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.

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.

scholarly journals Asymptotic Expansion of the Multi-Orientable Random Tensor Model

10.37236/4629 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Eric Fusy ◽  
Adrian Tanasa
Keyword(s):  
Asymptotic Expansion ◽  
Matrix Models ◽  
Three Dimensional ◽  
Natural Generalization ◽  
General Term ◽  
Tensor Model ◽  
Half Integer ◽  
Tensor Models ◽  
Random Tensor

Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expanion in $N$, the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree.

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 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 W-representation of Rainbow tensor model

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Bei Kang ◽  
Lu-Yao Wang ◽  
Ke Wu ◽  
Jie Yang ◽  
Wei-Zhong Zhao
Keyword(s):  
Partition Function ◽  
Matrix Model ◽  
Crucial Role ◽  
Elementary Function ◽  
Tensor Model ◽  
Witt Algebra ◽  
Virasoro Constraints ◽  
Compact Expression ◽  
Non Gaussian ◽  
Dual Expression

Abstract We analyze the rainbow tensor model and present the Virasoro constraints, where the constraint operators obey the Witt algebra and null 3-algebra. We generalize the method of W-representation in matrix model to the rainbow tensor model, where the operators preserving and increasing the grading play a crucial role. It is shown that the rainbow tensor model can be realized by acting on elementary function with exponent of the operator increasing the grading. We derive the compact expression of correlators and apply it to several models, i.e., the red tensor model, Aristotelian tensor model and r = 4 rainbow tensor model. Furthermore, we discuss the case of the non-Gaussian red tensor model and present a dual expression for partition function through differentiation.

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