Blobbed topological recursion: properties and applications

TL;DR

This paper introduces blobbed topological recursion, a generalization of topological recursion that incorporates holomorphic forms, providing new graphical representations, variational formulas, and applications to matrix models and map enumeration.

Contribution

It develops the theory of blobbed topological recursion, extending classical topological recursion by including holomorphic forms and establishing new representations and applications.

Findings

Graphical representation of solutions in terms of holomorphic forms

Connection to intersection numbers on moduli space

Application to multi-trace matrix models and stuffed maps

Abstract

We study the set of solutions $(\omega_{g,n})_{g \geq 0,n \geq 1}$ 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 $(\phi_{g,n})_{2g - 2 + n > 0}$. 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 formulae under infinitesimal transformation of $\phi_{g,n}$ ; (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.