Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices

TL;DR

This work advances the classification and testing of Seiberg dualities for 4d ${\mathcal N}=1$ theories with orthogonal gauge groups by formulating their superconformal indices as elliptic hypergeometric integrals and proving, or conjecturing, a wide variety of dualities including many spinor-matter cases. It shows how dualities reduce from $SP(2N)$ to $SO(N)$ through precise parameter restrictions in EHIs, and analyzes the Witten anomaly via SCI behavior, with $SO$ theories avoiding the anomaly while $SP$ cases reveal subtle reductions. The paper also connects these higher-dimensional dualities to lower-dimensional physics: reductions to 3d hyperbolic integrals yield knot-state integrals and 2d vortex partition functions, and a two-parameter elliptic Selberg framework hints at elliptic deformations of 2d CFT and related matrix models. Overall, the results provide a rigorous, mathematically rich blueprint linking 4d dualities, special functions, knot theory, and 2d/3d correspondences, while outlining numerous conjectural EHI identities needing rigorous proofs.

Abstract

We consider Seiberg electric-magnetic dualities for 4d $\mathcal{N}=1$ SYM theories with SO(N) gauge group. For all such known theories we construct superconformal indices (SCIs) in terms of elliptic hypergeometric integrals. Equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals. In particular, we describe a number of new elliptic beta integrals associated with the s-confining theories with the spinor matter fields. Reductions of some dualities from SP(2N) to SO(2N) or SO(2N+1) gauge groups are described. Interrelation of SCIs and the Witten anomaly is briefly discussed. Possible applications of the elliptic hypergeometric integrals to a two-parameter deformation of 2d conformal field theory and related matrix models are indicated. Connections of the reduced SCIs with the state integrals of the knot theory, generalized AGT duality for (3+3)d theories, and a 2d vortex partition function are described.