# On the counting of $O(N)$ tensor invariants

## TL;DR

This paper develops a combinatorial and algebraic framework for counting and analyzing $O(N)$ tensor invariants using graph theory, permutation groups, and topological field theory, revealing new structural insights.

## Contribution

It introduces a novel enumeration method for $O(N)$ invariants via regular colored graphs and permutation group techniques, and uncovers their algebraic and topological properties.

## Key findings

- Enumeration of invariants using permutation group techniques
- Identification of a new algebraic structure for $O(N)$ invariants
- Connection to topological coverings and representation theory

## Abstract

$O(N)$ invariants are the observables of real tensor models. We use regular colored graphs to represent these invariants, the valence of the vertices of the graphs relates to the tensor rank. We enumerate $O(N)$ invariants as $d$-regular graphs, using permutation group techniques. We also list their generating functions and give (software) algorithms computing their number at an arbitrary rank and an arbitrary number of vertices. As an interesting property, we reveal that the algebraic structure which organizes these invariants differs from that of the unitary invariants. The underlying topological field theory formulation of the rank $d$ counting shows that it corresponds to counting of coverings of the $d-1$ cylinders sharing the same boundary circle and with $d$ defects. At fixed rank and fixed number of vertices, an associative semi-simple algebra with dimension the number of invariants naturally emerges from the formulation. Using the representation theory of the symmetric group, we enlighten a few crucial facts: the enumeration of $O(N)$ invariants gives a sum of constrained Kronecker coefficients; there is a representation theoretic orthogonal base of the algebra that reflects its dimension; normal ordered 2-pt correlators of the Gaussian models evaluate using permutation group language, and further, via representation theory, these functions provide other representation theoretic orthogonal bases of the algebra.