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.
@article{1903.12526,
author = {Grosse, Harald and Hock, Alexander and Wulkenhaar, Raimar},
title = {A Laplacian to compute intersection numbers on
$\overline{\mathcal{M}}\_{g,n}$ and correlation functions in NCQFT},
journal = {arXiv},
volume = {2019},
number = {0},
year = {2019},
language = {en},
url = {http://dml.mathdoc.fr/item/1903.12526}
}
Grosse, Harald; Hock, Alexander; Wulkenhaar, Raimar. A Laplacian to compute intersection numbers on
$\overline{\mathcal{M}}_{g,n}$ and correlation functions in NCQFT. arXiv, Tome 2019 (2019) no. 0, . http://gdmltest.u-ga.fr/item/1903.12526/