On the refined counting of graphs on surfaces

TL;DR

This paper reviews and extends mathematical methods for counting bi-partite graphs embedded on surfaces, with applications to quantum field theory calculations involving ribbon graphs and topological field theories.

Contribution

It introduces new counting results for classes of bi-partite graphs, refined by physical features, and connects these to topological field theory observables.

Findings

Extended mathematical literature on graph counting

Refined counting based on vertices, faces, and genus

Connection to topological membrane interpretation

Abstract

Ribbon graphs embedded on a Riemann surface provide a useful way to describe the double line Feynman diagrams of large N computations and a variety of other QFT correlator and scattering amplitude calculations, e.g in MHV rules for scattering amplitudes, as well as in ordinary QED. Their counting is a special case of the counting of bi-partite embedded graphs. We review and extend relevant mathematical literature and present results on the counting of some infinite classes of bi-partite graphs. Permutation groups and representations as well as double cosets and quotients of graphs are useful mathematical tools. The counting results are refined according to data of physical relevance, such as the structure of the vertices, faces and genus of the embedded graph. These counting problems can be expressed in terms of observables in three-dimensional topological field theory with S_d gauge group which gives them a topological membrane interpretation.