Gauge invariant computable Gauge invariant computable quantities in Timelike Liouville quantities in Timelike Liouville

theory theory Jonathan Maltz

KEKMarch 12, 2014

Arxiv: 1210.2398v3 Published in JHEP, DOI: 10.1007/JHEP03(2013)097 NSF Grant # PHY-0969448, PHY- 0756174

Introduction

Quantum Liouville theory

Recent Motivations

• Non-critical String theory• A model for higher dimensional Euclidean Gravity• A non-compact conformal field theory• A Dilaton background for String theory

• 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.

A review of Liouville theoryClassical 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.

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.

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]

Computing the Greens Function

Computing the Greens Function

Computing the Geodesic Distance

We will be computing this to second order in b,for initially north south trajectories.

Computing the Geodesic Distance

Both correlators diverge when the points are coincident, the expression must be regulated

Computing the Geodesic Distance

SUCCESS!!!

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.

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