A tale of mirror symmetry and 3d N ≤2 CS theories

This thesis investigates new directions in three-dimensional supersymmetric (SUSY) Chern--Simons (CS) theories, with a focus on the rich interplay between dualities, moduli spaces, and infrared (IR) physics. In the first part, we develop a systematic algorithmic framework for deriving the mirror duals of 3d $\mathcal{N}=2$ SQCD-CS theories with matter. This approach not only streamlines the construction of mirror pairs under 3d mirror symmetry but also yields non-trivial predictions for the IR dynamics of the theories under study. We also analyze the renormalization group (RG) flows resulting from mass deformations of the newly found dualities and further explore the vast landscape of $\mathcal{N}=2$ SQCD-CS matter theories. In the second part, we venture beyond supersymmetry and propose a family of new bosonization dualities for non-supersymmetric 3d CS-QCD theories with both bosonic and fermionic matter. We support these conjectures with a broad range of explicit examples and independent consistency checks. The third part is devoted to the moduli space geometry of \textit{improved bifundamentals}, a distinguished class of 3d $\mathcal{N}=2$ theories that arise naturally in the context of mirror symmetry with four supercharges. We perform a detailed analysis of these moduli spaces using the Hilbert series (HS) and the Superconformal Index (SCI) as primary computational tools.