Hints of unitarity at large N in the O ( N ) 3 tensor field theory
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Springer
Received: October 5, 2019 Accepted: January 28, 2020 Published: February 12, 2020
Dario Benedetti,a Razvan Gurau,a,b Sabine Harribeya and Kenta Suzukia a
CPHT, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau, France b Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, ON, N2L 2Y5, Canada
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We compute the OPE coefficients of the bosonic tensor model of [1] for three point functions with two fields and a bilinear with zero and non-zero spin. We find that all the OPE coefficients are real in the case of an imaginary tetrahedral coupling constant, while one of them is not real in the case of a real coupling. We also discuss the operator spectrum of the free theory based on the character decomposition of the partition function. Keywords: 1/N Expansion, Conformal Field Theory, Field Theories in Lower Dimensions, Global Symmetries ArXiv ePrint: 1909.07767
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP02(2020)072
JHEP02(2020)072
Hints of unitarity at large N in the O(N )3 tensor field theory
Contents 1 Introduction
1
2 The model and the operator product expansion
4 7 10 11 12 12
4 A group-theoretic derivation of the spectrum of bilinear operators in the free theory 14 4.1 d = 3 16 4.2 d = 2 17 4.3 d = 1 18 5 Conclusion
19
A Measure and residue
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B The free theory for ζ ≤ 1
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C The SYK model
23
1
Introduction
Recently there has been extensive interest in tensor models because they admit a new kind of large N limit, the melonic limit [2–5]. The melonic limit is different from both the large N limit of vector models [6, 7] (dominated by bubble diagrams) and the one of matrix models [8–10] (dominated by planar diagrams). Although as algebraic objects tensors are more complicated than matrices, their large N limit is simpler because the melonic graphs are a subset of the planar graphs. The melonic limit is also obtained as a large D limit of planar diagrams, or at large N in matrix-tensor models [11–13]. Tensor models were initially studied in zero dimension in the context of quantum gravity and random geometry [3, 14–17]. They were then studied in one dimension [18–30] (see also [4, 31] for reviews) as a generalization of the Sachdev-Ye-Kitaev model [32–39] without quenched disorder. Tensor models can also be generalized in d dimensions to proper field theories. In this setting they give rise to a new family of conformal field theories (CFTs) at large N which are analytically accessible [40–45]. We call these conformal field theories melonic.
–1–
JHEP02(2020)072
3 Primary operators and OPE coefficients 3.1 The d = 3 case 3.2 The d = 2 case 3.3 Discontinuity at d = 2 3.4 The d = 1 case
Oh,J ∼ [(∂ 2 )... ∂(µ1 . . . ∂µi φ][∂µi+1 . . . ∂µJ ) (∂ 2 )... φ] − traces .
(1.1)
We then compute an infinite (sub)set of OPE coefficients, namely those o
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