Cluster Colloquium

Thursday 09.Apr.2026, 17:00 Central European Time

BFN Coulomb branches and cluster varieties

Alexander Shapiro
(University of Edinburgh)

Abstract. Given a quiver Γ with a dimension vector, one can consider the corresponding quantized K-theoretic Coulomb branch, a noncommutative algebra defined by Braverman, Finkelberg, and Nakajima. These algebras include many important examples, such as quantum groups, quantized Slodowy slices, and spherical DAHA.

In this talk, for each Γ I construct a quiver Q(Γ) such that the associated universally Laurent algebra is isomorphic to the quantized Coulomb branch corresponding to Γ, provided that Γ has no 1-cycles. In a sense, this construction is analogous to that of cluster varieties from moduli spaces of local systems on surfaces: in the latter, each triangulation of a surface gives rise to a quiver, and flips correspond to sequences of mutations; in our setting, the roles of triangulations and flips are played by orientations of Γ and arrow reversals.

This talk is based on the most recent version of the preprint arXiv:1910.03186, joint with Gus Schrader.