The modular geometry of Random Regge Triangulations

TL;DR

This paper introduces Random Regge Triangulations with variable connectivity and edge lengths, establishing a link to moduli space and providing a new approach to 2D quantum gravity analysis.

Contribution

It presents a novel bijection between random Regge triangulations and decorated moduli space, enabling analysis of their geometric and quantum gravity properties.

Findings

Established a bijection with moduli space of Riemann surfaces

Defined a Weil-Petersson metric on triangulations

Connected triangulation volume to quantum gravity partition function

Abstract

We show that the introduction of triangulations with variable connectivity and fluctuating egde-lengths (Random Regge Triangulations) allows for a relatively simple and direct analyisis of the modular properties of 2 dimensional simplicial quantum gravity. In particular, we discuss in detail an explicit bijection between the space of possible random Regge triangulations (of given genus g and with N vertices) and a suitable decorated version of the (compactified) moduli space of genus g Riemann surfaces with N punctures. Such an analysis allows us to associate a Weil-Petersson metric with the set of random Regge triangulations and prove that the corresponding volume provides the dynamical triangulation partition function for pure gravity.