The SL(2, ℤ) dualization algorithm at work

Recently an algorithm to dualize a theory into its mirror dual has been proposed, both for 3d N = 4 linear quivers and for their 4d N = 1 uplift. This mimics the manipulations done at the level of the Type IIB brane setup that engineers the 3d theories, where mirror symmetry is realized as S-duality, but it is enirely field-theoretic and based on the application of genuine infra-red dualities that implement the local action of S-duality on the quiver. In this paper, we generalize the algorithm to the full duality group, which is SL(2, ℤ) in 3d and PSL(2, ℤ) in 4d. This also produces dualities for 3d N = 3 theories with Chern-Simons couplings, some of which have enhanced N = 4 supersymmetry, and their new 4d N = 1 counterpart. In addition, we propose three ways to study the RG flows triggered by possible VEVs appearing at the last step of the algorithm, one of which uses a new duality that implements the Hanany-Witten move in field theory.