Coulomb Branches of Noncotangent Type: a Physics Perspective

TL;DR

This paper investigates the Coulomb branch of 3D $ ext{N}=4$ gauge theories with noncotangent matter, extending partition function methods to include half-hypermultiplets and analyzing the resulting algebraic structures and specific examples.

Contribution

It extends hemisphere partition function techniques to noncotangent matter theories, addressing parity anomalies and boundary conditions, and derives Coulomb branch operator algebras for complex gauge theories.

Findings

Derived generators and relations for Coulomb-branch algebra $\

Analyzed Coulomb branch for theories with half-integer spin representations

Validated monopole correlators and algebraic structures in specific models

Abstract

We study the Coulomb-branch sector of 3D $\mathcal{N}=4$ gauge theories with half-hypermultiplets in general pseudoreal representations $\mathbf{R}$ ("noncotangent" theories). This yields (short) quantization of the Coulomb branch and correlators of the Coulomb branch operators captured by the 1d topological sector. This is done by extending the hemisphere partition function approach to noncotangent matter. In this setting one must first cancel the parity anomaly, and overcome an obstacle that $(2,2)$ boundary conditions for half-hypers are generically incompatible with gauge symmetry. Using the Dirichlet boundary conditions for the gauge fields and a careful treatment of half-hypermultiplet boundary data, we describe the resulting shift/difference operators implementing monopole insertions (including bubbling effects) on $HS^3$, and use the $HS^3$ partition function as a natural module on which the Coulomb-branch operator algebra $\mathcal{A}_C$ is represented. As applications we derive generators and relations of $\mathcal{A}_C$ for $SU(2)$ theories with general matter (including half-integer spin representations), analyze theories with Coulomb branch $y^2=z(x^2-1)$, compute the Coulomb branch of an $A_n$ quiver with spin-$\frac32$ half-hypers, and check consistency of a general monopole-antimonopole two-point function.