Perturbations of Dirac operators

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 semiclassical 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 ${\mathrm{spin}}^c$ manifold to maps between its even and odd spinor bundles.