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Superconformal indices of three-dimensional theories related by mirror symmetry

C. Krattenthaler, V. P. Spiridonov, G. S. Vartanov

TL;DR

The paper establishes analytic proofs of the equality of superconformal indices for mirror-symmetric 3d $\mathcal{N}=2$ theories with abelian gauge groups, starting from the $N_f=1$ case and extending to arbitrary $N_f$. It employs $q$-hypergeometric identities, including Ramanujan summation and Heine transformations, to transform and equate electric and magnetic index expressions that factor into monopole sums and gauge integrals. A key feature is the treatment of the general $R$-charge parameter $h$ (via $a=q^{1-h}$) and the inclusion of the $U(1)_J$ symmetry, with results that hold beyond specific physical choices of $h$. The work reveals a structured, factorized kernel in 3d SCIs and points toward deeper connections with 4d elliptic hypergeometric integrals, while highlighting open questions about non-abelian generalizations and a full geometric interpretation of monopole contributions. Overall, it advances analytical understanding of 3d mirror symmetry through exact index equivalences grounded in $q$-special functions.

Abstract

Recently, Kim and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by the mirror symmetry. The proofs are based on the well known identities of the theory of $q$-special functions. We also suggest the general index formula taking into account the $U(1)_J$ global symmetry present for abelian theories.

Superconformal indices of three-dimensional theories related by mirror symmetry

TL;DR

The paper establishes analytic proofs of the equality of superconformal indices for mirror-symmetric 3d theories with abelian gauge groups, starting from the case and extending to arbitrary . It employs -hypergeometric identities, including Ramanujan summation and Heine transformations, to transform and equate electric and magnetic index expressions that factor into monopole sums and gauge integrals. A key feature is the treatment of the general -charge parameter (via ) and the inclusion of the symmetry, with results that hold beyond specific physical choices of . The work reveals a structured, factorized kernel in 3d SCIs and points toward deeper connections with 4d elliptic hypergeometric integrals, while highlighting open questions about non-abelian generalizations and a full geometric interpretation of monopole contributions. Overall, it advances analytical understanding of 3d mirror symmetry through exact index equivalences grounded in -special functions.

Abstract

Recently, Kim and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by the mirror symmetry. The proofs are based on the well known identities of the theory of -special functions. We also suggest the general index formula taking into account the global symmetry present for abelian theories.

Paper Structure

This paper contains 8 sections, 3 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Mirror symmetry for $d=3$ $\mathcal{N}=2$ field theory with $U(1)$-gauge group and $N_f=1$
  3. Mirror symmetry for $d=3$ $\mathcal{N}=2$ theory with $U(1)$ gauge group and $N_f=2$.
  4. Generalization to arbitrary $N_f$
  5. Concluding remarks
  6. General SCIs
  7. $N_f=1$ case
  8. $N_f=2$ case

Key Result

Theorem 1

The equality $I_{e,N_f=1}=I_{m,N_f=1}$ holds true.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3