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4d $S$-duality wall and $SL(2,\mathbb{Z})$ relations

In this paper we present various $4d$ $\mathcal{N}=1$ dualities involving theories obtained by gluing two $E[USp(2N)]$ blocks via the gauging of a common $USp(2N)$ symmetry with the addition of $2L$ fundamental matter chiral fields. For $L=0$ in particular the theory has a quantum deformed moduli space with chiral symmetry breaking and its index takes the form of a delta-function. We interpret it as the Identity wall which identifies the two surviving $USp(2N)$ of each $E[USp(2N)]$ block. All the dualities are derived from iterative applications of the Intriligator--Pouliot duality. This plays for us the role of the fundamental duality, from which we derive all others. We then focus on the $3d$ version of our $4d$ dualities, which now involve the $\mathcal{N}=4$ $T[SU(N)]$ quiver theory that is known to correspond to the $3d$ $S$-wall. We show how these $3d$ dualities correspond to the relations $S^2=-1$, $S^{-1}S=1$ and $T^{-1} S T=S^{-1} T S$ for the $S$ and $T$ generators of $SL(2,\mathbb{Z})$. These observations lead us to conjecture that $E[USp(2N)]$ can also be interpreted as a $4d$ $S$-wall.