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

James E. Tener

Published 2018-10-18Version 1

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 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. This article extends our 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, and apply this to the local equivalence problem for representations of loop groups at positive integral level.