Seiberg-Witten Equations on Three-Manifolds with Euclidean Ends

TL;DR

This paper develops Seiberg-Witten theory on 3-manifolds with Euclidean ends, analyzing moduli spaces and perturbations that approximate specific forms at infinity, with implications for relating 3D invariants to 4D theories.

Contribution

It introduces a framework for Seiberg-Witten equations on non-compact 3-manifolds with Euclidean ends, extending the theory and exploring moduli space structures.

Findings

Construction of Seiberg-Witten theory on Euclidean-ended 3-manifolds

Description of moduli space structures under specific perturbations

Connections to Taubes's program and singular Gromov invariants

Abstract

We construct the Seiberg-Witten theory on 3-manifolds with Euclidean ends (connected sums of $\R^3$ and a compact manifold) with perturbations which approximate $*dx_3$ at infinity, and describe the structure of the moduli spaces. The setup is inspired by Taubes's program of relating the 4-dimensional Seiberg-Witten invariant with `singular Gromov invariants' and has related applications.