On the analytical continuation of lattice Liouville theory

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.

28 pages, 9 figures; v2: minor changes, accepted version

References in corpus (2)

• Notes on Liouville Theory at c<=1
• Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model

Cited by in corpus (1)

• Probabilistic construction of non compactified imaginary Liouville field theory