Abstract
We study the Euclidean path integral of two-dimensional quantum gravity with positive cosmological constant coupled to conformal matter with large and positive central charge. The problem is considered in a semiclassical expansion about a round two-sphere saddle. We work in the Weyl gauge whereby the computation reduces to that for a (timelike) Liouville theory. We present results up to two-loops, including a discussion of contributions stemming from the gauge fixing procedure. We exhibit cancelations of ultraviolet divergences and provide a path integral computation of the central charge for timelike Liouville theory. Combining our analysis with insights from the DOZZ formula we are led to a proposal for an all orders result for the two-dimensional gravitational partition function on the two-sphere.
Highlights
The specific class of gravitational theories we consider are ones equipped with a positive cosmological constant Λ
We study the Euclidean path integral of two-dimensional quantum gravity with positive cosmological constant coupled to conformal matter with large and positive central charge
We work in the Weyl gauge whereby the computation reduces to that for a Liouville theory
Summary
We will consider the problem in the Weyl gauge (1.2). In doing so, we employ a hypothesis of Distler-Kawai [6] and David [7] that fixes the path-integration measure over the Weyl mode φ. The residual PSL(2, C) symmetry acts non-trivially on the Liouville field φ(z, z) We can study this in the semiclassical limit where β ≈ 1/q 1. Given the invariance of StL, and assuming it persists at the quantum level, the path-integral over the Liouville field φ will produce a term proportional to the volume PSL(2, C). As a gauge-fixing condition we follow [6] and impose that φ1m = 0 with m = {−1, 0, 1}.5 The reason this is a good gauge fixing procedure, as delineated in appendix C, is that the variation of δφ under the three non-compact generators of PSL(2, C) is precisely equal to the l = 1 modes. This condition will remain unchanged under the action of the SO(3) subgroup of PSL(2, C), fixing three of the six parameters of PSL(2, C) These are given by (2.13) where we must deform in directions αn that are outside of (2.17). We recall the exact relations q= 1 −β, β cm
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