Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks

TL;DR

This paper derives a differential equation for four-point correlation functions in Liouville field theory involving a degenerate field, introduces exactly calculable elliptic conformal blocks, and explores their bootstrap equations and relations to torus correlation functions.

Contribution

It provides explicit differential equations and finite integral representations for four-point conformal blocks in Liouville theory, advancing exact computational methods.

Findings

Derived differential equations for four-point functions with degenerate fields

Introduced finite-dimensional integral representations of elliptic conformal blocks

Established relations between sphere and torus correlation functions

Abstract

Liouville field theory on a sphere is considered. We explicitly derive a differential equation for four-point correlation functions with one degenerate field $V_{-\frac{mb}{2}}$. We introduce and study also a class of four-point conformal blocks which can be calculated exactly and represented by finite dimensional integrals of elliptic theta-functions for arbitrary intermediate dimension. We study also the bootstrap equations for these conformal blocks and derive integral representations for corresponding four-point correlation functions. A relation between the one-point correlation function of a primary field on a torus and a special four-point correlation function on a sphere is proposed.