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

  • James E. TenerEmail author
Article
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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 (Tener in Adv Math 349:488–563, 2019), 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.

Mathematics Subject Classification

81T40 81T05 17B69 46L37 46L60 

Notes

Acknowledgements

A significant portion of the research for this article was undertaken at the Max Planck Institute for Mathematics, Bonn, between 2014 and 2016, and I would like to gratefully acknowledge their hospitality and support during this period. This work was also supported in part by an AMS-Simons travel grant. I am grateful to many people for enlightening conversations which improved this article, including Marcel Bischoff, Sebastiano Carpi, Thomas Creutzig, Terry Gannon, André Henriques, and Robert McRae.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.CanberraAustralia

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