Theoretical High Energy Seminar by L. Chekhov – Niels Bohr Institute - University of Copenhagen

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Theoretical High Energy Seminar by L. Chekhov

Theoretical High Energy Seminar by L. Chekhov (Steklov):

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

Abstract

We re-examine the relation between Gaussian ensembles and Hermitian
one-matrix models. After a simple combinatorics, we present
all multi-trace means for the generating functions that are the standard
loop means in the Gaussian model in terms of the one-matrix model with
logarithmic potential and with the external field, the so-called
Kontsevich--Penner model. On one hand, this model is related by the Miwa
transformation to the general Hermitian matrix model and, on the other
hand, it describes the discretized moduli spaces and the reductions of the
standard moduli spaces. For the one-loop mean we find the new,
advantageous representation of the Harer-Zagier recursion relation, which
enables us to evaluate reduction coefficients of Riemann surfaces with one
marked point for all genera.

If time will allow, I would like also to show how one can enumerate chord
diagrams using the specific one-matrix model with rational potential,
which provides at once the number of RNA complexes of a given topology and
the number of cells in Riemann's moduli spaces for bordered surfaces. We
use the topological recursion for calculating the free energy of this
model. (Based on joint paper(s) with J.Andersen, R.C.Penner, C.M.Reidys
and P.Sulkowski).