Perturbations of Dirac operators

Igor Prokhorenkov; Ken Richardson

doi:10.1016/j.geomphys.2006.03.004

paper

Perturbations of Dirac operators

arXiv:math/0307251 · doi:10.1016/j.geomphys.2006.03.004

Abstract

We study general conditions under which the computations of the index of a perturbed Dirac operator $D_{s} = D + s Z$ localize to the singular set of the bundle endomorphism $Z$ in the semi-classical limit $s \to \infty$ . 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.

34 pages, improved results, new applications, literature list updated