Reconstruction. I: The classical part of a vertex operator algebra. (English) Zbl 1440.17022

Gannon, Terry

Izumi, Masaki (ed.) et al., Operator algebras and mathematical physics. Proceedings of the 9th international conference on operator algebras and mathematical physics, Tohoku University, Sendai, Japan, August 1–12, 2016. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 80, 71-107 (2019).

The representation category of a strongly rational (i.e. rational, \(C_2\)-cofinite/lisse, simple, self-contragredient/self-dual, CFT-type) vertex operator algebra (VOA) \(V\) is known to be a modular tensor category (MTC) by the work of Huang.
Reconstruction addresses the converse problem: given a modular tensor category \(\mathcal{C}\), does there exist a (strongly rational) vertex operator algebra \(V\) whose representation category is \(\mathcal{C}\)? This is conjectured to be the case, at least for unitary modular tensor categories.
The present paper collects evidence for this conjecture by studying the possible classical parts of vertex operator algebras \(V\) with a given modular tensor category \(\mathcal{C}\). The classical part of a strongly rational vertex operator algebra \(V\) is the largest vertex operator subalgebra of \(V\) of lattice and affine type, i.e. the maximal vertex operator subalgebra of \(V\) that is a conformal extension of a tensor product of a lattice vertex operator algebra and affine vertex operator algebras at positive integer levels.
Whether a classical part exists or not is a first test (but not sufficient) for the existence of a vertex operator algebra associated with \(\mathcal{C}\). With the classical part determined, the reconstruction of the vertex operator algebra then reduces to identifying the complementary part, the commutant/centraliser, which is in general a much harder problem.
The example studied in this text is the modular tensor category associated with the Haagerup subfactor. The result is that it seems to pass the test.
For the entire collection see [Zbl 1420.46001].

Reviewer: Sven Möller (Piscataway)

MSC:

17B69	Vertex operators; vertex operator algebras and related structures
18M15	Braided monoidal categories and ribbon categories

Keywords:

vertex operator algebra; modular tensor category; conformal field theory; Haagerup subfactor; reconstruction