# Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures

August 27, 2020

We provide strong evidence for the conjecture that the analogue of
Kontsevich's matrix Airy function, with the cubic potential
$\mathrm{Tr}(\Phi^3)$ replaced by a quartic term $\mathrm{Tr}(\Phi^4)$, obeys
the blobbed topological recursion of Borot and Shadrin. We identify in the
quartic Kontsevich model three families of correlation functions for which we
establish interwoven loop equations. One family consists of symmetric
meromorphic differential forms $\omega_{g,n}$ labelled by genus and number of
marked points of a complex curve. We reduce the solution of all loop equations
to a straightforward but lengthy evaluation of residues. In all evaluated
cases, the $\omega_{g,n}$ consist of a part with poles at ramification points
which satisfies the universal formula of topological recursion, and of a part
holomorphic at ramification points for which we provide an explicit residue
formula.

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