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

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    Recently an algorithm to dualize a theory into its mirror dual has been proposed, both for 3d3d N=4\mathcal{N}=4 linear quivers and for their 4d4d N=1\mathcal{N}=1 uplift. This mimics the manipulations done at the level of the Type IIB brane setup that engineers the 3d3d theories, where mirror symmetry is realized as SS-duality, but it is enirely field-theoretic and based on the application of genuine infra-red dualities that implement the local action of SS-duality on the quiver. In this paper, we generalize the algorithm to the full duality group, which is SL(2,Z)SL(2,\mathbb{Z}) in 3d3d and PSL(2,Z)PSL(2,\mathbb{Z}) in 4d4d. This also produces dualities for 3d3d N=3\mathcal{N}=3 theories with Chern--Simons couplings, some of which have enhanced N=4\mathcal{N}=4 supersymmetry, and their new 4d4d N=1\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.Comment: 79 plus 27 pages, 95 figures; v2: paragraph added to the introduction, a few figures modified, figure with an example of application of the algorithm in 3d added, references adde
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