Convergence in conformal field theory

TL;DR

This paper reviews key convergence and analytic extension results in conformal field theory, emphasizing the role of vertex operator algebras and discussing open problems and conjectures in the field.

Contribution

It provides a comprehensive overview of convergence results, conjectures, and problems in conformal field theory using vertex operator algebra representation theory.

Findings

Summarizes convergence results for products of intertwining operators.

Discusses analytic extension results and related conjectures.

Highlights open problems in orbifold conformal field theory.

Abstract

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. We review some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. We also review the related analytic extension results, conjectures and problems. We discuss the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of $q$-traces and pseudo-$q$-traces of products of intertwining operators. We also discuss the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. We then explain conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.