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

Abstract

For a compact spin^{c} 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 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.