The $SL(2,\mathbb{Z})$ dualization algorithm at work

Recently an algorithm to dualize a theory into its mirror dual has been proposed, both for $3d$ $\mathcal{N}=4$ linear quivers and for their $4d$ $\mathcal{N}=1$ 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 $SL(2,\mathbb{Z})$ in $3d$ and $PSL(2,\mathbb{Z})$ in $4d$. This also produces dualities for $3d$ $\mathcal{N}=3$ theories with Chern--Simons couplings, some of which have enhanced $\mathcal{N}=4$ supersymmetry, and their new $4d$ $\mathcal{N}=1$ 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.