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
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