Appears in collection : Coulomb branches and affine quantum groups / Branches de Coulomb et groupes quantiques affines

The Coulomb branches of 3d N=4 gauge theories are certain "singular hyper-kahler" manifolds associated to a reductive group G (over complex numbers) and a finite-dimensional representation of G. These were extensively studied from the algebraic point of view in a series of my joint works with M. Finkelberg and H. Nakajima. In particular, in any complex structure they come from an affine algebraic variety, whose algebra of functions can be constructed as the equivariant Borel-Moore homology of a certain ind-scheme closely related to the affine Grassmannian of G. A somewhat similar (but more complicated) story should exist for 4d N=2 gauge theories. In this case the Coulomb branches (associated to the 4d cylinder) should again be some singular hyper-kahler manifolds, but contrary to the 3d case they look differently in different complex structures: generically they still come from an affine algebraic variety (whose algebra of functions can be realized as the equivariant K-theory of the above ind-schemes), but in some special complex structure they become projective over a half dimensional affine base (this morphism is supposed to be a classical completely integrable system whose existence was predicted by Donagi and Witten). Thus it makes no sense to talk about its algebra of functions, but we can talk about its homogeneous coordinate ring. To the best of our knowledge there is no general mathematical construction of these varieties at the moment. We propose a conjectural framework for the construction of the above ring; roughly speaking one has to apply the 3d construction to the affine Kac-Moody group associated with G. We explain how to make this construction rigorous in the case when G is a torus (in this case the answer is closely related to the so-called "Dolbeault hypertoric varieties"). This is a joint work in progress with M. Semenyakin.