On the analytical continuation of lattice Liouville theory

Abstract

The path integral of Liouville theory is well understood only when the central charge c ∈ [25, ∞). 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 ∈ (−∞, 1]. We show that the c ∈ [25, ∞) 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 the one focused on Lefschetz thimbles, by solving a two-site toy model in detail. As the parameter equivalent to c varies from [25, ∞) to (−∞, 1], we find an infinite number of Stokes walls (where the thimbles undergo topological rearrangements), accumulating at the destination point c ∈ (−∞, 1], where the thimbles become equivalent to the elementary cycles.