2016/11/30 by Igor R. Klebanov, Grigory Tarnopolsky
Physics and Astronomy · #hep-th #cond-mat.str-el
paper · pdf · doi:10.1103/physrevd.95.046004
published as Phys. Rev. D 95, 046004 (2017) · 26 pages, 16 figures, v2: sections 3 and 5 expanded, minor corrections, references added, v3: minor corrections, a reference added, v4: minor corrections, v5: spectrum of the complex model corrected; a note added about "uncolored" higher rank tensors
arxiv created 2017/01/31 · arxiv updated 2017/02/22
Certain models with rank-3 tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large N limit, where g2 N3 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 indexes 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 anti-commuting 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 non-singlet 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 anti-commuting tensor, which has U(N)2× 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-ε expansion.