1.1. INTRODUCTION 3
18.104.22.168. The most natural interpretation of the index is as an obstruction class.
Let (Db)b∈B be a family of fiber-wise generalized Dirac operator on a fiber bundle
E → B with closed nonzero-dimensional fibers over a compact base B. Then it is
a natural question if there exists a family (Qb)b∈B of selfadjoint integral operators
with smooth integral kernels (which are odd with respect to the Z/2Z-grading in the
even-dimensional case) such that the perturbed family (Db + Qb)b∈B is invertible
for every b. We call such a family (Qb)b∈B a taming.
We have the following assertion (see Lemma 2.2.6): The family (Db)b∈B admits
a taming if and only if index((Db)b∈B) = 0.
1.1.2. Local index theory for families.
22.214.171.124. Let D be a generalized Dirac operator on an even-dimensional closed man-
ifold. By the McKean-Singer formula we can write
where t 0. The heat operator
has a smooth integral kernel
so that we can express the trace by an integral
(x, x) .
(x, x) ∈ End(Vx) ⊗ Λx, where Vx and Λx are the fibers of V and the
density bundle Λ → M over x ∈ M, and trs is the supertrace on End(Vx).
It is well known (see , Ch. 2.) that the local trace of the heat operator
admits an asymptotic expansion
where n = dim(M) and the coeﬃcients a2k−n(x) are locally determined by the
operator D. Thus we can write
This reduces the computation of the index to the determination of the coeﬃcient
a0 in the local heat trace asymptotic.
126.96.36.199. The determination of a0 is particularly easy for compatible Dirac operators.
These are Dirac operators which are associated to a Dirac bundle structure V on
V (see 2.1.1 for a definition). For compatible Dirac operators we have ak = 0 for
k 0, and a0 is given as follows.
Assume that M is oriented so that the density bundle is trivialized and a0 is a
highest degree form on M. If M admits a spin structure, then the Dirac bundle V is
isomorphic to a twisted spinor bundle S(M)⊗W, where S(M) is a spinor bundle of
M and W = (W, hW , ∇W ) is an auxiliary hermitian Z/2Z-graded hermitian vector
bundle with connection called the twisting bundle. Under these assumptions we
have the equality (see , Ch. 4.)
a0 = [
a formula which is usually called the local index theorem. The forms on the right-
hand side are the Chern-Weyl representatives of the corresponding characteristic
classes of TM and W associated to the Levi-Civita connection