Models of discretized moduli spaces, cohomological field theories, and Gaussian means

TL;DR

This paper establishes a combinatorial link between Gaussian matrix model loop means and the genus expansion of a matrix model generating discretized moduli space volumes, revealing their topological recursion and cohomological field theory structure.

Contribution

It provides a new combinatorial proof connecting Gaussian matrix models with discretized moduli space volumes and demonstrates their topological recursion and Givental decomposition.

Findings

Explicit relation between Gaussian matrix model loop means and KPMM genus expansion

Proof of topological recursion for discretized moduli space volumes

Finite graph sum representation of asymptotic expansion terms

Abstract

We prove combinatorially the explicit relation between genus filtrated $s$-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM). The latter is the generating function for volumes of discretized (open) moduli spaces $M_{g,s}^{\mathrm{disc}}$ given by $N_{g,s}(P_1,\dots,P_s)$ for $(P_1,\dots,P_s)\in{\mathbb Z}_+^s$. This generating function therefore enjoys the topological recursion, and we prove that it is simultaneously the generating function for ancestor invariants of a cohomological field theory thus enjoying the Givental decomposition. We use another Givental-type decomposition obtained for this model by the second authors in 1995 in terms of special times related to the discretisation of moduli spaces thus representing its asymptotic expansion terms (and therefore those of the Gaussian means) as finite sums over graphs weighted by lower-order monomials in times thus giving another proof of (quasi)polynomiality of the discrete volumes. As an application, we find the coefficients in the first subleading order for ${\mathcal M}_{g,1}$ in two ways: using the refined Harer--Zagier recursion and by exploiting the above Givental-type transformation. We put forward the conjecture that the above graph expansions can be used for probing the reduction structure of the Delgne--Mumford compactification $\overline{\mathcal M}_{g,s}$ of moduli spaces of punctured Riemann surfaces.