The modular geometry of random Regge triangulations

Abstract

We show that the introduction of triangulations with variable connectivity and fluctuating edge lengths (random Regge triangulations) allows for a relatively simple and direct analysis of the modular properties of two-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 N0 vertices) and a suitable decorated version of the (compactified) moduli space of genus g Riemann surfaces with N0 punctures \U0001d510g,N0. 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.