Tree Algebras: An algebraic axiomatization of intertwining vertex operators

TL;DR

This paper introduces a new algebraic framework called tree algebras to axiomatize intertwining operators in vertex algebra modules, connecting algebraic flat connections with vertex tensor categories and specific models.

Contribution

It formulates a purely algebraic axiom system for intertwining operators and demonstrates its application to vertex tensor categories and the chiral WZW model.

Findings

Tree algebras provide an algebraic axiomatization of intertwining operators.

Vertex tensor categories induce tree algebras over complex numbers.

The chiral WZW model induces a tree algebra over rational numbers.

Abstract

We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a {\em tree algebra}. Using the Riemann-Hilbert correspondence, we further prove that a vertex tensor category in the sense of Huang and Lepowsky gives rise to a tree algebra over $\C$. We also show that the chiral WZW model of a simply connected simple compact Lie group gives rise to a tree algebra over $\Q$.