Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ gauge theories, II

TL;DR

This paper provides a rigorous mathematical definition of Coulomb branches for 3D $\\mathcal N=4$ gauge theories, focusing on cases where the representation decomposes into a sum of a space and its dual, advancing the mathematical understanding of these physical objects.

Contribution

It formalizes the Coulomb branch as an affine algebraic variety with a $\\mathbb C^\times$-action for specific representations, building on previous proposals.

Findings

Defines Coulomb branches as affine algebraic varieties with $\\mathbb C^\times$-action

Applies to cases where the representation is of the form $\mathbf N \oplus \mathbf N^*$

Advances the mathematical foundation of Coulomb branches in gauge theories

Abstract

Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G_c$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch, which is a noncompact hyper-K\"ahler manifold with an $\mathrm{SU}(2)$-action, possibly with singularities. We give a mathematical definition of the Coulomb branch as an affine algebraic variety with $\mathbb C^\times$-action when $\mathbf M$ is of a form $\mathbf N\oplus\mathbf N^*$, as the second step of the proposal given in arXiv:1503.03676.