Hopf Algebraic Structure for Tagged Graphs and Topological Recursion

TL;DR

This paper introduces a novel Hopf algebraic framework for tagged graphs that facilitates the reconstruction of topological recursion on spectral curves, unifying various matrix integral equations.

Contribution

It develops a new Hopf algebraic structure based on shuffle graphs, enabling a unified approach to topological recursion and matrix integrals.

Findings

Reconstructed topological recursion using the new algebraic structure

Unified various matrix integral equations as special cases

Established non-commutative product in the Hopf algebra

Abstract

Using the shuffle structure of the graphs, we introduce a new kind of the Hopf algebraic structure for tagged graphs with, or without loops. Like a quantum group structure, its product is non-commutative. With the help of the Hopf algebraic structure, after taking account symmetry of the tagged graphs, we reconstruct the topological recursion on spectral curves proposed by B. Eynard and N. Orantin, which includes the one-loop equations of various matrix integrals as special cases.