Spinor Pairs and the Concentration Principle for Dirac operators

Abstract

We study perturbed Dirac operators of the form $ D_s= D + s{\cal A} :\Gamma(E)\rightarrow \Gamma(F)$ over a compact Riemannian manifold $(X, g)$ with symbol $c$ and special bundle maps ${\cal A} : E\rightarrow F$ for $s>>0$. Under a simple algebraic criterion on the pair $(c, {\cal A})$, solutions of $D_s\psi=0$ concentrate as $s\to\infty$ around the singular set $Z_\A\subset X$ of ${\cal A}$. We give many examples, the most interesting ones arising from a general ``spinor pair'' construction.