Loops on surfaces, Feynman diagrams, and trees

TL;DR

This paper explores the mathematical relationship between loops on surfaces, Feynman diagrams, and trees, introducing algebraic structures like pre-Lie coalgebras and Hopf algebras to connect these concepts.

Contribution

It establishes a novel connection between the Lie cobracket on loops and the Connes-Kreimer Lie bracket on trees, with new algebraic structures for loops on surfaces.

Findings

Introduces a pre-Lie coalgebra for loops on surfaces

Defines a Hopf algebra of pointed loops

Relates Lie cobracket to Connes-Kreimer Lie bracket

Abstract

We relate the author's Lie cobracket in the module additively generated by loops on a surface with the Connes-Kreimer Lie bracket in the module additively generated by trees. To this end we introduce a pre-Lie coalgebra and a (commutative) Hopf algebra of pointed loops on a surface. In the last version I added sections on Wilson loops and knot diagrams.