In a talk I will discuss how interesting cases of Coulomb branches arise as Hamiltonian reductions of cluster varieties. The Coulomb branches are spaces of vacua for supersymmetric gauge theories with 8 supercharges. It's been long know that for so-called class S 4d N=2 theories compactified on a circle those Coulomb branches are Fock-Goncharov cluster varieties. Recently a set of examples of Coulomb branches which have cluster structure on them was extended by both new 4d N=2 theories described through Braverman-Finkelberg-Nakajima construction and by Coulomb branches of 5d N=1 theories compactified on a torus being identified with Goncharov-Kenyon cluster integrable systems. In some cases, though, the known cluster varieties do not provide a satisfying description. So it is more natural to use Hamiltonian reductions of known cluster varieties, and show that the result is a cluster variety itself.

In my talk I will discuss in details the simplest example for 4d N=2 theory - abelian gauge theory - being a simplest possible cluster variety. And show that it is a multiplicative hypertoric variety using cluster Hamiltonian reduction. Then I will present more complicated examples, coming from 5d N=1 theories Coulomb branches being cluster Hamiltonian reductions of Goncharov-Kenyon integrable systems.