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

V. A. Fateev^{+*}, A. V. Litvinov^{+}

^{+}L. D. Landau Institute for Theoretical Physics RAS, 142432 Chernogolovka, Russia

^{*}Laboratoire de Physique Théorique et Astroparticules, UniversitéMontpelier II, Pl.E. Bataillon, 34095 Montpelier, France

PACS: 11.25.Hf

**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.