Correlators in tensor models from character calculus

TL;DR

The paper extends the Hurwitz/character calculus from matrix models to tensor models, focusing on rainbow and Aristotelian constructions, to efficiently compute Gaussian correlators. It shows that correlators are multi-linear combinations of dimensions $D_R(N_s)$ with all $R_s$ of the same size $m$, with coefficients given by symmetric-group characters $\psi_R$ and their generalizations; explicit results are provided for Hermitian, complex rectangular, and Aristotelian tensor models, including tri-linear dimension structures such as those for $\langle \mathcal{K}_2\rangle$ and $\langle \mathcal{K}_3\rangle$. The approach reveals deep connections to Hurwitz theory and knot-theoretic arborescent calculus, suggesting further avenues like recursion relations and topological recurrences, and sets the stage for incorporating Racah matrices in more complex correlators.

Abstract

We explain how the calculations of arXiv:1704.08648, which provided the first evidence for non-trivial structures of Gaussian correlators in tensor models, are efficiently performed with the help of the (Hurwitz) character calculus. This emphasizes a close similarity between technical methods in matrix and tensor models and supports a hope to understand the emerging structures in very similar terms. We claim that the $2m$-fold Gaussian correlators of rank $r$ tensors are given by $r$-linear combinations of dimensions with the Young diagrams of size $m$. The coefficients are made from the characters of the symmetric group $S_m$ and their exact form depends on the symmetries of the model. As the simplest application of this new knowledge, we provide simple expressions for correlators in the Aristotelian tensor model as tri-linear combinations of dimensions.