Blobbed topological recursion Properties and applications

We study the set of solutions (ωg,n)g≥0,n≥1 of abstract loop equations. We prove that ωg,n is determined by its purely holomorphic part: this results in a decomposition that we call "blobbed topological recursion". This is a generalisation of the theory of the topological recursion, in which the initial data (ω0,1, ω0,2) is enriched by non-zero symmetric holomorphic forms in n variables (φg,n)2g-2+n>0. In particular, we establish for any solution of abstract loop equations: (1) a graphical representation of ωg,n in terms of φg,n; (2) a graphical representation of ωg,n in terms of intersection numbers on the moduli space of curves; (3) variational formulas under infinitesimal transformation of φg,n; (4) a definition for the free energies ωg,0 = Fg respecting the variational formulas. We discuss in detail the application to the multi-trace matrix model and enumeration of stuffed maps.