We consider topologically twisted $N=4$ supersymmetric Yang-Mills theory on a four-manifold of the form $V=W\times {\mathbb{R}}_{+}$ 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.

• Received 5 June 2012

DOI:https://doi.org/10.1103/PhysRevD.86.085003