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2d TQFT structure of the superconformal indices with outer-automorphism twists

Noppadol Mekareeya, Jaewon Song, Yuji Tachikawa

TL;DR

This work extends the 2d TQFT interpretation of 4d class S superconformal indices to include outer-automorphism twists, showing that the twisted indices equal partition functions of deformed 2d Yang-Mills with gauge group G, S-dual to the σ-fixed subalgebra of Γ. In the two-parameter deformation, the organizing mathematical objects shift from standard Macdonald polynomials to those associated with twisted affine root systems, i.e., Koornwinder-type polynomials, reflecting the folded symmetry. The paper provides explicit constructions for twisted Haar measures, vector multiplet measures, and matter contributions, and tests the framework via free hypermultiplets and the Argyres-Seiberg duality, including nontrivial checks against dual frames. The results illuminate how outer automorphisms restructure the spectrum, puncture data, and dualities in class S theories, with potential applications to non-simply-laced gauge theories and twisted TQFTs.

Abstract

We study the superconformal indices of 4d theories coming from 6d N=(2,0) theory of type Γon a Riemann surface, with the action of the outer-automorphism σin the trace. We find that the indices are given by the partition function of a deformed 2d Yang-Mills on the Riemann surface with gauge group G which is S-dual to the subgroup of Γfixed by σ. In the 2-parameter deformed version, we find that it is governed not by Macdonald polynomials of type G, but by Macdonald polynomials associated to twisted affine root systems.

2d TQFT structure of the superconformal indices with outer-automorphism twists

TL;DR

This work extends the 2d TQFT interpretation of 4d class S superconformal indices to include outer-automorphism twists, showing that the twisted indices equal partition functions of deformed 2d Yang-Mills with gauge group G, S-dual to the σ-fixed subalgebra of Γ. In the two-parameter deformation, the organizing mathematical objects shift from standard Macdonald polynomials to those associated with twisted affine root systems, i.e., Koornwinder-type polynomials, reflecting the folded symmetry. The paper provides explicit constructions for twisted Haar measures, vector multiplet measures, and matter contributions, and tests the framework via free hypermultiplets and the Argyres-Seiberg duality, including nontrivial checks against dual frames. The results illuminate how outer automorphisms restructure the spectrum, puncture data, and dualities in class S theories, with potential applications to non-simply-laced gauge theories and twisted TQFTs.

Abstract

We study the superconformal indices of 4d theories coming from 6d N=(2,0) theory of type Γon a Riemann surface, with the action of the outer-automorphism σin the trace. We find that the indices are given by the partition function of a deformed 2d Yang-Mills on the Riemann surface with gauge group G which is S-dual to the subgroup of Γfixed by σ. In the 2-parameter deformed version, we find that it is governed not by Macdonald polynomials of type G, but by Macdonald polynomials associated to twisted affine root systems.

Paper Structure

This paper contains 37 sections, 118 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Twisted superconformal indices and TQFT
  3. Superconformal indices and topological field theory
  4. Superconformal indices with outer-automorphism twist
  5. For $\Gamma\neq A_{2\ell}$:
  6. For $\Gamma=A_{2\ell}$ :
  7. Haar measures
  8. For $\Gamma\neq A_{2\ell}$:
  9. For $\Gamma=A_{2\ell}$ :
  10. Contributions from vector multiplets
  11. Without outer-automorphism twist
  12. With outer-automorphism twist
  13. Macdonald measures
  14. Vector multiplet measures
  15. For $\Gamma\neq A_{2\ell}$:
  16. ...and 22 more sections

Figures (1)

  • Figure 1: The $\mathbb{Z}_2$ action \ref{['explicitZ2actionAodd']} and \ref{['explicitZ2actionAeven']} on the matrix. The sign is plus or minus if the entry is white or black, respectively.