Critical Phenomena for Riemannian Manifolds: Simple Homotopy and Simplicial Quantum Gravity

TL;DR

This paper develops a mathematical framework using Gromov's spaces to analyze 3D simplicial quantum gravity, establishing entropy estimates, connecting to Gaussian models, and exploring phase transitions related to homotopy types.

Contribution

It introduces a novel approach linking simplicial quantum gravity to simple homotopy types via Gaussian measures and Reidemeister torsion, with rigorous entropy and phase transition analysis.

Findings

Partition function is well-defined in the thermodynamic limit.

Critical behavior leads to exact evaluation of the partition function.

Evidence of phase transitions between different homotopy types.

Abstract

We show how Gromov's spaces of bounded geometries provide a general mathematical framework for addressing and solving many of the issues of $3D$-simplicial quantum gravity. In particular, we establish entropy estimates characterizing the asymptotic distribution of combinatorially inequivalent triangulated $3$-manifolds, as the number of tetrahedra diverges. Moreover, we offer a rather detailed presentation of how spaces of three-dimensional riemannian manifolds with natural bounds on curvatures, diameter, and volume can be used to prove that three-dimensional simplicial quantum gravity is connected to a Gaussian model determined by the simple homotopy types of the underlying manifolds. This connection is determined by a Gaussian measure defined over the general linear group $GL({\bf R},\infty)$. It is shown that the partition function of three-dimensional simplicial quantum gravity is well-defined, in the thermodynamic limit, for a suitable range of values of the gravitational and cosmological coupling constants. Such values are determined by the Reidemeister-Franz torsion invariants associated with an orthogonal representation of the fundamental groups of the set of manifolds considered. The geometrical system considered shows also critical behavior, and in such a case the partition function is exactly evaluated and shown to be equal to the Reidemeister-Franz torsion. The phase structure in the thermodynamical limit is also discussed. In particular, there are either phase transitions describing the passage from a simple homotopy type to another, and (first order) phase transitions within a given simple homotopy type which seem to confirm, on an analytical ground, the picture suggested by numerical simulations.