Three-point Functions in Sine-Liouville Theory

TL;DR

This paper explicitly calculates three-point functions in sine-Liouville theory, confirming dualities with coset models, deriving new integral formulas, and analyzing winding number violations in string theory contexts.

Contribution

It introduces a new integral formula for three-point functions and demonstrates winding number violation patterns, advancing understanding of sine-Liouville and related theories.

Findings

Winding number conservation is violated up to (+-)1 in three-point functions.

Derived a new Dotsenko-Fateev type integral formula for generic three-point functions.

Confirmed that correlators match coset model results when winding number is conserved.

Abstract

We calculate the three-point functions in the sine-Liouville theory explicitly. The same calculation was done in the (unpublished) work of Fateev, Zamolodchikov and Zamolodchikov to check the conjectured duality between the sine-Liouville and the SL(2,R)/U(1) coset CFTs. The evaluation of correlators boils down to that of a free-field theory with a certain number of insertion of screening operators. We prove that the winding number conservation is violated up to (+-)1 in three-point functions, which is in agreement with the result of FZZ that in generic N-point correlators the winding number conservation is violated up to N-2 units. A new integral formula of Dotsenko-Fateev type is derived, using which we write down the generic three-point functions of tachyons explicitly. When the winding number is conserved, the resultant expression is shown to reproduce the correlators in the coset model correctly, including the group-theoretical factor. As an application, we also study the superstring theory on linear dilaton background which is described by super-Liouville theory. We obtain the three-point amplitude of tachyons in which the winding number conservation is violated.