Combinatorial Physics: Combinatorics, Quantum Field Theory, and Quantum Gravity Models

✍ Scribed by Adrian Tanasa

• Tongue: English
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✦ Table of Contents

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Combinatorial Physics: Combinatorics, quantum field theory, and quantum gravity models
Copyright
Dedication
Contents
Chapter 1: Introduction
Chapter 2: Graphs, ribbon graphs, and polynomials
2.1 Graph theory: The Tutte polynomial
2.2 Ribbon graphs; the Bollobás–Riordan polynomial
2.3 Selected further reading
Chapter 3: Quantum field theory (QFT)—built-in combinatorics
3.1 Definition of the scalar Φ4 model
3.2 Perturbative expansion—Feynman graphs and their combinatorial weights
3.3 Fourier transform—the momentum space
3.4 Parametric representation of Feynman integrands
3.5 The propagator and the heat kernel
3.6 A glimpse of perturbative renormalization
3.6.1 The power counting theorem
3.6.2 Locality
3.6.3 Multi-scale analysis
3.6.4 The subtraction operator for a general Feynman graph
3.6.5 Dimensional renormalization
3.7 Dyson–Schwinger equation
3.8 Combinatorial (or 0-dimensional) QFT and the intermediate field method
3.8.1 Combinatorial (or 0-dimensional) QFT
3.8.2 The intermediate field method
3.9 Selected further reading
Chapter 4: Tree weights and renormalization in QFT
4.1 Preliminary results
4.2 Partition tree weights
4.3 Selected further reading
Chapter 5 Combinatorial QFT and the Jacobian Conjecture
5.1 The Jacobian Conjecture as combinatorial QFT model (the Abdesselam–Rivasseau model)
5.2 The intermediate field method for the Abdesselam–Rivasseau model
5.3 Selected further reading
Chapter 6: Fermionic QFT, Grassmann calculus, and combinatorics
6.1 Grassmann algebras and Grassmann calculus
6.1.1 The Grassmann algebra
6.1.2 Grassmann calculus; Pfaffians as Grassmann integrals
6.2 On Grassmann Gaussian measures
6.3 Lingström–Gessel–Viennot (LGV) formula for graphs with cycles
6.4 Stembridge’s formulas for graphs with cycles
6.5 A generalization
6.6 Tutte polynomial and the parametric representation in QFT
6.7 Selected further reading
Chapter 7: Analytic combinatorics and QFT
7.1 The Mellin transform technique
7.2 The saddle point method
7.3 Selected further reading
Chapter 8: Algebraic combinatorics and QFT
8.1 Algebraic reminder; Combinatorial Hopf Algebras (CHAs)
8.2 The Connes–Kreimer Hopf algebra of Feynman graphs
8.3 The B+ operator, Hochschild cohomology of the Connes–Kreimer algebra
8.4 Multi-scale renormalization, CHA description
8.5 Selected further reading
Chapter 9: QFT on the non-commutative Moyal space and combinatorics
9.1 Mathematical setting: Renormalizability
9.2 The Mehler kernel and the Grosse–Wulkenhaar model
9.3 Parametric representation of Grosse–Wulkenhaar-like models
9.4 The Mellin transform and the Grosse–Wulkenhaar model
9.5 Dimensional renormalization for the Grosse–Wulkenhaar model
9.6 A heat kernel–based renormalizable model
9.7 Parametric representation and the Bollobás–Riordan polynomial
9.7.1 Parametric representation
9.7.2 Relation between the multi-variate Bollobás–Riordan and the polynomials of the parametric representation
9.8 Combinatorial Connes–Kreimer Hopf algebra and its Hochschild cohomology
9.8.1 Combinatorial Connes–Kreimer Hopf algebra
9.8.2 Hochschild cohomology and the combinatorial DSE
9.9 Selected further reading
Chapter 10: Quantum gravity, group field theory (GFT), and combinatorics
10.1 Quantum gravity
10.2 Main candidates for a theory of quantum gravity: The holographic principle
10.3 GFT models: the Boulatov and the colourable models
10.4 The multi-orientable GFT model
10.4.1 Tadpoles and generalized tadpoles
10.4.2 Tadfaces
10.5 Saddle point method for GFT Feynman integrals
10.6 Algebraic combinatorics and tensorial GFT
10.6.1 The Ben Geloun–Rivasseau (BGR) model
10.6.2 Cones–Kreimer Hopf algebraic description of the combinatorics of the renormalizability of the BGR model
10.6.3 Hochschild cohomology and the combinatorial DSE for tensorial GFT
10.7 Selected further reading
Chapter 11: From random matrices torandom tensors
11.1 The large N limit
11.2 The double-scaling limit
11.3 From matrices to tensors
11.4 Tensor graph polynomials—a generalization of the Bollobás–Riordan polynomial
11.5 Selected further reading
Chapter 12: Random tensor models—the U(N)D-invariant model
12.1 Definition of the model and its DSE
12.1.1 U(N)D-invariant bubble interactions
12.1.2 Bubble observables
12.1.3 The DSE for the model
12.1.4 Navigating the following sections of the chapter
12.2 The DSE beyond the large N limit
12.2.1 The LO
12.2.2 Moments and Cumulants
12.2.3 Gaussian and non-Gaussian contributions
12.2.4 The DSE at NLO
12.2.5 The order 1/ND in the quartic model
12.3 The double-scaling limit
12.3.1 Double-scaling limit in the DSE
12.3.2 From the quartic model to a generic model
12.4 Selected further reading
Chapter 13: Random tensor models—the multi-orientable (MO) model
13.1 Definition of the model
13.2 The 1/N expansion and the large N limit
13.2.1 Feynman amplitudes; the 1/N expansion
13.2.2 The large N limit—the LO (melonic graphs)
13.2.3 The large N limit—the NLO
13.2.4 Leading and NLO series
13.3 Combinatorial analysis of the general term of the large N expansion
13.3.1 Dipoles, chains, schemes, and all that
13.3.2 Generating functions, asymptotic enumeration, and dominant schemes
13.4 The double-scaling limit
13.4.1 The two-point function
13.4.2 The four-point function
13.4.3 The 2r-point function
13.5 Selected further reading
Chapter 14: Random tensor models—the O(N)3-invariant model
14.1 General model and large N expansion
14.2 Quartic model, large N expansion
14.2.1 Large N expansion: LO
14.2.2 NLO
14.3 General quartic model: Critical behaviour
14.3.1 Explicit counting of melonic graphs
14.3.2 Diagrammatic equations, LO and NLO
14.3.3 Singularity analysis
14.3.4 Critical exponents
14.4 Selected further reading
Chapter 15: The Sachdev–Ye–Kitaev (SYK) holographic model
15.1 Definition of the SYK model: Its Feynman graphs
15.2 Diagrammatic proof of the large N melonic dominance
15.3 The coloured SYK model
15.3.1 Definition of the model, real, and complex versions
15.3.2 Diagrammatics of the real and complex model
15.3.3 More on the coloured SYK Feynman graphs
15.3.4 Non-Gaussian disorder average in the complex model
15.4 Selected further reading
Chapter 16: SYK-like tensor models
16.1 The Gurau–Witten model and its diagrammatics
16.1.1 Two-point functions: LO, NLO, and so on
16.1.2 Four-point function: LO, NLO, and so on
16.2 The O(N)3-invariant SYK-like tensor model
16.3 The MO SYK-like tensor model
16.4 Relating MO graphs to O(N)3-invariant graphs
16.5 Diagrammatic techniques for O(N)3-invariant graphs
16.5.1 Two-edge-cuts
16.5.2 Dipole removals
16.5.3 Dipole insertions
16.5.4 Chains of dipoles
16.5.5 Face length
16.5.6 The strategy
16.6 Degree 1 graphs of the O(N)3-invariant SYK-like tensor model
16.6.1 2PI, dipole-free graph of degree one
16.6.2 The graphs of degree 1
16.7 Degree 3/2 graphs of the O(N)3-invariant SYK-like tensor model
Appendix A: Examples of tree weights
A.1 Symmetric weights—complete partition
A.2 One singleton partition—rooted graph
A.3 Two singleton partition—multi-rooted graph
Appendix B: Renormalization of the Grosse–Wulkenhaar model, one-loop examples
Appendix C: The B+ operator in Moyal QFT,two-loop examples
C.1 One-loop analysis
C.2 Two-loop analysis
Appendix D: Explicit examples of GFT tensor Feynman integral computations
D.1 A non-colourable, MO tensor graph integral
D.2 A colourable, multi-orientable tensor graph integral
D.3 A non-colourable, non-multi-orientable tensor graph integral
Appendix E: Coherent states of SU(2)
Appendix F: Proof of the double-scaling limit of the U(N)D-invariant tensor model
Appendix G: Proof of Theorem 15.3.2
G.1 Bijection with constellations
G.1.1 Bijection in the bipartite case
G.1.2 The non-bipartite case
G.2 Enumeration of coloured graphs of fixed order
G.2.1 Exact enumeration
G.2.2 Singularity analysis
G.3 The connectivity condition and SYK graphs
G.3.1 Preliminary conditions
G.3.2 The case q > 3
G.3.3 The case q = 3
G.3.4 The non-bipartite case
Appendix H: Proof of Theorem 16.1.1
Appendix I: Summary of results on the diagrammatics of the coloured SYK model and of the Gur˘ au–Witten model
Bibliography
Index