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

Understand

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.