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arXiv:1707.09347 · Physical review. D/Physical review. D.

Spectra of operators in large N tensor models

Ksenia Bulycheva, Igor R. Klebanov, Alexey Milekhin, Grigory Tarnopolsky

Published Jan 29, 2018• 89 citations
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Abstract

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.

DOI
10.1103/physrevd.97.026016

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We study the operators in the large NNN tensor models, focusing mostly on the fermionic quantum mechanics with O(N)3O(N{)}^{3}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)O(N)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=2h=2h=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=0h=0h=0 eigenvalue corresponds to the conserved O(N)O(N)O(N) charges. A class of singlet operators may be constructed from contracted combinations of mmm symmetric traceless or antisymmetric two-particle operators. Their two-point functions receive contributions from mmm 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 2k2k2k-particle operators do not receive any ladder corrections and have quantized large NNN scaling dimensions k/2k/2k/2. We construct pictorial representations of various singlet operators with low kkk. For larger kkk, we use available techniques to count the operators and show that their number grows as 2kk!{2}^{k}k!2kk!. As a consequence, the theory has a Hagedorn phase transition at the temperature which approaches zero in the large NNN limit. We also study the large NNN spectrum of low-lying operators in the Gurau-Witten model, which has O(N)6O(N{)}^{6}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 NNN tensor integrals with various symmetries.
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The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large NNN 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 NNN 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)O(N), O(N)2O(N)^2O(N)2, or O(N)3O(N)^3O(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 NNN 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 NNN limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(N)3O(N)^3O(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×O(N)×U(1)SU(N)^2\times O(N)\times U(1)SU(N)2×O(N)×U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large NNN tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.
Certain models with rank-3 tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large NNN limit, where g2N3{g}^{2}{N}^{3}g2N3 is held fixed. In this limit the perturbative expansion in the quartic coupling constant, ggg, 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)3O(N{)}^{3}O(N)3 symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large NNN limit dominated by the melon diagrams. The quantum mechanics of a real anticommuting tensor therefore has a similar large NNN 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)3O(N{)}^{3}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 NNN limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension ddd. While the quartic interaction is not positive definite, we construct the large NNN 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.