The Geometry of Dynamical Triangulations

Lecture Notes in Physics: Monographs 50 ......Page 1
The Geometry of Dynamical Triangulations ......Page 3
Preface ......Page 5
Contents ......Page 7
1. Introduction ......Page 9
1.1 The Model: Simplicial Quantum Gravity ......Page 11
1.1.1 Summing over Topologies ......Page 13
1.1.2 Simplicial Quantum Gravity ......Page 15
1.2 Summary of Results ......Page 18
2. Triangulations ......Page 25
2.1 Preliminaries: Simplicial Manifolds and Pseudo-manifolds ......Page 26
2.1.1 Piecewise-Linear Manifolds ......Page 28
2.1.2 Dehn-Sommerville Relations ......Page 32
2.2 Distinct Triangulations of the Same PL Manifold ......Page 41
3. Dynamical Triangulations ......Page 47
3.1 Dynamical Triangulations as Length-Spaces ......Page 48
3.1.1 PL Connections and the Incidence Matrix ......Page 50
3.1.2 Curvature Assignments ......Page 52
3.1.3 The Einstein-Hilbert Action for Dynamical Triangulations ......Page 54
3.2 Dynamical Triangulations as Singular Metric Spaces ......Page 57
3.2.1 Geodesic Triangles ......Page 59
3.3 Approximating Riemannian Manifolds Through Dynamical Triangulations ......Page 64
3.4 Topological Finiteness Theorems for Dynamical Triangulations of Bounded Geometry ......Page 73
4. Moduli Spaces for Dynamically Triangulated Manifolds ......Page 77
4.1 Romer-Zahringer Deformations of Dynamically Triangulated Manifolds ......Page 79
4.2 Dynamical Triangulations and Locally Homogeneous Geometries ......Page 83
4.2.1 Locally Homogeneous Geometries ......Page 84
4.2.2 Moduli of Locally Homogeneous Geometries ......Page 87
4.3 Moduli of Dynamical Triangulations ......Page 90
4.4 Gauge-Fixing of the Moduli of a Dynamical Triangulation ......Page 94
4.5 A Measure on the Moduli Space ......Page 96
4.5.1 Moduli Asymptotics ......Page 98
4.5.2 Moduli Asymptotics for DT-Surfaces ......Page 100
4.5.3 A Compact Formula for Surfaces ......Page 101
5.1 Partitions of Integers and Curvature Assignments ......Page 103
5.2 Distinct Dynamical Triangulations with Given Curvature Assignments ......Page 110
5.3 The Counting Principle ......Page 128
5.4 A Remark on SU(2) Holonomy ......Page 129
6.1 The Asymptotic Generating Functions for the Enumeration of Dynamical Triangulation ......Page 133
6.2 Gauss Polynomials and Dynamical Triangulations ......Page 134
6.3 Dynamical Triangulations and oo-Dimensional Operators ......Page 135
6.4 Asymptotics and Entropy Estimates ......Page 137
6.6 The n > 3-Dimensional Case ......Page 140
6.6.1 The Infinite Volume Limit ......Page 150
6.7 Distinct Asymptotic Regimes ......Page 152
6.7.1 Strong Coupling ......Page 153
6.7.2 Critical and Weak Coupling ......Page 159
6.7.3 Weak Coupling and Complete Polymerization ......Page 161
7. Analytical vs. Numerical Data ......Page 165
7.1 The 4-Dimensional Case ......Page 166
7.2 Polymerization ......Page 168
7.3 Summing over Simply-Connected 4-Dimensional Manifolds ......Page 175
7.4 The 3-Dimensional Case ......Page 178
7.5 Concluding Remarks: the Order of the Phase Transition ......Page 181
A.1 Pachner Moves ......Page 183
A.1.2 The 3-Dimensional Case ......Page 186
A.1.3 The 4-Dimensional Case ......Page 188
A.2 The Tangent Space to the Representation Variety ......Page 190
References ......Page 195
Index ......Page 201