The Seiberg-Witten Equations on Manifolds with Boundary I: The Space of Monopoles and Their Boundary Values

TL;DR

This paper investigates the structure of solutions to the Seiberg-Witten equations on 3-manifolds with boundary, showing the solution space forms a Banach manifold and analyzing boundary restrictions, laying groundwork for monopole Floer theory.

Contribution

It establishes that the monopole solution space is a Banach manifold and that boundary restrictions form a Lagrangian submanifold, advancing the understanding of boundary value problems in Seiberg-Witten theory.

Findings

Solution space of monopoles is a Banach manifold.

Boundary restriction map is a submersion onto a Lagrangian submanifold.

Framework for monopole Floer theory on manifolds with boundary.

Abstract

In this paper, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Solutions to these equations are called monopoles. Under some simple topological assumptions, we show that the solution space of all monopoles is a Banach manifold in suitable function space topologies. We then prove that the restriction of the space of monopoles to the boundary is a submersion onto a Lagrangian submanifold of the space of connections and spinors on the boundary. Both these spaces are infinite dimensional, even modulo gauge, since no boundary conditions are specified for the Seiberg-Witten equations on the 3-manifold. We study the analytic properties of these monopole spaces with an eye towards developing a monopole Floer theory for 3-manifolds with boundary, which we pursue in Part II.