Gauge invariant computable quantities in timelike Liouville theory

Abstract

Timelike Liouville theory admits the sphere ${{\mathbb{S}}^2}$ as a real saddle point, about which quantum fluctuations can occur. An issue occurs when computing the expectation values of specific types of quantities, like the distance between points. The problem being that the gauge redundancy of the path integral over metrics is not completely fixed even after fixing to conformal gauge by imposing ${g_{{\mu \nu }}}={e^{{2\widehat{b}\phi }}}{{\widetilde{g}}_{{\mu \nu }}}$, where ϕ is the Liouville field and ${{\widetilde{g}}_{{\mu \nu }}}$ is a reference metric. The physical metric ${g_{{\mu \nu }}}$, and therefore the path integral over metrics still possesses a gauge redundancy due to invariance under SL ₂($\mathbb{C}$) coordinate transformations of the reference coordinates. This zero mode of the action must be dealt with before a perturbative analysis can be made.

This paper shows that after fixing to conformal gauge, the remaining zero mode of the linearized Liouville action due to SL ₂($\mathbb{C}$) coordinate transformations can be dealt with by using standard Fadeev-Popov methods. Employing the gauge condition that the “dipole” of the reference coordinate system is a fixed vector, and then integrating over all values of this dipole vector. The “dipole” vector referring to how coordinate area is concentrated about the sphere; assuming the sphere is embedded in ${{\mathbb{R}}^3}$ and centered at the origin, and the coordinate area is thought of as a charge density on the sphere. The vector points along the ray from the origin of ${{\mathbb{R}}^3}$ to the direction of greatest coordinate area.

A Green’s function is obtained and used to compute the expectation value of the geodesic length between two points on the ${{\mathbb{S}}^2}$ to second order in the Timelike Liouville coupling $\widehat{b}$. This quantity doesn’t suffer from any power law or logarithmic divergences as a na¨ıve power counting argument might suggest.