Abstract
Certain models with rank-3 tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large limit, where is held fixed. In this limit the perturbative expansion in the quartic coupling constant, , 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 symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large limit dominated by the melon diagrams. The quantum mechanics of a real anticommuting tensor therefore has a similar large 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 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 symmetry and argue that it is equivalent in the large limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension . While the quartic interaction is not positive definite, we construct the large 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 expansion.
9 More- Received 19 December 2016
DOI:https://doi.org/10.1103/PhysRevD.95.046004
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