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Elliptic hypergeometry of supersymmetric dualities

V. P. Spiridonov, G. S. Vartanov

TL;DR

This work connects Seiberg-like electric-magnetic dualities with the theory of elliptic hypergeometric integrals, providing a comprehensive catalog of N=1 dualities for SU(N), SP(2N), and G2 gauge groups and expressing their superconformal indices as elliptic integrals. It proves and conjectures a host of elliptic integral identities I_E = I_M that mirror dualities, and proposes two universal conjectures linking total ellipticity to both duality identities and ’t Hooft anomaly matching. The paper not only extends known results (Römelsberger, Dolan-Osborn) but also introduces numerous new dualities, including SP-SP and SU-SU mixes, and many conjectured elliptic beta integrals on root systems. By mapping dualities to integral transformations, it reveals deep structure connecting quantum field theory with the modular properties of elliptic functions, with potential for discovering additional dualities and exact solvable identities. The work thus provides a framework for systematic exploration of dualities through elliptic hypergeometry, with implications for confinement, anomaly matching, and the algebraic structure of supersymmetric theories.

Abstract

We give a full list of known $\mathcal{N}=1$ supersymmetric quantum field theories related by the Seiberg electric-magnetic duality conjectures for $SU(N), SP(2N)$ and $G_2$ gauge groups. Many of the presented dualities are new, not considered earlier in the literature. For all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals. This gives a systematic extension of the related Romelsberger and Dolan-Osborn results. Equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals. About half of them were proven earlier, and another half represents new challenging conjectures. In particular, we conjecture a dozen new elliptic beta integrals on root systems extending the univariate elliptic beta integral discovered by the first author.

Elliptic hypergeometry of supersymmetric dualities

TL;DR

This work connects Seiberg-like electric-magnetic dualities with the theory of elliptic hypergeometric integrals, providing a comprehensive catalog of N=1 dualities for SU(N), SP(2N), and G2 gauge groups and expressing their superconformal indices as elliptic integrals. It proves and conjectures a host of elliptic integral identities I_E = I_M that mirror dualities, and proposes two universal conjectures linking total ellipticity to both duality identities and ’t Hooft anomaly matching. The paper not only extends known results (Römelsberger, Dolan-Osborn) but also introduces numerous new dualities, including SP-SP and SU-SU mixes, and many conjectured elliptic beta integrals on root systems. By mapping dualities to integral transformations, it reveals deep structure connecting quantum field theory with the modular properties of elliptic functions, with potential for discovering additional dualities and exact solvable identities. The work thus provides a framework for systematic exploration of dualities through elliptic hypergeometry, with implications for confinement, anomaly matching, and the algebraic structure of supersymmetric theories.

Abstract

We give a full list of known supersymmetric quantum field theories related by the Seiberg electric-magnetic duality conjectures for and gauge groups. Many of the presented dualities are new, not considered earlier in the literature. For all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals. This gives a systematic extension of the related Romelsberger and Dolan-Osborn results. Equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals. About half of them were proven earlier, and another half represents new challenging conjectures. In particular, we conjecture a dozen new elliptic beta integrals on root systems extending the univariate elliptic beta integral discovered by the first author.

Paper Structure

This paper contains 66 sections, 5 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. General structure of the elliptic hypergeometric integrals
  3. Superconformal index
  4. $\mathcal{N}=1$ superconformal algebra
  5. The index
  6. Calculation of the index
  7. Seiberg duality for unitary gauge groups
  8. Intriligator-Pouliot duality for symplectic gauge groups
  9. Multiple duality for $SP(2N)$ gauge group
  10. A new $SP(2N) \leftrightarrow SP(2M)$ groups duality
  11. Multiple duality for $SU(2N)$ gauge group
  12. Kutasov-Schwimmer type dualities for the unitary gauge group
  13. $SU(N)$ gauge group with the adjoint matter field
  14. Two adjoint matter fields case
  15. Generalized KS type dualities
  16. ...and 51 more sections

Key Result

Theorem 1

Given $\mathbb Z^n\to \mathbb Z$ maps $\epsilon(m^{(a)})=\epsilon(m^{(a)}_1,\dots,m^{(a)}_n)$, $a=1,\dots,M$, with finite support, define the meromorphic function Suppose $\Delta$ is a totally elliptic hypergeometric term, i.e. its $q$-certificates are $p$-elliptic functions of $q$ and $x_1,\ldots,x_n$. Then these certificates are also modular invariant.

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1: Rains, Spiridonov, 2004
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Conjecture
  • Conjecture
  • Theorem 5