SYK-like tensor models

2021 ◽  
pp. 291-330
Author(s):  
Adrian Tanasa
Keyword(s):  
Perturbative Expansion ◽  
Invariant Tensor ◽  
Tensor Models ◽  
Feynman Graphs

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.

scholarly journals Double scaling limit for the O(N)3-invariant tensor model

2022 ◽  
Author(s):  
Valentin Bonzom ◽  
Victor Nador ◽  
Adrian Tanasa
Keyword(s):  
Scaling Limit ◽  
Tensor Model ◽  
Invariant Tensor ◽  
Generating Series ◽  
Tensor Models ◽  
Feynman Graphs ◽  
Diagrammatic Analysis ◽  
Finite Sum ◽  
Double Scaling Limit ◽  
The Continuum

Abstract We study the double scaling limit of the O(N)3-invariant tensor model, initially introduced in Carrozza and Tanasa, Lett. Math. Phys. (2016). This model has an interacting part containing two types of quartic invariants, the tetrahedric and the pillow one. For the 2-point function, we rewrite the sum over Feynman graphs at each order in the 1/N expansion as a finite sum, where the summand is a function of the generating series of melons and chains (a.k.a. ladders). The graphs which are the most singular in the continuum limit are characterized at each order in the 1/N expansion. This leads to a double scaling limit which picks up contributions from all orders in the 1/N expansion. In contrast with matrix models, but similarly to previous double scaling limits in tensor models, this double scaling limit is summable. The tools used in order to prove our results are combinatorial, namely a thorough diagrammatic analysis of the Feynman graphs, as well as an analytic analysis of the singularities of the relevant generating series.

Random tensor models—the O(N)3-invariant model

2021 ◽  
pp. 234-259
Author(s):  
Adrian Tanasa
Keyword(s):  
Critical Exponents ◽  
Perturbative Expansion ◽  
Critical Regime ◽  
Leading Order ◽  
Invariant Model ◽  
Analytic Combinatorics ◽  
Tensor Models ◽  
Feynman Graphs ◽  
Random Tensor

We define in this chapter a class of tensor models endowed with O(N)3-invariance, N being again the size of the tensor. This allows to generate, via the usual QFT perturbative expansion, a class of Feynman tensor graphs which is strictly larger than the class of Feynman graphs of both the multi-orientable model and the U(N)3-invariant models treated in the previous two chapters. We first exhibit the existence of a large N expansion for such a model with general interactions (non-necessary quartic). We then focus on the quartic model and we identify the leading order and next-to-leading Feynman graphs of the large N expansion. Finally, we prove the existence of a critical regime and we compute the so-called critical exponents. This is achieved through the use of various analytic combinatorics techniques.

From random matrices to random tensors

2021 ◽  
pp. 166-177
Author(s):  
Adrian Tanasa
Keyword(s):  
Matrix Models ◽  
Random Matrices ◽  
Mathematical Physics ◽  
Natural Generalization ◽  
Random Surface ◽  
The Third ◽  
Planar Surfaces ◽  
Tensor Models ◽  
Feynman Graphs ◽  
The Matrix

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).

Random tensor models—the U(N)D-invariant model

2021 ◽  
pp. 178-208
Author(s):  
Adrian Tanasa
Keyword(s):  
Scaling Limit ◽  
Point Function ◽  
Invariant Model ◽  
Invariant Tensor ◽  
The Third ◽  
Tensor Models ◽  
Random Tensor ◽  
Combinatorial Methods ◽  
Double Scaling Limit ◽  
The Way

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.

The Sachdev–Ye–Kitaev (SYK) holographic model

2021 ◽  
pp. 260-290
Author(s):  
Adrian Tanasa
Keyword(s):  
Gaussian Distribution ◽  
Mechanical Model ◽  
Quantum Mechanical ◽  
Tensor Model ◽  
Holographic Model ◽  
Leading Order ◽  
Tensor Models ◽  
Feynman Graphs ◽  
Non Gaussian ◽  
Random Tensor

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.

Tree weights and renormalization in QFT

2021 ◽  
pp. 39-49
Author(s):  
Adrian Tanasa
Keyword(s):  
Partition Function ◽  
Fundamental Theorem ◽  
Functional Integral ◽  
Perturbative Expansion ◽  
Functional Integrals ◽  
Perturbative Series ◽  
Fundamental Step ◽  
Feynman Graphs ◽  
Perturbative Renormalization ◽  
Feynman Amplitudes

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.

Random tensor models—the multi-orientable (MO) model

2021 ◽  
pp. 209-233
Author(s):  
Adrian Tanasa
Keyword(s):  
Feynman Graph ◽  
General Term ◽  
Intermediate Step ◽  
Tensor Model ◽  
Invariant Model ◽  
Tensor Models ◽  
Feynman Graphs ◽  
The One ◽  
Definition Of

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.

scholarly journals Gauge invariants, correlators and holography in bosonic and fermionic tensor models

2017 ◽  
Vol 2017 (9) ◽  
Author(s):  
Robert de Mello Koch ◽  
David Gossman ◽  
Laila Tribelhorn
Keyword(s):  
Tensor Models

scholarly journals ANALYTIC INVARIANT CHARGE IN QCD

2003 ◽  
Vol 18 (30) ◽  
pp. 5475-5519 ◽  
Author(s):  
A. V. NESTERENKO
Keyword(s):  
Perturbative Expansion ◽  
Quantum Field ◽  
Perturbative Approach ◽  
Interaction Processes ◽  
Running Coupling ◽  
Wide Range ◽  
Local Quantum ◽  
Concrete Realization ◽  
Β Function ◽  
Analytic Invariant

This paper gives an overview of recently developed model for the QCD analytic invariant charge. Its underlying idea is to bring the analyticity condition, which follows from the general principles of local Quantum Field Theory, in perturbative approach to renormalization group (RG) method. The concrete realization of the latter consists in explicit imposition of analyticity requirement on the perturbative expansion of β function for the strong running coupling, with subsequent solution of the corresponding RG equation. In turn, this allows one to avoid the known difficulties originated in perturbative approximation of the RG functions. Ultimately, the proposed approach results in qualitatively new properties of the QCD invariant charge. The latter enables one to describe a wide range of the strong interaction processes both of perturbative and intrinsically nonperturbative nature.

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