Dirac Operators in Riemannian
Geometry Thomas Friedrich
Graduate Studies in Mathematics Volume 25
American Mathematical Society
Dirac Operators in Riemannian
Geometry
Dirac Operators in Riemannian
Geometry Thomas Friedrich Translated by
Andreas Nestke
Graduate Studies in Mathematics Volume 25
American Mathematical Society Providence, Rhode Island
Editorial Board Humphreys (Chair) David Saltman David Sattinger Ronald Stern 2000 Mathematics Subject Classification. Primary 58Jxx; Secondary 53C27, 53C28, 57R57, 58J05, 58J20, 58J50, 81R25.
Originally published in the German language by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, D65189 Wiesbaden, Germany, as "Thomas Friedrich: DiracOperatoren in der Riemannschen Geometrie. 1. Auflage (1st edition)" © by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1997 Translated from the German by Andreas Nestke ABSTRACT. This text examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The presentation includes a review of preliminary material, including spin and spin0 structures.
An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature. Considerations of Killing spinors and solutions of the twistor equation on M lead to results about whether M is an Einstein manifold or conformally equivalent to one. An appendix contains a concise introduction to the SeibergWitten invariants, which are a powerful tool for the study of fourmanifolds.
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 CataloginginPublication Data Friedrich, Thomas, 1949[DiracOperatoren in der Riemannschen Geometrie. English] Dirac operators in Riemannian geometry / Thomas Friedrich ; translated by Andreas Nestke. p. cm.  (Graduate studies in mathematics, ISSN 10657339; v. 25) Includes bibliographical references and index. ISBN 0821820559 (alk. paper) 1. Geometry, Riemannian. 2. Dirac equation. I. Title. II. Series. QA649.F68513 516.3'73dc2l
2000 00038614
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Contents
Introduction
xi
Chapter 1. Clifford Algebras and Spin Representation 1.1.
Linear algebra of quadratic forms
1.2.
The Clifford algebra of a quadratic form
1.3.
Clifford algebras of real negative definite quadratic forms
1.4.
The pin and the spin group
1.5.
The spin representation
1.6.
The group Spin
1.7.
Real and quaternionic structures in the space of nspinors
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 Gprincipal bundles
2.4.
Existence of spin structures
2.5.
Associated spinor bundles
2.6.
References and exercises
vii
Contents
viii
Chapter 3.
Dirac Operators
57
3.1.
Connections in spinor bundles
57
3.2.
The Dirac and the Laplace operator in the spinor bundle
67
3.3.
The SchrodingerLichnerowicz 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 selfadjointness of the Dirac operator in L2
91
4.2.
The spectrum of Dirac operators over compact manifolds
98
4.3.
Dirac operators are Fredholm operators
107
4.4.
References and Exercises
111
Chapter 5.
Eigenvalue Estimates for the Dirac Operator and Twistor Spinors
113
5.1.
Lower estimates for the eigenvalues of the Dirac operator
113
5.2.
Riemannian manifolds with Killing spinors
116
5.3.
The twistor equation
121
5.4.
Upper estimates for the eigenvalues of the Dirac operator
125
5.5.
References and Exercises
127
Appendix A.
SeibergWitten Invariants
129
A.1.
On the topology of 4dimensional manifolds
129
A.2.
The SeibergWitten equation
134
A.3.
The SeibergWitten invariant
138
A.4.
Vanishing theorems
144
A.S.
The case dim ML (g) = 0
146
A.6.
The Kahler case
147
A.7.
References
153
Appendix B. B.1.
Principal Bundles and Connections
Principal fibre bundles
155 155
Contents
ix
B.2.
The classification of principal bundles
162
B.3.
Connections in principal bundles
163
B.4.
Absolute differential and curvature
166
B.5.
Connections in U(1)principal bundles and the Weyl theorem 169
B.6.
Reductions of connections
173
B.7.
Frobenius' theorem
174
B.8.
The FreudenthalYamabe theorem
177
B.9.
Holonomy theory
177
B.10.
References
178
Bibliography
179
Index
193
Introduction It is wellknown that a smooth complexvalued function f : 0 * C defined
on an open subset 0 C R2 is holomorphic if and only if it satisfies the CauchyRiemann equation
8z
0
with
a
2
(ax
+Z
ay
Geometrically, we consider R2 here as flat Euclidean space with fixed orien
tation. Changing this orientation results in replacing the operator j by the = (T  i3 Taking both operators together we differential operator 2 F obtain a differential operator P : C'(] 2; C2) > C°°(R2; C2) acting via
P(f
=2i
x
Of 8x
on pairs of complexvalued functions. An easy calculation leads to the following alternative formula for P:
P=(0i
i
0)8x01
01
)ay69
.
Denoting the matrices occurring in this formula by /_ and y., Yx=
i 0) 0
>
'Yy= ( 01
1
0),
yields
P = yx 8x
+ yy ey xi
Introduction
xii
as well as 'Yx 2 E _ 'Yy,
'Yx'Yy + 'Yy'Yx = O.
The square of the operator P coincides with the Laplacian A on If82: P2
=
a2 axe aye = A. 192
of the Laplacian within the class Thus we have found a square root P = of first order differential operators, and its kernel is, moreover, the space of holomorphic (antiholomorphic) functions.
In higherdimensional Euclidean spaces the question whether there exists a of the Laplacian was raised in the following discussion by square root P.A.M. Dirac (1928). Let T be a free classical particle in II83 with spin 2 whose motion is to be studied in special relativity. Denoting its mass by m, we have v"'' its energy by E and its momentum by p = 12/c2
E=
c2p2 m2 c4.
In quantum mechanics T is described by a state function %(t, x) defined on RI x 1R3, and energy as well as momentum are to be replaced by the differential operators
E + ihat and p i+ ih grad, respectively. The state function 0 then becomes a solution of the equation
ih at =
c2h2A + m2c4
 jr. Mathematia cally speaking we now move to an ndimensional Euclidean space and look involving the 3dimensional Laplacian A =  a for a square root P =
of the Laplacian A = 
i1
axti
The obvious as
sumption that P should be a first order differential operator with constant coefficients leads to the ansatz
Pyi n
i=1
Now the equation p2 = A
axi
a holds if and only if the coefficients ryi ti
of P satisfy the conditions
y2=E, i=1,...,n;
'Yi7j + 7j'Yi = 0,
i 0j.
Introduction
xiii
For n = 3, there is an obvious solution to these equations. The vector space C2
can be identified with the set of quaternions via C2
I
\
zl I = z1 + jz2, 2
= H * )F3[ = C2 then correspond to multiplication by the quaternions i, j, k E H , respectively. Writing these as complex (2 x 2)matrices, we obtain and yl, y2, y3
y1=
(o 0i),
y2=
(0 1),
73=
(0 o).
The algebra multiplicatively generated by n elements y1, ... , yn satisfying the relations
y2 = E,
yiyj + y;yi = 0 (i 0 j),
is called the Clifford algebra Cn (W.K. Clifford, 18451879) of the negative definite quadratic form (Rn, xi  ...  x,2n). Thus, the question whether there is a square root of the Laplacian leads to the study of complex representations K : Cn * End (V) of the Clifford algebra. It turns out that Cn has a smallest representation of dimension dime V = 2[2]. The corresponding vector space is denoted by On and its elements are the Dirac spinors. Moreover, v"A is a constant coefficient first order differential operator acting on the space C°O(IIBn; An) of smooth Anvalued functions on I[81.
Spinors can be multiplied by vectors from Euclidean space. In order to define this product we represent a vector x E an as a linear combination with respect to an orthonormal basis el,... , en, n
x=
xzei, i=1
and then define its product x V by a spinor 0 E An as n
x
_ E xzr, (yi) (0) i=1
From the defining relations of the Clifford algebra one immediately deduces the formula
x (x . 0) = I Ixl l2V) In particular, the product vanishes if and only if either the vector x E W or the spinor b E On is equal to zero. There is no nontrivial representation e of the linear or the orthogonal group in the space An of spinors that is compatible with Clifford multiplication, i.e. which satisfies the relation A(x) . e(A) (0) = e(A) (x . V))
Introduction
xiv
for every A E SO (n; R), x E RI and E Z. Hence spinors on Riemannian manifolds cannot be defined as sections of a vector bundle that is associated
with the frame bundle of the manifold. It is for this reason that in differential geometry the question to what extent the concept of spinors could be transferred from flat space to general Riemannian manifolds remained open for decades. In 1938 Elie Cartan expressed this difficulty in his book "Lecons sur la theorie des spineurs" with the following words:
"With the geometric sense we have given to the word `spinor' it is impossible to introduce fields of spinors into the classical Riemannian technique." Only the development of the framework of principal fibre bundles and their associated bundles as well as the general theory of connections within differential geometry at the end of the forties made it possible to overcome
this difficulty. The group SO(n; R) is not simply connected. For n > 3 its universal covering, the group denoted by Spin(n), is compact and covers SO(n; R) twice. On the other hand, there exists a representation e Spin(n) > GL(On) of the spin group which is compatible with Clifford :
multiplication. Considering now those special Riemannian manifolds Mn, today called spin manifolds, the frame bundle of which allows a reduction to the double cover Spin(n) of the structure group SO(n; R), we can define the vector bundle S associated with this reduction via the representation
e : Spin(n) > GL(An), the socalled spinor bundle of Mn. Then spinor fields over Mn are sections of the bundle S and, as in the Euclidean case, the Dirac operator D can be introduced by the formula n
Do_ ti=1
Here 17 denotes the covariant derivative corresponding to the LeviCivita connection of the Riemannian manifold. Therefore, spinor fields and Dirac operators cannot be introduced on every Riemannian space; but, nevertheless, they can be introduced for a large class. The existence of a Spin(n)reduction of the frame bundle of Mn
translates into a topological condition on the manifold, i.e. the first two StiefelWhitney classes have to vanish:
wi(M') = 0 = w2(Mn). In dimension n = 4, for a compact simply connected manifold M4, this topological condition is equivalent to the condition that the intersection form on H2(M4;Z), considered as a quadratic form over the ring Z, is even and unimodular. The algebraic theory of quadratic Zforms then implies that the signature v is divisible by 8. Surprisingly, in 1952 Rokhlin proved
Introduction
xv
a further divisibility by 2: the signature o (M') of a smooth compact 4dimensional spin manifold M4 is divisible by 16: a(M4)/16 E Z.
This additional divisibility of the signature of a 4dimensional spin manifol which does not result from purely algebraic considerations, was an essential aspect for the introduction of spinor fields and Dirac operators into mathematics. The consideration behind that may be outlined as follows. Could it be possible that there exists an elliptic operator P on every compact smooth 4dimensional manifold with even intersection form on H2 (M4; Z), the index of which coincides with v/16? Today we know the answer to that question: it is essentially given by the Dirac operator on a spin manifold, eventually introduced for Riemannian manifolds by M.F. Atiyah in 1962 in connection with his elaboration of the index theory for elliptic operators. Since then it
has occured in many branches of mathematics and has become one of the basic elliptic differential operators in analysis and geometry. This book was written after a onesemester course held at HumboldtUniversity in Berlin during 1996/97. It contains an introduction into the theory of spinors and Dirac operators on Riemannian manifolds. The reader is assumed to have only basic knowledge of algebra and geometry, such as a two or three year study in mathematics or physics should provide. The presentation starts with an algebraic part comprising Clifford algebras, spin groups and the spin representation. The topological aspects concerning the existence and classification of spin reductions of principal SO(n)bundles are discussed in Chapter 2. Here the approach essentially requires only elementary covering theory of topological spaces. At the same time, each result will also be translated into the cohomological language of characteristic classes. The subsequent Chapter 3 deals with analysis in the spinor bundle, the twistor operator and the Dirac operator in detail. Here the general techniques of principal bundles and the theory of connections are applied systematically. To make the book more selfcontained, these results of modern differential geometry are presented without proof in Appendix B. Chapter 4 contains special proofs for the analytic properties of Dirac operators (essential selfadjointness, Fredholm property) avoiding the general theory for elliptic pseudodifferential operators. Eigenvalue estimates and solution spaces of special spinorial field equations (Killing spinors, twistor spinors) are the topic of Chapter 5. We mainly discuss the general approach, referring to the literature for detailed investigations of these problems. The book is concluded in Appendix A with an extended version of a talk on
xvi
Introduction
SeibergWitten theory given by the author in the seminar of the Sonderforschungsbereich 288 "Differentialgeometrie and Quantenphysik" in Berlin on February 9, 1995. Since the eighties a group of younger mathematicians at HumboldtUniver
sity in Berlin has been working on spectral properties of Dirac operators and solution spaces of spinorial field equations. Many of the results from this period are collected in the references. On the other hand, the present book may serve as an introduction for a closer study. I would like to thank all those students and colleagues whose remarks and hints had an impact on the contents of this text in various ways. I am particularly grateful to Dr. Ines Kath for her careful and detailed corrections of the text, and to Heike Pahlisch, whose typing of the manuscript took into account every single wish. Thomas Friedrich Berlin, March 1997
The English translation of this book has been prepared in the beginning of the year 2000. It does not differ essentially from the original text, although I made many changes in details which are not worth listing. During the last three years many new results have been published in this still dynamic area of mathematics. I included the corresponding references in the bibliography of the translation. During the academic year 1996/97 Dr. Andreas Nestke provided the exercises for students of my lectures at Humboldt University which furnished the starting point for this book. Two years later he had to leave the University. I thank him, as well as Dr. Ilka Agricola and Heike Pahlisch, for all the work and help related with the preparation of the English edition of this book. Thomas Friedrich Berlin, March 2000
Chapter 1
Clifford Algebras and Spin Representation
1.1. Linear algebra of quadratic forms We will start by recalling some facts from linear algebra. Let K be a field of characteristic 0 2. A bilinear form is a pair (V, B) consisting of a Kvector space V and a symmetric bilinear map
B:VxV*K. The function Q : V > K defined by Q(v) = B(v, v) is the corresponding quadratic form. Thus,
Q(Av)=A2Q(v) vEV, AEK. On the other hand, B can also be expressed by Q: B(vi, V2)
=
2
[Q(vl + v2)  Q(vl)  Q(v2)]
hence we will often identify B and Q. A bilinear form (V, B) determines a linear map from the vector space V to its dual V*, VE)
(V, B) is called a nondegenerate bilinear form, if this map is injective. So, (V, B) is a nondegenerate bilinear form if and only if
b'0; vEV 2wEV:B(v,w); 0. Let V be a finitedimensional Kvector space and vl,... , v, (n = dimK V) a basis. Then the matrix M(V, B) _ (B (vi, vj))Z j.1 1
1. Clifford Algebras and Spin Representation
2
is symmetric. It is called the matrix of the form.
Definition. The rank of the bilinear form (V, B) is defined to be the rank of the matrix M(V, B), rank (V, B) := rank (M(V, B)).
Theorem (Lagrange). Let (V, B) be a finitedimensional bilinear form. Then there exists a basis v1, ... , v,z for the vector space V with the property that the matrix M(V, B) is diagonal (bases with this property are called canonical):
/ Ai
0
Ar
M(V, B) =
0
0
r = rank(V, B).
,
0
)
If the field is K = R or C, then even more holds: Theorem (Sylvester). For every quadratic form over R there exists a canonical basis with respect to which the form has the following matrix:
/1
0
1
M(V,B) = 0
The number p of (+1)entries and the number q of (1)entries in this diagonal matrix do not depend on the choice of this basis. The pair (p, q) is called the signature of the form, the number q its index.
1.1. Linear algebra of quadratic forms
3
Theorem. For every quadratic Cform there exists a canonical basis with respect to which the form has the following matrix:
/1
0
1
M(V, B) _
0
0
0)
We now consider a finitedimensional nondegenerate bilinear form (V, B). If W C V is a subspace, we define the Borthogonal complement W1 as
W1={vEV:B(v,w)=0 bwEW}. W is called an isotropic subspace if W f1 W1 $ {0}. W is called a nullsubspace, if W C W1 holds. Obviously, W is a nullsubspace if and only if Qjw = 0 (or Blwxw = 0), i.e. Q (B) vanishes identically on W. Every nullsubspace is isotropic but not vice versa.
Theorem (Witt Decomposition). Let (V, B) be a finitedimensional nondegenerate quadratic Kform and W C V a maximal nullsubspace. Then there exists a maximal nullsubspace U C V such that
a) dimU=dimW, UnW = {0}. b) V=WeU®(W®U)1. c) For every vector 0 $ v E (W ®U)1 Q(v) $ 0. Moreover, for every basis w1 ... Wk in W there exists a basis u1 ... uk in U satisfying
B(uz7wj) = 8zj,
1 < i, j < k
(Witt basis).
Corollary. Let K be an algebraically closed field and (V, B) a finitedimensional nondegenerate form. Then for every maximal nullsubspace W
dimW
T(V) the natural embedding of the vector space into its tensor algebra, then
j =7roi determines a linear map j : V > C(Q) for which, by construction, j (V)2 = Q(v) 1. Moreover, each linear map u : V + A into any algebra extends via U(vl (9 ... ® vk) = u(V1)
...
u(vk)
to an algebra homomorphism U : T(V) + A. If, in addition, u(v)2 = Q(V) 1, then we have _T(Q) C ker(U), and thus U induces a homomorphism
U : C(Q) > A satisfying the relation we were looking for. Given another homomorphism U1 : C(Q) + A satisfying C(Q)
V
X
A
then u = U1 o j = Uo j. Hence U and U1 coincide on V C C(Q). On the other hand, the vectors from V generate the tensor algebra T(V) multiplicatively and hence the algebra C(Q) as well. Thus we immediately conclude U = U1. Uniqueness is a straightforward consequence of the third defining property of a Clifford algebra.
Corollary. The linear map j : V > C(Q) from the vector space V of the quadratic form into its Clifford algebra is injective. The set j(V) C C(Q) generates the algebra multiplicatively.
For this reason we will often view the vector space V as a linear subspace of C(Q).
Proposition. The Clifford algebra C(Q) of a quadratic form is equipped with an involution /3 : C(Q)  C(Q) such that a) /3 is an algebra homomorphism and an involution, i.e. /32 = Id. b) Setting C°(Q) = {x E C(Q) : /3(x) = x}, C'(Q) = {x E C(Q) /3(x) = x} we have the splitting C(Q) = C°(Q) ® C1(Q)
1. Clifford Algebras and Spin Representation
6
and the relations C°(Q) C°(Q) C C°(Q) , C°(Q) C'(Q) C C'(Q) as well as C'(Q) C1(Q) C C°(Q). In particular, C°(Q) C C(Q) is a subalgebra.
Proof. Consider the linear map u : V 3 C(Q), u2(v)
u(v) = j(v). Since
= (j(v))2 = j(v)2 = Q(v)
. 1,
there exists an algebra homomorphism 0: C(Q) 3 C(Q) satisfying
poj(v)=g(v) for all v E V. Since, by construction,
(3o0oj)(v) =3H(v)) = Oi(v) =j(v), ,
2 is the identity on the set j (V) C C(Q). Thus 02 = Id holds in general.
In addition to the involution : C(Q) + C(Q) each Clifford algebra also carries an antiinvolution y : C(Q) > C(Q). To construct it, we begin with a preliminary remark: If A is a Ilkalgebra, then one can define a new algebra
A on the set A by introducing the product
Now let (V, Q) be a quadratic form and C(Q) its Clifford algebra. Consider the algebra A = C(Q). For the linear map
V ')A=C(Q) the relation j (v) * j (v) = j (v) j (v) = Q (v) 1 holds in the algebra A. Thus there exists a unique algebra homomorphism 'y
: C(Q) ' C(Q)
such that
j(v) = yj(v),
v E V.
The properties of y as a map from C(Q) to itself are stated in the following
Proposition. For every
Clifford algebra there
C(Q) > C(Q) with the following properties:
1) y is linear. 2) y o y = Id (y is an involution). 3) y(v) = v for all v E V C C(Q). 4) y (x  y) = y(y)'y(x),
x, y E C(Q)
exists a linear map
1.2. The Clifford algebra of a quadratic form
7
Given two bilinear forms (V,, B,) and (V2, B2), their direct sum is defined as the bilinear form (V, B) with V = V1 ®V2,
B(Vi,V2) = 0,
B1v1xvl = B1,
Blv2xv2 = B2.
Correspondingly, we will consider the direct sum Q1 ® Q2 of two quadratic forms.
Proposition. The Clifford algebra C(Q1(9 Q2) is isomorphic to the tensor product C(Q,)®C(Q2),
c(Q, ® Q2) = C(Q,)®c(Q2)
Remark. The tensor product C(Q1)®C(Q2) is to be understood as a tensor product of 7Z2graded algebras. For two Z2graded algebras A = A° A', B = B°6 B1 the tensor product A®B is the following Z2graded 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®v2v1®v2+1®v2, since 1 E C°(Q,), v1 E C'(Q,), and V2 E C1(Q2) belong to the corresponding sets of the Z2grading. 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 ndimensional vector space. Then the vector space C(Q) has dimension 2n, dimK C(Q) = 2n.
1. Clifford Algebras and Spin Representation
8
Proof. By the Lagrange theorem the quadratic form is the sum of n onedimensional quadratic forms:
Q=Q1®...®Qn. But the Clifford algebra of a onedimensional quadratic form B : K x K > K is easily computed,
and e2=Q(1).
C1(Q) = K e,
CO(Q)=K,
Hence, dimK C(Q) = 2 for a onedimensional quadratic form. From the last proposition we conclude C(Q) = C(Q1)®... ®C(Qn)
and hence dimx C(Q) = 21.
Proposition. Let (V, B) be a quadratic form and v1, ... , vn a basis of V such that
B(vi, vj) = 0,
i
j.
Then the Clifford algebra C(Q) is multiplicatively generated by the elements , vn E V C C(Q) which satisfy the relations
vi, ...
v2 = Q(vi),
vivj + vjvi = 0,
i 0 j.
A particular basis of the vector space C(Q) is formed by the elements 1 and vil .... via, where 1 < it < i2 < < is < n with 1 < s < n.
Proof. V C C(Q) generates C(Q) multiplicatively, and vl,... , vn are a basis of the vector space V. Hence the vectors vi, ... , vn generate the algebra C(Q), too. Moreover, v? = Q(vi) for trivial reasons, and (vi + vj)2 = Q(vi + vj) = Q(vi) + Q(vj) = v? + v immediately implies
vivj + vjvi = 0,
i # j.
These 2n elements generate C(Q) linearly. Finally, since dim C(Q) = 2n, this system has to be a basis.
Example. Let us consider the trivial quadratic form Q  0 on the vector space V. Then C(Q) = A * (V) is nothing but the exterior algebra of V.
Example. Let IK = IE8 and take V = R2 with the quadratic form Q = x2 y2. Then C(Q) = } is the algebra of quaternions.

1.2. The Clifford algebra of a quadratic form
9
Proof. C(I[82, Q) is generated by el, e2 E R2. Thus 1, ei, e2, el e2 are a basis of the vector space C(Q). With the notation
is=e1i
j:=e2, k:=el e2,
the fundamental relations take the form
k.i=j, j2=j2=k2=1,
i j=k, since e2 = 1 = e2,
e1e2 = e2e1.
El
Example. Consider K = R, V = II81 and Q = x2. In this case, C(Q) coincides with the algebra of complex numbers.
Proof. 1 and el E R1 form a special linear basis of C(Q) subject to the single relation ei = 1. Next we want to determine the centers of C(Q) and C°(Q), respectively, in the case of a nondegenerate form. In order to achieve this, consider a basis V1.... , vn of the vector space V such that
B(vi, vi) = Ai 00 and B(vi, vj) =0 for i
j,
i = 1,...
, n.
Let P be the set of all strictly ordered subsets of {1, ... n}: an element of ,
P is a subset P = {pl,... pk} where 1 18n, x1  ...  xn)  Clifford algebra of the ndimensional real
Cn = C(1
, xi +... + xn)
negative definite form,  Clifford algebra of the ndimensional real positive definite quadratic form,
1.3. Clifford algebras of real negative definite quadratic forms
Cn = C(Cn, zi + ... + zn)
11
 Clifford algebra of the ndimensional complex quadratic form.
Remark. If Q is a nondegenerate complex quadratic form of dimension n (e.g. Q = zi  ... zn), then Q is equivalent to the form (Cn, z1 +...+zn) over the complex numbers and hence both their Clifford algebras coincide, C(Q) = Cn. Let us begin with the following general consideration concerning complexification. For a real quadratic form (V, B) the complexification (V OR C, BC) is defined as
Bc(vl 0 z, v2 ®w) = B(vl, v2)zw. On the other hand, if A is any real algebra, then its complexification A OR C
carries the product structure
(al 0 z) (a2 0 w) = (ala2) 0 (zw) turning A ® C into a complex algebra.
Proposition. Let (V, B) be a real quadratic form and (Vt, BC) its complexification. Then, in the sense of isomorphic Calgebras, C(VV, Bc) = C(V, B) OR C.
Proof. Define u : V OR C > C(V, B) 0 C by u(v 0 z) = j(v) 0 z, where j : V + C(V, B) is the embedding of the real vector space into the Clifford algebra. Then, u(v ®z)2 = (j(v) (&z)2
=
j(v)2 ®z2
= B(v, v)z2 101 = Bc(v ®z, v ®z)
1.
Hence u extends to a homomorphism u : C(VV, BC) > C(V, B) OR C of the
complex algebras. It is easy to check that u is an isomorphism. Corollary. One has the following identity of complexifications:
Cn = CnOR C = Cn,®RC.
Proof. The complexifications of the real forms xi+...+xn and xi...xn are equivalent.
Proposition. The following graded real algebras are isomorphic: Cn+2 = Cn, OR C2
C1
n+2 = Cn ®C2.
The tensor product is to be understood as the usual one for algebras.
1. Clifford Algebras and Spin Representation
12
Proof. Choose an orthonormal basis of the vector space R'n+2 consisting of the vectors el, ... , en+2. The first n vectors then generate the algebras C, and Cn, where e', ... , e;n denote these same vectors considered this time as generators of C. Now define u : Rn+2  Cn, OR C2 by u(el) = 1 ®el, u(e2) = 1 ®e2i u(ei) = ei2 ®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'i2 (3 ele2)(42 (D ele2) = ei2 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+2pC'n®II8 C2
fitting into the commutative diagram Cn+2 U
Rn+2
 Cn OR C2
Now u preserves the Z2grading and is the desired isomorphism.
Example. The algebra C2 = C2 OR C is the complexification of C2. The latter algebra is generated by el, e2 satisfying the relations e2 =  1 = e2,
ele2 + e2e1 = 0.
On the other hand, the underlying vector space of the complex algebra M2(C) has the basis E
0
_
(10
1)
,
\i
91 =
0
iJ
92 =
(i
0T
(i
0
\
with the relations 91
=
_I = 92,
9192 + 9291 = 0.
This implies C2 = C2 OR C = M2 (C) .
Making repeated use of this example, we arrive at the following isomorphism.
Proposition. There is an isomorphism Cn+2 = Cn ®C M2(C). Proof. Cn+2 = Cn+2 OR C = (C;, OR C2) OR C = (Cn OR C) ®r (C2 0 C) _ C.' ®CC2=C; ®CM2(C)
1.3. Clifford algebras of real negative definite quadratic forms
13
This isomorphism explicitly involves the one described in the proof of the previous proposition, i.e. Cn+2 = C,,, OR C2. Thus we obtain an explicit isomorphism C,' +2 = C° 0 M2 (C) which we spell out once again.
Corollary. Let e1, ...
, en+2 be the generating elements of the algebra Cn+2
... en those for the algebra C. Moreover, denote
and, correspondingly, ei,
,
by
i
91=
0i
),
92
=
\
O
i/
the generating elements of the algebra M2 (C) The isomorphism .
Cn+2 ' Cn OC M2 (C)
is then given by
e1*1®g1,
e2r+1®g2,
3<jCn+2.
ei''(Set2)®9192,
We will now apply these isomorphisms repeatedly and take into account the descriptions of the Clifford algebras involved, i.e.
C1'=C1®RC=C®R(C=C®C,
C2c=M2(C).
In summary, we immediately obtain the following proposition:
Proposition. a) If n = 2k is even, then Cn = M2 (C) ®... ® M2 (C) = .End (C2 (& ... ® C2) = End(C2"). k times
k times
b) If n = 2k + 1 is odd, then Cn = {M2(C)®...®M2(C)}®{M2(C)®...®M2(C)} = End(C2k)®End(C2k). These isomorphisms are explicitly described by the following formulas: The case n = 2k even:
ei >E®...®E®9«(i)®7
. (9
T
times
where
if j is odd, 2 if j is even. The case n = 2k + 1 odd: If 1 < j < 2k, ej is mapped to ej i> (E®...®E®9a(9)®T T, E®...®E(3 9a(j)®T 0 1
a (7)
[ 2 ] times
®T). times
The endomorphism e2k+1 is realized by e2k+1 H (iT ®... ®T, iT ®... (S T).
1. Clifford Algebras and Spin Representation
14
Definition (complex nspinors, Dirac spinors). The vector space of complex nspinors is
forn=2k,2k+1.
pn:=cC2k =C2®...®C2 k times
The elements of An are called complex spinors.
Using this notation we obtain
if n = 2k is even,
Cn = End(An),
Cn = End(On) ® End(An), if n = 2k + 1 is odd. Moreover, the following diagram is commutative: cc
cc
2k+1
2k
End(A2k) `P
End(A2k+1) ® End(A2k+1)
Here cp is the diagonal map cp(A) _ (A, A) and the vector spaces A2k = A2k+1 coincide.
Denote by Kn the socalled spin representation of the Clifford algebra C. In the case of an even dimension, n = 2k, this is nothing but the isomorphism explained before, Kn : Cn
 End(An).
If n = 2k + 1 is odd, then kn consists of the isomorphism Cn = End(An) End(An) followed by the projection onto the first component, Kn : Cn
) End(An) ® End(An) p4 End(An).
The vector space of complex nspinors is thus turned into a module over the Clifford algebra C,',.
1.4. The pin and the spin group Consider the real vector space Rn and the Clifford algebra Cn of the quadratic form xi  ...  x,2n. W' itself is a linear subspace of Cn, R C Cn. For every vector x E Rn, the equality
x'x=IIxIl2 holds in Cn and hence the inverse element x1 is given by X
1
x
T1x,2.
1.4. The pin and the spin group
15
Definition. Pin(n) C Cn is the group which is multiplicatively generated by all vectors x E Sn1. Therefore, the elements of Pin(n) are the products x1 ... x,,,, with xi E IISn, I xi 1. The spin group, Spin(n), is defined as I
Spin(n) = Pin(n) n CO,. Now we will define a homomorphism
A : Pin(n)  O(n) from the group Pin(n) onto the group O(n) of orthogonal transformations of the Euclidean space Rn. To this end, recall the antiinvolution
'Y:Cn*Cn existing in every Clifford algebra. It has the particular property that y(x) _ x holds for all vectors x E ]l C Cn.
Lemma. If y E IEBn C Cn and x E Pin(n) C Cn, then the element x y y(x) belongs to the subspace R C Cn.
Proof. x is, by definition, the product x = xl ... x,,,, of vectors from the sphere Sn1. 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 Sn1. Next we choose an orthonormal basis in R' with x = el as the first vector. Then, writing
n
y=
yieie i=1
we get
n
n
E yiei
x . y . y(x) = e1
e1 = y1e1 + E yiei,
(i=1
i=2
and hence x y y(x) is the vector in R which is the image of y under the reflection in the plane perpendicular to x. For x E Pin(n) we now define A(x) : Rn
Rn,
.\(x)y = x . y .'Y(x)
Then, obviously,
A(xlx2)y = x1x2YY(x1x2) = xlx2YY(x2)y(xl) = A(xi)(A(x2)y)
and hence A : Pin(n) > GL(n) is a group homomorphism. Every element
x of Pin(n) is the product x = xl ... x,n of vectors xi E S"1, and the calculation above shows that A(xi) is a reflection. Therefore, A(x) itself is the superposition of reflections, so, in particular, an orthogonal transformation.
1. Clifford Algebras and Spin Representation
16
Proposition. a) A : Pin(n) > 0(n) is a continuous surjective group homomorphism.
b) A1(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 wellknown fact that every orthogonal linear map A : I[8n * Rn is the superposition of reflections. Let 0 : C, + Cn be the involution of the Clifford algebra which defines the decomposition Cn = C° ® C. For every vector x E IlBn we have 0(x) x. Now apply /3 to any element x = xi ... x,,,,: O(x) = )(XI) ...O(xm,) _ (1)mxl ... x,n.
From this we see that x = xi ... x,,,, belongs to Spin(n) if and only if m is even. On the other hand, A(x) = A(xi) ... A(x,,,,,) is the superposition of m reflections and, hence, A(x) lies in SO(n) if and only if m is even. This
proves b). Now suppose that A(x) = 1 holds in 0(n). Then xyy(x) = y for all y E IlBn. Moreover, since \(x) E SO(n), x is the product of an even number of vectors, x = x1... x,n and m  0 mod 2. Thus, x1...xmyxm...x1 = Y.
Multiplying this equation from the right by x1 ... x,,,, and taking into account that xi = ... = xm = 1 as well as m 0 mod 2, we obtain
x1...xmy = yx1...xm. The element x = x1 xm (m  0 mod 2) of the Clifford algebra Cn thus commutes with every vector from IRE. Therefore, x belongs to the center of Cn and, at the same time, to the center of CO,. But, for every Clifford algebra,
Z(C(Q)) n Z(C°(Q)) _ K.
Hence, x E R', and from xj = 1 we see that x = ±1. It remains to be shown that Spin(n) is a connected group. As
A : Spin(n) ; SO(n) is surjective with ker A _ {1, 1}, it suffices to find a path in Spin(n) connecting the element (1) E Spin(n) with the neutral element 1 E Spin(n). In the case n > 2, one such path is given by (0 < t < 1) y(t) =  cos(7rt)  sin(7rt)e1e2.
1.4. The pin and the spin group
17
y(t) indeed lies in Spin(n), since
tlelsin y(t)= (cos(it) el+sin(it) e2I/ Icos1 \ \ /
it/ e2/ I
\(
in Cn.
Now we will describe the Lie algebra of the group Spin(n). For this we need the following general remark: Let A be a finitedimensional associative real
algebra and A* C A the group of its invertible elements. A* is an open subset of A and, furthermore, a Lie group. Its Lie algebra a*, which can be identified with Ti(A*) A, thus coincides with A, a* = A.
The commutator of two elements al, a2 E A = a* is given by [al, a21 ala2  a2al, and the exponential map
exp:a*=A>A* is defined by the power series of the exponential function: °O
exp(a)
a an
 1:
mt'
n=O
We are going to apply this observation in the case of the real Clifford algebra Cn. The spin group is a subgroup of Cn, Spin(n) C Cn,
and hence, to describe the Lie algebra spin(n), it suffices to determine the tangent space T1(Spin(n)) C Cn. We will proceed as follows: Let y(t) = xi(t) ... X2, (t) be a path in Spin(n) with xi(t) E Sn1 and y(0) = 1. The tangent to y at t = 0 is Tt (0)
=
dtl (0) ' x2 (0)
...
x2n, (0) + xi (0)x2 (0)
dtm (o).
Let M2 C Cn be the subspace of Cn spanned by the Clifford products ei ej, 1 < i < j < n. We will prove that each summand of dt (0) belongs to m2. Since y(O) = 1, the first summand is equal to d
dt1(0) x11(0) But xi (t) xi (t)  1 implies
dtl (0) x1(0)
dt1(0)xi(0) + xi(0)
dti(0) = 0
and, because of the relations in the Clifford algebra, ddt (0) and xi (0) are perpendicular as vectors in R1. Therefore, the first summand belongs to m2.
1. Clifford Algebras and Spin Representation
18
Analogously, the second summand coincides with x1(0) dt (O)x2 1(0)x1 1(0) _x1(0) d2 (O)x2 (O)x1 1(0)
_ {x1(0) dt (0)x1 1(0)} {x1(0)x2(0)XI 1(0)}. '
As dt (0) and x2(0) are perpendicular, the vectors x1(0) dt (0)xi 1(0) and xl(0)x2(0)xi 1(0) are perpendicular, too, and the second summand lies in m2. Altogether we arrive at the
Proposition. a) The linear subspace m2 C Cn
m2 = Lin(eiej : 1 < i < j < n) equipped with the commutator
[x, y] = xy  yx
is a Lie algebra which coincides with the Lie algebra of the group Spin(n) C C. b) The exponential map exp : m2 + Spin(n) is given by °°
exp(x)
i
_Y, x
v.
i=o
c) If a : Cn + End(W) is a (real or complex) representation of the Clifford algebra and 0ISpin(n)
: Spin(n) > Aut(W),
the group homomorphism defined by restriction, then its differential (UjSpin(n))* : spin(n) = m2 > End(W) is given by the formula (aaISpin(n))* = QJm2
Proof. The above calculation, first of all, shows that the Lie algebra spin(n) is contained in the subspace m2. Since dimR(m2) =
n(n  1) 2
and dim(Spin(n)) = dim(SO(n)) = n(n  1)/2, the two spaces have to coincide. This proves a) and b). c) follows from general facts concerning the differential of a homomorphism f : H ; G between two Lie groups.
1.4. The pin and the spin group Namely, f* :
19
hB is uniquely determined by the commutative diagram exp
HG as a homomorphism of Lie algebras. But in our situation the diagram
End(W)
m2 exp
exp
Aut(W)
Spin(n)
commutes, because u : Cn > End (W) is a homomorphism of algebras. This completes the proof.
Consider now the universal covering
A : Spin(n)  SO(n) and let us compute its differential
A. : spin(n) > so(n). As before, we identify spin(n) with the vector space
m2 = Lin(ejej : 1 < i < j < n) and so (n) with the set of all skewsymmetric matrices. A particular basis of the vector space so(n) is formed by the matrices Ezj (i < j) defined by
0
j
i
...
...
0
0
...
1
0
0 0
EZj _
j
1
0
...
...
0
...
0 0
Now we want to prove the formula '\*(ezej) = 2E,;j
for X, : spin(n) = m2 3 so(n). The path
y(t) = cos(t)+sin(t)eiej = (cos(t/2)e;,+sin(t/2)ej)(cos(t/2)eisin(t/2)ei) is a subgroup y C Spin(n) with at (0) = eiej. Then, for k A('Y(t))ek = 616
i, j,
1. Clifford Algebras and Spin Representation
20
and, e.g. in the case k = i, we have
A(y(t))ei = (cos(t) + sin(t)eiej)ei(cos(t) + sin(t)ejei) = cos2(t)ei + 2sin(t) cos(t)ej  sin2(t)ei = cos(2t)ei + sin(2t)ej. g(A(y(t))ei) = lei. This computation proves the formula A (eiei) _ Thus 2Eij . The result can also be written invariantly.
Proposition. If z E spin(n) is an element of the Lie algebra, then for the differential A,, : spin(n) + so(n) the relation
A*(z)x = zx  xz holds for every x E Rn. In particular, the Clifford product zx  xz belongs to Rn(z E m2i x E Rn).
1.5. The spin representation Consider the Cnmodule of nspinors On. Via
Spin(n) C Cn C Cn  End(An), we obtain a representation rc of the group Spin(n) by restriction: K = KnjSpin(n) : Spin(n) * Aut(On),
the spin representation of the group Spin(n).
Proposition. The spin representation is a faithful representation of the group Spin(n).
Proof. If n = 2k is an even number, then Cn = End(On) and the statement is trivial. This leaves the case n = 2k + 1. As vector spaces, O2k+1 = A2k, and the diagram Spin(2k)
Spin(2k + 1)
k2k
11k+1
GL(O2k) 1
GL(A2k+1)
commutes. This implies that the normal subgroup H := ker(K2k+1) has only the neutral element in common with Spin(2k),
H n Spin(2k) = {1}.
The subgroup A(H) C SO(2k + 1) is normal, since A : Spin(2k + 1) SO(2k + 1) is surjective. Moreover,
A(H) n SO(2k) = {E}.
Let A E \(H) C SO(2k + 1) be an element from this normal subgroup. The characteristic polynomial of A has odd degree, hence there exists a
1.5. The spin representation
21
unit vector vo such that A(vo) = vo. This means there is an element B E SO(2k + 1) for which BAB1 E SO(2k). As A(H) is normal, this implies BAB1 E A(H) f1 SO(2k), i.e. BAB1 = E. Thus we proved that A(H) is the trivial subgroup of SO(2k+1). For H itself this leaves two possibilities, either H = {1} or H = {1, 1}. But (1) E Spin(2k + 1) does not belong to the kernel of the spin representation. End(On,) can be considered as an endomorphism of On. This leads to the socalled Clifford multiplication of vectors and spinors, which is a linear map A vector x E Il8n C Cn C Cn
p: IR OR On ) On. Here p(x ®z)) is defined for x E JR and 0 E On by
µ(x (9 0) = K.(x)(0)Instead of p(x 0 v'), we will often simply write x cation extends to a homomorphism
This Clifford multipli
p : A(II8n) OR An  On
as follows: Using the orthonormal basis ei, ...
, en
of Rn, each element of
the exterior algebra A(IEB') can be written as
wk =
U
wili2 ...ikeil A ... A eik.
it GL(An) is the representation of a compact group in a complex vector space. Thus there exists a Spin(n)invariant Hermitian scalar product in An. We will construct one such product satisfying an even stronger invariance property.
Proposition. In the vector space of nspinors, On, there exists a positive definite Hermitian scalar product ( , ) with the invariance property
(x0, x E I[8n, co, 0 E On. The spin representation rc : Spin(n)  GL(On) is a unitary representation with respect to this scalar product.
Proof. Let M2 = Lin(ezej i < j) C Cn be the Lie algebra of the group Spin(n). Consider g = an ® m2 C Cn. A simple calculation shows that g is a Lie algebra with the commutator z,WEg. :
The map cp : g  Cn+1 given by
OIm2 = Id,
lo(ei) = eien+1 for 1 < i < n
is the restriction of an algebra homomorphism 1 : Cn } Cn+l.
is induced
by the map 1) : an > Cn+l, l(ei) = eien+l, as can be seen from the equations
1 < i < n,
D(ei)2 = 1,
l(ei)D(ej) +
0,
1 < i < j < n.
Thus co : g p Cn+1 maps the Lie algebra g bijectively onto the Lie algebra
of the group Spin(n + 1) and is, moreover, an isomorphism of these Lie algebras. Hence g is a compact Lie algebra. In view of the following remark this proves the assertion.
Remark. In the previous proof we made use of the following general result: Let g be a compact real Lie algebra and rc : g * End(W) a representation in a complex vector space. Then in W there is a positive definite Hermitian
scalar product (,) with the invariance property (I(x)wl,W2) + (Wi,,(x)w2) = 0
for x E 9,Wi,W2 E W. To prove this we consider the compact group G corresponding to the Lie algebra g, an arbitrary positive definite product (, ) * in W, and we set (wl, W2) =
f(gwl,gw2)*dg, G
where dg is the Haar measure of the group G.
1.6. The group Spin
25
Proposition. If ic : Spin(n)  U(On) is the spin representation, then det (,(g)) = 1
for every group element g E Spin(n). In other words, the spin representation is a representation into the special unitary group SU(On) of the space of nspinors.
Proof. Basically, this is not a property special to the spin representation, but a consequence of the following observation: Consider the group homomorphism
f : Spin(n) * S',
f (g) = det (s (g)).
As Spin(n) is simply connected, there exists a lift F : Spin(n) + R to the universal covering of S', f (g) = elriF(g) ,
which is a group homomorphism as well. Spin(n) is compact. Hence F(Spin(n)) C II8 is a subgroup which is contained in a bounded interval. This implies that F  0 and thus f (g) = det (n(g))  1.
1.6. The group Spine The complex Clifford algebra Cam, comprises the group Spin(n) as well as the group Si of all complex numbers of modulus 1. Together they generate a
group which we want to denote by Spinc (n). Since Spin(n) fl Sl = {1, 1}, the group Spine (n) is apparently given by
Spine(n) = (Spin (n) x Sl)/{fl} = Spin(n)
xZ2
Sl.
The elements of Spine (n) are thus classes [g, z] of pairs (g, z) E Spin(n) x Sl
under the equivalence relation (g, z)  (g, z). We define several homomorphisms:
a) Let A : Spine(n) + SO(n) be given by A[g, z] _ .fi(g). b) i : Spin(n) > Spine(n) is the natural inclusion, i(g) = [g, 1]. c) j : Sl + Spine(n) is the natural inclusion, j (z) = [1, z]. d) Let l : Spine (n) + S' be given by 1 [g, z] = z2. e) p : Spine (n) + SO(n) x Sl is given by p([g, z]) = (A(g), z2). Hence,
p=Axl.
1. Clifford Algebras and Spin Representation
26
Then the following diagram commutes:
Ii Spin(n)
1
Z
I
Spinc(n)
1
A
SO(n)
1
and all its rows and columns are exact. Moreover, p is a 2fold covering of the group SpinC(n) over SO(n) x Sl. We will use this diagram to compute the fundamental group.
Proposition. Let n > 3. Then, a) The fundamental group ir1(SpinC(n)) is isomorphic to Z, and 1 : 7rl (SpinC (n)) > 7ri (Sl) = Z is an isomorphism. b) Choose generators for the following groups: a E 7r1(SpinC(n)),
,(3 E 7rl(SO(n)),
y E 7rl(Sl)
with 1p(a) = y. Then for the homomorphism induced by the 2fold covering pp : ir1(SpinC(n)) + rr1(SO(n)) x 7r1(Sl) the following formula holds: pp (a) = /3 + y.
Proof. a) follows directly from the exact sequence
1 ; Spin(n) > Spinc (n) > S1 > 1 combined with 7rl (Spin(n)) = 1 for n > 3. Asp = A x 1, we still have to prove
A (a) = . 7r1(SO(n)) is isomorphic to Z2, and hence this is equivalent to A#(a) 0. Suppose that An(a) = 0. From the corresponding exact column sequence we see that there exists an element S E 7r1(Sl) with j (S) = a. This implies y = 1q(a) = ldjd(S) = 25
and y (as well as a) is not the generating element of the group 7r1(Sl).
1.6. The group Spin
27
Let n = 2k be an even number. The unitary group U(k) is a subgroup of SO(2k). Consider the homomorphism
f : U(k)  > SO(2k) x S',
f (A) _ (A, det A).
Proposition. There exists a homomorphism F : U(k) * Spin(2k) such that the diagram
Spin (2k) p
U(k) ' SO(2k) x S1 commutes.
Proof. We have to prove that the group fp(7r1(U(k)) is contained in the If we choose a generating element d E irl(U(k)) = Z in set addition to the generating elements a, 3, y, then we have f f (6) _ l3 + y, and ,
the assertion follows by covering theory.
The same argument yields a lift of the homomorphism
f1 : U(k) ; SO(2k) x Sl,
f1(A) = I\A, det1 A /I
Proposition. There exists a homomorphism F1 : U(k) + Spin(2k) such that the diagram
Spin (2k) IP
U(k) fl
SO(2k) X Sl
commutes.
Remark. The homomorphism F : U(k) > Spin(2k) can be explicitly described. Let A E U(k). Then there is a unitary basis fl, ... , fk in Ck with respect to which A has diagonal form eiO1
0
0
ei0k
A= If J : Ck > Ck is the complex structure of Ck, then fj and J(fj),1 < j < k, are elements of the complex Clifford algebra C. We then define a
1. Clifford Algebras and Spin Representation
28
homomorphism F : U(k) > Spin(2k) = Spin(2k) XZ2 S' by the formula k
F(A) = fl( cos C
Bl 2
l+ sin
g1
(2
fjJ(fj))
I
x e2
E a,
J
Since Spin(n) is contained in the Clifford algebra Cn, the spin representation of the group Spin(n) extends to a Spin (n) representation. For an element [g, z] from Spin (n) and any spinor 0 E On we then have i[g, z]ib = z . ,c(g)(,0).
Hence the space On of nspinors becomes a Spin (n)representation. The determinant of the endomorphism /c [g, z] : On  + On is given by the formula det ,c[g, z] = zdim(On)
In the case of an even dimension, n = 2k, the splitting 02k = 02k ®A2k is Spin(2k)invariant and for the corresponding representations we have det rc±[g, z] = zdim(Ok)
This implies that the Spin (n)representations det(A ) = and
(l)dimon±/2=l®
...®l
Adim(On) (A±)
(dim On /2 times)
are equivalent. For the particular case n = 4 we obtain the equation
A2(A)
A2(A) = l 4
= in the sense of Spin (4)representations. 4
Proposition. There exists an injective homomorphism f : Spin (n) Spin(n + 2) such that the diagram
Spin (n) p
f
Spin(n + 2) A
SO(n) x Sl = SO(n) x SO(2)  SO(n + 2) commutes.
Proof. Spin(n) is a subgroup of Spin(n + 2), and S1 can be realized as a subgroup of Spin(n + 2) by the elements cost + sinters+len+2
= (cos (t/2) en+1 + sin (t/2) en+2) (cos (t/2) en+1  sin (t/2) en+2)
The intersection of these subsets of Spin(n + 2) is {±1}, and so we obtain a subgroup of Spin(n + 2) isomorphic to Spin' (n).
1.7. Real and quaternionic structures in the space of nspinors
29
The Lie algebra of the group Spinc (n) C CI, is the direct sum spine (n) = m2 ® iIR
and the differential p* : spine (n) + so (n) x iR of the covering p is described by
p* (e«ep, it) = (2Eaa, 2it),
1 < a < R < n.
1.7. Real and quaternionic structures in the space of nspinors Recall the following definitions of a real and a quaternionic structure in a complex vector space.
Definition. Let V be a complex vector space. A real structure is an liblinear map a : V + V with the properties
a2 = Id,
a(iv) = ia(v).
Definition. Let V be a complex vector space. A quaternionic structure is an JRlinear map a : V + V with the properties
a2 = Id,
a(iv) = ia(v).
First, we want to study real and quaternionic structures in the 3dimensional spin representation A3 = C2 and their invariance properties under Clifford multiplication.
Proposition. a) In A3 there exists a quaternionic structure a : 03
A3 which commutes with /Clifford multiplication by each vector x E 1R3:
a p A3 3 03 which anticommutes In with Clifford multiplication by each vector from a 2dimensional subspace x E R2 C R3, ,8(x
b) = x/(c),
X E J2 C R3,
and commutes with e3: /3(e3 . ') = e3 . /3('0)
,3 is not Spin(3)equivariant.
V)E A3,
1. Clifford Algebras and Spin Representation
30
Proof. We realize 03 as the vector space A3 = C2 and make use of the realization of the spin representation of the Clifford algebra C3 described in Section 1.3:
/O i
i 0 e191(0 i),
e2=92=1 i
0),
e3=iT=
0
1
1
0
A3 by the formulas
Define a, /3 : A3
a(z2)=( z12), )3(z2)=(z2 A straightforward calculation then leads to the result, e.g. /3 ( e 2
N(e3 ' b)
) = /3
)
(
=
(
)
( Z2 ) = ( zzl ) ,
e2 ( ) = e2 (z2 ) e3 ' 0(0) = e3 (z2
=
(izl )
2 ) _ ( zzl ).
Hence, /3(e2  0) = e2/3(?) as well as /3(e3  0) = e3/(L).
In order to continue the construction, we will need the following algebraic preparation. If a : V > V and /3 : W * W are real or quaternionic structures, then their tensor product
a®/3:V®CW+V®cW can be defined by (a (3 /3) (v (9 w) = a(v) ® /3(w).
a ®/3 is Ii linear and anticommutes 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 welldefined. a ®,Q again is a real (quaternionic) structure, since (a ® /3)2 = a2 ® /32 = ±Id .
Recall the real and quaternionic structures a, /3 : C2 > C2 defined above and their commutation relations (note that e3 = iT is replaced by T) a91 = 91U7
a92 = 92a,
aT = T a,
/3T = T/3. Now we will construct one of these structures in each vector space Z nspinors a,, : A, k A,,. 091 = 910,
092 = 920,
of
1.7. Real and quaternionic structures in the space of nspinors
31
Proposition. 1) Let n = 8k, 8k + 1. In On there exists a real Spin (n)equivariant structure an : On * On which anticommutes with Clifford multiplication:
xERn and ')EOn. 2) Let n = 8k + 2, 8k + 3. In On there exists a quaternionic Spin(n)equivariant structure an : On * On which commutes with Clifford multiplication:
an(x 0) = x an(O), x E R
and V) E L.
3) Let n = 8k + 4, 8k + 5. In On there exists a quaternionic Spin(n)equivariant structure an : On  On which anticommutes with Clifford multiplication:
an(x
XEll Bn
and 0EOn.
4) Let n = 8k + 6, 8k + 7. In On there exists a real Spin(n)equivariant structure an : On > On which commutes with Clifford multiplication:
xEl8n and 2b EOn. Proof. We define an case by case. First case. n = 8k, 8k + 1. We have On = C2 ® ... ® C2 (4k times), and we set
an (a(3,3)®...®(a(9,3) Second case. n = 8k + 2, 8k + 3.
(2k times).
We have On = C2 ® ... ® C2 (4k+1 times), and we set
an = a ®(Q ®a) ®... ®(0 (3 a)
(2k times).
Third case. n = 8k + 4,8k + 5. We have A. = C2 ® ... ® C2 (4k+2 times), and we set
an = (a (& ,6) ®... ®(a ®0)
(2k + 1 times).
Fourth case n = 8k + 6, 8k + 7.
We have On = C2 ® ... ® C2 (4k+3 times), and we set
an=a® (/3(9 a)® ...0(,c3(9 a)
(2k+1times).
Using the presentation of Clifford multiplication and the commutation relations between a, 0 and gi, 92, T, respectively, described in Section 1.3, the properties listed above are easily checked.
1. Clifford Algebras and Spin Representation
32
Summarizing, we arrive at the following table for the real or quaternionic structures in An: an
quaternionic structures
real structures
n commutes with Clifford multiplication
6,7 mod 8
n
2, 3 mod 8
n anticommutes with Clifford multiplication
0, 1 mod 8
n
4, 5 mod 8
We now ask whether, in the case of an even dimension n, the structures just constructed are compatible with the decomposition of Dirac spinors On into the sum of Weyl spinors On ®On . Let n = 8k + 2e (e = 0, 1, 2, or 3). The decomposition is defined by the operator
f = i 4k+e el ... e8k+2e = ieel
e8k+2e
For e = 1, 3, Clifford multiplication commutes with an. Since an is complex antilinear, in these cases, an anticommutes with f = ±iel ... e8k+2e For
dimensions n = 8k + 2, 8k + 6 this implies that the real (or quaternionic) structure an : An  On interchanges the summands in the decomposition On=An ED An:
an (On) C On For e = 0, 2, Clifford multiplication anticommutes with an. Thus an com.
mutes with f = ±el ...
e8k+2e,
an f = f an, and an preserves the
decomposition On = An ®On In summary, we obtain the .
Proposition. 1) The representation A k admits a real Spin(8k) equivari ant structure. 2) The representation A8k+4 admits a quaternionic Spin(8k + 4)equivariant structure.
1.8. References and exercises E. Artin. Geometric Algebra, Princeton University Press 1957. H. Baum. SpinStrukturen and DiracOperatoren fiber pseudoRiemannschen Mannigfaltigkeiten, Teubner Verlag Leipzig 1981.
H. Baum, Th. Friedrich, R. Grunewald, I. Kath. Twistors and Killing Spinors on Riemannian Manifolds, Teubner Verlag 1991.
1.8. References and exercises
33
B. Budinich, A. Trautman. The Spinorial Chessboard, SpringerVerlag 1988.
D. Husemoller. Fibre Bundles, McGrawHill 1966. H.B. Lawson, M.L. Michelsohn. Spin Geometry, Princeton University Press 1989.
A.L. Onishchik, R. Sulanke. Algebra and Geometrie, Teil II, Deutscher Verlag der Wissenschaften, Berlin 1988.
Exercise 1. Let (V, Q) be a nondegenerate quadratic form. Determine all elements in the Clifford algebra C(Q) which anticommute with every element of V.
Exercise 2. Let (Vi, Ql) and (V2, Q2) be two quadratic forms and f : V1 > V2 a linear map such that Q1 (VI) = Q2 (f (v1))
for all vectors vi E V1. Prove that there is a homomorphism C(f) : C(Q1) > C(Q2) of Clifford algebras such that the diagram C(Q1)
C(Q2)
commutes.
Exercise 3. In the text, the equations C1 = C, C2 = IHI were proved. Prove the following additional isomorphisms: ED
C6 = M8 (I),
C4 = M2 (H),
C5 = M4 (C)
C7 = M8 (I) ®M8 (R),
C8 = Ml6 (R)
Exercise 4. Prove the isomorphisms
Ci=I®R, C2'=M2(I), Cs=M2(C), C4=M2(H), = M2(H) ®M2(H),
Cs
= M4(H), Exercise 5. Prove the equation Ck_1 = Ck. C5
C81=M8(C)
Hint: If Ck_1 = Ck1 6 C1k1 is the decomposition of the algebra Ck_1 and ek E IRk the last vector, then setting
f (xo + xl) = x0 + ekxl defines a homomorphism f : Ck_1 4 Ck.
1. Clifford Algebras and Spin Representation
34
Exercise 6. Prove that Spin(3) Spin(4) Spin(5) Spin(6)
SU(2) = {q E IHI : IIgII =1},
SU(2) x SU(2), Sp(2), SU(4).
Exercise 7. Prove that the group Spinc (4) is isomorphic to the following subgroup H of U(2) x U(2): H = {(A, B) E U(2) x U(2) : det (A) = det (B)}.
Exercise 8. Let el,...
, en be an orthonormal basis of Rn and A. : spin(n) so(n) the differential of the 2fold covering .1 : Spin(n) + SO(n). Prove that for every element z E spin(n) the following equation holds:
z = 2 (A*(z)ez, ej)ezej = 4 (A*(z)ei, ej)ei, ej. i<j
z,j
Chapter 2
Spin Structures 2.1. Spin structures on SO(n)principal bundles Let X be a connected CWcomplex and let (Q, ir, X; SO(n)) denote an SO(n)principal bundle over X.
Definition. A spin structure on the principal bundle Q is a pair (P, A) where
a) P is a Spin(n)principal bundle over X, b) A : P > Q is a 2fold covering for which the diagram
commutes. Here the rows contain the action of the respective group on the corresponding principal bundle.
Definition. Two spin structures (PI, Al) and (P2, A2) are called equivalent if there exists a Spin(n)equivariant map f : Pl > P2 compatible with the coverings Al and A2:
Pl f P2
/2
Al
Q 35
2. Spin Structures
36
Denote by F a fibre of the SO(n)principal bundle Q. F is diffeomorphic to the group SO(n) and, since n > 3, the fundamental group irl(F) consists of two elements: 7r1(F) = 7L2.
Let a E 7rl (F) be the nontrivial element. Denote by i : F ; Q the embedding of F into the space Q. Then
aF := i#(a) is an element of the fundamental group 7r1(Q). Consider a spin structure A : P ; Q on Q. To this covering there corresponds the subgroup
H(P,A) := A#(ir1(P)) C ir1(Q) H(P, A) C 7r1 (Q) is a subgroup of index 2.
Proposition. The element aF does not belong to H(P, A), aF ¢ H(P, A).
Proof. Suppose that aF E H(P, A). Then the inclusion i : F 3 Q lifts to a continuous map I : F  P such that the diagram
P A
F zQ commutes. I(F) C P is contained in one fibre F' of the Spin(n)principal bundle P, and hence we obtain a map
I : F = SO(n) f F' = Spin(n)
with
A o I = Idso(,,,)
For the induced homomorphisms of fundamental groups this implies A#I# _ Id,.1(so(,,,)) Since 7r1(SO(n)) = Z2 and 7r1(Spin(n)) = 1, this is a contradiction.
Proposition. The equivalence classes of spin structures on an SO(n)principal bundle Q over a connected CWcomplex X are in bijective correspondence with those subgroups H C irl (Q) of index 2 which do not contain aF,
aF 0 H. Proof. Let a subgroup H C 7r1(Q) of index 2 with aF 0 H be given. This subgroup defines a 2fold covering
2.1. Spin structures on SO(n)principal bundles
37
with connected total space P. Fix a point po E P and denote by y Q x SO(n) * Q the action of the group SO(n) on Q. The map
P
P x Spin(n)
A
AxA
QxSO(n) µ Q induces the homomorphism p# o (A x A)#,
71(P) = 7rl(P x Spin(n))
(n
#
7ri(Q) ®7l(SO(n))
N#
i1(Q),
the image in 7rl (Q) of which coincides with H. Hence there exists a unique
continuous map µ : P x Spin(n) > P with µ(po, 1) = po fitting into the commutative diagram P x Spin(n) µ P IAxA
IA
QxSO(n) µ Q It is easy to show that µ is an action of Spin(n) on P. For example, the map f (p) = µ(p, 1) : P * P is the lift of the map A : P > Q,
P...f.. P QlA
IA Id
Q
with initial condition f (po) = po. Uniqueness of the lift then implies f = Idp, i.e.
µ(p,1)=p
for all p E P. It remains to be proved that Spin(n) acts simply transitively on each fibre
of the map 7r o A : P + X over X. To show this, it suffices to check that (1) E Spin(n) does not have any fixed point. Suppose that
µ(po, (1)) = Choose a path 'y(t) (0 < t < 1) from 1 to (1) in Spin(n). Then y*(t) _ µ(po, 7(t)) is a loop in P; hence it defines an element of the fundamental group irl (P). The equivalence class of the loop Ay* (t) thus belongs to the subgroup H C ,7rl (Q) of the covering A P + Q. On the other hand, Ay* (t) = AA(po, y(t)) = A(A(po), A o 'y(t))
2. Spin Structures
38
Now A o y(t) is a loop in SO(n) representing the nontrivial element a E 7rl(SO(n)). Finally, for the fibre F = \(po) SO(n) C Q we obtain the inclusion aF = Ay* (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 nontrivial homomorphism
fH : 7r1(Q)  iri(Q)/H = Z2 and vice versa. The condition aF
H is equivalent to requiring that the
homomorphism fH 0 i# : 7L2 = 7r1(F)
ir1(Q) ' 7r1(Q)/H = Z2
is the identity. Hence we obtain the
Corollary. The spin structures on an SO(n)principal bundle Q over a connected CWcomplex X are in onetoone correspondence with the homomorphisms splitting the sequence (*):
f : 7r1(Q) ' 7r1 (F)
and
f o i# = Id,,(F).
Corollary. If the SO(n)principal bundle has a spin structure, then the following groups are isomorphic: a) 7r1(Q) = ir1(F)
7r1(X),
b) 7r2(Q) = 7r2(X)
Corollary. Let X be a simply connected CWcomplex. An SO (n) principal bundle Q has a spin structure if and only if 71 (Q) = Z2.
In this case, the spin structure is uniquely determined.
The last corollary can be generalized using the extension theory of finite groups. The preceding considerations show that an SO(n)principal bundle Q over a CWcomplex X has a spin structure if a) i# :7r1(F) = Z2 + 7r1(Q) is injective;
b) the exact sequence splits.
1 + 7ri (F) #) 7r1(Q)  7ri (X)  1
2.1. Spin structures on SO(n)principal bundles
39
We now suppose that irl (X) is a finite group and, moreover, that condition a) is satisfied. Then 7r1 (Q) is an extension of the group Z2 = 7r1 (F) by 7r1(X). Recall the notion of a 2Sylow subgroup of a finite group G. If G1 denotes the order of the group G and 2k is the largest power of 2 dividing IGJ, then each subgroup of G of order 2k is called a 2Sylow subgroup. It is wellknown that there exists at least one such subgroup in every group G. If
1iZ2>G*IF >1 is an extension of Z2 by r, then we have the following criterion for the splitting of this extension.
Proposition (SchurZassenhaus, Gaschiitz). The extension G of the group Z2 by the finite group r splits if and only if for every 2Sylow subgroup G2 C G the extension
1> splits.
Proof. Compare for example R. Kochendorffer, Lehrbuch der Gruppentheorie unter besonderer Beriicksichtigung der endlichen Gruppen, TeubnerVerlag, Leipzig 1966, or the translation, Group theory, McGrawHill, 1970.
A first conclusion is that for 7r1(X) finite, the principal bundle Q admits a spin structure if and only if the following conditions are satisfied:
a') i# : 7rl (F) > ir1(Q) is injective; b') for each 2Sylow subgroup G2 C 7rl (Q), the corresponding extension
1 p (G2nZ2) ) G2 pG2/(G2nZ2)
1
splits.
If G2 C 71 (Q) is a 2Sylow subgroup of 7r1 (Q), then G2 = p# (G2) C 71 (X)
is a 2Sylow subgroup of 7r1 (X) and vice versa. Thus, if every extension of Z2 by G2 splits (or, equivalently, H2(G2; Z2) = 0), then condition b') is automatically satisfied. We obtain the Corollary. Let X be a CWcomplex with finite fundamental group satisfying the condition H2 (G2; Z2) = 0
for every 2Sylow subgroup G2 C 7r1(X). Then an SO(n)principal bundle Q over X has a spin structure if
i# : 7rl (F) * in1(Q) is injective.
2. Spin Structures
40
In particular, this corollary applies in the case where the fundamental group 7r1(X) is of odd order. The only condition for the existence of a spin structure then consists in requiring the injectivity of the homomorphism i# : 7r1 (F) _k 7ri (Q)
We will now reformulate the description of spin structures thus obtained in the language of cohomology. For every CWcomplex Y we have H1(Y; Z2) = Hom (Hl (Y); Z2)
= Hom (7r1(Y)/[7r1(Y), 7r1 (Y)] ; Z2) = Hom (7r1(Y) ; Z2)
Hence the homomorphism f : 7r1 (Q) > 7r1 (F) = Z2 defines an element in
the cohomology group, f E H'(Q; Z2). On the other hand, H'(F; Z2) = Hom (Z2, Z2) = Z2, and the condition f o i# = Id,, (F) is equivalent to the requirement that the element f E H1 (Q; Z2) remains nontrivial after restriction to the fibre i* : H1 (Q; Z2) > H1 (F; Z2) = Z2 . From this we obtain the Proposition. The spin structures on an SO (n) principal bundle over a connected CWcomplex X are in onetoone correspondence with those elements f E H1 (Q; Z2) for which i*(f) 0 holds in H1 (F; Z2) = Z2 
The SO(n)fibration 7r : Q + X induces the following exact sequence of cohomology groups:
1 ) H' (X; Z2) ' H1(Q; Z2)  H' (F; Z2) = Z2
H2 (X; Z2)
* .. .
If 1 E Z2 = H1(F;Z2) denotes the nontrivial element, then w2(Q) := 8(1) E H2(X; Z2)
is called the second Stiefel Whitney class of the SO(n)principal bundle. The above sequence, moreover, immediately implies the
Proposition. An SO(n)principal bundle over a connected CWcomplex X has a spin structure if and only if w2 (Q) vanishes, W2(Q) = 0.
In this case, the spin structures are classified by H' (X ; Z2) .
Example. Consider the complex projective space Cpn. The group SU(n + 1) acts transitively on CP :
cpn = SU(n + 1)/S(U(n) x U(1)). The isotropy representation a
: S(U(n) x U(1)) > U(n) C SO(2n)
2.1. Spin structures on SO(n)principal bundles
41
is given by the formula
(B
0
l
I
= det B B.
det B
Let R = SU(n + 1) x, SO(2n) be the frame bundle. As CIPh is simply connected, the fundamental group iri (R) has at most two elements; it is the surjective image of iri(SO(n)) = Z2. In order to decide whether R admits a spin structure, we first compute the homomorphism o# between fundamental groups induced from the isotropy representation a. The generating element of the fundamental group iri(S(U(n) x U(1))) is represented by the loop
(eit 1
0
0 Z2 = 7r1(SO(n))
+1
splits. Considering the covering A : Spin(n + 1) * SO(n + 1), we obtain
F\SO(n + 1) _ .\1(F)\Spin(n + 1) as well as 7r1(A1(r)\Spin(n+1)) _ A' (r). Hence the sequence in question is 1
) Z2
) A1(r)
)r
)
1,
and the manifold X* = r\Sn is a spin manifold if and only if this sequence splits. For a group F of odd order JFI this sequence always splits. Otherwise, the splitting of the above exact sequence is equivalent to the splitting of the sequence 1
, (A1(F2) n Z2)
; A1(F2)
> F2
) 1,
where F2 C F is a 2Sylow subgroup. Summarizing, we obtain the
Proposition. Let F C SO(n + 1) be a finite subgroup acting freely on the sphere S. The manifold F\Sn has a spin structure if and only if for every 2Sylow subgroup r2 C F the sequence
1 f (.\'(r2) n z2) _, \1(F2) > F2 > 1 splits.
Let us consider the case n = 4k + 1 = 1 mod 4 separately.
Proposition. Let n = 4k + 1, and let r C SO(n + 1) be a finite subgroup acting without fixed points on the sphere Sn. Then F\Sn has a spin structure if and only if r contains no elements of order 2. In this case, H1(F\Sn; 7G2) = 0,
and hence the spin structure of F\Sn is unique.
Proof. Suppose that r does not contain any elements of order 2. Then F has no proper 2Sylow subgroups, too, i.e. on F\Sn there exists a spin structure. Let A E F be an element of order 2. Since (Ax, Ay) = (x, y) for x, y E Rn+1 and A2 = 1, the matrix A is symmetric: (Ax, y) = (x, Ay). Hence A is diagonalizable with eigenvalues Al = ... = An = ±1. If 1 is an eigenvalue, then A has a fixed point on the sphere. This implies A = E.
2.3. Spin structures on Gprincipal bundles
45
Set ro = {E, E}. This leads to the following commutative diagram of coverings:
84k+1 __ Fo\s4k+1 = Rp4k+1
r\S4k+1
If F\S4k+1 has a spin structure, then pulling it back we construct a spin structure on the real projective space 1E8p4k+1 Thus we arrive at a contradiction, since R? admits a spin structure only for n  3 mod 4. Finally, we will prove that H1(F\S'; Z2) = Hom (F; 7L2) = 1,
if F contains no elements of order 2. Actually, in this case, the order IFl
of the group F is odd, i.e. F has an odd number of elements. A nontrivial homomorphism f : F 3 7Z2 defines by means of Fo = ker(f) and any yo 0 ker(f) a partition of the set r into r = Fo U yo Fo, and hence IFI = 2IFo1  0 mod 2, in contradiction with the assumption.
2.3. Spin structures on Gprincipal bundles Let G C SO(n) be a connected compact subgroup for which, moreover, in this section, it will generally be assumed that the inclusion induces an epimorphism
i# : 7r1(G) ' 7rl(SO(n)). Then the homogeneous space SO(n)/G is simply connected, 7rl(SO(n)/G) _ {1}. Consider a Gprincipal bundle (Q, in, X; G) over a CWcomplex X. Then the associated bundle Q* = Q xG SO(n) is an SO(n)principal bundle over X. Definition. The Gprincipal bundle Q admits a spin structure if the SO(n)principal bundle Q* admits one. Now we will derive a condition for the existence of a spin structure in this sense. To do so, denote by F = G and F* = SO(n) the fibres in the bundles Q and Q*, respectively. Consider the commutative diagram
F=G
 SO(n)=F*
z
i
Q* I SO(n)/G
46
2. Spin Structures
where the maps j, 1, m are defined as follows for q E Q, A E SO(n):
j(q) = [q, l],
m(A) = A mod G.
1[q, A] = A mod G,
The row Q + Q* i SO(n)/G as well as the columns are fibrations. Hence we obtain the following commutative diagram of homotopy groups:
ir2(X)
1r2(X)
71(F*)
71(F)
ir2(SO(n)/G)
E'
7r1(Q)
3
7r1 (Q*) #
4
7rl(SO(n)/G)
Suppose that Q* admits a spin structure, and let f * : 7rl(Q*) , 7rl(F*) be a homomorphism for which 7r1(F*) 7r1 (Q*) * rrl(F*) is the identity. Then consider the homomorphism
f = f* o j# : 7r1 (Q) ' 7r1(F*) = rrl(SO(n)) and conclude that
f o6= f*oj#os= f*os*oi#

Hence the following diagram commutes: 1r1 (F)
irl (F* )
Conversely, let f with this commutative diagram be given. Define f * irl(Q*) + 7rl(F*) as follows: For each element x E 7rl(Q*) there exists an element y E 7r1 (Q) such that j# (y) = x. Set f*(x) = f(y). If yl is another element from 7r1(Q) with j# (yl) = x, then there exists an element z E rr2(SO(n)/G) such that yl = y a(z). But this implies :
f (yl) = f (y)f (a(z)) = f (y)f (sa(z)) = f (y)i#a(z) = f (y) 1 in 7r1 (F*). Thus f* : 7rl(Q*) > 7rl(F*) is uniquely defined. We still have to check the condition f* o s* = Id on 7r1(F*). Choose a E 7r1(F) with
i#(a) = a*
1 in 7r1 (F*). Then,
a* = i# (a) = f s(a) = f *j#s(a) = f *s*i# (a) = f *e* (a*).
2.4. Existence of spine structures
47
Summarizing, we obtain the
Proposition. Let G C SO(n) be a connected compact subgroup with
7r,(50(n)/G) = 1. A Gprincipal bundle Q over a connected CWcomplex X has a spin structure if and only if there exists a homomorphism f : 7rl(Q) , ir1(SO(n)) for which the diagram iri (F) = 7rl (G)
7r1(SO (n) )
I
i1(Q) commutes.
This condition can again be reformulated cohomologically. The homomorphism f defines an element f E H1(Q;Z2) = Hom (ir1(Q),Z2) whose restriction to the fibre F, i* (f) E H'(F; Z2) = Horn (7r1(G), Z2), has to coincide with i# : 7r, (G) f 7rl(SO(n)) = Z2. Hence we have the
Proposition. Let G C SO(n) be a connected compact group with 7rl (SO(n)/G) = 1.
A Gprincipal bundle Q over a connected CWcomplex has a spin structure if and only if there is an element f E Hl (Q; Z2) for which the restriction to the fibre F coincides with i#, i* (f) = i# .
2.4. Existence of spine structures Consider an SO(n)principal bundle Q with base space X. In analogy with a spin structure, we define
Definition. A spine structure on Q is a pair (P, A) consisting of a Spincprincipal bundle P over the space X and a map A : P  Q such that the diagram
P x Spin's(n)  P I
IAxA
A
Q x SO(n)
Q
commutes.
Example. Because of the inclusion i Spin(n) > Spinc(n), each spin structure on Q induces a spine structure. :
2. Spin Structures
48
Example. Suppose that the SO(n)bundle Q (n = 2k) has a U(k)reduction, i.e. there exists a U(k)bundle R with Q=R
XU(k) SO(2k).
In Section 1.6 we constructed a homomorphism F : U(k) + Spin(2k) for which the diagram Spin (2k)
F'
U(k)
SO(2k)
commutes. Hence
P := R XU(k) Spin (2k)
is a spin structure on Q. In other words: Every U(k)reduction of the SO(n)bundle Q induces a spin structure in Q. The groups Spin(n) and SI are subgroups of Spin (n) whose elements commute with each other. Hence, if (P, A) is a spin structure, then
1) PIS' is a Spin(n)/{f1} = SO(n)bundle isomorphic to Q,
P/S,=Q 2) PI := P/Spin(n) is an S'/{f1} = Slbundle over X, and the combination of bundle morphisms over the base space P + QxPI is a 2fold covering. Here QxPI denotes the fibreproduct of the SO(n)principal bundle Q with the S1 = SO(2)principal bundle PI over X. QxPI is an (SO(n) x SO(2))principal bundle over X. Because of the diagram (compare Section 1.6)
Spin (n)
Spin (n + 2)
SO(n) x SO(2) i SO(n + 2) the (SO(n) x SO(2))principal bundle QxPI admits a spin structure in the sense of Section 2.3. All in all, we obtain the
Proposition. If the SO(n)principal bundle Q admits a spin structure, then there exists an S'principal bundle PI over X such that the fibre product QxPI has a spin structure. Conversely, if such a bundle Pi with the given
property exists, then Q has a spin structure. Remark. Making use of characteristic classes, we can formulate this equivalently as follows:
a) Q has a spin structure;
2.4. Existence of spin structures
49
b) there exists an S'bundle Pl such that w2 (Q x Pi) = 0; c) there exists an S1bundle 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 StiefelWhitney class w2(Q) E H2(X; Z) is the Z2reduction of an integral class z E H2(X; Z).
Corollary. If H2(X; Z) * H2(X; Z2) is surjective, then every SO(n)bundle Q over X admits a spin structure. We will now discuss the existence question for a spin structure on an SO(n)principal bundle over a special class of base spaces X, and show that, sometimes, one can obtain a complete answer. Concerning the base space X we assume that
and 7r2 (X) is a finite group. Let Q be an SO(n)principal bundle and P1 an Slbundle over X. On the one hand, the fibreproduct QxP1 is an (SO(n) x SO(2))principal bundle over X, and, on the other, a fibration over Q with fibre Si. Together, this yields the diagram 7r1(X) = 1
7r2 (X)
la Z2ED z ce  (0,a)
Z = 7r1(Sl)  7r1(QOn
or
µ:A(Rn)®II2An
On,
is a homomorphism of the spin or spine representation, respectively, and, since
T=Q XSO(n)Rn=Px)118n, p induces a bundle morphism of the associated bundles
µ:TOS>S or
u:A(T)OS
) S.
2. Spin Structures
54
The real or quaternionic structures in On, respectively, pass on, in the case of a spin structure (not for a spin structure !), to the corresponding bundles.
The spin representations det(An) = Adim(on)(A,) and ldim(°m)/2 are equivalent where 1 : Spin (n) + S' is the homomorphism constructed above. This implies for the determinant bundle L of the spin structure the formula Ldim(S)/2
= Adim(S) (S)
and, in the case of even dimension n = 2k, we analogously obtain £dim(S±)/2 = A dim (St) (S±).
In forming the associated spin bundle S starting from a spin or spin structure, it may well happen that different spin structures lead to isomorphic spin bundles. We will now study this question in the case of a spin structure in greater detail. Let (Q, 7r, X; SO(n)) be a principal bundle. We think of Q as being given by
a) a covering X = U Uj of X by open sets Uj C X, iEI
b) a system of transition functions gzj : U2 n Uj  SO (n) such that 9ijgjk = 9ik,
9ii = 1.
Then, a spin structure (P, A) on Q is completely determined by a system of transition functions gzj : Uj n Uj * Spin(n) satisfying the conditions A 0 gij = 9ij,
gijgjk = 9jk,
g22 = 1.
Hence two spin structures (P, A) and (P*, A*) are described by maps gzj, gij*
UznUj > Spin(n) with A o gij = gzj = \ o gzj*.
Set eij = gzj [gzj*]1. Then ezj is a map from UznUj to ker (A) = Z2 C Spin(n),
ezj:UinUj>7L2, and eij ejk = eik. Thus the system of transition functions ezj defines a real 1dimensional bundle E. Since
for the spinor bundle S of the spin structure (P, A) (S* for the spin structure (P*, A*)) there are isomorphisms
S = S* OR E = S® ®(c E°,
2.5. Associated spinor bundles
55
where E° = E OR C is the complexification of E. As (±1) belongs to the centre of the Clifford algebra, this isomorphism is compatible with Clifford multiplication, i.e. the following diagram commutes:
T®S µ  S T®S .0
E
WgIdES *
®E
From these considerations we obtain, e.g., the following
Proposition. Let X be a CWcomplex whose second integral cohomology has no 2torsion. Then the spinor bundles corresponding to possibly different spin structures on an SO(n)principal bundle are isomorphic.
Proof. The complex line bundle E° is the complexification of a real line bundle, and hence, for the first Chern class, 2c1 (E°) = 0 in H2 (X; Z). Since
H2 (X; Z) has no 2torsion, this implies ci (E°) = 0, i.e. E° is a trivial bundle.
Remark. The spin structures (P, A) and (P*, A*) are described by elements f f * E H'(Q; Z2) whose restrictions to H' (fibre; Z2) = 7G2 are nontrivial. Thus, f  f * vanishes after restriction to the fibre. From the exact sequence of the SO(n)principal bundle, ,
0 > H1 (X; Z2)
) H' (Q; Z2)
H' (SO (n); Z2)
we thus conclude that f  f * is an element of H' (X ; Z2). This implies
wi(E) = f  f*, where wl is the first StiefelWhitney class of the real bundle E.
Example. Let X = II p5 be the real projective space of dimension five and Q = IE8p5 x SO(3) the trivial SO(3)principal bundle. Since Hi(R?5; Z2) = Z2 on Q, there are two spin structures (P, A) and (P*, A*). The bundle E is uniquely determined by wi (E) 0 in Hi (II81P5; 7L2). The first spin structure is trivial, and hence so is the corresponding spin bundle S. This implies for S*
S* = 2E° = Ec ®Ec. But, in general, c2(2E°) = ci(E°)2.
However, the bundle Ec is the associated bundle E° = S5 X Z2 c.
2. Spin Structures
56
If E° is trivial, then there exists a mapping f : S5 + S' with f (x) _ f (x), in contradiction with the BorsukUlam theorem. Thus E° is nontrivial, i.e.
cl(E°)
0
in
H2(RP5;7L).
Since the map Z = H2(If TP5; Z)
is injective, this implies c2 (2E°)
a > a2 E H4(II8IP5; Z)
0. Hence S* is not the trivial bundle.
2.6. References and exercises Th. Friedrich. Zur Abhangigkeit des DiracOperators von der SpinStruktur, Colloq. Math. 48 (1984), 5762. J. Milnor. Remarks concerning Spinmanifolds, Differential and combinatorial topology (in honour of Marsten Morse), Princeton Univ. Press, 1965, 5562.
R.C. Kirby and L.R. Taylor. Pin structures on lowdimensional manifolds, in Geometry of LowDimensional Manifolds, Part 2 (Durham, England, 1989; ed. by S.K. Donaldson), London Math. Soc., Lect. Note Series 151, Cambridge University Press 1990, 177242.
Exercise 1. Let RP' be the ndimensional real projective space. Prove that a) IMP' is orientable n = 1 mod 2. b) RPn has a spin structure Q be a local section of the frame bundle Q. e consists of an orthonormal frame e = (el, ... , e,) of vector fields defined on the open set U C Mn. The local connection form Ze = e*(Z) : TU > so(n) is given by the formula
Ze = E wijEij i<j where the 1forms wij denote the forms defining the LeviCivita connection,
wij = g(Vei, ej), and Eij E so(n) are the standard basis matrices of the Lie algebra so (n). Analogously, we fix a section s : U * P1 of the U(1)principal bundle and obtain the local connection form
AS=s*(A):TU>ill8. AS is an imaginaryvalued 1form defined on the set U, and e x s : U  Q x Pl
is a local section of the principal bundle Q x Pl. Let
xe
be a lift of this
section to the 2fold covering ?r : P + Q x P1. Since e
sx
wjjEjj, A'
)*(p*(Z x A)) _ (e x s)*7r*(Z x A) = (Z', A') i<j
and
T (P) Z. spin (n) = m2 ®iR d(exs) _1 dTr
T(U)
d ex^)
iii
Pw
T(Q xPi)"Z ` so(n) ®iIR
3. Dirac Operators
60
the local connection form Z x A
exs)
. exs) =
1
ZxA
2
is given by the formula 1
E wijeiej, 2As i<j
With respect to the section e x s, the section 0 E r(US) of the spinor bundle over U is described by a function' : U  On,, and its covariant derivative is computed according to the formula VA 2J=
do
1
1
S
i<j
Remark (Special case of a spin structure). If (P, A) is a spin structure on Q, then P x spin(,) Spin`C (n) is an induced spin0 structure. The U(1)bundle P1, in this case, is trivial with a canonical global section s : Mn + Pl. If we choose the connection A0 in Pl for which Ao 0 holds, all the formulas simplify correspondingly. Given a spin structure, we will simply denote the covariant derivative in the spinor bundle S with respect to this canonical connection A0 by V.
Remark (Special case of a U(k)reduction). Let n = 2k be even, and let a topological U(k)reduction R of the SO(2k)principal bundle Q be given, R C Q. This is equivalent to saying that there is an almostcomplex structure J : T(M2n) > T(M2n) which is compatible with the metric g: 9(J(tl), J(t2)) = 9(ti, t2). The lift F : U(k)  Spin0(2k) fitting into the commutative diagram Spin" (2k)
U(k) ' SO(2k) x S1 together with f (A) = (A, det (A)), A E U(k), induces a spin0 structure on Q via P = R xU(k) Spinc(2k). The corresponding U(1)bundle Pl is
Pi=RxdetS' with the 1dimensional complex vector bundle E = R X &t C. This vector bundle can also be described differently. T(M2k) becomes a kdimensional complex vector bundle by means of the almostcomplex structure J and, by construction,
E=Ak(T).
3.1. Connections in spinor bundles
61
Summarizing, we conclude that each connection A in the U(1)bundle P1 = R X det S1 (or in the vector bundle Ak (T) ) induces a covariant derivative 7A
in the spinor bundle. Particularly important is the case that a connection can be distinguished in the principal bundle P1 in an "obvious" way. This happens, e.g., if the LeviCivita connection Z reduces to the U(k)reduction R to a connection Z*. Then (M2k, g, j) is a Kahler manifold. In this case, Z* in turn induces a special connection AO in the associated bundle Pi = R Xdet Si, and we again obtain a distinguished covariant derivative which only depends upon the geometry of the base space. Now we return to the general situation and consider two connections A and A' in P1. The difference A  A' is an iI valued 1form on the manifold M' which will be denoted by 11:
AA'=ri. From the local formula for the covariant derivative VA we immediately conclude that 1
2ri(X). for all spinor fields E F(S) and all vectors X E T(MT). A gauge transformation f : P1 > Pi of the U(1)bundle P1 is described by a mapping p f : M7z > S1 satisfying
f(pi) = Pi of (7r(pi)) , where 7r : Pi j Mn is the projection in the bundle. The connection f *(A) is given by
f*(A) =A+7r*A*(O) with the MaurerCartan form O = zx of the group U(1) = S'. This implies the formula V f *(A) %
 Ox
1
=2
dpµ X
f
for the corresponding covariant derivatives.
We now turn to the description of the curvature form of the connection Z x A. Let SZZ : TQ x TQ > so (n) be the curvature form of the LeviCivita connection with the components
OZ=EQijEzj, Qij:TQxTQ>IR. i<j
The curvature !QA as a 2form on P1 is simply QA = dA. The commutative diagram defining the connection Z x A immediately implies the formula QZXA
= 2 1: 7r*(SZtij)eiej ®27r*(dA) i<j
3. Dirac Operators
62
The 2form dA is a form on the base space MI. From the general equation DZDZQf = p*(fZ)q, for the 2fold absolute differential with respect to the connection Z we obtain the formula VA(VAO)
= 2
EQijeiej 0+
2dA
i<j
Here e = (ei, ... , eI) is a local orthonormal frame and (S1 )e = e* (Qz) _ Ei<j 52ijEij defines the corresponding components of the curvature form of the LeviCivita connection. Using the structure equations of the Riemannian space (or, more generally, the formula for the curvature form, Q Z = dZ + 2 [Z, Z]) we can express the 2forms Iij in terms of the forms wij = g(Vei, ej) of the LeviCivita 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 Gprincipal 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 2form p. (AZ) on the manifold with values in the endomorphisms of the associated
vector bundle V = P x,, Vo. On the other hand, a section 0 of this bundle can be identified with a function 0 : P * Vo obeying the transformation rule 0(p g) = p(g')O(p). Then DZcb is a tensorial 1form of type p and hence a 1form on the manifold M with values in V. The induced covariant derivative in V is thus given by '7XO
= DZq(X*),
where X* is a (horizontal) lift of X. The equation
RZ(X, Y) =VXVZOVZVZ0VZ
y] 0
determines the curvature tensor Rz, which is a 2form with values in End(V), too. The curvature form and the curvature tensor are related by the following wellknown formula.
Lemma. One has the identity Rz = P. (Qz) Proof. To prove this, consider vector fields X, Y on the manifold M and denote by X*,Y* the corresponding Zhorizontal lifts. The section VZO is given by DZc(Y*) = do(Y*) +Z(Y*)o = do(Y*). An analogous calculation shows that V Z V ZO coincides with X *Y*O, and thus
X*Y*(0) Y*X*(0) = [X*, Y*] (O)  [X = [X*, Y*]vertw.

[X,Y]hor(cb) *, Y*]hor (0)
3.1. Connections in spinor bundles
63
On the other hand, the structure equation of the connection immediately implies [X*,Y*]vert = Z[X*,Y*] _ SZ(X*,Y*). Hence,
Finally, if W E g is an element of the Lie algebra and IV the corresponding fundamental vector field, then W (0) (p)
cb(p.etW)O(p) = lim t*o t (P(et,)  1)
O(p) =
to
This implies the formula we wanted to prove, RZ(X, Y) 0 = P* (Q (X, Y))
Applying this to the components of the LeviCivita connection, we obtain SZij (X, Y)
I:Rklij0,k(X)a,(Y)
= (f2Z(X, Y)ei,, ej) = (R(X, Y)ei, ej) = k,l
9
E Rijkl (dk A a,l) (X, Y), k,l
where ad, ... , vn is the frame dual to el, ... , en. Thus we arrive at the local formula for the curvature form SZZxA of the connection Z x A, QZxA
= 4i<jE
RijklO' k A o.1
eiej + 2 dA,
k,l
and the 2form VAVA with values in the spinor bundle is calculated as follows:
VAVA,p = Here DADA
1
4
Rijklak A o'
i<j
k,l
eiej 0 + 12 dA
is the 2fold absolute differential of the spinor field V) and hence
a section of I'(A2 (& S). Starting from this 2form with values in the spinor bundle, we construct a spinorvalued 1form HA by a suitable contraction:
Definition. For a vector X E T(Mn) the 1form HA is defined by n
H,A(X)
Then the following holds.
(VAVAO)(X,ea)
3. Dirac Operators
64
Proposition. Let Ric : T(MT) > T(Mn) be the Ricci tensor of the Riemannian space considered as a symmetric endomorphism of the tangent bundle. Then one has the relation 2 Ric
H,p (X)
(X) z + 2 (X idA)
Proof. We are going to use the formula for V AV AO stated above, and we have to prove the following two relations: n
1) E ea (dA(X, ea)) V = (XidA) 0, a=1
2) E > E Rijklo'k A Qi(X, ea)eaeiej a i<j k,l
= 2Ric(X)
V).
The first one is trivial since the sum Ea=1 dA(X, ea) ea represents the decomposition of the form X_jdA with respect to the basis e1, following calculation takes place in the Clifford algebra Cn: Rijklok Aol(X,
...
, en. The
ea)eaeiej
a,k,l
i<j
Rijalo1(X)eaeiej a,k,i<j
a,l,i<j
= 2 E RijkaQk(X)eaeiej a,k,i<j
= 2 E Rijkick(X )ej + 2 k,i<j
2
k,i<j
2 E Rijkiok (X)ej + 2 i,j,k
E RjkaUk(X)eaeiej k,i<j
a54i , j
Rjkao'k(X)eaeiej k,i<j
a0i,j
2Ric(X) + 2
RjkaQk(X)eaeiej. k,i<j a9`i,j
However, the second summand vanishes. Namely, for a fixed index k this sum contains the Clifford product epeger, p < q < r, precisely three times, for the triples (a, i, j) = (p, q, r), (q, p, r) and (r, p, q). Hence the coefficient at epeger is proportional to the sum Rqrkp  Rprkq + Rpgkr = Rqrkp + Rrpkq + Rpqkr = 0
(Bianchi identity for the curvature tensor).
Remark. Taking into account that (VAVAO) (X, ea) = RS(X, e,,)0
= oX o a  V a0X  [x ea]
,
3.1. Connections in spinor bundles
65
the formula
e« (VAVAV)(X, e«) _ 2Ric(X)O + 1(XidA)
V)
can also be written as n
RS(X, e«) _ 2Ric(X)o + (XidA) 2
where RS(X,Y) = oXoY in the spinor bundle.

VYVX

V
is the curvature tensor
A spinor field 0 E F(S) is called VAparallel (or simply parallel) if VA 1 = 0. Since
x2 =
(vXO, ' ) +
) = 0,
(V), V
the length of a parallel spinor field is constant. In the following we suppose that 0 does not vanish identically. DAB = 0 implies V AV Ab = 0, and hence the 1form H,, vanishes identically. Thus, for every vector X E T(Mn),
Ric(X) 0 = (XjdA)
V).
Ric(X) is a real vector while X_jdA is purely imaginary. For the evaluation of the last equation we need the following
Lemma. Let 0 E On be a nontrivial spinor and Z1, Z2 E I(Sn two real vectors. If
(Z1+iZ2) '=0, then for the vectors Z1, Z2 1) IZ1I = IZ2I, 2) (ZI, Z2) = 0.
Proof. Multiply the equation (ZI +iZ2)0 = 0 once again by (Zi +iZ2). In the complexified Clifford algebra Cc,
(Z1 + iZ2)(Zl + iZ2) = Zl  Z2 + i(ZIZ2 + Z2Z1) = {IZ1I2 + IZ2I2} + i{2(ZI, Z2)}. From (Z1 +iZ2)(Zl + iZ2)' = 0 and O 0 we then obtain Z112 = IZ2I2 as well as (Z1, Z2) = 0.
Consequently, the condition Ric(X) ip = (XjdA) 1)
Ric(X)I = I z (X_jdA)I for all X E T(MT)
,
2) (Ric(X), (X_jdA)) = 0 for all X E T(M').
b implies
3. Dirac Operators
66
For the sake of brevity we denote by S the symmetric endomorphism S(X) _ Ric(X), and by A the antisymmetric endomorphism A(X) = 2 (XjdA) of the tangent bundle. The second equation implies
(S (X), A(X)) = 0 for every vector X. Inserting X + Y now leads to
(S(X), A(Y)) + (S(Y), A(X)) = 0 and hence
(X, SA(Y))  (X, AS(Y)) = 0.
Thus, SA = AS, i.e. A and S commute. Hence, at a fixed point m E Mn, A and S can be diagonalized simultaneously: A has the form o
WI
W1
0
0 o
W2
W2
0
A= 0
Wk
Wk
0 0
0
0
and S is similar to Al
0
S= t 0
An l
The condition I S(X) I = A(X) 1, X E T, now leads to the equations 1\1 = A2 = ±W1,
...
,
A2k1 = A2k = ±Wk,
A2k+1 =
... _ An = 0.
For the lengths of the endomorphisms, n IIS112
= tr(S o ST) = tr(S2) i=1 k
IIAII2
= tr(AAT) = tr(A2) =
2Tw2, i=1
3.2. The Dirac and the Laplace operator in the spinor bundle
67
we then obtain IIRicHI = IIAII.
However, the square of the length of A as an antisymmetric mapping is twice the square of the length of the 2form dA, IIAI12 = 211dAII2.
To summarize:
Proposition. Let (Ma, g) be a Riemannian manifold with a spine structure and A a connection in the U(1)principal bundle P1 induced from this spine structure. VA denotes the induced covariant derivative in the spinor bundle. If there exists a VA parallel spinor,0, VAO = 0,
then the following necessary conditions are satisfied: 1) JIRic1I2
= 2IIcAII2, where S1A = dA is the curvature form of the
connection,
2) rank(QA) = rank(Ric), and 3) the endomorphisms Ric and QA of the tangent bundle commute.
Corollary. Let (Mn, g) be a connected Riemannian manifold with a fixed spin structure. V denotes the canonical covariant derivative in the spinor bundle s (A = 0). If there exists a nontrivial parallel spinor 0, then the Ricci tensor of Mn vanishes identically, Ric  0.
Remark. The conditions for the existence of parallel spinor fields listed above are only necessary. These conditions are not even locally sufficient compare the exercises.
Corollary. Let (M', g) be a connected Riemannian manifold which admits a spine structure. If the Ricci tensor has odd rank at least at one point, then Mn has no VAparallel spinor fields for any spine structure and for any connection A in the U(1)bundle of the spine structure.
Remark. In 1997 A. Moroianu studied the classification of Riemannian manifolds with parallel spine spinors in greater detail.
3.2. The Dirac and the Laplace operator in the spinor bundle We start from a Riemannian manifold (Mn, g) with a fixed spine structure and a connection A in the U(1)principal bundle P1. These induce a covariant derivative VA : r(s)
) r(T* 9 s) = r(T (9 s)
3. Dirac Operators
68
in the associated spinor bundle S. Here and in what follows the cotangent bundle T* of Mn and the tangent bundle T will be identified by means of the metric g. Clifford multiplication and the Hermitian metric (,) in S behave as follows with respect to the covariant derivative VA (X, Y E r(T), 01, 2 E
r(S)) : VA((X./1 )=(DYX)'/1'0+X.VA
,
1,VXY'2)=X(/1,02) As in every vector bundle with connection, we can define the Laplace operator A in the bundle S.
E r(S) is a spinor field,
Definition (Laplace operator on spinors). If then 0 (') Lis defined by n
AA( )_
ve

e
n
div(ei)Ve
Using Stokes' theorem one then derives  as for every Laplace operator in a Hermitian vector bundle  the formula f (AA (01), 02) =
/'
J
/ (VAW1, VAW2)
_
Mn
f
/
/'
(01, DAMn
(02))
Mn
for two spinor fields 01, 02 with compact support contained in the interior of the manifold Mn. Here, (VAV)1, VA 02) is the scalar product on 1forms, i.e. n
(DA
E(V 1, DA02) = i=1
1,
VA
e 02)
The Dirac operator in turn results from the composition of the canonical derivative with Clifford multiplication.
Definition (Dirac operator). The composition
DA = µoVA:
r(s)
) r(T*(9 S) = r(T(9 S) r (s)
is called the Dirac operator. With respect to a (local) orthonormal frame e = (el, ... , en) on the manifold Mn, n
ei De
DA's _ i=1
Obviously, DA is a first order differential operator. Moreover, DA is an elliptic operator. Its symbol o(DA) (X) : S + S, for a vector X E T, is given by Clifford multiplication:
a(DA)(X)(') = X 0.
3.2. The Dirac and the Laplace operator in the spinor bundle
69
This immediately follows from the formula n
DA(f 0) =
ei De (f
n
)=
i=1
ei
{df (ei) 4' +
f7e}
i=1
= grad(f) V)+ fDA(') We compute (DA'O, 01) : n
n
(,7
/,
(DAb, 0l) _ (ei Ve O, 4'1) i=1

e Y' , ei
/'
'Y1)
i=1 n
/
/
/1
/
{ei(`b, ei Yb1)  (Y , (Ve,ei) 4'1)  (Y , ei De 4 1)} i=1 n
n
ei(4', ei
01) 
i=1
div(ei)(0, ei 01) + (0, DA01) i=1
Considering the 1form MO,01(X) = (V), X b1), we see that the first two summands are just its divergence, 6M"""1. Thus, (DA0, 01) = (V, DA'b1) + JMO>01
This implies that the Dirac operator is a symmetric operator with respect to the L2product.
Proposition. Let
and ?P1 be spinor fields with compact support (contained
in the interior of the manifold). Then,
f (DAO, 01) = f ( DAbl). ,
Mn
Mn
Remark. If the dimension n = 2k is even, then the spinor bundle splits into the sum S = S+ ® S of Dirac spinors. Since Clifford multiplication by vectors interchanges these summands, the Dirac operator decomposes into the sum of two operators, DA : F(St) * r(SF).
Now we want to discuss a third operator acting on spinor fields, the socalled twistor operator TA. To this end, we need several preparations: First, Clifford multiplication µ : T ® S + S is a surjective homomorphism. Let ker(a) C T ® S denote its kernel.
Lemma. The formula n 1
P(X(94b) =X®0+ n i=1 defines a projection from the bundle T ® S onto the bundle ker(a) C T ® S.
3. Dirac Operators
70
Proof. A straightforward calculation shows that the image of P is contained in ker(A):
X
0.
i=1
Analogously, one shows that P acts as the identity on ker(µ).
Definition. The twistor operator TA = P o VA is defined as the superposition of the covariant derivative with the projection onto the kernel of Clifford multiplication,
TA : F(S) > F(ker(/2)). n
From DAB _
ei ® V AO we obtain the following formula for TA:
i=1 TA(b)
ei ®De +
= i=1
1
n
ei ®ei DA(Y') i=1
n 1
DA()) ei ® oe + ei n Z
Corollary. A spinor field 0 E F(S) belongs to the kernel of the twistor operator TA if and only if, for every vector X E T, 1
+ nX DA(s) = 0.
Vx
Example. Consider a2 with its Euclidean metric and coordinates x, y. is an orthonormal frame. For the forms wii of Then e1 = j, e2 = the LeviCivita connection we have wi,7 = 0. A spinor field is simply a mapR2  f A2 = C2. The covariant derivative V coincides with the ping differential do, since wi.7 = 0. Contrary to the convention valid up to now, :
this time we will employ an equivalent though different realization of the Clifford algebra described by the matrices e1 =
If
=
( 01 0)'
e2=
(0 o)
(f) : a2 * C2 is a spinor field, then of
of 0
1
DO _ (1 0)
ax a ax
+
0
i
ay
\i
0
a
af
ay
az
2
3.3. The SchrodingerLichnerowicz formula
71
where a
a
1
a
a
a
a
1
az2 axZayaz2Cax+zay The kernel of the Dirac operator (DV) = 0) thus consists of the pairs of complexvalued functions f, g : R2 > C which satisfy the CauchyRiemann equations
ofag=0. az az (BA)
Fix a vector by
E C2 and consider the spinor field
(x'y) = x (01 0)
i
(BA)
0/ \B/
+y \ .
Then
0)
B1
A)
ax
: R2 __+ C2 defined
(iA J
ay
and hence
D( ) = ( O1 0)
ax +
(0i
i
0'
90
= 2 I B
I
We want to show that 0 is a solution to the twistor equation. To this end, we have to check that a'O
ax+2 ( 1 0) D(O) =0
and y
+2
I
\
0
i)
0.
0
But this immediately follows from
ax+2 (01 ao
,
0
) D(
)=
1 (0 i D(O) = i 0)
ay 7 2
1
( A)  (01 0) (B) 0, iB 0 i ) (A)= 0. iA
i
0
B
3.3. The SchrodingerLichnerowicz formula The square DA of the Dirac operator as well as that of the Laplace operator DA are second order differential operators. We compare these operators computing their difference D29  OA:
3. Dirac Operators
72
DA  AA'O
E ei De (ej VA O) + EVA VA e ij i AV) Eei {(Deyej) V + ej Ve
+ E div(ei) 7 o Ve.,0}
i, j
b+ Eeiej V eV
E g(Deiej, ek)eiek V
i,j, k
)VAI
+ EVe De O +
e
+E DeVe + 'div(ei)Ve z i
EE g(Vei ej, ek)eiek' Ve + >eiej . De Ve j ilk
ij
The latter of these equations is a consequence of the definition of the divergence, i.e.
j:g(Veiej,ek)eiekVej211 = 
j
div(ej)DejO.
i=k
Now rewrite the following endomorphism:
57 g(ej, V e2ek)eiek
Eg(Veiej, ek)eiek
i0k
i56k
 E g(ej, V eiek  V ekei)eiek i S is Clifford multiplication by the vector X. Without loss of generality we choose the Hermitian basis el, J(e1),... e2k1, J(e2k_1) with X = el. Now compute k
k
v(a1 DAa) (X) : E A0,r ®So  } E A0,r ®So. r=0
r=0
To this end, first decompose a given (0, r)form 77O,r into 77O,r
el_Jr7*0,r
= (e1 + ie2) A r70,r1 + 77*0,r with
= e2.J
*O,r
77*o'r
V)o
= 0.
Then, U(DA) (X)a(T1o'r
=
/2/2
22
®,00)
= 21 el .
70,r . 00
el (e1 + ie2) r7o r1 'po +
2r/2 el
(1 + iele2) 71°'r1 'bo + 2 (e1 A 7,*O,r)
Clifford multiplication by e1e2 commutes with
770,r', and
. 00
e1e200 = i00
Moreover, e1Jr7*O,r = 0. Hence, o(DA) (X)a(770'r (& 00)
770,r1 00
2222
+
2
{(el
+ ie2) A 77*O,r} 00.
3.4. Hermitian manifolds and spinors
81
Applying a1, we conclude that a1o(DA)(X)a(?7"' ®00) I
_
V'2_77 O,rI + v'2
(_X1Or +
(el + ie2)
A 17*0'r) ®1P0
2
J
11
(X +2JX) A770,r/ ®,00
Thus the symbol of a1DAa is computed. For the symbols of aAo and aAo we have
a(5A0)(X) _ o(a)(X) ®Ids0,
o,(5A0)(X) _ c(a*)(X) ®Ids0,
and the mappings o(a)(X) : A°,r + AO,r+l and v(a)*(X) : AO,r  AO,r1 respectively, are given by 0.(5)(X)(771'r)
=
X +2 JX
A77O,r'
te(a)*(X)(rl°'r) _ XJ77°,r.
Consider the special case of a Kahler manifold (M2k, j, g). If Q is the bun of SO(2k)frames and R its U(k)reduction, then the LeviCivita connection
reduces to the U(k)principal bundle R. Choosing, furthermore, the anti
canonical spin structure, we know that So = 81, and L = Ak(T*) = R X (det)1 C. Thus the LeviCivita connection induces a connection A in L.
As the connection AO in So = 191 we take the trivial one. In this case, the corresponding Dirac operator D just agrees with ,l'2(a + a*):
Proposition. Let (M2k, J, g) be a Kahler manifold with the anticanonical
spin structure. Then, 1) S
AO,O +
... + A°,k, and
2) the Dirac operator defined by the LeviCivita connection coincides with v (a + a*).
Corollary. The space of harmonic spinors, {, E r(S) : Do = 0}, of a compact Kahler manifold (with respect to the anticanonical spin structure) is isomorphic to k
E Hr(M2k;
0).
r=0
Finally, we discuss the case of a Kahler manifold (M2k, j, g) with fixed spin
structure. Then,
L = Ol,
S0
= A°'k,
Sk2
= Ak'0.
3. Dirac Operators
82
Thus Sk is a square root of the canonical bundle K = Ak(T*) = Ak,o of the Kahler manifold, and the spinor bundle is isomorphic to
S= (A0,0+A0,1 +...+AOk) ®Sk. Choose the trivial connection A in L = O1. As Sk = Ak,0 the connection in Sk is induced from the LeviCivita connection. Summarizing, we have for the corresponding Dirac operator D the Proposition. Let (M2k, J, g) be a Kahler manifold with fixed spin structure. Then the following assertions hold: 1) Sk is a square root of the canonical bundle, Sk = K = Ak,o k
2) The spinor bundle is isomorphic to S = > AO,r ® Sk. r=0 3) The Dirac operator coincides with v"2(6 + 8*).
Proposition. The space of harmonic spinors, {0 E I'(S) : DV = 0}, on a compact Kdhler manifold with spin structure is isomorphic to k
Hr(M2k, O(Sk)), r=0
where Sk is a line bundle for which Sk = K = Ak,o (describing the spin structure).
3.5. The Dirac operator of a Riemannian symmetric space Consider a Riemannian symmetric space Mm with isotropy group G. If K is the isotropy group of a fixed point mo E Mm, then Mm can be identified with the homogeneous space G/K. The Lie algebra g of the group G splits into
g=t+m and the following commutation relations hold: [e, f] C e,
[t, m] C m,
[m, m] C f.
Moreover, m is an Ad(K)invariant subspace of the Lie algebra g, Ad (k) (m) C m
for
k E K.
Let (,) be a scalar product in the vector space g which is positive definite on m and has the invariance property ([X, Y1, Z) + (Y, [X, Z]) = 0
3.5. The Dirac operator of a Riemannian symmetric space
83
for all X, Y, Z E g. This scalar product defines a Riemannian metric on Mn which we also want to denote by (,). Right and left translations in the group G will be denoted by Rg and Lg, respectively, R9(91) = gig,
L9(91) = 991
The projection 7r : G > G/K = Mm is a Kprincipal bundle. This principal
bundle has a canonical connection Z. The group K acts on G from the right. For X E t, the fundamental vector field of the Kaction at the point g E G is given by (9) = dt (9 etx)t=0.
But this is precisely the left invariant vector field X determined by the vector X E t. Thus the vertical tangent space of the Kprincipal bundle 7r : G + G/K at the point g E G coincides with the space dLg(t), TT(G) = dLg(t).
Define the connection in the Kprincipal 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 Kaction. However, T9 (G) = dLgdLk(m) = dRkdRkidLgdLk(m).
Each right translation commutes with every left translation, and thus Ty (G) = dRkdLgdRkidLk(m) = dRkdL9Ad(k)(m) = dRk(TT (G)).
The canonical connection in the principal bundle (G, 7r, G/K; K) as a 1form
Z : TG + t is easily described. Let O be the MaurerCartan form of the Lie group G, O : T(G) p g,
O(t) = dLg(tg).
Then Z = pre o O. Indeed, for X E t, the fundamental vector field X of the Kaction is described by the left invariant vector field X, and this implies
O(X) = X, i.e. Z(X) = X. On the other hand, the kernel of Z is exactly Th(G). We also compute the curvature form of this canonical connection:
cZ = dZ + 2 [Z, Z] = prt(d0) + 2 [prte, prt0] If, however, O = Ot + Gm is the decomposition of the MaurerCartan form, then [O, O] = [Ot, Ot] + [Ot, Om] + [Om, Ot] + [em, Om] and the commutator relations imply pre [O, O] = [O', Of] + [Om, 9m].
3. Dirac Operators
84
Thus, S2z
= pre
(de+ 2 [O, O]) 
2 [19m, am]
and the structure equation dO + 2 [O, O] = 0 of the Lie group G leads to S2
2
_  12 [prmE), prma]
Let Q be the bundle of orthonormal frames on M. Then there exists an inclusion i : G > Q such that the diagram
G
Q
G/K + Mm commutes. To this end, fix an orthonormal basis el,... , e,,,, at the point mo and define i(g) = (dly(e1), ... , dl9(e, )), where 19 : Mm > Mm is the action of g E G on Mm. Now we want to see that the LeviCivita connection reduces to the Kprincipal bundle (G, 7r, G/K; K) and coincides there with
the constructed connection Z. Note that the tangent bundle T(Mm) of Mm = G/K is T(Mm) = G XAd(K) m,
the bundle associated with the representation Ad : K * 0(m). Hence a vector field T on the manifold Mm is a mapping T : G p m with the invariance property T(gk) = Ad(k1)T(g). Z induces a covariant derivative Vz in T(Mm) = G XAd(K) m, and VZT = dT + [prf@,T].
By the invariance property of (,) this implies (OZT,T1) + (T, VZT1)
_ (dT,T1) + (T, dTl) + ([prtO, T], Tl) + (T, [prtO, T1])
(dT,T1)+(T,dT1) =d(T,T1), i.e.
Vz preserves the Riemannian metric. Analogously, one shows that Vz
is torsionfree. However, these two conditions uniquely determine the LeviCivita connection of Mm.
3.5. The Dirac operator of a Riemannian symmetric space
85
Fix a homogeneous spin structure of the symmetric space G/K, i.e. a homomorphism Ad : K  Spin(m) such that the diagram Spin(m) IA
A
K  So (m) commutes. Let n : Spin(m) f GL(A) be the spin representation. Then a spinor field 7P is identified with a function : G * A satisfying the invariance condition V (gk)
= r.Xd(k1)O(.4')
Let X be a left invariant vector field on the group G with X E f. Then, Xzb(g)
dtrcAd(etx),(s) =  *Ad*(X) (g),
=
and hence
Xzb = ic Ad(X)O = Ad*(X) 0, where Ad(X) zb is Clifford multiplication of the spinor 0 E A = A(m) by the element Ad*(X) E spin(m) C Viff (m) of the Clifford algebra. Consider the 1form DZc with values in A. Then, (DZ?G)(X) = DO(X) + r,*Ad*(Z(X))V).
For X E f, the last computation (Z(X) = X!) immediately shows that DZ'(X) = 0. For X E m we obtain Z(X) = 0, and thus DZO(X) = X (V)). Choosing an orthonormal basis Xl,... , X,,,, in m, we obtain m
DZb _
Xi ®Xi(0), i=1
and, for the Dirac operator, this implies the simple formula m
i=1
We will now compute D2: m "/' D20 =
i,j=1
m
m
i=1
i,j=1
/ /' X,.Xj . (XXj(')) _ EXi (0) '+ 2 E Xi'Xj . ([X,,Xj]o)
However, the vector field [Xi, Xj] belongs to t, and hence the above calculation shows that [Xi, Xj] (0) = Ad*([Xi, Xj]) . 0
3. Dirac Operators
86
Inserting this yields M 2
X2(0)
D2
i,j=1
i=1
Choose an orthonormal basis in the Lie algebra f, Y1, ... , Y. Then, from Y. ('b) = Ad* (Y«)
'
we immediately conclude that Y« (0) = Ad.(Ya) ' Ad* (Y.)
m and, finally, using the Casimir operator S2c
k
.qb _
= QG(V))
 > Ya of the group a=1
i=1
G, we arrive at the formula D20
k
X2
1 2
a=1
To treat the remaining term, we compute in the Clifford algebra Jif f (m) as follows. Let Ad. (Y,,) E spin (m), and let X1, ... Xm be an orthonormal basis in in. Then, ,
m
Ad, (Y,) =
4
i,j=1 m
A
E (Ad. (Y.) (Xi), Xj) Xi ' Xj
i,j=1 m
d
1: ([Y., Xi], Xj)Xi Xi i,j=1
and, analogously, m
Ad.([Xi, Xj]) = 4
([[Xi, Xj], Xp], Xq) Xp ' Xq. p,q=1
The remainder term thus coincides with 1R
E E E ([YY, Xi], Xj) ([Ya, Xp], Xq)XiXjXpXq a=1 i,j=1 p,q=1 in
in
1: E i,j=1 p,q=1
Xi,X ,X ,X XiX X X
3.5. The Dirac operator of a Riemannian symmetric space
87
Because of the invariance property of the scalar product (,) in g we also have k
T, ([Y., XiJ, Xj) ([Y., Xp],Xq) a=1
(Y., [Xi, Xj]) (Y., [Xp, XqJ) _ ([Xi, Xj], [Xp, Xq]) a=1
Hence the term under consideration simplifies to the expression
16 i,j,p,q ([Xi, Xj], [Xp, Xq])XiXjXpXq The LeviCivita connection is induced from the connection Z in the Kprincipal bundle. Thus for the curvature tensor R of the Riemannian space the following formula holds:
R(Xi, Xj) = [QZ(Xi, Xj), i.e.
(R(Xi, Xj)Xp, Xq)
([1Z(Xi, Xj), Xp], Xq) = ([[Xi, Xj], Xp], Xq) ([Xi, Xj], [XP, Xq])
In the Clifford algebra Vif f (m) = Cm, the remaining term has CO and Cmas well as C4Lparts. The Cm and C4,,parts vanish, and for the C° part we have
 d E([X" Xj], [Xi, Xjl)X'XjXiXj i<j
=4EII[Xi,Xj]II2=
8
I[Xi,Xj]112 = 8R,
where R is the scalar curvature of the space G/K. Summarizing, we arrive at the
Proposition. Let Mm = G/K be a compact Riemannian symmetric space with a homogeneous spin structure. Let SZG denote the Casimir operator of the Lie group G. Then,
D2=SlG+8R. Remark. The importance of this formula lies in the fact that it allows us to compute the eigenvalues of D2 purely by means of representation theory. Let A : G > GL(VA) be an irreducible complex representation and
3. Dirac Operators
88
A.
gt(V,\) its differential. Then, m
A*(QG) =

E '\.(X,)1 z=1
k
 a=1 E) *(Y«)2
is an operator in VA. It is easy to show that this operator commutes with all automorphisms .(g) : Va i VA (g E G). This follows immediately from the formula A(g)A*(X)A(g1) = a*(Ad(g)X), X E g, g E G, which itself is the result of a straightforward calculation using the fact that Ad(g) g > g maps the orthonormal basis consisting of {X1i ... , Xm, Y1, ... ,Yk} again onto an orthonormal basis of g. Let u be an eigenvalue of )*(c)G) and W. C V, the corresponding eigensubspace. Then W. is invariant under all .(g), g E G. The irreducibility of VA implies WN0 = 0 or VA. But WN, = 0 is excluded, as ). (cla) indeed has eigenvalues over C. Hence we have WN, = Va, i.e. A*(1 ) is a multiple of the identity.
Finally, consider the Hilbert space L2 (S) = L2 (G/K; S) of all squareintegrable sections of the spinor bundle over the compact Riemannian sym
metric space. Then G acts as a group of unitary transformations on L2. Decompose L2 into the direct sum L2(S)
_
VA
AEA
of finitedimensional irreducible Grepresentations VA, and compute each time the number c(A) with \*(I1G) = The spectrum of D2 is then given by
Spec (D2) _ {c(A) + 8R : A E A } . These computations were, e.g.
carried out in the case of spheres, cer
tain Graf3mannian manifolds, and for the complex projective spaces CP271+1 (compare the references).
3.6. References and Exercises M. Cahen, A. Franc et S. Gutt. Spectrum of the Dirac Operator on Complex Projective Space Cp2q1, Letters in Math. Phys., 18, 1989, 165176.
Th. Friedrich. Der erste Eigenwert des DiracOperators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkriimmung, Math. Nachr. 97 (1980), 117146.
Th. Friedrich. Zur Existenz paralleler Spinorfelder fiber Riemannschen Mannigfaltigkeiten, Coll. Math. XLIV (1981), 277290.
3.6. References and Exercises
89
N. Hitchin. Harmonic spinors, Adv. in Math. 14 (1974), 155. K.D. Kirchberg. Compact sixdimensional Kahler spin manifolds of positive scalar curvature with the smallest possible first eigenvalue of the Dirac operator, Math. Ann. 282 (1988), 157176. A. Lichnerowicz. Spineurs harmoniques, C.R. Acad. Sci. Paris 257 (1963), 79.
J.L. Milhorat. Spectre de 1'operateur de Dirac sur les espaces projectifs quaternioniens, C.R. Acad. Sci. Paris Ser. I 314 (1992), 6972. A. Moroianu. Parallel and Killing spinors on Spincmanifolds, Comm. Math Phys. 187 (1997), 417428.
E. Schrodinger. Diracsches Elektron im Schwerfeld I, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Phys.Math. Klasse 1932, Verlag der Akademie der Wissenschaften, Berlin 1932, 436460.
S. Seifarth, U. Semmelmann. The spectrum of the Dirac operator on the odddimensional complex projective space CP2m1, Preprint des SFB 288 "Differentialgeometrie and Quantenphysik" No. 95 (1993). H. Strese. Uber den DiracOperator auf Graf3mannschen Mannigfaltigkeiten, Math. Nachr. 98 (1980), 5358. S. Sulanke. Berechnung des Spektrums des Quadrates des DiracOperators D2 auf der Sphare and Untersuchungen zum ersten Eigenwert von D auf 5dimensionalen Raumen konstanter positiver Schnittkrummung, Dissertation, HumboldtUniversitat zu Berlin 1981. Exercise 1. Let (M4, g) be a 4dimensional Riemannian spin manifold with nontrivial parallel spinors V+, L in the bundles S+ and S, respectively. Prove that (M4, g) is flat.
Exercise 2. Prove that every parallel spinor 0+ in the bundle S+ over a 4dimensional Riemannian manifold (M4, g) induces a complex structure J : TM4 * TM4 such that (M4, g, J) is a Kahler manifold. Exercise 3. Prove that, in 2dimensional Euclidian space i 2, the general solution to the twistor equation T() = 0 is given by
tl>(x,y) _ (D) with arbitrary vectors

(XI+iY x
(BA)
02y)
(A) (C) D E CZ = A2D B ,
Exercise 4. The metric on M4 C II84,
M4 = {(xl, ... x4) E L ,
4
:x1 > 0, 0 < x2 < 7r},
3. Dirac Operators
90
defined by ds2
=
xl
(dxl)2 + xI(dx2)2 + xlsin2(x2)(dx3)2 +
xl +
xl + c xl (c > 0) is Ricciflat but does not admit any parallel spinors. Exercise 5. Let (M4, g) be a Riemannian spin manifold and
c(dx4)2
E F(S) a
spinor field. If
7XV) =w(X)'') with a realvalued 1form w, then
a) Ric  0, b) dw = 0. Prove by examples that this does not hold for general complexvalued forms.
Chapter 4
Analytical Properties of Dirac Operators 4.1. The essential selfadjointness of the Dirac operator in L2 Let us recall some notions from the spectral theory of linear operators in complex Hilbert spaces. Let A be an (in general unbounded) operator with dense domain of definition, D(A), in the complex Hilbert space H. Denote the range of A by R(A). The graph, r(A) C H x H, consists of all pairs (x, Ax), x E D(A). In the sequel we will assume that its closed hull F(A) C H x H again is the graph of an operator A which then is called the closure of A. Then A acts via the formula A(x) = rli n A(xn) and its domain of definition, D(A), consists of all vectors x E H for which there exists a sequence xn E D(A) withnroo lim xn = x such that, moreover, the sequence A(xn) converges in H. Differential operators always have a closure in this sense. The spectrum of an operator consists of three parts. First, there are the eigenvalues of A which form the socalled point spectrum 0p (A):
Qp(A) = {A E C: ker(A  A)
{0}}.
Furthermore, there are the residual spectrum, vr(A), as well as the continuous spectrum, Q,(A):
ar(A) _{AEC: ker(A  A) = 0,
R(A  A) 0 H}, 91
4. Analytical Properties of Dirac Operators
92
o (A) = {A E C : ker(A  A) = 0, R(A  A) = H, (A  A)1 is unbounded}.
The remaining complex numbers form the resolvent set p(A):
p(A) = {A E C : (A  \)1 is a bounded operator defined on R(A  \) = H}. To each operator A there corresponds an adjoint operator A* with the domain of definition
D(A*)={xEH:EyEH `dzED(A):(Az,x)=(z,y)} and A*(x) = y. This implies the relation (Az,x) = (z, A* (x))
for all vectors z E D(A), x E D(A*). Let A be a symmetric operator, i.e. (Ax, y) _ (x, Ay),
x, y E D(A).
A. (We will Then D(A) is contained in D(A*) and, in addition, abbreviate this as A C A*.) The double adjoint operator A** coincides with the closure A (von Neumann theorem):
A=A
CA.
Definition. The operator A is called selfadjoint if A = A*. In particular, selfadjoint operators are closed, A = A.
Definition. The operator A is called essentially selfadjoint if its closure A is selfadjoint, i.e. A = A*. The spectrum of essentially selfadjoint operators is real, a(A) = a(A) = a(A*) C 1181.
Finally, recall the notion of spectral measure. If S C C is a set of complex numbers and B(S) the oalgebra of all its Borel subsets, then a spectral measure F is a mapping
F : 13(S) + Proj(H) from the oalgebra B(S) into the set Proj (H) of all projectors in the Hilbert space with the following properties:
a) F(S) = IdH, b) for every x E H the equality Mx(B) = (F(B)x, x), B E B(S), defines a measure on the oalgebra B(S).
4.1. The essential selfadjointness of the Dirac operator in L2
93
Given two vectors x, y E H, then M.,y(B) = (F(B)x, y)
is a complexvalued measure. Any measurable function f : S  C can be integrated against a spectral measure:
f f (s)dF(s). S
The value of this integral is an operator in the Hilbert space H which is bounded for a bounded function. Moreover, we have the formula
((I f (s)dF(s)
ff(s)d/2xv(s).
x, y
s S The spectral theorem for selfadjoint operators can now be formulated as I
follows:
Theorem. Let A be a selfadjoint operator in H with spectrum o(A) C R. Then there exists exactly one spectral measure F on the Qalgebra 13(v(A)) with
f \dF(,\).
A=
v(A)
We now turn to the situation for the Dirac operator DA on a Riemannian manifold (Mn, g) with fixed spin structure and fixed connection A in the determinant bundle of the spin structure. The space 1T (S) of all sections of the spinor bundle with compact support carries the scalar product (01, 02) =
f(i(x),2(x))dMm. Mn
Let L2 (S) be the completion of this space. DA is a symmetric operator in L2(S) with domain D(DA) = Fe(S) (compare Section 3.2). In this section, we will prove that DA is essentially selfadjoint whenever (Mn, g) is a complete Riemannian manifold. To start with, we state the following formula for DA:
Proposition. If f is a smooth function defined on the manifold Mn, grad(f) its gradient field and V) a spinor field, then
DA(f . 0) = fDA(0) + grad(f) DA (0) .
4. Analytical Properties of Dirac Operators
94
Proof. This formula is obtained by a straightforward computation:
DA(f
)=
ei '
VA
(f Y) _
+f
Ve
4)
i=1
e
i=1
ei (ei (f) 4
grad(f) ,+fDA(ib) Now we start to prove the essential selfadjointness of the Dirac operator DA. The argument will proceed along the lines of the the article by J. Wolf (see the references at the end of this chapter). The proof is subdivided into several steps. Let DA be the adjoint to DA. In the domain D(DA) we introduce the norm
N(b) where
110112 + I IDAI12,
denotes the norm in the Hilbert space L2.
Lemma 1. Let I'e(S) C D(DA) be dense with respect to the Nnorm. Then DA is essentially selfadjoint.
Proof. Under the assumptions above we have to prove that D(DA) C D(DA). Let 0 E D(DA). By assumption, there exists a sequence on E Fe(S)
N(0  On) = 0. This implies that rli On = 0 in L2 (S) and, n moreover, that DAWn) converges to DA(s) in L2. However, n is a smooth with 1li
spinor field with compact support, and hence DAW ) = DA(2b ). Thus the sequence DA('On) converges in L2. But the latter just means that b E D(DA).
Next we introduce the linear subspace D,(DA)
E D(DA) : V) has compact support}.
Lemma 2. F,(S) is dense in D,(DA) with respect to the Nnorm. Proof. Choose a locally finite covering of the manifold M1 by charts (Ui, hi)
indexed by the set i E I such that each Ui C Mn is a compact subset. Let {fi}iEI be a partition of unity subordinate to this covering with supp(fi) C Ui. For a given spinor such that
E D,(DA), there are only finitely many indices i E I
supp(fi) n supp('')
0.
Denote those by i1, ..., ii E I. Then, 0 = b1 + ... + 0e with 'bj = fj 0, where 1 < j < 1. The spinor field O j can be considered as a 2[,/2] tuple of functions with compact support defined on the space R1. Approximate 0j
4.1. The essential selfadjointness of the Dirac operator in L2
95
by the convolutions with an approximation of the deltadistribution. Let h : IIBn  R1 be given by
h(x)
0
forlxl > 1,
e 11x1'
for IxI < 1,
and let hE : R'  R1 be defined by hE = E h (1). Then bj * h, is a smooth function with compact support approximating bj in L2. Since DA (0i) belongs to L2, the sequence DA( j * h,) thus converges to DA( j). Applying the chart mapping again, we obtain smooth spinor fields with compact support i,l, j,2i ... defined in the Riemannian space with
k N(Oj  'i,k) = 0. 1,k + ... +'1,k, we now conclude that k belongs to Pe(S) and lim N(b  L1k) = 0. Forming 'Ok = k*oo
Lemma 3. If (Mn, g) is a complete Riemannian manifold, then D°(DA) is dense in D(DA) with respect to the Nnorm.
Proof. Denote the interior distance between two points ml and m2 in the Riemannian manifold by d(ml, m2). Let mo E Mn be a fixed point and p(m) = d(mo, m) the distance fromm to mo. The triangle inequality implies Ip(m1)  P(M2)1 < d(ml, m2), i.e. p is a Lipschitz continuous function. Hence p(m) is differentiable almost
everywhere and the gradient grad(p) exists a.e., too. Furthermore, at each of these points I
I grad(p) I 0
f
I
f (b2r DA (0), DA (0))
I brDA(O) I2 =
K(2r+E)
(DA (b2
K(2r+e)
1C(2r+E)
2
f
(br grad(br)
f (brDA(0),)
' DA(s),)
K(2r+E)
1C(2r+E)
Here we used the fact that the support of the spinor in )Q2r). For e + 0 this implies
f
f (DA (,0), br'p)  f (b, DA (0), 2grad (b
II brDA(O)II2 =
IC(2r)
is contained
IC(2r)
IC(2r)
Now we will apply Schwarz' inequality, I (x, y) I < IxI IYI 0. This allows us to estimate the last term in the equation (with t = 1): f (brDA(O), 2grad (br)
V))
fHbrDA()II2+2 f I Igrad (br) I I2
< 2
IC(2r)
IC(2r)
< 2
f
IC(2r) 2
IIbrDA(
)II2+2M f
K(2r)
11
112.
K(2r)
From 0 < br < 1 and another application of Schwarz' inequality, this time to the first term, we obtain
f (DA(),br)
2
K(2r)
f JIDA()II2+ 2t f IIII2. K(2r)
IC(2r)
Combining both relations yields
fHDA()H2 0}
.
In general, vp(A) U o, (A) C o (A) C o(A).
Applying this inclusion to the case of a Dirac operator over a compact manifold, we at once conclude from the facts proved so far that Ca(DA) = 0a(DA) = o(DA) _ 0 (DA)
Finally, it remains to be shown that o ,,(DA) = up(DA). Let \ E o (DA). Then there is a sequence of spinor fields yin E F(S) with IIInIIL2 = 1,
IIDA(On)  4nIIL2  0.
4. Analytical Properties of Dirac Operators
100
On is smooth and we can thus apply the SchrodingerLichnerowicz formula: 1
2
II DA (0n)  AV).II L2 < I IDA('On)II2
=
I0.I I2
f(dAn,n)+A2Hnl.
f
IIVAV)nHIL2+
+'\2I
Mn
Mn
Since Mn is compact and I Yin I L2 = 1 as well as I DA (bn) the L2lengths I I VA On I I L2 of the 1forms DAn, I
DAY nI IL2
I
=
I
f
VA 1:(oe'On,
e
Mn i=1

AV), 112
L2 3 0,
0n)dMn,
are bounded. Hence On is a bounded sequence in the Sobolev space Hl (S). By the Rellich lemma, the embedding H'(S) > L2 (S) is compact. Thus we can assume that on converges to a spinor field V)o in L2, and this immediately implies DA(' o) = A'0o So we have proved that A is an eigenvalue of DA, and hence a, (DA) = up (DA)11 We still suppose that (Mn, g) is a compact Riemannian manifold with fixed spin`s structure. The first Sobolev norm of a smooth spinor field '0 E F(S) is given by
H' =
112
IIL2+IIoA and the corresponding Sobolev space H'(S) is the completion of F(S) with respect to this norm. Since Mn is compact, different connections A induce equivalent norms in the determinant bundle of the spinC structure. Moreover, the embedding II
II
I
H'(S) , L2(S) is a compact operator (Rellich lemma). The Dirac operator DA is a contin
uous operator DA : H'(S) , L2(S). This follows, e.g., from the estimate below for the norm of DA(s):
IIDAOIIL2 = ,
f(eiV,ejV)
S is bounded, i.e. there exists a constant c > 0 such that for every point of the manifold Mn and any spinor 0 we have the pointwise inequality cIIV)II2
 C ICI IL2
holds.
Inequality (*) admits still another interpretation. I
I DA'' I I L2
n
,
Since II11 '0 1122
>
we have
{II0IIL2 + IIDA0IIL2} < II IIHI 0). Thus D,k9 o St is continuous in L2. The argument above now proves that the image of each operator St is contained in F(S),
St(L2(S)) C n Hk(S) = F(S). k=0
Moreover, as an operator from L2 to Hk(S), St is continuous. Choosing k > dim M', we obtain 2
L2 (S) St
Hk (S)  L2(S),
where Hk(S) > L2(S) is a HilbertSchmidt operator. But then St : L2(S) L2(S) is a HilbertSchmidt operator, too.
E
Corollary. The function
etae :_ CD22 (t) is finite for t > 0.
AEO(DA)
Proof. The HilbertSchmidt norm of the operator St : L2(S) > L2(S) is IStI1Hs
ISt(
=
e2t 2 = D2 (2t),
)11i2 = AEQ(DA)
AEQ(DA)
where V),\ is a complete orthonormal basis in L2 (S) consisting of eigenspinors of DA.
By the same method we can define other spectral functions of the operator DA. Particularly important here is the socalled nfunction. To define it, suppose that the kernel of DA is trivial. Then zero has positive distance to the spectrum o(DA). Let z E C be a complex number with positive real part, Re(z) > 0. The function sgn(.A)
1 I
4. Analytical Properties of Dirac Operators
106
is bounded on the set a(DA). The corresponding operator
T(z) =
Ja
sgn
1
dF(A)
lAlz
is bounded in L2(S). The superposition DA o T(z) is given by the function dim(M) sgn(A)IT x , which is bounded if k < Re(z). Now assume Re(z) > 2 dim(M Then the operators DA oT(z), ..., Dkk o Choose k with Re(z) > k > T(z) are bounded in L2. Hence T(z) maps the space L2(S) into the space
Hk(S). The embedding Hk(S) 3 L2(S) is a HilbertSchmidt operator. Eventually, we obtain the
I
Mn) and ker(DA) = 0, then the operator Proposition. If Re(z) > 2 dim(fsgn()dF(A)
AI
T(z) is a HilbertSchmidt operator in L2(S).
The HilbertSchmidt norm/ is now computed by '\I2Re(z)
IIT(z)IHS = AEa
Define the socalled 77function of the Dirac operator DA as
77DA(z) _ E sgn (A)IA
.
00AEa
Then we obtain the following result.
Proposition. Suppose that ker(DA) = 0. Then the function r7DA(z) is analytic in the halfplane Re(z) > dim(Mn). Remark. 77DA (z) has a meromorphic continuation onto the complex plane and is, in particular, analytic at the point z = 0. The 77invariant of DA is then defined to be 77DA(0).
We also want to discuss the boundary case z = dim(Mn). Since T. r n/2 DA/2 o = J I,\In/Z dF(A) = f(sgn())f12dF(A), a
a
the operator
T* _
f a
IAIn/ZdF(A)
Hn/2 _._, L2 maps the space L2(S) into H,/2 (S) and DA/2 o T* : L2 is invertible, (DA'2 o T*)2 = IdL2. The kernel of DA is trivial and hence .
4.3. Dirac operators are Fredholm operators DA/2
107
H,12(S) + L2(S) is bijective, since the index of DA is equal to zero (compare the next section). Thus T* : L2 * Hn/2 is bijective and its HilbertSchmidt norm as an operator in L2 coincides with the HilbertSchmidt norm of the embedding Hn/2 * L2. The latter is infinite, and we obtain the :
Proposition. The 71function 77DA(z) of a Dirac operator has a singularity at the point z = dim(Mn).
4.3. Dirac operators are Fredholm operators Proposition. The Dirac operator DA : H'(S) > L2(S) over a compact Riemannian manifold is a Fredholm operator of index zero.
Proof. We have to prove that ker(DA) and L2/im(DA) are finitedimensional vector spaces of equal dimension. In the vector space ker(DA) consider the balls
K1 = {z E H'(S) : DA (V)) = 0,
II0IIH1 = IIIIL2 + IDAbIIL2 < 1},
K° = {b E H'(S) : DA(b) = 0, II0IIL2 < 1}. Obviously, K° = K1. On the other hand, H1(S) > L2(S) is a compact operator and hence K° = K' C L2(S) is compact. Considering ker(DA) now as a subspace of L2, the balls are compact in the L2norm. Hence ker(DA) is a finitedimensional vector space. We determine the orthogonal
complement of DA(Hl) in L2. A spinor cp from this space satisfies the condition (DA (0), co) L2 = 0
for all 0 E H'(S). Similarly to the proof of the fact that the residual spectrum or (DA) of DA in L2 is empty, we conclude first the smoothness of cp and then DA(W) = 0. Thus,
(DA(H1))1 = ker(DA)
and, last, to complete the proof it remains to be shown that DA(H') C L2 is a closed subspace. Assume that the sequence DA(On) converges to E L2 (S) in L2. Without loss of generality we can suppose that 0, is orthogonal to the kernel ker(DA). But then, IIDA(0n)II L2 >_ C'IIjjJL2
(compare the corollary in Section 4.2) and V),, is a Cauchy sequence in L2. Inequality (*) implies that 0n is a Cauchy sequence in H1(S) as well. Thus the limitn+oo lim 0n = exists in H1(S). The operator DA : H'(S) + L2(S) is continuous, and 2b
noc DA( n) = DA(n
4
n) = DA(w*)
4. Analytical Properties of Dirac Operators
108
for 0* E H'(S). This means that 0 belongs to the image DA(H'). In the case of a manifold M2k of even dimension, n = 2k, consider the Dirac operators
DA : r(s:':) } r(S ). They are Fredholm operators DA : H1(S±)  L2(S:F).
The index of DA will be denoted by Index(DA). Since DA is a selfadjoint operator, Index (DA) = dim ker(DA)  dim ker(DA ). The index of DA depends on the characteristic classes of the manifold M2k as well as on the first Chern class of the determinant bundle L of the spinC structure. We quote the corresponding formula without proof. The power series of the even function
_
t/2
t et/2  et/2
sinh (t/2) can be represented in the form t 1 + A2t2 + A4 t4 + .... et/2  et/2 = An easy calculation shows, e.g., that 1
A2
24,
A4
10 4.24
5760
Denote the Pontrjagin classes of a 4kdimensional compact manifold M4k by pl, p2, ..., A. The class pj, 1 < j < k, is an element of the 4jth cohomology group Hi (M4k). Introduce k formal variables x1, ..., xk and represent P1, ..., Pk as the elementary symmetric functions in the squares of these variables:
xi+...+xk=p1 k
Then 2rj
=i 2
X .i/2 is a symmetric power series in the variables xi, ...
,
x
and hence defines a polynomial in the Pontrjagin classes. Denote this cohomology class by A(M41,): k
,A(M4k) = 1f
x2/2
sinh (xi/2)
For example, for k = 1, 2 we obtain the formulas
A(M4)=1pi, 4
k
=1,
4.3. Dirac operators are Redholm operators
A(Mg) = 1
 24pi + 5760pl
109
k = 2. A manifold of dimension 4k + 2 also has k Pontrjagin classes, and we define A(M4k+2) by the same formulas. The index theorem for Dirac operators now reads as follows: 1740p2,
Proposition. Let (M2k, g) be a compact oriented Riemannian manifold with spin structure, and let c = ci(L) denote the first Chern class of the determinant bundle of the spin structure. The index of the Dirac operator D+ associated with a connection A in the U(1) bundle of the spin structure is equal to Index (DA) =
e2°A(M2k).
J
M2k
As a consequence of this one can prove that certain characteristic numbers are integers.
Corollary. Let M2k be an oriented compact smooth manifold and take c E H2(M2k; Z) to be a cohomology class whose Z2reduction coincides with the second Stiefel Whitney class of M2k, c = w2(M2k) mod 2. Then
f e2°A(M2k) M2k
is an integer. As an example, we discuss the case of a 4dimensional manifold M4 in greater
detail. The intersection form of M4 is defined by the cupproduct in H2: H2 (M4; Z) x H2 (M4; Z) + Z,
(a, Q) * (a U'3) [M4] .
Considering this quadratic form over the ring Z of integers as a real quadratic form, the resulting form has a signature (p, q). By Poincare duality we have p + q = dim H2 (M4; R). The number 0(M4) = p  q is called the signature
of the 4dimensional manifold M4. This signature is closely related to the Pontrjagin numbers and, by the Hirzebruch signature theorem, o (M4) = 3
f
pi.
M4
The A(M4)class can thus be written as A(M4) = 1  $a, and the polynomial e 2 A = (1 + c + c2) (1 ! a) yields the formula 2
$

Index (DA) _ $ (c2  Q).
As an application of the formula just stated we prove the following result, originally due to Rokhlin.
110
4. Analytical Properties of Dirac Operators
Proposition. Let M4 be a smooth compact orientable 4dimensional manifold with spin structure, w2(M4) = 0. Then the signature o(M4) is divisible by 16.
Proof. Since W2 = 0, the manifold M4 has a Spin (4)structure. Consider the corresponding Dirac operator in the spinor bundle S. Then, 810 (M4).
Index (D+)
But S is a vector bundle associated with the group Spin(4). From the considerations in Section 1.7 we know that the spin representations 04 have equivariant quaternionic structures. These in turn induce parallel quaternionic structures in the spinor bundles S±, and hence the complex vector spaces ker(D+), ker(D) also carry quaternionic structures. Thus, dime ker(D::) 0 mod 2, which immediately implies Index (D+) = 0 mod 2. This means that a(M4) O mod 16.
Remark. The condition w2(M4) = 0 is equivalent to the integral intersection form in H2 (M4; Z) being an even Zform, Va E H2(M4; Z). a2 = 0 mod 2 However, it is a wellknown algebraic fact that the signature o of even forms
over the ring Z is divisible by 8. The Rokhlin theorem thus states an additional divisibility of the signature by 2 in the case of smooth manifolds with even intersection form! The smoothness of M4 is indeed necessary. There exist simply connected topological manifolds Mtp with w2(Mt p) = 0 and o(Mtop = 8. Remark. The parallel quaternionic structures in the spinor bundle S± used in the proof of the Rokhlin theorem exist in all dimensions n = 8k + 4 4 mod 8, if St is associated with the group Spin (trivial spinC structure). Thus we have e.g.: Let M8k+4 be an orientable compact smooth spin manifold. Then, Msk+4
is an integer.
Apart from results concerning the integrality of special characteristic numbers, a further application of the index formula for Dirac operators consists in deriving topological obstructions for the existence of Riemannian metrics with positive scalar curvature. The SchrodingerLichnerowicz formula, 1
1
DA=DA+4 R+ 1dA,
4.4. References and Exercises
111
immediately implies ker(DA) = 0 and hence Index(DA) = 0, if all eigenvalues of the selfadjoint endomorphism R + dA : S  S are positive. Thus a 4 we have the
Proposition. Let (M2k, g) be a compact oriented Riemannian manifold with spine structure, and denote by c = cl(L) the first Chern class of the determinant bundle L of the spine structure. If L has an Hermitian connection A such that all eigenvalues of the endomorphism
4R+1dA=4R+21lA:S+S are positive, then
f e,
A(M2k) = 0.
M2k
Corollary. Let M4k be a compact oriented spin manifold (w2(M4k) = 0), and assume that M4k
Then M4k admits no Riemannian metric of positive scalar curvature.
Example. In the last corollary the assumption W2 (M4k) = 0 is necessary. The complexprojective plane M4 = C]P2 with the FubiniStudy metric has a Riemannian metric of positive scalar curvature, and
A(ci2)8o(Cp2)_8#0. However, C p2 is not a spin manifold.
4.4. References and Exercises M.F. Atiyah, V.K. Patodi, I.M. Singer. Spectral asymmetry and Riemannian geometry Part I, Math. Proc. Cambridge Phil. Soc. 77 (1975) 4369; Part II, ibid. 78 (1975), 405432; Part III, ibid. 79 (1976), 7179.
M.F. Atiyah, I.M. Singer. The index of elliptic operators III, Ann. of Math. 87 (1968), 546604. M.H. Freedman. The topology of fourdimensional manifolds, Journ. Diff. Geom. 17 (1982), 357453.
P.B. Gilkey. Invariance theory, the heat equation and the AtiyahSinger index theorem, Publish or Perish 1984. K. Maurin. Methods of Hilbert spaces, PWN, Warsaw, 1965. K.H. Mayer. Elliptische Differentialoperatoren and Ganzzahligkeitssatze fur charakteristische Klassen, Topology 4 (1965), 295313.
4. Analytical Properties of Dirac Operators
112
J. Wolf. Essential selfadjointness for the Dirac operator and its square, Indiana Univ. Math. J. 22 (1972/73), 611640.
Exercise 1. Let (M4, g) be a compact Riemannian manifold with spine structure and DA the Dirac operator. Consider the heat equation,
m) = DA
t > 0,
m),
with the Cauchy initial condition 24'(0, m) = 00 (m). Prove:
1) This Cauchy problem has at most one solution. 2) If St : L2(S) > L2(S) is defined by
St = fe_tA2dF(A), a
then 0(t, m) = St(0o(m)) is the unique solution to the Cauchy problem.
Exercise 2. Let 0a, A E v(DA)), be a complete basis of L2(S) consisting of eigenspinors of the Dirac operator. Prove that the section E(t, m1, m2) in the bundle S ® S over M' x M" defined by e_t.\20a(ml)
E(t, m1, m2) =
(9 b (m2)
AEa
for t > 0 is smooth. Show, in addition, the following properties: 1) Ft
(t, m1, m2) = D22A (E(t, m1, m2)),
2) 0(m) = lira f E(t, m1i m2)04'(m2)dm2 for ?G E L2(S). t}OMn
Exercise 3. The operator St : L2(S) > L2(S) is an integral operator with kernel E(t, m1, m2), i.e. AV)) (M) _
f
'I'/
Mn
Exercise 4. Let 0 < A
oc: N Ck2/n. k
00
Hint: E
converges for Re(z) > n/2 and has a singularity at the point
z = n/2. Exercise 5. Let D be the Dirac operator of a compact Riemannian spin manifold (Ma, g). Prove that in the cases n # 3,7 mod 8 the 77function of D vanishes identically.
Chapter 5
Eigenvalue Estimates for the Dirac Operator and Twistor Spinors
5.1. Lower estimates for the eigenvalues of the Dirac operator In this chapter we will consider a compact Riemannian manifold (M', g) with fixed spin structure and its Dirac operator D which, in this case, is exclusively determined by the LeviCivita connection. By integration, from the SchrodingerLichnerowicz formula,
D2=0+41R, 4R0 for every eigenvalue A of we immediately obtain the inequality A2 > the Dirac operator, where Ro = min{R(m) : m E M' j is the minimum of the scalar curvature. However, this estimate is not optimal. We have (Th.
Friedrich, 1980)
Proposition. Let (M n, g) be a compact Riemannian manifold with spin structure, and A an eigenvalue of the Dirac operator D. Then, A2
>
1 n Ro. 4n1
113
114
5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors
Moreover, if A = ±a nn1 Ro is an eigenvalue of the Dirac operator and b is a solution of the field equation
a corresponding eigenspinor, then
=T_
VX
2
n(nRo
1)X0
and the scalar curvature R is constant.
Proof. The idea of the proof is based on not using the LeviCivita connection but, instead, considering a suitably modified covariant derivative in the spinor bundle. To this end, fix a realvalued function f : Mn + R1 and introduce the covariant derivative Vf in the spinor bundle S by the formula The algebraic properties of Clifford multiplication imply that Vf is a metric covariant derivative in the spinor bundle S: ,/
X (V,, Y'1) = (V
Let Of i=1
,''b1) + ('0, Vf '01).
V Ve  > div(ei)V be the corresponding Laplace operi=1
ator, and denote by
n
of,012
n
I2IoejO+ fei .012
IV i=1
i=1
the length of the 1form Vf 0. We will compute the operator (D  f )2. First,
(D f)2 = (D  f)(D  f) = D 2  2fD  grad(f) + f2, and the SchrodingerLichnerowicz formula implies
(Df)2 =0+
1
R  2f D  grad(f) + f2.
On the other hand,
Of
n
n
i=1
i=1
=  E(oej+fei) (oe + fei) 
div(ei) (Vex + fei)
02fDgrad(f)+nf2. Summing up, this yields
(D f)2=Of+4R+(1n)f2, and, by integration over Mn, we obtain the formula
f
Mn
((Df)2gp,V))= f {Iof
Mn
012+4RIOI2+(1n)f2IV12}
5.1. Lower estimates for the eigenvalues of the Dirac operator
115
Suppose now that DV = AV). Then we can insert the function f = n into the last formula and obtain A2
(n_
11011L2
n 1)
= 11V nI
IL2 + \21
n
1
I ICI IL2 +
4
f
RIV)I2.
Mn
An algebraic transformation yields
.2n
n2n
IL2 = Ilon 12 +
I
4
f
RI
12 >_
Mn
\2 >
n
1
Discussing the boundary case in this estimate, we immediately obtain the remaining assertions of the proposition. i.e.
 4n1 Ro.
The method of proof applied here may be refined in various ways. Consider,
for example, for a fixed smooth realvalued function f
:
M'z + R1 the
(nonmetric) covariant derivative
txv = Ox + AX n
V) +,aX grad(f)
z/) + vdf (X)V)
V=  n and perform a 1 n 1' n 1' calculation with the length I I eµf V bI 1L22 similar to the one in the proof above. Then one obtains the inequality (0. Hijazi, 1986) with the "optimal" parameters µ = 

Proposition. A2
>
grad f12}.  n2l n min{1R+0(f) 4 n1 n1
In particular, in dimension 2 the summand (grad f 12 drops out. Then the formula simplifies to
,\2 > min{1R+20(f)}. The Gauf3 curvature K of the Riemann surface (M2, 9) is equal to K = 2R, and we can choose f as a solution to the differential equation
20(f) = K + Thus
vol(M2 g) (M2))
2
R + 2A(f) = Vol(M ol(M
)
fK=
K +
M2
is constant, and we obtain
A2 > 2irX (M2)
 vol(M2)
2vo(M2))
116
5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors
Of course, the last inequality is interesting only for 2dimensional Riemannian manifolds which, topologically, are spheres. Summarizing, we obtain the following proposition, originally due to Lott, Hijazi, and Bar.
Proposition. If (S2, g) is a Riemannian metric on S2, then, for the first eigenvalue of the Dirac operator, we have 47r A2
vol(S2, g)
The method we have outlined for estimating the eigenvalues of the Dirac operator may be refined even further when the Riemannian manifold carries additional geometric structures. Let us consider e.g. the case of a Kahler manifold (M2k, J, g) with complex structure J : T(M2k) + T(M2k). In this situation, consider the covariant derivative depending on two parameters f and h which can be chosen freely. Elaborating on the Weitzenbock formulas for Riemannian manifolds with additional geometric structures, one will in general obtain better estimates than in the general case of a Riemannian manifold. For example, the following inequality, first proved by K.D. Kirchberg, holds for Kahler manifolds:
Proposition. Let (M2k, J, g) be a compact Kahler spin manifold and A an eigenvalue of the Dirac operator. Then, if k = dime M is odd, 11 Ro A2 >
4
I k&1Ro
if k = dime M is even.
Remark. The quaternionic Kahler case has been investigated by Kramer, Semmelmann, and Weingart in 1997/98.
5.2. Riemannian manifolds with Killing spinors By the proposition proved in Section 5.1, a spinor field which is an eigenspinor for the eigenvalue 2 n1 Ro solves the stronger field equation
Vxz/i==F2\/n(no This leads to the general notion of Killing spinors. Definition. A spinor field 0 defined on a Riemannian spin manifold (Mn, g) is called a Killing spinor, if there exists a complex number µ such that Ox0=µX.0
for all vectors X E T. µ itself is called the Killing number of '.
5.2. Riemannian manifolds with Killing spinors
117
We begin by listing a few elementary properties of Killing spinors.
Proposition. Let (Mn, g) be a connected Riemannian manifold. 1) A not identically vanishing Killing spinor has no zeroes. 2) Every Killing spinor ' belongs to the kernel of the twistor operator T. Moreover, b is an eigenspinor of the Dirac operator, D( ,O) = nµ 3) If b is a Killing spinor corresponding to a real Killing number µ E R1, then the vector field n
V" =
E (ei
,
4')ei
i=1
is a Killing vector field of the Riemannian manifold (Mn, g).
Proof. A Killing spinor restricted to the curve ry(t), fi(t) _ 0(y(t)), satisfies the following first order ordinary differential equation along this curve: dt (t) = µ Y(t) fi(t)
Now z/'(0) = 0 immediately implies 4(ry(t))  0, and this in turn yield property (1). Starting from Vx = µX 0, we compute n
=nµo i=1
i=1
and thus obtain (O)
T
ei ® (VejO + 1 ei DO) _
ei ® (µeii  µei 0) = 0. n i=1 For a fixed point mo E Mn and a local orthonormal frame el,... , en with Vei(mo) = 0 we compute the covariant derivative VxV''P:
_
i=1
n
n
/ (ei
'7xV0
"/'
Vx4', w)ei + I1
i=1 n
'/1
Y', Vx )ei
i=1
(ei X
µ
(ei

µ(eiO, X  Y')ea
Y, 'Y)ei +
i=1
i=1
n
E((ei  X Xei) 0,O)ei i=1
This implies g(VxVO,Y) = µ((YXXY)
, v); hence g(VxVO,Y) is antisymmetric in X, Y. But this property characterizes Killing vector fields on a Riemannian manifold.
5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors
118
Not every Riemannian manifold allows Killing spinors 0 0 0, and not every
number µ E C occurs as a Killing number. We now derive a series of necessary conditions. To this end, recall the Weyl tensor of a Riemannian manifold. Let Rijkl = g(VeiVejek  VejVeiek  V[ei,ej]ek, el)
be the components of the curvature tensor and n
Rij = E Raija a=1
those of the Ricci tensor. Then define two new tensors K and W by 1 R
Ki. =
n2 2(n1) gij  Rij
,
Wa,Qya = Ra,OyS  gp5Kay  gayKQS + g,3,Ka5 + ga5KKy.
W is called the Weyl tensor of the Riemannian manifold. Because of its symmetry properties the Weyl tensor can be considered as a bundle morphism defined on the 2forms of (Mn, g): W : A2(Mn) > A2(Mn),
Wijklek A el.
W(ei A ej) _ k_ 2mi (mo + ... + Mk1) +
1,
then the kth eigenvalue A2 is bounded by I) kI 4 the category Top(n) of ndimensional topological manifolds does not coincide with the category Diff (n) of smooth manifolds, i.e. the natural mapping Diff (n) j Top(n)
forgetting the differential structure is neither injective nor surjective. In connection with the solution of the Poincare conjecture in dimensions n > 5 and the proof of the hcobordism theorem (Smale), the socalled surgery techniques were developed in the sixties (Wall, Browder, Novikov). They led to a farreaching classification theory for certain classes of smooth compact manifolds in dimensions n > 5.
129
A. SeibergWitten Invariants
130
The situation in low dimensions, n = 3, 4, is rather special. On the one hand, the possible variety of forms is considerably larger already in dimension n = 3 than for surfaces, and the classification question is much more difficult. On the other hand, even in a 4dimensional manifold there is
not enough room to apply the surgery techniques which were so successful
in higher dimensions. Dimension n = 3 turned out to be the last one in which Top and Diff coincide: every compact 3dimensional manifold is triangulizable, every two triangulizations are combinatorially equivalent and, moreover, every such manifold admits exactly one differential structure. The Poincare conjecture in this dimension is still unsettled. Apart from the fundamental group 7rl (M4), the integral intersection form H2 (M4; Z) is the most important invariant of an orientable compact 4dimensional manifold. In 1982, M. Freedman proved that for simply connected compact topological 4manifolds this intersection form almost determines the manifold itself in
Top(4). In particular, each unimodular quadratic form over the ring Z of integers can be realized as the intersection form of a compact simply connected topological manifold M4. On the other hand, it was already known for a long time (Rokhlin 1952) that if a unimodular form of even type can be realized by a closed smooth 4manifold, then its signature is divisible by 16. Take, e.g., the positive definite unimodular Zform E8 of dimension 8, 2
1
0
0
0
0
0
1
2
1
0
0
0
0
2
1
0
0
0
0
0
1
2
1
0
0
0 0 0 0
0
0
0
1
2
1
0
1
0
0
0
0
1
2
1
0
0
0
0
0
0
1
2
0
0
0
0
0
1
0
0
2
0 1
E8 =
E8 is of even type and has signature 8, o (E8) = dim E8 = 8. Hence there exists a simply connected topological manifold Mo with intersection form E8, and Mo is by no means smooth, i.e. the mapping Diff(4) > Top(4)
is not surjective (and not injective as well). Thus the smooth topology in dimension 4 is already a completely different topic than the continuous.
Likewise, at the beginning of the eighties, S.K. Donaldson introduced a method for the study of Diff(4) based on associating with every smooth 4dimensional manifold the moduli space of solutions of the selfdual YangMills equation in a nonabelian gauge field theory as an invariant, and on deriving new invariants from it. In this way, he was able to exclude further unimodular quadratic forms over the ring Z as intersection forms of smooth
A.1. On the topology of 4dimensional manifolds
131
simply connected and closed 4manifolds M4. For example, if H2 (M4; Z) is positive definite, then this intersection form has to be trivial. In particular, E8 ® E8 cannot occur as the intersection form of a smooth manifold even though the Rokhlin condition o116 E Z is satisfied. This obstruction eventually led to the proof that there exist exotic differential structures in R4. On the other hand, using known algebraic surfaces, many unimodular Zforms can be realized as intersection forms. Computations for connected sums of K3surfaces with (S2 X S2) resulted in the conjecture that the form
(2k(E8) ®m
0
0 )J
does occur as the intersection form of a smooth 4manifold if and only if m > 3k (the socalled 11/8 conjecture; this inequality is equivalent to b2(M4)/1o(M4)1 > 11). Within the framework of Donaldson theory, this
a
inequality could only be proved for small k. Another application of the invariants constructed by means of nonabelian gauge field theory concerns splitting questions. A compact simply connected complex surface S with b2 (S) > 3 cannot be smoothly represented as a connected sum S = Xi#X2 with b2 (Xi) > 0. This led to the construction of different differential structures on compact simply connected 4manifolds.
In autumn 1994, E. Witten suggested that all these results, and others reaching even further, could be obtained by considering the moduli space of a system of equations for a pair consisting of a spinor and an abelian con
nection. The spinor has to be harmonic with respect to the abelian gauge field and, on the other hand, algebraically related to the curvature form of the abelian connection (SeibergWitten equation). This system of equations is the 4dimensional analogue to the 2dimensional GinzburgLandau model (1950) of superconductivity. Witten's claim was elaborated by many mathematicians in the following months and turned out to be right. In contrast to nonabelian gauge field theories with their nonlinear equations, one could now return to an abelian theory, and the analytical theory of smooth 4dimensional topology gets drastically simplified !
The first observation forming the base of SeibergWitten theory is that each orientable compact 4dimensional manifold M4 has a spinC structure (though, possibly, no spin structure!). Hence spinors may be defined on it. We briefly sketch a proof: The universal coefficient theorem implies the formula
H3(M4; Z) = {H3 (M4; Z)/Tor(H3(M4; Z))} ® Tor(H2(M4; 7L)),
A. SeibergWitten Invariants
132
and, from Poincare duality, H2 (M4; Z) = H2(M4; Z), we conclude the relations Tor (H3 (M4; Z)) = Tor (H2 (M4; Z)) = Tor (H2 (M4; Z)).
Let T C H2 (M4; Z) be the torsion subgroup. Consider the exact sequence H2 (M4, Z)
H2 (M4, 7G)  H2 (M4, Z2)
H3 (M4, Z) ,...
H3 (M4, Z)
.
Then, im(/3*)
= {a3 E Tor(H3(M4; Z)) : 2ca3 = O}
{y2 E T : 2y2 = 0}.
2
The sequence {y2 E T : 2ry2 = 0} + T 4 T + T/2T is an exact sequence of 7L2vector spaces. Thus, dimz2 (T/2T) = dimz2 {y2 E T : 2y2 = 0} and,
since r(T) = T/2T, we obtain dimz2 (H2(M4; Z2)) = dimz2 (im(r)) + dimz2 ('m(8 )) = dimz2 (im(r)) + dimz2 (r(T)). The following inclusion is obvious:
r(T) C im(r) c H2(M4; Z2).
Consider x E im(r) with x = r(a) and a E H2(M4; Z). If y E r(T), then there exists an element /3 E T C H2(M4; Z) with r(6) = y. 0 is torsion and hence a U (3 in H4 (M4; Z) Z vanishes. This in turn implies
xUy=O for xEim(r),yEr(T). The set IT = {y E H2(M4; Z) : V y E r(T) y U y = 0} thus contains im(r). On the other hand, dimz2 M= dimz2 (H2(M4; Z2))  dimz2 (r(T)) = dimz2 (im(r))
by Z2Poincare 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 Z2reduction r : H2 (M4; Z) f H2 (M4; Z2) . We are going to use this to prove that
M4 has a spinC structure. The necessary and sufficient condition to be considered is that the second StiefelWhitney class w2 belongs to the image
Im(r). We thus have to check whether w2 U y = 0 for all y E r(T). By the Wu formulas for an orientable 4dimensional manifold, w2 is the only cohomology class w2 E H2(M4; Z2) satisfying the condition x2 = W2 U x for all x E H2(M4;Z2). Now for y E r(T) we have y2 = 0, since y is the Z2reduction of a torsion element from H2(M4; Z). This implies
W2Uy=y2=0
A.1. On the topology of 4dimensional manifolds
133
for all y E r(T), i.e. the second StiefelWhitney class w2(M4) is the Z2reduction of an integral cohomology class. Altogether we obtain the
Proposition (Wu 1950; Hirzebruch and Hopf 1958). Every compact orientable 4dimensional manifold M4 has a spinc(4) structure. Next we will collect a few special formulas from 4dimensional spin algebra. The Hodge operator * : A' (R4)  A2 (1184) acting on 2forms splits A2 into the selfdual and the antiselfdual 2forms: A2(I[84)
= A2 (II84) ® A2 (184).
The 4dimensional spin representation A4 also decomposes into A4 = Al A4 with 04 C2. The endomorphisms ej ej : 04 04 induced by Clifford multiplication have the following matrices: ele2 e2e3
_ (i i0 0 _
e1e3
i
0
i
0
1
= (1
0
0
e2e4
0
)'
i
1
= 1 0)
'
i
0
0)
ele4 = (i e3e4 = (0
0
i
In particular, as endomorphisms in 04 eie2  e3e4 = 0,
ele4  e2e3 = 0.
ele3 + e2e4 = 0,
For a spinor c E 04 we define a 2form w" by the formula W,11
(X,
y) =
CD)
(X . Y
+ (X,
Y)14)12,
where X, Y E 1184. wD is a 2form with imaginary values. This results from the following calculations: w111(X,Y)
=
(X'Y'.D'.j))+(X,Y)I(DI2
= ((YX  2(X,Y))D, (D) + (X,Y)1j)12
= (YXc,(D) (X, y)
= (X.Y.
,')+(X,Y)I(D I2=(W,X.Y.
)+(X,Y)Iq I2
(Y X', (D) + (X, y)1.1) 12 = w'' (Y, X) = w' (X, Y).
Using the explicitly described spin representation, we easily obtain the proof of the next
Proposition. 1) Let
E O4 and w2 E A2 . Then, w2
2) (&D ' , (D) _ 21D 14 and 1w'
2
=
(D
21'14.
= 0.
A. SeibergWitten Invariants
134
E C2
Proof. Setting
= 04 yields
2
^el, e2) =
i{I.,pl12
 I'1)2I2} = 0'(e3, 64),
^ei, e3) = b2C + '1)1'52 = W = iDAi
w'D(ei, e4)
ID
(e2, e4),
wD(e2i e3),
and the formulas simply result from inserting these entities.
A.2. The SeibergWitten equation Let M4 be an oriented (compact) 4dimensional manifold. Fix a Spine (4)structure and a connection A E C(P) in the U(1)principal bundle associated with the spinC structure. For D E F(S+), consider the equations (SeibergWitten equation): DA (D =0,
Q+y4
To understand the SeibergWitten equation we compute, for an arbitrary section E F(S+) and any connection A E C(P), the integral 2
+IDA1)
J
I2.
M4
Since QA and w' are forms with purely imaginary values, calculating their length amounts to 2
2
[QA+(ei,
i<j
I
14T 21: QA(ei,ej)(eiejD,4))
= Icrj2+ IQAI2
ej) + 4w"' (ei, ej)
iw
i<j
1IW D12
+
QA I2 + 8
4

 2 (Q++
On the other hand, by the SchrodingerLichnerowicz formula we have
f M4
ID9.1)12
=
f
I(DI2
+ (Q1), P)]
M4
This implies
f
M4
f [Ic2
M4
+
+
I2+8I,DI4].
4
A.2. The SeibergWitten equation
135
Hence the functional E(4), A) = fM4 IIQ+12 +
1 I(DI4]
4 solutions of the Seibergis nonnegative and its zeroes are precisely the Witten equation. Proposition. Let (1), A) be a solution of DA = 0, SZA = 41w4' over a compact Riemannian manifold (M4, g) with scalar curvature R. Then, at each point, where
14)(x)12 < Rmin,
Rmin = min{R(m) : m E M4}.
Proof. At a point x where 14) (x) 12 attains its maximum we have 0 < L I (p 12. However,
o < AI4)I2 = 2((VA)*vA(D, 4)) 2 (VA(p,
} RITZ 
2 f ( 4 (1), 1))  2 (QA 1), 4))
_RIcl;I2 2
If now I(D max >
0,
+
then 0
S1 acting via f (p) = p f (7r(p)). Let O = zx = zdz be the 1form O : TS' i1181
Then the action of the gauge group C; (P) = Map (M', S1) on C(P) is described by
f*(A) =A+7r*f*(O) and the curvature forms of A and f *(A), respectively, are QA=dA, f2f*(A)=f2A.
(L) = Map (M4, SI) in greater detail. Consider f : M4 > S1. Then f acts on C(P) by We will describe the action of the gauge group !; (P)
f*(A)=A+7r*f
.
For connections in the fibre product R x P of the frame bundle with the U(1)bundle of the spinC structure we thus obtain
LC®A
(A+rr*).
Here LC denotes the LeviCivita connection of R. Hence, LC ® (A+7r*
)
LC ® A is a 1form on R x P with values in 81 vanishing on TR. Lifting
A. SeibergWitten Invariants
136
both connections to the SpinC(4)structure, we obtain, for the covariant derivatives in S,
Vt *(A)4)  Ot = df
(1)
f
.
This implies that for the Dirac operators DA, D f*(A) : F(S) * F(S) the following formula holds:
Df*(A)(D  DAB= fgrad(f).
As the group !;(P) _ { f : M4 * S'} acts on F(S) x C(P) by f ((D, A) _ (.1
 4), f* (A)), we have
Df*(A) ('Dlf) = DA(`I'/f) + f grad(f) . (elf) =
f2 grad(f) (D + f2 grad(f) (D = fDA("D). 
This implies that if (,D, A) is a solution of the SeibergWitten equation, so
is f
((D, A)
_
(1'lf,f*A).
Now we turn to the definition of the moduli space for SeibergWitten theory. If M4 is a compact oriented Riemannian spinC manifold, P the U(1)bundle of the spinCrstructure and L the line bundle, then define
ML = { (4),A) E F(S+) x C (P) :
DAI) = 0,
ci =
1 JJJ
Proposition. J)tL is compact.
Proof. Let F(L) = F = {w2 E A2(M4) : and let
dw2
[w2]DR = cl(L)},
be the only harmonic form in F(L). Since the curvature form E C(P) is gauge invariant, we obtain a mapping
wharm
!QA of A
P:fitLNF(L), First Step. on F(L)). Proof.
= 0,
P[A,(D]=SZA.
P(J)tL) C F(L) is a compact subset (relative to the L2topology
Suppose that DA) = 0 and Q++
Write the curvature form
QA as
QA = (1lA)+ + (IA) = Warm + dij1.
A.2. The SeibergWitten equation
137
Denote by x1 the harmonic 1forms and im(d°) = im(d : A° + A'). Then we can choose 771 to be orthogonal to im(d°) E )'HI r(Al). From oI,D I2
= 2((VA)*VA(D, (b)2(VAI),VAS) =
VA(D)
and f A I I2 = 0 we obtain M4 2IIVA4)IIL2
f (_2_1/24)
=
M4
2 for the SeibergWitten invariant to be defined independently of the metric).
Proof. Let (D, A) be an arbitrary solution of the equation DAI) = 0,
ul = 4w
Then, Iq) (M) 12 < Rmin
= R
and
81
IQAI2 =
8
Since p+ = E Q and I S2I 2 = 2, this implies (*)
f IpAI2 < J
M4
M4
8 R2
16 = = f R2I6I2
M4
I
M4
(RIIII) Ip+I2.
4
2
M4
A. SeibergWitten Invariants
150
On the other hand, since p is the curvature form of the LeviCivita connection in the bundle A2T, it follows that cl(A2T)
47x2
f (Ip+12  Ip
12),
M4
and hence f Ip+I2 = 27r2ci(A2T) +
2
M4
f
IAI2.
M4
The scalar curvature of the Kahler manifold is constant. Therefore, the Ricci form p is a harmonic 2form, a consequence of the Bianchi identity. The harmonic form realizes the minimum of the L2norm in each cohomology class. A is a connection in A2T, and hence
f
f 1P12 M4
AI2.
M4
Thus, (**)
f Ip+I2 < 27r2ci(L) + M4
f IQAI2 = f IpAI2. M4
M4
Combining (*) and (**) yields S2A = ip and 1.1) 12 = R. The proof of the estimate I4)I2 _< Rmin then immediately implies VA p __ 0. Decompose D according to the splitting of the spinor bundel S+ = A°'° ® A°'2 into (P =0,0®4,0,2
°,2 is a VAparallel section in A°'2. If it is nontrivial, then we conclude that cl (A°'2) = cl (A2T) = 0. However, the curvature form of the LeviCivita connection A0 in A2T is
52+90=ip+=i4S2#0, since R < 0. Hence (D°,2 vanishes and 4) is proportional to the standard spinor, ' = f 4D°. The length If I2 = R is constant, since I.DI2 is constant. The connection A differs from the LeviCivita connection A0 by an imaginaryvalued 1form 171, A  A0 = 771. As QA = ip, we have dr71 = 0. But
0=DA
1P =grad (f)'11°+771f,D°.
This implies grad f + 771 f = 0. Now consider the gauge transformation 9
+ :M4
1.
A.6. The Kahler case
151
Locally we can write f as f = ,/ReiF and dg = Rdf . Then, g = ezF = Rdf Since grad(f) + f 771 = 0, this implies 771 =  s . In addition, we have .
found a gauge transformation g : M4 f T1 with
A=AO  gg, i.e. (cD, A) is equivalent to ( /
=g.(
c1)o),
0,A0).
Corollary. Let M4 be a compact oriented manifold with b2 > 2. If M4 has a Kdhler structure with constant negative scalar curvature, then M4 admits no Riemannian metric of positive scalar curvature. Remark. The preceding corollary is the special case of a more general fact. Consider a 4dimensional compact and oriented manifold M4 with a symplectic structure w and assume that w A w > 0 defines the fixed orientation.
Choose an almostcomplex structure J : TM4 , TM4, J2 = Id, for which g(X, Y) = w(X, JY) is a Riemannian metric. From w A w > 0 one easily concludes that J is a section in the bundle 'j+(M4, D). Let c = c(w) E H2 (M4; Z) be the first Chern class of the complex vector bundle (TM4; J). Then, by the observations above,
v  dim i)lL = 0,
i.e. for a generic metric the moduli space is discrete.
Theorem (Taubes, 1994). Let (M4, s7) be an oriented compact 4dimensional manifold and w a symplectic form with w A w > 0. Moreover, assume that b2 (M4) > 2, i.e. the SeibergWitten invariant is defined. If c = c(w) E H2 (M4; Z) is the Chern class of an almostcomplex structure associated with w, then nc(M4) = 1 mod 2.
Corollary. Let (M4, sD) be an oriented compact 4dimensional manifold satisfying the following two conditions:
1) b2(M4)>2, 2) M4 has a symplectic structure w with w A w > 0.
Then M4 admits no Riemannian metric of positive scalar curvature.
We will now turn to the case of a Kahler manifold (M4, g, j) where the Spin' (4) structure c E H2 (M4; Z) is not necessarily the canonical one of M4. As before, denote the line bundle by L. The Kahler form SZ acts as an endomorphism on the spinor bundle and has the eigenvalues ±2i there. Hence the spinor bundle splits, S+ = S+ (2i) ® S+(2i).
A. SeibergWitten Invariants
152
L is isomorphic to A2S+ for each Spincc(4) structure, and hence we obtain the isomorphism S+ (2i) ® S+(2i) = L. For every connection A E C(L) the Kahler form SZ is parallel, and hence
the decomposition S+ = S+(i) ® S+(i) is
VAparallel,
too. For a spinor
4D = (D+ +' the integral formula now takes the following form:
f IAA + 4wY + IDA J2 M4
f {I'qA12+IDAj
4
(11)+12+II2)+ I (ICD +I2+IpI2)2}
M4
This implies that for a given solution
(D+ + (D, A) the pair cD+  cD_, A) also solves the SeibergWitten equation. This remark, in turn, allows us to reduce the SeibergWitten equation considerably. Start from a solution of the equation,
DAB=O,
S2+A=4W
with ( _ (D+ + 4)_, and note that, moreover,
DAB= 0,
Q+A=4w4)
with 1)+  1)_. In a local orthonormal frame Kahler form ci has the form
el,... e4 on M4 the
i = el A e2 + e3 A e4
and the spinor P = (45+, 4D_) is given by its components. By the definition of w w'D
=
i(I.1)+I2
 I(pI2)(el n e2 + e3 A e4) 5+D)(el A e3  e2 A e4) + (D+1)_)(el n e4 + e2 A e3).
Therefore,
w(D +w = 2i(ID+I2  I.1)_I2)n.
Since Q++ = 4w` _
w
we obtain
= i(I.D+I2  j)_I2)SZ and 4'+ . ' = 0. Thus, either (D+  0 or (D_ = 0. Moreover, S2 is a Allform, and hence, 41l,+9
since A°'2 n A2 _ {0} = A2'° n A? , we at once arrive at QO, A
2=0=QA °.
A. 7. References
153
The curvature form of the connection A in L is thus a (1, 1)form. Hence A defines a holomorphic structure in L, and, therefore, in S+(±2i) as well. In this case, the Dirac equation, DA1+ = 0 (or 0, respectively), means that 4)t is holomorphic. The Chern class of L is given by
C+(L) = QA
ci(L)
and hence, from 12 A cl = S2 A c1 , we conclude that
fc2Aci(L) = 1 fi+i2  I`)12)12 A c2
J :def M4
M4
J is the cup product of S2 and cl(L), hence a topological invariant. If J < 0, then 41)+  0; in case J > 0 we thus have (D_ = 0. Taking into account, in additon, the action of the gauge group leads to the
Theorem (Witten, 1994). Every solution of the SeibergWitten equation over a Kahler manifold M4 for the spinc structure c E H2 (M4; Z) corresponds to a pair consisting of a holomorphic structure in the bundle S+ (±2i) and an element of
PH°(M4; S+(2i)) if S2 A c < 0, PH°(M4;S+(2i)) if S2Ac> 0. For bi (M4) = 0, the holomorphic structure in S+(±2i) is unique and )tL(g) IPH°(M4; S+ (±2i)). Remark. In general, O7tL(g) cannot be used to determine the SeibergWitten invariant S  W (M4, c) E J V,,, since the Kahler metric g is not generic.
A.7. References J. Eichhorn, Th. Friedrich. SeibergWitten theory, in Symplectic Singularities and Geometry of Gauge Fields (Warsaw; 1995), Banach Center Publ., vol. 39, Polish Acad. Sci., Warsaw, 1997, pp. 231267.
M. Furuta. An invariant of spin 4manifolds and the 11/8conjecture, Preprint 1995.
P. B. Kronheimer, T. S. Mrowka. The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), 797808.
D. Kotschick, J.W. Morgan, C.H. Taubes. Fourmanifolds without symplectic structures, but with nontrivial SeibergWitten invariants, Math. Res. Lett. 2 (1995), 119124.
C. LeBrun. Einstein metrics and Mostow rigidity, Math. Res. Lett. 2 (1995), 18.
154
A. SeibergWitten Invariants
C. LeBrun. On the scalar curvature of complex surfaces, Geom. Funct. An. 5 (1995), 619628.
C. LeBrun. Polarized 4manifolds, extremal Kahler metrics and SeibergWitten theory, Math. Res. Lett. 2 (1995), 653662.
C. LeBrun. 4manifolds without Einstein metrics, Math. Res. Lett. 3 (1996), 133147.
J.W. Morgan. The SeibergWitten equation and applications to the topology of smooth fourmanifolds, Mathematical Notes, Princeton University Press 1996.
J.W. Morgan, Z. Szabo, C.H. Taubes. A product formula for the SeibergWitten invariants and the generalized Thom conjecture, J. Diff. Geom. 44 (1996), 706788.
C. H. Taubes. The SeibergWitten invariants and symplectic forms, Math. Res. Lett. 1 (1994), 809822. C. H. Taubes. More constraints on symplectic forms from SeibergWitten invariants, Math. Res. Lett. 2 (1995), 914. C.H. Taubes. The SeibergWitten and the Gromov invariants, Math. Res. Lett. 2 (1995), 221238. C.H. Taubes. From the SeibergWitten equations to pseudoholomorphic curves, J. Amer. Math. Soc. 9 (1996), 845918. E. Witten. Monopoles and fourmanifolds, Math. Res. Lett. 1 (1994), 769796.
Appendix B
Principal Bundles and Connections B.1. Principal fibre bundles Definition. Let E, X and F be three topological spaces. A mapping 7r E + X is called a locally trivial fibration with fibre F if for each point xo E X there exist a neighborhood U(xo) and a homeomorphism DU(xo)
p1(U(xo)) , U(xo) x F such that the diagram
7r1(U)"
UxF
commutes. Then E is called the total space of the fibration, X its base and F = Ex = 7r1(x) the fibre over x E X.
Example 1. Consider E = X x F and the projection 7r : E * X. Then (E, 7r, X; F) is a locally trivial fibration.
Example 2. Let 7r : E * X be an unramified covering, and let F be a discrete space whose points can be mapped bijectively onto 7r1(x). Then 7r : E + X is a locally trivial fibration with fibre F.
Example 3. Let X = M" be a smooth manifold, E = T(MT) and 7r E > X the projection of the tangent bundle. Then (T (MT ), 7r, Mn; R') is a locally trivial fibration with fibre F = IlBn.
Example 4. Let G be a Lie group, H C G a closed subgroup and E = G, X = G/H. Then the projection 7r : G * G/H is a smooth locally trivial fibration with fibre H. 155
B. Principal Bundles and Connections
156
Definition. Let E and E* be two fibrations over X with projections it and 7r*, respectively. These fibrations are called equivalent if there is a homeomorphism f : E * E* fitting into the commutative diagram
E
f
E*
Definition. A locally trivial fibration (E, it, X; F) is called trivial if it is equivalent to (X x F, 7r, X; F).
Example 5. Not every fibration is trivial. Let E = T(S2) be the tangent bundle of the sphere X = S2. Then T(S2) p S2 cannot be the trivial fibration, since its Euler characteristic is two, X (S2) = 2 $ 0, and hence, by the Hopf theorem, there are no vector fields without zeroes.
Definition. Let p : E  p X be a fibration. A mapping s : X * E is called a section if it o s = Idx. In case s is defined on an open set U C X only, it is called a local section.
Let (E, it, X; F) be a fibration over X and f : Y 4 X a (e.g. continuous or smooth) mapping. Then define a bundle f*E over Y by
f*E={(y,e)EYxE: f(y)=7r(e)} with projection 7r* (y, e) = y. The fibration 7r* fibration induced by f.
Proposition. p* : f *E
f *E
Y is called the
Y is a locally trivial fibration with fibre F.
Definition. A 4tuple (P, it, X; G) is called a Gprincipal bundle if 1) P is a topological space and G a topological group acting freely from the right on P; 2) 7r : P > X is continuous and surjective, and ir(pl) = 7r(p2) if and
only if there exists an element g E G such that plg = P2 holds; and 3) it : P s X is a locally trivial fibration in the sense of principal bundles, i.e. for every x E X there exist U, x E U C X and 1)U 7r1(U) + U x G such that '11 U(p) = Or (p), cou(p))
and cpu :7r'(U) 3 G has the property cpu(p g) = cou(p) . 9.
B.1. Principal fibre bundles
157
Remark. a) Every Gprincipal bundle is a locally trivial fibration with fibre F = G.
b) If (P, it, X; G) is a Gprincipal bundle and f : Y > X is continuous, then (f *P, 7r*, Y; G) is again a Gprincipal bundle. c) If P, G, X and all mappings are smooth, then the principal bundle is also smooth.
Definition. Let (P, it, X; G) and (P1, 7r1, X; G) be two principal bundles over the same base X with the same structure group G. These two principal bundles are called isomorphic if there exists a homeomorphism f : P + P1, satisfying the following conditions:
1) The diagram
P
f P1 X
commutes.
2) f (p g) = f (p) g, i.e. f is compatible with the action of G. Proposition. If the principal bundle (P, it, X; G) has a section, then this principal bundle is isomorphic to the trivial Gprincipal bundle (X x G, 7r, X; G).
Example 1. Let G be a Lie group and H a closed subgroup. Then, setting
P = G, X = G/H and G = H, we see that (G, it, G/H; H) defines an Hprincipal bundle over G/H. Remark. Nonisomorphic principal bundles may well be equivalent as fibrations.
Example 2. Take X = S2 = C]P1 and P = S3 = {(w1, w2) E C2 :
Iw1I2 +
IW212 = 1}. Consider the fibration
7r : S3 * CP',
lr(wl, w2) = [w1 : w2],
and two principal bundles 1 = (S3, 7r, Ce'; Sl) and 6 = (S3, it, CPI; S1) differing only in the action of the group S1 on S3. In i;'1 let the group S1 act on S3 by (w1, w2) . z = (wlz, w2z)
and in 2 by (WI, W2) . z =
(w1zi,w2z1).
Then S1 as well as 6 are principal bundles. 61 and 62 are not isomorphic as Slprincipal bundles. The principal bundle S1 is called the Hopf fibration.
B. Principal Bundles and Connections
158
Example 3. Let Mn be a smooth manifold and Lx(MT) _ {(VI, ... , v"n) E TXMn I det(vl, ... ,,0n) 54 0} the set of all frames at the point x E Mn.
The union L(Mn) _ U Lx(Mn) is called the frame bundle of M. Let xEMn
GL(n;1i8) act on L(Mn) by All
A1n
_ Anl
n
n
(vl,... ,vn)'
Ann
iAin
vjAil,...
i1
i=1
Then (L(Mn), 7r, Mn; GL(n, R)) is a GL(n,118)principal bundle over Mn.
Example 4. Let (Mn, g) be a Riemannian manifold and O(Mn; g) _{(411, ... vn) E L(Mn) : g(vi, v9) = Then O(Mn; g) is an O(n; R)principal bundle over Mn.
a.7}.
Example 5. Let (M2n, w) be a symplectic manifold and Sp(M2n; w) (v1
... vn,, 191 ... 2Un) E L(M2n) :
(vi, vj) = w(wi, wj) = 0 1 w(vi, w.9) = ail
Then Sp(M2n, w) is a Sp(2n;118)principal bundle over
.
J
M2n.
Example 6. Let Mn be a manifold with a fixed orientation O. Set
P=
{(v"1,
...
,v"n
) E L(M') :
{v"1,
...
,
v"n}
= O}.
Then P is a GL+(n;1[8)principal bundle.
Definition. Let p = (P, 7r, X; G) be a Gprincipal bundle and A : G1 > G a continuous (smooth) group homomorphism. A Areduction is a pair (µ, f) consisting of a G1principal bundle µ = (Q, ir, X; G1) over X and a mapping f : Q + P such that 1) the diagram Q
f
P
1 X X
commutes, and 2) f (9' ' g1) = f (4') ' A(g1)
Examples 4, 5, 6 describe different reductions of the frame bundle to the groups 0(n;118), Sp(2n; R) and GL+(n; R), respectively.
B.1. Principal fibre bundles
159
Definition. Two Areductions (µ, f ), (µ, f) of the principal bundle p are called equivalent if there exists an isomorphism D : Q f Q of the G1principal bundles such that the diagram
Q
f
Q
it
P commutes.
Proposition. Let A : O(n; R) } GL(n; R) be the canonical embedding of groups and Mn a smooth manifold. The set of all Areductions of the frame bundle L(M') is in bijective correspondence with the set of all Riemannian metrics on Mn. Proposition. Let A : GL+(n; IIi) > GL(n; l[8) be the canonical embedding of groups and Mn a smooth manifold. The set of all Areductions of the frame bundle L(Mn) is in bijective correspondence with the set of all orientations on Mn.
Remark. For a given Men, each symplectic structure w defines an Sp(2n; R)reduction of L(M2n); compare Example 5. Conversely, given an Sp(2n)
reduction, one can define at each point x E Men a 2form wx : TM n X TxMn > R. The resulting 2form w will be nondegenerate. However, in general, dw = 0 will not hold. Thus not every reduction of the frame bundle L(M2n) to the subgroup Sp(2n; I[8) can be identified with a symplectic structure on Men.
Consider now a Gprincipal bundle (P, 7r, X; G) and a topological space F
on which G acts from the left, G x F > F. Let G act from the right on
PxFby
Set
E=PxF/G:=PXGF and take the projection 7r : E > X defined by
7r(e)_7r[p,f] _ir(p) The 4tuple (E, p, X; F) is a locally trivial fibration, i.e. we have the
Proposition. If (P, 7r, X; G) is a principal bundle and F a space on which G acts from the left, then (E, p, X; F) with E = P xG F is a locally trivial fibration. The fibration (E, 7r, X; F) is called the bundle with fibre F associated to the principal bundle. Proposition. Let (P, 7r, X; G) be a Gprincipal bundle and F a Gspace defining the associated bundle E = P xG F. Then there exists a bijection between the sections s in the bundle (E, 7r, X; F) and the maps s* : P + F with s*(p g) = g1s*(p)
B. Principal Bundles and Connections
160
Now let G be a group and H C G a closed subgroup. Consider the Gspace
G/H = F and the bundle E = P xG (G/H) associated to the principal bundle (P, 7r, X; G).
Proposition. The bundle (E, 7r, X; G/H) has a section if and only if the Gprincipal bundle (P, 7r, X; G) has a reduction to the subgroup H 4 G.
Example 7. Let G be a group and H a closed subgroup. Consider the trivial Gprincipal bundle P = X x G. A section in P XG (G/H) ^ X x G/H is then simply a maps : X 3 G/H, and the Hprincipal bundle corresponding to this section is
Q={(x,g) EX with the projection (x, g) * x. Now fix
G = S0(3), H = S0(2) =
/ C
l 0)
S0(3)
and X = S2 = SO(3)/SO(2), where the identification SO(3)/SO(2) ^ S2 is defined by A  A(e3); e3 is the third basis vector of the Euclidean space 1[83. Since e  H e3 under this identification, we conclude that, for every mapping f : X = S2 > SO(3)/SO(2) S2, Q = {(x, g) E S2 x SO(3) : 9If (x) = e3}
is an S1principal bundle which is a reduction of the trivial bundle P = X x G = S2 x SO(3). Take, e.g., f : S2 + S2 to be the identity. Then the resulting Sl = SO(2)principal bundle is Q = {(x,g) E S2 x SO(3) : glx = e3}. Though Q is the reduction of the trivial SO(3)principal bundle over S2, Q itself is not a trivial Slprincipal bundle over S2. This example shows that reductions of trivial bundles may well be nontrivial principal bundles.
As the last topic in this section we want to discuss vector bundles. To this end, consider again a Gprincipal bundle (P, ir, X; G) and, in addition, a vector space F = V (complex or real). Let G act on V via a representation
p : G > GL(V). Then we obtain the associated bundle E = P xG V = P X PV with projection 7r : E  X. Now define ll8 x E  E or C x E > E, respectively, by
EE) e=[p,v] A.e=[p,Av] EE and an addition of two elements el, e2 satisfying 7r(el) = 7r(e2) by el = [p, v1],
e2 = [p, v2] > el + e2 = [p, V1 + v2].
As G acts linearly on V, these operations are uniquely defined. This results in the following structure: Each fibre of P x P V = E is a vector space over
B.1. Principal fibre bundles
161
II8 or C, respectively, and for every x E X, there exist a neighborhood U, x E U C X, and a mapping cDu : p1 (U) ; U x V which is linear in each fibre.
Definition. A (real, complex) vector bundle over the space X is a fibration (E, 7r, X; Vn) such that each fibre is a vector space and for every xo E X
there exist an open set U, xo E U C X, and a homeomorphism bu p1(U) * U X R, (U X (Cn) which is linear in each fibre.
Example 8. Let Mn be a manifold and (L(Mn), ir, Mn; GL(n; Ia)) its frame bundle. Moreover, let p : GL(n) ; GL(118n) be the usual representation. Then the associated vector bundle is isomorphic to the tangent bundle: L(M') XGL(n) ll _ T(Mn). To show this, define a mapping f : L(Mn) XGL(n) R' + T(MT) by
f A, ...
> vn; Cl,
..
,
Cn) = 1:vici = v"  C.
For A E GL(n; I[8) we have (v, c) = (VA, Alc) and, at the same time, vAAlc = v c. Hence, f : L(Mn) XGL I[8n i T(Mn) is uniquely defined and, in addition, an isomorphism of vector bundles. Example 9. T*Mn ^_ L(Mn) xp* (Rn)*, where the mapping p* : GL(n; II8) GL((I[8n)*) is the dual representation in the dual space (R)*.
Example 10. Let Pk : GL(n; Il) * GL(Ak((?n)*) be the representation in the space of kforms of RI. Then, Ak(Mn) = L(Mn) xp, Ak((I[8n)*).
Example 11. Let = (S3, 7r, ClP 1; S1) be the Hopf fibration and H = S3 x p C the associated bundle, where p : S1  GL(C) is given by p(z)w = zw. H consists of the equivalence classes [(wl, w2), w] of pairs of complex numbers under the identification {(wl, w2), w}  {(wlz, w2z), z1w}.
The mapping f : H > H = {(1, e) E C?1 X C2 f (wl, w2; w) _ ([WI : ECP1
:
E l} defined by
(wlwW2W)) EC2
obviously defines an isomorphism between H and A. Hence H ). C?I, as a vector bundle, is equal to
H={(1,1;) EC?1 >C2El} with projection p : H * C?1, p(l, l;) = 1. H (H) is the socalled Hopf bundle (tautological bundle over Cl?').
B. Principal Bundles and Connections
162
B.2. The classification of principal bundles In this section, we will study the following question:
Let a space X be given. How many isomorphism classes of Gprincipal bundles (P, 7r, X; G) with structure group G exist over X? Theorem (First homotopy classification theorem). Let = (P, 7r, Y; G) be a Gprincipal bundle over the topological space Y, X a paracompact space and fl, f2 : X + Y two homotopic mappings, fl  f2. Then the induced principal bundles fl , f2 over X are isomorphic.
Definition. If X is a paracompact space, then denote by HFBG(X) the set of all isomorphism classes of principal bundles over X with structure group G.
Definition. Let G be a topological group. A universal Gbundle is a Gprincipal bundle c = (EG, 7, BG; G) such that for every CWcomplex X the assignment
[X; BG] E [f]  f *G E HFBG (X) is a bijection. In other words, the following two conditions have to be satisfied:
1) For every Gprincipal bundle l; over X there exists a mapping f X f BG with f*
2) For fo,f1:X>BGsuch that f c fl* Gwehave fo fl. BG is called the classifying space of the topological group G.
Obviously, the classifying space of a topological group is not uniquely determined. Namely, we have the
Proposition. If ec = (EG, 7r, BG; G) is a universal bundle and B a topological space which is homotopy equivalent to BG, then over B there is also a Gprincipal bundle which is universal.
Proposition. Let 6c = (EG, 7r, BG; G) andG = (EG, 7r, PG, G) be two universal Gprincipal bundles with the CWcomplexes BG, BG. Then there exist homotopy equivalences 0 : BG > PG and : PG > BG with
2/) o0IdgG, 0o0Id and ec='* c,
c=cb*6c
In the class of CWcomplexes, the homotopy type of the classifying space of a group, if one exists, is uniquely determined. Now we can formulate the second homotopy classification theorem for principal bundles:
B.3. Connections in principal bundles
163
Theorem (Second homotopy classification theorem). 1) For every topological group G there exists a universal Gprincipal bundle Z;G = (EG, 7r, BG; G). This bundle, moreover, has the property that for every paracompact space X the assignment
[X; BG] E) [.f] + f*G E HFBG(X) is bijective.
2) For every topological group G there exists a universal Gprincipal bundle
(EG, ir, BG; G) such that BG is a CWcomplex.
Example (G = Z2). The set PBz2 (X) = [X; B(Z2)] = [X; I[8IP°°] (X a CWcomplex) is in bijective correspondence with the elements of H1(X; Z2). If is a Z2principal bundle, then the element in H1 (X; Z2) corresponding
to this principal bundle is called the first StiefelWhitney class wl
E
HI (X; Z2).
Example (G = S'). If X is a CWcomplex, then PBS, (X) = [X; B(S')] _ [X; C?'] is in bijective correspondence with the elements of H2(X; Z). For an S'principal bundle 6, the element in H2 (X; Z) corresponding to this fibration is called the first Chern class of and will be denoted by cl (l;) E H2(X;Z).
B.3. Connections in principal bundles Consider a smooth principal bundle (P, ir, M'n; G). The vertical space Tp (P) of the projection 7r is the space Tp (P) = {X E Tp(P) : d7r(X) = 0}.
Lemma 1. For X E g denote by X (p) = at (p exp(tX)) t=0 the fundamental vector field of the Gaction on P. Then
gEX>X(p)ETT(P) is a linear isomorphism.
Definition. The assignment Th : p E P > Tph(P) C TpP (geometric distribution, Pfaff system) is called a connection on (P, 7r, M; G) if 1) TpP = Th(P) G Tp (P), 2) dR9(Tph(P)) =Tp9(P), and 3) Th is smooth. The projection onto the vertical space X (p) ®Y E Tp (P) ®Tph (P) , X E g defines a gvalued 1form Z on P, Z : TP  g. Then we have the
B. Principal Bundles and Connections
164
Proposition. a) Z(X) = X for every X E g, b) (Rg)*Z = Ad(g1)Z. Conversely, given a gvalued 1form Z : TP * g satisfying a) and b), then Tp (P) = {X E TpP : Z(X) = 0} is a connection. Now we will describe the local characterization of a connection: Let s : U C M > P be a section. ZS = Z o ds = s* (Z) : TU g is the local connection define form. For two sections si Ui * P, s j : Uj r P and Ui n Uj :
gij:UinU1Gby si (x) = sj (x)  gij (x)
Denote the MaurerCartan form of the group G by O : TG > g, O(tg) _ dLg1 (tg). The corresponding 1forms on Ui n Uj are Oij = gz(O):
Oij =
dLgidgij.
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 1forms such that
Zi = Ad(gzj')Zj + Oij. Then there exists precisely one connection Z : TP i g with Zsi = Zi.
Example. Let (Mn, g) be a Riemannian manifold, V : TM * T*M ® TM the LeviCivita connection, and O(M, g) the bundle of orthonormal frames. In the Lie algebra so(n) we choose the basis {Xij}i<j, where Xij =Eij Eji and Eij denotes the matrix with 1 in the ith row and jth column (and zero otherwise). Let s : U * O(M, g), s = (Si,... , sn) be a local section and set ZS = E g(Vsi, sj)Xij. Then {ZS, s} defines a connection Z in O(M, g) i<j
and the 1forms wij = g(Vei, ej) are the connection forms of the LeviCivita connection.
Example. Let Mn be a differentiable manifold. The set of connections on the frame bundle L(Mn) is in bijective correspondence with the set of affine connections V : TM + T*M ® TM. Example. X = G/H is called reductive if there exists a decomposition g = hem with Ad(H) (m) C m. Consider the principal bundle t; = (G; ir, G/H; H) and the distribution G E) g > Ty G = dLg(m). This is a connection in with dL,,, (T9 G) = Tag G.
Let a principal bundle (P, 7r, M; G) and a representation p : G + GL(V) be given.
B.3. Connections in principal bundles
165
Definition. A qform w E Al (P, V) with values in V is called tensorial of type p if
1) w2(tl, , tq) = 0 if one of the vectors tz E TT is vertical, and 2) R*w = p(g1)w, g E G.
Example. If Z, Z : TP > g are two connections, then Z  2 is a tensorial 1form of type Ad on P with values in g. Conversely, if Z is a connection and w : T P > g a tensorial 1form of type Ad, then Z + w is again a connection.
Proposition. The vector space of tensorial qforms of type p on P with values in V is isomorphic to the vector space Aq(M; E) of qforms on M with values in the associated vector bundle E = P x P V V.
Corollary. Let C(P) be the set of all connections on P. C(P) is an of ne space with vector space A'(M;g). Here g denotes the bundle g = P XAd 9 associated by means of the representation Ad.
Definition. Let (P,,7r, M; G) be a principal bundle with connection. If X E X (M) is a vector field on M and X* E X (P) a vector field on P, then X* is called a horizontal lift of X if a) X * (p) is horizontal at every point p E P, and b) d7r(X*(p)) = X (7r (p)).
Proposition. 1) Let X E X(M) be given. Then there exists a uniquely determined horizontal lift X* E X (P) of X. X * is right invariant. If, on the other hand, Y E X (P) is a right invariant horizontal vector field, then there exists a vector field X E X (M) with X* = Y.
2) X,Y E X(M)
X* +Y* = (X +Y)*,
(fX)* _ (f
07r)X*,
[X, Y]* = projhor. [X*, Y*]
3) If Y is a horizontal vector field on P, then [X, Y] is horizontal for all X E g. 4) In particular, [X, Z*] = 0 for X E g, Z E X (Mn). Let y : [a, b] > M be a curve (continuous, piecewise C2).
Definition. A curve y* : [a, b] > P is called a horizontal lift of y if 1) lry* = y, and 2) y* is horizontal.
Proposition. Let y : I > M be a curve in M and u E Py(o) a fixed point. Then there exists precisely one horizontal lift yu of y with y,*,(0) = u.
B. Principal Bundles and Connections
166
Definition. Let Ty : Py(a)  Py(b) be defined by ry(u) = yo(b). ry is called the parallel transport along y.
Proposition. 1) ry does not depend on the parametrization of y.
2) ryRg=R9Ty VgEG. 3) ry is bijective.
Proposition. Let (P, 7r, M; G) be a principal bundle with connection Z, and suppose that M is connected. If the parallel transport r does not depend on
the curve, then there exists a horizontal section in P and, moreover, the principal bundle is isomorphic to the trivial bundle (M x G, prl, M; G) with flat connection.
For E = P XG F (F an arbitrary space), the parallel transport in E, rE
Ey(a) 3 Ey(b), is defined by rE[p, v] = [ry(p), v]. A connection in
the principal bundle induces a parallel transport in every associated bundle.
B.4. Absolute differential and curvature Definition. Let (P, 7r, M; G) be a principal bundle with connection Z, V an arbitrary vector space, and let w E Aq(P, V) denote a qform on P with values in V. Define Dw E Aq+l(P,V) by (Dw)p(to, ,tq) = dw(prhto, ,prhtq) Dw is called the absolute differential of w. It defines a linear mapping
D : Aq(p, V)  Aq+i(P, V).
Proposition. 1) If w is a q form of type p, then Dw is a tensorial (q+1)form of type P.
2) Suppose that w E Aq(P; V) is tensorial of type p. Then,
Dw =
dw + p. (Z) A w with p* : p * gl(V) and q
(p*(Z) A w) (to, ... , tq) = L(1)op*(Z(ta))w(to, ... , ta, ... , tq). a=o
Now let p : G + GL(V) be a representation and set E = P xP V. The tensorial forms of type p coincide with the forms AP(M, E) on M with values in the vector bundle E. Hence the absolute differential can be viewed as an operator D : Aq (M; E) r Aq+1(M; E). Definition. Let (E, 7r, M) be a vector bundle over M and F(E) the space of smooth sections. A mapping V : r(E) + r(T*M ® E) is called a covariant derivative in E if the following conditions are satisfied:
B.4. Absolute differential and curvature
167
1) V is linear, V(ei + e2) = Vel + Del. 2) V (f e) = df ®e I f Ve for f E C°°(M), e E F(E). The 1form V e is also written as (V e) (X) = Vie.
Proposition. Let (P, 7r, M; G) be a principal bundle with connection, p G > GL(V) a representation and E = P x p V the associated vector bundle. Then the absolute differential D : r(E) 3 A1(M;E) = P(T*M (9 E) is a covariant derivative.
D is called the covariant derivative in E associated with Z. It is occasionally also denoted by Vz, VE
Proposition. Let s E F(E) be a smooth section. Then, (Ds) (X)
dt(TtO(s('Y(t)))t=o,
X E TpM,
where y(t) C M is a curve with y(O) = p,,y(0) = X, and T% : Ey(t) > Ey(o) is the parallel transport.
Corollary. Let y(t) C M be a curve and s(t) C E a curve over y(t). Then s(t) is the parallel transport of s(O) along y(t) if and only if dt Ds('y(t)) = 0.
Definition. Let (P, 7r, M; G) be a principal bundle and Z : TP  g a connection. Then Z is a 1form on P of type Ad (i.e. Rg**Z = Ad(g1)Z). By the preceding theorem,
Q:=DZ is a 2form on P with values in g which is tensorial and of type Ad. SZ is called the curvature form of the connection. By the general identification {tensorialqform of type p}
Aq (M; P x p V)
,
S2 can also be considered as. a 2form on M with values in g = P XAd g, S2 E A2(M; g). We introduce a few notations: Let w E Ai(P; g), ,r E Aj (P; g) be two forms on P with values in g (or else, w E AZ(M; g), ,T E Aj (M; g)
two forms on M with values in the bundle g). Then define a form [w, T] E Ai+j (P; g) (or [w, r] E Ai+j (M; g) }, respectively) by
[w,r](X1,... ,Xi+j)
= iiji E (1)9[w(Xg(1), ...
, X9(i)), r(X9(i+l), ... , Xg(i+i))]
9ESi+1
If Al,
,
Ae is a basis in g and w = wiAj, r = ri Aj, then, obviously, [w,r] = wZ Arj [Ai,Aj].
B. Principal Bundles and Connections
168
This bracket has the following properties:
a) [w,,r] = (1)zj"[7, w]. b) For w E Ai(P, g), T E Aj (P; g), cp E A' (P; g),
7]+(1)ii[[T,coj, w] =0. c) d[w, 7] = [dw, T] + (1)i[w, dr]. d) If w is a 1form and w E A' (P, g) (or w E A' (M; g)), then 2
[w, w] (X, Y) = [w(X ), w(Y)]
Proposition. Let (P, 7r, M; G) be a principal bundle, Z : TP > g a connection and Il = DZ its curvature form. Then we have: 1) The structure equation: Q = dZ + [Z, Z] 2 2) The Bianchi identity: D11 = 0. 3) If w E Aq(P; V) is tensorial of type p : G  GL(V), then .
DDw = p* (Sl) A w.
4) If w E Aq(P, g) is tensorial of type Ad : G * GL(g), then
Dw=dw+[Z,w]. Corollary. Let X, Y be horizontal vector fields. Then,
Z([X,Y]) = f2(X,Y). Proof. SZ = dZ+ 2 [Z, Z]. Inserting horizontal fields we obtain Z(X) = 0 = Z(Y), hence Q (X' Y) = Z[X, Y].
Proposition. Let (P, ir, M; G) be a principal bundle and Z, Z two connections with curvatures SZ = DZ, = DZ. Then, 77 = Z  Z is a tensorial 1form of type Ad and 1 fl=St+Dr7+ 2[77,r1]
Considering S2,1 as sforms on M and 77 as a 1form on M with values in g = P xAd g 0, n E A2 (M, g), r7 E S2' (M, g), we have the same formula, this time with the operator D : A' (M; g)
A2 (M;
Definition. A connection Z on (P, ir, M; G) is called (locally) flat if there
exists an open covering Ui of M such that (PU,, Z) is isomorphic to (Ui x G, prl, Ui; G) with the canonical connection.
B.S. Connections in U(1)principal bundles and the Weyl theorem
Proposition. Z is a locally flat connection
)0
169
the bundle
Th(P) C TP of horizontal vectors is involutive.
Proposition. Let 7rl (M) = 0 and let (P, 7r, M; G) be a principal bundle with a locally flat connection Z. Then (P; Z) is isomorphic to (M x G) with the canonical connection. Definition. Let (P, 7r, M; G) be a principal bundle. A gauge transformation is a diffeomorphism f : P + P with
f = 7r, and 1) oTr,
2) f(p.g)=f(p).g Denote by 9(P) the group of all gauge transformations.
Proposition. 1) If Z E C(P) is a connection and f E Q(P) a gauge transformation, then f * Z E C (P) is again a connection. In other words, the group of gauge transformations acts on the set of all connections. 2) Each gauge transformation f is given by a mapping A f : P > G,
f (p) = p µf (p). Then, (f*Z)p = Ad(µf(p)1)Zp+dLµf(p)idl flp
=Ad(/tf1(p))Zp+µp0.
Proposition. Let the gauge transformation f : P  P be given by ,a f P  G, and let Z be a connection. Then, for the curvature forms S2z and Sjf*z
Qf*z
= Ad(µ f 1)QZ.
B.S. Connections in U(1)principal bundles and the Weyl theorem In this section, we deal with the group G = S1 = U(1) = {z E C : zj = 1}.
If y(t) is a curve in G with y(0) = 1, then y(0) E T1S1 = C71. On the other hand, 1y(t)12  1 implies ' (0) E i118. Hence we obtain an identification 61 D '(0) , y(0) E iIi. The Lie algebra E51 can be identified with iR in
such a way that the diagram
/e
exp\ S1
with e : i]R p S1, e(ix) = eix, commutes. Consider now the canonical form (MaurerCartan form) O : TS' * 61 iIIB of the group. We will show that dz 0==zdz. z
B. Principal Bundles and Connections
170
In fact, if z E S1, F E TZS1 and y is a curve with 'y(0) = z, 'y(0) = t, then
8(1 _ (dLz1(t)) =
d ('t) to = z dt It=o = 1tz = dt z 1
dry
z
i.e. e = z Moreover, f S = f dz/z = 27ri. Hence, go := 27rZ19: TS1 R S1
S1
is a realvalued 1form on S' with f cp = 1. Si
Now let (P, 7r, M'; S1) be an Slprincipal bundle over M. If f : P > P is a gauge transformation, then set f (p) = p Izf(p), j if : P > S'. Since
f(p.z)=f(p).z,
=p.µf(p) z.
Hence, z7i f (p z) = u f (p) z and, since S' is abelian, we have 1.1f (p z) _ Ft f (p). Thus A f : P i S' is constant on the fibres and induces a mapping
9f : M'1 p S1. Conversely, if a mapping µ : Mn > S' is given, then
f(p)=p.µ(7r(p)) defines a gauge transformation. The group of gauge transformations thus coincides with the group of all mappings p : M' + Sl: )S1}.
{A:M' Fix a connection Z : TP > iR on P. If f : P > P is the gauge transformation corresponding to T if : Mn > S1, then
f*Z=Ad(pfl)Z+µf0=Z+Pfe=Z+7r*µf8=Z+27rµfcp. Consider SZ = Qz : TP x TP > iR, the connection form of Z. Since
R*f2 = Ad(z1)t2 = SZ,
SZ is a tensorial 2form on TP invariant under the action of right translations. Thus SZ is simply a 2form on M'z with values in iR,
t2Z:TM' xTM' + iR. As 0 = Dt2Z = dt2Z + Ad* (Z) A Qz = dt2Z, QZ is a closed 2form, i.e. =0
dttZ=0. If 2 is another connection, then
QZ =QZ+Dr7+ with 77 =
1
2[77,77] =t2Z+Dr7
Z. Now, 77 again is a tensorial 1form with RZ77 = Ad(Z1)77 =
77, hence a 1form on M' with values in iR. This implies:
1) The curvature form S1Z of an arbitrary connection in P is a closed 2form on M' with values in iR.
B.5. Connections in U(1)principal bundles and the Weyl theorem
171
2) If SZZ, StZ are the curvature forms of two connections, then there exists a 1form on Mn with values in iR such that SIZ
 Qz=drl.
The de Rham cohomology of a compact manifold is defined by
H2 (Mn;R) = Z2(Mn)/B2(Mn), where
Z2(Mn) _ {w2 : w2 is a 2form and dw2 = 01, B2(Mn) = {w2 : there exists a 1form µl with w2 = dµ1}. 1) and 2) imply that the class [27rC2SZZ] E H2nR(Mn; I[8) is a uniquely determined element of the de Rham cohomology of Mn not depending on the choice of Z, but only on the principal bundle. This class will be denoted by
cl(P) E HDR(M;R) It is called the real Chern class of the SI principal bundle P. Set .F(P) = {w2 : w2 is a 2form with dw2 = 0, [w2] = cl(P)}. Then, C(P)
obviously defines a mapping
Z
Qz
27r.
E.F(P)
: C(P) +.F(P). We state some of its prop
erties:
1) If Z and 2 are gauge equivalent connections, then 2) 0 is surjective.
(Z)
From 1) and 2) we obtain a surjective mapping
0: C(P)/G(P)
.P(P).
The first de Rham cohomology is defined by HDR(Mn; I[8) = Z' (M) 1B1 (M),
where Z' and B1 are the following spaces:
Z' (M) = {w' : wlis a closed 1form, dwl = 01, B1(M) = {w' : there exists a function f on M with wl = df }. Let, moreover,
ZI(M;Z) =
fw' E Z for all closed curves y
wl E ZI(M) : ly
Since f df = f f = 0, we obviously have ZI (M; Z) D BI(M). Let 1
09Y R(Mn;
HD'
Z) = Z1(Mn; Z)/BI (Mn)
B. Principal Bundles and Connections
172
be the socalled integral de Rham cohomology. Again consider w2 E .P(P) and denote by C,,2 (P) the set
C.2(P) =,O1(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 socalled Weyl theorem.
Theorem (Weyl theorem). Let (P, 7r, M; Sl) be an S' principal bundle over the compact manifold Mn with first Chern class cl(P) E HDR(M;lR), and set
,r(P) = {w2 :dw2 = 0,
[w2]
= cl(P)}.
Define a surjective mapping b : C(P)/G(P) 3 .P(P) by the assignment
Z . nZ =
27ri
QZ.
Then each fibre r1(w2) of 0 is diffeomorphic to the Picard manifold
Pic(Mn) = HDR(Mn; R)IHDR(Mn; Z) of Mn. In particular, HDR(Mn; lib) = 0 (e.g. for simply connected Mn) implies that V is bijective.
Example. Consider the Hopf fibration it : S3 > CP1 = S2 with S3 = {(wl, w2) E C2 : Iw112 + IW212 = 1}
and the S'action S3 X Sl  S3, ((wl, w2); z) = (WI Z, w2z)
We will construct a connection in this S'principal bundle. Set 1
Z = {wldwl  wldwl + w2dw2  w2dw2}. 2
Since z  2 E i]R for every z E C, it follows that Z is a 1form on S3 with values in iR. It has the following properties:
1) Z is invariant under the Slaction, i.e. (Rx)*Z = Z, z E S'. 2) For ix E ilR and the corresponding fundamental vector field (ix) on S3 we have Z(ix) = ix. Thus Z is a connection in the bundle 7r : S3 + S2. We are going to compute its curvature. As S' is abelian, S2 = dZ, and thus, 0 = dwl A dwl  dw2 A d1702
B.6. Reductions of connections
173
as a 2form on S3 with values in M. Since S2 is a curvature form, S2 = 7r*1 for a 2form f2 on CP' = S2. Let y : S2\{north pole} > C denote stereographic projection, and let SZ be a 2form on C with Then ry o 7r : S3\{(w1, w2) : w2 5L 0} f C is given by ry o 7r(w1, w2) = w2
and S2 has the form

dz A dz (1 + IzI2)2'
Hence, for the curvature form Il of the connection Z, we have f SZ/2iri = 1. S2
This equation means that cl(Hopf fibration)= 1. Remark. For the Slprincipal bundle (S3, 7r, CP'; S1) with Slaction ((w1, w2), z) _ (W1z1, W2z1),
a similar argument shows that Z* _ Z : TS3 i]R is a connection in that bundle. This implies SZ* = dZ* = 12 = dw1 A dw1 + dw2 A dzu2i SZ
dzAdz (1 + Iz12)2 '
1
27ri S2
Hence, c1(6) = +1, and we have once again proved that this S'principal bundle is not isomorphic to the Hopf fibration.
B.6. Reductions of connections Let (P, 7r, M; G) be a Gprincipal bundle over M'n, and let (P', 7r, M'z; G') together with f : P' > P be a Areduction of this bundle, i.e.:
f
1) A : G' * G is a group homomorphism,
2) f : P' > P is smooth and the diagram
P' commutes,
3) f (p'9') = f (p')A(9').
" M fI
P
B. Principal Bundles and Connections
174
Proposition. Let Z' : TP' (P',ir,M;G').
be a connection in the G'principal bundle
1) There exists one and only one connection Z : TP j g such that df : TP' > TP maps the horizontal spaces with respect to Z' onto the horizontal spaces with respect to Z. 2) A*Z' = f *Z and, for the curvature forms, A*SZ' = f *Q.
Remark. 1) The connection Z constructed starting from the connection Z' is called the induced connection or the Aextension of Z'.
2) If G' is a subgroup of G and A G' 3 G the embedding, then Z' is called a reduction of the connection Z onto the subbundle :
(P', 7r, M; G).
Consider now a principal bundle (P, 7r, M; G) with connection, as well as a subgroup H C G and a subbundle (Q; nr, M; H). We ask the following question: When does a connection Z' exist in (Q, 7r, M; H) such that Z' is a reduction of Z? A provisional answer to this question is contained in the following:
Proposition. In the above notation, with the additional assumption that there exists a decomposition of the Lie algebra
g= lj em with Ad(H)(m) C m, we have: If Z : TP + g is a connection in (P, 7r, M; G), then Z' = prb o ZJTQ : TQ > tj is a connection in Q.
If, in particular, Z : TP + g takes only values in the subalgebra [ , then Z reduces to a connection Z' : TQ + .
B.7. Frobenius' theorem The local Frobenius theorem in Euclidean space can be formulated as follows:
Theorem (Local Frobenius theorem in R n). Let U C R1 be an open subset and w1, ... wr 1forms on U, n = r + s. Moreover, suppose that wr are linearly independent at every point and a) wl, . b) there exist 1forms E on U with ,
.
,
r
dwz=EG'Awi. j=1
B.7. Frobenius' theorem
175
For x E U define
Es(x) = {t E TTU : w'(i) =
= wr'(t = 0}.
Then for every point xO E U there exists a regular sdimensional surface piece FS with
1) xo E Fs, 2) y E Fs TyFs = Es(y)by an ndimensional manifold, this leads to the notion of a
Replacing Ift
distribution:
Definition. Let Mh be a manifold. A differential system or a distribution on Mn is a selection of kdimensional subspaces E. C TM' in every tangent space such that Ex depends smoothly on the point x in the following sense: For every x E Mn there exist a neighborhood U(x) and vector for tk on U(x) with Ey = Lin(tl (y), , Fk (y)) fields tl, ,
allyEU(x). Then Ek = U Ex is a smooth subvector bundle of the tangent bundle. X
Definition. Let Ek C TMn be a kdimensional distribution. Ek is called integrable if the following condition is satisfied: For two vector fields fl, F2 on Mn with values in Ek, the commutator [tl, t2] also has values in Ek.
Theorem (Local Frobenius theorem on manifolds). Let Ek c TMn be an integrable kdimensional distribution. For every point x E M there exist a neighborhood Ux and a submanifold x E Fk C Ux with TyFk = Ey for all
yEFk. Before turning to the global version of the Frobenius theorem we have to explain or extend a few general notions. To this end, recall the following definitions:
Definition. A smooth manifold without boundary is a pair (M, D), where 1) M is a topological T2space with countable basis, and 2) D is a differentiable structure on M, i.e. a family D = {(UU, hi)}iET where Ui C M is open, hi : Ui + Vi C Rn are homeomorphisms and the hihi 1 are smooth.
Definition. Let (M, D) be a smooth manifold without boundary. A subset A C M is called a kdimensional submanifold if
VaEA2 (U,cp)ED,aEU,cp:U>VCllgn: cp(A fl U) is an open subset of I8k x {0}.
B. Principal Bundles and Connections
176
One then shows that (M, D(M)) is a manifold in the induced topology and with the atlas D(A) = {(A fl U, cPjAnu)}. Moreover, the embedding i : A > M is smooth and di : TA > TM is injective.
Definition. Let (M, D(M)) be a smooth manifold. A subset A C M is called a weak submanifold if there exist a smooth manifold (N, D(N)) and a differentiable mapping f : N ; M with the following properties: 1) f is injective.
2) f (N) = A. 3) df : Tn,N 3 Tf(n)M is injective.
If a E A is a point in the weak submanifold, then there is one and only one
n E N with f (n) = a. The space dfn(TnN) =: TaA is called the tangent space of A at the point a E A.
Example. Take M = T2 and let cp(t) = (eiat eiat), a//3 irrational, A = cp(R'). Then A C T2 is a (dense) weak submanifold which is not a submanifold.
Proposition. Let A C M be a weak submanifold and f: N j A a model. For every point n E N there exists a neighborhood u E U(n) C N with the following properties:
1) f (U(n)) is a submanifold of M. 2) f : U(n)  f(U(n)) is a diffeomorphism.
Corollary. Let A C M be a weak submanifold, and let f : N + A, fl Ni > A be two models. Then fi' o f : N  Nl is a diffeomorphism. Definition. Let Ek C TM be a distribution. An integral manifold of Ek is a weak submanifold A C M with TA = Ey for all y E A. Theorem (Global Frobenius theorem on manifolds). Let Ek C TM be an integrable distribution on a manifold M. Then for every point x E M there exists a weak submanifold A(x) with the following properties:
1) A(x) is an integral manifold of Ek, i.e.
TTA(x) = Ey V y E A(x). 2) A(x) is connected.
3) A(x) is maximal, i.e., if B is a connected integral manifold of Ek with A(x) C B, then A(x) = B.
B.9. Holonomy theory
177
B.8. The FreudenthalYamabe theorem Definition. Let G be a Lie group. A subset H C G is called a (weak) Lie subgroup if the following conditions are satisfied.
1) H is a subgroup. 2) H is a weak submanifold respecting the group structure, i.e. there exist a Lie group H and a smooth mapping f : H * G such that a) f is injective,
b) f (H) = H, c) df is injective,
d) f is a group homomorphism.
Theorem (FreudenthalYamabe). Let G be a Lie group and H C G a subgroup with the following property: Each element of H can be connected with the neutral element e E G by a piecewise smooth curve, and this curve lies in H. Then H is a weak Lie subgroup.
B.9. Holonomy theory Let (P,,7r, M; G) be a principal bundle and Z : TP > g a connection. (In this section, we suppose that M is connected.) Take, moreover, p E P and x = ir(p). If y is a curve in M (piecewise smooth) starting and ending at x, then we can consider the parallel displacement ryPx  *Px. Let ry(p) = p gy. Since ry(p h) = Ty(p) h = p gy h for all h, it P5 is completely described by gy E G. The {g E G : there exists a loopy at x with ry(p) = p g} is an set 0(p) (algebraic) subgroup of G, obviously follows that Ty : Px
O(p) CG, p E P. In fact, if yl, 72 are two loops at x, then 7yl*y2 = Tyl o rye, and hence gy1 *72 = g
' gy2
Definition. The group O(p) is called the holonomy group of the connection Z with respect to the base point p E P. Furthermore, define
0° (p) = {g E G : 3 a loop y at x which is nullhomotopic to ry (p) = p g}.
For trivial reasons, 0°(p) C O(p) C G is a subgroup.
B. Principal Bundles and Connections
178
Proposition. 1) 0°(p) is a weak Lie subgroup of G.
2) 0°(p) is normal in /(p), and 0(p)/0°(p) is countable. Theorem (Reduction theorem of holonomy theory). Let (P, 7r, M; G) be a principal bundle with connected base Mn and Z : TP > g a connection. For a fixed point p0 E P, denote by c(po) the holonomy group and by P(po) the set
P(po) = {p E P : there exists a horizontal path from po to p }. Then (P(po), 7r, M; O(po)) is a reduction of the principal bundle (P, 7r, M; G), and the connection Z reduces to this bundle.
Theorem (AmbroseSinger, 1953). Let (P, 7r, M; G) be a principal bundle, M connected, and let Z : TP * g be a connection with curvature form SZ = DZ. Let p0 E P be a fixed point and (P(po), ir, M, c5(po)) the reduction. Then the Lie algebra of the holonomy group ¢(p0) is generated by the elements SZ(X, Y) with X, Y E (TP)p and p E P(po) .
B.10. References S. Kobayashi, K. Nomizu, Foundations of differential geometry, Volume 1, Wiley 1963.
R. Sulanke, P. Wintgen, Differentialgeometrie and Faserbiindel, Deutscher Verlag der Wissenschaften, 1972. D. Husemoller, Fibre bundles, McGrawHill, 1966.
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Index absolute differential, 58, 166, 167 adjoint operator, 92 algebra of complex numbers, 9 algebra of quaternions, 8 almostcomplex manifold, 73, 74 almostcomplex structure, 60, 80, 146, 147, 151
AmbroseSinger theorem, 178
anticanonical spin structure, 79, 81 associated fibration of a principle bundle, 159
associated spinor bundle, 75 associated vector bundle, 161 Bianchi identity, 168 bilinear form, 1, 7 direct sum of, 7 index of, 2 nondegenerate, 1 rank of, 2 signature of, 2 canonical basis, 2 canonical connection, 83
canonical spin structure, 79 of an Hermitian manifold, 77, 78 canonical spin(4) structure, 147, 149 Casimir operator, 86, 87 CauchyRiemann equations, 71 center of an algebra, 9 characteristic class, 108 Chern class, 108, 141, 163, 171, 172 Clifford algebra, 4, 10, 11
Clifford multiplication, 21, 32, 53, 68, 70, 133
complete Riemannian manifold, 98 complex nspinors, 14
complex projective space, 40, 42, 48, 161 complexification of a real algebra, 11 of a real quadratic form, 11 conjecture, 8 , 131 connection, 163, 165, 169 holonomy group of, 177 locally flat, 168 reduction of, 174 continuous spectrum, 91 covariant derivative, 58, 60, 67, 68, 70, 166, 167
covering spaces, 40 curvature form, 62, 135, 167, 169 curvature tensor, 62 de Rham cohomology, 171 integral, 172
determinant bundle of spin structure, 5254, 108, 113
Dirac operator, 68, 69, 71, 93, 96, 101, 107, 127, 136, 148 eigenvalues of, 113, 116, 126, 128 Gfunction for, 103 index formula for, 110 index theorem for, 109 spectrum of, 99 Dirac spinors, 14, 69, 113, 115 direct sum of bilinear forms, 7 distribution, 175 integral, 175
eigenspinor, 102, 103, 112, 114, 126 eigenvalues, 91, 126 of the Dirac operator, 113, 116, 126, 128 Einstein space, 118
8 conjecture, 131 193
Index
194
equivalent fibrations, 166 equivalent Areductions, 158 equivalent spin structures, 35 equivalent spinC structures, 51 essentially selfadjoint operator, 9294, 96 nfunction, 105, 112 for the Dirac operator, 103 exponential map, 18 exterior differential, 73
fibration, 156, 159 equivalence of, 156 locally trivial, 155 first integral, 124 frame bundle, 158 Fredholm operator, 107 FreudenthalYamabe theorem, 177 Frobenius theorem, 174 global, 176 local, 174, 175 fundamental group, 26
Kahler manifold, 61, 81, 82, 89, 116, 147, 150, 151, 153 Killing number, 116, 118, 119 Killing spinor, 116, 118, 119, 121, 124, 125, 128
imaginary, 120 real, 120
Lagrange theorem, 2 Areduction, 158, 173 equivalent, 158 Laplace operator, 71 on spinors, 68 LeviCivita connection, 57, 61, 81, 84, 113, 125, 135, 148, 164 Lie algebra of Spin(n), 17, 18 of Spinc(n), 29 linear operator, 91 locally flat connection, 168 locally trivial fibration, 155
Gaschiitz proposition, 39 gauge field theory, 130, 131 gauge group, 135 gauge transformation, 135, 169 GinzburgLandau model, 131 Gprincipal bundle, 156 Grafmannian manifold, 56, 88
manifold without boundary, 175 MaurerCartan form, 83, 164 moduli space, 140, 147 for SeibergWitten theory, 136
harmonic spinors, 81, 82 heat equation, 112 Hermitian manifold, 75, 78, 147 canonical spinC structure, 77, 78 spinor bundle of, 79 Hermitian metric, 53, 68, 74, 80 Hermitian scalar product, 24 HilbertSchmidt operator, 103, 104 HirzebruchHopf proposition, 133, 146 Hirzebruch signature theorem, 109 holonomy group of a connection, 177 homogeneous spin structure, 85, 87 homotopy classification theorem, 162 homotopy theory, reduction theorem of, 178 Hopf bundle, 161 Hopf fibration, 157, 161, 172, 173 horizontal lift, 165
orientation, 159
index, 108
of a form, 2 index formula for Dirac operators, 110 index theorem for Dirac operators, 109 integrable distribution, 175 integral de Rham cohomology, 172 integral manifold, 176 intersection form, 109, 130 isomorphic principal bundles, 157 isotropic subspace, 3
nondegenerate bilinear form, 1 null subspace, 3
parallel spine spinor, 67 parallel spinor, 67, 89 parallel spinor field, 67 parallel transport, 166, 167 Picard manifold, 172 Pin(n), 15 point spectrum, 91 Pontrjagin class, 108 principal bundle, 157 associated fibration of, 159 G, 156 isomorphic, 157
Si, 163 Z2, 163 projective space complex, 40, 42, 48, 161 real, 55, 56
qform of type p, tensorial, 165 quadratic form, 1 quaternionic structure, 29, 30, 110 in A ,,, 32, 54
rank of a bilinear form, 2 real projective space, 55, 56 real structure, 29, 30 in An, 32, 54
Index reducible solution of the SeibergWitten equation, 140, 142 reduction of a connection, 174 A, 158, 173
equivalent, 158 U(k), 48, 60, 61, 81 reduction theorem of homotopy theory, 178 Rellich lemma, 100 residual spectrum, 91 resolvent set, 92 Ricci tensor, 64, 118 Riemannian manifold, complete, 98 Riemannian metric, 141, 159 Riemannian symmetric space, 82, 87 Rokhlin's theorem, 110
S1principal bundle, 163 scalar curvature, 111, 113, 118, 135, 144,
145, 148, 149, 151 Schrodinger operator, 127 SchrodingerLichnerowicz formula, 73, 100, 110, 113, 134, 145 SchurZassenhaus proposition, 39 second StiefelWhitney class, 40 section, 156 SeibergWitten equation, 131, 134, 136, 138, 140, 153
reducible solution of, 140, 142 SeibergWitten invariant, 144146, 149, 151 SeibergWitten theory, moduli space for, 136 selfadjoint operator, 92 essentially, 9294, 96 spectral theorem for, 93 signature, 109 of a form, 2 spectral measure, 92, 93 spectral theorem for selfadjoint operators, 93
spectrum of a Dirac operator, 99 of an operator, 91, 92 sphere, 43, 88, 116, 125, 128 spin bundle, 54 spin representation, 14, 23, 25, 54, 58, 75,
133
of Spin(n), 20 spin structure, 35, 36, 3840, 4245, 47, 50, 5355, 60, 79, 113 equivalence of, 35 homogeneous, 85, 87
spinC structure, 47, 48, 50, 51, 53, 57, 60, 78, 93, 96, 111, 131, 153 canonical, 79 of an Hermitian manifold, 77, 78 determinant bundle of, 5254, 108, 113 equivalence of, 51
195
Spine (4) structure, 134, 141, 146 canonical, 147, 149 Spinc(n), 25, 26 Spinc(12) representation, 28
Spin(n), 15 spin representation of, 20
spinor bundle, 53, 78 of an Hermitian manifold, 77 spinor derivative, 59 spinor field, 67 parallel, 67 StiefelWhitney class, 163 second, 40 structure identity, 168 submanifold, 175 weak, 176 Sylow subgroup, 2, 39, 44 Sylvester's theorem, 2 symmetric operator, 69, 92, 93 symmetric space, Riemannian, 82, 87 symplectic manifold, 158 symplectic structure, 151, 159
tangent bundle, 155 tautological bundle over C?1, 161 tensor product of Z2graded algebras, 7 tensorial 1form, 62 tensorial qform of type p, 165 twistor equation, 128 twistor operator, 69, 70, 121 twistor spinor, 121, 123 2Sylow subgroup, 39, 44 U(k)reduction, 48, 60, 61, 81 unitary group, 27 universal covering, 19 of SO(n), 16 universal Gbundle, 162 vanishing theorem, 140 vector bundle, 161 associated, 161
von Neumann theorem, 92 weak submanifold, 176 Weyl spinors, 22, 32 Weyl tensor, 118, 121 Weyl theorem, 138, 172 Witt decomposition theorem, 3 Wu's proposition, 133 YangMills equation, 130
Z2principal bundle, 163 (function for the Dirac operator, 103
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