Correlators in two rainbow tensor and complex multi-matrix models

TL;DR

This paper addresses counting gauge-invariant operators in rainbow tensor models of rank-3 and their exact correlators by leveraging a $W$-representation framework and a novel operator-to-colored-Dessin correspondence. It introduces two rainbow tensor constructions realized by $W$-operators, yielding two compact, equivalent expressions for correlators: one via colored Dessins and one via $W$-operators; these results specialize to known models such as the red rainbow and Aristotelian cases. A key contribution is the explicit counting formula for independent operators at each level $l$ in terms of Hurwitz numbers, together with the one-to-one link between connected operators and colored Dessins with $ ilde{ ilde{L}}^2$ colors, which provides a combinatorial path to correlators. The degradations to complex multi-matrix models extend the formalism to matrix settings, enabling a broader class of exact results and connecting tensor model techniques to conventional matrix-model analyses. Overall, the work unifies algebraic, combinatorial, and representation-theoretic methods to obtain exact, scalable expressions for correlators in rainbow tensor and complex multi-matrix theories.

Abstract

We construct two rainbow tensor models with multi-tensors of rank-$3$ and present their $W$-representations. We give the formula of counting number of independent gauge-invariant operators in terms of Hurwitz numbers and establish a one-to-one correspondence between connected operators and colored Dessins. By means of the colored Dessins and $W$-representations, respectively, we derive two compact expressions of correlators for each of rainbow tensor models. Furthermore, two complex multi-matrix models from the degradations of the constructed rainbow tensor models are also discussed.

Figures (6)

• Figure 1: Correspondence between some connected operators and colored Dessins. Number above edge represents the color.
• Figure 2: Correspondence between $\mathcal{\tilde{R}}_{(id,id,id)}^{(1,1,1,1),(12)}$ and the Dessin.
• Figure 3: The Dessin $D_1$ of $\langle\langle\mathcal{\tilde{R}}_{(id,id,id)}^{(1,1,1,1),(12)}\rangle\rangle_{I,\sharp_L=2}$.
• Figure 4: The Dessin $D_2$ of $\langle\langle\mathcal{\tilde{R}}_{(id,id,id)}^{(1,1,1,1),(12)}\rangle\rangle_{I,\sharp_L=2}$.
• Figure 5: The Dessin $D_3$ of $\langle\langle\mathcal{\tilde{R}}_{(id,id,id)}^{(1,1,1,1),(12)}\rangle\rangle_{I,\sharp_L=2}$.