Spectra of Operators in Large $N$ Tensor Models

TL;DR

This work analyzes the spectrum and operator content of large-N tensor models with O(N)^3 (and related Gurau–Witten) symmetries, focusing on both global and gauged formulations. By solving Schwinger–Dyson equations for melonic ladders and explicitly constructing invariant operators, the authors show SYK-like scaling for symmetric-traceless bilinears, a distinct antisymmetric sector, and a rich class of multi-particle singlets renormalized by multiple ladders. They demonstrate a factorial growth in the number of singlet invariants (n_{2k} ~ 2^k k! in many cases) leading to a Hagedorn transition at temperature T that scales as 1/log N, and they connect Gurau–Witten spectra to generalized SYK models of Gross–Rosenhaus. The paper also provides detailed counts of invariants in d=0 and d=1, and develops pictorial representations of invariant operators, expanding the toolkit for analyzing melonic tensor theories.

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.

Figures (19)

• Figure 1: A ladder contribution to the two-point function of a bilinear operator with two pairs of indices contracted, $\mathcal{O}^{c_1 c_2}$. It is not suppressed in the large $N$ limit.
• Figure 2: Different contributions to the two-point function of a bilinear operator with one pair of indices contracted, ${\mathcal{O}}_m^{b_1 c_1 b_2 c_2}$. The ladder diagrams, such as the rightmost figure, are suppressed in the large $N$ limit.
• Figure 3: The $O(N)_1$, $O(N)_2$ and $O(N)_3$ charges.
• Figure 4: All the four-particle operators, the tetrahedron and the three pillows, with the index contractions shown explicitly.
• Figure 5: All six-particle operators. They are present in the scalar model but vanish in the fermionic model.