Low-regularity Seiberg-Witten moduli spaces on manifolds with boundary

TL;DR

This paper develops a framework for analyzing low-regularity solutions to Seiberg-Witten equations on manifolds with boundary, establishing manifold structures, boundary properties, and a gluing theorem, with new regularity and unique continuation results.

Contribution

It introduces a novel approach to Seiberg-Witten moduli spaces with low Sobolev regularity, proving they form Hilbert manifolds and establishing key boundary and gluing properties.

Findings

Moduli spaces are Hilbert manifolds in $L^2_1$ regularity.

Denseness and semi-infinite-dimensionality of boundary restrictions.

A new gluing theorem for low-regularity solutions.

Abstract

For a compact spinc manifold $X$ with boundary $b_1(\partial X)=0$, we consider moduli spaces of solutions to the Seiberg-Witten equations in a generalized double Coulomb slice in $L^2_1$ (i.e., $W^{1,2}$) Sobolev regularity. We prove they are Hilbert manifolds, prove denseness and "semi-infinite-dimensionality" properties of the restriction to $\partial X$, and establish a gluing theorem. To achieve these, we prove a general regularity theorem and a strong unique continuation principle for Dirac operators, and smoothness of a restriction map to configurations of higher regularity on the interior, all of which are of independent interest.