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On Irregular Singularity Wave Functions and Superconformal Indices

Matthew Buican, Takahiro Nishinaka

TL;DR

The paper provides a manifestly Weyl-invariant SU(N) generalization of irregular puncture wave functions in 2D q-deformed Yang-Mills theory and uses this to derive closed-form Schur indices for all (A_{N-1}, A_{N(n-1)-1}) Argyres-Douglas theories, thereby completing the indices for the (A_N, A_M) family. The construction expresses the Schur index as I_{(A_{N-1},A_{N(n-1)-1})}(q;{f x}) = \sum_R C_R(q)\widetilde{f}^{(n)}_R(q;{f x}) with Weyl-invariant irregular wave functions \widetilde{f}^{(n)}_R(q;{f x}) and coefficients C_R(q), enabling analyses of discrete symmetries, RG flows via index residues, and S-duality actions. Consistency checks include lower-rank verifications, detailed RG-flow comparisons between 4D indices and 3D mirrors, and Cardy-like behavior at q→1; the framework also yields Type IV indices and suggests possible extensions to other Lie algebras. The results offer a compact, Lie-algebra-based computational handle on AD theories, their conformal manifolds, and duality structures, and point to rich future avenues in chiral algebras and broader algebraic generalizations.

Abstract

We generalize, in a manifestly Weyl-invariant way, our previous expressions for irregular singularity wave functions in two-dimensional SU(2) q-deformed Yang-Mills theory to SU(N). As an application, we give closed-form expressions for the Schur indices of all (A_{N-1}, A_{N(n-1)-1}) Argyres-Douglas (AD) superconformal field theories (SCFTs), thus completing the computation of these quantities for the (A_N, A_M) SCFTs. With minimal effort, our wave functions also give new Schur indices of various infinite sets of "Type IV" AD theories. We explore the discrete symmetries of these indices and also show how highly intricate renormalization group (RG) flows from isolated theories and conformal manifolds in the ultraviolet to isolated theories and (products of) conformal manifolds in the infrared are encoded in these indices. We compare our flows with dimensionally reduced flows via a simple "monopole vev RG" formalism. Finally, since our expressions are given in terms of concise Lie algebra data, we speculate on extensions of our results that might be useful for probing the existence of hypothetical SCFTs based on other Lie algebras. We conclude with a discussion of some open problems.

On Irregular Singularity Wave Functions and Superconformal Indices

TL;DR

The paper provides a manifestly Weyl-invariant SU(N) generalization of irregular puncture wave functions in 2D q-deformed Yang-Mills theory and uses this to derive closed-form Schur indices for all (A_{N-1}, A_{N(n-1)-1}) Argyres-Douglas theories, thereby completing the indices for the (A_N, A_M) family. The construction expresses the Schur index as I_{(A_{N-1},A_{N(n-1)-1})}(q;{f x}) = \sum_R C_R(q)\widetilde{f}^{(n)}_R(q;{f x}) with Weyl-invariant irregular wave functions \widetilde{f}^{(n)}_R(q;{f x}) and coefficients C_R(q), enabling analyses of discrete symmetries, RG flows via index residues, and S-duality actions. Consistency checks include lower-rank verifications, detailed RG-flow comparisons between 4D indices and 3D mirrors, and Cardy-like behavior at q→1; the framework also yields Type IV indices and suggests possible extensions to other Lie algebras. The results offer a compact, Lie-algebra-based computational handle on AD theories, their conformal manifolds, and duality structures, and point to rich future avenues in chiral algebras and broader algebraic generalizations.

Abstract

We generalize, in a manifestly Weyl-invariant way, our previous expressions for irregular singularity wave functions in two-dimensional SU(2) q-deformed Yang-Mills theory to SU(N). As an application, we give closed-form expressions for the Schur indices of all (A_{N-1}, A_{N(n-1)-1}) Argyres-Douglas (AD) superconformal field theories (SCFTs), thus completing the computation of these quantities for the (A_N, A_M) SCFTs. With minimal effort, our wave functions also give new Schur indices of various infinite sets of "Type IV" AD theories. We explore the discrete symmetries of these indices and also show how highly intricate renormalization group (RG) flows from isolated theories and conformal manifolds in the ultraviolet to isolated theories and (products of) conformal manifolds in the infrared are encoded in these indices. We compare our flows with dimensionally reduced flows via a simple "monopole vev RG" formalism. Finally, since our expressions are given in terms of concise Lie algebra data, we speculate on extensions of our results that might be useful for probing the existence of hypothetical SCFTs based on other Lie algebras. We conclude with a discussion of some open problems.

Paper Structure

This paper contains 22 sections, 84 equations, 2 figures.

Table of Contents

  1. Introduction
  2. AD theories from 6d (2,0) $A_{N-1}$ theories
  3. Irregular singularities of type I
  4. Seiberg-Witten curves of the AD theories and exactly marginal couplings
  5. Reduction to three dimensions
  6. The Schur index
  7. Motivating our generalization
  8. Some general properties
  9. Consistency checks
  10. Lower rank checks
  11. The $(A_2,A_2)$ index
  12. $(A_3,A_3)$ index
  13. $(A_2,A_5)$ theory
  14. RG flows
  15. The 3D picture and the mirror RG flow
  16. ...and 7 more sections

Figures (2)

  • Figure 1: A quiver diagram describing a weak coupling limit of the $(A_3,A_3)$ theory.
  • Figure 2: A quiver diagram describing a weak coupling limit of the $(A_2,A_5)$ theory.