Dirac Operators in Riemannian Geometry...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

Dirac Operatorsin RiemannianGeometry

Thomas Friedrich

Graduate Studiesin MathematicsVolume 25

American Mathematical Society



Thomas Friedrich

Translated by

Andreas Nestke

Graduate Studiesin Mathematics

Volume 25

American Mathematical SocietyProvidence, Rhode Island

Editorial BoardHumphreys (Chair)

David SaltmanDavid 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 Verlagsge-sellschaft mbH, D-65189 Wiesbaden, Germany, as "Thomas Friedrich: Dirac-Operatorenin der Riemannschen Geometrie. 1. Auflage (1st edition)" © by Friedr. Vieweg & SohnVerlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1997

Translated from the German by Andreas Nestke

ABSTRACT. This text examines the Dirac operator on Riemannian manifolds, especially its con-nection with the underlying geometry and topology of the manifold. The presentation includes areview of preliminary material, including spin and spin0 structures.

An important link between the geometry and the analysis is provided by estimates for theeigenvalues 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 aboutwhether M is an Einstein manifold or conformally equivalent to one. An appendix contains aconcise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study offour-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 DataFriedrich, 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.

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Contents

Introduction

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


xi

vii

viii Contents

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


Chapter 5. Eigenvalue Estimates for the Dirac Operator and TwistorSpinors 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


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. Principal Bundles and Connections 155

B.1. Principal fibre bundles 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 definedon an open subset 0 C R2 is holomorphic if and only if it satisfies theCauchy-Riemann equation

- 0 with +Z

2 (ax8z a -ay

Geometrically, we consider R2 here as flat Euclidean space with fixed orien-tation. Changing this orientation results in replacing the operator j by thedifferential operator

F=

2(T - i3 Taking both operators together we

obtain a differential operator P : C'(] 2; C2) --> C°°(R2; C2) acting via

P(f =2ix

Of8x-

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

iP=(0i 0)8x0101 )ay69

.

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

0 1Yx= i 0) > 'Yy= ( 01 0),

yields

P = yx 8x + yy ey

xi

xii Introduction

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:

192 a2P2 = axe -aye = A.

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

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

1-2/c2

E = c2p2 -m2 c4.

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

E --+ ihat and p i--+ -ih grad,

respectively. The state function 0 then becomes a solution of the equation

ih at = c2h2A + m2c4

involving the 3-dimensional Laplacian A = - a - a - jr. Mathemati-cally speaking we now move to an n-dimensional Euclidean space and look

for a square root P = of the Laplacian A = - The obvious as-axtii-1

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

nPyiaxi

i=1

Now the equation p2 = A a holds if and only if the coefficients ryiti

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 spaceC2 can be identified with the set of quaternions via I zl I = z1 + jz2,\ 2

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

y1= (o 0i), y2= (0 -1), 73= (0 o).

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

y2 = -E, yiyj + y;yi = 0 (i 0 j),

is called the Clifford algebra Cn (W.K. Clifford, 1845-1879) of the negativedefinite quadratic form (Rn, -xi - ... - x,2n). Thus, the question whetherthere is a square root of the Laplacian leads to the study of complex rep-resentations K : Cn -* End (V) of the Clifford algebra. It turns out that Cnhas a smallest representation of dimension dime V = 2[2]. The correspond-ing vector space is denoted by On and its elements are the Dirac spinors.Moreover, v"A is a constant coefficient first order differential operator actingon 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 todefine this product we represent a vector x E an as a linear combinationwith respect to an orthonormal basis el,... , en,

nx = xzei,

i=1

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

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

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

x (x . 0) = -I Ixl l2V)

In particular, the product vanishes if and only if either the vector x E Wor the spinor b E On is equal to zero. There is no non-trivial representatione of the linear or the orthogonal group in the space An of spinors that iscompatible with Clifford multiplication, i.e. which satisfies the relation

A(x) . e(A) (0) = e(A) (x . V))

xiv Introduction

for every A E SO (n; R), x E RI and E Z. Hence spinors on Riemannianmanifolds cannot be defined as sections of a vector bundle that is associatedwith the frame bundle of the manifold. It is for this reason that in differ-ential geometry the question to what extent the concept of spinors couldbe transferred from flat space to general Riemannian manifolds remainedopen 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 classicalRiemannian technique."

Only the development of the framework of principal fibre bundles and theirassociated bundles as well as the general theory of connections within dif-ferential geometry at the end of the forties made it possible to overcomethis difficulty. The group SO(n; R) is not simply connected. For n > 3its universal covering, the group denoted by Spin(n), is compact and cov-ers 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 Cliffordmultiplication. Considering now those special Riemannian manifolds Mn,today called spin manifolds, the frame bundle of which allows a reductionto the double cover Spin(n) of the structure group SO(n; R), we can definethe vector bundle S associated with this reduction via the representatione : Spin(n) -> GL(An), the so-called spinor bundle of Mn. Then spinorfields 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-Civitaconnection of the Riemannian manifold.

Therefore, spinor fields and Dirac operators cannot be introduced on everyRiemannian space; but, nevertheless, they can be introduced for a largeclass. The existence of a Spin(n)-reduction of the frame bundle of Mntranslates into a topological condition on the manifold, i.e. the first twoStiefel-Whitney classes have to vanish:

wi(M') = 0 = w2(Mn).

In dimension n = 4, for a compact simply connected manifold M4, thistopological condition is equivalent to the condition that the intersectionform on H2(M4;Z), considered as a quadratic form over the ring Z, is evenand unimodular. The algebraic theory of quadratic Z-forms then impliesthat 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 4-dimensional spin manifold M4 is divisible by 16:

a(M4)/16 E Z.

This additional divisibility of the signature of a 4-dimensional spin manifold,which does not result from purely algebraic considerations, was an essentialaspect for the introduction of spinor fields and Dirac operators into mathe-matics. The consideration behind that may be outlined as follows. Could itbe possible that there exists an elliptic operator P on every compact smooth4-dimensional manifold with even intersection form on H2 (M4; Z), the indexof 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, eventuallyintroduced for Riemannian manifolds by M.F. Atiyah in 1962 in connectionwith his elaboration of the index theory for elliptic operators. Since then ithas occured in many branches of mathematics and has become one of thebasic elliptic differential operators in analysis and geometry.

This book was written after a one-semester course held at Humboldt-Uni-versity in Berlin during 1996/97. It contains an introduction into the theoryof spinors and Dirac operators on Riemannian manifolds. The reader is as-sumed to have only basic knowledge of algebra and geometry, such as a twoor three year study in mathematics or physics should provide. The pre-sentation starts with an algebraic part comprising Clifford algebras, spingroups and the spin representation. The topological aspects concerning theexistence and classification of spin reductions of principal SO(n)-bundlesare discussed in Chapter 2. Here the approach essentially requires only el-ementary covering theory of topological spaces. At the same time, eachresult will also be translated into the cohomological language of character-istic classes. The subsequent Chapter 3 deals with analysis in the spinorbundle, the twistor operator and the Dirac operator in detail. Here thegeneral techniques of principal bundles and the theory of connections areapplied systematically. To make the book more self-contained, these resultsof modern differential geometry are presented without proof in AppendixB. Chapter 4 contains special proofs for the analytic properties of Dirac op-erators (essential self-adjointness, Fredholm property) avoiding the generaltheory for elliptic pseudo-differential operators. Eigenvalue estimates andsolution spaces of special spinorial field equations (Killing spinors, twistorspinors) are the topic of Chapter 5. We mainly discuss the general approach,referring to the literature for detailed investigations of these problems. Thebook 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 Sonder-forschungsbereich 288 "Differentialgeometrie and Quantenphysik" in Berlinon 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 operatorsand solution spaces of spinorial field equations. Many of the results fromthis period are collected in the references. On the other hand, the presentbook may serve as an introduction for a closer study. I would like to thankall those students and colleagues whose remarks and hints had an impacton the contents of this text in various ways.

I am particularly grateful to Dr. Ines Kath for her careful and detailed cor-rections of the text, and to Heike Pahlisch, whose typing of the manuscripttook into account every single wish.

Thomas Friedrich

Berlin, March 1997

The English translation of this book has been prepared in the beginning ofthe year 2000. It does not differ essentially from the original text, althoughI made many changes in details which are not worth listing. During the lastthree years many new results have been published in this still dynamic areaof mathematics. I included the corresponding references in the bibliographyof the translation. During the academic year 1996/97 Dr. Andreas Nestkeprovided the exercises for students of my lectures at Humboldt Universitywhich furnished the starting point for this book. Two years later he had toleave the University. I thank him, as well as Dr. Ilka Agricola and HeikePahlisch, for all the work and help related with the preparation of the Eng-lish edition of this book.

Thomas Friedrich

Berlin, March 2000

Chapter 1

Clifford Algebras andSpin Representation

1.1. Linear algebra of quadratic forms

We will start by recalling some facts from linear algebra. Let K be a field ofcharacteristic 0 2. A bilinear form is a pair (V, B) consisting of a K-vectorspace V and a symmetric bilinear map

B:VxV--*K.The function Q : V --> K defined by Q(v) = B(v, v) is the correspondingquadratic 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 alinear 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

2 1. Clifford Algebras and Spin Representation

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

Definition. The rank of the bilinear form (V, B) is defined to be the rankof 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 prop-erty that the matrix M(V, B) is diagonal (bases with this property are calledcanonical):

/ Ai 0

M(V, B) = Ar

0, r = rank(V, B).

0 0 )

If the field is K = R or C, then even more holds:

Theorem (Sylvester). For every quadratic form over R there exists a canon-ical 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 thisdiagonal matrix do not depend on the choice of this basis. The pair (p, q) iscalled 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 withrespect to which the form has the following matrix:

/ 1 0

M(V, B) _ 1

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 null-subspace, if W C W1 holds. Obviously, W is a null-subspace if and onlyif Qjw = 0 (or Blwxw = 0), i.e. Q (B) vanishes identically on W. Everynull-subspace is isotropic but not vice versa.

Theorem (Witt Decomposition). Let (V, B) be a finite-dimensional non-degenerate quadratic K-form and W C V a maximal null-subspace. Thenthere 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 Usatisfying

B(uz7wj) = 8zj, 1 < i, j < k (Witt basis).

Corollary. Let K be an algebraically closed field and (V, B) a finite-dimen-sional 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 twonon-trivial vectors. According to the preceding proposition, Part c), we havefor .,p E K

Av+µv'=04=#- Q(Av+µv') =0.


For fixed 0 p E K the quadratic equation in \

Q(Av + E.tv') _ A2 + 2tB(v,v') I\ + A2Q(v') = 0

Q(v) Q(v) Q(v)

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

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 sat-isfying u(v)2 = Q(v) 1, then there exists one and only one algebrahomomorphism U : C(Q) -> A such that u = U o j.

V -A

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 samequadratic form (V, Q), then there exists an isomorphism f : C(Q) -->C'(Q) of the algebras satisfying f o j = j'

C(Q) C'(Q)

\j /i I

V

1.2. The Clifford algebra of a quadratic form 5

Proof. To show existence, consider the tensor algebra T(V) = K ® V(V (9 V) EB ... of the vector space V, and denote by I (Q) the two-sided idealgenerated by all elements of the form

{v ®v - Q(v) : v E V}.

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

j =7roidetermines 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 homomorphismU : C(Q) -> A satisfying the relation we were looking for. Given anotherhomomorphism U1 : C(Q) -+ A satisfying

C(Q)

XV A

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

Corollary. The linear map j : V -> C(Q) from the vector space V of thequadratic 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 subspaceof C(Q).

Proposition. The Clifford algebra C(Q) of a quadratic form is equippedwith 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)


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

u2(v) = (-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 alsocarries an anti-involution y : C(Q) -> C(Q). To construct it, we begin witha preliminary remark: If A is a Ilk-algebra, then one can define a new algebraA on the set A by introducing the product

Now let (V, Q) be a quadratic form and C(Q) its Clifford algebra. Considerthe 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. Thusthere 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 exists a linear mapC(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)


Given two bilinear forms (V,, B,) and (V2, B2), their direct sum is definedas 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 quadraticforms.

Proposition. The Clifford algebra C(Q1(9 Q2) is isomorphic to the tensorproduct 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 ten-sor 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 correspond-ing 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 vectorspace C(Q) has dimension 2n,

dimK C(Q) = 2n.


Proof. By the Lagrange theorem the quadratic form is the sum of n one-dimensional quadratic forms:

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

CO(Q)=K, C1(Q) = K e, and e2=Q(1).

Hence, dimK C(Q) = 2 for a one-dimensional quadratic form. From the lastproposition 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 Vsuch that

B(vi, vj) = 0, i j.

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

v2 = Q(vi), vivj + vjvi = 0, i 0 j.

A particular basis of the vector space C(Q) is formed by the elements 1 andvil .... 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 ba-sis of the vector space V. Hence the vectors vi, ... , vn generate the algebraC(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 vectorspace 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.


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

is=e1i j:=e2, k:=el e2,the fundamental relations take the form

i j=k, k.i=j, j2=j2=k2=-1,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 thesingle relation ei = -1.

Next we want to determine the centers of C(Q) and C°(Q), respectively, inthe case of a non-degenerate form. In order to achieve this, consider a basisV1.... , 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 ofP is a subset P = {pl,... pk} where 1 -<P1 < p2 < ... < pk < n. As anabbreviation we set

ZIP = VP1 ... VPk, P = {Pi < p2 < ... < Pk}

1, P = 0

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 givenby

-1 _ 1 1VP

P1... 5Pk vPk ... VP1.

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 for dim V even,K®K[vl ... vn] for dim V odd.


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

(C0(Q)) _ K ®K[vl ... vn] for dim V even,K for dim V odd.

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 = avRnecessarily holds for each set R E P with RI = 2. This implies

1: apvRVpvR-1 = apvp.PEP PEP

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

so that (-1)IPnRlap = ap for each set R C P with SRI = 2. This relationis a necessary condition to be satisfied by any central element. From this itfollows 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 ... vn commutes withall elements of the algebra C(Q), and we arrive at

Z(C(Q)) = K ® K[vl ... vn].

The case dim V = 0 mod 2 is treated analogously.

1.3. Clifford algebras of real negative definitequadratic forms

Let us first fix notations for some special Clifford algebras which will playan eminent role through the rest of the book:

Cn = C(>18n, -x1 - ... - xn) - Clifford algebra of the n-dimensional realnegative definite form,

Cn = C(1 , xi +... + xn) - Clifford algebra of the n-dimensional realpositive definite quadratic form,

1.3. Clifford algebras of real negative definite quadratic forms 11

Cn = C(Cn, zi + ... + zn) - Clifford algebra of the n-dimensionalcomplex 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 complexi-fication. 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 Ccarries 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 complex-ification. 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, wherej : V -+ C(V, B) is the embedding of the real vector space into the Cliffordalgebra. 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 thecomplex 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-...-xnare equivalent.

Proposition. The following graded real algebras are isomorphic:

Cn+2 = Cn, OR C2C1

n+2 = Cn ®C2.

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


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 asgenerators 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

Rn+2 U- 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. Thelatter 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 algebraM2(C) has the basis

(100

E _ 1) , 91 =\i

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 theprevious proposition, i.e. Cn+2 = C,,, OR C2. Thus we obtain an explicitisomorphism 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+2and, correspondingly, ei, ... , en those for the algebra C. Moreover, denoteby

91= i 0i ), 92O

= \ 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, ei'-'(Set-2)®9192, 3<jCn+2.We will now apply these isomorphisms repeatedly and take into account thedescriptions 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 Ttimes

where1 if j is odd,

a (7) 2 if j is even.

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

[ 2 ] times times

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


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

pn:=cC2k =C2®...®C2 forn=2k,2k+1.k times

The elements of An are called complex spinors.

Using this notation we obtain

Cn = End(An), if n = 2k is even,

Cn = End(On) ® End(An), if n = 2k + 1 is odd.

Moreover, the following diagram is commutative:

cc2k

cc2k+1

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. Inthe case of an even dimension, n = 2k, this is nothing but the isomorphismexplained 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 theClifford algebra C,',.

1.4. The pin and the spin group

Consider the real vector space Rn and the Clifford algebra Cn of the quadraticform -xi - ... - x,2n. W' itself is a linear subspace of Cn, R C Cn. For everyvector x E Rn, the equality

x'x=-IIxIl2holds in Cn and hence the inverse element x-1 is given by

-1 -xX T1x,2.

1.4. The pin and the spin group 15

Definition. Pin(n) C Cn is the group which is multiplicatively generatedby all vectors x E Sn-1. Therefore, the elements of Pin(n) are the productsx1 ... x,,,, with xi E IISn, I I xi 1. The spin group, Spin(n), is defined asSpin(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 transformationsof the Euclidean space Rn. To this end, recall the anti-involution

'Y:Cn-*Cnexisting 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 thesphere 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. Nextwe choose an orthonormal basis in R' with x = el as the first vector. Then,writing

n

y = yieiei=1

we getn n

x . y . y(x) = e1 E yiei 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 thereflection 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 elementx of Pin(n) is the product x = xl ... x,n of vectors xi E S"-1, and thecalculation above shows that A(xi) is a reflection. Therefore, A(x) itself is thesuperposition of reflections, so, in particular, an orthogonal transformation.


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 orthogonallinear map A : I[8n -* Rn is the superposition of reflections. Let 0 : C, -+ Cnbe the involution of the Clifford algebra which defines the decompositionCn = C° ® C. For every vector x E IlBn we have 0(x) -x. Now apply /3 toany 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 mis even. On the other hand, A(x) = A(xi) ... A(x,,,,,) is the superposition ofm reflections and, hence, A(x) lies in SO(n) if and only if m is even. Thisproves b). Now suppose that A(x) = 1 holds in 0(n). Then xy-y(x) = yfor all y E IlBn. Moreover, since \(x) E SO(n), x is the product of an evennumber 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 intoaccount 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 thuscommutes with every vector from IRE. Therefore, x belongs to the centerof Cn and, at the same time, to the center of CO,. But, for every Cliffordalgebra,

Z(C(Q)) n Z(C°(Q)) _ K.

Hence, x E R', and from xj = 1 we see that x = ±1. It remains to beshown 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) con-necting 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.


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

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

it/ e2/

in Cn.

Now we will describe the Lie algebra of the group Spin(n). For this we needthe following general remark: Let A be a finite-dimensional associative realalgebra and A* C A the group of its invertible elements. A* is an opensubset of A and, furthermore, a Lie group. Its Lie algebra a*, which can beidentified with Ti(A*) A, thus coincides with A,

a* = A.

The commutator of two elements al, a2 E A = a* is given by [al, a21ala2 - a2al, and the exponential map

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

°O anexp(a)

a

- 1: mt'n=O

We are going to apply this observation in the case of the real Clifford algebraCn. 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 thetangent 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. Thetangent 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 eiej, 1 < i < j < n. We will prove that each summand of dt (0) belongs tom2. Since -y(O) = 1, the first summand is equal to

ddt1(0) x11(0) dtl (0) x1(0)

But xi (t) xi (t) - 1 implies

dt1(0)xi(0) + xi(0) dti(0) = 0

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


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) andxl(0)x2(0)xi 1(0) are perpendicular, too, and the second summand lies inm2. 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 groupSpin(n) C C.

b) The exponential map exp : m2 -+ Spin(n) is given by

°° i

exp(x)x_Y, -v.

i=o

c) If a : Cn -+ End(W) is a (real or complex) representation of theClifford 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 tocoincide. This proves a) and b). c) follows from general facts concerningthe differential of a homomorphism f : H --; G between two Lie groups.


Namely, f* : is uniquely determined by the commutative diagramh-B

exp

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

m2 End(W)

exp exp

Spin(n) Aut(W)

commutes, because u : Cn ---> End (W) is a homomorphism of algebras. Thiscompletes 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 ofthe vector space so(n) is formed by the matrices Ezj (i < j) defined by

i j0 ... ... 0 0 ... 0

0 -1 0

EZj _1 j

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 i, j,

A('Y(t))ek = 616


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.

Thus g(A(y(t))ei) = lei. This computation proves the formula A (eiei) _2Eij . The result can also be written invariantly.

Proposition. If z E spin(n) is an element of the Lie algebra, then for thedifferential 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 belongsto 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 thegroup Spin(n).

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

Spin(2k) k2k GL(O2k)

1

Spin(2k + 1)11k+1

GL(A2k+1)

commutes. This implies that the normal subgroup H := ker(K2k+1) has onlythe 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 ESO(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 itselfthis leaves two possibilities, either H = {1} or H = {1, -1}. But (-1) ESpin(2k + 1) does not belong to the kernel of the spin representation.

A vector x E Il8n C Cn C Cn End(On,) can be considered as an endomor-phism of On. This leads to the so-called Clifford multiplication of vectorsand spinors, which is a linear map

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 This Clifford multipli-cation extends to a homomorphism

p : A(II8n) OR An -- On

as follows: Using the orthonormal basis ei, ... , en of Rn, each element ofthe exterior algebra A(IEB') can be written as

wk = U wili2 ...ikeil A ... A eik.it<...<ik

Define

Y(wk ®4,) = wk . Y = E wil...iklil<...<ik

.. eik `Y7

where, as before, ei,, cp denotes Clifford multiplication of the vector eic bythe 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 homo-morphism of Spin(n) -representations.

Proof. For every element g E Spin(n) we have to check the formula

K(g)(W1. ) _ (,\(g)wk)

'(r

We will do this by induction on the degree k of the multi-vector wk. Fork = 1, wk is an ordinary vector x E II8n and the equation results from the


following computation:

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

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

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

K(9)((xAwk) )

K(9) (x ' (wk ' )) + (g)((x-j wk) .

)

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

(9) (x wk) ' K(9)'b

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

((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 multipliedby spinors. The product is always a spinor, and this multiplication is equi-variant for the actions of the group Spin(n) on corresponding spaces. Nextwe will consider the case n = 2k in greater detail. In this case, the elementel ... e2k belongs to the center of the algebra CO. As Spin(n) C CO, this el-ement 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 in-volution, f 2 = Idon . Thus the spin representation A2k decomposes into theeigensubspaces of f :

A2k = A2k ®02k, A2k = {2O J E 02k : f ( ) _

Definition (Weyl spinors). The spinors belonging to the subspaces 02k arecalled (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 belongsto L. Thus Clifford multiplication induces homomorphisms

m : R2k OR Ak A2k'

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

x (el... e2k) = -(el ... e2k) X.

1.5. The spin representation 23

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

We will now prove that the spin representations A2k, A2k and A2k+1 allare irreducible representations of the spin group. To do this, we need thefollowing preparation from linear algebra: Let V be a complex vector spaceand A = End(V) the algebra of all endomorphisms. Then A is a simplealgebra, i.e. A does not contain any proper ideal. From this we immediatelyconclude 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+ ®A-2k 2k

and assume that {0} # W g 02k is a Spin(2k)-invariant subspace. Theproducts ei ej (i < j) belong to the group Spin(2k), and hence W is leftinvariant by them. On the other hand, the products eiej (i < j) multiplica-tively generate the algebra (C20°. Thus we obtain a representation of thisalgebra in the vector space W:

(C2k)° --> End(W).

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

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 givesrise to a representation

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


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

Proposition. In the vector space of n-spinors, On, there exists a positivedefinite 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 aunitary representation with respect to this scalar product.

Proof. Let M2 = Lin(ezej : i < j) C Cn be the Lie algebra of the groupSpin(n). Consider g = an ® m2 C Cn. A simple calculation shows that g isa 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 inducedby the map 1) : an -> Cn+l, l(ei) = eien+l, as can be seen from theequations

-D(ei)2 = -1, 1 < i < n,

l(ei)-D(ej) + 0, 1 < i < j < n.

Thus co : g --p Cn+1 maps the Lie algebra g bijectively onto the Lie algebraof the group Spin(n + 1) and is, moreover, an isomorphism of these Liealgebras. Hence g is a compact Lie algebra. In view of the following remarkthis 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 representationin a complex vector space. Then in W there is a positive definite Hermitianscalar 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 Gcorresponding 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 representationis a representation into the special unitary group SU(On) of the space of n-spinors.

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

f : Spin(n) -* S', f (g) = det (s (g)).

As Spin(n) is simply connected, there exists a lift F : Spin(n) --+ R to theuniversal covering of S',

f (g) = elriF(g) ,

which is a group homomorphism as well. Spin(n) is compact. HenceF(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 thegroup Si of all complex numbers of modulus 1. Together they generate agroup 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 Slunder 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.


Then the following diagram commutes:

Ii

1 Spin(n)Z

Spinc(n)

A

SO(n)

1

I1

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

Proposition. Let n > 3. Then,

a) The fundamental group ir1(SpinC(n)) is isomorphic to Z, and1 : 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-foldcovering pp : ir1(SpinC(n)) -+ rr1(SO(n)) x 7r1(Sl) the following for-mula 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 proveA (a) = . 7r1(SO(n)) is isomorphic to Z2, and hence this is equivalent toA#(a) 0. Suppose that An(a) = 0. From the corresponding exact columnsequence we see that there exists an element S E 7r1(Sl) with j (S) = a.This implies

y = 1q(a) = ldjd(S) = 25and 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 ofSO(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) suchthat the diagram

Spin (2k)

p

U(k) ' SO(2k) x S1commutes.

Proof. We have to prove that the group fp(7r1(U(k)) is contained in theset If we choose a generating element d E irl(U(k)) = Z inaddition to the generating elements a, ,3, y, then we have f f (6) _ l3 + y, andthe 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, 1 I\ det A /

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

Spin (2k)

IP

U(k) fl SO(2k) X Sl

commutes.

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

eiO1 0

A=0 ei0k

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


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

k B l g 12

E a,F(A) = fl( cos C 2 l+ sin ( 2 I

fjJ(fj))x e

J

Since Spin(n) is contained in the Clifford algebra Cn, the spin representationof 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. Thedeterminant 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 isSpin(2k)-invariant and for the corresponding representations we have

det rc±[g, z] = zdim(Ok)

This implies that the Spin (n)-representations det(A ) = Adim(On) (A±)

and(l)dimon±/2=l® ...®l (dim On /2 times)

are equivalent. For the particular case n = 4 we obtain the equationA2(A)

=A2(A) = l4 4

in the sense of Spin (4)-representations.

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

Spin (n) f Spin(n + 2)

p

commutes.

A

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

Proof. Spin(n) is a subgroup of Spin(n + 2), and S1 can be realized as asubgroup 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 obtaina 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 describedby

p* (e«ep, it) = (2Eaa, 2it), 1 < a < R < n.

1.7. Real and quaternionic structures in thespace of n-spinors

Recall the following definitions of a real and a quaternionic structure in acomplex vector space.

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

a2 = Id, a(iv) = -ia(v).

Definition. Let V be a complex vector space. A quaternionic structure isan 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-dimensionalspin representation A3 = C2 and their invariance properties under Cliffordmultiplication.

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

mutes with /Clifford multiplication by each vector x E 1R3:

aIn p A3 --3 03 which anti-commuteswith Clifford multiplication by each vector from a 2-dimensional sub-space x E R2 C R3,

,8(x b) = -x/(c), X E J2 C R3, V)E A3,

and commutes with e3:

/3(e3 . ') = e3 . /3('0)-

,3 is not Spin(3)-equivariant.


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

i 0 / O i 0 1e1-91-(0 -i), e2=92=1 i 0), e3=iT= -1 0

Define a, /3 : A3 A3 by the formulas

a(z2)=( z12), )3(z2)=(z2A straightforward calculation then leads to the result, e.g.

/3 ( e 2 ) = /3 ( ) = ( ) e2 ( ) = e2 (z2 ) = (izl )2

N(e3 ' b) ( Z2 ) = ( zzl ) , e3 ' 0(0) = e3 (z2 ) _ (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 algebraicpreparation. If a : V -> V and /3 : W -* W are real or quaternionicstructures, then their tensor product

a®/3:V®CW-+V®cWcan 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 aboveand their commutation relations (note that e3 = iT is replaced by T)

a91 = 91U7 a92 = 92a, aT = -T a,

091 = -910, 092 = -920, /3T = -T/3.Now we will construct one of these structures in each vector space Z ofn-spinors a,, : A, --k A,,.

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 multipli-cation:

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 Cliffordmultiplication:

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 Clif-ford 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) (2k times).

Second case. n = 8k + 2, 8k + 3.

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 rela-tions between a, 0 and gi, 92, T, respectively, described in Section 1.3, theproperties listed above are easily checked.


Summarizing, we arrive at the following table for the real or quaternionicstructures in An:

an real structures quaternionic structures

commutes with n 6,7 mod 8 n 2, 3 mod 8Clifford multiplication

anti-commutes with n 0, 1 mod 8 n 4, 5 mod 8Clifford multiplication

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

4k+e ef = i el ... e8k+2e = i el e8k+2e

For e = 1, 3, Clifford multiplication commutes with an. Since an is complexanti-linear, in these cases, an anti-commutes with f = ±iel ... e8k+2e Fordimensions n = 8k + 2, 8k + 6 this implies that the real (or quaternionic)structure an : An - On interchanges the summands in the decompositionOn=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 thedecomposition 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)-equi-variant structure.


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

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

1.8. References and exercises 33

B. Budinich, A. Trautman. The Spinorial Chessboard, Springer-Verlag1988.

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

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

Exercise 1. Let (V, Q) be a non-degenerate quadratic form. Determineall elements in the Clifford algebra C(Q) which anti-commute with everyelement 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. Provethe following additional isomorphisms:

E D C4 = M2 (H), C5 = M4 (C)

C6 = M8 (I), 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),

C 5 = M2(H) ®M2(H), Cs = M4(H), C81=M8(C)-

Exercise 5. Prove the equation Ck_1 = Ck.

k-1 is the decomposition of the algebra Ck_1 andHint: If Ck_1 = Ck-1 6 C1ek E IRk the last vector, then setting

f (xo + xl) = x0 + ekxl

defines a homomorphism f : Ck_1 -4 Ck.


Exercise 6. Prove that

Spin(3) SU(2) = {q E IHI : IIgII =1},Spin(4) SU(2) x SU(2),

Spin(5) Sp(2),

Spin(6) SU(4).

Exercise 7. Prove that the group Spinc (4) is isomorphic to the followingsubgroup 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 thatfor 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 anSO(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 groupon the corresponding principal bundle.

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

Pl f P2Al /2

Q

35

36 2. Spin Structures

Denote by F a fibre of the SO(n)-principal bundle Q. F is diffeomorphic tothe group SO(n) and, since n > 3, the fundamental group irl(F) consists oftwo elements:

7r1(F) = 7L2.

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

aF := i#(a)

is an element of the fundamental group 7r1(Q). Consider a spin structureA : 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 toa continuous map I : F - P such that the diagram

P

A

F zQcommutes. I(F) C P is contained in one fibre F' of the Spin(n)-principalbundle 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 contradic-tion.

Proposition. The equivalence classes of spin structures on an SO(n)-prin-cipal bundle Q over a connected CW-complex X are in bijective correspon-dence 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. Thissubgroup 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 yQ x SO(n) -* Q the action of the group SO(n) on Q. The map

P x Spin(n) P

AxA A

QxSO(n) µ- Qinduces 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 uniquecontinuous map µ : P x Spin(n) -> P with µ(po, 1) = po fitting into thecommutative diagram

P x Spin(n) µ P

IAxAI A

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

P...f.. P

lA I A

Q 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 fibreof 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 fundamentalgroup irl (P). The equivalence class of the loop Ay* (t) thus belongs to thesubgroup 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))


Now A o -y(t) is a loop in SO(n) representing the non-trivial element a E7rl(SO(n)). Finally, for the fibre F = \(po) SO(n) C Q we obtain theinclusion

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 thehomomorphism

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 aconnected CW-complex X are in one-to-one correspondence with the homo-morphisms 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 thefollowing 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) -principalbundle 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 finitegroups. The preceding considerations show that an SO(n)-principal bundleQ over a CW-complex X has a spin structure if

a) i# :7r1(F) = Z2 --+ 7r1(Q) is injective;

b) the exact sequence 1 -+ 7ri (F) -#-) 7r1(Q) -- 7ri (X) -- 1splits.


We now suppose that irl (X) is a finite group and, moreover, that conditiona) is satisfied. Then 7r1 (Q) is an extension of the group Z2 = 7r1 (F) by7r1(X). Recall the notion of a 2-Sylow subgroup of a finite group G. If G1denotes the order of the group G and 2k is the largest power of 2 dividingIGJ, then each subgroup of G of order 2k is called a 2-Sylow subgroup. It iswell-known that there exists at least one such subgroup in every group G.If

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

Proposition (Schur-Zassenhaus, Gaschiitz). The extension G of the groupZ2 by the finite group r splits if and only if for every 2-Sylow subgroupG2 C G the extension

1-->splits.

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

A first conclusion is that for 7r1(X) finite, the principal bundle Q admits aspin 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 extensionof Z2 by G2 splits (or, equivalently, H2(G2; Z2) = 0), then condition b') isautomatically satisfied. We obtain the

Corollary. Let X be a CW-complex with finite fundamental group satisfyingthe condition

H2 (G2; Z2) = 0

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

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

is injective.


In particular, this corollary applies in the case where the fundamental group7r1(X) is of odd order. The only condition for the existence of a spin structurethen 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 inthe 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 inthe 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 tothe requirement that the element f E H1 (Q; Z2) remains non-trivial afterrestriction to the fibre i* : H1 (Q; Z2) --> H1 (F; Z2) = Z2 . From this weobtain the

Proposition. The spin structures on an SO (n) -principal bundle over a con-nected CW-complex X are in one-to-one correspondence with those elementsf 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 ofcohomology 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. Theabove sequence, moreover, immediately implies the

Proposition. An SO(n)-principal bundle over a connected CW-complex Xhas 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 groupSU(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)


is given by the formula

(B 0I = det B B.l

det B

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

(eit01

0<t<2ir.

0

1e-it j

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

e2it

0Ieit

0'('Y(t)) _eit

0 eit

and hence is equal to the (n + 1)-fold power of the generating element of7ri(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 isodd. 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 spingroup. Hence

P := SU(n + 1) x& Spin(2n)is a spin structure on the frame bundle R. Finally, we also discuss the casethat 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 theformula

[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 isSU(n + 1)/Z +i. Hence the resulting exact sequence has the form

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


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

7G2 if n is odd,i (R)

1 if n is even.

Since CIPn is simply connected, we can apply the corresponding criterion forthe existence of a spin structure on R. The bundle R has a spin structure ifand only if irl (R) = Z2. This leads to the

Proposition. The frame bundle R of the complex projective space C?n ad-mits 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 andthe same SO(n)-principal bundle may not be equivalent even if the cor-responding 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). BecauseH'(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 boththese spin structures are isomorphic as principal bundles over X = II ]?2. Thesame 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 Qas a group of SO(n)-bundle morphisms. Set

x* = r\x, Q* = r\Qand 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 ofwhether Q* admits a spin structure.

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

Q7- Q

If e is a left action of r on P with e(y) = then P* = r\P obviouslybecomes 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 spinstructure on Q on which F acts as a group of automorphisms. But, sinceirl(X) = 1, the spin structure q*(P*) is equivalent to (P,A). Thus, insummary, we conclude:

Proposition. The spin structures in the principal bundle Q* over X* arein 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* bylooking at the fundamental group. From

Q - Q*

IX *

we obtain the following commutative diagram:

Z2 = 7ri (F) 7rl(F*)

1

1 -' 7r1(Q) = 7. (Q*) F

1 = 7rl (X) -. 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 spinstructure 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 sphereand 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)


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 ifand only if the sequence

1 -> Z2 = 7r1(SO(n)) ir1(F\SO(n + 1)) - r - + 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 questionis

1 ) Z2 ) A-1(r) ) r ) 1,

and the manifold X* = r\Sn is a spin manifold if and only if this sequencesplits. 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 thesequence

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 thesphere S. The manifold F\Sn has a spin structure if and only if for every2-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 subgroupacting without fixed points on the sphere Sn. Then F\Sn has a spin structureif 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. ThenF has no proper 2-Sylow subgroups, too, i.e. on F\Sn there exists a spinstructure. 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 aneigenvalue, 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 ofcoverings:

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 spinstructure on the real projective space 1E8p4k+1 Thus we arrive at a contra-diction, 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 IFlof the group F is odd, i.e. F has an odd number of elements. A non-trivial homomorphism f : F ---3 7Z2 defines by means of Fo = ker(f) andany yo 0 ker(f) a partition of the set r into r = Fo U yo Fo, and henceIFI = 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 anepimorphism

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 bundleover 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 thissense. To do so, denote by F = G and F* = SO(n) the fibres in the bundlesQ and Q*, respectively. Consider the commutative diagram

F=G z - SO(n)=F*

i Q* I SO(n)/G


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

j(q) = [q, l], 1[q, A] = A mod G, m(A) = A mod G.

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

1r2(X) ir2(X)

71(F)71(F*)

ir2(SO(n)/G) E'

7r1(Q)3 7r1 (Q*) # 7rl(SO(n)/G)

4Suppose 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. Thenconsider 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 existsan element y E 7r1 (Q) such that j# (y) = x. Set f*(x) = f(y). If yl isanother element from 7r1(Q) with j# (yl) = x, then there exists an elementz 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 haveto check the condition f* o s* = Id on 7r1(F*). Choose a E 7r1(F) withi#(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 struc-ture if and only if there exists a homomorphism f : 7rl(Q) , ir1(SO(n)) forwhich the diagram

iri (F) = 7rl (G) 7r1(SO (n) )

Ii1(Q)

commutes.

This condition can again be reformulated cohomologically. The homomor-phism 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), hasto 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 structureif and only if there is an element f E Hl (Q; Z2) for which the restriction tothe 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 witha spin structure, we define

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

P x Spin's(n) - PIAxA A

I

Q x SO(n) Q

commutes.

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


Example. Suppose that the SO(n)-bundle Q (n = 2k) has a U(k)-reduc-tion, 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) forwhich the diagram

Spin (2k)F'U(k) SO(2k)

commutes. HenceP := R XU(k) Spin (2k)

is a spin structure on Q. In other words: Every U(k)-reduction of theSO(n)-bundle Q induces a spin structure in Q.

The groups Spin(n) and SI are subgroups of Spin (n) whose elements com-mute 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,=Q2) PI := P/Spin(n) is an S'/{f1} = Sl-bundle over X,

and the combination of bundle morphisms over the base space P --+ QxPIis 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. QxPIis 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 thesense 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 productQxPI has a spin structure. Conversely, if such a bundle Pi with the givenproperty exists, then Q has a spin structure.

Remark. Making use of characteristic classes, we can formulate this equiv-alently 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 classw2(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, some-times, one can obtain a complete answer. Concerning the base space X weassume that

7r1(X) = 1 and 7r2 (X) is a finite group.

Let Q be an SO(n)-principal bundle and P1 an Sl-bundle over X. On theone hand, the fibre-product QxP1 is an (SO(n) x SO(2))-principal bundleover X, and, on the other, a fibration over Q with fibre Si. Together, thisyields the diagram

7r2 (X)

laZ2ED z

ce - (0,a)

Z = 7r1(Sl) -- 7r1(Q<Pl) ---------

1I

7r1(Q)=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, thereare two cases.

First case: 8 = 0.Then Z2®9L = 7ri(SO(n) xSO(2)) f--; 7r1(Q<P1) is an isomorphism. Hencethe bundle Q PI has a spin structure, i.e. Q itself admits a spin structure.Moreover, the exact row immediately implies 7r1(Q) = Z2.


Second case: im(8) = ?L2.

Then the generating element of the group Z2 = 7ri(SO(n)) belongs to thekernel 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. Qitself has no spin structure. Finally, 7ri (Q x Pi) = Z, and, therefore, thehomomorphism 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 theexistence of a spin structure on an SO(n)-principal bundle over a simplyconnected base space, we conclude:

Proposition. Let X be a CW-complex with 7ri (X) = 1 and 7r2 (X) a fi-nite group. If Q is an SO(n)-principal bundle over X, then the followingconditions 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). Theisotropy representation cp : SO(3) -+ SO(5) of this space induces an iso-morphism 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 bun-dle over X5. We will show that Q has no spin structure. To do so, start bycomputing the first and the second homotopy groups of X5 = SU(3)/SO(3)from the exact sequence

(S 3 -1 X5 7LSO 37r2 U( )) = -- 7r2( ) 7ri( 2( )) =--> 7ri(SU(3)) = 1 --f 7ri(X5) -* 1.

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 spinstructure.

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

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

1) There exists a bundle isomorphism

:Pi 'Piof the S'-principal bundles Pl = P/Spin(n) and Pl = P*/Spin(n).

2) There exists a spin-bundle isomorphism 1 : P -> P* fitting intothe commutative diagram

P P*J IdxV) j *

QxPi QXP1

Remark. Given a bundle isomorphism : P ---p P* with the commutativediagram

P P*

AX 4Q

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

P 4) P*IdxO

Q;P1-''QXPIcommutes. The equality of two spin structures can thus be formulatedequivalently by requiring the existence of a bundle isomorphism (D : PP* with the commutative diagram

P P*

A /AQ


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 charac-teristic 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 theexistence of a spin structure on Q is equivalent to the existence of anelement a E H2(X; Z) with w2(Q) = a mod 2, we conclude that the mapfrom Spin(X;Z) to {a E H2(X;7G) : a = w2(Q) mod 2} is surjective. Forfixed a E H2(X; Z) we have, on the other hand, the SO(n) x S'-bundleQ x P1, and a spin structure on Q is then only a reduction of this (SO (n) xS')-bundle onto the group Spin (n) - ) SO(n) x S'. These reductions arelabeled by the elements of H'(X; Z2). However, in this last step we stillallow for a gauge transformation of the S1-bundle P1. But the ambiguityresulting thereby may be controlled. Let P1 be an S1-bundle and Pl areduction onto the 2-fold covering z -+ z2 of S1. A gauge transformation Fof 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 onlyif the map

f# :7r1(X) -' 7r1(S1) = Ztakes values in 2Z c Z (f has a square root). The gauge transformationitself 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, thespin 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 ontothe 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=RXdetCOn the other hand, if E = R X U(k) Ck is the associated complex vectorbundle, 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 orspine structure. The spin representation

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

S=Px,, An-S 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 casen = 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.


The real or quaternionic structures in On, respectively, pass on, in the caseof 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 equiv-alent where 1 : Spin (n) -+ S' is the homomorphism constructed above.This implies for the determinant bundle L of the spin structure the for-mula

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 struc-ture, it may well happen that different spin structures lead to isomorphicspin bundles. We will now study this question in the case of a spin structurein greater detail.

Let (Q, 7r, X; SO(n)) be a principal bundle. We think of Q as being givenby

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 oftransition 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 CSpin(n),

ezj:UinUj->7L2,and eij ejk = eik. Thus the system of transition functions ezj defines a real1-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 thecentre of the Clifford algebra, this isomorphism is compatible with Cliffordmultiplication, i.e. the following diagram commutes:

T®S µ -- ST®S .0

EWgIdES *

®E

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

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

Proof. The complex line bundle E° is the complexification of a real linebundle, and hence, for the first Chern class, 2c1 (E°) = 0 in H2 (X; Z). SinceH2 (X; Z) has no 2-torsion, this implies ci (E°) = 0, i.e. E° is a trivialbundle.

Remark. The spin structures (P, A) and (P*, A*) are described by elementsf , 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 sequenceof 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 andQ = 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 isuniquely determined by wi (E) 0 in Hi (II81P5; 7L2). The first spin structureis trivial, and hence so is the corresponding spin bundle S. This implies forS*

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.


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 non-trivial, i.e.

cl(E°) 0 in H2(RP5;7L).

Since the map

Z = H2(If TP5; Z) a --> a2 E H4(II8IP5; Z)

is injective, this implies c2 (2E°) 0. Hence S* is not the trivial bundle.


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

R.C. Kirby and L.R. Taylor. Pin structures on low-dimensional man-ifolds, in Geometry of Low-Dimensional Manifolds, Part 2 (Durham, Eng-land, 1989; ed. by S.K. Donaldson), London Math. Soc., Lect. Note Series151, Cambridge University Press 1990, 177-242.

Exercise 1. Let RP' be the n-dimensional real projective space. Provethat

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. Thenevery SO(n)-principal bundle Q over M3 has a spine structure.

Hint: If M3 is orientable, then by Poincar6 duality we have371

H2(M3;7G) = H1(M3;9Z) =

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 -+ Mnthe SO(n)-principal bundle of positively oriented orthonormal frames. TheRiemannian manifold has a uniquely determined torsion-free metric con-nection. Considering it as a covariant derivative on vector fields, we willdenote this Levi-Civita connection by V. Viewed as a connection in theSO(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 correspondingU(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 purelyimaginary numbers. The connections Z and A together define a connection

ZxA:T(Q<Pi) --->.so(n)®iR

in the fibre-product Q x P1. It is not difficult to see that this connection liftsto the 2-fold covering 7r : P --+ Q x PI as a connection Z x A in the spine

57

58 3. Dirac Operators

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 withrespect 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 consideredas a function X : P - an with X (p g) = A(g-1)X (p), since the tangentbundle is an associated vector bundle to the principal bundle P. Then theClifford 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 avector 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 = y X + X (iso).However, since y E M2 and X E an, the following formula holds in theClifford 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 NK*(ZxA)(X X{k*(ZxA)zb}+(A*(Z)X).


Inserting this implies

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

All in all, we have thus proved:

Proposition. Let X, Y be vector fields on M7z and E P(S) a spinorfield. Then, for the spinor derivative with respect to any connection A inthe U(1)-bundle P1,

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

The spinor derivative VA is metric with respect to the Hermitian product inS, i.e.

(VX ,01) + (,V1).

Proof. The last formula is a consequence of the fact that the spin repre-sentation n : Spin -* GL(An) is unitary.

We also want to specify local formulas for the connections. Let e : U CMn --> Q be a local section of the frame bundle Q. e consists of an or-thonormal frame e = (el, ... , e,) of vector fields defined on the open setU C Mn. The local connection form Ze = e*(Z) : TU --> so(n) is given bythe formula

Ze = E wijEiji<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 Liealgebra so (n). Analogously, we fix a section s : U -* P1 of the U(1)-principalbundle 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 Plis a local section of the principal bundle Q x Pl. Let xe be a lift of thissection to the 2-fold covering ?r : P -+ Q x P1. Since

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

and

T(U)

T (P) Z-.- spin(n) = m2 ®iRd(exs) _-1

dTrPwiiid ex^) xPi)-" ` so(n) ®iIRZT(Q


exs)the local connection form Z x A is given by the formula

. exs) 1 1Z x A = 2 E wijeiej, 2As

i<j

With respect to the section e x s, the section 0 E r(US) of the spinorbundle over U is described by a function' : U -- On,, and its covariantderivative is computed according to the formula

VA do 1 1S2J=

i<j

Remark (Special case of a spin structure). If (P, A) is a spin structure onQ, then P x spin(,) Spin`C (n) is an induced spin0 structure. The U(1)-bundleP1, in this case, is trivial with a canonical global section s : Mn -+ Pl. Ifwe choose the connection A0 in Pl for which Ao 0 holds, all the formulassimplify correspondingly. Given a spin structure, we will simply denote thecovariant derivative in the spinor bundle S with respect to this canonicalconnection A0 by V.

Remark (Special case of a U(k)-reduction). Let n = 2k be even, and leta 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 struc-ture 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 onQ 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 vectorbundle can also be described differently. T(M2k) becomes a k-dimensionalcomplex vector bundle by means of the almost-complex structure J and, byconstruction,

E=Ak(T).


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 connectioncan be distinguished in the principal bundle P1 in an "obvious" way. Thishappens, e.g., if the Levi-Civita connection Z reduces to the U(k)-reductionR to a connection Z*. Then (M2k, g, j) is a Kahler manifold. In thiscase, Z* in turn induces a special connection AO in the associated bundlePi = R Xdet Si, and we again obtain a distinguished covariant derivativewhich only depends upon the geometry of the base space.

Now we return to the general situation and consider two connections A andA' 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 con-clude that

1

2ri(X).for all spinor fields E F(S) and all vectors X E T(MT). A gauge trans-formation f : P1 -> Pi of the U(1)-bundle P1 is described by a mappingp 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 impliesthe formula

*V f (A) % - Ox =1

2dpµ X

ffor the corresponding covariant derivatives.

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

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

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

QZXA = 21: 7r*(SZtij)eiej ®27r*(dA)i<j


The 2-form dA is a form on the base space MI. From the general equationDZDZQf = p*(fZ)q, for the 2-fold absolute differential with respect to theconnection 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 ofthe Levi-Civita connection. Using the structure equations of the Riemannianspace (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-formp. (AZ) on the manifold with values in the endomorphisms of the associatedvector bundle V = P x,, Vo. On the other hand, a section 0 of this bundlecan be identified with a function 0 : P -* Vo obeying the transformationrule 0(p g) = p(g-')O(p). Then DZcb is a tensorial 1-form of type p andhence a 1-form on the manifold M with values in V. The induced covariantderivative 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 follow-ing well-known formula.

Lemma. One has the identity Rz = P. (Qz)

Proof. To prove this, consider vector fields X, Y on the manifold M anddenote by X*,Y* the corresponding Z-horizontal lifts. The section VZO isgiven by DZc(Y*) = do(Y*) +Z(Y*)o = do(Y*). An analogous calculationshows that V Z V ZO coincides with X *Y*O, and thus

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

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

= [X*, Y*]vertw.


On the other hand, the structure equation of the connection immediatelyimplies [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 correspondingfundamental vector field, then

W (0) (p) = lim cb(p.etW)-O(p)t-*o t

t o(P(e-t,) - 1)

O(p) =

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) = (f2Z(X, Y)ei,, ej) = (R(X, Y)ei, ej) = I:Rklij0,k(X)a,(Y)

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

k,l

k,l

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

QZxA= 4 E RijklO' k A o.1 eiej + 2 dA,

i<j k,l

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

VAVA,p =1

Rijklak A o-' eiej 0 + 1 dA4 i<j k,l

2

Here DADA is the 2-fold absolute differential of the spinor field V) and hencea section of I'(A2 (& S). Starting from this 2-form with values in the spinorbundle, 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 byn

H,A(X) (VAVAO)(X,ea)

Then the following holds.


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

H,p (X) 2 Ric (X) z + 2 (X idA)

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

n1) E ea (dA(X, ea)) V = (XidA) 0,

a=1

2) E > E Rijklo'k A Qi(X, ea)eaeiej = -2Ric(X) V).a i<j k,l

The first one is trivial since the sum Ea=1 dA(X, ea) ea represents thedecomposition of the form X_jdA with respect to the basis e1, ... , en. Thefollowing calculation takes place in the Clifford algebra Cn:

Rijklo-k Aol(X, ea)eaeieja,k,li<j

Rijalo-1(X)eaeiej

a,k,i<j a,l,i<j

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

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

2

k,i<jE RjkaUk(X)eaeiej

k,i<ja54i j

-2 E Rijkiok (X)ej + 2 Rjkao'k(X)eaeieji,j,k k,i<j

a0i,j

,

-2Ric(X) + 2 RjkaQk(X)eaeiej.k,i<ja9-ì,j

However, the second summand vanishes. Namely, for a fixed index k thissum 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 coefficientat 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] ,


the formula

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

can also be written asn

RS(X, e«) _ -2Ric(X)o +2

(XidA)

where RS(X,Y) = oXoY - VYVX - V is the curvature tensorin the spinor bundle.

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

VA 1 = 0.

Since x2 = (vXO, ' ) + (V), V ) = 0,the length of a parallel spinor field is constant. In the following we supposethat 0 does not vanish identically. DAB = 0 implies V AV Ab = 0, and hencethe 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 evaluationof the last equation we need the following

Lemma. Let 0 E On be a non-trivial spinor and Z1, Z2 E I(Sn two realvectors. 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). Inthe 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 aswell as (Z1, Z2) = 0.

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

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').


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) ofthe 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 -WIW1 0

o -W2W2 0

A=0 -Wk

Wk 0

0

0

and S is similar to

Al 0

S=

0

0

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

kIIAII2 = tr(AAT) = -tr(A2) = 2Tw2,

i=1


we then obtainIIRicHI = IIAII.

However, the square of the length of A as an anti-symmetric mapping istwice 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 structureand A a connection in the U(1)-principal bundle P1 induced from this spinestructure. 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 theconnection,

2) rank(QA) = rank(Ric), and3) the endomorphisms Ric and QA of the tangent bundle commute.

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

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

Corollary. Let (M', g) be a connected Riemannian manifold which admitsa 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 forany connection A in the U(1)-bundle of the spine structure.

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

3.2. The Dirac and the Laplace operator in thespinor bundle

We start from a Riemannian manifold (Mn, g) with a fixed spine struc-ture and a connection A in the U(1)-principal bundle P1. These induce acovariant derivative

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


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

Definition (Laplace operator on spinors). If E r(S) is a spinor field,then 0 (') Lis defined by

AA( )_n n

eve - div(ei)Ve

Using Stokes' theorem one then derives - as for every Laplace operator ina Hermitian vector bundle - the formula

/' / /' (02))f (AA (01), 02) = J / (VAW1, VAW2) _ f (01, DAMn

Mn Mn

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

n

(DA 1, DA02) = E(V1, VAe02)i=1

The Dirac operator in turn results from the composition of the canonicalderivative 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 framee = (el, ... , en) on the manifold Mn,

n

DA's _ ei Dei=1

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

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


This immediately follows from the formulan n

DA(f 0) = ei De (f ) = ei {df (ei) 4' + f7e}i=1 i=1

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

n/,

n/'(DAb, 0l) _ (ei Ve O, 4'1) (,7

Y'e , ei 'Y1)i=1 i=1

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

ei(4', ei 01) - div(ei)(0, ei 01) + (0, DA01)i=1 i=1

Considering the 1-form MO,01(X) = (V), X b1), we see that the first twosummands 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 respectto the L2-product.

Proposition. Let and ?P1 be spinor fields with compact support (containedin 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 splitsinto the sum S = S+ ® S- of Dirac spinors. Since Clifford multiplication byvectors interchanges these summands, the Dirac operator decomposes intothe sum of two operators, DA : F(St) -* r(SF).

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

Lemma. The formula

1n

P(X(94b) =X®0+n

i=1

defines a projection from the bundle T ® S onto the bundle ker(a) C T ® S.


Proof. A straightforward calculation shows that the image of P is containedin 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 super-position of the covariant derivative with the projection onto the kernel ofClifford multiplication,

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

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

i=1

TA(b) =i=1 i=1

1ei ®De + ei ®ei DA(Y')n

n

ei ® oe + 1-ei DA())

Z n

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

V x1

+ nX DA(s) = 0.

Example. Consider a2 with its Euclidean metric and coordinates x, y.Then e1 = j, e2 = is an orthonormal frame. For the forms wii ofthe Levi-Civita connection we have wi,7 = 0. A spinor field is simply a map-ping : R2 - f A2 = C2. The covariant derivative V coincides with thedifferential do, since wi.7 = 0. Contrary to the convention valid up to now,this time we will employ an equivalent though different realization of theClifford algebra described by the matrices

e1 = ( 01 0)' e2= (0 o)

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

of of0 1 ax 0 i ay

DO _ (-1 0)a + \ i 0 aax ay

2

afaz

3.3. The Schrodinger-Lichnerowicz formula 71

where

a 1 a a a 1 a aaz-2 ax-Zayaz2Cax+zay

The kernel of the Dirac operator (DV) = 0) thus consists of the pairs ofcomplex-valued functions f, g : R2 --> C which satisfy the Cauchy-Riemannequations

ofag=0.az az

Fix a vector(BA)

E C2 and consider the spinor field : R2 __+ C2 definedby

i(x'y) = x (01 0)

(BA)+y \ . 0/ \B/

Then

0)B1(iAax A) ay

and hence

J

i = -2 I B I90D( ) = ( O1 0) ax + (0i 0'

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 + 2 I 0 i) 0.y \ 0

But this immediately follows from

1

ax+2 (01 0 ) D ( ) = ( A) - (01 0) (B) 0,

ao , 1 (0 iD(O) = iB 0 i ) (A)= 0.

ay -7 -2 i 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 opera-tor DA are second order differential operators. We compare these operatorscomputing their difference D29 - OA:


DA - AA'OE ei De (ej VA O) + EVA VA

e+ E div(ei) 7 o

i j i

Eei {(Deyej) VAV) + ej Ve Ve.,0} + EVe De O + )VAIi, j

E g(Deiej, ek)eiek V b+ Eeiej V eVei,j, k

+E DeVe + 'div(ei)Ve zi

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

The latter of these equations is a consequence of the definition of the diver-gence, i.e.

j:g(Veiej,ek)eiekVej211 = - div(ej)DejO.j i=k

Now rewrite the following endomorphism:

Eg(Veiej, ek)eieki56k

This implies

DA - DAO

-57 g(ej, V e2ek)eieki0k

- E g(ej, V eiek - V ekei)eieki<k

1:g(ej, [6k, ei])eieki<k

g(ej, [ek, ei])eiekVej w + eiej (De Ve - De, De ) Y'j i<k i<j

eiej(Deve - oeoe - veie,]) = 2 eiejR'(ei,ej)i<j 2, j

In Section 3.2, the identity

ej ' RS(ei, ej)zb = - IRic(ei) + 2 (ei_jdA)

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

eiej Rs(ei, ej) _ - 1 12 ej Ric(ei) +

2 E ej (ei_jdA) '.i,j i i


But, in the Clifford algebra,

eiRic (ei) _ Rjezei = - Rii = -R.ij

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 +2dA-0.D2AO-DA =2 ('-

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

Proposition (Schrodinger-Lichnerowicz formula). Denote by R the scalarcurvature of the Riemannian manifold and let dA = Q be the imaginary-valued curvature 2-form of the connection A in the U(1)-bundle associatedwith the spin structure. Then one has

DA AAO +R0

+ 2dA 0.

F-I

3.4. Hermitian manifolds and spinors

Consider an almost-complex manifold (M2k, j) with almost-complex struc-ture 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)-eigensubspacesof J:

Al = T*(M2k) ® C = A1'0 ® A°,1,

where

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

A°'1 = {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

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,


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

C7Qp,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 canshow the following.

Lemma. The exterior differential d maps P P,7 into the sumcp-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 actson 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-complexmanifold (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 vectorfields

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 linearcombination of exterior products of r forms of this type. Next we willcompute some algebraic identities. First,

e2,,-1J(e2o A (e2,3-1 + ie20)) + e2a-1 A (e2a-J(e20-1 + ie2Q))

0 ifa /3,

i(e20-1 + ie20) if a = /t,


and

e2«- (e2a-1 A (e20-1 + ie20)) + e2a A (e2a-1-j(e2,3-1 +ie2Q))

f 0 if cti '3'ie2Q) if a = ,3.

This implies the

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

E e2a-1- (e2a A,0'r) + E e2a-1 A (e2aJ770'r) = 2r770a=1 a=1k k

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

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

52 : S )S.

We compute the eigenvalues of this endomorphism.

Proposition. Q : S -+ S has the eigenvalues i(k(Ic),- 2r), 0 < r < k, andthe corresponding eigensubspaces have dimension respectively. Therspinor 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 Section1.3. In the space of Dirac spinors O2k = CC2 ®... ®C2 (k times) the operatore2.-1e2a, 1 < a < k, is given by the matrix

e2a-le2a=E0 ...0 E0 91920 E9 ...®E

with 0 19192 = 1 0 ). 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 C2consisting 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

k

1(v(61) ®... ®v(ek)) = i (e) v(sl) ®... ®v(sk).

a=1

From this we conclude the assertion.


Now we will present an explicit isomorphism from the (0, r)-forms withvalues in So onto the bundle Sr,

A°'r®So Sr, 0<r<k.Proposition. The mapping ar : A°,r (9 So Sr defined by

?7°'r (g V)0 i)1

2

77O,r 0022

induces an isomorphism between the bundles A°,r ® So and Sr preservinginner products.

Proof. We first have to prove that the product i 0,r '50 belongs to Sr. Todo so, we will use the formula

(x A wk) = x (wk V)) + (xJwk) 0for x E Rn, wk E Ak and % E On. Next we obtain

e2a-le2a(770'r - Oo) = e2.-1 . {(e2a A 77°'r) - ,/010 - (e2aJ?7°'r) - 001

_ (71°,r A e2a-1 A e2a) ' `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 rewritethe following expression:

(77o'r A e2,-1 A e2a) 00_ (77O,r A e2a-1) . e2a - 00 - (-1)r+1(e2a-J(?701r A e2a-1)) 00

_ ,O,r 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

n/1

k/,= 77O,r

(QV)O) - E e2,,-1J(e2a A ?70'r) . 4'0 - E e2a-1 A (e2a_J770'r)bo

a=1 a=1k k

+ 1 e2aJ(e2a-1 A71°'r)00 + T e2a A (e2a-1J?70,r)V)o

a=1 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 followdirectly from algebraic calculations.

Remark. The factor2 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 = it0we obtain I(e1 + ie2)Jo12 = 4I0oI2.


Remark. In the isomorphism specified above we have to use the complex-conjugate bundle A°'r, since to apply Clifford multiplication we first have tochange the r-form into the corresponding r-vector. However, an Hermitianmetric induces a complex anti-linear identification of vectors with covectors.

An analogous calculation leads to the following

Proposition. The mapping /3r : A°,r,r ® Sk --+ Sk-r defined by rlr 0 ® k -1r,02 21] 'Ok is an isometry.

Corollary. The spinor bundle S of an Hermitian manifold (with respect toan 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 Her-mitian manifold (M2k, j, g), i.e. the one defined by the lift of the grouphomomorphism 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,

thenk

2 Z 9 i -k7 / / \ / \F(A) = e j=1 11 (cos 12 I+ sin I l l e2j_1e2i

7

So corresponds to the basis vector v(1) ® ... 0 v(1) and Sk to the basisvector v(-1) ® ... ® v(-1). Hence the endomorphism e2j_1e2j acts on Soas multiplication by (+i) and on Sk as multiplication by (-i). Thus thebundle So coincides with the highest power of the complex tangent bundle(TM2k, j), while Sk is trivial. On the other hand, the determinant bundleG of the spin structure is again Ak(TM2k, J); hence

So=G=Ak(T), Sk=®1


in the case of the canonical spin structure. Now we turn to the followingcalculation involving first Chern classes. In general, for every spin structurewe 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 Hermitianmanifold, L the determinant bundle of this spin structure and So the cor-responding subbundle of the spinor bundle. Then for the Chern classes thefollowing 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 canonicalspin 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) toSpin (2k) related to the homomorphism fl(A) = (A, det a)

.We will call the

spin structure corresponding to this the anti-canonical spin structure. Inthis case, L = Ak(T*) and So = 81. Furthermore, we will discuss the casethat the given spin structure on (M2k, j, g) originates from a spin structure(P, A) . 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 thespinor 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 thegroup Spin(2k). On the other hand, A2k has a real (quaternionic) structurej : A2k -' A2k which is spin equivariant and commutes (or anti-commutes,


respectively) with Clifford multiplication (compare Section 1.7). j inducesa morphism in the spinor bundle which is complex anti-linear and, since

clj(O) = j(i(k - 2r)0) = -(k - 2r)ij (o) = (k - 2(k - r))ij(O),

it maps the subbundle Sr into Sk_T. Going to the conjugate bundle 8k_, jcan 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'0Hence,

So = Ako = Ak(Ao'1) = Aok.

Altogether, we obtain the following table.

G So Sk

canonical spin structure Ak(T) Ak(T) 01

anti-canonical spin structure Ak (T*) 61 Ak (T* )

spin structure ®1 So = AO k Sk = Ak,O

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 vectorbundle 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") (9i=O

So. Here 8A0 and aAo are defined by

2k 0,r+1

19A. (1"7O'T ®VO) = (5,70,r) (g 0C) + ei A 770,T

i=1

® 0A °0o,

2k5A" (,70,r ® 00) _

(6*7,O,T) ® 0 - (E T1 ®ve °o0i=1


Note that for every vector X E T (M2k) and each (0, r)-form r70'r, the formX-Jr70,r is a (0, r - 1)-form. Thus in the formula for aAo the projection ontothe (0, r - 1)-component is no longer necessary. The isomorphisms alreadydefined,

ar:AO,r(9 So )Sr,k k _

combine to an isomorphism a = L Car between the bundle j AO"r ® So andr=0 r=O

S. Hence we can compare DA with aAo + aAo . Note that a-'DAa becomesk

a complex linear operator in the bundle L Au," ® S.r=O

Proposition. There exists an endomorphismk k

E : (A0?') ®SO --> A0,r ®SOr=0 r=0

such thata-1DAa = v(&0 + aAo) + E-

E depends on the almost-complex structure J, on the Hermitian metric gand 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 bythe vector X. Without loss of generality we choose the Hermitian basisel, 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 into77O,r = (e1 + ie2) A r70,r-1 + 77*0,r with el_Jr7*0,r = e2.J *O,r = 0.

Then,

U(DA) (X)a(T1o'r®,00) = 21 el . 70,r . 00

=/2/2

el (e1 + ie2) r7o r-1 'po +2r/2 el 77*o'r V)o

22 (-1 + iele2) 71°'r-1 'bo + 2 (e1 A 7,*O,r) . 00

Clifford multiplication by e1e2 commutes with 770,r-', and e1e200 = i00Moreover, e1Jr7*O,r = 0. Hence,

o-(DA) (X)a(770'r (& 00) 2222770,r-1 00 + 2 {(el + ie2) A 77*O,r} 00.


Applying a-1, we conclude that

a-1o-(DA)(X)a(?7"' ®00)I

V'2_77 O,r-I + v'2- (el + ie2) A 17*0'r) ®1P0

2 11 J

_ (_X1Or + (X +2JX) A770,r/ ®,00

Thus the symbol of a-1DAa is computed. For the symbols of aAo and aAowe 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-1respectively, 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 bundleof SO(2k)-frames and R its U(k)-reduction, then the Levi-Civita connectionreduces 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, thecorresponding Dirac operator D just agrees with ,l'2-(a + a*):

Proposition. Let (M2k, J, g) be a Kahler manifold with the anti-canonicalspin structure. Then,

1) S AO,O + ... + A°,k, and

2) the Dirac operator defined by the Levi-Civita connection coincideswith v (a + a*).

Corollary. The space of harmonic spinors, {, E r(S) : Do = 0}, of acompact 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 spinstructure. Then,

L = Ol, S0 = A°'k, Sk2 = Ak'0.


Thus Sk is a square root of the canonical bundle K = Ak(T*) = Ak,o of theKahler 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 inSk is induced from the Levi-Civita connection. Summarizing, we have forthe 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,ok

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 acompact 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 spinstructure).

3.5. The Dirac operator of a Riemanniansymmetric space

Consider a Riemannian symmetric space Mm with isotropy group G. If Kis the isotropy group of a fixed point mo E Mm, then Mm can be identifiedwith the homogeneous space G/K. The Lie algebra g of the group G splitsinto

g=t+mand 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 definiteon m and has the invariance property

([X, Y1, Z) + (Y, [X, Z]) = 0


for all X, Y, Z E g. This scalar product defines a Riemannian metric onMn which we also want to denote by (,). Right and left translations in thegroup G will be denoted by Rg and Lg, respectively,

R9(91) = gig, L9(91) = 991The projection 7r : G -> G/K = Mm is a K-principal bundle. This principalbundle 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 Gis given by

(9) = dt(9 etx)t=0.

But this is precisely the left invariant vector field X determined by thevector X E t. Thus the vertical tangent space of the K-principal bundle7r : 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-formZ : TG -+ t is easily described. Let O be the Maurer-Cartan form of theLie 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 theK-action is described by the left invariant vector field X, and this impliesO(X) = X, i.e. Z(X) = X. On the other hand, the kernel of Z is exactlyTh(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].


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

2 1S2 _ - 2

[prmE), prma]

Let Q be the bundle of orthonormal frames on M. Then there exists aninclusion i : G --> Q such that the diagram

G Q

G/K +- Mm

commutes. To this end, fix an orthonormal basis el,... , e,,,, at the pointmo and define i(g) = (dly(e1), ... , dl9(e, )), where 19 : Mm --> Mm is theaction of g E G on Mm. Now we want to see that the Levi-Civita connectionreduces to the K-principal bundle (G, 7r, G/K; K) and coincides there withthe constructed connection Z. Note that the tangent bundle T(Mm) ofMm = G/K is

T(Mm) = G XAd(K) m,

the bundle associated with the representation Ad : K -* 0(m). Hence avector field T on the manifold Mm is a mapping T : G -p m with theinvariance property T(gk) = Ad(k-1)T(g). Z induces a covariant derivativeVz 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 Vzis torsion-free. However, these two conditions uniquely determine the Levi-Civita connection of Mm.


Fix a homogeneous spin structure of the symmetric space G/K, i.e. a ho-momorphism Ad : K -- Spin(m) such that the diagram

Spin(m)

A I A

K - So (m)

commutes. Let n : Spin(m) --f GL(A) be the spin representation. Thena spinor field 7P is identified with a function : G -* A satisfying theinvariance 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) bythe element Ad*(X) E spin(m) C Viff (m) of the Clifford algebra. Considerthe 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 thatDZ'(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 formulam

i=1

We will now compute D2:

"/'m m m

/ /'D20 = X,.Xj . (XXj(')) _ -EXi (0) '+ 2E Xi'Xj . ([X,,Xj]o)

i,j=1 i=1 i,j=1

However, the vector field [Xi, Xj] belongs to t, and hence the above calcu-lation shows that

[Xi, Xj] (0) = -Ad*([Xi, Xj]) . 0-

86

Inserting this yields

3. Dirac Operators

M

D2 X2(0) 2i=1 i,j=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 k

and, finally, using the Casimir operator S2c X2 - > Ya of the groupi=1 a=1

G, we arrive at the formula

D20 = QG(V))

k.qb _ 1

a=12

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 orthonormalbasis in in. Then,

mAd, (Y,) = 4 i,j=1

m

A E (Ad. (Y.) (Xi), Xj) Xi ' Xji,j=1

m

d 1: ([Y., Xi], Xj)Xi Xii,j=1

and, analogously,

m

Ad.([Xi, Xj]) = 4 ([[Xi, Xj], Xp], Xq) Xp ' Xq.p,q=1

The remainder term thus coincides with

1 R E E E ([YY, Xi], Xj) ([Ya, Xp], Xq)XiXjXpXqa=1 i,j=1 p,q=1

in

1: Ein

Xi,X ,X ,X XiX X Xi,j=1 p,q=1


Because of the invariance property of the scalar product (,) in g we alsohave

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 ([Xi, Xj], [Xp, Xq])XiXjXpXqi,j,p,q

The Levi-Civita connection is induced from the connection Z in the K-principal bundle. Thus for the curvature tensor R of the Riemannian spacethe 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 Cm-as well as C4L-parts. The Cm- and C4,,-parts vanish, and for the C° -part wehave

- d E([X" Xj], [Xi, Xjl)X'XjXiXji<j

=4EII[Xi,Xj]II2=8

I[Xi,Xj]112 = 8R,

where R is the scalar curvature of the space G/K. Summarizing, we arriveat the

Proposition. Let Mm = G/K be a compact Riemannian symmetric spacewith a homogeneous spin structure. Let SZG denote the Casimir operator ofthe Lie group G. Then,

D2=SlG+8R.

Remark. The importance of this formula lies in the fact that it allowsus to compute the eigenvalues of D2 purely by means of representationtheory. Let A : G -> GL(VA) be an irreducible complex representation and


A. gt(V,\) its differential. Then,m k

A*(QG) = - E '\.(X,)1 - E) *(Y«)2z=1 a=1

is an operator in VA. It is easy to show that this operator commutes withall automorphisms .(g) : Va -i VA (g E G). This follows immediately fromthe 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 calculationusing the fact that Ad(g) g -> g maps the orthonormal basis consistingof {X1i ... , Xm, Y1, ... ,Yk} again onto an orthonormal basis of g. Let ube 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 impliesWN0 = 0 or VA. But WN, = 0 is excluded, as ). (cla) indeed has eigenvaluesover 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 square-integrable 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 eachtime the number c(A) with \*(I1G) = The spectrum of D2 is thengiven 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 onComplex Projective Space Cp2q-1, Letters in Math. Phys., 18, 1989, 165-176.

Th. Friedrich. Der erste Eigenwert des Dirac-Operators einer kompak-ten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkriimmung, Math.Nachr. 97 (1980), 117-146.

Th. Friedrich. Zur Existenz paralleler Spinorfelder fiber RiemannschenMannigfaltigkeiten, Coll. Math. XLIV (1981), 277-290.


N. Hitchin. Harmonic spinors, Adv. in Math. 14 (1974), 1-55.K.-D. Kirchberg. Compact six-dimensional Kahler spin manifolds of pos-itive scalar curvature with the smallest possible first eigenvalue of the Diracoperator, 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 projectifsquaternioniens, 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, Sitzungsberichteder 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 onthe odd-dimensional complex projective space CP2m-1, Preprint des SFB288 "Differentialgeometrie and Quantenphysik" No. 95 (1993).H. Strese. Uber den Dirac-Operator auf Graf3mannschen Mannigfaltig-keiten, Math. Nachr. 98 (1980), 53-58.S. Sulanke. Berechnung des Spektrums des Quadrates des Dirac-OperatorsD2 auf der Sphare and Untersuchungen zum ersten Eigenwert von D auf5-dimensionalen Raumen konstanter positiver Schnittkrummung, Disserta-tion, Humboldt-Universitat zu Berlin 1981.

Exercise 1. Let (M4, g) be a 4-dimensional Riemannian spin manifold withnon-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 a4-dimensional Riemannian manifold (M4, g) induces a complex structureJ : TM4 -* TM4 such that (M4, g, J) is a Kahler manifold.

Exercise 3. Prove that, in 2-dimensional Euclidian space i 2, the generalsolution to the twistor equation T() = 0 is given by

tl>(x,y) _ (D) -(-XI+iY x

02y)(BA)

with arbitrary vectors (A), (C)D E CZ = A2-B D

Exercise 4. The metric on M4 C II84,

M4 = {(xl, ... , x4) E L 4 :x1 > 0, 0 < x2 < 7r},


defined by

ds2 =xl

(dxl)2 + xI(dx2)2 + xlsin2(x2)(dx3)2 +xl +

c(dx4)2xl + 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 E F(S) aspinor 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 Propertiesof Dirac Operators

4.1. The essential self-adjointness of the Diracoperator in L2

Let us recall some notions from the spectral theory of linear operators incomplex Hilbert spaces. Let A be an (in general unbounded) operator withdense domain of definition, D(A), in the complex Hilbert space H. Denotethe 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) CH x H again is the graph of an operator A which then is called the closureof 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 whichthere exists a sequence xn E D(A) with lim xn = x such that, moreover,

n-roothe sequence A(xn) converges in H. Differential operators always have aclosure 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 spectrum0p (A):

Qp(A) = {A E C: ker(A - A) {0}}.

Furthermore, there are the residual spectrum, vr(A), as well as the contin-uous spectrum, Q,(A):

ar(A) _{AEC: ker(A - A) = 0, R(A - A) 0 H},

91

92 4. Analytical Properties of Dirac Operators

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 do-main 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).

Then D(A) is contained in D(A*) and, in addition, A. (We willabbreviate this as A C A*.) The double adjoint operator A** coincides withthe 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 Ais 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 complexnumbers and B(S) the o-algebra of all its Borel subsets, then a spectralmeasure 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 Hilbertspace 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).


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 beintegrated against a spectral measure:

f f (s)dF(s).S

The value of this integral is an operator in the Hilbert space H which isbounded for a bounded function. Moreover, we have the formula

((I f (s)dF(s) x, y

sff(s)d/2xv(s).

I S

The spectral theorem for self-adjoint operators can now be formulated asfollows:

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

A = f \dF(,\).v(A)

We now turn to the situation for the Dirac operator DA on a Riemannianmanifold (Mn, g) with fixed spin structure and fixed connection A in thedeterminant bundle of the spin structure. The space 1T (S) of all sectionsof 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 inL2(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 com-plete Riemannian manifold. To start with, we state the following formulafor 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)


Proof. This formula is obtained by a straightforward computation:

DA(f ) = ei 'VA

e(f Y) _ ei (ei (f) 4 + f

Ve4 )

i=1 i=1

grad(f) ,+fDA(ib)

Now we start to prove the essential self-adjointness of the Dirac operatorDA. 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 intoseveral steps. Let DA be the adjoint to DA. In the domain D(DA) weintroduce the norm

N(b) 110112 + I IDAI12,

where 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. ThenDA is essentially self-adjoint.

Proof. Under the assumptions above we have to prove that D(DA) CD(DA). Let 0 E D(DA). By assumption, there exists a sequence on E Fe(S)with 1linN(0 - On) = 0. This implies that rli On = 0 in L2 (S) and,moreover, that DAWn) converges to DA(s) in L2. However, n is a smoothspinor field with compact support, and hence DAW ) = DA(2b ). Thusthe sequence DA('On) converges in L2. But the latter just means thatb 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 E D,(DA), there are only finitely many indices i E Isuch that

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 offunctions with compact support defined on the space R1. Approximate 0j


by the convolutions with an approximation of the delta-distribution. Leth : IIBn -- R1 be given by

0 forlxl > 1,h(x)

e 11-x1' for IxI < 1,

and let hE : R' - R1 be defined by hE = E h (1). Then bj * h, is a smoothfunction with compact support approximating bj in L2. Since DA (0i) be-longs to L2, the sequence DA( j * h,) thus converges to DA( j). Applyingthe chart mapping again, we obtain smooth spinor fields with compact sup-port i,l, j,2i ... defined in the Riemannian space with

k N(Oj - 'i,k) = 0.

Forming 'Ok = 1,k + ... +'1,k, we now conclude that k belongs to Pe(S)and lim N(b - L1k) = 0.

k-*oo

Lemma 3. If (Mn, g) is a complete Riemannian manifold, then D°(DA) isdense in D(DA) with respect to the N-norm.

Proof. Denote the interior distance between two points ml and m2 in theRiemannian manifold by d(ml, m2). Let mo E Mn be a fixed point andp(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 almosteverywhere and the gradient grad(p) exists a.e., too. Furthermore, at eachof these points

I I grad(p) I I << 1.

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 com-pact subset of Mn. Choose a C°°-function, a : II81 --* [0, 1], with the follow-ing properties:

(i)

(ii)

a(t)-1 for -oo<t<1,a(t) =_0 for 2 < t < oo.

Let the constant M be the maximum of the derivative I a' (t) I:

M= sup a'(t)j.1<t<2

The function b,.: M' -> [0, 1] is defined by

b, (m) = a (P(m))r J


Then, br - 1 on JC(r) and supp (br) C 1C(2r). Moreover, br is Lipschitzcontinuous and the following inequality holds a.e.:

1

IIgrad(br)112 = r a' 2 Ilgrad(p)II2 <2

r2

Let E D, (DA) be given. Then 0, = belongs to D(DA). Furthermore,

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\IC(r) Mn

Thus we obtain the following estimate for the N-norm:

N2(b- b) = II0-wrUI2+IIDA(0 -br)112

< f IIV)I12 +Mn\IC(r)

f 2 I IDA( ) I 1 2 +2M2 f I

V)I I2.

Mn\IC(r) Mn

From f 110112 < oc and fI

I DA () I I2 < oo we then immediately concludeMn Mn

thatlim N2(b - fir) = 0.

r-roo

In particular, we have proved the following:

Proposition. Let (M'n, g) be a complete Riemannian manifold with spinstructure. 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 aspinor field of class C2. Then, for the L2-norm I ' I

I and any number t > 0,

IDA( )I12 < tIID2 (V)) 112 + t 11 112.

Proof. We will use the same balls 1C(r) and functions br as in the proof ofthe previous proposition. The equality

DA(br,0) = 2br grad(br) + b2DA(' )


implies that for every positive number e > 0

f I I brDA(O) I2 = f (b2r DA (0), DA (0)) (DA (b2

K(2r+E) 1C(2r+E) K(2r+e)

2 f (br grad(br) ' DA(s),) f (brDA(0),)1C(2r+E) K(2r+E)

Here we used the fact that the support of the spinor is containedin )Q2r). For e -+ 0 this implies

f II brDA(O)II2 = f (DA(,0), br'p) - f (b, DA (0), 2grad (b

IC(2r) 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 (witht = 1):

f (brDA(O), 2grad (br) V)) < 2 fHbrDA()II2+2 f I Igrad (br) I I2IC(2r) IC(2r) IC(2r)

2

< 2 f IIbrDA( )II2+2M f 11 112.

K(2r) K(2r)

From 0 < br < 1 and another application of Schwarz' inequality, this timeto the first term, we obtain

f (DA(),br) 2 f JIDA()II2+ 2t f IIII2.K(2r) IC(2r) K(2r)

Combining both relations yields

fHDA()H2<f IIbrDA(V))II2 = 2 f IlbrDA(V))II2 - f IIbrDA(V))II2K(r) K(2r) K(2r) K(2r)

< 2 f { 2M2

II0II2+2IID2(4)II2+2II0II2Jr2 A

IC(2r)

- f IIbrDA('O)II2K(2r)

f 4M2 ) 110112 }{JI2+ Ct

+

K(2r) I


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,

IDA()112 < tIID2 (0)II2 + t IIV) 112.

Corollary. For a complete Riemannian manifold (Mn, g), the kernels ofthe 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 forsolutions of elliptic differential equations we first conclude that '0 is smooth.Hence we can apply the inequality from the previous proposition and thusobtain

IIDA(b)II2 _<tHID2(O)II2+111

112= tII0112.

II II2 < oc then implies for t - oo that DA(b) - 0-

4.2. The spectrum of Dirac operators overcompact 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). Inthe 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 continuousspectrum, a (DA):

o-(DA) = o-(DA) = o'p(DA) U Qc(DA):

For a compact Riemannian manifold the domain of definition of the operatorDA coincides with r(S). If \ E o-p(DA) is an eigenvalue of the closure, thenthere exists a spinor field 0 E L2(S) with

DA(V)) _ Aw, 0 E L2.

The regularity theorem for elliptic differential operators then implies that 'is smooth. Hence we have 0 E F(S) = D(DA), i.e. the point spectrum ofDA coincides with the point spectrum of the closure:

o-p(DA) = op(DA).


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 spinstructure. For the spectrum of the Dirac operator the following relationshold:

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 residualspectrum of DA is empty. Indeed, for \ E Qr(DA), there exists a spinor fieldcp E L2(S) such that

(DA(0)-AW,c0)L2=Ofor all ' E F(S). Choosing '' with support in a chart and transferringthis equation to Euclidean space, we obtain an elliptic differential operatorP (= DA - A) as well as a function cp E L2 (R') such that

(P (0), cp) L2 = 0

for all E C'°(R'). By the regularity theorem for elliptic operators, cp issmooth. 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) _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 compactmanifold, 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.


On is

1

2

smooth and we can thus apply the Schrodinger-Lichnerowicz formula:

II DA (0n) - AV).II L2 < I IDA('On)II2+'\2I

I0.I I2

= IIVAV)nHIL2+ f f(dAn,n)+A2Hnl.Mn Mn

Since Mn is compact and I I Yin I I L2 = 1 as well as I I DA (bn) - AV), 112L2 -3 0,

the L2-lengths I I VA On I I L2 of the 1-forms DAn,

DAY nI IL2 = f 1:(oe'On,VA

e0n)dMn,

Mn i=1

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 wecan assume that on converges to a spinor field V)o in L2, and this immediatelyimplies 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 fixedspin`s structure. The first Sobolev norm of a smooth spinor field '0 E F(S)is given by

II II H' = I IIL2+IIoA112

and the corresponding Sobolev space H'(S) is the completion of F(S) withrespect to this norm. Since Mn is compact, different connections A induceequivalent norms in the determinant bundle of the spinC structure. More-over, the embedding

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 estimatebelow for the norm of DA(s):

IIDAOIIL2 = , f(eiV,ejV)< fivtvii,j=1Mn id =1Mn

n/1

2f(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 theother hand, rewrite IIDA(0)IIL2 as follows:

IIDA()IIL2 = (DA1 ),0)L2 = IIVA IIL2 + f 4II0II2+ f 1(dA'0,0).Mn Mn


The endomorphism 1 dA : S -> S is bounded, i.e. there exists a constantc > 0 such that for every point of the manifold Mn and any spinor 0 wehave the pointwise inequality

-cIIV)II2 < dA- 0,0) cII0II2.(Inserting this, we obtain the inequality

(*) ICI IH1 +(Rrnin

- C - l l 112 2IL2 < II DAV)II L2

110 1I1+

Rax+ C - 1 I 110112H(4 L2,

where Rmax = max {R(m) : m E Mn} is the maximum of the scalar cur-vature and Rmin its minimum. We will use this inequality to prove thefollowing:

Proposition. Let DA be a Dirac operator over a compact manifold. Theclosure DA = DA of the Dirac operator DA is defined on the subspaceH'(S) C L2(S).

Proof. If V E D(DA) belongs to the domain of DA, then there is a sequenceOn E IT(S) such that On 0 in L2(S) and DA(',) converges in L2. Frominequality (*) 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) --+L is continuous, hence b* coincides with ib. Thus 0 belongs to H'(S).The converse is trivial.

Proposition. Let A ¢ o'(DA) be a number which is not in the spectrum ofthe 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 IIL2.

Setting cp = (DA - A)b E im(DA - A), we have /II (DA - IICI122 +CII (DA - A)-1;pIIL2'

However, the operator (DA - A) is continuous in L2. Hence, for a suitableconstant C*, we obtain /the estimate

II (DA - '\)_'(P1 1H1 < C*IIsoIIL2.


Thus the image of the operator (DA - A)-' (A ¢ v(f)A)) is contained in theSobolev space H1(S). The assertion then follows from the compactness ofthe embedding HI(S) -+ L2(S).

Proposition. There exists a complete orthonormal basis zbj, V)2.... of theHilbert space L2 (S) consisting of eigenspinors of the Dirac operator DA,

DA()n) = AnOn-

Moreover, lim I An I = 00-n- oo

Proof. Choose the real number A 0 in such a way that A is not inthe spectrum of DA. The operator (DA - A)-' : L2(S) -* L2(S) is compactand self-adjoint. The spectral theory of these operators leads to a completeorthonormal basis '01 i V)2, ... in L2 (S) with

(DA - A)-12Nn = AnOn

and lim An = 0. Thus (u 0),n- oo

1 + A On)DA(V)n) = (An/

i.e. the spinor field On is an eigenspinor of the Dirac operator for the eigen-value An = + n

Corollary. There exists a constant C > 0 such that for all spinor fieldsco 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. Since II11 '0 1122 ->

I I DA'' I I L2 , we have

n {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 respectto the norm

k2

11011k - E IIDA( )IIL2.i=0


c*If E r(S) is a smooth spinor field and = j Anon its decomposition inn=1

L2 with respect to the complete orthonormal system 01, 02, ... consisting ofeigenspinors of DA, one easily computes

IIDAk O11L2 =00

n=1

A 2,\2kn n

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 theSobolev space Hk(S) if and only if the sum

00

A,,2,\nk<o0

is finite.

n=1

We will use the previous proposition to define the (- and the 77-functionfor the Dirac operator. First, we prove that the embedding Hk --> L2 is aHilbert-Schmidt operator for k > 2 dim(Mn).

Lemma. Let E be a complex vector bundle with Hermitian metric and met-ric 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 containedin C° (E) and that this embedding is continuous for k >

2in. Fix a point

mo E M' and an orthonormal basis el(m°), ..., el(mo) in the fibre Emo. Letcp : 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 = sup1s(MO)11 < sup I IS =: C.

SEHk I1SIIHk 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)


The norm IIWII2 is then computed to be

sup supIIs(m2)II2 = I(S'si2Hk I2IkoHI2 =Hk ISHk EHk i=1 SIHk

and hence, for each index 1 < i < 1,

11 112 > Sup

H

((1181122 kI2

= IIsjIIHk.

Now let f1, f2, ... be an orthonormal basis in Hk(E). Then,

00 00

E Ifj(mo)I2 = E IV(fj)I2j=1 j=1

0C l l

= EEI(fj,si)Hk12=EII8,112k < III(PII2 <iC2.

j=1 i=1 i=1

Integrating this inequality over the compact manifold M' yields

I I f j1 1 2

L 2 < l C2 vol (Mn).

The spectrum a(DA) of a Dirac operator only consists of eigenvalues, and theclosure DA defined on H'(S) is self-adjoint. Let F : 5(o-) -> Proj(L2(S))be the spectral measure. Then,

rDA =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.


Proof. Only the last point needs to be proved. First, note that for allco E L2 (S) the H'-norm I St (cp) I lHk is finite. Indeed, since

kI 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 Stis given by the integral

AkdF(A) , f e-t,\2dF(A) _ fAke_tdF(A)i0" a

and Ake-ta2 is a bounded function of A (t > 0). Thus D,k9 o St is continuousin L2. The argument above now proves that the image of each operator Stis 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. Choosingk >

2

dim M', we obtain

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.

Corollary. The function E 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( )11i2 = e-2t 2 = D2 (2t),

AEQ(DA) AEQ(DA)

where V),\ is a complete orthonormal basis in L2 (S) consisting of eigenspinorsof DA.

By the same method we can define other spectral functions of the operatorDA. 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 tothe spectrum o-(DA). Let z E C be a complex number with positive realpart, Re(z) > 0. The function

sgn(.A) I1


is bounded on the set a(DA). The corresponding operator

dF(A)T(z) =J

sgn 1lAlz

a

is bounded in L2(S). The superposition DA o T(z) is given by the function

sgn(A) x , which is bounded if k < Re(z). Now assume Re(z) > dim(M)IT 2

Choose k with Re(z) > k > dim(M Then the operators DA oT(z), ..., Dkk o

T(z) are bounded in L2. Hence T(z) maps the space L2(S) into the spaceHk(S). The embedding Hk(S) -3 L2(S) is a Hilbert-Schmidt operator.Eventually, we obtain the

Mn) and ker(DA) = 0, then the operatorProposition. If Re(z) > 2 dim(fsgn()dF(A)

T(z) AIIis a Hilbert-Schmidt operator in L2(S).

The Hilbert-Schmidt norm/ is now computed by

IIT(z)IH-S ='\I-2Re(z)

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) isanalytic in the half-plane Re(z) > dim(Mn).

Remark. 77DA (z) has a meromorphic continuation onto the complex planeand is, in particular, analytic at the point z = 0. The 77-invariant of DA isthen defined to be 77DA(0).

We also want to discuss the boundary case z = dim(Mn). Sincer n/2

DA/2 oT.

= J I,\In/Z dF(A) = f(sgn())f12dF(A),

the operator

a a

T* _ f IAIn/ZdF(A)a

maps the space L2(S) into H,/2 (S) and DA/2 o T* : L2 .Hn/2 _._, L2

is invertible, (DA'2 o T*)2 = IdL2. The kernel of DA is trivial and hence

4.3. Dirac operators are Fredholm operators 107

DA/2 : H,12(S) --+ L2(S) is bijective, since the index of DA is equal tozero (compare the next section). Thus T* : L2 -* Hn/2 is bijective andits Hilbert-Schmidt norm as an operator in L2 coincides with the Hilbert-Schmidt norm of the embedding Hn/2 -* L2. The latter is infinite, and weobtain the

Proposition. The 71-function 77DA(z) of a Dirac operator has a singularityat the point z = dim(Mn).

4.3. Dirac operators are Fredholm operators

Proposition. The Dirac operator DA : H'(S) -> L2(S) over a compactRiemannian manifold is a Fredholm operator of index zero.

Proof. We have to prove that ker(DA) and L2/im(DA) are finite-dimen-sional vector spaces of equal dimension. In the vector space ker(DA) considerthe 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 compactoperator 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. Henceker(DA) is a finite-dimensional vector space. We determine the orthogonalcomplement of DA(Hl) in L2. A spinor cp from this space satisfies thecondition

(DA (0), co) L2 = 0for all 0 E H'(S). Similarly to the proof of the fact that the residualspectrum or (DA) of DA in L2 is empty, we conclude first the smoothness ofcp 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') CL2 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, isorthogonal 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. Thusthe limit lim 0n = exists in H1(S). The operator DA : H'(S) --+ L2(S)

n-+oois continuous, and

2b DA( n) = DA(n 4 n) = DA(w*)noc


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 Diracoperators

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-adjointoperator,

Index (DA) = dim ker(DA) - dim ker(DA ).

The index of DA depends on the characteristic classes of the manifold M2kas well as on the first Chern class of the determinant bundle L of the spinCstructure. We quote the corresponding formula without proof.

The power series of the even functiontt/2 _

sinh (t/2)

can be represented in the formt

et/2 - e-t/2

+ ....et/2 - e-t/2 = 1 + A2t2 + A4 t4

An easy calculation shows, e.g., that1

A224,

A4 10 4.24 5760

Denote the Pontrjagin classes of a 4k-dimensional compact manifold M4kby pl, p2, ..., A. The class pj, 1 < j < k, is an element of the 4j-th cohomol-ogy group Hi (M4k). Introduce k formal variables x1, ..., xk and representP1, ..., Pk as the elementary symmetric functions in the squares of these vari-ables:

xi+...+xk=p1k

Then 2rj =i 2 X -.i/2 is a symmetric power series in the variables xi, ... , xand hence defines a polynomial in the Pontrjagin classes. Denote this coho-mology class by A(M41,):

,A(M4k) =k

x2/21-fsinh (xi/2)

For example, for k = 1, 2 we obtain the formulas

A(M4)=1-pi, k4 =1,

4.3. Dirac operators are Redholm operators 109

A(Mg) = 1 - 24pi + 5760pl 1740p2, k = 2.A manifold of dimension 4k + 2 also has k Pontrjagin classes, and we defineA(M4k+2) by the same formulas. The index theorem for Dirac operatorsnow reads as follows:

Proposition. Let (M2k, g) be a compact oriented Riemannian manifoldwith spin structure, and let c = ci(L) denote the first Chern class of thedeterminant bundle of the spin structure. The index of the Dirac operatorD+ associated with a connection A in the U(1) -bundle of the spin structureis equal to

Index (DA) = Je2°A(M2k).

M2k

As a consequence of this one can prove that certain characteristic numbersare integers.

Corollary. Let M2k be an oriented compact smooth manifold and take c EH2(M2k; Z) to be a cohomology class whose Z2-reduction coincides with thesecond 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 greaterdetail. 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 quadraticform, the resulting form has a signature (p, q). By Poincare duality we havep + q = dim H2 (M4; R). The number 0-(M4) = p - q is called the signatureof the 4-dimensional manifold M4. This signature is closely related to thePontrjagin 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 polyno-mial e2 A = (1 +

2c +

$c2) (1 - ! a) yields the formula

Index (DA) _ $ (c2 - Q).

As an application of the formula just stated we prove the following result,originally due to Rokhlin.


Proposition. Let M4 be a smooth compact orientable 4-dimensional mani-fold with spin structure, w2(M4) = 0. Then the signature o(M4) is divisibleby 16.

Proof. Since W2 = 0, the manifold M4 has a Spin (4)-structure. Considerthe corresponding Dirac operator in the spinor bundle S. Then,

Index (D+)810- (M4).

But S is a vector bundle associated with the group Spin(4). From the con-siderations in Section 1.7 we know that the spin representations 04 haveequivariant quaternionic structures. These in turn induce parallel quater-nionic structures in the spinor bundles S±, and hence the complex vectorspaces 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 intersec-tion form in H2 (M4; Z) being an even Z-form,

a2 = 0 mod 2 Va E H2(M4; Z).

However, it is a well-known algebraic fact that the signature o- of even formsover the ring Z is divisible by 8. The Rokhlin theorem thus states an addi-tional divisibility of the signature by 2 in the case of smooth manifolds witheven intersection form! The smoothness of M4 is indeed necessary. Thereexist simply connected topological manifolds Mtp with w2(Mt p) = 0 and

= 8.o(Mtop

Remark. The parallel quaternionic structures in the spinor bundle S± usedin the proof of the Rokhlin theorem exist in all dimensions n = 8k + 44 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 man-ifold. Then,

is an integer.Msk+4

Apart from results concerning the integrality of special characteristic num-bers, a further application of the index formula for Dirac operators consistsin deriving topological obstructions for the existence of Riemannian metricswith positive scalar curvature. The Schrodinger-Lichnerowicz formula,

1 1DA=DA+4 R+ 1dA,


immediately implies ker(DA) = 0 and hence Index(DA) = 0, if all eigenval-ues of the self-adjoint endomorphism

4R +

adA : S -- S are positive. Thus

we have the

Proposition. Let (M2k, g) be a compact oriented Riemannian manifoldwith spine structure, and denote by c = cl(L) the first Chern class of the de-terminant bundle L of the spine structure. If L has an Hermitian connectionA such that all eigenvalues of the endomorphism

4R+1dA=4R+21lA:S-+Sare 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 hasa Riemannian metric of positive scalar curvature, and

A(ci2)8o-(Cp2)_-8#0.However, C p2 is not a spin manifold.


M.F. Atiyah, V.K. Patodi, I.M. Singer. Spectral asymmetry and Rie-mannian 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. ofMath. 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-Singerindex theorem, Publish or Perish 1984.K. Maurin. Methods of Hilbert spaces, PWN, Warsaw, 1965.K.H. Mayer. Elliptische Differentialoperatoren and Ganzzahligkeitssatzefur charakteristische Klassen, Topology 4 (1965), 295-313.


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 spinestructure and DA the Dirac operator. Consider the heat equation,

m), t > 0,m) = -DA

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 prob-lem.

Exercise 2. Let 0a, A E v(DA)), be a complete basis of L2(S) consistingof 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, m1, m2) = e_t.\20a(ml)(9 b (m2)

AEa

for t > 0 is smooth. Show, in addition, the following properties:

1) (t, m1, m2) = -D22 (E(t, m1, m2)),Ft A

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 withkernel E(t, m1, m2), i.e.

'I'/AV)) (M) _ fMn

Exercise 4. Let 0 < A < be the eigenvalues of D. Prove thefollowing asymptotic formula for k --> oc:

kN Ck2/n.

00Hint: 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 spinmanifold (Ma, g). Prove that in the cases n # 3,7 mod 8 the 77-functionof D vanishes identically.

Chapter 5

Eigenvalue Estimatesfor the Dirac Operatorand Twistor Spinors

5.1. Lower estimates for the eigenvalues of theDirac 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, isexclusively determined by the Levi-Civita connection. By integration, fromthe Schrodinger-Lichnerowicz formula,

D2=0+41R,

we immediately obtain the inequality A2 > 4R0 for every eigenvalue A ofthe Dirac operator, where Ro = min{R(m) : m E M' j is the minimum ofthe scalar curvature. However, this estimate is not optimal. We have (Th.Friedrich, 1980)

Proposition. Let (M n, g) be a compact Riemannian manifold with spinstructure, 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 ba corresponding eigenspinor, then is a solution of the field equation

VX =T_ Ro 1)X-02 n(n -

and the scalar curvature R is constant.

Proof. The idea of the proof is based on not using the Levi-Civita connec-tion but, instead, considering a suitably modified covariant derivative in thespinor bundle. To this end, fix a real-valued function f : Mn -+ R1 andintroduce the covariant derivative Vf in the spinor bundle S by the formula

The algebraic properties of Clifford multiplication imply that Vf is a metriccovariant derivative in the spinor bundle S:

X (V,, Y'1) = (V,/

,''b1) + ('0, Vf '01).

Let Of V Ve - > div(ei)V be the corresponding Laplace oper-i=1 i=1

ator, and denote byn nof,012

IV I2IoejO+ fei .012i=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) = D2 - 2fD - grad(f) + f2,

and the Schrodinger-Lichnerowicz formula implies

(D-f)2 =0+ 1R - 2f D - grad(f) + f2.

On the other hand,n n

Of = - E(oej+fei) (oe + fei) - div(ei) (Vex + fei)i=1 i=1

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 ((D-f)2gp,V))= f {Iof012+4RIOI2+(1-n)f2IV12}Mn Mn

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 intothe last formula and obtain

A2(n_

n 1) 11011L2= 11V nI IL2 + \21 n2n

I ICI IL2 + 4 f RIV)I2.

Mn

An algebraic transformation yields

.2nn 1 I IL2 = Ilon 12 + 4

f RI 12 >_

Mn

i.e. \2 > 1 nRo. Discussing the boundary case in this estimate, we- 4n-1

immediately obtain the remaining assertions of the proposition.

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 V) +,aX grad(f) z/) + vdf (X)V)n

with the "optimal" parameters µ = - 1n

V= -n

n1'

and perform a-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)

Proposition.

A2 > n min{1R+0(f) - n-2l grad f12}.n-1 4 n-1In particular, in dimension 2 the summand (grad f 12 drops out. Then theformula 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 +vol(M2 g) f K =

-K + 2vo(M2))

M2

(M2)Thus

2 ol(MR + 2A(f) = Vol(M

)

)is constant, and we obtain

A2 > 2irX (M2)- vol(M2)


Of course, the last inequality is interesting only for 2-dimensional Riemann-ian manifolds which, topologically, are spheres. Summarizing, we obtain thefollowing proposition, originally due to Lott, Hijazi, and Bar.

Proposition. If (S2, g) is a Riemannian metric on S2, then, for the firsteigenvalue of the Dirac operator, we have

A247r

vol(S2, g)

The method we have outlined for estimating the eigenvalues of the Diracoperator may be refined even further when the Riemannian manifold carriesadditional geometric structures. Let us consider e.g. the case of a Kahlermanifold (M2k, J, g) with complex structure J : T(M2k) -+ T(M2k). In thissituation, consider the covariant derivative

depending on two parameters f and h which can be chosen freely. Elaborat-ing on the Weitzenbock formulas for Riemannian manifolds with additionalgeometric structures, one will in general obtain better estimates than in thegeneral case of a Riemannian manifold. For example, the following inequal-ity, first proved by K.-D. Kirchberg, holds for Kahler manifolds:

Proposition. Let (M2k, J, g) be a compact Kahler spin manifold and A aneigenvalue of the Dirac operator. Then,

A2 > 411 Ro if k = dime M is odd,

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 eigen-spinor 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 '.


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

nV" = 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)), satisfiesthe 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 yieldsproperty (1). Starting from Vx = µX 0, we compute

n

=-nµoi=1 i=1

and thus obtain

T(O)

_ ei ® (VejO + 1ei DO) _ ei ® (µeii - µei 0) = 0.i=1 n i=1

For a fixed point mo E Mn and a local orthonormal frame el,... , en withVei(mo) = 0 we compute the covariant derivative VxV''P:

'7xV0n n

/ I1 "/' '/1(ei Vx4', w)ei + (ei Y', Vx )ei

i=1 i=1n

µ (ei X - Y, 'Y)ei + µ(eiO, X - Y')eai=1 i=1

n

E((ei - X -Xei) 0,O)eii=1

This implies g(VxVO,Y) = µ((YX-XY) , v); hence g(VxVO,Y) is anti-symmetric in X, Y. But this property characterizes Killing vector fields ona Riemannian manifold.


Not every Riemannian manifold allows Killing spinors 0 0 0, and not everynumber µ E C occurs as a Killing number. We now derive a series ofnecessary conditions. To this end, recall the Weyl tensor of a Riemannianmanifold. Let

Rijkl = g(VeiVejek - VejVeiek - V[ei,ej]ek, el)

be the components of the curvature tensor andn

Rij = E Raijaa=1

those of the Ricci tensor. Then define two new tensors K and W by

1 RKi. = gij - Rij ,n-2 2(n-1)

Wa,Qya = Ra,OyS - gp5Kay - gayKQS + g,3-,Ka5 + ga5KKy.

W is called the Weyl tensor of the Riemannian manifold. Because of its sym-metry properties the Weyl tensor can be considered as a bundle morphismdefined on the 2-forms of (Mn, g):

W : A2(Mn) -> A2(Mn), W(ei A ej) _ Wijklek A el.k<l

With these notations we have the following

Proposition. Let (Mn, g) be a connected Riemannian spin manifold witha non-trivial Killing spinor 0 for the Killing number µ. Then:

1) µ2 = 1 1R at each point. In particular, the scalar curvature4n(n- 1)

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. andhence

(VXVy - VyVX - V [X,y])V = µ2(Y X - X. Y)b.n

Computing £ ea R(X, ea) now yieldsa=1

n n

E ea R(X, e.),0 =µ2E ea(edX - Xea) z(i = 2(1 - n)µ2zb.a=1 a=1

On the other hand, in Section 3.1 we proved the formulan

ea R(X, ea)O Ric(X) V).

a=1


Hence, Ric(X) 0 = 4(n - 1),a2X , and, since 0 does not vanish at anypoint, this implies

Ric(X) = 4(n - 1)µ2X.Thus (Mn, g) is an Einstein space of scalar curvature R = 4n(n -1)p2. Thecurvature tensor R(X, Y) in the spinor bundle S is related to the curvaturetensor R(X, Y) Z of the Riemannian manifold (Mn, g) via the formula

n

R(X, Y)O =4

E ea R(X, Y)ea .Hence, because 4µ2 = n R

1 , the equation

VXVy' - VyV b - V[X,y]b =,u2(YX - XY) 0

can also be written as

{Ee« R(X,Y)e«+n(R 0=0

and, for an Einstein space,n

Re,, AR(X,Y)ea+n( 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 followinggeometrical property of manifolds with Killing spinors:

Proposition. A Riemannian spin manifold admitting a Killing spinor 'O0 with Killing number µ 54 0 is locally irreducible.

Proof. If Mn is locally the Riemannian product Mn = Mi X M2 -k, thenwe may consider vectors X, Y tangent to Mi and M2 -k, respectively. Thisimplies R(X,Y)Z = 0, and from

we obtain

n

Ee,,R(X,Y)ea+n(R 1)(XY-YX) 0=0

Since a 0, the scalar curvature is different from zero. Moreover, X and Yare orthogonal vectors. But this implies = 0, hence a contradiction.


The next-to-last proposition shows that Killing spinors are divided into twotypes, depending on whether the Killing number p is real or imaginaryO/t 0):

real Killing spinors p E R 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 eigen-spinors of the Dirac operator for the eigenvalue ±1 nni1

R. The field equa-tion V xV) = pX 0 could be generalized by allowing µ : M' -> C to be acomplex-valued function. However, due to a result by A. Lichnerowicz, thisdoes not lead to an actual generalization:

Proposition. Let (M', g) be a connected spin manifold, µ : Mn -> C asmooth 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 exis-tence of real or imaginary Killing spinors, respectively, are rather restrictive,and for n < 4 only Riemannian spaces of constant sectional curvature admitthis kind of spinor fields. Consider e.g. the case n = 3. Then (M3, g) isnecessarily a 3-dimensional Einstein space, hence a space form. We meetthe same situation in dimension n = 4.

Proposition. Let (M4, g) be a connected Riemannian spin manifold with anon-trivial Killing spinor 0 for the Killing number p 0. Then (M4, g) isa space of constant sectional curvature.

Proof. Decompose the Killing spinor V) =,0++0- according to the splittingof the spinor bundle, S = S+ ® S-. The equation for the Killing spinor thentakes the form

Vx'0+ = µX'0-, Vxb- = Ax'b+

Define the set

N = {m E M4 : V)+(m) = 0 or-(m) = 0}.


N C M4 is a closed subset without inner points. Indeed, if N has innerpoints, 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 Killingequation we obtain 0 0++r- vanishes identically on thesubset U, a contradiction to the fact proved above that non-trivial Killingspinors 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 C4-module A4 = A+ (D 04 explicitly given in Sections 1.3 and 1.5 proves thefollowing fact.

If 772 E A2 (R4) is a 2-form and + E A+, V)- E 04 are twonon-trivial spinors, then 172. 0+ = 0 = 772. 0- implies that the2-form 772 is trivial, 772 = 0.

Applying this, we immediately conclude that the Weyl tensor W vanishes onthe set M4\N. However, this set is dense. Hence (M4, g) is a 4-dimensionalEinstein space with vanishing Weyl tensor, i.e. a space of constant sectionalcurvature.

In view of this proposition, the search for necessary and sufficient condi-tions on a Riemannian space to have (real or imaginary) Killing spinors isinteresting only in dimensions n > 5. There are extensive series of examplesand studies, for which we refer to the book [BFGK] and the supplementaryarticle [Bar] using the holonomy theory. In the case of real Killing spinors(e.g. in the important dimension n = 7) this classification problem has notbeen 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 onlyif

vx'b + 1X D(O) = 0

for all vectors X E T(M'n). Here D() denotes the application of the Diracoperator to 0 (compare Section 3.2). Solutions ' of this field equation arecalled twistor spinors. Killing spinors are special solutions of this twistorequation. The essential difference between the field equation for Killingspinors and the twistor equation consists in the fact that the latter is con-formally invariant. By a straightforward and elementary calculation oneeasily proves the following


Proposition. Let g and g* be two conformally equivalent Riemannian met-rics 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 conformalinvariance, because the (not conformally invariant) Einstein condition isnecessary for the existence of Killing spinors. On the other hand, if (Mn, g)is a Riemannian spin manifold with Killing spinors and g* a Riemannianmetric conformally equivalent to g, then, in general, (Mn, g*) carries noKilling spinors although it admits twistor spinors. In the compact case, thisis the only difference between these two equations.

Proposition. Let (Mn, g) be a compact connected Riemannian spin mani-fold with ker(T) 0. Then there exists an Einstein metric g* conformallyequivalent 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 compactRiemannian manifolds implies that the metric g can be replaced by a con-formally equivalent metric g* of constant scalar curvature. Hence, withoutloss of generality, we can assume that g itself has constant scalar curvatureand that ker(T) ; 0. We have to prove that, in this case, every solution ofthe twistor equation is representable as the sum of real Killing spinors. Theequation

vX,O + 1XD(O)=0implies

0 = Ve.Vea0+ n E Vea(ea . D(O)) = -0(O) + nD2( )a=1 a=1

Applying the Schrodinger-Lichnerowicz formula, D2 = A + 4R, we obtain

2 nD ()=4n-1RIf the (constant) scalar curvature vanishes, R = 0, then D2(o) = 0 and,since

0 = f(D2(),) = f Iv I2,Mn Mn

is a parallel section. In case R ; 0, decompose into

_ n+nR (P+ -)


1 / nRwith cp f =

2v n 1 ± 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

2

V7n

RiFrom the proposition proved in Section 5.1 we conclude that

cp f are Killing spinors.

Now we want to prove a local version of the preceding proposition. To thisend, 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) = 0the equations

DeaVx0 + 12(V e«X) D(V)) + nX . Ve«(D(b)) = 0.

VxVea + -(Vxe«) D( ) + -e« Vx(D( )) = 0.nThese imply

R(X, ea) + -e,,, Vx(D(o)) - nX - Vea(D(o)) = 0.

Multiplying by e« and adding up yieldsn

ea R(X, e.) V) = Vx(D(10)) - nX D2( ) - Vx(D(U=1

Inserting D2( ) = 1 Rn 0 and E ea R(X, ea) _ -2Ric(X) fib, we4n-1 a=1immediately 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 firstintegrals:


Proposition. Let 0 be a twistor spinor defined on a connected Riemannianmanifold. Then the functions

C(b) = Re(z(', D ),

Q() =10121D 12 - C2(0)

n

[Re (DO, ei 0)]2

are constant.Q=1

Proof. Differentiating C(0) yields

X (C(')) = Re(Vx , D'0) + Re(', V (D0))

+0=0.(_xAn analogous, if a little longer, computation using the formula of the previ-ous 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 thefollowing local result which shows that, locally outside its zero set, a giventwistor spinor V can always be transformed conformally into the sum of tworeal Killing spinors. The proof relies on using the conformal weight of thetwistor operator and can be found in [F2] (compare also [BFGK]).

Proposition. Let (M', g) be a Riemannian spin manifold with non-trivialtwistor 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 curvatureR* _ 4(n - 1)

(C2(V))

+ Q(0)).n

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 solu-tions 0 of the twistor equation admitting zeroes are known (see Kiihneland Rademacher). The classification of Riemannian manifolds with thesesolutions 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 theDirac operator

Estimates from above for the eigenvalues of the Dirac operator dependingupon the geometry of the base space can be attained in various ways. By con-structing suitable test spinors and considering the corresponding Rayleighquotients one obtains bounds depending on the injectivity radius and curva-ture data of the Riemannian manifold (compare [Bal]). A further method,based on a comparison with the sphere, was suggested by Vafa and Wittenin 1984 and geometrically elaborated by H. Baum (compare [Ba]). In thissection, we will describe the results obtained along these lines, but refer tothe original papers for the details. The starting point of this method is theobservation 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 S2mby spinors V)1, ... , zp2- whose covariant derivatives V i (1 < i < 2m) areknown (take, e.g. real Killing spinors on S2m). Then the space F(So) of allspinors 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 theformula

Vx(u) = X(u) +2

(-1),2X . (u+ - U-)

holds, where u = u+ + u- is the decomposition of u E C, (S2,; A2m) ac-cording to the splitting A2m =

'®D2m.

Now let f : M2m -* S2m be a smooth mapping from the compact connectedRiemannian spin manifold M2m, into the sphere S2m. If S is the spinorbundle on M2" and f *(So) its pull-back by the mapping f , then there aretwo operators of Dirac-type in the bundle S ® f*(So). The Levi-Civitaconnection of the sphere S2m defines a connection Vf in the induced bundlef*(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 thusf*(So) as well, has a flat connection V° defined by requiring the constantfunctions 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 Dof 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),


and its L2-norm as an operator in the Hilbert space L2(S ® f *(So)) can becontrolled:

I ILf I I<_ 2m-1Vm__I

I df I I ..Here II df I I . = max{I

I : x E M2m}. Since Do = D f -L f, a perturbationargument shows that there are at least as many eigenvalues of Do in theinterval [- I ILf 11, I I Lf I I ]

as the dimension dim ker(D f) of the kernel of theoperator Df prescribes. But this dimension in turn can be computed usingindex theory. We have

Index(D f) = 2--IA(M2,) ± deg(f ),

where deg(f) is the degree of the mapping f : M2m, S2m. This immedi-ately implies the inequality

dim ker(D f) > I2m-1A(M2m) + deg(f) I + I2m-lA(M2m) - deg(f)> 21

deg(fl)I.

Assuming now that the mapping degree deg(f) is greater than

2m-1(mo + ... + Mk-1) + 1,

where 0 < '\o < '\i < ... denote the eigenvalues of D2 over M2m andmo, ml,... the dimension of the corresponding eigensubspaces, we see thatthe operator Do= 2m D has at least 2m (mo + ... + mk_1) +2 eigenvaluesin 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-theigenvalue still in this interval. Summarizing this argument leads to thefollowing result.

Proposition (compare [Bau]). Let f : M2m --> S2m be a smooth mappingfrom the compact connected Riemannian spin manifold M2m, into the sphere.Denote by mo, ml,. . . the dimension of the spaces of eigenspinors of D2 forthe 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 toderive geometrical bounds on the eigenvalues of Dirac operators from thisproposition. For example, one obtains the


Corollary. Let (M2m, g) be a compact connected Riemannian spin manifoldof even dimension with positive sectional curvature K. Denote by Kmax themaximum of the sectional curvature and by A the first eigenvalue of the Diracoperator. Then

AI < 2m-1Kmax

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. Forsurfaces 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 if the genus of M2 is g = 0,C(M2) = 3 if the genus of M2 is g = 2 or 3,

2 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 firsteigenvalue of the Dirac operator to the first eigenvalue of a Schrodingeroperator in the case of a surface M2 C R3. The detailed discussion can befound in the cited paper.

Proposition (compare [Agr/Fri]). Let M2 C 183 be a closed oriented sur-face with mean curvature H, and denote by D the Dirac operator on M2corresponding 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 firsteigenvalue of the Schrodinger operator A + H2 on the surface M2.


I. Agricola, Th. Friedrich. Upper bounds for the first eigenvalue of theDirac operator on surfaces, Journ. Geom. Phys. 30 (1999), 1-22.H. Baum, Th. Friedrich, R. Grunewald, I. Kath. Twistors andKilling spinors on Riemannian manifolds, Teubner 1991.H. Baum. An upper bound for the first eigenvalue of the Dirac operatoron 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.


Chr. Bar. Real Killing spinors and holonomy, Comm. Math. Phys. 154(1993), 509-521.

Th. Friedrich. Der erste Eigenwert des Dirac-Operators einer kompak-ten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrummung, Math.Nachr. 97 (1980), 117-146.Th. Friedrich. On the conformal relation between twistors and Killingspinors, Supplemento de Rendiconti des Circole Mathematico de Palermo,Serie II, No. 22 (1989), 59-75.Th. Friedrich, I. Kath, A. Moroianu, U. Semmelmann. On nearlyparallel G2 structures, Journ. Geom. Phys. 23 (1997), 259-286.Th. Friedrich, E. C. Kim Some remarks on the Hijazi inequality andgeneralizations of the Killing equation for spinors, to appear in Journ. Geom.Phys.0. Hijazi. A conformal lower bound for the smallest eigenvalue of the Diracoperator and Killing spinors, Comm. Math. Phys. 104 (1986), 151-162.K.-D. Kirchberg. An estimation for the first eigenvalue of the Diracoperator 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 eigen-value of the Dirac operator, Journ. Geom. Phys. 7 (1990), 449-468.A. Lichnerowicz. Spin manifolds, Killing spinors and the universality ofthe Hijazi inequality, Lett. Math. Phys. 13 (1987), 331-344.

Exercise 1. Let (Mn, g) be a compact connected Riemannian spin manifoldand 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 Snand for Euclidean space Rn. Determine all solutions of the twistor equationfor hyperbolic space H n.

Exercise 3. 118?3 has two spin structures. Each of the numbers ±2 2 isan eigenvalue of the Dirac operator associated with exactly one of these spinstructures.

Exercise 4. Let (Mn, g) be a Riemannian spin manifold and Nn-1 C Mnan umbilic submanifold. Prove that the restriction of a Killing spinor onMn to the submanifold Nn-1 is a solution of the twistor equation on N'

Appendix A

Seiberg-WittenInvariants

A.1. On the topology of 4-dimensional manifolds

The topology of manifolds looks back at a long history which began in thelast century (Riemann). In the works of many mathematicians (Poincare,Brouwer, Hopf, Morse etc.) during a first period lasting until the middle ofthe thirties of this century homological properties of manifolds were studied,the calculus of variations was developed and, in particular, complete proofswere given for the classification of compact 2-dimensional manifolds. Thecharacteristic features in the topology of manifolds between 1935 and 1960were the theory of characteristic classes (Whitney, Pontryagin), the calcu-lation of the bordism ring (Thom) and the discovery of exotic differentialstructures (Milnor). In particular, it became obvious that in dimensionsn > 4 the category Top(n) of n-dimensional topological manifolds does notcoincide with the category Diff (n) of smooth manifolds, i.e. the naturalmapping

Diff (n) --j Top(n)

forgetting the differential structure is neither injective nor surjective. Inconnection with the solution of the Poincare conjecture in dimensions n > 5and the proof of the h-cobordism theorem (Smale), the so-called surgerytechniques were developed in the sixties (Wall, Browder, Novikov). Theyled to a far-reaching classification theory for certain classes of smooth com-pact manifolds in dimensions n > 5.

129

130 A. Seiberg-Witten Invariants

The situation in low dimensions, n = 3, 4, is rather special. On the onehand, the possible variety of forms is considerably larger already in dimen-sion n = 3 than for surfaces, and the classification question is much moredifficult. On the other hand, even in a 4-dimensional manifold there isnot enough room to apply the surgery techniques which were so successfulin higher dimensions. Dimension n = 3 turned out to be the last one inwhich Top and Diff coincide: every compact 3-dimensional manifold is tri-angulizable, every two triangulizations are combinatorially equivalent and,moreover, every such manifold admits exactly one differential structure. ThePoincare conjecture in this dimension is still unsettled. Apart from the fun-damental group 7rl (M4), the integral intersection form H2 (M4; Z) is themost important invariant of an orientable compact 4-dimensional manifold.In 1982, M. Freedman proved that for simply connected compact topological4-manifolds this intersection form almost determines the manifold itself inTop(4). In particular, each unimodular quadratic form over the ring Z ofintegers can be realized as the intersection form of a compact simply con-nected topological manifold M4. On the other hand, it was already knownfor a long time (Rokhlin 1952) that if a unimodular form of even type canbe realized by a closed smooth 4-manifold, then its signature is divisible by16. Take, e.g., the positive definite unimodular Z-form E8 of dimension 8,

2 -1 0 0 0 0 0 0-1 2 -1 0 0 0 0 00 -1 2 -1 0 0 0 00 0 -1 2 -1 0 0 0E8 =0 0 0 -1 2 -1 0 -10 0 0 0 -1 2 -1 00 0 0 0 0 -1 2 00 0 0 0 -1 0 0 2

E8 is of even type and has signature 8, o- (E8) = dim E8 = 8. Hence thereexists a simply connected topological manifold Mo with intersection formE8, 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 indimension 4 is already a completely different topic than the continuous.

Likewise, at the beginning of the eighties, S.K. Donaldson introduced amethod for the study of Diff(4) based on associating with every smooth4-dimensional manifold the moduli space of solutions of the self-dual Yang-Mills equation in a non-abelian gauge field theory as an invariant, and onderiving new invariants from it. In this way, he was able to exclude furtherunimodular quadratic forms over the ring Z as intersection forms of smooth


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 partic-ular, E8 ® E8 cannot occur as the intersection form of a smooth manifoldeven though the Rokhlin condition o-116 E Z is satisfied. This obstructioneventually led to the proof that there exist exotic differential structures inR4. On the other hand, using known algebraic surfaces, many unimodularZ-forms can be realized as intersection forms. Computations for connectedsums of K3-surfaces with (S2 X S2) resulted in the conjecture that the form

®m0 0 J(2k(-E8) )

does occur as the intersection form of a smooth 4-manifold if and onlyif m > 3k (the so-called 11/8 conjecture; this inequality is equivalent tob2(M4)/1o(M4)1 >

a11). Within the framework of Donaldson theory, this

inequality could only be proved for small k. Another application of the in-variants constructed by means of non-abelian gauge field theory concernssplitting questions. A compact simply connected complex surface S withb2 (S) > 3 cannot be smoothly represented as a connected sum S = Xi#X2with b2 (Xi) > 0. This led to the construction of different differential struc-tures on compact simply connected 4-manifolds.

In autumn 1994, E. Witten suggested that all these results, and othersreaching even further, could be obtained by considering the moduli space ofa 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 gaugefield and, on the other hand, algebraically related to the curvature formof the abelian connection (Seiberg-Witten equation). This system of equa-tions is the 4-dimensional analogue to the 2-dimensional Ginzburg-Landaumodel (1950) of superconductivity. Witten's claim was elaborated by manymathematicians in the following months and turned out to be right. In con-trast to non-abelian gauge field theories with their non-linear equations, onecould now return to an abelian theory, and the analytical theory of smooth4-dimensional topology gets drastically simplified !

The first observation forming the base of Seiberg-Witten theory is thateach orientable compact 4-dimensional manifold M4 has a spinC structure(though, possibly, no spin structure!). Hence spinors may be defined onit. We briefly sketch a proof: The universal coefficient theorem implies theformula

H3(M4; Z) = {H3 (M4; Z)/Tor(H3(M4; Z))} ® Tor(H2(M4; 7L)),


and, from Poincare duality, H2 (M4; Z) = H2(M4; Z), we conclude the rela-tions

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}.

The sequence2

{y2 E T : 2ry2 = 0} --+ T -4 T -+ T/2T is an exact sequenceof 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), thenthere exists an element /3 E T C H2(M4; Z) with r(6) = y. 0 is torsion andhence 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-reductionr : H2 (M4; Z) --f H2 (M4; Z2) . We are going to use this to prove thatM4 has a spinC structure. The necessary and sufficient condition to beconsidered is that the second Stiefel-Whitney class w2 belongs to the imageIm(r). We thus have to check whether w2 U y = 0 for all y E r(T). Bythe Wu formulas for an orientable 4-dimensional manifold, w2 is the onlycohomology class w2 E H2(M4; Z2) satisfying the condition x2 = W2 U xfor all x E H2(M4;Z2). Now for y E r(T) we have y2 = 0, since y is theZ2-reduction of a torsion element from H2(M4; Z). This implies

W2Uy=y2=0


for all y E r(T), i.e. the second Stiefel-Whitney class w2(M4) is the Z2-reduction of an integral cohomology class. Altogether we obtain the

Proposition (Wu 1950; Hirzebruch and Hopf 1958). Every compact orien-table 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 intothe 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 = AlA4 with 04 C2. The endomorphisms ej ej : 04 04 induced byClifford multiplication have the following matrices:

_ (i 0 0 1 0 iele2

0 -i e1e3 = (1 0 )' ele4 = (-i 0)_ 0 i

=0 1 i 0

e2e3i 0 e2e4 -1 0) ' e3e4 = (0 i

In particular, as endomorphisms in 04

eie2 - e3e4 = 0, ele3 + e2e4 = 0, ele4 - e2e3 = 0.

For a spinor c E 04 we define a 2-form w" by the formula

W,11

(X,y) =

(X . YCD)

+ (X,Y)14)12,

where X, Y E 1184. wD is a 2-form with imaginary values. This results fromthe 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 proofof the next

Proposition.1) Let E O4 and w2 E A2 . Then, w2 (D = 0.

2) (&D ' , (D) _ -21-D 14 and 1w' 2 = 21'14.


Proof. Setting E C2 = 04 yields2

êl, e2) = i{I.,pl12 - I'1)2I2} = 0'(e3, 64),

êi, e3) = --b2C + '1)1'52 = -WID

(e2, e4),

w'D(ei, e4) = -iDAi 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 associatedwith the spinC structure. For -D E F(S+), consider the equations (Seiberg-Witten equation):

DA (D =0, Q+y4To understand the Seiberg-Witten equation we compute, for an arbitrarysection E F(S+) and any connection A E C(P), the integral

J2

+IDA-1)I2.

M4

Since QA and w' are forms with purely imaginary values, calculating theirlength amounts to

2 2

[QA+(ei, ej) + 4w"' (ei, ej)Ii<j

= Icrj2+iw14T -21: QA(ei,ej)(eiej-D,4))

i<j

IQAI2+

1IW D12 -QA I2 + 8 4 - 2 (Q++

On the other hand, by the Schrodinger-Lichnerowicz formula we have

f ID9.1)12 = f I(DI2 + (Q1), P)]M4 M4

This implies

f f [Ic2 + +4

I2+8I,DI4].

M4 M4


Hence the functional

E(4), A) = fM4 IIQ+12 +

41 I(DI4]

is non-negative and its zeroes are precisely the solutions of the Seiberg-Witten equation.

Proposition. Let (1), A) be a solution of DA = 0, SZA = --41w4' over acompact Riemannian manifold (M4, g) with scalar curvature R. Then, ateach point,

14)(x)12 < -Rmin, where 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, VA.D) < 2((VA)*VA(D, (D)

2 f (- 4 (1), -1)) - 2 (QA -1), 4)) }=

-RITZ - (D)

_RIcl;I2 + (4(D) = -RI.1) I2 _ 1 I.,D I4.2 2 2

If now I(D max > 0, then 0 < -R - 1/2I m and -Rmin

Let P - M be a U(1)-principal bundle and A : TP -. E51 = iR1 a connec-tion. A gauge transformation on P is given by a function f : Mn --> S1 actingvia 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 de-scribed by

f*(A) =A+7r*f*(O)and the curvature forms of A and f *(A), respectively, are

QA=dA, f2f*(A)=f2A.

We will describe the action of the gauge group !; (P) (L) = Map (M4, SI)in greater detail. Consider f : M4 --> S1. Then f acts on C(P) by

f*(A)=A+7r*f .

For connections in the fibre product R x P of the frame bundle with theU(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


both connections to the SpinC(4)-structure, we obtain, for the covariantderivatives in S,

Vt *(A)4) - Ot = df(1) .f

This implies that for the Dirac operators DA, D f*(A) : F(S) -* F(S) thefollowing 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(Ì'/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, sois 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)-bundleof the spinCrstructure and L the line bundle, then define

ML = { (4),A) E F(S+) x C (P) : DAI) = 0, ci = 1

Proposition. J)tL is compact.JJJ

Proof. Let

F(L) = F = {w2 E A2(M4) : dw2 = 0, [w2]DR = cl(L)},

and let wharm be the only harmonic form in F(L). Since the curvature form!QA of A E C(P) is gauge invariant, we obtain a mapping

P:fitL-NF(L), P[A,(D]=SZA.

First Step. P(J)tL) C F(L) is a compact subset (relative to the L2-topologyon F(L)).

Proof. Suppose that DA) = 0 and Q++ Write the curvature formQA as

QA = (1lA)+ + (IA) = Warm + dij1.


Denote by x1 the harmonic 1-forms and im(d°) = im(d : A° --+ A'). Thenwe 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 obtainM4

2IIVA4)IIL2 = f (_2_1/24)M4

< f (-R/2) I4)I2 < f (-R/2. (-Rmin)) Cl (R).

{R<0} R<O

Hence there is an L2-bound IIVA(DIIL2 < Cl. Moreover,

SS2A = *d* (cA+*clA) = *d*cA,b,QA = *d * (QA - *QA = *d * QA,

i.e. 811A = 2SSZA = Now we compute 5w ':

28(wd)) = SfA(t = 2 ((', VA(D)) - tThis implies

I ISQAI12 < C2 I

I

p1IL2IIVA.D IIL2

and, by the C°-bound forI I ,

IIscAIIL2 <- C22 2arm + dr71, this implies Iladii1Ili2 < C2. Now 771 was chosenSince Q2 = wh

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, wehave

II?7 HL2 < C3II o 111L2

Thus,

II"? IIL2 <C3i IIo7711IL2 <C2.

So far we have shown that the set of 1-forms 771 in question is bounded inthe 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 ofP : 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.


Second Step. Consider Pi : 'mt -3 C(P)/g(P), A] _ [A]. ThenPI(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

fitLP,

C(P)/9(P)

F(L)

commutes and P(9RL) C F(L) is compact, the same also holds for P1(! L) CC(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. consistsof the solutions of

DAI)= 0, maxj<D(x)j< -Rmin

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 Seiberg-Witten 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, fromw'D(X, Y) = (XY(j), (I)) + (X, Y) I<I,I2 we obtain

(dtwjt` o)(X, Y)

_ (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)


= (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) (Ì', ) + (X, Y) (W 1)) + (X, Y) (1),W)

_ (XYW, ) - (XY, ) + (X,Y){(, ) - def w `y(X,Y).

Hence Q++ _ -4w( is linearized by (dr7')+ _ Analogously, DAB0 linearizes to 771(D+ DAY = 0.

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 1 d -dt 1 ) = dt (T,(1)) = h) _ -h-t

where h := dftit=o and ft *A - A = t . Thus,

dt(ft *A - A) t=0 = dh.

Hence P(0,11,A) : 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(,D, -1))}

= h{(XY41), -1)) + (-cD, XY-15) + 0.

Next we want to compute the index of the elliptic complex (over the realnumbers)

(*) r(A°)P A4

r(s+) ®r(A') P r(S-) ®r(A+),


Since operators of lower order than that determining the principal symbol donot 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 indexis additive:

IndexR = IndexR(r(A°) r(A') p r(A+))

+ IndexR (r (s+) r(S-))

= IndexR(r(A°) -L r(A') pr+d r(A+)) - 2 Indexc (D+).

By the index formula we have

Index (DA+) = -c2 -1(7.

8 8

This implies

IndexR = IndexR(r(A°) d- r(A') -> r(A+)) - 4c2 + 4a.

The index of the complex r(A°) r(Al) ---+ r(A+) equals2X+

2v. Thus,

Index *1 2 1 1 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 if0. 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 modulispace.

Proposition. One has v - dimR L = 4c2 - 4(2X + 3Q).

Now we want to prove the following generic vanishing theorem for the secondcohomology 7-12 of the complex (*).


Proposition. Let M4 be a compact oriented manifold and c E H2(M4; Z) aSpinc(4) structure. For a generic set of metrics, g E Met (M4), the secondcohomology 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 perturbationsof the metric in a neighbourhood U, g E U C Met (M4), so that we mayidentify the spinor bundles corresponding to different metrics. Then regardthe Seiberg-Witten equation DAB = 0, (c A)+(g) _ -owl) as depending alsoupon the metric g E U. If gt = g + t h is a variation of the metric, then thevariation 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, E, h) E,D + DA (D +dt

(D+t)jt=o

2 d h1P(-P,A,9) h) )(1 A)+(9+twc,W +(dE)+(9) -2 dt

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 assume0(h) = 0. Hence the variation of the Dirac operator involves, among oth-ers, the Clifford product d(tr(h)) and, on the other hand, the variation ofthe decomposition A2 = A+ ® A? only depends on the traceless part of thesymmetric tensor h, since the Hodge operator * of a 4-dimensional manifoldacting on A2 is conformally invariant. If now (Q, rl) E r(S-) er(A2) belongsto 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 ellipticdifferential equation D9A 45 = 0 ((D $ 0), does not vanish on a dense set inM4.

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 Chernclass 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;


2) *w2 = -w2;

3) [W2]DR = ci(L).

We first discuss under which conditions there exists a "nice" metric. TheA-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 thereis an isomorphism

H2(g) HDR(M4; IIS).

The Hodge operator * : H2 (g) -* H2 (g) is an involution. Hence each metricg 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 eigensubspacesfor +1 of *9 . Then, dim E+ (g) = b2 .

First case. If b2 = 0 and cl (L) 0, then there exists a g-harmonic form w2in cl (L) such that Lw2 = 0. As b2 = 0, we have *w2 = -w2. Hence, in thiscase, there is no nice metric for cl (L) at all.

Second case. b2 = 1. In this case, HDR is a pseudo-Euclidean space withindex (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 transversalityreasons, it is obvious that an L-nice metric g can be chosen. However, twosuch metrics cannot necessarily be joined, since E- (gt) has to stay in therange 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 notconnected. (If c2 (L) > 0, then every metric is L-nice.)

Third case. b2 > 2. It is clear that any two metrics in E- (g) can bedeformed within x 2 + x2 + +

x22+

- yi - . yb- < 0 in such a way thatci (L) is not met. This implies

2 2

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 aSpinC(4) structure with line bundle L. Let g be an L-nice metric. Thenthere are no reducible solutions of the Seiberg-Witten equation.


Proof. DA(P = 0, i i = -4w and 1) - 0 together immediately implyQ+y = 0. Hence QA is an anti-self-dual 2-form, *QA = -QA. Thus QA is alsoharmonic, LQA = 0, since dSlA = 0 and [Z,,QAI DR = cl(L). However, bythe assumption on the metric, there is no such form QA-

For the moduli space YtL = DDtL(g) the following picture results.

First case. b2 (M4) > 2. Then, for a generic metric g on M4, we havethat g is an L-nice metric, 7-12 ker((PI

(15A))*) = 0, and the space of these

metrics is pathwise connected. Consequently, for such a metric, ML (g) isempty or a compact smooth manifold of dimension

dimfitL(g) =14c

2 - 14(2X+3ov)

and the bordism class in the unoriented bordism ring N* is independent ofthe 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-niceand, generically, 712 ker((P(4, A))*) = 0. As above, the bordism class9)tL (g) is uniquely determined.

Case 2.2. c2 (L) < 0, b2 (M4) = 1. A generic metric g is L-nice andwe have H2 ker((P(P A))*) = 0. Hence 9JtL(g) is empty or a compactsmooth manifold, but the space of the metrics in question is not necessarilyconnected. A definition of

S - W(M4, c) = [9XL(g)] E N*is only possible for a chosen connected component in the corresponding spaceof metrics (or following a different line of argument).

Third case. b2 (M4) = 0. In this case, we have 712 Pz ker((Pl Al)*) = 0for 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 incl (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 sin-gular set Pic(M4) = Tb1(1''14). Its "bordism class" is again unique.


Remark. To define the Seiberg-Witten invariant within the unoriented bor-dism ring by

S - W(M4, c) = [k(g)] E 9t*is the simplest possibility. More refined invariants are obtained by the ob-servations to follow.

Observation 1. On 9X L(9) there exists a universal S'-principal bundle. Ifm° E M4 is a fixed point, then we consider the subgroup go C 9(P) of thegauge group given by

go ={f :M4-}S': f(mo)=1}.Then i)tz,(g) _ {(', A) : DA (D = 0, Q++ _ -4w' }/Go is a principal bundlewith structure group c/cro = Sl over 9 RL(g). Hence 9RL(g) has a dis-tinguished cohomology class e E H2 (9 L (g), Z). Thus the Seiberg-Witteninvariant can be understood as a pair ([TZL(g)], e) consisting of a bordismclass 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 acanonical 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+). Thusthe 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) canbe 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-Witteninvariant 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 M4has 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 spaceof 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 havethe

Proposition. Let M4 admit a metric of positive scalar curvature and sup-pose 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, consideras S - W (M4, C) the Seiberg- Witten invariant computed with respect to thecomponent containing the metric of positive scalar curvature.

Let g be a fixed Riemannian metric and ((D, A) a solution of

DAB = 0, SZA =

1.1We already know that at each point )(x)12 < -Rmin. By the Schrodinger-Lichnerowicz formula,

oI.CpI2 = -1 -2and hence

This implies

I4)I4 < f(-ii .1)2< Rminvol(M4).

M4 M4

f p-+12 6 f W-DI2 = 8f Iq)I4 <- 1-R 2 n vol (M4).

M4 M4 M4

As QA is a curvature in L,

ci(L) =f(cI2 - InA2)

M4

If 9RL (g) 0, this implies ci (L) - (2X + 3o) > 0, and hence

2X + 3Q < ci(L) = f(c2 - InAI2),

M4

i.e.

f ISZAI2 < f IS2AI2 -M4 M4

Thus f I cl I2 as well as f I1 I2 is bounded, if ML is not empty. This

M4 M4implies the


Proposition. Let (M4, g) be a compact oriented manifold with fixed Rie-mannian metric g. Then there exist at most finitely many Spinc(4) struc-tures 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 becomesa numerical invariant for dimTZL(g) = 0. If M4 is a compact oriented4-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 determinedfor a generic metric; for b2 = 1 a "component" in the space of L-nicemetrics has to be fixed additionally. If, moreover, an orientation 0 inHl (M4, IR) ® H+ (M4; R) is given, then no (g) E Z is defined as an in-tegral invariant to be the alternating sum of the points in 9RL (g) whichnow carry a sign. However, this numerical Seiberg-Witten invariant doesnot exist for every manifold M4. Namely, the existence of a cohomologyelement c E H2(M4; IR) with c2 = 2X + 3v and c =_ w2(M4) mod 2 isa necessary condition. The number 2X + 3cr _ (2e + pl)[M4] has to berepresentable by an element c E H2(M4; ]I8). Here e E H4(M4; I[8) de-notes the Euler class and pl the first Pontryagin class. In other words,the resulting necessary condition on M4 requires that the cohomology class2e + pi E H4(M4; I[8) has to belong to the image of the quadratic inter-section form H2(M4; I[8) E c -+ C2 E H4(M4; II8). Suppose that M4 hasan almost-complex structure J : TM4 --+ TM4 inducing the orientation.Then TM4 is a complex 2-dimensional vector bundle over M4 having Chernclasses cl E H2 (M4; Z) and C2 E H2 (M4; Z). Since the complex structure Jis compatible with the orientation, the real characteristic classes w2, e andpi 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 oriented4-dimensional manifold. Then there exists an almost-complex structure onM4 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

Let (M4, g, J) be a Kahler manifold (or, for the moment, just an Hermitianmanifold) and denote by Pj C P(M4, g) the corresponding U(2)-principalbundle. 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 signatureand 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 Kahlermanifold with respect to the canonical Spin' (4) structure.

Remark. This computation of the virtual dimension can also be performedin a more general situation. Consider an oriented manifold M4 and analmost-complex structure J : TM4 --+ TM4, j2 = -Id. Then, det (J) = 1.The set of almost-complex structures has two components. One componentconsists 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 1V - 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 useof 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 anendomorphism 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 givenby the matrix


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, onecomputes wD = -ij 2l for spinors <P E S+(-2i). The bundles S+(2i) andS+(-2i) are isomorphic to Ao'2 and A°'0, respectively (compare Section 3.4).

Denote by cl° the spinor in S+(-2i) _- A°'0 corresponding to the function1. Then, in the chosen coordinates,

4(D° = I I and w"0 = -in.

The bundle L = A2T of the canonical spinC structure has a distinguishedconnection A0, the Levi-Civita connection. Considering the correspondingDirac operator

DA0 : r(S+) - P(S )

and identifying as above

S+ A°'0 ®A°'2 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 inS+, and

DAo 1 = 0,

w11, _ -I,1)I2ic _ -(-R)ic = RiQ.

On the other hand, the curvature 11A0 in the line bundle L = A2T is givenby 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 thematrices

0 -1 0 0 R11 R12 R13 R14_J 1 0 0 0

Ric =R12 R22 R23 R24

0 0 0 -1 R13 R13 R33 R340 0 1 0 R14 R24 R34 R44


Since J and Ric commute, we obtain the following special shape for theskew-symmetric endomorphism:

0 -A D -CJ o Ric = -D -C 0 -B , A = R11 = R22, B = R33 = R44,

C -D B 0

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 projectionp+ of the form p onto the subbundle A+ this implies the formula

pA +BB

(el A e2 + e3 A e4).2

Since A + B = R11 + R33 =2

(R11 + R22 + R33 + R44), we thus obtain thefollowing relation, holding in every 4-dimensional Kahler manifold:

p+= 4As

QAo = ip = i4Q = 4

the pair ('k, Ao) = (V-R-cDo, Ao) is a solution of the Seiberg-Witten equa-tion.

Theorem (LeBrun 1994). Let (M4, J, g) be a compact Kahler manifold ofconstant 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 themetric).

Proof. Let (-D, A) be an arbitrary solution of the equation

DAI) = 0, ul = 4w

Then,

Iq) (M) 12 < -Rmin = -R and IQAI2 =81

8Since p+ = E Q and I S2I 2 = 2, this implies

(*) f IpAI2 < J R2 = f R2I6I2 = I (RIIII)

2

Ip+I2.

8 16 4M4 M4 M4 M4 M4


On the other hand, since p is the curvature form of the Levi-Civita connec-tion in the bundle A2T, it follows that

cl(A2T)47x2

f (Ip+12 - Ip 12),

M4

and hence

f Ip+I2 = 27r2ci(A2T) +2f IAI2.

M4 M4

The scalar curvature of the Kahler manifold is constant. Therefore, theRicci 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 cohomologyclass. A is a connection in A2T, and hence

f 1P12 f AI2.

M4 M4

Thus,

(**) f Ip+I2 < 27r2ci(L) + f IQAI2 = f IpAI2.M4 M4 M4

Combining (*) and (**) yields S2A = ip and 1.1) 12 = -R. The proof of theestimate I4)I2 _< -Rmin then immediately implies VA p __ 0. Decompose Daccording 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 concludethat cl (A°'2) = cl (A2T) = 0. However, the curvature form of the Levi-Civitaconnection A0 in A2T is

52+90=ip+=i4S2#0,

since R < 0. Hence (D°,2 vanishes and 4) is proportional to the standardspinor, ' = f 4D°. The length If I2 = -R is constant, since I.DI2 is con-stant. The connection A differs from the Levi-Civita connection A0 by animaginary-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 + : M 4 1 .


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 havefound a gauge transformation g : M4 -f T1 with

A=AO - gg, =g.( c1)o),

i.e. (-cD, A) is equivalent to ( / 0 , A 0 ) .

Corollary. Let M4 be a compact oriented manifold with b2 > 2. If M4 hasa Kdhler structure with constant negative scalar curvature, then M4 admitsno 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 sym-plectic structure w and assume that w A w > 0 defines the fixed orientation.Choose an almost-complex structure J : TM4 , TM4, J2 = -Id, forwhich g(X, Y) = w(X, JY) is a Riemannian metric. From w A w > 0one easily concludes that J is a section in the bundle 'j+(M4, D). Letc = 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-dimen-sional manifold and w a symplectic form with w A w > 0. Moreover, assumethat b2 (M4) > 2, i.e. the Seiberg-Witten invariant is defined. If c = c(w) EH2 (M4; Z) is the Chern class of an almost-complex structure associated withw, then

nc(M4) = 1 mod 2.

Corollary. Let (M4, sD) be an oriented compact 4-dimensional manifold sat-isfying 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 theSpin' (4) structure c E H2 (M4; Z) is not necessarily the canonical one ofM4. As before, denote the line bundle by L. The Kahler form SZ acts asan endomorphism on the spinor bundle and has the eigenvalues ±2i there.Hence the spinor bundle splits,

S+ = S+ (2i) ® S+(-2i).


L is isomorphic to A2S+ for each Spincc(4) structure, and hence we obtainthe isomorphism

S+ (2i) ® S+(-2i) = L.For every connection A E C(L) the Kahler form SZ is parallel, and hencethe decomposition S+ = S+(i) ® S+(-i) is VA-parallel, too. For a spinor4D = (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 asolution 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 frameKahler 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 definitionof w

w'D =

Therefore,

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).

w(D +w = 2i(I-D+I2 - I.1)_I2)n.

Since Q++ = -4w` _ w we obtain

-41l,+9 = 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,since A°'2 n A2 _ {0} = A2'° n A? , we at once arrive at

QO,2=0=QA°.A

A. 7. References 153

The curvature form of the connection A in L is thus a (1, 1)-form. HenceA 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

ci(L) C+(L) = -QA

and hence, from 12 A cl = S2 A c1 , we conclude that

J :def fc2Aci(L) = 1 fi+i2 - I`)-12)12 A c2

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, inadditon, the action of the gauge group leads to the

Theorem (Witten, 1994). Every solution of the Seiberg-Witten equationover a Kahler manifold M4 for the spinc structure c E H2 (M4; Z) corre-sponds 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 Seiberg-Witten invariant S - W (M4, c) E J V,,, since the Kahler metric g is notgeneric.

A.7. References

J. Eichhorn, Th. Friedrich. Seiberg-Witten theory, in Symplectic Sin-gularities and Geometry of Gauge Fields (Warsaw; 1995), Banach CenterPubl., 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 inthe projective plane, Math. Res. Lett. 1 (1994), 797-808.D. Kotschick, J.W. Morgan, C.H. Taubes. Four-manifolds withoutsymplectic 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.


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 Seiberg-Witten 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 topol-ogy of smooth four-manifolds, Mathematical Notes, Princeton UniversityPress 1996.

J.W. Morgan, Z. Szabo, C.H. Taubes. A product formula for theSeiberg-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-Witteninvariants, 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-holomorphiccurves, 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 andConnections

B.1. Principal fibre bundles

Definition. Let E, X and F be three topological spaces. A mapping 7rE -+ X is called a locally trivial fibration with fibre F if for each pointxo 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)" U x F

commutes. Then E is called the total space of the fibration, X its base andF = 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 adiscrete space whose points can be mapped bijectively onto 7r-1(x). Then7r : E -+ X is a locally trivial fibration with fibre F.

Example 3. Let X = M" be a smooth manifold, E = T(MT) and 7rE -> X the projection of the tangent bundle. Then (T (MT ), 7r, Mn; R') isa 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 trivialfibration with fibre H.

155

156 B. Principal Bundles and Connections

Definition. Let E and E* be two fibrations over X with projections itand 7r*, respectively. These fibrations are called equivalent if there is ahomeomorphism f : E -* E* fitting into the commutative diagram

E fE*

Definition. A locally trivial fibration (E, it, X; F) is called trivial if it isequivalent to (X x F, 7r, X; F).

Example 5. Not every fibration is trivial. Let E = T(S2) be the tangentbundle of the sphere X = S2. Then T(S2) -p S2 cannot be the trivialfibration, since its Euler characteristic is two, X (S2) = 2 $ 0, and hence, bythe Hopf theorem, there are no vector fields without zeroes.

Definition. Let p : E -- p X be a fibration. A mapping s : X --* E is calleda section if it o s = Idx. In case s is defined on an open set U C X only, itis called a local section.

Let (E, it, X; F) be a fibration over X and f : Y -4 X a (e.g. continuous orsmooth) 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* f *E Y is called thefibration induced by f.

Proposition. p* : f *E 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 fromthe right on P;

2) 7r : P -> X is continuous and surjective, and ir(pl) = 7r(p2) if andonly 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 principalbundles, i.e. for every x E X there exist U, x E U C X and -1)U7r-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.


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 isalso smooth.

Definition. Let (P, it, X; G) and (P1, 7r1, X; G) be two principal bundlesover the same base X with the same structure group G. These two principalbundles are called isomorphic if there exists a homeomorphism f : P ---+ P1,satisfying the following conditions:

1) The diagram

P f P1X

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 thisprincipal 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, settingP = G, X = G/H and G = H, we see that (G, it, G/H; H) defines anH-principal bundle over G/H.

Remark. Non-isomorphic principal bundles may well be equivalent as fi-brations.

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 S1act 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 asSl-principal bundles. The principal bundle S1 is called the Hopf fibration.


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

xEMnGL(n;1i8) act on L(Mn) by

All A1n n n

(vl,... ,vn)' _ vjAil,... iAin

Anl Ann 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) = a.7}.

Then O(Mn; g) is an O(n; R)-principal bundle over Mn.

Example 5. Let (M2n, w) be a symplectic manifold andSp(M2n; w)

(v1 ... vn,, 191 ... 2Un) E L(M2n) : (vi, vj) = w(wi, wj) = 0 1.

w(vi, w.9) = ail J

Then Sp(M2n, w) is a Sp(2n;118)-principal bundle over 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 -> Ga 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 mappingf : Q -+ P such that

1) the diagramQ f P

-1 XX

commutes, and

2) f (9' ' g1) = f (4') ' A(g1)

Examples 4, 5, 6 describe different reductions of the frame bundle to thegroups 0(n;118), Sp(2n; R) and GL+(n; R), respectively.


Definition. Two A-reductions (µ, f ), (µ, f) of the principal bundle p arecalled equivalent if there exists an isomorphism D : Q --f Q of the G1-principal bundles such that the diagram

Q Q

fitP

commutes.

Proposition. Let A : O(n; R) -} GL(n; R) be the canonical embedding ofgroups and Mn a smooth manifold. The set of all A-reductions of the framebundle L(M') is in bijective correspondence with the set of all Riemannianmetrics on Mn.

Proposition. Let A : GL+(n; IIi) --> GL(n; l[8) be the canonical embedding ofgroups and Mn a smooth manifold. The set of all A-reductions of the framebundle L(Mn) is in bijective correspondence with the set of all orientationson 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 XTxMn -> R. The resulting 2-form w will be non-degenerate. However, ingeneral, dw = 0 will not hold. Thus not every reduction of the frame bun-dle L(M2n) to the subgroup Sp(2n; I[8) can be identified with a symplecticstructure on Men.

Consider now a G-principal bundle (P, 7r, X; G) and a topological space Fon which G acts from the left, G x F -> F. Let G act from the right onPxFby Set

E=PxF/G:=PXGFand 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 whichG acts from the left, then (E, p, X; F) with E = P xG F is a locally triv-ial fibration. The fibration (E, 7r, X; F) is called the bundle with fibre Fassociated to the principal bundle.

Proposition. Let (P, 7r, X; G) be a G-principal bundle and F a G-spacedefining the associated bundle E = P xG F. Then there exists a bijectionbetween the sections s in the bundle (E, 7r, X; F) and the maps s* : P --+ Fwith s*(p g) = g-1s*(p)


Now let G be a group and H C G a closed subgroup. Consider the G-spaceG/H = F and the bundle E = P xG (G/H) associated to the principalbundle (P, 7r, X; G).

Proposition. The bundle (E, 7r, X; G/H) has a section if and only if theG-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 thetrivial G-principal bundle P = X x G. A section in P XG (G/H) ^ X xG/H is then simply a maps : X -3 G/H, and the H-principal bundlecorresponding to this section is

Q={(x,g) EXwith the projection (x, g) -* x. Now fix

/ lG = S0(3), H = S0(2) =

C 0) S0(3)

and X = S2 = SO(3)/SO(2), where the identification SO(3)/SO(2) ^ S2is defined by A -- A(e3); e3 is the third basis vector of the Euclidean space1[83. Since e - H <--> e3 under this identification, we conclude that, for everymapping 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 theresulting 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, Qitself is not a trivial Sl-principal bundle over S2. This example shows thatreductions of trivial bundles may well be non-trivial principal bundles.

As the last topic in this section we want to discuss vector bundles. To thisend, consider again a G-principal bundle (P, ir, X; G) and, in addition, avector space F = V (complex or real). Let G act on V via a representationp : 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] EEand 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 resultsin the following structure: Each fibre of P xP V = E is a vector space over


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 eachfibre.

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 Xthere exist an open set U, xo E U C X, and a homeomorphism bup-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 framebundle. 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 definedand, 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 inthe 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 pC the associated bundle, where p : S1 -- GL(C) is given by p(z)w = zw. Hconsists of the equivalence classes [(wl, w2), w] of pairs of complex numbersunder the identification

{(wl, w2), w} - {(wlz, w2z), z-1w}.

The mapping f : H --> H = {(1, e) E C?1 X C2: E l} defined by

f (wl, w2; w) _ ([WI : (wlwW2W))

ECP1 EC2

obviously defines an isomorphism between H and A. Hence H ---). C?I, asa 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 Hopfbundle (tautological bundle over Cl?').


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 ofG-principal bundles (P, 7r, X; G) with structure group G existover X?

Theorem (First homotopy classification theorem). Let = (P, 7r, Y; G) bea G-principal bundle over the topological space Y, X a paracompact spaceand fl, f2 : X -+ Y two homotopic mappings, fl - f2. Then the inducedprincipal bundles fl , f2 over X are isomorphic.

Definition. If X is a paracompact space, then denote by HFBG(X) theset of all isomorphism classes of principal bundles over X with structuregroup G.

Definition. Let G be a topological group. A universal G-bundle is a G-principal bundle c = (EG, 7, BG; G) such that for every CW-complex Xthe assignment

[X; BG] E [f] - f *G E HFBG (X)is a bijection. In other words, the following two conditions have to besatisfied:

1) For every G-principal bundle l; over X there exists a mapping fX -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 de-termined. Namely, we have the

Proposition. If ec = (EG, 7r, BG; G) is a universal bundle and B a topo-logical space which is homotopy equivalent to BG, then over B there is alsoa G-principal bundle which is universal.

Proposition. Let 6c = (EG, 7r, BG; G) andG = (EG, 7r, PG, G) be twouniversal G-principal bundles with the CW-complexes BG, BG. Then thereexist 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 spaceof a group, if one exists, is uniquely determined. Now we can formulate thesecond homotopy classification theorem for principal bundles:


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 propertythat 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-principalbundle (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 aCW-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) correspondingto 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). Foran S'-principal bundle 6, the element in H2 (X; Z) corresponding to thisfibration is called the first Chern class of and will be denoted by cl (l;) EH2(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 fundamentalvector 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 distri-bution, 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


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), thenTp (P) = {X E TpP : Z(X) = 0} is a connection.

Now we will describe the local characterization of a connection: Let s : U CM -> P be a section. ZS = Z o ds = s* (Z) : TU g is the local connectionform. For two sections si : Ui -* P, s j : Uj --r P and Ui n Uj define

gij:UinU1-Gbysi (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 ® TMthe 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 -Ejiand Eij denotes the matrix with 1 in the i-th row and j-th column (and zerootherwise). Let s : U -* O(M, g), s = (Si,... , sn) be a local section andset ZS = E g(Vsi, sj)Xij. Then {ZS, s} defines a connection Z in O(M, g)

i<jand the 1-forms wij = g(Vei, ej) are the connection forms of the Levi-Civitaconnection.

Example. Let Mn be a differentiable manifold. The set of connections onthe frame bundle L(Mn) is in bijective correspondence with the set of affineconnections 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 inwith dL,,, (T9 G) = Tag G.

Let a principal bundle (P, 7r, M; G) and a representation p : G -+ GL(V) begiven.


Definition. A q-form w E Al (P, V) with values in V is called tensorial oftype 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 tensorial1-form of type Ad on P with values in g. Conversely, if Z is a connection andw : 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 withvalues in V is isomorphic to the vector space Aq(M; E) of q-forms on Mwith 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 nespace with vector space A'(M;g). Here g denotes the bundle g = P XAd 9associated by means of the representation Ad.

Definition. Let (P,,7r, M; G) be a principal bundle with connection. If X EX (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 theother 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 forall 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, and2) 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.


Definition. Let Ty : Py(a) - Py(b) be defined by ry(u) = yo(b). -ry is calledthe 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, andsuppose that M is connected. If the parallel transport r does not depend onthe curve, then there exists a horizontal section in P and, moreover, theprincipal bundle is isomorphic to the trivial bundle (M x G, prl, M; G) withflat 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 inthe 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, Van arbitrary vector space, and let w E Aq(P, V) denote a q-form on P withvalues 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. Thetensorial forms of type p coincide with the forms AP(M, E) on M withvalues in the vector bundle E. Hence the absolute differential can be viewedas 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 ofsmooth sections. A mapping V : r(E) -+ r(T*M ® E) is called a covariantderivative 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, pG -> 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 acovariant derivative.

D is called the covariant derivative in E associated with Z. It is occasionallyalso 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 dtDs('y(t)) = 0.

Definition. Let (P, 7r, M; G) be a principal bundle and Z : TP - g aconnection. Then Z is a 1-form on P of type Ad (i.e. Rg**Z = Ad(g-1)Z).By the preceding theorem,

Q:=DZis a 2-form on P with values in g which is tensorial and of type Ad. SZ iscalled 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] EAi+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].


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 con-nection and Il = DZ its curvature form. Then we have:

1) The structure equation: Q = dZ +2

[Z, Z] .

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 connec-tions with curvatures SZ = DZ, = DZ. Then, 77 = Z - Z is a tensorial1-form of type Ad and

fl=St+Dr7+ 12[77,r1]-

Considering S2,1 as s-forms on M and 77 as a 1-form on M with values ing = P xAd g 0, n E A2 (M, g), r7 E S2' (M, g), we have the same formula, thistime with the operator

D : A' (M; g) A2 (M;

Definition. A connection Z on (P, ir, M; G) is called (locally) flat if thereexists 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 169

Proposition. Z is a locally flat connection ) - 0 the bundleTh(P) C TP of horizontal vectors is involutive.

Proposition. Let 7rl (M) = 0 and let (P, 7r, M; G) be a principal bundlewith a locally flat connection Z. Then (P; Z) is isomorphic to (M x G) withthe canonical connection.

Definition. Let (P, 7r, M; G) be a principal bundle. A gauge transformationis a diffeomorphism f : P -+ P with

1) Tr,o f = 7r, and

2) f(p.g)=f(p).gDenote 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 ofgauge 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 fP - G, and let Z be a connection. Then, for the curvature forms S2z andSjf*z

Qf*z= Ad(µ f1)QZ.

B.S. Connections in U(1)-principal bundles andthe 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 theother hand, 1y(t)12 - 1 implies ' (0) E i118. Hence we obtain an identification61

D '(0) , y(0) E iIi. The Lie algebra E51 can be identified with iR insuch a way that the diagram

exp\ /eS1

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

dz0=-=zdz.z


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) =1

dry = 1t =dt z t-o z dt It=o z z

i.e. e = z Moreover, f S = f dz/z = 27ri. Hence, go := 27rZ19: TS1 -RS1 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 -> Pis a gauge transformation, then set f (p) = p Izf(p), j if : P -> S'. Sincef(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 mapping9f : 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 thuscoincides with the group of all mappings p : M' --+ Sl:

{A:M')S1}.

Fix a connection Z : TP -> iR on P. If f : P -> P is the gauge transforma-tion 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+ 12[77,77] =t2Z+Dr7

with 77 = 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 closed2-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 thereexists 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) _B2(Mn) =

{w2 : w2 is a 2-form and dw2 = 01,

{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 de-termined element of the de Rham cohomology of Mn not depending on thechoice 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,-Qz

C(P) Z -27r.

E.F(P)

obviously defines a mapping : C(P) -+.F(P). We state some of its prop-erties:

1) If Z and 2 are gauge equivalent connections, then (Z)

2) 0 is surjective.

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) = wl E ZI(M) : fw' E Z for all closed curves y

ly

Since f df = f f = 0, we obviously have ZI (M; Z) D BI(M). Let1 09-Y

HD'R(Mn; Z) = Z1(Mn; Z)/BI (Mn)


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 overthe compact manifold Mn with first Chern class cl(P) E HDR(M;lR), andset

,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

Z =1

2{wldwl - wldwl + w2dw2 - w2dw2}.

Since z - 2 E i]R for every z E C, it follows that Z is a 1-form on S3 withvalues 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) onS3 we have Z(ix) = ix.

Thus Z is a connection in the bundle 7r : S3 -+ S2. We are going to computeits 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 fora 2-form f2 on CP' = S2. Let -y : S2\{north pole} --> C denote stereographicprojection, 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) = w2and 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 thatbundle. This implies

SZ* = dZ* = -12 = dw1 A dw1 + dw2 A dzu2idzAdz 1

-SZ(1 + Iz12)2 ' 27ri

S2

Hence, c1(6) = +1, and we have once again proved that this S'-principalbundle 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.:

1) A : G' -* G is a group homomorphism,2) f : P' -> P is smooth and the diagram

P'f P" fIM

commutes,

3) f (p'9') = f (p')A(9').


Proposition. Let Z' : TP' be a connection in the G'-principal bundle(P',ir,M;G').

1) There exists one and only one connection Z : TP -j g such thatdf : TP' -> TP maps the horizontal spaces with respect to Z' ontothe 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, thenZ' 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 asa subgroup H C G and a subbundle (Q; nr, M; H). We ask the followingquestion:

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 thatthere 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 oZJTQ : TQ -> tj is a connection in Q.

If, in particular, Z : TP -+ g takes only values in the subalgebra [ , then Zreduces 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 subsetand w1, ... , wr 1-forms on U, n = r + s. Moreover, suppose that

a) wl, . . , wr are linearly independent at every point and

b) there exist 1-forms E on U withr

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 surfacepiece FS with

1) xo E Fs,2) y E Fs TyFs = Es(y)-

Replacing Ift by an n-dimensional manifold, this leads to the notion of adistribution:

Definition. Let Mh be a manifold. A differential system or a distributionon Mn is a selection of k-dimensional subspaces E. C TM' in every tangentspace 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 vectorfields tl, , tk on U(x) with Ey = Lin(tl (y), , Fk (y)) forallyEU(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 calledintegrable if the following condition is satisfied: For two vector fields fl, F2on 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 anintegrable k-dimensional distribution. For every point x E M there exist aneighborhood Ux and a submanifold x E Fk C Ux with TyFk = Ey for allyEFk.

Before turning to the global version of the Frobenius theorem we have toexplain or extend a few general notions. To this end, recall the followingdefinitions:

Definition. A smooth manifold without boundary is a pair (M, D), where

1) M is a topological T2-space with countable basis, and2) 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 andthe hihi 1 are smooth.

Definition. Let (M, D) be a smooth manifold without boundary. A subsetA 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}.


One then shows that (M, D(M)) is a manifold in the induced topologyand with the atlas D(A) = {(A fl U, cPjAnu)}. Moreover, the embeddingi : A --> M is smooth and di : TA --> TM is injective.

Definition. Let (M, D(M)) be a smooth manifold. A subset A C M iscalled a weak submanifold if there exist a smooth manifold (N, D(N)) anda 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 onen E N with f (n) = a. The space dfn(TnN) =: TaA is called the tangentspace 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 subman-ifold.

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 thefollowing 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, flNi -> 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 Ekis 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 anintegrable distribution on a manifold M. Then for every point x E M thereexists 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 Ekwith 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) Liesubgroup 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 thata) 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 sub-group with the following property: Each element of H can be connected withthe neutral element e E G by a piecewise smooth curve, and this curve liesin 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. (Inthis section, we suppose that M is connected.) Take, moreover, p E P andx = 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, itobviously follows that Ty : Px P5 is completely described by gy E G. Theset 0(p) {g E G : there exists a loopy at x with ry(p) = p g} is an(algebraic) subgroup of G,

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 connectionZ 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.


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 aprincipal bundle with connected base Mn and Z : TP -> g a connection. Fora fixed point p0 E P, denote by c(po) the holonomy group and by P(po) theset

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. Thenthe Lie algebra of the holonomy group ¢(p0) is generated by the elementsSZ(X, Y) with X, Y E (TP)p and p E P(po) .

B.10. References

S. Kobayashi, K. Nomizu, Foundations of differential geometry, Volume1, Wiley 1963.

R. Sulanke, P. Wintgen, Differentialgeometrie and Faserbiindel, Deu-tscher Verlag der Wissenschaften, 1972.

D. Husemoller, Fibre bundles, McGraw-Hill, 1966.

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Index

absolute differential, 58, 166, 167adjoint operator, 92algebra of complex numbers, 9algebra of quaternions, 8almost-complex manifold, 73, 74almost-complex structure, 60, 80, 146, 147,

151Ambrose-Singer theorem, 178anti-canonical spin structure, 79, 81associated fibration of a principle bundle,

159associated spinor bundle, 75associated vector bundle, 161

Bianchi identity, 168bilinear form, 1, 7

direct sum of, 7index of, 2nondegenerate, 1rank of, 2signature of, 2

canonical basis, 2canonical connection, 83canonical spin structure, 79

of an Hermitian manifold, 77, 78canonical spin(4) structure, 147, 149Casimir operator, 86, 87Cauchy-Riemann equations, 71center of an algebra, 9characteristic class, 108Chern class, 108, 141, 163, 171, 172Clifford algebra, 4, 10, 11Clifford multiplication, 21, 32, 53, 68, 70,

133complete Riemannian manifold, 98complex n-spinors, 14

complex projective space, 40, 42, 48, 161complexification

of a real algebra, 11of a real quadratic form, 11

conjecture, 8 , 131connection, 163, 165, 169

holonomy group of, 177locally flat, 168reduction of, 174

continuous spectrum, 91covariant derivative, 58, 60, 67, 68, 70, 166,

167covering spaces, 40curvature form, 62, 135, 167, 169curvature tensor, 62

de Rham cohomology, 171integral, 172

determinant bundle of spin structure, 52-54, 108, 113

Dirac operator, 68, 69, 71, 93, 96, 101, 107,127, 136, 148

eigenvalues of, 113, 116, 126, 128G-function for, 103index formula for, 110index theorem for, 109spectrum of, 99

Dirac spinors, 14, 69, 113, 115direct sum of bilinear forms, 7distribution, 175

integral, 175

eigenspinor, 102, 103, 112, 114, 126eigenvalues, 91, 126

of the Dirac operator, 113, 116, 126, 128Einstein space, 1188 conjecture, 131

193

194

equivalent fibrations, 166equivalent A-reductions, 158equivalent spin structures, 35equivalent spinC structures, 51essentially self-adjoint operator, 92-94, 96n-function, 105, 112

for the Dirac operator, 103exponential map, 18exterior differential, 73

fibration, 156, 159equivalence of, 156locally trivial, 155

first integral, 124frame bundle, 158Fredholm operator, 107Freudenthal-Yamabe theorem, 177Frobenius theorem, 174

global, 176local, 174, 175

fundamental group, 26

Gaschiitz proposition, 39gauge field theory, 130, 131gauge group, 135gauge transformation, 135, 169Ginzburg-Landau model, 131G-principal bundle, 156Grafmannian manifold, 56, 88

harmonic spinors, 81, 82heat equation, 112Hermitian manifold, 75, 78, 147

canonical spinC structure, 77, 78spinor bundle of, 79

Hermitian metric, 53, 68, 74, 80Hermitian scalar product, 24Hilbert-Schmidt operator, 103, 104Hirzebruch-Hopf proposition, 133, 146Hirzebruch signature theorem, 109holonomy group of a connection, 177homogeneous spin structure, 85, 87homotopy classification theorem, 162homotopy theory, reduction theorem of, 178Hopf bundle, 161Hopf fibration, 157, 161, 172, 173horizontal lift, 165

index, 108

of a form, 2index formula for Dirac operators, 110index theorem for Dirac operators, 109integrable distribution, 175integral de Rham cohomology, 172integral manifold, 176intersection form, 109, 130isomorphic principal bundles, 157isotropic subspace, 3

Index

Kahler manifold, 61, 81, 82, 89, 116, 147,150, 151, 153

Killing number, 116, 118, 119Killing spinor, 116, 118, 119, 121, 124, 125,

128imaginary, 120real, 120

Lagrange theorem, 2A-reduction, 158, 173

equivalent, 158Laplace operator, 71

on spinors, 68Levi-Civita connection, 57, 61, 81, 84, 113,

125, 135, 148, 164Lie algebra

of Spin(n), 17, 18of Spinc(n), 29

linear operator, 91locally flat connection, 168locally trivial fibration, 155

manifold without boundary, 175Maurer-Cartan form, 83, 164moduli space, 140, 147

for Seiberg-Witten theory, 136

nondegenerate bilinear form, 1null subspace, 3

orientation, 159

parallel spine spinor, 67parallel spinor, 67, 89parallel spinor field, 67parallel transport, 166, 167Picard manifold, 172Pin(n), 15point spectrum, 91Pontrjagin class, 108principal bundle, 157

associated fibration of, 159G-, 156isomorphic, 157Si-, 163Z2-, 163

projective spacecomplex, 40, 42, 48, 161real, 55, 56

q-form of type p, tensorial, 165quadratic form, 1quaternionic structure, 29, 30, 110

in A ,,, 32, 54

rank of a bilinear form, 2real projective space, 55, 56real structure, 29, 30

in An, 32, 54

Index

reducible solution of the Seiberg-Wittenequation, 140, 142

reductionof a connection, 174A-, 158, 173

equivalent, 158U(k)-, 48, 60, 61, 81

reduction theorem of homotopy theory, 178Rellich lemma, 100residual spectrum, 91resolvent set, 92Ricci tensor, 64, 118Riemannian manifold, complete, 98Riemannian metric, 141, 159Riemannian symmetric space, 82, 87Rokhlin's theorem, 110

S1-principal bundle, 163scalar curvature, 111, 113, 118, 135, 144,

145, 148, 149, 151Schrodinger operator, 127Schrodinger-Lichnerowicz formula, 73, 100,

110, 113, 134, 145Schur-Zassenhaus proposition, 39second Stiefel-Whitney class, 40section, 156Seiberg-Witten equation, 131, 134, 136, 138,

140, 153reducible solution of, 140, 142

Seiberg-Witten invariant, 144-146, 149, 151Seiberg-Witten theory, moduli space for, 136self-adjoint operator, 92

essentially, 92-94, 96spectral theorem for, 93

signature, 109of a form, 2

spectral measure, 92, 93spectral theorem for self-adjoint operators,

93spectrum

of a Dirac operator, 99of an operator, 91, 92

sphere, 43, 88, 116, 125, 128spin bundle, 54spin representation, 14, 23, 25, 54, 58, 75,

133

of Spin(n), 20spin structure, 35, 36, 38-40, 42-45, 47, 50,

53--55, 60, 79, 113equivalence of, 35homogeneous, 85, 87

spinC structure, 47, 48, 50, 51, 53, 57, 60,78, 93, 96, 111, 131, 153

canonical, 79of an Hermitian manifold, 77, 78

determinant bundle of, 52-54, 108, 113equivalence of, 51

195

Spine (4) structure, 134, 141, 146canonical, 147, 149

Spinc(n), 25, 26Spinc(12) representation, 28

Spin(n), 15

spin representation of, 20

spinor bundle, 53, 78of an Hermitian manifold, 77

spinor derivative, 59spinor field, 67

parallel, 67Stiefel-Whitney class, 163

second, 40structure identity, 168submanifold, 175

weak, 176Sylow subgroup, 2-, 39, 44Sylvester's theorem, 2symmetric operator, 69, 92, 93symmetric space, Riemannian, 82, 87symplectic manifold, 158symplectic structure, 151, 159

tangent bundle, 155tautological bundle over C?1, 161tensor product of Z2-graded algebras, 7tensorial 1-form, 62tensorial q-form of type p, 165twistor equation, 128twistor operator, 69, 70, 121twistor spinor, 121, 1232-Sylow subgroup, 39, 44

U(k)-reduction, 48, 60, 61, 81unitary group, 27universal covering, 19

of SO(n), 16universal G-bundle, 162

vanishing theorem, 140vector bundle, 161

associated, 161

von Neumann theorem, 92

weak submanifold, 176Weyl spinors, 22, 32Weyl tensor, 118, 121Weyl theorem, 138, 172Witt decomposition theorem, 3Wu's proposition, 133

Yang-Mills equation, 130

Z2-principal bundle, 163(-function for the Dirac operator, 103

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Dirac Operators in Riemannian Geometry...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