mannigfaltigkeiten, Math. Nachr. 98 (1980), 53-58. [260] H. Strese, Zur harmonischen Analyse des Paares (SO(n),SO(k) x SO(n-k)) and (Sp(n), Sp(k) x Sp(n - k)), Math. Nachr. 98 (1980), 61-73. [261] H. Strese, Spektren symmetrischer Raume, Math. Nachr. 98 (1980), 75-82. [262] H. Strese, Zum Spektrum des Laplace-Operators auf p-Formen, Math. Nachr. 106 (1982), 35-40.

[263] S. Sulanke, Berechnung des Spektrums des Quadrates des Dirac-Operators auf der Sphare and Untersuchungen zum ersten Eigenwert von D auf 5dimensionalen Raumen konstanter positiver Schnittkrummung, Dissertation, Humboldt-Universitat zu Berlin 1981. [264] S. Sulanke, Der erste Eigenwert des Dirac-Operators auf S5, Math. Nachr. 99 (1980), 259-271.

[265] A. Trautman, Spinors and the Dirac operator on hypersurfaces. I. General theory, Journ. Math. Phys. 33 (1992), 4011-4019. [266] A. Trautman, The Dirac operator on hypersurfaces, Acta Phys. Polon. B 26 (1995), 1283-1310.

[267] A. Trautman, Reflections and spinors on manifolds, Proceedings of the conference "Particles, Fields and Gravitation", Lddz 14.-18. May 1998, American Institute of Physics. [268] A. Trautman, K. Trautman, Generalized pure spinors, Journ. Geom. Phys. 15 (1994), 1-22.

[269] C. Vafa, E. Witten, Eigenvalue inequalities for fermions in gauge theories, Commun. Math. Phys. 95 (1984), 257-276. [270] M. Wang, Parallel spinors and parallel forms, Ann. Glob. Anal. Geom. 7 (1989), 59-68.

[271] M. Wang, Preserving parallel spinors under metric deformations, Indiana Univ. Math. Journ. 40 (1991), 815-844. [272] M. Wang, On non-simply connected manifolds with non-trivial parallel spinor, Ann. Glob. Anal. Geom. 13 (1995), 31-42.

Bibliography

192

[273] E. Witten, Monopols and four-manifolds, Math. Res. Letters 1 (1994), 769796.

[274] J. Wolf, Essential self-adjointness for the Dirac operator and its square, Indiana Univ. Math. Journ. 22 (1972/73), 611-640. [275] Wu Wen-Tsun, Classes caracteristiques et i-carres d'une variete, C.R. Acad. Sci. Paris 230 (1950), 508-511.

Index absolute differential, 58, 166, 167 adjoint operator, 92 algebra of complex numbers, 9 algebra of quaternions, 8 almost-complex manifold, 73, 74 almost-complex structure, 60, 80, 146, 147, 151

Ambrose-Singer theorem, 178

anti-canonical spin structure, 79, 81 associated fibration of a principle bundle, 159

associated spinor bundle, 75 associated vector bundle, 161 Bianchi identity, 168 bilinear form, 1, 7 direct sum of, 7 index of, 2 nondegenerate, 1 rank of, 2 signature of, 2 canonical basis, 2 canonical connection, 83

canonical spin structure, 79 of an Hermitian manifold, 77, 78 canonical spin(4) structure, 147, 149 Casimir operator, 86, 87 Cauchy-Riemann equations, 71 center of an algebra, 9 characteristic class, 108 Chern class, 108, 141, 163, 171, 172 Clifford algebra, 4, 10, 11

Clifford multiplication, 21, 32, 53, 68, 70, 133

complete Riemannian manifold, 98 complex n-spinors, 14

complex projective space, 40, 42, 48, 161 complexification of a real algebra, 11 of a real quadratic form, 11 conjecture, 8 , 131 connection, 163, 165, 169 holonomy group of, 177 locally flat, 168 reduction of, 174 continuous spectrum, 91 covariant derivative, 58, 60, 67, 68, 70, 166, 167

covering spaces, 40 curvature form, 62, 135, 167, 169 curvature tensor, 62 de Rham cohomology, 171 integral, 172

determinant bundle of spin structure, 5254, 108, 113

Dirac operator, 68, 69, 71, 93, 96, 101, 107, 127, 136, 148 eigenvalues of, 113, 116, 126, 128 G-function for, 103 index formula for, 110 index theorem for, 109 spectrum of, 99 Dirac spinors, 14, 69, 113, 115 direct sum of bilinear forms, 7 distribution, 175 integral, 175

eigenspinor, 102, 103, 112, 114, 126 eigenvalues, 91, 126 of the Dirac operator, 113, 116, 126, 128 Einstein space, 118

8 conjecture, 131 193

Index

194

equivalent fibrations, 166 equivalent A-reductions, 158 equivalent spin structures, 35 equivalent spinC structures, 51 essentially self-adjoint operator, 92-94, 96 n-function, 105, 112 for the Dirac operator, 103 exponential map, 18 exterior differential, 73

fibration, 156, 159 equivalence of, 156 locally trivial, 155 first integral, 124 frame bundle, 158 Fredholm operator, 107 Freudenthal-Yamabe theorem, 177 Frobenius theorem, 174 global, 176 local, 174, 175 fundamental group, 26

Kahler manifold, 61, 81, 82, 89, 116, 147, 150, 151, 153 Killing number, 116, 118, 119 Killing spinor, 116, 118, 119, 121, 124, 125, 128

imaginary, 120 real, 120

Lagrange theorem, 2 A-reduction, 158, 173 equivalent, 158 Laplace operator, 71 on spinors, 68 Levi-Civita connection, 57, 61, 81, 84, 113, 125, 135, 148, 164 Lie algebra of Spin(n), 17, 18 of Spinc(n), 29 linear operator, 91 locally flat connection, 168 locally trivial fibration, 155

Gaschiitz proposition, 39 gauge field theory, 130, 131 gauge group, 135 gauge transformation, 135, 169 Ginzburg-Landau model, 131 G-principal bundle, 156 Grafmannian manifold, 56, 88

manifold without boundary, 175 Maurer-Cartan form, 83, 164 moduli space, 140, 147 for Seiberg-Witten theory, 136

harmonic spinors, 81, 82 heat equation, 112 Hermitian manifold, 75, 78, 147 canonical spinC structure, 77, 78 spinor bundle of, 79 Hermitian metric, 53, 68, 74, 80 Hermitian scalar product, 24 Hilbert-Schmidt operator, 103, 104 Hirzebruch-Hopf proposition, 133, 146 Hirzebruch signature theorem, 109 holonomy group of a connection, 177 homogeneous spin structure, 85, 87 homotopy classification theorem, 162 homotopy theory, reduction theorem of, 178 Hopf bundle, 161 Hopf fibration, 157, 161, 172, 173 horizontal lift, 165

orientation, 159

index, 108

of a form, 2 index formula for Dirac operators, 110 index theorem for Dirac operators, 109 integrable distribution, 175 integral de Rham cohomology, 172 integral manifold, 176 intersection form, 109, 130 isomorphic principal bundles, 157 isotropic subspace, 3

nondegenerate bilinear form, 1 null subspace, 3

parallel spine spinor, 67 parallel spinor, 67, 89 parallel spinor field, 67 parallel transport, 166, 167 Picard manifold, 172 Pin(n), 15 point spectrum, 91 Pontrjagin class, 108 principal bundle, 157 associated fibration of, 159 G-, 156 isomorphic, 157

Si-, 163 Z2-, 163 projective space complex, 40, 42, 48, 161 real, 55, 56

q-form of type p, tensorial, 165 quadratic form, 1 quaternionic structure, 29, 30, 110 in A ,,, 32, 54

rank of a bilinear form, 2 real projective space, 55, 56 real structure, 29, 30 in An, 32, 54

Index reducible solution of the Seiberg-Witten equation, 140, 142 reduction of a connection, 174 A-, 158, 173

equivalent, 158 U(k)-, 48, 60, 61, 81 reduction theorem of homotopy theory, 178 Rellich lemma, 100 residual spectrum, 91 resolvent set, 92 Ricci tensor, 64, 118 Riemannian manifold, complete, 98 Riemannian metric, 141, 159 Riemannian symmetric space, 82, 87 Rokhlin's theorem, 110

S1-principal bundle, 163 scalar curvature, 111, 113, 118, 135, 144,

145, 148, 149, 151 Schrodinger operator, 127 Schrodinger-Lichnerowicz formula, 73, 100, 110, 113, 134, 145 Schur-Zassenhaus proposition, 39 second Stiefel-Whitney class, 40 section, 156 Seiberg-Witten equation, 131, 134, 136, 138, 140, 153

reducible solution of, 140, 142 Seiberg-Witten invariant, 144-146, 149, 151 Seiberg-Witten theory, moduli space for, 136 self-adjoint operator, 92 essentially, 92-94, 96 spectral theorem for, 93 signature, 109 of a form, 2 spectral measure, 92, 93 spectral theorem for self-adjoint operators, 93

spectrum of a Dirac operator, 99 of an operator, 91, 92 sphere, 43, 88, 116, 125, 128 spin bundle, 54 spin representation, 14, 23, 25, 54, 58, 75,

133

of Spin(n), 20 spin structure, 35, 36, 38-40, 42-45, 47, 50, 53--55, 60, 79, 113 equivalence of, 35 homogeneous, 85, 87

spinC structure, 47, 48, 50, 51, 53, 57, 60, 78, 93, 96, 111, 131, 153 canonical, 79 of an Hermitian manifold, 77, 78 determinant bundle of, 52-54, 108, 113 equivalence of, 51

195

Spine (4) structure, 134, 141, 146 canonical, 147, 149 Spinc(n), 25, 26 Spinc(12) representation, 28

Spin(n), 15 spin representation of, 20

spinor bundle, 53, 78 of an Hermitian manifold, 77 spinor derivative, 59 spinor field, 67 parallel, 67 Stiefel-Whitney class, 163 second, 40 structure identity, 168 submanifold, 175 weak, 176 Sylow subgroup, 2-, 39, 44 Sylvester's theorem, 2 symmetric operator, 69, 92, 93 symmetric space, Riemannian, 82, 87 symplectic manifold, 158 symplectic structure, 151, 159

tangent bundle, 155 tautological bundle over C?1, 161 tensor product of Z2-graded algebras, 7 tensorial 1-form, 62 tensorial q-form of type p, 165 twistor equation, 128 twistor operator, 69, 70, 121 twistor spinor, 121, 123 2-Sylow subgroup, 39, 44 U(k)-reduction, 48, 60, 61, 81 unitary group, 27 universal covering, 19 of SO(n), 16 universal G-bundle, 162 vanishing theorem, 140 vector bundle, 161 associated, 161

von Neumann theorem, 92 weak submanifold, 176 Weyl spinors, 22, 32 Weyl tensor, 118, 121 Weyl theorem, 138, 172 Witt decomposition theorem, 3 Wu's proposition, 133 Yang-Mills equation, 130

Z2-principal bundle, 163 (-function for the Dirac operator, 103

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