The Coulomb branch of 3d N=4 theories

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.