Analytic Continuation of Liouville Theory

TL;DR

This paper explores the analytic continuation of Liouville theory beyond the physical region, revealing the need for multivalued solutions or reformulation as Chern-Simons theory to understand semiclassical behavior.

Contribution

It demonstrates that standard complex solutions are insufficient for semiclassical limits and proposes reformulating Liouville theory as a Chern-Simons theory or including multivalued solutions.

Findings

Semiclassical limits involve complex saddle points in some regions.

Liouville equations lack enough complex solutions outside the physical region.

Reformulation as Chern-Simons theory provides the necessary solutions.

Abstract

Correlation functions in Liouville theory are meromorphic functions of the Liouville momenta, as is shown explicitly by the DOZZ formula for the three-point function on the sphere. In a certain physical region, where a real classical solution exists, the semiclassical limit of the DOZZ formula is known to agree with what one would expect from the action of the classical solution. In this paper, we ask what happens outside of this physical region. Perhaps surprisingly we find that, while in some range of the Liouville momenta the semiclassical limit is associated to complex saddle points, in general Liouville's equations do not have enough complex-valued solutions to account for the semiclassical behavior. For a full picture, we either must include "solutions" of Liouville's equations in which the Liouville field is multivalued (as well as being complex-valued), or else we can reformulate Liouville theory as a Chern-Simons theory in three dimensions, in which the requisite solutions exist in a more conventional sense. We also study the case of "timelike" Liouville theory, where we show that a proposal of Al. B. Zamolodchikov for the exact three-point function on the sphere can be computed by the original Liouville path integral evaluated on a new integration cycle.