The Geometry of Dynamical Triangulations

Abstract

The express purpose of these Lecture Notes is to go through some aspects of the simplicial quantum gravity model known as the Dynamical Triangulations approach. Emphasis has been on lying the foundations of the theory and on illustrating its subtle and often unexplored connections with many distinct mathematical fields ranging from global riemannian geometry, moduli theory, number theory, and topology. Our exposition will concentrate on these points so that graduate students may find in these notes a useful exposition of some of the rigorous results one can establish in this field and hopefully a source of inspiration for new exciting problems. We also illustrate the deep and beautiful interplay between the analytical aspects of dynamical triangulations and the results of MonteCarlo simulations. The techniques described here are rather novel and allow us to address successfully many high points of great current interest in the subject of simplicial quantum gravity while requiring very lit-1 tle in the way of fancy field theoretical arguments. As a consequence, these notes contains mostly original and yet unpublished material of great potential interest both to the expert practitioner and to graduate students entering in the field. Among the topic addressed here in considerable details there are:(i) An analytical discussion of the geometry of dynamical triangulations in dimension n = 3 and n = 4; (ii) A constructive characterization of entropy estimates for dynamical triangulations in dimension n = 3, n = 4, and a comparision of the analytical results we obtain with tha data coming from Monte Carlo simulations for the 3-sphere S 3 and the 4-sphere S 4 ; (iii) A proof that in the four-dimensional case the analytical data and the numerical data provide the same critical line k 4 (k 2 ) characterizing the infinite-volume limit of simplicial quantum gravity; (iv) An analytical characterization of the critical point