Topological recursion for chord diagrams, RNA complexes, and cells in moduli spaces

TL;DR

This paper applies topological recursion to a Hermitian matrix model to enumerate chord diagrams, RNA complexes, and cells in moduli spaces, providing explicit calculations for low genera and a general framework for all genera.

Contribution

It introduces a novel matrix model and applies topological recursion to enumerate complex topological structures in biology and geometry.

Findings

Explicit enumeration formulas for low-genus cases

Connection between chord diagrams and moduli space cells

Perturbative expansion of the partition function

Abstract

We introduce and study the Hermitian matrix model with potential V(x)=x^2/2-stx/(1-tx), which enumerates the number of linear chord diagrams of fixed genus with specified numbers of backbones generated by s and chords generated by t. For the one-cut solution, the partition function, correlators and free energies are convergent for small t and all s as a perturbation of the Gaussian potential, which arises for st=0. This perturbation is computed using the formalism of the topological recursion. The corresponding enumeration of chord diagrams gives at once the number of RNA complexes of a given topology as well as the number of cells in Riemann's moduli spaces for bordered surfaces. The free energies are computed here in principle for all genera and explicitly for genera less than four.