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

V. A. Fateev, A. V. Litvinov, A. Neveu, E. Onofri

TL;DR

The paper derives an explicit differential equation of order $m+1$ for four-point Liouville correlators with a degenerate insertion $V_{- rac{mb}{2}}$, using a five-point setup with a smaller degenerate field and a torus mapping. It then constructs a family of elliptic conformal blocks with exact finite-dimensional integral representations, and shows how the four-point functions can be expressed as torus integrals after integrating over intermediate momentum, revealing a sphere–torus duality and unitarized monodromy. A parallel development identifies integrable Treibich–Verdier-type potentials leading to generalized Lamé equations, whose solutions yield elliptic blocks as multi-dimensional integrals and connect to the Liouville bootstrap via explicit integral representations. These results provide new, exact tools for Liouville four-point amplitudes, with potential applications to Liouville gravity and related CFTs, and open avenues for studying accessory parameters and higher-rank generalizations.

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.

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

TL;DR

The paper derives an explicit differential equation of order for four-point Liouville correlators with a degenerate insertion , using a five-point setup with a smaller degenerate field and a torus mapping. It then constructs a family of elliptic conformal blocks with exact finite-dimensional integral representations, and shows how the four-point functions can be expressed as torus integrals after integrating over intermediate momentum, revealing a sphere–torus duality and unitarized monodromy. A parallel development identifies integrable Treibich–Verdier-type potentials leading to generalized Lamé equations, whose solutions yield elliptic blocks as multi-dimensional integrals and connect to the Liouville bootstrap via explicit integral representations. These results provide new, exact tools for Liouville four-point amplitudes, with potential applications to Liouville gravity and related CFTs, and open avenues for studying accessory parameters and higher-rank generalizations.

Abstract

Liouville field theory on a sphere is considered. We explicitly derive a differential equation for four-point correlation functions with one degenerate field . 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.

Paper Structure

This paper contains 12 sections, 187 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Differential equation for four-point correlation function
  3. Integrable potentials and conformal blocks
  4. Concluding remarks
  5. Useful formulae
  6. $\Upsilon(x)$ function.
  7. Theta function.
  8. Weierstraß function and Lamé equation.
  9. Covariant differential operators
  10. Integral representation for solutions to the generalized Lamé heat equation
  11. Normalization integrals
  12. Integrals for conformal blocks

Figures (2)

  • Figure 1: Transformation \ref{['u-map']} maps a two-sheeted covering of the sphere with four marked points onto a torus which is represented by the parallelogram with the periods $\pi$ and $\pi\tau$. Points $x$, $0$, $1$ and $\infty$ are mapped to the points $0$, $\frac{\pi}{2}$, $\frac{\pi\tau}{2}$ and $\frac{\pi+\pi\tau}{2}$ on a torus.
  • Figure 2: Contour of integration $\mathcal{C}$ in \ref{['4point-contour-integration']}. This picture is drawn with the assumption that $b^2\ll1$. Otherwise the contour has to be deformed.