Rainbow tensor model with enhanced symmetry and extreme melonic dominance

Abstract

We introduce and briefly analyze the rainbow tensor model where all planar diagrams are melonic. This leads to considerable simplification of the large N limit as compared to that of the matrix model: in particular, what are dressed in this limit are propagators only, which leads to an oversimplified closed set of Schwinger-Dyson equations for multi-point correlators. We briefly touch upon the Ward identities, the substitute of the spectral curve and the AMM/EO topological recursion and their possible connections to Connes-Kreimer theory and forest formulas.

Ward identities and combinatorics of rainbow tensor models

“…Similar considerations can be also found in [83,84,174,175]. Though we do not go that far in this paper, the first really interesting tensor model to analyze within this context is the rainbow model of [25] (see also [77,78] for earlier works). In the rainbow model, each index of the rank-r tensor field belongs to the representation of its own unitary group and, as a consequence, all the r + 1 fields merging at the hypertetrahedron (simplex) vertex are different.…”

“…Revealing and exploiting such symmetries is the main idea of studying the rainbow tensor models, their enhanced symmetries make properties of the RG-completeness much simpler and explicit. In [25], we already mentioned one example of this kind: in the rainbow models melonic are the only existing among the planar diagrams, and this is what explains this their dominance at large N in all tensor models. However, the examples are…”

“…Moreover, when considered from the right perspective, these sets come with their own hierarchy, which also provides a useful approach to the RG-evolution problem. As already mentioned, only the tree descendants of the keystone operators matter quasiclassically and already this reduces the set, when the keystone is carefully selected, especially in the theories with high symmetry like the rainbow tensor models of [25]. The question is what happens at the loop operator level.…”

SYK-like tensor models on the lattice

“…Numerical analysis for finite N tensor model has been done in [45,46], and an abelian tensor model on the lattice will be analyzed in an adjoining paper [47]. As an aside, we mention here that tensor models have also been studied from the perspective of matrix models, where it arises in the limit of large number of matrices [48][49][50]. Also, for a discussion of how tensor models could possibly arise in string theories, see [51,52].…”

Correlators in the N = 2 $$ \mathcal{N}=2 $$ supersymmetric SYK model

“…Of course, the model is also of great interest in condensed matter physics [51,54,[57][58][59][60][61][62][63][64][65][66][67]. Other related recent work can be found in [68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83].…”