Reconstruction I. The classical part of a vertex operator algebra

Abstract

It is known that the category of representations of a rational vertex operator algebra (RVOA) is a modular tensor category. An interesting and important question is the converse, reconstruction: given a modular tensor category $C$, is there a RVOA whose category of representations is $C$? In this paper we explain how, given $C$, to recover the affine algebra and lattice vertex operator subalgebras possible inside a RVOA realizing $C$. We apply this to the most celebrated 'exotic' modular tensor category: that associated to the Haagerup subfactor. Whether this can be done consistently is a highly nontrivial test of the existence of the still-hypothetical Haagerup VOA.