A Laplacian to Compute Intersection Numbers on overline{{{mathcal {M}}}}_{g,n} and Correlation Functions in NCQFT

Abstract

Let F_g(t) be the generating function of intersection numbers of psi -classes on the moduli spaces overline{{{mathcal {M}}}}_{g,n} of stable complex curves of genus g. As by-product of a complete solution of all non-planar correlation functions of the renormalised Phi ^3-matrical QFT model, we explicitly construct a Laplacian Delta _t on a space of formal parameters t_i which satisfies exp (sum _{gge 2} N^{2-2g}F_g(t))=exp ((-Delta _t+F_2(t))/N^2)1 as formal power series in 1/N^2. The result is achieved via Dyson-Schwinger equations from noncommutative quantum field theory combined with residue techniques from topological recursion. The genus-g correlation functions of the Phi ^3-matricial QFT model are obtained by repeated application of another differential operator to F_g(t) and taking for t_i the renormalised moments of a measure constructed from the covariance of the model.