Lie algebras, Fuchsian differential equations and CFT correlation functions

TL;DR

This paper explores the connection between affine Kac-Moody algebra-derived differential equations, their monodromy, and conformal field theory correlation functions, proving their existence on arbitrary topologies.

Contribution

It establishes a link between solutions of Knizhnik-Zamolodchikov equations and CFT correlation functions, including a proof of their existence on any world sheet topology.

Findings

Solutions encode monodromy via modular tensor categories.

Correlation functions exist on world sheets of arbitrary topology.

Provides a mathematical framework connecting algebraic and physical aspects.

Abstract

Affine Kac-Moody algebras give rise to interesting systems of differential equations, so-called Knizhnik-Zamolodchikov equations. The monodromy properties of their solutions can be encoded in the structure of a modular tensor category on (a subcategory of) the representation category of the affine Lie algebra. We discuss the relation between these solutions and physical correlation functions in two-dimensional conformal field theory. In particular we report on a proof for the existence of the latter on world sheets of arbitrary topology.