Coulomb branches of 3d N=4 gauge theories and the affine Grassmannian

We propose a mathematically rigorous definition of Coulomb branch of a 3d N=4 SUSY gauge theory, as an affine algebraic variety, based on the homology group of a variant of the affine Grassmannian. In particular, coordinate rings of various hyper-Kaehler manifolds, such as instanton moduli spaces on ALE spaces, nilpotent orbits, etc, are conjecturally given by such a construction.