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

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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.

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Notes

  1. See [22, §6] for a discussion of some of the difficulties involved.

  2. The definition of generalized annulus is introduced in order to avoid certain scalar ambiguities arising from the projective nature of the representations of \({\text {Diff}}_c(S^1)\) under consideration. These ambiguities were not problematic in [45], but they make it quite difficult to formulate Definition 5.1 capturing the relationship between VOA modules and conformal net representations. While the introduction of generalized annuli accounts for increased complexity up front in Sect. 3, it significantly simplifies the statements of results for the remainder of the article.

  3. The condition on the rationality of the lowest eigenvalues in the cited result is not essential, and is only present because rationality of eigenvalues is included in the definition of module in [20].

  4. \({\text {Vir}}_c\) is a Lie algebra equipped with a distinguished central copy of \(\mathbb {R}\), and by a representation of \({\text {Vir}}_c\) we mean a representation of the Lie algebra in which the distinguished copy of \(\mathbb {R}\) acts standardly. This is the same as a central charge c representation of \({\text {Vir}}\).

  5. More precisely, we pick a single X for each orbit \(U(r_{\theta })\pi (X)U(\gamma )^*\) under reparametrization, and use this to define the interior for the entire orbit. When possible, we choose X with non-empty interior.

  6. The Möbius transformation \(\psi \) does not appear in [45, Thm. 3.21], but the argument only requires that \(\Sigma \) be a degenerate annulus inside the unit disk with outgoing boundary parametrized by the identity. This is why it is important to work with \(\hat{\Sigma }_X\) instead of \(\Sigma _X\) (see the discussion before Lemma 4.24). We are using crucially here that \(\psi \) is a Möbius transformation and not just an arbitrary diffeomorphism.

  7. To be more precise, we should write \(\pi ^{\tilde{M}}_{I^\prime }(\pi ^0_{I^\prime }(\tilde{B}Y^W(\tilde{a},\tilde{z})\tilde{A}))\), where \(\pi ^0_{I^\prime }:\mathcal {A}_W(I^\prime ) \rightarrow \mathcal {B}(\mathcal {H}_V)\) is the representation of Proposition 4.16 exhibiting \(\mathcal {A}_W\) as a subnet of \(\mathcal {A}_V\).

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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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    James E. Tener

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Tener, J.E. Representation theory in chiral conformal field theory: from fields to observables. Sel. Math. New Ser. 25, 76 (2019). https://doi.org/10.1007/s00029-019-0526-3

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