Correlators in tensor models from character calculus

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.