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Higher equations of motion for boundary Liouville Conformal Field Theory from the Ward identities

Baptiste Cerclé

TL;DR

The work advances a rigorous probabilistic formulation of boundary Liouville CFT by deriving Ward identities and introducing descendant fields directly from correlation functions. It proves higher equations of motion at level 2 and recovers BPZ differential equations without relying on mating-of-trees methods, instead leveraging a detailed analysis of derivatives with respect to boundary insertions and the KPZ identity. The paper constructs a robust framework for correlation functions via Gaussian Free Field regularization and Gaussian Multiplicative Chaos, establishing existence, analytic extension, and delicate remainder estimates for derivative and Ward-identity computations. These results provide intrinsic, field-theoretic justification for the relationship between boundary cosmological constants and degeneracy conditions and open pathways to higher-order BPZ equations and boundary Toda CFT extensions, with potential implications for boundary Liouville gravity and conformal bootstrap in probabilistic settings.

Abstract

In this document we prove higher equations of motion at the level 2 for boundary Liouville Conformal Field Theory. As a corollary we present a new derivation of the Belavin-Polyakov-Zamolodchikov differential equations. Our method of proof does not rely on the mating of trees machinery but rather exploits the symmetries of the model through the Ward identities it satisfies. To do so we provide a definition of derivatives of the correlation functions with respect to a boundary insertion which was lacking in the existing literature, and introduce a new notion of descendant fields related to these Ward identities.

Higher equations of motion for boundary Liouville Conformal Field Theory from the Ward identities

TL;DR

The work advances a rigorous probabilistic formulation of boundary Liouville CFT by deriving Ward identities and introducing descendant fields directly from correlation functions. It proves higher equations of motion at level 2 and recovers BPZ differential equations without relying on mating-of-trees methods, instead leveraging a detailed analysis of derivatives with respect to boundary insertions and the KPZ identity. The paper constructs a robust framework for correlation functions via Gaussian Free Field regularization and Gaussian Multiplicative Chaos, establishing existence, analytic extension, and delicate remainder estimates for derivative and Ward-identity computations. These results provide intrinsic, field-theoretic justification for the relationship between boundary cosmological constants and degeneracy conditions and open pathways to higher-order BPZ equations and boundary Toda CFT extensions, with potential implications for boundary Liouville gravity and conformal bootstrap in probabilistic settings.

Abstract

In this document we prove higher equations of motion at the level 2 for boundary Liouville Conformal Field Theory. As a corollary we present a new derivation of the Belavin-Polyakov-Zamolodchikov differential equations. Our method of proof does not rely on the mating of trees machinery but rather exploits the symmetries of the model through the Ward identities it satisfies. To do so we provide a definition of derivatives of the correlation functions with respect to a boundary insertion which was lacking in the existing literature, and introduce a new notion of descendant fields related to these Ward identities.
Paper Structure (52 sections, 31 theorems, 179 equations, 1 figure)

This paper contains 52 sections, 31 theorems, 179 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The setting: boundary Liouville theory
  3. Liouville Conformal Field Theory
  4. Structure constants and BPZ differential equations
  5. Probabilistic approaches to boundary Liouville conformal field theory
  6. Mathematical derivation of higher equations of motion and BPZ differential equations
  7. Contribution of the present paper
  8. Method of proof and main results
  9. Definition of the correlation functions
  10. Derivatives of the correlation functions
  11. Ward identities and descendant fields
  12. Degenerate fields and higher equations of motion
  13. BPZ differential equations
  14. Some perspectives
  15. Higher-order BPZ equations
  16. ...and 37 more sections

Key Result

Lemma 1.1

Assume that the weights satisfy $\bm\alpha\in \mathcal{A}_{N,M}$. Then as $\rho$, $\varepsilon$ and then $\delta\to0$, the following limit exists and is finite: Moreover we have the equality in the sense of weak derivatives :

Figures (1)

  • Figure 1: The domains of integration

Theorems & Definitions (56)

  • Lemma 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Corollary 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • ...and 46 more