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Higher-spin symmetry in the $\mathfrak{sl}_3$ boundary Toda conformal field theory I: Ward identities

Baptiste Cerclé, Nathan Huguenin

TL;DR

This work rigorously establishes higher-spin symmetry for the boundary $ rak{sl}_3$ Toda CFT within a probabilistic framework based on Gaussian Free Fields and Gaussian Multiplicative Chaos. By constructing regularized descendant fields and proving both Virasoro and $W_3$ Ward identities, it shows the local and global constraints that govern correlation functions with vertex insertions on the upper half-plane. The paper lays down the mathematical machinery to translate Ward identities into differential relations among correlators and sets the stage for the computation of singular vectors and BPZ-type equations in a companion paper, ultimately contributing to the integrability program for boundary Toda theories. The results provide a rigorous foundation for higher-spin symmetry in a nontrivial boundary CFT, with implications for exact structure constants, reflection coefficients, and potential connections to boundary integrable systems. Overall, the work advances the mathematical understanding of boundary Toda CFTs and their rich symmetry structure, bridging probabilistic constructions with conformal and higher-spin algebraic frameworks.

Abstract

This article is the first of a two-part series dedicated to studying the symmetries enjoyed by the probabilistic construction of the $\mathfrak{sl}_3$ boundary Toda Conformal Field Theory. Namely in the present document we show that this model enjoys higher-spin symmetry in the form of Ward identities, both local and global. To do so we consider the $\mathfrak{sl}_3$ Toda theory on the upper-half plane and rigorously define the descendant fields associated to the Vertex Operators. We then show that we can express local as well as global Ward identities based on them, for both the stress-energy tensor and the higher-spin current that encodes this enhanced level of symmetry. This answers a question raised in the physics literature as to whether Toda theory still enjoys higher-spin symmetry in the boundary case. The second part of this series will be dedicated to computing the singular vectors of the theory and showing that they give rise to higher equations of motion as well as, under additional assumptions, BPZ-type differential equations for the correlation functions.

Higher-spin symmetry in the $\mathfrak{sl}_3$ boundary Toda conformal field theory I: Ward identities

TL;DR

This work rigorously establishes higher-spin symmetry for the boundary Toda CFT within a probabilistic framework based on Gaussian Free Fields and Gaussian Multiplicative Chaos. By constructing regularized descendant fields and proving both Virasoro and Ward identities, it shows the local and global constraints that govern correlation functions with vertex insertions on the upper half-plane. The paper lays down the mathematical machinery to translate Ward identities into differential relations among correlators and sets the stage for the computation of singular vectors and BPZ-type equations in a companion paper, ultimately contributing to the integrability program for boundary Toda theories. The results provide a rigorous foundation for higher-spin symmetry in a nontrivial boundary CFT, with implications for exact structure constants, reflection coefficients, and potential connections to boundary integrable systems. Overall, the work advances the mathematical understanding of boundary Toda CFTs and their rich symmetry structure, bridging probabilistic constructions with conformal and higher-spin algebraic frameworks.

Abstract

This article is the first of a two-part series dedicated to studying the symmetries enjoyed by the probabilistic construction of the boundary Toda Conformal Field Theory. Namely in the present document we show that this model enjoys higher-spin symmetry in the form of Ward identities, both local and global. To do so we consider the Toda theory on the upper-half plane and rigorously define the descendant fields associated to the Vertex Operators. We then show that we can express local as well as global Ward identities based on them, for both the stress-energy tensor and the higher-spin current that encodes this enhanced level of symmetry. This answers a question raised in the physics literature as to whether Toda theory still enjoys higher-spin symmetry in the boundary case. The second part of this series will be dedicated to computing the singular vectors of the theory and showing that they give rise to higher equations of motion as well as, under additional assumptions, BPZ-type differential equations for the correlation functions.

Paper Structure

This paper contains 40 sections, 19 theorems, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Higher-spin symmetry for boundary Toda CFT
  3. Two-dimensional conformal field theory
  4. Toda conformal field theories
  5. Probabilistic approaches to conformal field theory
  6. Singular vectors and towards integrability
  7. From Ward identities to differential equations
  8. Towards integrability for the boundary $\sl_3$ Toda CFT
  9. Description of the method and some technical estimates
  10. The general framework
  11. Gaussian Free Fields
  12. Gaussian Multiplicative Chaos
  13. On the Lie algebra $\mathfrak{sl}_3$
  14. Definition of the (regularized) correlation functions and descendant fields
  15. Toda correlation functions on $\mathbb{H}$
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $t\in \mathbb{R}$. For any $n\ge 2$, in the sense of weak derivatives, the conformal Ward identity holds: Moreover, for any $n\;\geqslant\;3$, the higher-spin Ward identity is valid: where for $j=1,2$, $\bm{\mathcal{W}}_{-j}^{(k)}\langle V_\beta(t)\bm{\mathrm V} \rangle\coloneqq \langle \bm{\mathrm W}_{-j}V_{\alpha_k}(z_k)\prod_{l\neq k}V_{\alpha_l}(z_l) \rangle$, and $\bm{\mathcal{W}}_{-j}^

Figures (3)

  • Figure 1: The domains $\mathbb{H}_{\delta,\varepsilon}$ and $\mathbb{R}_\varepsilon$
  • Figure 2: The contours of integration in $I_{reg}$
  • Figure 3: Integration around $t$

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: KPZ identity
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 26 more