Fisher Room

While the Higgs branch of a 3d N=4 gauge theory is protected from quantum corrections and its metric is easily computable, the Coulomb branch suffers both perturbative and nonperturbative corrections, and has long remained mysterious. I will present a construction of the Coulomb branch as a complex manifold, and (in principle) as a hyperkahler manifold. In particular, holomorphic functions on the Coulomb branch come from vevs of monopole operators in a chiral ring, and it turns out that this ring has a simple, quasi-abelian description. Applying the construction to linear quiver gauge theories, one finds new descriptions of singular monopole moduli spaces. I may also touch upon relations to equivariant vortex counting, geometric representation theory, and symplectic duality.

The spectrum of BPS states in four-dimensional gauge theories and string vacua with N=2 supersymmetry is well-known to be jump across certain walls in moduli space, where bound states can decay. In this talk I will survey how the discontinuity can be understood in terms of the supersymmetric quantum mechanics of mutually non-local point particles. This physical picture 

suggests that, at any point in moduli space, the BPS spectrum can be viewed as a sum of bound states of absolutely stable `single-centered' constituents. This idea appears to be vindicated in the context of quiver moduli spaces. Finally, I shall explain how the discontinuous BPS indices can be combined into a `new' supersymmetric index, a function which sums up multi-particle state contributions and is continuous across the wall.