Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models

Abstract

Certain models with rank-3 tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large N limit, where g(2)N(3) is held fixed. In this limit the perturbative expansion in the quartic coupling constant, g, 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)(3) symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large N limit dominated by the melon diagrams. The quantum mechanics of a real anticommuting tensor therefore has a similar large N 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)(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) x O(N) symmetry and argue that it is equivalent in the large N limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension d. While the quartic interaction is not positive definite, we construct the large N 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 - epsilon expansion.