The Seiberg-Witten equations on manifolds with boundary. I: The space of monopoles and their boundary values (Q1932343)

There are several families of gauge-theoretic invariants including Donaldson theory, Seiberg-Witten theory, and Heegard-Floer theory. These three families are conjecturally equivalent, so it is natural to mimic constructions from one theory in another. One relatively new direction in Heegard-Floer theory is the development of Bordered Heegard-Floer theory. In the paper under review, T. Nguyen starts to develop the foundations for a bordered version of the Seiberg-Witten monopole homology. On a \(3\)-manifold with non-empty boundary with no boundary conditions, the space of Seiberg-Witten monopoles modulo gauge is infinite dimensional. Nevertheless it is a natural space to study. Even with a suitable gauge fixing condition the relevant equations for this study are not elliptic and are non-linear. Nguyen addresses this by studying a closely related elliptic system. There is, however, a bigger problem: one usually uses Sobolev spaces in gauge theory. For manifolds with bounday, there is no way to restrict elements of a Sobolev space to the boundary in the regularity regime studied here. For this reason, Nguyen uses Besov spaces. The main result of this paper is that for suitable function spaces, the spaces of Seiberg-Witten monopoles, monopoles satisfying a gauge fixing condition and the space of boundary values of monopoles are all Banach manifolds, and the space of boundary values is a Lagrangian subspace of the space of configurations over a surface in the sense that it is isotropic and has an isotropic complement in a natural symplectic structure. Nguyen carefuly explains why all of the technical details are required, and provides clear exposition of fairly technical material.