Coulomb branches of 3-dimensional gauge theories and related structures

TL;DR

This paper reviews the mathematical construction of Coulomb branches in 3d N=4 gauge theories, explores related categorical structures, and discusses connections to the local geometric Langlands program.

Contribution

It provides a rigorous mathematical definition of Coulomb branches for 3d N=4 theories and discusses their categorical and geometric structures, extending prior physical and mathematical work.

Findings

Mathematical definition of Coulomb branches for cotangent type theories

Construction of categories of line operators in topologically twisted theories

Connection between Coulomb branches and local geometric Langlands correspondence

Abstract

These are (somewhat informal) lectures notes for the CIME summer school "Geometric Representation Theory and Gauge Theory" in June 2018. In these notes we review the results and constructions of a series of our joint papers with H.Nakajima where a mathematical definition of Coulomb branches of 3d N=4 quantum gauge theories (of cotangent type) is given. We also discuss some further constructions (such as categories of line operators in the corresponding topologically twisted theories) and briefly discuss a connection to local geometric Langlands correspondence.