Spinor pairs and the concentration principle for Dirac operators

Abstract:

We study perturbed Dirac operators of the form $D_s= D + s\mathcal {A} :\Gamma (E)\rightarrow \Gamma (F)$ over a compact Riemannian manifold $(X, g)$ with symbol $c$ and special bundle maps $\mathcal {A} : E\rightarrow F$ for $s \gg 0$. 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 $Z_{\mathcal {A}}\subset X$ of $\mathcal {A}$. We give many examples, the most interesting ones arising from a general “spinor pair” construction.