The ABCD's of Mirror Symmetry: Novel Dualities in Three Dimensions

This thesis investigates mirror symmetry in three-dimensional quantum field theories with reduced supersymmetry, focusing on theories with $\mathcal N\leq 2$. While mirror symmetry for $\mathcal N =4$ theories admits a clear geometric interpretation in terms of Type IIB brane constructions and $SL(2;\mathbb Z)$ duality, this intuition breaks down once supersymmetry is reduced since the moduli space may be lifted, and canonical brane realizations are often unavailable. Despite the breakdown of the geometric intuition afforded to us by $\mathcal N=4$ supersymmetry, we show that mirror-like dualities persist beyond the conventional $\mathcal N=4$ setting. Guided by symmetry considerations -- most notably the exchange of flavor and topological symmetries -- we develop a systematic framework to construct mirror duals of non-Abelian $\mathcal N=2$ Chern-Simons-matter theories. We analyze unitary and symplectic gauge groups, study their renormalization group flows under real mass deformations, and map gauge-invariant operators across dual descriptions. The framework further admits extensions to non-supersymmetric theories, revealing a broader notion of three-dimensional mirror symmetry as a duality between order and disorder operators.