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

TL;DR

The paper formulates ${\mathcal N}=4$ SCI for all simple gauge groups as elliptic hypergeometric integrals, and uses $S$-duality as a stringent test by comparing corresponding electric/magnetic integral pairs. It proves duality in several classical and even exceptional cases (e.g., ${\rm G}_2$, ${\rm F}_4$, ${\rm SP}(2N)$ vs ${\rm SO}(2N+1)$) and derives exact degenerations to computable 3d ${\mathcal N}=2$ partition functions via hyperbolic limits, including cases with adjoint matter. The work extends to exceptional groups ${E_6},{E_7},{E_8}$ where it presents the first known elliptic hypergeometric integrals on these root systems and exact $p=0$ evaluations, reinforcing duality through explicit manipulations of the integrals. It further connects to quiver and marginal deformation contexts, suggesting deep links between SCIs, total ellipticity, and potential biorthogonal function frameworks, with implications for AdS/CFT and 3d/4d dualities. Overall, the paper provides a comprehensive, mathematically rich treatment of SCI dualities across a broad spectrum of gauge groups, including new exceptional-case identities and their physical interpretations.

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.