Spinor Pairs and the Concentration Principle for Dirac operators

TL;DR

This paper investigates the behavior of perturbed Dirac operators on compact manifolds, showing solutions concentrate near singular sets under certain algebraic conditions, with many examples from spinor pair constructions.

Contribution

It introduces a simple algebraic criterion for solution concentration of perturbed Dirac operators and explores numerous examples, especially from spinor pair constructions.

Findings

Solutions concentrate around singular sets as the parameter grows large.

The algebraic criterion effectively predicts concentration behavior.

Many explicit examples from spinor pair constructions are provided.

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.