Topological recursion for Gaussian means and cohomological field theories

TL;DR

This paper connects Gaussian matrix model means with cohomological field theories using topological recursion, revealing their polynomial structure and integrality, and provides explicit calculations for moduli space coefficients.

Contribution

It introduces a novel method to express Gaussian means as polynomials weighted by cohomological invariants, linking matrix models with topological recursion and cohomological field theories.

Findings

Gaussian means are expressed as polynomials in special times.

Proved integrality and positivity of expansion coefficients.

Explicitly computed coefficients for moduli spaces ${ m M}_{g,1}$ for all g.

Abstract

We use 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), which is the generating function for volumes of discretized (open) moduli spaces $M_{g,s}^{disc}$ (discrete volumes), to express Gaussian means in all genera as polynomials in special times weighted by ancestor invariants of an underlying cohomological field theory. We translate topological recursion of the Gaussian model into recurrent relations for coefficients of this expansion proving their integrality and positivity. As an application, we find the coefficients in the first subleading order for ${\mathcal M}_{g,1}$ for all $g$ in three ways: by using the refined Harer--Zagier recursion, by exploiting the Givental-type decomposition of KPMM, and by an explicit diagram counting.