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Chiral algebras of class S

Christopher Beem, Wolfger Peelaers, Leonardo Rastelli, Balt C. van Rees

TL;DR

The paper develops a comprehensive framework for chiral algebras arising from class S 4d N=2 SCFTs, recasting their protected sectors as two-dimensional chiral algebras organized by a generalized TQFT on the UV curve. It establishes a concrete dictionary between 4d Higgs-branch data and 2d chiral generators, and shows that puncture reductions are realized by quantum Drinfeld–Sokolov reduction, with precise central-charge shifts and Schur-index behavior. The work provides explicit constructions for trinions at low rank (T₂ and T₃), conjectures a general χ[Tₙ] structure as a W-algebra, and outlines a theory-space bootstrap guided by associativity and S-duality. It also extends the formalism to cylinders and decorated caps, offering a robust algebraic handle on the full class S chiral-algebra landscape and consistency checks via dualities and index computations.

Abstract

Four-dimensional N=2 superconformal field theories have families of protected correlation functions that possess the structure of two-dimensional chiral algebras. In this paper, we explore the chiral algebras that arise in this manner in the context of theories of class S. The class S duality web implies nontrivial associativity properties for the corresponding chiral algebras, the structure of which is best summarized in the language of generalized topological quantum field theory. We make a number of conjectures regarding the chiral algebras associated to various strongly coupled fixed points.

Chiral algebras of class S

TL;DR

The paper develops a comprehensive framework for chiral algebras arising from class S 4d N=2 SCFTs, recasting their protected sectors as two-dimensional chiral algebras organized by a generalized TQFT on the UV curve. It establishes a concrete dictionary between 4d Higgs-branch data and 2d chiral generators, and shows that puncture reductions are realized by quantum Drinfeld–Sokolov reduction, with precise central-charge shifts and Schur-index behavior. The work provides explicit constructions for trinions at low rank (T₂ and T₃), conjectures a general χ[Tₙ] structure as a W-algebra, and outlines a theory-space bootstrap guided by associativity and S-duality. It also extends the formalism to cylinders and decorated caps, offering a robust algebraic handle on the full class S chiral-algebra landscape and consistency checks via dualities and index computations.

Abstract

Four-dimensional N=2 superconformal field theories have families of protected correlation functions that possess the structure of two-dimensional chiral algebras. In this paper, we explore the chiral algebras that arise in this manner in the context of theories of class S. The class S duality web implies nontrivial associativity properties for the corresponding chiral algebras, the structure of which is best summarized in the language of generalized topological quantum field theory. We make a number of conjectures regarding the chiral algebras associated to various strongly coupled fixed points.

Paper Structure

This paper contains 34 sections, 171 equations, 9 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Review of protected chiral algebras
  4. Taxonomy of Schur operators
  5. The $4d/2d$ dictionary
  6. Review of theories of class S
  7. Chiral algebras of class S
  8. A TQFT valued in chiral algebras
  9. Lagrangian class S building blocks
  10. Trinion chiral algebras
  11. The Chi[T2] chiral algebra
  12. The Chi[T3] chiral algebra
  13. A proposal for Chi[Tn]
  14. A theory space bootstrap?
  15. Reduced punctures
  16. ...and 19 more sections

Figures (9)

  • Figure 1: Elementary building blocks of a two-dimensional TQFT.
  • Figure 2: Additional building blocks of a class $\mathcal{S}$ TQFT.
  • Figure 3: Associativity of composition of $T_n$ chiral algebras.
  • Figure 4: Duality and the S-diagram.
  • Figure 5: Gluing together maximal punctures.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Conjecture 1: $T_{n\geqslant3}$ chiral algebras
  • Conjecture 2: Genus zero chiral algebras
  • Conjecture 3
  • Conjecture 4: Cylinder chiral algebra