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Published By Oxford University Press

9780192895493, 9780191914973

2021 ◽  
pp. 76-94
Author(s):  
Adrian Tanasa

We have seen in the previous chapter some of the non-trivial interplay between analytic combinatorics and QFT. In this chapter, we illustrate how yet another branch of combinatorics, algebraic combinatorics, interferes with QFT. In this chapter, after a brief algebraic reminder in the first section, we introduce in the second section the Connes–Kreimer Hopf algebra of Feynman graphs and we show its relation with the combinatorics of QFT perturbative renormalization. We then study the algebra's Hochschild cohomology in relation with the combinatorial Dyson–Schwinger equation in QFT. In the fourth section we present a Hopf algebraic description of the so-called multi-scale renormalization (the multi-scale approach to the perturbative renormalization being the starting point for the constructive renormalization programme).


2021 ◽  
pp. 7-16
Author(s):  
Adrian Tanasa

In this chapter we present some notions of graph theory that will be useful in the rest of the book. It is worth emphasizing that graph theorists and theoretical physicists adopt, unfortunately, different terminologies. We present here both terminologies, such that a sort of dictionary between these two communities can be established. We then extend the notion of graph to that of maps (or of ribbon graphs). Moreover, graph polynomials encoding these structures (the Tutte polynomial for graphs and the Bollobás–Riordan polynomial for ribbon graphs) are presented.


2021 ◽  
pp. 121-165
Author(s):  
Adrian Tanasa

This chapter is the first chapter of the book dedicated to the study of the combinatorics of various quantum gravity approaches. After a brief introductory section to quantum gravity, we shortly mention the main candidates for a quantum theory of gravity: string theory, loop quantum gravity, and group field theory (GFT), causal dynamical triangulations, matrix models. The next sections introduce some GFT models such as the Boulatov model, the colourable and the multi-orientable model. The saddle point method for some specific GFT Feynman integrals is presented in the fifth section. Finally, some algebraic combinatorics results are presented: definition of an appropriate Conne–Kreimer Hopf algebra describing the combinatorics of the renormalization of a certain tensor GFT model (the so-called Ben Geloun–Rivasseau model) and the use of its Hochschild cohomology for the study of the combinatorial Dyson–Schwinger equation of this specific model.


2021 ◽  
pp. 166-177
Author(s):  
Adrian Tanasa

After a brief presentation of random matrices as a random surface QFT approach to 2D quantum gravity, we focus on two crucial mathematical physics results: the implementation of the large N limit (N being here the size of the matrix) and of the double-scaling mechanism for matrix models. It is worth emphasizing that, in the large N limit, it is the planar surfaces which dominate. In the third section of the chapter we introduce tensor models, seen as a natural generalization, in dimension higher than two, of matrix models. The last section of the chapter presents a potential generalisation of the Bollobás–Riordan polynomial for tensor graphs (which are the Feynman graphs of the perturbative expansion of QFT tensor models).


2021 ◽  
pp. 72-75
Author(s):  
Adrian Tanasa

In this chapter we present how several analytic techniques, often used in combinatorics, appear naturally in various QFT issues. In the first section we show how one can use the Mellin transform technique to re-express Feynman integrals in a useful way for the mathematical physicist. Finally, we briefly present how the saddle point approximation technique can be also used in QFT. The first phrase of Philippe Flajolet and Robert Sedgewick's encyclopaedic book on analytic combinatorics gives the reader a first glimpse of what analytic combinatorics deals. In the following chapter, we present how several analytic techniques, often used in combinatorics,appear naturally in various QFT issues.


2021 ◽  
pp. 178-208
Author(s):  
Adrian Tanasa

In the first section we give a briefly presentation of the U(N)D-invariant tensor models (N being again the size of the tensor, and D being the dimension). The next section is then dedicated to the analysis of the Dyson–Schwinger equations (DSE) in the large N limit. These results are essential to implement the double scaling limit mechanism of the DSEs, which is done in the third section. The main result of this chapter is the doubly-scaled 2-point function for a model with generic melonic interactions. However, several assumptions on the large N scaling of cumulants are made along the way. They are proved using various combinatorial methods.


2021 ◽  
pp. 260-290
Author(s):  
Adrian Tanasa

In this chapter, we first review the Sachdev–Ye–Kitaev (SYK) model, which is a quantum mechanical model of N fermions. The model is a quenched model, which means that the coupling constant is a random tensor with Gaussian distribution. The SYK model is dominated in the large N limit by melonic graphs, in the same way the tensor models presented in the previous three chapters are dominated by melonic graphs. We then present a purely graph theoretical proof of the melonic dominance of the SYK model. It is this property which led E. Witten to relate the SYK model to the coloured tensor model. In the rest of the chapter we deal with the so-called coloured SYK model, which is a particular case of the generalisation of the SYK model introduced by D. Gross and V. Rosenhaus. We first analyse in detail the leading order and next-to-leading order vacuum, two- and four-point Feynman graphs of this model. We then exhibit a thorough asymptotic combinatorial analysis of the Feynman graphs at an arbitrary order in the large N expansion. We end the chapter by an analysis of the effect of non-Gaussian distribution for the coupling of the model.


2021 ◽  
pp. 39-49
Author(s):  
Adrian Tanasa

In this chapter we define specific tree weights which appear natural when considering a certain approach to non-perturbative renormalization in QFT, namely the constructive renormalization. Several examples of such tree weights are explicitly given in Appendix A. A fundamental step in QFT is to compute the logarithm of functional integrals used to define the partition function of a given model This comes from a fundamental theorem of enumerative combinatorics, stating the logarithm counts the connected objects. The main advantage of the perturbative expansion of a QFT into a sum of Feynman amplitudes is to perform this computation explicitly: the logarithm of the functional integral is the sum of Feynman amplitudes restricted to connected graphs. The main disadvantage is that the perturbative series indexed by Feynman graphs typically diverges.


2021 ◽  
pp. 291-330
Author(s):  
Adrian Tanasa

In this chapter we analyse in detail the diagrammatics of various Sachdev–Ye–Kitaev-like tensor models: the Gurau–Witten model (in the first section), and the multi-orientable and O(N)3-invariant tensor models, in the rest of the chapter. Various explicit graph theoretical techniques are used. The Feynman graphs obtained through perturbative expansion are stranded graphs where each strand represents the propagation of an index nij, alternating stranded edges of colours i and j. However, it is important to emphasize here that since no twists among the strands are allowed, one can easily represent the Feynman tensor graphs as standard Feynman graphs with additional colours on the edges.


2021 ◽  
pp. 209-233
Author(s):  
Adrian Tanasa

In its first section, this chapter presents the definition of the multi-orientable tensor model. The 1/N expansion and the large N limit of this model are exposed in the second section of the chapter. In the third section, a thorough enumerative combinatorial analysis of the general term of the 1/N expansion is presented. The implementation of the double scaling mechanism is then exhibited in the fourth section. This chapter presents the multi-orientable (MO) tensor model and it follows the review article. This rank three model, having O(N) U(N) O(N) symetry, can be seen as an intermediate step between the U(N) invariant model presented in the previous chapter, and the O(N) invariant model presented in the following chapter. The class of Feynman graph generated by perturbative expansion of MO model is strictly larger than the class of Feynman graphs of the U(N) invariant model and strictly smaller than the one of the O(N) invariant model.


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