Higher melonic theories

Abstract

We classify a large set of melonic theories with arbitrary q-fold interactions, demonstrating that the interaction vertices exhibit a range of symmetries, always of the form Z n 2 for some n, which may be 0. The number of different theories proliferates quickly as q increases above 8 and is related to the problem of counting one-factorizations of complete graphs. The symmetries of the interaction vertex lead to an effective interaction strength that enters into the Schwinger-Dyson equation for the two-point function as well as the kernel used for constructing higher-point functions. June 2018 3 1 5 7 2 6 4 0 Figure 1: A melonic insertion.

Tensor network and (p-adic) AdS/CFT

“…With recent developments [61,62], the situation is quickly evolving. One of the main difficulties that remain JHEP01 (2018)139 intriguing is that the ordinary derivative ∂ z of the real case does not exist for the p-adic field Q p .…”

Majorana fermion quantum mechanics for higher rank tensors

“…The more symmetry our interaction has, the larger the order of the automorphism group will be. It follows that the q = 8 fully symmetric diagram has the largest group order, with Aut = Z 3 2 [29]. As noted in section 7, the |Aut| factor cancels out in the spectra calculation.…”

“…In contrast to the q = 6 case, there are six Note that the fluctuations for the bottom subfigure are much smaller than the two above -this is because the SFF is calculated at a lower temperature. distinct such q = 8 interactions [28,29]. However, only one of these interactions has the property that it stays connected whenever any 5 colors are erased.…”

“…One can think of a coloring automorphism as a permutation of the vertices of the interaction graph in a way that preserves the colors of the edges. Paper [29] explores these symmetries in more detail, and shows that the group of coloring automorphisms is Z n 2 . Furthermore, [29] proves that for q = u2 v , u odd, melonic tensor models, the group of coloring automorphisms, which we will denote as Aut, can be at most Z v 2 for u = 1 and Z v−1 2 for u > 1.…”

Supersymmetric Landau-Ginzburg tensor models

“…We summarize some of the relevant data for these models in Table 4. We will not write out the superpotential explicitly, apart from noting that for q > 6 there may potentially be multiple index contraction structures (cf., [34]) that guarantee melonic dominance in the suitable large N limit. We will assume for the sake of simplicity that we have picked one such term in writing the superpotential (eg., the maximally single-trace term of [35]).…”