On differential equation on four-point correlation function in the Conformal Toda Field Theory

TL;DR

This paper investigates four-point correlation functions in Conformal Toda Field Theory, revealing they generally do not satisfy differential equations unless additional assumptions are made, and derives explicit solutions consistent with semiclassical results.

Contribution

It demonstrates the conditions under which differential equations hold for four-point functions in Conformal Toda Theory and provides explicit solutions using fusion properties.

Findings

Four-point functions with one degenerate field generally do not satisfy differential equations.

Additional assumptions are necessary for other fields to derive differential equations.

Derived three-point functions agree with semiclassical calculations.

Abstract

The properties of completely degenerate fields in the Conformal Toda Field Theory are studied. It is shown that a generic four-point correlation function that contains only one such field does not satisfy ordinary differential equation in contrast to the Liouville Field Theory. Some additional assumptions for other fields are required. Under these assumptions we write such a differential equation and solve it explicitly. We use the fusion properties of the operator algebra to derive a special set of three-point correlation function. The result agrees with the semiclassical calculations.