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Fusion and positivity in chiral conformal field theory

James E. Tener

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.

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.

Paper Structure

This paper contains 30 sections, 39 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Unitary VOAs and their representations
  4. Unitary VOAs
  5. Unitary modules
  6. Intertwining operators
  7. Modes of intertwining operators as unbounded operators
  8. Conformal nets and Connes' fusion
  9. Bounded localized vertex operators
  10. Systems of generalized annuli
  11. VOAs with bounded localized vertex operators
  12. Representation theory and (local) intertwiners
  13. Transport modules and positivity for unitary VOAs
  14. The transport forms and transport modules
  15. Regular VOAs and positivity of transport matrices
  16. ...and 15 more sections

Key Result

Theorem A

The WZW conformal nets are rational for all finite-dimensional simple complex Lie algebras and all positive integer levels. In particular, the Jones-Wassermann subfactors corresponding to irreducible representations of these nets have finite index.

Theorems & Definitions (110)

  • Theorem A
  • Theorem B
  • Corollary C
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 100 more