Higher order BPZ equations for Liouville conformal field theory

TL;DR

This paper develops a new algebraic method to prove higher order BPZ equations in probabilistic Liouville conformal field theory, covering various parameter ranges and extending to boundary cases.

Contribution

Introduces a general algebraic mechanism to establish higher order BPZ equations in probabilistic Liouville CFT, applicable to sphere and boundary cases across parameter ranges.

Findings

BPZ equations hold on the sphere for specific gamma ranges

Method applies to boundary Liouville field theory with zero bulk cosmological constant

Proves higher order BPZ equations of orders (r,1) and (1,r)

Abstract

Inspired by some intrinsic relations between Coulomb gas integrals and Gaussian multiplicative chaos, this article introduces a general mechanism to prove BPZ equations of order $(r,1)$ and $(1,r)$ in the setting of probabilistic Liouville conformal field theory, a family of conformal field theory which depends on a parameter $\gamma \in (0,2)$. The method consists in regrouping singularities on the degenerate insertion, and transforming the proof into an algebraic problem. With this method we show that BPZ equations hold on the sphere for the parameter $\gamma \in [\sqrt{2},2)$ in the case $(r,1)$ and for $\gamma \in (0,2)$ in the case $(1,r)$. The same technique applies to the boundary Liouville field theory when the bulk cosmological constant $\mu_{\text{bulk}} = 0$, where we prove BPZ equations of order $(r,1)$ and $(1,r)$ for $\gamma \in (0,2)$.