# The Space of Vacua of 3d $\mathcal{N}=3$ Abelian Theories

## TL;DR

This paper employs brane techniques to analyze the algebraic structure of vacua in 3d $ ext{N}=3$ abelian gauge theories, revealing various branches and their operator relations without extensive computations.

## Contribution

It introduces a graphical method using brane setups to determine chiral operator bases and relations in the vacua of 3d $ ext{N}=3$ abelian theories, including new branches for circular quivers.

## Key findings

- Identified Coulomb, Higgs, and mixed branches in linear quivers.
- Discovered an extra geometric branch in circular quivers.
- Provided explicit algebraic descriptions of vacua in various examples.

## Abstract

We use brane techniques to study the space of vacua of abelian 3d $\mathcal{N}=3$ gauge theories. The coordinates on these spaces are the vevs of chiral monopole and meson operators, which are realized in the type IIB brane configuration of the theory by adding semi-infinite $(1,k)$ strings or F1 strings. The study of various brane setups allows us to determine a basis of chiral operators and chiral ring relations relevant to each branch of vacua, leading to the algebraic description of these branches. The method is mostly graphical and does not require actual computations. We apply it and provide explicit results in various examples. For linear quivers we find that the space of vacua has in general a collection of Coulomb-like branches, a Higgs branch and mixed branches. For circular quivers we find an extra branch, the geometric branch, parametrized by monopoles with equal magnetic charges in all $U(1)$ nodes and meson operators. We explain how to include FI and mass deformations. We also study $\mathcal{N}=3$ theories realized with $(p,q)$ 5-branes.