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.

arXiv comment28 pages, 9 figures; v2: minor changes, accepted version

References in corpus (24)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Structure Constants and Conformal Bootstrap in Liouville Field Theory
• Biorthogonal Quantum Mechanics
• Two and three-point functions in Liouville theory
• Analytic Continuation of Liouville Theory
• On the Liouville three-point function
• Rolling Tachyons from Liouville theory
• Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models
• Liouville theory with a central charge less than one
• On three-point connectivity in two-dimensional percolation
• A conformal bootstrap approach to critical percolation in two dimensions
• Three-point functions in c <= 1 Liouville theory and conformal loop ensembles
• Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator
• Liouville Field Theory of Fluctuating Loops
• Geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: The interchiral conformal bootstrap
• Notes on Liouville Theory at c<=1
• Non-Rational 2D Quantum Gravity: I. World Sheet CFT
• On Minisuperspace Models of S-branes
• Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model
• Factorization of correlations in two-dimensional percolation on the plane and torus
• Four-point geometrical correlation functions in the two-dimensional $Q$-state Potts model: connections with the RSOS models
• Comments on the Quantum Field Theory of the Coulomb Gas Formalism
• Liouville field theory and log-correlated Random Energy Models
• Operator Product Expansion in Liouville Field Theory and Seiberg type transitions in log-correlated Random Energy Models

Cited by in corpus (1)

• Probabilistic construction of non compactified imaginary Liouville field theory