Formal multidimensional integrals, stuffed maps, and topological recursion

TL;DR

This paper demonstrates that the large N expansion of a multi-trace hermitian matrix model is governed by topological recursion, linking combinatorial enumeration of complex maps with advanced mathematical physics techniques.

Contribution

It establishes the connection between multi-trace matrix models and topological recursion, characterizes initial data via Riemann-Hilbert problems, and introduces the concept of stuffed maps.

Findings

Topological recursion governs the large N expansion.

Initial data characterized by Riemann-Hilbert problems.

Enumeration of complex maps with elementary cells of any topology.

Abstract

We show that the large N expansion in the multi-trace 1 formal hermitian matrix model is governed by the topological recursion of [Eynard and Orantin, 2007] with initial conditions. In terms of a 1d gas of eigenvalues, this model includes - on top of the squared Vandermonde - multilinear interactions of any order between the eigenvalues. In this problem, the initial data (W10,W20) of the topological recursion is characterized: for W10, by a non-linear, non-local Riemann-Hilbert problem on a discontinuity locus to determine ; for W20, by a related but linear, non-local Riemann-Hilbert problem on the discontinuity locus. In combinatorics, this model enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology - W10 being the generating series of disks and W20 that of cylinders. In particular, by substitution one may consider maps whose elementary cells are themselves maps, for which we propose the name "stuffed maps". In a sense, our results complete the program of the "moment method" initiated in the 90s to compute the formal 1/N in the one hermitian matrix model.