Perturbations of Dirac operators
Abstract
We study general conditions under which the computations of the index of a perturbed Dirac operator
D
s
=D+sZ
localize to the singular set of the bundle endomorphism
Z
in the semi-classical limit
s→∞
. We show how to use Witten's method to compute the index of
D
by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator
Z
. In particular, we provide examples of novel deformations of the de Rham operator to establish new results relating the Euler characteristic of a spin
c
manifold to maps between its even and odd spinor bundles. The paper contains a list of the current literature on the subject.