The Seiberg-Witten equations on manifolds with boundary

Abstract

In this thesis, we undertake an in-depth study of the Seiberg-Witten equations on manifolds with boundary. We divide our study into three parts. In Part One, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Here, we study the solution space of these equations without imposing any boundary conditions. We show that the boundary values of this solution space yield an infinite dimensional Lagrangian in the symplectic configuration space on the boundary. One of the main difficulties in this setup is that the three-dimensional Seiberg-Witten equations, being a dimensional reduction of an elliptic system, fail to be elliptic, and so there are resulting technical difficulties intertwining gauge-fixing, elliptic boundary value problems, and symplectic functional analysis. In Part Two, we study the Seiberg-Witten equations on a 3-manifold with cylindrical ends. Here, Morse-Bott techniques adapted to the infinite-dimensional setting allow us to understand topologically the space of solutions to the Seiberg-Witten equations on a semiinfinite cylinder in terms of the finite dimensional moduli space of vortices at the limiting end. By combining this work with the work of Part One, we make progress in understanding how cobordisms between Riemann surfaces may provide Lagrangian correspondences between their respective vortex moduli spaces. Moreover, we apply our results to provide analytic groundwork for Donaldson's TQFT approach to the Seiberg-Witten invariants of closed 3-manifolds. Finally, in Part Three, we study analytic aspects of the Seiberg-Witten equations on a cylindrical 4-manifold supplied with Lagrangian boundary conditions of the type coming from the first part of this thesis. The resulting system of equations constitute a nonlinear infinite-dimensional nonlocal boundary value problem and is highly nontrivial. We prove fundamental elliptic regularity and compactness type results for the corresponding equations, so that these results may therefore serve as foundational analysis for constructing a monopole Floer theory on 3-manifolds with boundary.