0 A ug 2 00 5 PERTURBATIONS OF DIRAC OPERATORS

We study general conditions under which the computations of the index of a perturbed Dirac operator Ds = 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 manifold to maps between its even and odd spinor bundles. The paper contains a list of the current literature on the subject.