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Higher-spin symmetry in the $\mathfrak{sl}_3$ boundary Toda conformal field theory II: Singular vectors and BPZ equations

Baptiste Cerclé, Nathan Huguenin

TL;DR

This article provides a rigorous, probabilistic treatment of the $ frak{sl}_3$ boundary Toda CFT and proves the existence of higher-spin ($W_3$) symmetry by constructing explicit singular vectors at levels 1–3. By expressing descendants via a Gaussian free field and Gaussian multiplicative chaos framework, the authors derive higher equations of motion and, under suitable boundary-cosmological-constant tuning, BPZ-type differential equations for correlation functions. The work connects $W$-algebra representation theory with probabilistic Toda correlators, clarifying when degeneracies yield null relations and how these constrain correlators and potential structure constants. The results offer a rigorous path toward determining boundary Toda CFT structure constants and provide a detailed, multi-level analysis of degeneracies (semi-degenerate and fully-degenerate fields) within the probabilistic setting, potentially informing integrability inquiries and reflection coefficient analyses for boundary theories.

Abstract

This article is the second chapter of a two-part series dedicated to the mathematical study of the higher-spin symmetry enjoyed by the $\mathfrak{sl}_3$ boundary Toda Conformal Field Theory. Namely, based on a probabilistic definition of this model and building on the framework introduced in the first article of this series, we compute some singular vectors of the theory which at the level of correlation functions give rise to higher equations of motion. Under additional assumptions these become BPZ-type differential equations satisfied by the correlation functions, a key feature in the perspective of a rigorous derivation of the structure constants of the theory. Such equations of motion and differential equations were previously unknown in the physics literature.

Higher-spin symmetry in the $\mathfrak{sl}_3$ boundary Toda conformal field theory II: Singular vectors and BPZ equations

TL;DR

This article provides a rigorous, probabilistic treatment of the boundary Toda CFT and proves the existence of higher-spin () symmetry by constructing explicit singular vectors at levels 1–3. By expressing descendants via a Gaussian free field and Gaussian multiplicative chaos framework, the authors derive higher equations of motion and, under suitable boundary-cosmological-constant tuning, BPZ-type differential equations for correlation functions. The work connects -algebra representation theory with probabilistic Toda correlators, clarifying when degeneracies yield null relations and how these constrain correlators and potential structure constants. The results offer a rigorous path toward determining boundary Toda CFT structure constants and provide a detailed, multi-level analysis of degeneracies (semi-degenerate and fully-degenerate fields) within the probabilistic setting, potentially informing integrability inquiries and reflection coefficient analyses for boundary theories.

Abstract

This article is the second chapter of a two-part series dedicated to the mathematical study of the higher-spin symmetry enjoyed by the boundary Toda Conformal Field Theory. Namely, based on a probabilistic definition of this model and building on the framework introduced in the first article of this series, we compute some singular vectors of the theory which at the level of correlation functions give rise to higher equations of motion. Under additional assumptions these become BPZ-type differential equations satisfied by the correlation functions, a key feature in the perspective of a rigorous derivation of the structure constants of the theory. Such equations of motion and differential equations were previously unknown in the physics literature.

Paper Structure

This paper contains 38 sections, 19 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Boundary Toda CFT and its symmetries
  3. Liouville and Toda conformal field theories
  4. Ward identities for the $\sl_3$ boundary Toda CFT
  5. Singular vectors and representation theory of $W$-algebras
  6. Representation theory of $W$-algebras
  7. Probabilistic implications
  8. Differential equations for the correlation functions
  9. From higher equations of motion to BPZ differential equations
  10. Perspectives of research: integrability of bouyndary Toda CFT
  11. Other directions of work
  12. Background and preliminary reminders
  13. Probabilistic definition of the correlation functions
  14. The probabilistic background
  15. On the Lie algebra $\mathfrak{sl}_3$
  16. ...and 23 more sections

Key Result

Theorem 1

Let $t\in \mathbb{R}$. For any $n\ge 1$ the local Ward identities hold true for the stress-energy tensor, while for the higher-spin current: Moreover the following global Ward identities are valid, for $0\;\leqslant\; n\;\leqslant\; 2$ and $0\;\leqslant\; m\;\leqslant\; 4$:

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Lemma 2
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • ...and 26 more