Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Infinite Chiral Symmetry in Four Dimensions

Christopher Beem, Madalena Lemos, Pedro Liendo, Wolfger Peelaers, Leonardo Rastelli, Balt C. van Rees

TL;DR

The paper establishes a robust 4d/2d correspondence: a protected, Schur-sector of any four-dimensional N=2 SCFT organizes into a two-dimensional chiral algebra, with Virasoro and affine flavor enhancements tying 4d conformal and flavor anomalies to 2d central charges and levels. This leads to universal constraints on spectra and correlators, concrete realizations in free and Lagrangian theories via BRST reduction, and a framework for understanding non-Lagrangian theories such as class S through chiral algebras and W-algebras. The work also connects chiral-algebra data to the Schur limit of the superconformal index, providing powerful tests in explicit examples (SU(2) SQCD, N=4 SYM, and higher-rank theories) and offering conjectures for the generating structures in various cases. Collectively, the results suggest a deep, algebraic handle on the landscape of N=2 SCFTs and open pathways toward classification via chiral-algebra methods, extending to class S and beyond.

Abstract

We describe a new correspondence between four-dimensional conformal field theories with extended supersymmetry and two-dimensional chiral algebras. The meromorphic correlators of the chiral algebra compute correlators in a protected sector of the four-dimensional theory. Infinite chiral symmetry has far-reaching consequences for the spectral data, correlation functions, and central charges of any four-dimensional theory with ${\mathcal N}=2$ superconformal symmetry.

Infinite Chiral Symmetry in Four Dimensions

TL;DR

The paper establishes a robust 4d/2d correspondence: a protected, Schur-sector of any four-dimensional N=2 SCFT organizes into a two-dimensional chiral algebra, with Virasoro and affine flavor enhancements tying 4d conformal and flavor anomalies to 2d central charges and levels. This leads to universal constraints on spectra and correlators, concrete realizations in free and Lagrangian theories via BRST reduction, and a framework for understanding non-Lagrangian theories such as class S through chiral algebras and W-algebras. The work also connects chiral-algebra data to the Schur limit of the superconformal index, providing powerful tests in explicit examples (SU(2) SQCD, N=4 SYM, and higher-rank theories) and offering conjectures for the generating structures in various cases. Collectively, the results suggest a deep, algebraic handle on the landscape of N=2 SCFTs and open pathways toward classification via chiral-algebra methods, extending to class S and beyond.

Abstract

We describe a new correspondence between four-dimensional conformal field theories with extended supersymmetry and two-dimensional chiral algebras. The meromorphic correlators of the chiral algebra compute correlators in a protected sector of the four-dimensional theory. Infinite chiral symmetry has far-reaching consequences for the spectral data, correlation functions, and central charges of any four-dimensional theory with superconformal symmetry.

Paper Structure

This paper contains 53 sections, 230 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Chiral symmetry algebras in four dimensions
  3. A brief review of chiral symmetry in two dimensions
  4. Twisted conformal subalgebras
  5. Intermezzo: translation invariance from cohomology
  6. Holomorphy from cohomology
  7. The cohomology classes of local operators
  8. A chiral operator product expansion
  9. The SCFT/chiral algebra correspondence
  10. Schur operators
  11. Supermultiplets of Schur type
  12. Notable elements of the chiral algebra
  13. Virasoro enhancement of the $\mathfrak{sl}(2)$ symmetry
  14. Affine enhancement of the flavor symmetry
  15. The Hall-Littlewood chiral ring and chiral algebra generators
  16. ...and 38 more sections

Figures (1)

  • Figure 1: Weak coupling limits of the genus two class $\mathcal{S}$ theory.

Theorems & Definitions (4)

  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Conjecture 4