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

TL;DR

The paper introduces a comprehensive framework for the quantum-corrected Coulomb branch of 3d N=4 gauge theories by constructing an abelianized patch where monopole operators and vectormultiplet scalars generate the holomorphic functions on the Coulomb branch. It establishes a nonrenormalization-based abelianization map, develops a canonical Omega-background quantization, and builds a twistor-space description that unifies all complex structures to recover the full hyperkähler metric. The authors verify the construction across diverse theories, including SQCD and linear unitary quivers, showing consistency with Bogomolnyi/Nahm descriptions and with known monopole moduli spaces. They further extend the formalism to nonabelian theories, predicted monopole operators, and higher-dimensional generalizations, linking to geometric structures such as affine Grassmannian slices and Yangians. The work provides a practical, structurally rich toolkit for analyzing Coulomb branches, their quantization, and their deep connections to integrable systems and string-theoretic dualities, with explicit results for key theories like SQED, Atiyah–Hitchin, and T[U(n+1)].

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 hyperkähler 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.

Figures (6)

• Figure 1: The quiver gauge theory $T^\nu_\mu$.
• Figure 2: A type IIB brane construction for a 3d quiver gauge theory, and the transition that leads to monopoles and to Nahm equations (Section \ref{['sec:slices']}). The $x^3-x^7$ plane is shown, with D3 branes has horizontal line segments, NS5 branes as vertical lines, and D5 branes as crosses. Here $\mu$ and $\nu$ are written as partitions (Young diagrams).
• Figure 3: The balanced abelian quiver, with $\nu=(2,1,...,1)$ and $\mu=0$
• Figure 4: The $T[U(n+1)]$ quiver, with $\nu=(n+1)$ and $\mu=0$
• Figure 5: Quivers whose Coulomb branches are intermediate nilpotent orbits.