A Localization Theorem for Dirac operators

We study perturbed Dirac operators of the form $D_s= D + s\A :\Gamma(E^0)\rightarrow \Gamma(E^1)$ 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 $D_s\psi=0$ 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.