A Localization Theorem for Dirac operators

Manousos Maridakis

paper

A Localization Theorem for Dirac operators

arXiv:2110.11654

Abstract

We study perturbed Dirac operators of the form $ D_s= D + s\A :Γ(E^0)\rightarrow Γ(E^1)$ over a compact Riemannian manifold $(X, g)$ with symbol $c$ and special bundle maps $\A : E^0\rightarrow E^1$ for $s >> 0$ . Under a simple algebraic criterion on the pair $(c, \A)$, solutions of $D_{s} ψ = 0$ concentrate as $s \to \infty$ around the singular set $Z_\A$ of $\A$. We prove a spectral separation property of the deformed Laplacians $D_{s}^{*} D_{s}$ and $D_{s} D_{s}^{*}$ , for $s >> 0$ . As a corollary we prove an index localization theorem.