Two perspectives of the 2D unit area quantum sphere and their equivalence

TL;DR

This paper compares two probabilistic definitions of the 2D unit area quantum sphere in Liouville quantum gravity and proves their equivalence through a unified limiting approach.

Contribution

It establishes the equivalence of two different probabilistic constructions of the 2D unit area quantum sphere in LQG.

Findings

Both definitions of the unit area quantum sphere are shown to be the same.

A unified limiting procedure links the two perspectives.

The result enhances understanding of LQG's mathematical foundations.

Abstract

2D Liouville quantum gravity (LQG) is used as a toy model for 4D quantum gravity and is the theory of world-sheet in string theory. Recently there has been growing interest in studying LQG in the realm of probability theory: David, Kupiainen, Rhodes, Vargas (2014) and Duplantier, Miller, Sheffield (2014) both provide a probabilistic perspective of the LQG on the 2D sphere. In particular, in each of them one may find a definition of the so-called unit area quantum sphere. We examine these two perspectives and prove their equivalence by showing that the respective unit area quantum spheres are the same. This is done by considering a unified limiting procedure for defining both objects.