Analytic continuation of Liouville theory

Daniel Harlow, Jonathan Maltz, Edward Witten

Research output: Contribution to journalArticlepeer-review

163 Scopus citations

Abstract

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 S2. 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 S2 can be computed by the original Liouville path integral evaluated on a new integration cycle.

Original languageEnglish (US)
Article number71
JournalJournal of High Energy Physics
Volume2011
Issue number12
DOIs
StatePublished - 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics

Keywords

  • 2D Gravity
  • Chern-Simons Theorie
  • Conformal and W Symmetry
  • Tachyon Condensation

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