Unitarity of the Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, I

Abstract

This is the first part in a two-part series of papers constructing a unitary structure for the modular tensor category (MTC) associated to a unitary rational vertex operator algebra (VOA). Given a rational VOA, we know that its MTC is constructed using the (finite dimensional) vector spaces of intertwining operators of this VOA. Moreover, the tensor-categorical structures can be described by the monodromy behaviors of the intertwining operators. Thus, constructing a unitary structure for the MTC of a unitary rational VOA amounts to defining an inner product on each (finite dimensional) vector space of intertwining operators, and showing that the monodromy matrices of the intertwining operators (e.g. braiding matrices, fusion matrices) are unitary under these inner products. In this paper, we develop necessary tools and techniques for constructing our unitary structures. This includes giving a systematic treatment of one of the most important functional analytic properties of the intertwining operators: the energy bounds condition. On the one side, we give some useful criteria for proving the energy bounds condition of intertwining operators. On the other side, we show that energy bounded intertwining operators can be smeared to give rise to (unbounded) closed operators. We prove that the (well-known) braid relations and adjoint relations for unsmeared intertwining operators have the corresponding smeared versions. We also give criteria on the strong commutativity between smeared intertwining operators and smeared vertex operators localized in disjoint open intervals of S1 (the strong intertwining property). Besides investigating the energy bounds condition, we also study certain genus 0 geometric properties of intertwining operators. Most importantly, we prove the convergence of certain mixed products-iterations of intertwining operators. Many useful braid and fusion relations will also be discussed.