  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.
