Superconformal indices for N=1N=1 theories with multiple duals

Following a recent work of Dolan and Osborn, we consider superconformal indices of four-dimensional N=1N=1 supersymmetric field theories related by an electric–magnetic duality with the SP(2N)SP(2N) gauge group and fixed rank   flavour groups. For the SP(2)SP(2) (or SU(2)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 E7E7-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(E7)/S8W(E7)/S8. For the general SP(2N)SP(2N), N>1N>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 BCNBCN-root system.