4d S-duality wall and SL(2, Z) relations

In this paper we present various 4dN = 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 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 S2 = −1, S−1S = 1 and STS = T−1S−1T−1 for the S and T generators of SL(2, ℤ). These observations lead us to conjecture that E[USp(2N)] can also be interpreted as a 4d S-wall.