L²-topology and Lagrangians in the space of connections over a Riemann surface

Abstract

We examine the L²-topology of the gauge orbits over a closed Riemann surface. We prove a subtle local slice theorem based on the div-curl lemma of harmonic analysis, and deduce local pathwise connectedness of the gauge orbits. Based on a quantitative version of the connectivity, we generalize compactness results for anti-self-dual instantons with Lagrangian boundary conditions to general gauge-invariant Lagrangian submanifolds. This provides the foundation for the construction of instanton Floer homology for pairs of a 3-manifold with boundary and a Lagrangian in the configuration space over the boundary.