CHAPTER 1

1.1. Introduction

1.1.1. The index of families of Dirac operators.

1.1.1.1. Let M be a smooth manifold. A generalized Dirac operator D on M is

a first order differential operator acting on sections of a complex vector bundle

V → M. It is characterized among all first order differential operators by the

property that the symbol of its square has the form

σ(D2)(ξ)

=

gM

(ξ, ξ)id + O(ξ) , ξ ∈ T

∗M

,

where gM is a Riemannian metric on the underlying manifold.

1.1.1.2. In the framework of index theory the operators have an additional sym-

metry. We assume that the bundle V has a hermitian metric. Then we can define

a L2-scalar product between compactly supported sections of V . It is generally

assumed that D is formally selfadjoint, i.e. it is symmetric on the space of smooth

sections with compact support in the interior of M.

If the dimension of M is even, then in addition we require that V has a self-

adjoint involution z ∈ End(V ) (i.e. a Z/2Z-grading) which anti-commutes with D.

Then we can decompose V = V

+

⊕ V

−

into the ±1-eigenspaces of z and write

D =

0

D−

D+

0

.

1.1.1.3. Assume that M is even-dimensional and closed. Then

D+

:

C∞(M,

V

+)

→

C∞(M,

V

−)

has a finite dimensional kernel and cokernel. By definition

index(D) :=

dim(ker(D+))

−

dim(ker(D−))

.

This number can also be written as

index(D) = TrsP ,

where P is the orthogonal projection onto the kernel of D and TrsA := TrzA.

The question of classical index theory is to compute index(D) ∈ Z in terms of

the symbol of D. It was solved by the index theorem of Atiyah-Singer [4].

1.1.1.4. Let B be some auxiliary compact topological space. Let us consider a

family (Db)b∈B of Dirac operators which is continuously parameterized by B.

Assume that M is compact and odd-dimensional. Then we can form the family

(Fb)b∈B of selfadjoint Fredholm operators on

L2(M,

V ), where Fb := Db(Db

2+1)−1/2

is defined by functional calculus. The family (Fb)b∈B is not continuous in the norm

topology of bounded operators. But for all ψ ∈

L2(M,

V ) the family (Fbψ)b∈B is a

continuous family of vectors in the Hilbert space, and the family (1 − Fb

2)b∈B

is a

norm continuous family of compact operators.

1