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On the Superconformal Index of Argyres-Douglas Theories

Matthew Buican, Takahiro Nishinaka

TL;DR

This work conjectures exact closed-form expressions for the Schur limit of the superconformal index of two infinite families of Argyres-Douglas theories, (A_1,A_{2n-3}) and (A_1,D_{2n}), by leveraging a generalized 2d q-deformed Yang-Mills framework within the class S construction. The indices are built from SU(2) representation data with irregular-puncture wavefunctions tilde f_R^{(n)}(q;x) and regular-puncture factors, capturing the emergent R-symmetry and non-Lagrangian nature of these theories. The authors perform thorough consistency checks: matching low-rank cases to known free hypermultiplet results, aligning with the Beem et al. chiral algebra correspondence in rank-one examples, validating S-duality invariance, and verifying RG flows and Cardy-like behavior in the small S^1 limit. Collectively, the results illuminate the structure of Schur operators in AD theories, hint at constrained chiral algebras, and point to broader implications for dualities and holography in strongly coupled N=2 systems.

Abstract

We conjecture a closed-form expression for the Schur limit of the superconformal index of two infinite series of Argyres-Douglas (AD) superconformal field theories (SCFTs): the (A_1,A_{2n-3}) and the (A_1,D_{2n}) theories. While these SCFTs can be realized at special points on the Coulomb branch of certain N=2 gauge theories, their superconformal R symmetries are emergent, and hence their indices cannot be evaluated by localization. Instead, we construct the (A_1, A_{2n-3}) and (A_1, D_{2n}) indices by using a relation to two-dimensional q-deformed Yang-Mills theory and data from the class S construction. Our results generalize the indices derived from the torus partition functions of the two-dimensional chiral algebras associated with the (A_1, A_3) and (A_1, D_4) SCFTs. As checks of our conjectures, we study the consistency of our results with an S-duality recently discussed by us in collaboration with Giacomelli and Papageorgakis, we reproduce known Higgs branch relations, we check consistency with a series of renormalization group flows, and we verify that the small S^1 limits of our indices reproduce expected Cardy-like behavior. We will discuss the S^1 reduction of our indices in a separate paper.

On the Superconformal Index of Argyres-Douglas Theories

TL;DR

This work conjectures exact closed-form expressions for the Schur limit of the superconformal index of two infinite families of Argyres-Douglas theories, (A_1,A_{2n-3}) and (A_1,D_{2n}), by leveraging a generalized 2d q-deformed Yang-Mills framework within the class S construction. The indices are built from SU(2) representation data with irregular-puncture wavefunctions tilde f_R^{(n)}(q;x) and regular-puncture factors, capturing the emergent R-symmetry and non-Lagrangian nature of these theories. The authors perform thorough consistency checks: matching low-rank cases to known free hypermultiplet results, aligning with the Beem et al. chiral algebra correspondence in rank-one examples, validating S-duality invariance, and verifying RG flows and Cardy-like behavior in the small S^1 limit. Collectively, the results illuminate the structure of Schur operators in AD theories, hint at constrained chiral algebras, and point to broader implications for dualities and holography in strongly coupled N=2 systems.

Abstract

We conjecture a closed-form expression for the Schur limit of the superconformal index of two infinite series of Argyres-Douglas (AD) superconformal field theories (SCFTs): the (A_1,A_{2n-3}) and the (A_1,D_{2n}) theories. While these SCFTs can be realized at special points on the Coulomb branch of certain N=2 gauge theories, their superconformal R symmetries are emergent, and hence their indices cannot be evaluated by localization. Instead, we construct the (A_1, A_{2n-3}) and (A_1, D_{2n}) indices by using a relation to two-dimensional q-deformed Yang-Mills theory and data from the class S construction. Our results generalize the indices derived from the torus partition functions of the two-dimensional chiral algebras associated with the (A_1, A_3) and (A_1, D_4) SCFTs. As checks of our conjectures, we study the consistency of our results with an S-duality recently discussed by us in collaboration with Giacomelli and Papageorgakis, we reproduce known Higgs branch relations, we check consistency with a series of renormalization group flows, and we verify that the small S^1 limits of our indices reproduce expected Cardy-like behavior. We will discuss the S^1 reduction of our indices in a separate paper.

Paper Structure

This paper contains 27 sections, 111 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Argyres-Douglas theories from 6d $(2,0)$ theories
  3. The $(A_1,A_{2n-3})$ and $(A_1,D_{2n})$ theories
  4. Motivating our conjectures for the superconformal index
  5. Consistency with known results for lower-rank cases
  6. The $(A_1,A_1)$ and $(A_1,D_2)$ theories
  7. The $(A_1,A_3)$ and $(A_1,D_4)$ theories and two-dimensional chiral algebras
  8. The $(A_1,A_3)$ theory
  9. The $(A_1,D_4)$ theory
  10. Comments on the chiral algebras of general AD theories
  11. Consistency with S-duality
  12. General consistency
  13. Flavor symmetry and Higgs branch relations
  14. The $(A_1,A_{2n-3})$ theories
  15. The $(A_1,D_{2n})$ theories
  16. ...and 12 more sections

Figures (1)

  • Figure 1: The $(A_3,A_3)$ theory is equivalent to the above quiver gauge theory in which a diagonal $SU(2)$ flavor symmetry of the two $(A_1,D_4)$ theories are gauged by an $SU(2)$ vector multiplet coupled to a fundamental hypermultiplet.