On the analytical continuation of lattice Liouville theory

The path integral of Liouville theory is well understood only when the central charge c is an element of [25, infinity). 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 is an element of (-infinity, 1]. We show that the c is an element of [25, infinity) 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, infinity) to (-infinity, 1], we find an infinite number of Stokes walls (where the thimbles undergo topological rearrangements), accumulating at the destination point c is an element of (-infinity, 1], where the thimbles become equivalent to the elementary cycles.

Automated summary

We study the analytical continuation of the lattice Liouville path integral to generic values of the central charge c, with a specific focus on c in the range of (-infinity, 1]. By introducing a new integration cycle involving complex field configurations, we show that the lattice path integral, which initially requires c to be in the range of [25, infinity), can be extended. We provide an explicit formula for the new integration cycle, which is expressed as a discrete sum over elementary cycles. Our approach is compared to the Lefschetz thimbles method using a two-site toy model, revealing the accumulation of Stokes walls and the equivalence between the thimbles and elementary cycles when c is in the range of (-infinity, 1].