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Gauge invariant computable quantities in Timelike Liouville theory

Gauge invariant computable quantities in Timelike Liouville theory. Jonathan Maltz KEK March 12, 2014. Arxiv : 1210.2398v3 Published in JHEP, DOI: 10.1007/JHEP03(2013)097 NSF Grant # PHY-0969448, PHY- 0756174. Introduction. Quantum Liouville theory.

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Gauge invariant computable quantities in Timelike Liouville theory

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  1. Gauge invariant computable quantities in Timelike Liouville theory Jonathan Maltz KEK March 12, 2014 Arxiv: 1210.2398v3 Published in JHEP, DOI: 10.1007/JHEP03(2013)097 NSF Grant # PHY-0969448, PHY- 0756174

  2. Introduction Quantum Liouville theory • Non-critical String theory • A model for higher dimensional Euclidean Gravity • A non-compact conformal field theory • A Dilaton background for String theory Recent Motivations • It has a connection to four dimensional gauge theories with with extended SUSY • It is important component in Holographic duals of de Sitter space and the multiverse at large – FRW/CFT. In this model the dual of bulk gravity theory in an open FRW,Coleman-Deluccia bubble in a De Sitter background is a Matter CFT of large positive central charge coupled to ghosts and Liouville field of large negative central charge a.k.a. Timelike Liouville theory.

  3. Classical Liouville theory was first used in efforts to prove the Uniformization theorem, as it classically it maps a Riemann surface of genus g and n boundaries to a surface of constant curvature with the same g and n via a conformal rescaling of the metric. A review of Liouville theory Quantum mechanically it comes up as a factor multiplying the measure of the string path-integral in Non-Critical string theory and generically when gauge fixing a generic conformal field coupled to 2D gravity.

  4. Timelike Liouville • Timelike and spacelike liouville theory are described by the same path integral, evaluated on different integration cycles [D.Harlow, J.M. ,E.Witten - hep-th:1108.4417]

  5. Computing the Greens Function

  6. Computing the Greens Function

  7. Computing the Geodesic Distance We will be computing this to second order in b, for initially north south trajectories.

  8. Computing the Geodesic Distance Both correlators diverge when the points are coincident, the expression must be regulated

  9. Computing the Geodesic Distance SUCCESS!!!

  10. Conclusions • A perturbative expansion of small fluctuations around the spherical saddle-point in Timelike Liouville theory can be made using standard Fadeev-Popov methods, by integrating over SL(2,C) gauge redundancy of the sphere. • There exist gauge invariant quantities that can be computed in this expansion, the geodesic distance between two points is one of them. Future Speculations • When coupling T.L. Liouville to a matter theory, gauge invariant quantities could be constructed by integrating over the positions of correlation functions with the constraint that the distance between the points are fixed physical Lengths.

  11. The End

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