Correlation functions in conformal Toda field theory I

TL;DR

This paper derives explicit three-point correlation functions in conformal Toda field theory, explores special cases and differential equations, and compares semiclassical and minisuperspace approaches to analytical results.

Contribution

It provides explicit formulas for three-point functions with special fields and connects semiclassical and minisuperspace methods with exact conformal field theory results.

Findings

Explicit three-point correlation functions derived

Differential equations solved for degenerate fields

Semiclassical and minisuperspace approaches yield comparable results

Abstract

Two-dimensional sl(n) quantum Toda field theory on a sphere is considered. This theory provides an important example of conformal field theory with higher spin symmetry. We derive the three-point correlation functions of the exponential fields if one of the three fields has a special form. In this case it is possible to write down and solve explicitly the differential equation for the four-point correlation function if the fourth field is completely degenerate. We give also expressions for the three-point correlation functions in the cases, when they can be expressed in terms of known functions. The semiclassical and minisuperspace approaches in the conformal Toda field theory are studied and the results coming from these approaches are compared with the proposed analytical expression for the three-point correlation function. We show, that in the framework of semiclassical and minisuperspace approaches general three-point correlation function can be reduced to the finite-dimensional integral.