Fusion and positivity in chiral conformal field theory

Abstract

In this article we show that the conformal nets corresponding to WZW models are rational, resolving a long-standing open problem. Specifically, we show that the Jones-Wassermann subfactors associated with these models have finite index. This result was first conjectured in the early 90s but had previously only been proven in special cases, beginning with Wassermann's landmark results in type A. The proof relies on a new framework for the systematic comparison of tensor products (a.k.a. `fusion') of conformal net representations with the corresponding tensor product of vertex operator algebra modules. This framework is based on the geometric technique of `bounded localized vertex operators,' which realizes algebras of observables via insertion operators localized in partially thin Riemann surfaces. We obtain a general method for showing that Jones-Wassermann subfactors have finite index, and apply it to additional families of important examples beyond WZW models. We also consider applications to a class of positivity phenomena for VOAs, and use this to outline a program for identifying unitary tensor product theories of VOAs and conformal nets even for badly-behaved models.

66 pages. Version 3 has improvements to exposition and typo corrections to match published version. There are changes to the numbering of results in Section 2 (starting from Definition 2.14) and Section 3 (starting from the new Lemma 3.15)

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• W-algebras as coset vertex algebras
• Categorical extensions of conformal nets
• Unitarity of the modular tensor categories associated to unitary vertex operator algebras, I
• Conformal covariance and related properties of chiral QFT
• Representation theory in chiral conformal field theory: from fields to observables
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Cited by in corpus (6)

• Unbounded field operators in categorical extensions of conformal nets
• From vertex operator superalgebras to graded-local conformal nets and back
• Non-unitary Wightman CFTs and non-unitary vertex algebras
• Regular vertex operator subalgebras and compressions of intertwining operators
• Conformal nets from minimal W-algebras
• Weak quasi-Hopf algebras, C*-tensor categories and conformal field theory, and the Kazhdan-Lusztig-Finkelberg theorem