# A Laplacian to compute intersection numbers on $\overline{\mathcal{M}}_{g,n}$ and correlation functions in NCQFT

March 29, 2019

Let $F_g(t)$ be the generating function of intersection numbers on the moduli
spaces $\overline{\mathcal{M}}_{g,n}$ of 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 the space of formal parameters $t_i$ satisfying
$\exp(\sum_{g\geq 2} N^{2-2g}F_g(t))=\exp((-\Delta_t+F_2(t))/N^2)1$ for any
$N>0$. 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.

Keywords:

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