Superconformal indices for N=1 theories with multiple duals

TL;DR

This paper explores superconformal indices of N=1 supersymmetric theories with SP(2N) gauge groups, revealing multiple dual theories connected by Weyl group symmetries and demonstrating their index equivalence through elliptic hypergeometric identities.

Contribution

It constructs and analyzes multiple dual theories for SP(2N) gauge groups using elliptic hypergeometric functions and Weyl group symmetries, extending known dualities.

Findings

Identifies 72 dual theories for the SP(2) case with 8 flavors.

Shows superconformal indices coincide across duals via elliptic hypergeometric identities.

Extends duality framework to higher N with similar index equivalences.

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 Cs\'aki 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.