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

Understand

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.