Perturbations of Dirac operators

Details Perturbations of Dirac operators Perturbations of Dirac operators

2003-07-17

arxiv.org/abs/math/0307251v3

J.Geom.Phys. 57 (2006) 297-321

Mathematics

Differential Geometry

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

Scientific paper

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\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.

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