Andy Royston (Texas A&M University)

N=2 super Yang--Mills and the differential geometry of magnetic monopoles.

The past 10 years have seen tremendous developments in four dimensional gauge theory with N=2 supersymmetry. Some of the buzzwords include BPS states, wall crossing, line defects, and the no-exotics conjecture. Our goal is to understand what these phenomena and results mean and imply for the semiclassical description of the theory---specifically, of BPS states in the theory. The latter consists of a quantum mechanics for the collective coordinates associated with monopoles and dyons, in which states are represented by geometric objects on the moduli space of monopoles. We will see, for example, that "no-exotics" implies a broad generalization of Ashoke Sen's famous prediction from 1994 concerning the existence of harmonic forms on monopole moduli space. This is based on work done in collaboration with Greg Moore and Dieter Van den Bleeken.