This is a joint work with M.Finkelberg and
H.Nakajima
To a (reductive) group G and a symplectic representation M of G,
physicists associate an (N=4) super-symmetric gauge theory in 3
dimensions. Any such theory is supposed to give rise to two (affine)
singular symplectic moduli spaces - the Higgs branch and the Coulomb
branch. These spaces are related by a certain mysterious symplectic
duality (also called Gale duality) studied in mathematical literature
by many authors.
While the Higgs branch is easy to define (usually, it is just the
symplectic reduction of M by G), the Coulomb branch has no
mathematically rigorous definition.The purpose of the talk will be to
suggest such a definition in the case when M is of the form N⊕ N* for
some representation V of G, and to show that in many examples it gives
the right answer. The suggestion is based on the geometry of the
affine Grassmannian of G.