The FZZ-Duality Conjecture - A Proof

TL;DR

This paper proves the FZZ duality between the cigar conformal field theory and the Sine-Liouville model, confirming their equivalence through correlation functions and extending the duality to arbitrary surfaces.

Contribution

It provides a rigorous proof of the FZZ duality, connecting cigar and Sine-Liouville models via a geometric Langlands duality approach and path integral methods.

Findings

Correlation functions of tachyon vertex operators match in both models

Duality extends from genus zero to arbitrary closed surfaces

The proof leverages Liouville self-duality and geometric Langlands correspondence

Abstract

We prove that the cigar conformal field theory is dual to the Sine-Liouville model, as conjectured originally by Fateev, Zamolodchikov and Zamolodchikov. Since both models possess the same chiral algebra, our task is to show that correlations of all tachyon vertex operators agree. We accomplish this goal through an off-critical version of the geometric Langlands duality for sl(2). More explicitly, we combine the well-known self-duality of Liouville theory with an intriguing correspondence between the cigar and Liouville field theory. The latter is derived through a path integral treatment. After a very detailed discussion of genus zero amplitudes, we extend the duality to arbitrary closed surfaces.