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

A bstract In this paper we present various 4 d N N = 1 dualities involving theories obtained by gluing two E [USp(2 N )] blocks via the gauging of a common USp(2 N ) symmetry with the addition of 2 L 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(2 N ) of each E [USp(2 N )] 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 3 d version of our 4 d dualities, which now involve the N N = 4 T [SU( N )] quiver theory that is known to correspond to the 3 d S -wall. We show how these 3 d dualities correspond to the relations S 2 = − 1, S − 1 S = 1 and STS = T − 1 S − 1 T − 1 for the S and T generators of SL(2 , ℤ). These observations lead us to conjecture that E [USp(2 N )] can also be interpreted as a 4 d S -wall.