Spectra of operators in large N tensor models

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 SYK model with real fermions; it includes the h = 2 zero-mode. For the operators anti-symmetric 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. of bilinear operators [22,23,[25][26][27] have been shown to be identical in the tensor and SYK models [18].In this paper we continue exploration of the large N tensor models, in particular the O(N ) 3 symmetric model of [18], which appears to be the minimal quantum mechanical model possessing the melonic limit. 2 This model has N 3 anti-commuting degrees of freedom, ψ abc , where a, b, c = 1, . . . , N . In the model with global symmetry, the operators may be classified according to the group representations. In section 3 we study the spectra of two-particle operators, which are either symmetric traceless or antisymmetric under two indices belonging to the same O(N ) group. We find that the spectrum of symmetric traceless operators (3.5)is the same as that in the SYK model with real fermions; in particular it includes the h = 2 zero-mode which plays an important role in the dual gravitational dynamics [28][29][30]. While in the SYK model there is one h = 2 zero-mode, in the O(N ) 3 tensor model it appears with multiplicity 1 + 3 2 (N − 1)(N + 2). For the operators anti-symmetric in the two indices, (3.6), the spectrum is identical to the additional sector found in the complex tensor and SYK models [18,[31][32][33][34][35][...

Hints of unitarity at large N in the O(N )3 tensor field theory

“…We follow the set of ideas which have been developed in a number of papers, in connection to the Hagedorn transition in gauge theories [64,65] and the AdS/CFT duality between vector models and higher-spin theory [66][67][68]. Similar methods have also been applied to tensor models in [24,25,69]. Since we are here interested in the free theory, and its spectrum of bilinear operators, the O(N ) 3 symmetry of our model will play no role, and we can actually treat its free limit as a O(N 3 ) vector model.…”

“…Tensor models were initially studied in zero dimension in the context of quantum gravity and random geometry [3,[14][15][16][17]. They were then studied in one dimension [18][19][20][21][22][23][24][25][26][27][28][29][30] (see also [4,31] for reviews) as a generalization of the Sachdev-Ye-Kitaev model [32][33][34][35][36][37] without quenched disorder.…”

Notes on melonic O(N)q−1 tensor models

“…We have recently become aware of the preprint [43] which overlaps with this paper in multiple ways. We hope it will prove possible to combine the results of this paper with the methods of [43] to better understand the new light modes discussed earlier in this introduction.…”

“…Note that extra powers of beta β n−1 came from the integration because of evaluation of residue around a pole of order n. This contribution is to be multiplied with a factor of n due to freedom in choosing one external propagator on which x 1/2 is considered. 43 Here an extra factor of (n − 1)! comes as compared to n = 2 case because of freedom of joining n blobs with one another.…”

The Tensor Track V: Holographic Tensors

“…Secondly, and it was one the first things pointed out, [27] initiated the enumeration of multiparticle operators (more indices give much more freedom for contracting them) providing evidence for a Hagedorn transition 7 in tensor models at a temperature of order 1/ log N. Correlation functions…”