On the analytical continuation of lattice Liouville theory

TL;DR

This paper explores the analytical continuation of lattice Liouville theory's path integral from well-understood regimes to more general central charge values, introducing a new integration cycle and analyzing its properties.

Contribution

It provides an explicit formula for the new integration cycle in lattice Liouville theory and compares it with Lefschetz thimbles, revealing complex topological changes as the central charge varies.

Findings

The lattice path integral can be continued to complex configurations via a new cycle.

Explicit formula for the new integration cycle as a sum over elementary cycles.

Identification of Stokes walls and topological changes in thimbles as c varies.

Abstract

The path integral of Liouville theory is well understood only when the central charge $c\in [25, \infty)$. Here, we study the analytical continuation the lattice Liouville path integral to generic values of $c$, with a particular focus on the vicinity of $c\in (-\infty, 1]$. We show that the $c\in [25, \infty)$ lattice path integral can be continued to one over a new integration cycle of complex field configurations. We give an explicit formula for the new integration cycle in terms of a discrete sum over elementary cycles, which are a direct generalization of the inverse Gamma function contour. Possible statistical interpretations are discussed. We also compare our approach to one focused on Lefschetz thimbles, by solving a two-site toy model in detail. As the parameter equivalent to $c$ varies from $[25, \infty)$ to $(-\infty, 1]$, we find an infinite number of Stokes walls (where the thimbles undergo topological rearrangements), accumulating at the destination point $c \in (-\infty, 1]$, where the thimbles become equivalent to the elementary cycles.