Blobbed topological recursion: properties and applications

We study the set of solutions $({\omega }_{g,n}{)}_{g\ge 0,n\ge 1}$ of abstract loop equations. We prove that ${\omega }_{g,n}$ is determined by its purely holomorphic part: this results in a decomposition that we call “blobbed topological recursion”. This is a generalization of the theory of the topological recursion, in which the initial data $({\omega }_{0,1},{\omega }_{0,2})$ is enriched by non-zero symmetric holomorphic forms in $n$ variables $({\varphi }_{g,n}{)}_{2g-2+n>0}$. In particular, we establish for any solution of abstract loop equations: (1) a graphical representation of ${\omega }_{g,n}$ in terms of ${\varphi }_{g,n}$; (2) a graphical representation of ${\omega }_{g,n}$ in terms of intersection numbers on the moduli space of curves; (3) variational formulae under infinitesimal transformation of ${\varphi }_{g,n}$ ; (4) a definition for the free energies ${\omega }_{g,0}=F_g$ respecting the variational formulae. We discuss in detail the application to the multi-trace matrix model and enumeration of stuffed maps.