Representation theory in chiral conformal field theory: from fields to observables

TL;DR

This paper explores the deep relationship between different mathematical frameworks of chiral conformal field theory, establishing a correspondence between their representation theories and demonstrating the construction of WZW conformal nets via geometric interpolation.

Contribution

It extends the geometric interpolation method to relate the representation theories of conformal nets and VOAs, including the construction of conformal net representations from VOAs and their fusion products.

Findings

Established a correspondence between conformal net and VOA representations.

Constructed conformal net representations from VOAs.

Demonstrated all WZW conformal nets can be obtained through geometric interpolation.

Abstract

This article develops new techniques for understanding the relationship between the three different mathematical formulations of two-dimensional chiral conformal field theory: conformal nets (axiomatizing local observables), vertex operator algebras (axiomatizing fields), and Segal CFTs. It builds upon previous work which introduced a geometric interpolation procedure for constructing conformal nets from VOAs via Segal CFT, simultaneously relating all three frameworks. In this article, we extend this construction to study the relationship between the representation theory of conformal nets and the representation theory of vertex operator algebras. We define a correspondence between representations in the two contexts, and show how to construct representations of conformal nets from VOAs. We also show that this correspondence is rich enough to relate the respective 'fusion product' theories for conformal nets and VOAs, by constructing local intertwiners (in the sense of conformal nets) from intertwining operators (in the sense of VOAs). We use these techniques to show that all WZW conformal nets can be constructed using our geometric interpolation procedure.