The Coulomb Branch of 3d $\mathcal{N}=4$ Theories

TL;DR

This paper constructs and quantizes the Coulomb branch of 3d $ abla=4$ supersymmetric gauge theories, providing a geometric and algebraic framework that unifies complex structures and verifies results in key examples.

Contribution

It introduces a novel construction of the quantum-corrected Coulomb branch using abelianized coordinates and equivariant integration, unifying complex structures via twistor space.

Findings

Constructed the chiral ring of monopole operators.

Quantized the chiral ring in a 2d Omega background.

Verified the framework in SQCD and linear quiver examples.

Abstract

We propose a construction of the quantum-corrected Coulomb branch of a general 3d gauge theory with $\mathcal{N}=4$ supersymmetry, in terms of local coordinates associated with an abelianized theory. In a fixed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperk\"ahler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to the Bogomolnyi and/or Nahm equations.