The role of Coulomb branches in 2D gauge theory

TL;DR

This paper constructs Coulomb branches for 3D and 4D gauge theories, characterizing their structure and functions as operators on equivariant quantum cohomology, revealing new geometric and algebraic insights.

Contribution

It provides a simple construction of Coulomb branches for gauge theories, extending their interpretation as classifying spaces and linking functions to operators on quantum cohomology.

Findings

Coulomb branches are abelian group schemes over regular adjoint orbits.

Functions on Coulomb branches act as operators on equivariant quantum cohomology.

Non-commutative versions relate to the Gamma function of the index bundle.

Abstract

I give a simple construction of certain Coulomb branches $C_{3,4}(G;E)$ of gauge theory in 3 and 4 dimensions defined by Nakajima et al. for a compact Lie group $G$ and a polarisable quaternionic representation $E$. The manifolds $C(G; 0)$ are abelian group schemes (over the bases of regular adjoint $G_c$-orbits, respectively conjugacy classes), and $C(G;E)$ is glued together from two copies of $C(G;0)$ shifted by a rational Lagrangian section $\varepsilon_V$, the Euler class of the index bundle of a polarisation $V$ of $E$. Extending the interpretation of $C_3(G;0)$ as "classifying space" for topological 2D gauge theories, I characterise functions on $C_3(G;E)$ as operators on the equivariant quantum cohomologies of $M\times V$, for all compact symplectic $G$-manifolds $M$. The non-commutative version has an analogous description in terms of the $\Gamma$-function of $V$, appearing to play the role of Fourier transformed J-function of the gauged linear Sigma-model $V/G$.