limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large

N

limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of

O(N)

O(N)^2

, or

O(N)^3

symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large

N

limits. In the latter example we review how the dominance of melonic diagrams follows when a special "tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large

N

limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with

O(N)^3

symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with

SU(N)^2\times O(N)\times U(1)

symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large

N

tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.

DOI: 10.48550/arxiv.1808.09434

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The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large $N$ limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large $N$ limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of $O(N)$ , $O(N)^2$ , or $O(N)^3$ symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large $N$ limits. In the latter example we review how the dominance of melonic diagrams follows when a special ``tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large $N$ limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with $O(N)^3$ symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with $SU(N)^2\times O(N)\times U(1)$ symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large $N$ tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large $N$ limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large $N$ limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of $O(N)$, $O(N)^2$, or $O(N)^3$ symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large $N$ limits. In the latter example we review how the dominance of melonic diagrams follows when a special ``tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large $N$ limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with $O(N)^3$ symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with $SU(N)^2\times O(N)\times U(1)$ symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large $N$ tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.

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Tensor models and tensor field theories admit a 1/N expansion and a melonic large N limit which is simpler than the planar limit of random matrices and richer than the large N limit of vector models. They provide examples of analytically tractable but non-trivial strongly coupled quantum field theories and lead to a new class of conformal field theories. We present a compact introduction to the topic, covering both some of the classical results in the field, like the details of the 1/N expansion, as well as recent developments. These notes are loosely bases on four lectures given at the Journées de physique mathématique Lyon 2019: Random tensors and SYK models.Tensor models and tensor field theories admit a 1/N expansion and a melonic large N limit which is simpler than the planar limit of random matrices and richer than the large N limit of vector models. They provide examples of analytically tractable but non-trivial strongly coupled quantum field theories and lead to a new class of conformal field theories. We present a compact introduction to the topic, covering both some of the classical results in the field, like the details of the 1/N expansion, as well as recent developments. These notes are loosely bases on four lectures given at the Journées de physique mathématique Lyon 2019: Random tensors and SYK models.

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Large $N$ Tensor and SYK Models

Jae-Won Kim

The SYK model proposed by Sachdev, Ye, and Kitaev consists of Majorana fermions that interact randomly four at a time. The model develops a dense spectrum above the ground state, due to which the model becomes nearly conformal. This suggests that a holographic dual may exist, which makes the SYK model interesting in the study of quantum gravity. It has been found that the SYK model is similar to large $N$ tensor models: in both models, only the melonic diagrams survive in the large $N$ limit. In this thesis, we explore the large $N$ tensor model with $O(N)^3$ symmetry containing two flavors of Majorana fermions in the fundamental representation. Its quartic Hamiltonian depends on a real parameter $β$ . We derive the kernels of the four point functions. With the spectra that we find from the kernels, we calculate the scaling dimensions of several types of conformal primaries. We also find a duality relation between two Hamiltonians of different values of $β$ . This is not a perfect duality, because the normalization of energy scales with the transformation. Nevertheless, the ratios of the energies are the same, and the operator dimensions are preserved. In addition, we discover that for $β> 1$ or $β< 0$ the scaling dimensions of one of the conformal primaries become complex, rendering the model unstable.The SYK model proposed by Sachdev, Ye, and Kitaev consists of Majorana fermions that interact randomly four at a time. The model develops a dense spectrum above the ground state, due to which the model becomes nearly conformal. This suggests that a holographic dual may exist, which makes the SYK model interesting in the study of quantum gravity. It has been found that the SYK model is similar to large $N$ tensor models: in both models, only the melonic diagrams survive in the large $N$ limit. In this thesis, we explore the large $N$ tensor model with $O(N)^3$ symmetry containing two flavors of Majorana fermions in the fundamental representation. Its quartic Hamiltonian depends on a real parameter $β$. We derive the kernels of the four point functions. With the spectra that we find from the kernels, we calculate the scaling dimensions of several types of conformal primaries. We also find a duality relation between two Hamiltonians of different values of $β$. This is not a perfect duality, because the normalization of energy scales with the transformation. Nevertheless, the ratios of the energies are the same, and the operator dimensions are preserved. In addition, we discover that for $β> 1$ or $β< 0$ the scaling dimensions of one of the conformal primaries become complex, rendering the model unstable.

arXiv:1811.04330APSNov 11, 20182 citations

The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large

N

limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large

N

limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of

O(N)

O(N)^2

, or

O(N)^3

N

N

limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with

O(N)^3

SU(N)^2\times O(N)\times U(1)

symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large

N

tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.

The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large

N

limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large

N

limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of

O(N)

O(N)^2

, or

O(N)^3

N

limits. In the latter example we review how the dominance of melonic diagrams follows when a special ``tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large

N

limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with

O(N)^3

SU(N)^2\times O(N)\times U(1)

symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large

N

tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.

O(N{)}^{5}

symmetric sixth-order Hamiltonian leading to a solvable $\mathrm{large}\text{\ensuremath{-}}N$ limit dominated by the melonic diagrams. We solve for the complete energy spectrum of this model when

N=2

and deduce exact expressions for all the eigenvalues. The subset of states which are gauge invariant exhibits degeneracies related to the discrete symmetries of the gauged model. We also study quantum chaos properties of the tensor model and compare them with those of the

q=6

SYK model. For

q>6

, there is a rapidly growing number of $O(N{)}^{q\ensuremath{-}1}$ invariant tensor interactions. We focus on those of them that are maximally single trace---their stranded diagrams stay connected when any set of $q\ensuremath{-}3$ colors is erased. We present a general discussion of why the tensor models with maximally single-trace interactions have $\mathrm{large}\text{\ensuremath{-}}N$ limits dominated by the melonic diagrams. We solve the $\mathrm{large}\text{\ensuremath{-}}N$ Schwinger-Dyson equations for the higher rank Majorana tensor models and show that they match those of the corresponding SYK models exactly. We also study other gauge invariant operators present in the tensor models.

We study the operators in the large

N

tensor models, focusing mostly on the fermionic quantum mechanics with

O(N{)}^{3}

symmetry which may be either global or gauged. In the model with global symmetry, we study the spectra of bilinear operators, which are in either the symmetric traceless or the antisymmetric representation of one of the

O(N)

groups. In the symmetric traceless case, the spectrum of scaling dimensions is the same as in the Sachdev-Ye-Kitaev (SYK) model with real fermions; it includes the

h=2

zero mode. For the operators antisymmetric in the two indices, the scaling dimensions are the same as in the additional sector found in the complex tensor and SYK models; the lowest

h=0

eigenvalue corresponds to the conserved

O(N)

charges. A class of singlet operators may be constructed from contracted combinations of

m

symmetric traceless or antisymmetric two-particle operators. Their two-point functions receive contributions from

m

melonic ladders. Such multiple ladders are a new phenomenon in the tensor model, which does not seem to be present in the SYK model. The more typical

2k

-particle operators do not receive any ladder corrections and have quantized large

N

scaling dimensions

k/2

. We construct pictorial representations of various singlet operators with low

k

. For larger

k

, we use available techniques to count the operators and show that their number grows as

{2}^{k}k!

. As a consequence, the theory has a Hagedorn phase transition at the temperature which approaches zero in the large

N

limit. We also study the large

N

spectrum of low-lying operators in the Gurau-Witten model, which has

O(N{)}^{6}

symmetry. We argue that it corresponds to one of the generalized SYK models constructed by Gross and Rosenhaus. Our paper also includes studies of the invariants in large

N

tensor integrals with various symmetries.

Certain models with rank-3 tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large

N

limit, where

{g}^{2}{N}^{3}

is held fixed. In this limit the perturbative expansion in the quartic coupling constant,

g

, is dominated by a special class of ``melon'' diagrams. We study ``uncolored'' models of this type, which contain a single copy of real rank-3 tensor. Its three indices are distinguishable; therefore, the models possess

O(N{)}^{3}

symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large

N

limit dominated by the melon diagrams. The quantum mechanics of a real anticommuting tensor therefore has a similar large

N

limit to the model recently introduced by Witten as an implementation of the Sachdev-Ye-Kitaev (SYK) model which does not require disorder. Gauging the

O(N{)}^{3}

symmetry in our quantum mechanical model removes the nonsinglet states; therefore, one can search for its well-defined gravity dual. We point out, however, that the model possesses a vast number of gauge-invariant operators involving higher powers of the tensor field, suggesting that the complete gravity dual will be intricate. We also discuss the quantum mechanics of a complex 3-index anticommuting tensor, which has $U(N{)}^{2}\ifmmode\times\else\texttimes\fi{}O(N)$ symmetry and argue that it is equivalent in the large

N

limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension

d

. While the quartic interaction is not positive definite, we construct the large

N

Schwinger-Dyson equation for the two-point function and show that its solution is consistent with conformal invariance. We carry out a perturbative check of this result using the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion.

N

tensor models: in both models, only the melonic diagrams survive in the large

N

limit. In this thesis, we explore the large

N

tensor model with

O(N)^3

symmetry containing two flavors of Majorana fermions in the fundamental representation. Its quartic Hamiltonian depends on a real parameter

β

. We derive the kernels of the four point functions. With the spectra that we find from the kernels, we calculate the scaling dimensions of several types of conformal primaries. We also find a duality relation between two Hamiltonians of different values of

β

. This is not a perfect duality, because the normalization of energy scales with the transformation. Nevertheless, the ratios of the energies are the same, and the operator dimensions are preserved. In addition, we discover that for

β> 1

β< 0

the scaling dimensions of one of the conformal primaries become complex, rendering the model unstable.