3d mirror duality: From eight to zero supercharges

The low-energy dynamics of strongly coupled Quantum Field Theories displays a rich variety of interesting phenomena. These phenomena are notoriously difficult to study, but the introduction of Supersymmetry and/or Conformal symmetry offers valuable tools to probe them. Among non-perturbative phenomena, a central role is played by Infra-Red dualities where two distinct Ultra-Violet QFTs flow to the same interacting CFT in the IR. The first example of an IR duality is Seiberg duality, which relates two 4d N=1 theories and was formulated exploiting crucial properties of super- symmetric QFTs. Over time, many more IR dualities have been proposed, motivating the search for organizing principles. In this thesis we focus on mirror duality, which is an IR duality that is typical of 3d N=4 theories. Mirror duality exchanges the Higgs and the Coulomb branches of the moduli space of a 3d N=4 theory and, as a consequence, it maps local operators constructed from elementary fields into local disorder operators. Mirror duality for 3d N=4 theories was understood as being induced from S-duality acting on the Type IIB brane setups engineering these theories. It was also later understood that mirror duality for Abelian theories can be derived in a field-theoretical framework as a consequence of an elementary Abelian mirror duality, relating an SQED with one flavor to a free hypermultiplet. A first goal of this thesis is to extend this field-theoretical approach to the case of non-Abelian N=4 mirror dualities. This is achieved by developing an algorithmic strategy in which non-Abelian N=4 theories are first chopped into QFT blocks, that are collections of hypermultiplets in one-to-one correspondence with five-branes. Each block is then dualized using a set of basic dualities that mimic the action of SL(2,Z) on five-branes, and the resulting components are recombined to obtain the mirror dual of the original theory. A second objective of this thesis is the generalization of mirror dualities to N=2 theories. This problem is addressed along two complementary directions. First, the mirror dualization algorithm is extended introducing QFT blocks composing N=2 theories and their corresponding basic duality moves. Starting from the example of the N=2 adjoint SQCD, we discover that its mirror dual is a quasi-Lagrangian quiver, in the sense that it is obtained from the gauging of emergent symmetries of strongly coupled SCFTs. This feature is general to an infinite class of N=2 dualities, which we interpret as the field-theoretical realization of S-duality acting on Type IIB brane setups preserving four supercharges. In this context, we associate to an NS brane a strongly-coupled theory instead of a collection of free hypermultiplets, as in the case of eight supercharges. A second strategy to derive N=2 mirror dualities is to perform SUSY-breaking deformations of known N=4 dualities. This is a non-trivial task, as the mapping between N=2 vacua of the mirror theories is hard to achieve. We show that the result of this deformation is a novel type of mirror duality relating a N=2 Chern-Simons SQCD to a purely Abelian planar quiver theory. Finally, inspired by this N=2 mirror dualities, we propose novel N=0 mirror dualities for non-Abelian gauge theories coupled to scalars and fermions. These dualities are tested through the mapping of global symmetries, the spectrum of the lightest operators and massive phases.