TY - JOUR

T1 - Spectra of operators in large N tensor models

AU - Bulycheva, Ksenia

AU - Klebanov, Igor R.

AU - Milekhin, Alexey

AU - Tarnopolsky, Grigory

N1 - Funding Information:
We are very grateful to E. Witten for important input into many aspects of this paper. We are also grateful to S. Minwalla for important discussions and for sharing a draft of the paper [48] prior to publication. We also thank I. Danilenko, A. Jevicki, C. Krishnan, J. Maldacena, C. Peng, F. Popov, D. Roberts, S. Shenker, and D. Stanford for useful discussions. The work of I.R.K. and G.T. was supported in part by the U.S. NSF under Grant No.PHY-1620059. G.T. acknowledges the support of a Myhrvold-Havranek Innovative Thinking Fellowship.
Publisher Copyright:
© 2018 authors. Published by the American Physical Society.

PY - 2018/1/29

Y1 - 2018/1/29

N2 - 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 2kk!. 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.

AB - 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 2kk!. 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.

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U2 - 10.1103/PhysRevD.97.026016

DO - 10.1103/PhysRevD.97.026016

M3 - Article

AN - SCOPUS:85042177145

SN - 2470-0010

VL - 97

JO - Physical Review D

JF - Physical Review D

IS - 2

M1 - 026016

ER -