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Table of Contents
Intro; Supervisors' Foreword; Abstract; Acknowledgements; Contents; Symbols; 1 Introduction; References; 2 Colored Simplices and Edge-Colored Graphs; 2.1 Combinatorial Maps; 2.2 Colored Simplicial Pseudo-Complexes; 2.3 Edge-Colored Graphs; 2.3.1 Graph Encoded Manifolds; 2.3.2 Topology; 2.3.3 Gurau's Degree; 2.3.4 Melonic Graphs; 2.4 p-Angulations in Higher Dimension; 2.4.1 Building Blocks: Bubbles; 2.4.2 Gluings of Bubbles; 2.4.3 Bubble-Dependent Degree; 2.4.4 Pairings and Coverings; 2.4.5 Colored Polytopes in Three Dimensions; 2.5 Quantum Gravity and Random Tensor Models
2.5.1 Discretized Einstein-Hilbert Action2.5.2 Two Dimensions; 2.5.3 Guideline in Higher Dimension; 2.5.4 Colored Random Tensor Model; 2.5.5 Uncolored Random Tensor Models; 2.5.6 The Sachdev-Ye-Kitaev Model and Real Tensors; References; 3 Bijective Methods; 3.1 A First Bijection in the Simpler Case of Cyclic Bubbles; 3.2 Tutte's Bijection and Cyclic Bubbles; 3.2.1 Tutte's Bijection; 3.2.2 Cyclic Bubbles Again; 3.3 Bijection Between Colored Graphs and Stacked Maps; 3.3.1 Stacked Maps; 3.3.2 The Bijection; 3.3.3 Bubble-Restricted Gluings; 3.4 The Quartic Melonic Case
3.4.1 Quartic Melonic Gluings of Positive Degree3.4.2 The Boundary Case; 3.4.3 Locally-Orientable Quartic Melonic Gluings; 3.5 Unicellular Graphs and the Colored SYK Model; 3.5.1 Complex SYK: Bipartite Contributions; 3.5.2 Real SYK: Non-bipartite Contributions; 3.5.3 Counting Unicellular Graphs of a Given Order; 3.6 Intermediate Field Theory; 3.6.1 The Hubbard-Stratonovich Transformation; 3.6.2 Matrix Models for Quartic Tensor Models; 3.6.3 Matrix Models for Generic Tensor Models; References; 4 Properties of Stacked Maps; 4.1 Trees and the Bound on (D-2)-Cells; 4.1.1 Projected Maps and Trees
4.1.2 Generalized Unicellular Maps4.1.3 Edge-Unhooking; 4.1.4 Score of Trees and the Coefficient aB; 4.1.5 Maps of Positive Degree and the Choice of aB; 4.2 Maximal Maps; 4.2.1 Large h-Pairs; 4.2.2 Comparing Maps and Trees; 4.2.3 One-Cycle Maps; 4.2.4 Connected Sum; 4.2.5 Non-connected Bubbles; 4.2.6 Tree-Like Families and the Bound on (D-2)-Cells; 4.3 Examples; 4.3.1 D/2-Cyclic Bubbles of Different Kinds; 4.3.2 A Bubble with Toroidal Boundary in D=3; 4.3.3 Sextic Bubbles in D=4; 4.3.4 Gluings of Bi-Pyramids and More; 4.4 Fluctuations Around Non-melonic Vacuums
4.4.1 Leading Order Free Energy and Trees4.4.2 Effective Theory of Fluctuations: Pruning and Schemes; References; 5 Summary and Outlook; 5.1 Summary of the Studied Building Blocks; 5.1.1 Dimension Three; 5.1.2 Dimension Four; 5.1.3 Dimension D; 5.2 Towards More Interesting Behaviors?; Reference
2.5.1 Discretized Einstein-Hilbert Action2.5.2 Two Dimensions; 2.5.3 Guideline in Higher Dimension; 2.5.4 Colored Random Tensor Model; 2.5.5 Uncolored Random Tensor Models; 2.5.6 The Sachdev-Ye-Kitaev Model and Real Tensors; References; 3 Bijective Methods; 3.1 A First Bijection in the Simpler Case of Cyclic Bubbles; 3.2 Tutte's Bijection and Cyclic Bubbles; 3.2.1 Tutte's Bijection; 3.2.2 Cyclic Bubbles Again; 3.3 Bijection Between Colored Graphs and Stacked Maps; 3.3.1 Stacked Maps; 3.3.2 The Bijection; 3.3.3 Bubble-Restricted Gluings; 3.4 The Quartic Melonic Case
3.4.1 Quartic Melonic Gluings of Positive Degree3.4.2 The Boundary Case; 3.4.3 Locally-Orientable Quartic Melonic Gluings; 3.5 Unicellular Graphs and the Colored SYK Model; 3.5.1 Complex SYK: Bipartite Contributions; 3.5.2 Real SYK: Non-bipartite Contributions; 3.5.3 Counting Unicellular Graphs of a Given Order; 3.6 Intermediate Field Theory; 3.6.1 The Hubbard-Stratonovich Transformation; 3.6.2 Matrix Models for Quartic Tensor Models; 3.6.3 Matrix Models for Generic Tensor Models; References; 4 Properties of Stacked Maps; 4.1 Trees and the Bound on (D-2)-Cells; 4.1.1 Projected Maps and Trees
4.1.2 Generalized Unicellular Maps4.1.3 Edge-Unhooking; 4.1.4 Score of Trees and the Coefficient aB; 4.1.5 Maps of Positive Degree and the Choice of aB; 4.2 Maximal Maps; 4.2.1 Large h-Pairs; 4.2.2 Comparing Maps and Trees; 4.2.3 One-Cycle Maps; 4.2.4 Connected Sum; 4.2.5 Non-connected Bubbles; 4.2.6 Tree-Like Families and the Bound on (D-2)-Cells; 4.3 Examples; 4.3.1 D/2-Cyclic Bubbles of Different Kinds; 4.3.2 A Bubble with Toroidal Boundary in D=3; 4.3.3 Sextic Bubbles in D=4; 4.3.4 Gluings of Bi-Pyramids and More; 4.4 Fluctuations Around Non-melonic Vacuums
4.4.1 Leading Order Free Energy and Trees4.4.2 Effective Theory of Fluctuations: Pruning and Schemes; References; 5 Summary and Outlook; 5.1 Summary of the Studied Building Blocks; 5.1.1 Dimension Three; 5.1.2 Dimension Four; 5.1.3 Dimension D; 5.2 Towards More Interesting Behaviors?; Reference