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

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: ]

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