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

Abstract

We introduce and study the Hermitian matrix model with potential Vs,t(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.

Moduli spaces and macromolecules

“…A good reference for RNA chemistry and structure is [90] with [72,107,113,130] providing excellent and comprehensive treatments of the RNA bioinformatics and beyond. Most of the material in this section comes from [10,13,102,115].…”

“…The enumeration of partially matched RNA structures of fixed genus g was obtained in [126] using matrix models, and a closed-form expression for the number of such partial chord diagrams was given in [36] in terms of Stirling numbers of the first kind. Chord diagrams without isolated vertices of fixed genus g are likewise computed in [10]; cf. the Appendix.…”

“…We shall describe here an application of these methods to an effective computation of the c g,b (n) in §4.3 from [10], which is surveyed in [11]. These quantities also arise from the multiresolvents of the simpler Gaussian density, but our methods encode at once c g,b (n) for all b ≥ 1.…”

“…The free energies had previously been computed in the two cases g = 0 [25,83] and g = 1 [29]. The topological recursion is applied in [10] to compute F g explicitly for g ≤ 3 and in principle in complete generality for any g ≥ 0. For example in genus two,…”

“…This can be solved exactly in one of s, t and perturbatively in the other to extract specific C g,b (z) as in [10], for example,…”