The symbol map $\sigma$ of an elliptic operator carries essential topological and geometrical information about the underlying manifold. We investigate this connection by studying Dirac operators with a perturbation term. These operators have the form $ D_s= D + s\A :Gamma(E)\rightarrow Gamma(F)$ over a Riemannian manifold $(X, g)$ for special bundle maps $\A : E\rightarrow F$ and their behavior as $s\rightarrow \infty$ is interesting. We start with a simple algebraic criterion on the pair $(\sigma, \A)$ that insures that solutions of $D_s\psi=0$ localize as $s\to\infty$ around the singular set $Z_\A$ of $\A$. Under certain assumptions of $\A$, $Z_\A$ is a union of submanifolds, and this gives a new localization formula for the index of $D$ as a sum over contributions from the components of $Z_\A$. We give numerous examples.
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