Large Strebel graphs and $(3,2)$ Liouville CFT

TL;DR

This paper demonstrates that expectations of observables in large planar Strebel graphs with fixed face perimeters converge to the amplitudes of the (3,2) minimal model in Liouville CFT, linking discrete graph models to continuous quantum gravity.

Contribution

It proves the convergence of observables in fixed-perimeter planar Strebel graphs to (3,2) Liouville CFT amplitudes, establishing a rigorous connection between discrete and continuous models.

Findings

Convergence of graph observables to CFT amplitudes.

Validation of discrete models approximating quantum gravity.

Link between Strebel graphs and Liouville CFT.

Abstract

2D quantum gravity is the idea that a set of discretized surfaces (called map, a graph on a surface), equipped with a graph measure, converges in the large size limit (large number of faces) to a conformal field theory (CFT), and in the simplest case to the simplest CFT known as pure gravity, also known as the gravity dressed (3,2) minimal model. Here we consider the set of planar Strebel graphs (planar trivalent metric graphs) with fixed perimeter faces, with the measure product of Lebesgue measure of all edge lengths, submitted to the perimeter constraints. We prove that expectation values of a large class of observables indeed converge towards the CFT amplitudes of the (3,2) minimal model.