Blobbed topological recursion: properties and applications

We study the set of solutions (omega(g, n)) g >= 0, n >= 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 generalisation 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 (phi(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 phi(g, n); (2) a graphical representation of omega(g, n) in terms of intersection numbers on the moduli space of curves; (3) variational formulas under infinitesimal transformation of phi(g, n); (4) a definition for the free energies omega(g),(0) = F-g respecting the variational formulas. We discuss in detail the application to the multi-trace matrix model and enumeration of stuffed maps.