Lagrangian correspondences and Donaldson's TQFT construction of the Seiberg-Witten invariants of 3-manifolds

TL;DR

This paper develops a gauge-theoretic framework using Morse-Bott techniques to analyze monopoles on 3-manifolds, leading to a Lagrangian correspondence approach for defining Seiberg-Witten invariants via Donaldson's TQFT.

Contribution

It introduces a novel Morse-Bott analytic method to construct Lagrangian submanifolds in vortex moduli spaces, advancing the geometric understanding of Seiberg-Witten invariants.

Findings

Boundary values form immersed Lagrangians in vortex moduli space

Intersections of Lagrangians encode Seiberg-Witten invariants

Provides detailed analytic foundation for Donaldson's TQFT approach

Abstract

Using Morse-Bott techniques adapted to the gauge-theoretic setting, we show that the limiting boundary values of the space of finite energy monopoles on a connected 3-manifold with at least two cylindrical ends provides an immersed Lagrangian submanifold of the vortex moduli space at infinity. By studying the signed intersections of such Lagrangians, we supply the analytic details of Donaldson's TQFT construction of the Seiberg-Witten invariants of a closed 3-manifold.