Notes on Integral Identities for 3d Supersymmetric Dualities

TL;DR

This paper analytically derives integral identities confirming 3d mirror dualities between certain supersymmetric theories, extending previous numerical checks with explicit hyperbolic hypergeometric integral calculations.

Contribution

It provides an analytic derivation of 3d mirror dualities for Argyres-Douglas theories using hyperbolic hypergeometric integrals, including new identities involving higher monopole superpotentials.

Findings

Analytic proof of 3d mirror duality for $A_n$ Argyres-Douglas theories.

Extension of integral identities to the $D_4$ case.

Discussion of integral identities involving higher monopole superpotentials.

Abstract

Four dimensional $\mathcal{N}=2$ Argyres-Douglas theories have been recently conjectured to be described by $\mathcal{N}=1$ Lagrangian theories. Such models, once reduced to 3d, should be mirror dual to Lagrangian $\mathcal{N}=4$ theories. This has been numerically checked through the matching of the partition functions on the three sphere. In this article, we provide an analytic derivation for this result in the $A_n$ case via hyperbolic hypergeometric integrals. We study the $D_4$ case as well, commenting on some open questions and possible resolutions. In the second part of the paper we discuss other integral identities leading to the matching of the partition functions in 3d dual pairs involving higher monopole superpotentials.