Abstract: In this talk, we will first review some aspects of 2d gauged linear sigma models (GLSMs). Then, we investigate phases of 3d 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 N = 2 gauge theory is, in general, different from the phase of the 3d N = 2 theory on R ^2 × S^1, which means taking phases of a 3d gauge theory do not necessarily commute with compactification. Finally, if time permits, we will also talk about Givental-Lee's and KWRZ's (Kapustin, Willet, Ruan, Zhang) quantum K theories from 3d gauge theories.
This is a joint work w/ Du Pei and Ming Zhang.