Anti-self-dual instantons with Lagrangian boundary conditions

We consider a nonlocal boundary condition for anti-self-dual instantons on four-manifolds with a space-time splitting of the boundary. This boundary condition naturally arises from making the Chern-Simons functional on a three-manifold with boundary closed: The restriction of the instanton to each time-slice of the boundary is required to lie in a Lagrangian submanifold of the moduli space of flat connections. We establish the fundamental elliptic regularity and compactness properties of this boundary value problem. Firstly, every weak solution is gauge equivalent to a smooth solution. Secondly, all closed subsets of the moduli space of solutions with an L-bound on the curvature for p > 2 are compact. We moreover establish the Fredholm property of the linearized operator of this boundary value problem on compact four-manifolds. The proofs are based on a decomposition of the instantons near the boundary. Due to the global nature of the boundary condition the crucial regularity of one of these components has to be established by studying Cauchy-Riemann equations with totally real boundary conditions for functions with values in a complex Banach space. These results provide the basic analytic set-up for the definition of a Floer homology for pairs consisting of a compact three-manifold with boundary and a Lagrangian submanifold in the moduli space of flat connections over the boundary. Such a Floer homology lies at the center of the program for the proof of the Atiyah-Floer conjecture by Salamon. The program aims to use this Floer homology as intermediate step for the conjectured natural isomorphism between the instanton Floer homology of a homology threesphere and the symplectic Floer homology of two Lagrangian submanifolds in a moduli space of flat connections arising from a Heegard splitting of the homology three-sphere. These isomorphisms should result from adiabatic limits of the boundary value problem that is studied in this thesis.