$L^2$-topology and Lagrangians in the space of connections over a Riemann surface

TL;DR

This paper studies the $L^2$-topology of gauge orbits on a Riemann surface, establishing a local slice theorem and applying it to compactness results for instantons, aiding the construction of Floer homology.

Contribution

It introduces a new local slice theorem for gauge orbits using harmonic analysis, enabling advances in instanton theory with Lagrangian boundary conditions.

Findings

Proved a local slice theorem for gauge orbits.

Established local pathwise connectedness and quasiconvexity.

Extended compactness results to gauge invariant Lagrangian submanifolds.

Abstract

We examine the $L^2$-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 and local uniform quasiconvexity of the gauge orbits. Using these, we generalize compactness results for anti-self-dual instantons with Lagrangian boundary counditions 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.