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

TL;DR

This paper explores elliptic hypergeometric integrals derived from supersymmetric dualities in 4d SO(N) gauge theories, revealing new mathematical identities and potential links to knot theory, conformal field theory, and vortex models.

Contribution

It constructs superconformal indices for SO(N) theories, establishes new integral identities, and connects these to various physical and mathematical frameworks.

Findings

Derived new elliptic beta integrals for s-confining theories with spinor matter.

Established equalities of superconformal indices for dual theories, leading to new mathematical identities.

Connected superconformal indices to knot invariants, conformal field theories, and vortex partition functions.

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.