Hopf Algebras and Topological Recursion

TL;DR

This paper explores the algebraic structures underlying topological recursion, extending Hopf algebra models to include loops and analyzing the correlation functions' ring structures in different genera.

Contribution

It extends previous Hopf algebra models for topological recursion to incorporate loop graphs and studies the algebraic structures of correlation functions in various genera.

Findings

Extended Hopf algebra model includes loops for full recursion formula.

Identified ring structures in spaces of correlation functions.

Analyzed algebraic properties in both classical and quantum contexts.

Abstract

We first review our previous work arxiv:1503.02993 [math-ph] where we considered a model for topological recursion based on the Hopf Algebra of planar binary trees of Loday and Ronco and showed that extending this Hopf Algebra by identifying pairs of nearest neighbor leaves and thus producing graphs with loops we obtain the full recursion formula of Eynard and Orantin. Then we discuss the algebraic structure of the spaces of correlation functions in g = 0 and in g > 0. By taking a classical and a quantum product respectively we endow both spaces with a ring structure. This is an extended version of the contributed talk given at the 2016 von Neumann Symposium on Topological Recursion and its Influence in Analysis, Geometry and Topology, from 4 to 8 July 2016 at Hilton Charlotte University Place, USA.