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On the chiral algebra of Argyres-Douglas theories and S-duality

Jaewang Choi, Takahiro Nishinaka

TL;DR

This work identifies the chiral algebra for the marginal (A_3,A_3) Argyres-Douglas theory by proposing a minimal generating set from the Higgs branch and SU(2)_R current, then bootstraping the OPEs with Jacobi identities to a unique, consistent algebra. The construction is cross-validated against the Schur index, Higgs-branch relations, and BRST cohomology, and the automorphism group G is shown to realize S-duality through a map to $S_4\times\mathbb{Z}_2$ with an S_3 subgroup matching PSL(2,ℤ). These results provide a concrete link between four-dimensional S-duality and two-dimensional chiral algebras in a theory with an exactly marginal coupling, and they set the stage for generalizations to other Argyres-Douglas families via Hamiltonian reduction and BRST methods. The findings offer key insights into how duality acts on chiral algebras and pave the way for broader applications to AD theories and their conformal manifolds.

Abstract

We study the two-dimensional chiral algebra associated with the simplest Argyres-Douglas type theory with an exactly marginal coupling, i.e., the $(A_3,A_3)$ theory. Near a cusp in the space of the exactly marginal deformations (i.e., the conformal manifold), the theory is well-described by the $SU(2)$ gauge theory coupled to isolated Argyres-Douglas theories and a fundamental hypermultiplet. In this sense, the $(A_3,A_3)$ theory is an Argyres-Douglas version of the $\mathcal{N}=2$ $SU(2)$ conformal QCD. By studying its Higgs branch and Schur index, we identify the minimal possible set of chiral algebra generators for the $(A_3,A_3)$ theory, and show that there is a unique set of closed OPEs among these generators. The resulting OPEs are consistent with the Schur index, Higgs branch chiral ring relations, and the BRST cohomology conjecture. We then show that the automorphism group of the chiral algebra we constructed contains a discrete group $G$ with an $S_3$ subgroup and a homomorphism $G\to S_4 \times {\bf Z}_2$. This result is consistent with the S-duality of the $(A_3,A_3)$ theory.

On the chiral algebra of Argyres-Douglas theories and S-duality

TL;DR

This work identifies the chiral algebra for the marginal (A_3,A_3) Argyres-Douglas theory by proposing a minimal generating set from the Higgs branch and SU(2)_R current, then bootstraping the OPEs with Jacobi identities to a unique, consistent algebra. The construction is cross-validated against the Schur index, Higgs-branch relations, and BRST cohomology, and the automorphism group G is shown to realize S-duality through a map to with an S_3 subgroup matching PSL(2,ℤ). These results provide a concrete link between four-dimensional S-duality and two-dimensional chiral algebras in a theory with an exactly marginal coupling, and they set the stage for generalizations to other Argyres-Douglas families via Hamiltonian reduction and BRST methods. The findings offer key insights into how duality acts on chiral algebras and pave the way for broader applications to AD theories and their conformal manifolds.

Abstract

We study the two-dimensional chiral algebra associated with the simplest Argyres-Douglas type theory with an exactly marginal coupling, i.e., the theory. Near a cusp in the space of the exactly marginal deformations (i.e., the conformal manifold), the theory is well-described by the gauge theory coupled to isolated Argyres-Douglas theories and a fundamental hypermultiplet. In this sense, the theory is an Argyres-Douglas version of the conformal QCD. By studying its Higgs branch and Schur index, we identify the minimal possible set of chiral algebra generators for the theory, and show that there is a unique set of closed OPEs among these generators. The resulting OPEs are consistent with the Schur index, Higgs branch chiral ring relations, and the BRST cohomology conjecture. We then show that the automorphism group of the chiral algebra we constructed contains a discrete group with an subgroup and a homomorphism . This result is consistent with the S-duality of the theory.

Paper Structure

This paper contains 16 sections, 55 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. $(A_3,A_3)$ theory
  3. Quiver description and S-duality
  4. Schur index
  5. Higgs branch from 3d reduction
  6. Chiral algebra conjecture
  7. Review of the chiral algebra construction
  8. Generators
  9. OPEs among the baryonic generators
  10. Consistency with the Schur index
  11. Consistency with the Higgs branch chiral ring
  12. Consistency with BRST cohomology
  13. Automorphisms and S-duality
  14. Discussions
  15. Flavor central charges of the $(A_3,A_3)$ theory
  16. ...and 1 more sections

Figures (2)

  • Figure 1: The quiver diagram describing the $(A_3, A_3)$ theory near a cusp in the conformal manifold. The top box with $1$ in it stands for a fundamental hypermultiplet of $SU(2)$, and the left and right boxes stand for two copies of the $(A_1,D_4)$ theory. The middle circle stands for an $SU(2)$ vector multiplet which gauges the diagonal $SU(2)$ flavor symmetry of the three sectors.
  • Figure 2: The quiver diagram for the 3d reduction of the $(A_3,A_3)$ theory. Each circle stands for a $U(1)$ vector multiplet, and each edge stands for a hypermultiplet charged under the $U(1)$ gauge symmetries at its ends. Note that this quiver is self-mirror, namely its 3d mirror pair is itself. Therefore, its Coulomb and Higgs branches are isomorphic to each other.