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 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.

scholarly journals ON THE CONTINUUM LIMIT OF THE CONFORMAL MATRIX MODELS

1993 ◽  
Vol 08 (18) ◽  
pp. 3107-3137 ◽  
Author(s):  
A. MIRONOV ◽  
S. PAKULIAK
Keyword(s):  
Matrix Models ◽  
Continuum Limit ◽  
Scaling Limit ◽  
Universality Class ◽  
Partition Functions ◽  
Discrete Level ◽  
New Class ◽  
Double Scaling Limit ◽  
The Continuum

The double scaling limit of a new class of the multi-matrix models proposed in Ref. 1, which possess the W-symmetry at the discrete level, is investigated in detail. These models are demonstrated to fall into the same universality class as the standard multi-matrix models. In particular, the transformation of the W-algebra at the discrete level into the continuum one of the papers2 is proposed and the corresponding partition functions compared. All calculations are demonstrated in full in the first nontrivial case of W(3)-constraints.

scholarly journals SUPERLOOP EQUATIONS AND TWO-DIMENSIONAL SUPERGRAVITY

1992 ◽  
Vol 07 (21) ◽  
pp. 5337-5367 ◽  
Author(s):  
L. ALVAREZ-GAUMÉ ◽  
H. ITOYAMA ◽  
J.L. MAÑES ◽  
A. ZADRA
Keyword(s):  
Partition Function ◽  
Discrete Model ◽  
Continuum Limit ◽  
Basic Assumption ◽  
Scaling Limit ◽  
Integrable Hierarchy ◽  
Two Dimensional ◽  
Virasoro Constraints ◽  
Double Scaling Limit ◽  
The Continuum

We propose a discrete model whose continuum limit reproduces the string susceptibility and the scaling dimensions of (2, 4m) minimal superconformal models coupled to 2D supergravity. The basic assumption in our presentation is a set of super-Virasoro constraints imposed on the partition function. We recover the Neveu-Schwarz and Ramond sectors of the theory, and we are also able to evaluate all planar loop correlation functions in the continuum limit. We find evidence to identify the integrable hierarchy of nonlinear equations describing the double scaling limit as a supersymmetric generalization of KP studied by Rabin.

N = ½ SUPERSTRING IN ZERO DIMENSION AND THE SPONTANEOUS BREAKDOWN OF THE SUPERSYMMETRY

1992 ◽  
Vol 07 (32) ◽  
pp. 2979-2989 ◽  
Author(s):  
SHIN’ICHI NOJIRI
Keyword(s):  
Matrix Models ◽  
Random Matrix ◽  
Continuum Limit ◽  
Scaling Limit ◽  
Partition Functions ◽  
Spontaneous Breakdown ◽  
Random Matrix Models ◽  
Double Scaling Limit ◽  
The Continuum ◽  
String Theories

We propose random matrix models which have N=1/2 supersymmetry in zero dimension. The supersymmetry breaks down spontaneously. It is shown that the double scaling limit can be defined in these models and the breakdown of the supersymmetry remains in the continuum limit. The exact non-trivial partition functions of the string theories corresponding to these matrix models are also obtained.

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.

scholarly journals DOUBLE SCALING LIMIT OF THE SUPER-VIRASORO CONSTRAINTS

1993 ◽  
Vol 08 (13) ◽  
pp. 2297-2331 ◽  
Author(s):  
L. ALVAREZ-GAUMÉ ◽  
K. BECKER ◽  
M. BECKER ◽  
R. EMPARAN ◽  
J. MAÑES
Keyword(s):  
Scaling Limit ◽  
Majorana Fermion ◽  
Two Dimensional ◽  
Genus Zero ◽  
Heuristic Argument ◽  
Kdv Hierarchy ◽  
Virasoro Constraints ◽  
Bosonic Part ◽  
Double Scaling Limit ◽  
The Continuum

We obtain the double scaling limit of a set of superloop equations recently proposed to describe the coupling of two-dimensional supergravity to minimal superconformal matter of type (2,4m). The continuum loop equations are described in terms of a [Formula: see text] theory with a Z2-twisted scalar field and a Weyl–Majorana fermion in the Ramond sector. We have computed correlation functions in genus zero, one and partially in genus two. An integrable supersymmetric hierarchy describing our model has not yet been found. We present a heuristic argument showing that the purely bosonic part of our model is described by the KdV hierarchy.

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 Towards a double scaling limit for tensor models: probing sub-dominant orders

2014 ◽  
Vol 16 (6) ◽  
pp. 063048 ◽  
Author(s):  
Wojciech Kamiński ◽  
Daniele Oriti ◽  
James P Ryan
Keyword(s):  
Scaling Limit ◽  
Tensor Models ◽  
Double Scaling Limit

scholarly journals The double scaling limit of random tensor models

2014 ◽  
Vol 2014 (9) ◽  
Author(s):  
Valentin Bonzom ◽  
Razvan Gurau ◽  
James P. Ryan ◽  
Adrian Tanasa
Keyword(s):  
Scaling Limit ◽  
Tensor Models ◽  
Random Tensor ◽  
Double Scaling Limit

scholarly journals ONE-DIMENSIONAL STRING THEORY WITH VORTICES AS THE UPSIDE-DOWN MATRIX OSCILLATOR

1993 ◽  
Vol 08 (05) ◽  
pp. 809-851 ◽  
Author(s):  
DMITRI BOULATOV ◽  
VLADIMIR KAZAKOV
Keyword(s):  
Field Theory ◽  
Scaling Limit ◽  
Adjoint Representation ◽  
Analytical Continuation ◽  
Partition Functions ◽  
One Dimensional ◽  
Feynman Graphs ◽  
The One ◽  
Double Scaling Limit ◽  
Matrix Quantum

Matrix quantum mechanics at a finite temperature is considered, which is equivalent to the one-dimensional compactified string field theory with vortex excitations. It is explicitly demonstrated that states transforming under nontrivial U(N) representations describe different vortex-antivortex configurations. For example, for the adjoint representation, corresponding Feynman graphs always contain two big loops wrapping around the compactified t space, which corresponds to the vortex–antivortex pair. The technique is developed to calculate the partition functions in given representations for the standard matrix oscillator, and then the procedure of their analytical continuation to the upside-down case is worked out. This procedure enables us to obtain the partition function in the presence of the vortex–antivortex pair in the double scaling limit. Using this result, we calculate the critical temperature for the Berezinski-Kosterlitz–Thouless phase transition. A possible generalization of our technique for the D+1 dimensional matrix model is sketched out.

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