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

• Published in: arXiv.org
• Main Authors: Bottini, Lea E, Hwang, Chiung, Pasquetti, Sara, Sacchi, Matteo
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 10.03.2022
• Subjects: Broken symmetry; Gluing
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2110.08001

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.