Combinatorial physics: combinatorics, quantum field theory, and quantum gravity models

After briefly presenting (for the physicist) some notions frequently used in combinatorics (such as graphs or combinatorial maps) and after briefly presenting (for the combinatorialist) the main concepts of quantum field theory (QFT), the book shows how algebraic combinatorics can be used to deal with perturbative renormalisation (both in commutative and non-commutative quantum field theory), how analytic combinatorics can be used for QFT issues (again, for both commutative and non-commutative QFT), how Grassmann integrals (frequently used in QFT) can be used to proCve new combinatorial identities (generalizing the Lindström-Gessel-Viennot formula), how combinatorial QFT can bring a new insight on the celebrated Jacobian conjecture (which concerns global invertibility of polynomial systems) and so on. In the second part of the book, matrix models, and tensor models are presented to the reader as QFT models. Several tensor model results (such as the implementation of the large N limit and of the double-scaling limit for various such tensor models, N being here the size of the tensor) are then exposed. These results are natural generalizations of results extensively used by theoretical physicists in the study of matrix models and they are obtained through intensive use of combinatorial techniques (this time mainly enumerative techniques). The last part of the book is dedicated to the recently discovered relation between tensor models and the holographic Sachdev-Ye-Kitaev model, model which has been extensively studied in the last years by condensed matter and by high-energy physicists.

Introduction -- Graphs, ribbon graphs, and polynomials -- Quantum field theory (QFT)--built-in combinatorics -- Tree weights and renormalization in QFT -- Combinatorial QFT and the Jacobian Conjecture -- Fermionic QFT, Grassmann calculus, and combinatorics -- Analytic combinatorics and QFT -- Algebraic combinatorics and QFT -- QFT on the non-commutative Moyal space and combinatorics -- Quantum gravity, group field theory (GFT), and combinatorics -- From random matrices to random tensors --Random tensor models--the U(N)D-invariant model -- Random tensor models--the multi-orientable (MO) model -- Random tensor models--the O(N)3-invariant model -- The Sachdev-Ye-Kitaev (SYK) holographic model -- SYK-like tensor models.