Concentrating Dirac Operators and Generalized Seiberg-Witten Equations

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 \(C^\infty_{loc}\).