Perturbations of Dirac operators

We study general conditions under which the computations of the index of a perturbed Dirac operator Ds=D+sZDs=D+sZ localize to the singular set of the bundle endomorphism ZZ in the semiclassical limit s→∞s→∞. We show how to use Witten’s method to compute the index of DD by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator ZZ. In particular, we provide examples of novel deformations of the de Rham operator to establish new results relating the Euler characteristic of a spinc manifold to maps between its even and odd spinor bundles.