On Phases of 3d ${\cal N}=2$ Chern-Simons-Matter Theories

TL;DR

This paper explores the phase structure of 3d ${ m N}=2$ Chern-Simons-matter theories, revealing subtleties in how phases behave under circle compactification and their relation to 2d models, with implications for vacua and index calculations.

Contribution

It uncovers that phases of 3d ${ m N}=2$ theories do not always commute with compactification and demonstrates the decomposition into multiple 2d GLSMs when matter charges are uniform.

Findings

Effective theories differ before and after compactification.

Vacua behavior changes with the radius of $S^{1}$.

Witten index calculations support the phase structure differences.

Abstract

We investigate phases of 3d ${\cal N}=2$ Chern-Simons-matter theories, extending to three dimensions the celebrated correspondence between 2d gauged Wess-Zumino-Witten (GWZW) models and non-linear sigma models (NLSMs) with geometric targets. We find that although the correspondence in 3d and 2d are closely related by circle compactification, an important subtlety arises in this process, changing the phase structure of the 3d theory. Namely, the effective theory obtained from the circle compactification of a phase of a 3d ${\cal N}=2$ gauge theory is, in general, different from the phase of the 3d ${\cal N}=2$ theory on ${\mathbb R}^2\times S^{1}$, which means taking phases of a 3d gauge theory does not necessarily commute with compactification. We compute the Witten index of each effective theory to check this observation. Furthermore, when the matter fields have the same non-minimal charges, the 3d ${\cal N}=2$ Chern-Simons-matter theory with a proper Chern-Simons level will decompose into several identical 2d gauged linear sigma models (GLSMs) for the same target upon reduction to 2d. To illustrate this phenomenon, we investigate how vacua of the 3d gauge theory for a weighted projective space $W\mathbb{P}_{[l,\cdots,l]}$ move on the field space when we change the radius of $S^{1}$.