Notes on Integral Identities for 3d Supersymmetric Dualities

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.