Quiver gauge theories give two types of algebraic symplectic varieties, which are called quiver varieties and Coulomb branches respectively. The first ones were introduced by the speaker in 1994, and their homology groups are representations of Kac–Moody Lie algebras. The second ones were introduced by the speaker and Braverman, Finkelberg in 2016. The two types of varieties are very different (e.g., dimensions are different), but are expected to be related in rather mysterious ways. As an example of mysterious links, I would like to explain a conjectural realization of Kac–Moody Lie algebra representations on homology groups of Coulomb branches, which the speaker proves in affine type A. It nicely matches with geometric Satake correspondence for usual finite dimensional complex simple groups and loop groups

Cremonesi, Hanany and others recently compute Coulomb branch Hilbert series of 3-dimensional N=4 gauge theories, and write down explicit combinatorial expressions. I identify their expressions with certain motivic DT-type invariants, defined by moduli stack of bundles over the projective line together with their sections. In particular, characters of coordinate rings of many hyper-Kaehler manifolds (such as instanton moduli spaces on R^4) are conjecturally equal to those invariants.