Spinor Pairs and the Concentration Principle for Dirac operators

Abstract

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