2011/08/31 by Daniel Harlow, Jonathan S. Maltz, Jonathan Maltz +1 · 325 citations
Mathematics · Physics and Astronomy · #Action (physics) #Black Holes and Theoretical Physics #Function (biology) #Limit (mathematics) #Liouville equation #Liouville field theory #Liouville's formula #Mathematical analysis #Mathematical physics #Mathematics #Meromorphic function #Path integral formulation #Physics #Quantum Chromodynamics and Particle Interactions #Quantum chaos and dynamical systems #Quantum gravity #Quantum mechanics #Saddle point #Semiclassical physics #hep-th
paper · pdf · doi:10.1007/jhep12(2011)071
published in Journal of High Energy Physics 2011(12) (Springer Nature) · 86 pages plus appendices, 9 figures, minor typos fixed, references added, more discussion of the literature added
arxiv created 2011/10/07 · openalex publication_date 2011/12/01 · arxiv updated 2017/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Correlation functions in Liouville theory are meromorphic functions of the Liouville momenta, as is shown explicitly by the DOZZ formula for the three-point function on the sphere. In a certain physical region, where a real classical solution exists, the semiclassical limit of the DOZZ formula is known to agree with what one would expect from the action of the classical solution. In this paper, we ask what happens outside of this physical region. Perhaps surprisingly we find that, while in some range of the Liouville momenta the semiclassical limit is associated to complex saddle points, in general Liouville's equations do not have enough complex-valued solutions to account for the semiclassical behavior. For a full picture, we either must include "solutions" of Liouville's equations in which the Liouville field is multivalued (as well as being complex-valued), or else we can reformulate Liouville theory as a Chern-Simons theory in three dimensions, in which the requisite solutions exist in a more conventional sense. We also study the case of "timelike" Liouville theory, where we show that a proposal of Al. B. Zamolodchikov for the exact three-point function on the sphere can be computed by the original Liouville path integral evaluated on a new integration cycle.