Blobbed Topological Recursion of the Quartic Kontsevich Model I: Loop Equations and Conjectures

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