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We investigate phases of 3d ${\cal N}=2$ Chern-Simons-matter theories,extending to three dimensions the celebrated correspondence between 2d gaugedWess-Zumino-Witten (GWZW) models and non-linear sigma models (NLSMs) withgeometric targets. We find that although the correspondence in 3d and 2d areclosely related by circle compactification, an important subtlety arises inthis process, changing the phase structure of the 3d theory. Namely, theeffective 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 phasesof a 3d gauge theory does not necessarily commute with compactification. Wecompute 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 willdecompose into several identical 2d gauged linear sigma models (GLSMs) for thesame target upon reduction to 2d. To illustrate this phenomenon, we investigatehow 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 radiusof $S^{1}$.

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