Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters

Laura Fredrickson, Du Pei, Wenbin Yan, Ke Ye

TL;DR

This work establishes a deep link between the quantization of wild Hitchin moduli spaces and the Coulomb branch indices of Argyres-Douglas theories on lens spaces, producing explicit wild Hitchin characters for four infinite families via fixed-point localization. It demonstrates that these characters encode geometric data of the wild moduli spaces and, in suitable limits, reproduce matrix elements of ST^kS in associated 2d chiral algebras, thereby weaving together geometric Langlands, Higgs bundle moduli, and 4d SCFT chiral algebras. The authors systematically compute the Coulomb branch indices from N=1 Lagrangian UV completions, establish a TQFT-like structure in the index, and derive fixed-point decompositions for the wild Hitchin characters. The results provide a physical realization of the geometric Langlands triangle for wild ramification and illuminate how chiral algebras control Coulomb-branch counting, with explicit identifications: Virasoro minimal models for (A1,A2N), affine sl2 at admissible levels for (A1,D2N+1), W_N for (A1,D2N), and B_N for (A1,A2N-3). These insights advance the program linking 4d SCFT data, Hitchin moduli, and 2d chiral algebras, with potential implications for quantum geometry and Langlands duality.

Abstract

We use Coulomb branch indices of Argyres-Douglas theories on $S^1 \times L(k,1)$ to quantize moduli spaces ${\cal M}_H$ of wild/irregular Hitchin systems. In particular, we obtain formulae for the "wild Hitchin characters" -- the graded dimensions of the Hilbert spaces from quantization -- for four infinite families of ${\cal M}_H$, giving access to many interesting geometric and topological data of these moduli spaces. We observe that the wild Hitchin characters can always be written as a sum over fixed points in ${\cal M}_H$ under the $U(1)$ Hitchin action, and a limit of them can be identified with matrix elements of the modular transform $ST^kS$ in certain two-dimensional chiral algebras. Although naturally fitting into the geometric Langlands program, the appearance of chiral algebras, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising.

Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters

TL;DR

This work establishes a deep link between the quantization of wild Hitchin moduli spaces and the Coulomb branch indices of Argyres-Douglas theories on lens spaces, producing explicit wild Hitchin characters for four infinite families via fixed-point localization. It demonstrates that these characters encode geometric data of the wild moduli spaces and, in suitable limits, reproduce matrix elements of ST^kS in associated 2d chiral algebras, thereby weaving together geometric Langlands, Higgs bundle moduli, and 4d SCFT chiral algebras. The authors systematically compute the Coulomb branch indices from N=1 Lagrangian UV completions, establish a TQFT-like structure in the index, and derive fixed-point decompositions for the wild Hitchin characters. The results provide a physical realization of the geometric Langlands triangle for wild ramification and illuminate how chiral algebras control Coulomb-branch counting, with explicit identifications: Virasoro minimal models for (A1,A2N), affine sl2 at admissible levels for (A1,D2N+1), W_N for (A1,D2N), and B_N for (A1,A2N-3). These insights advance the program linking 4d SCFT data, Hitchin moduli, and 2d chiral algebras, with potential implications for quantum geometry and Langlands duality.

Abstract

We use Coulomb branch indices of Argyres-Douglas theories on to quantize moduli spaces of wild/irregular Hitchin systems. In particular, we obtain formulae for the "wild Hitchin characters" -- the graded dimensions of the Hilbert spaces from quantization -- for four infinite families of , giving access to many interesting geometric and topological data of these moduli spaces. We observe that the wild Hitchin characters can always be written as a sum over fixed points in under the Hitchin action, and a limit of them can be identified with matrix elements of the modular transform in certain two-dimensional chiral algebras. Although naturally fitting into the geometric Langlands program, the appearance of chiral algebras, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising.

Paper Structure

This paper contains 34 sections, 210 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Wild Hitchin moduli space and Argyres-Douglas theories
  3. Hitchin equations from six dimensions
  4. General Argyres-Douglas theories
  5. Geometry of the wild Hitchin moduli space
  6. Quantization of ${\mathcal{M}}_H$
  7. The Coulomb branch index of AD theories from ${\mathcal{N}} = 1$ Lagrangian
  8. The construction
  9. Coulomb branch index on lens spaces
  10. Wild Hitchin characters
  11. The wild Hitchin character as a fixed-point sum
  12. The moduli space ${\mathcal{M}}_{2, 2N+1}$
  13. The moduli space ${\widetilde{{\mathcal{M}}}}_{2, 2N-1}$
  14. The moduli space ${\mathcal{M}}_{2, 2N}$
  15. The moduli space ${\widetilde{{\mathcal{M}}}}_{2, 2N-2}$
  16. ...and 19 more sections

Figures (3)

  • Figure 1: Left: the affine $A_1$ Dynkin diagram. Right: the nilpotent cone of Hitchin fibration for ${\widetilde{{\mathcal{M}}}}_{2, 1}$, consisting of two $\mathbb{C}\mathbf{P}^1$ intersecting at $O$ with intersection number $2$. Together with $P_1$, $P_2$, they comprise the three fixed points of the Hitchin moduli space ${\widetilde{{\mathcal{M}}}}_{2, 1}$.
  • Figure 2: Left: the affine $A_2$ Dynkin diagram, with Dynkin label indicated at each node. Right: the nilpotent cone of singular fibration, consisting of three $\mathbb{C}\mathbf{P}^1$ intersecting at $O$. The spheres are distorted a little to accommodate the common intersection. Together with $P_1$, $P_2$ and $P_3$, they comprise the four fixed points of the Hitchin moduli space ${\widetilde{{\mathcal{M}}}}_{2, 2}$.
  • Figure 3: The 3d mirror of $(A_1, D_{2N})$ theories. There are $N-1$ hypermultiplet between two $U(1)$ gauge nodes, and there are additional one hypermultiplet charged under each node.