Boundary conditions for GL-twisted N=4 SYM

TL;DR

This paper studies boundary conditions in topologically twisted N=4 supersymmetric Yang-Mills theory on specific four-manifolds, analyzing solution spaces and explicitly finding spherically symmetric solutions on S^3.

Contribution

It classifies boundary conditions as middle-dimensional subspaces and explicitly constructs spherically symmetric solutions on S^3.

Findings

Boundary conditions form middle-dimensional subspaces of solution space

Explicit spherically symmetric solutions on S^3 are found

Solutions exhibit bifurcation behavior with boundary separation

Abstract

We consider topologically twisted N=4 supersymmetric Yang-Mills theory on a four-manifold of the form V = W \times R_+ or V = W \times I, where W is a Riemannian three-manifold. Different kinds of boundary conditions apply at infinity or at finite distance. We verify that each of these conditions defines a `middle-dimensional' subspace of the space of all bulk solutions. Taking the two boundaries of V into account should thus generically give a discrete set of solutions. We explicitly find the spherically symmetric solutions when W = S^3 endowed with the standard metric. For widely separated boundaries, these consist of a pair of solutions which coincide for a certain critical value of the boundary separation and disappear for even smaller separations.