Counting Tensor Model Observables and Branched Covers of the 2-Sphere

Lattice gauge theories of permutation groups with a simple topological action (henceforth permutation-TFTs) have recently found several applications in the combinatorics of quantum field theories (QFTs). They have been used to solve counting problems of Feynman graphs in QFTs and ribbon graphs of large $N$, often revealing inter-relations between different counting problems. In another recent development, tensor theories generalizing matrix theories have been actively developed as models of random geometry in three or more dimensions. Here, we apply permutation-TFT methods to count gauge invariants for tensor models (colored as well as non-colored), exhibiting a relationship with counting problems of branched covers of the 2-sphere, where the rank $d$ of the tensor gets related to a number of branch points. We give explicit generating functions for the relevant counting and describe algorithms for the enumeration of the invariants. As well as the classic count of Hurwitz equivalence classes of branched covers with fixed branch points, collecting these under an equivalence of permuting the branch points is relevant to the color-symmetrized tensor invariant counting. We also apply the permutation-TFT methods to obtain some formulae for correlators of the tensor model invariants.