Superconformal indices of ${\mathcal N}=4$ SYM field theories

TL;DR

This paper explores superconformal indices of 4d ${\mathcal N}=4$ SYM theories with various gauge groups, expressing them through elliptic hypergeometric integrals, and investigates their dualities, reductions, and new integral conjectures.

Contribution

It provides the first examples of elliptic hypergeometric integrals for certain gauge groups and studies dualities and reductions of superconformal indices in these theories.

Findings

First elliptic hypergeometric integrals for $F_4, E_6, E_7, E_8$ gauge groups.

S-duality for $G_2$ and $F_4$ indices corresponds to variable changes.

Proved equality of SCIs for $SP(2N)$ and $SO(2N+1)$ in key cases.

Abstract

Superconformal indices (SCIs) of 4d ${\mathcal N}=4$ SYM theories with simple gauge groups are described in terms of elliptic hypergeometric integrals. For $F_4, E_6, E_7, E_8$ gauge groups this yields first examples of integrals of such type. S-duality transformation for G_2 and F_4 SCIs is equivalent to a change of integration variables. Equality of SCIs for SP(2N) and SO(2N+1) group theories is proved in several important special cases. Reduction of SCIs to partition functions of 3d $\mathcal{N}=2$ SYM theories with one matter field in the adjoint representation is investigated, corresponding 3d dual partners are found, and some new related hyperbolic beta integrals are conjectured.