Abstract
We discuss two-dimensional quantum gravity coupled to conformal matter and fixed area in a semiclassical large and negative matter central charge limit. In this setup the gravity theory — otherwise highly fluctuating — admits a round two-sphere saddle. We discuss the two-sphere partition function up to two-loop order from the path integral perspective. This amounts to studying Feynman diagrams incorporating the fixed area constraint on the round two-sphere. In particular we find that all ultraviolet divergences cancel to this order. We compare our results with the two-sphere partition function obtained from the DOZZ formula.
Highlights
In addition to a theory of pure two-dimensional quantum gravity one can consider the addition of matter fields
We discuss two-dimensional quantum gravity coupled to conformal matter and fixed area in a semiclassical large and negative matter central charge limit
We compare our results with the two-sphere partition function obtained from the DOZZ formula
Summary
The theory we will focus on consists of a two-dimensional matter CFT of central charge cm. The pure gravity theory has no classical solutions in two-dimensions due to the topological nature of the Einstein term, upon fixing the area and coupling to a matter theory with cm → −∞ the effective gravitational action including the contribution from ZC(hF)T[gij] does [4]. Upon integrating out the matter and bc-ghost fields, our resulting gravitational path integral on a genus zero surface is given by. Given the invariance of SL, and assuming it persists at the quantum level, the path-integral over the Liouville field φ will produce a term proportional to the volume of PSL(2, C). For further details we refer to [15] At this stage, all but the SO(3) isometry group of the original two-dimensional diffeomorphisms has been gauge fixed. Imposing the area constraint the Liouville path integral on a genus zero surface is ZL[υ]. We recall that the semiclassical limit corresponds to b → 0
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