The SL(2, ℤ) dualization algorithm at work

Abstract

A bstract Recently an algorithm to dualize a theory into its mirror dual has been proposed, both for 3 d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 linear quivers and for their 4 d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 1 uplift. This mimics the manipulations done at the level of the Type IIB brane setup that engineers the 3 d 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 3 d and PSL(2 , ℤ) in 4 d . This also produces dualities for 3 d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 3 theories with Chern-Simons couplings, some of which have enhanced $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 supersymmetry, and their new 4 d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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.