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arXiv · 2301.07454

On the analytical continuation of lattice Liouville theory

Xiangyu Cao·Raoul Santachiara·Romain Usciati
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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.

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Xiangyu Cao, Raoul Santachiara, Romain Usciati. 2023-01-18. On the analytical continuation of lattice Liouville theory. https://doi.org/10.1007/jhep03(2023)061

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