Reconstruction. I: The classical part of a vertex operator algebra (Q2327159)

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. \textit{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 \textit{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. \textit{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 \textit{Haagerup subfactor}. The result is that it seems to pass the test. For the entire collection see [Zbl 1420.46001].