Concentrating Dirac Operators and Generalized Seiberg-Witten Equations

TL;DR

This paper investigates the concentration phenomena of solutions to a class of Dirac operators and applies these results to analyze the compactness and convergence properties of solutions to generalized Seiberg-Witten equations, especially near singular sets.

Contribution

It introduces a framework for understanding solution concentration in Dirac operators with perturbations and extends these results to non-linear equations, improving convergence theorems for Seiberg-Witten solutions.

Findings

Solutions exhibit exponential decay away from concentration loci.

Weak convergence of solutions is upgraded to smooth local convergence.

The framework applies to moduli spaces of generalized Seiberg-Witten solutions.

Abstract

This article studies a class of Dirac operators of the form $D_\varepsilon= D+\varepsilon^{-1}\mathcal A$, where $\mathcal A$ is a zeroth order perturbation vanishing on a subbundle. When $\mathcal A$ satisfies certain additional assumptions, solutions of the Dirac equation have a concentration property in the limit $\varepsilon\to 0$: components of the solution orthogonal to $\ker(\mathcal A)$ decay exponentially away from the locus $\mathcal Z$ where the rank of $\ker(\mathcal A)$ jumps up. These results are extended to a class of non-linear Dirac equations. This framework is then applied to study the compactness properties of moduli spaces of solutions to generalized Seiberg-Witten equations. In particular, it is shown that for sequences of solutions which converge weakly to a $\mathbb Z_2$-harmonic spinor, certain components of the solutions concentrate exponentially around the singular set of the $\mathbb Z_2$-harmonic spinor. Using these results, the weak convergence to $\mathbb Z_2$-harmonic spinors proved in existing convergence theorems is improved to $C^\infty_{loc}$.