Two Approaches For a Perturbative Expansion in Blobbed Topological Recursion

Abstract: In this paper we continue the perturbative analysis of the quartic Kontsevich model. We investigate meromorphic functions $\Omega^{(0)}_m$ with $m=1,2$, that obey blobbed topological recursion. We calculate their expansions and check their equivalence to sums of ribbon graph weights, which are obtained with common methods of perturbation theory in QFT, up to fifth order in the coupling using Mathematica. Furthermore, we provide a catalog of permutation pairs $(\alpha,\sigma)$, which encode all 5660 vacuum ribbon graphs that contribute to the free energy $\mathcal{F}^{(g)}$ with genus $g\geq 0$ up to fifth order and begin to expand upon the used methods to also consider ribbon graphs of general correlation functions $G_{\dots}$. This is a first step towards automation of the calculation of ribbon graph expansions in the quartic Kontsevich model.