The Seiberg-Witten Equations on Manifolds with Boundary II: Lagrangian Boundary Conditions for a Floer Theory

In this paper, we study the Seiberg-Witten equations on the product R x Y, where Y is a compact 3-manifold with boundary. Following the approach of Salamon and Wehrheim in [36] and [25] in the instanton case, we impose Lagrangian boundary conditions for the Seiberg-Witten equations. The resulting equations we obtain constitute a nonlinear, nonlocal boundary value problem. We establish regularity, compactness, and Fredholm properties for the Seiberg-Witten equations supplied with Lagrangian boundary conditions arising from the monopole spaces studied in [20]. This work therefore serves as an analytic foundation for the construction of a monopole Floer theory for 3-manifolds with boundary.