Functoriality of Coulomb branches

TL;DR

This paper proves that certain Coulomb branches related to the cotangent bundle of parabolic affine spaces for GL_n and SL_n are affine closures, confirming a conjecture and showing finite generation of their function algebras.

Contribution

It establishes the functoriality of Coulomb branches under group maps and representations, and demonstrates how Coulomb branches of quivers without loops can be constructed from simpler cases.

Findings

Affine closure of cotangent bundles are Coulomb branches.

Coulomb branches for quivers without loops can be built from two-vertex cases.

Function algebras on these Coulomb branches are finitely generated.

Abstract

We prove that the affine closure of the cotangent bundle of the parabolic base affine space for $\mathrm{GL}_n$ or $\mathrm{SL}_n$ is a Coulomb branch, which confirms a conjecture of Bourget-Dancer-Grimminger-Hanany-Zhong. In particular, we show that the algebra of functions on the cotangent bundle of the parabolic base affine space of $\mathrm{GL}_n$ or $\mathrm{SL}_n$ is finitely generated. We prove this by showing that, if we are given a map $H \to G$ of complex reductive groups and a representation of $G$ satisfying an assumption we call gluable, then the Coulomb branch for the induced representation of $H$ is obtained from the corresponding Coulomb branch for $G$ by a certain Hamiltonian reduction procedure. In particular, we show that the Coulomb branch associated to any quiver with no loops can be obtained from Coulomb branches associated to quivers with exactly two vertices using this procedure.