Analytic Continuation of Liouville Theory

TL;DR

The paper probes how Liouville theory correlators analytically continue beyond the physical region, revealing that DOZZ data necessitates complex or multivalued Liouville configurations and, in some cases, a Chern-Simons reformulation for a conventional saddle structure. It develops a detailed complex-saddle framework, including Stokes phenomena, to reproduce the DOZZ analytic continuation for two- and three-point functions and extends to four-point functions where singular saddles arise. The timelike Liouville analysis shows the exact timelike DOZZ data emerge from the same path integral on a different integration cycle, highlighting a nonunitary but conformally consistent sector. Overall, the work unifies semiclassical Liouville, complex saddles, and CS methods to understand analytic continuations and timelike regimes with implications for holography and cosmology.

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.

Figures (9)

• Figure 1: Spherical triangles.
• Figure 2: Monodromy defects, depicted as horizontal dotted lines, in $\Sigma\times I$. In the example shown, $\Sigma=\sf S^2$ and the number of monodromy defects is 3.
• Figure 3: Zeros of $H_b(x)$. The solid circles come from the zeros of $\Upsilon_b(x)$ while the empty circles come from zeros of $\Upsilon_{ib}(-ix+ib)$.
• Figure 4: The Pochhammer contour.
• Figure 5: For the Gamma function integral (\ref{['gamma1']}) to converge, the integration contour must begin and end in regions of the $\phi$ plane with $\mathrm{Re}(-z(e^{\phi} -\phi))\to-\infty$. These regions are shaded here for the case that $z$ is real and positive. In addition, we show the critical points at $\phi_n=2\pi i n$ (represented in the figure by dots) and the steepest descent cycles ${\mathcal{C}}_n$, which are the horizontal lines $\Im\,\phi=2\pi n$.