Superconformal indices for N=1 theories with multiple duals

TL;DR

This work connects 4D ${\cal N}=1$ superconformal indices to elliptic hypergeometric integrals, revealing a rich web of Seiberg dualities for SP(2N) gauge theories organized by the Weyl group of the exceptional root system $E_7$. By leveraging $W(E_7)$- and BC$_N$-root-system transformations (via Rains identities), the authors show that SP(2) with $N_f=8$ admits 72 dual theories with identical indices, and extend the construction to higher rank SP(2N) with similar dualities, including reductions to $N_f=6$ that yield 36 gauge-dual models plus an $s$-confining case. They verify index equality across duals and discuss anomaly-matching puzzles and potential nonlinear realizations that may reconcile UV and IR flavour symmetries, suggesting a deep algebraic structure behind Seiberg duality. The results illuminate new elliptic hypergeometric identities, connect to the BC$_N$ root system, and hint at broader implications for dualities and possibly AdS/CFT, motivating further exploration of superpotential distinctions and nonperturbative symmetry realizations.

Abstract

Following a recent work of Dolan and Osborn, we consider superconformal indices of four dimensional ${\mathcal N}=1$ supersymmetric field theories related by an electric-magnetic duality with the SP(2N) gauge group and fixed rank flavour groups. For the SP(2) (or SU(2)) case with 8 flavours, the electric theory has index described by an elliptic analogue of the Gauss hypergeometric function constructed earlier by the first author. Using the $E_7$-root system Weyl group transformations for this function, we build a number of dual magnetic theories. One of them was originally discovered by Seiberg, the second model was built by Intriligator and Pouliot, the third one was found by Csáki et al. We argue that there should be in total 72 theories dual to each other through the action of the coset group $W(E_7)/S_8$. For the general $SP(2N), N>1,$ gauge group, a similar multiple duality takes place for slightly more complicated flavour symmetry groups. Superconformal indices of the corresponding theories coincide due to the Rains identity for a multidimensional elliptic hypergeometric integral associated with the $BC_N$-root system.