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

Abstract

For a compact spin c manifold X with boundary b 1 (∂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 ∂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.