(Ebook PDF) Combinatorial Physics Combinatorics Quantum Field Theory and Quantum Gravity Models 1st edition by Adrian Tanasa ISBN 9780192648068 0192648063 full chapters

• 1: Introduction
• 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
• 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
• 4: Tree weights and renormalization in QFT
• 4.1 Preliminary results
• 4.2 Partition tree weights
• 4.3 Selected further reading
• 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
• 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
• 7: Analytic combinatorics and QFT
• 7.1 The Mellin transform technique
• 7.2 The saddle point method
• 7.3 Selected further reading
• 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
• 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 r
• 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
• 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 t
• 10.6.3 Hochschild cohomology and the combinatorial DSE for tensorial GFT
• 10.7 Selected further reading
• 11: From random matrices to random 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
• 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
• 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
• LO series
• 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
• 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
• LO
• NLO
• 14.4 Selected further reading
• 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
• 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
• The intermediate field representation
• LO of the four-point cumulants in the 1/N expansion
• Reduced maps
• 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–

Bibliography

Index