The Fusion Algebra of Bimodule Categories

We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of F . As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category. Introduction The Grothendieck ring K0(C) of a semisimple monoidal category C encodes a considerable amount of information about the structure of C. If C is braided, so that K0(C) is commutative, then upon complexification to K0(C)⊗ZC almost all of this information gets lost: what is left is just the number of isomorphism classes of simple objects. In contrast, in the non-braided (but still semisimple) case the complexified Grothendieck ring is no longer necessarily commutative and thus contains as additional information the dimensions of its simple direct summands. The following statement, which is a refined version of an assertion made in [O] as Claim 5.3, determines these dimensions for a particularly interesting class of categories: Theorem O. Let C be a modular tensor category, M a semisimple nondegenerate indecomposable module category over C, and C M the category of module endofunctors of M. Then there is an isomorphism of C-algebras F ∼= ⊕ i,j∈I EndC(HomC∗ M (α(Ui), α (Uj))) (1) between the complexified Grothendieck ring F =K0(C ∗ M)⊗ZC and the endomorphism algebra of the specified space of morphisms in C M. Here I is the (finite) set of isomorphism classes of simple objects of C, Ui and Uj are representatives of the classes i, j ∈ I, and α are the braided-induction functors from C to C M. For more details about the concepts appearing in the Theorem see section 1 below, e.g. the tensor functors α are given in formulas (5) and (6). In [O] the assertion of Theorem O was formulated with the help of the integers zi,j := dimCHomC∗ M (α(Ui), α (Uj)) , (2) in terms of which it states that F is isomorphic to the direct sum ⊕ i,j Matzi,j of full matrix algebras of sizes zi,j , i, j ∈ I. In this form the statement had been established previously for the particular case that the modular tensor category C is a category of endomorphisms of a type III factor (Theorem 6.8 of [BEK]), a result which directly motivated the formulation of the statement in [O]. Indeed, in [O] the additional assumption is made that the quantum dimension of any nonzero object of C is positive, a property that is automatically fulfilled for the categories arising in the framework of [BEK], but is violated in other categories (e.g. those relevant for the so-called non-unitary minimal Virasoro models) of interest in physical applications. In this note we derive the statement in the form of Theorem O, where this positivity requirement is replaced by the condition that M is nondegenerate. This property, to be explained in detail further below, is satisfied in particular in the situation studied in [BEK]. We present our proof in section 2. As an additional benefit, it provides a concrete expression for the structure constants of the Grothendieck ring of C M in terms of certain endomorphisms of the tensor unit of C, see formulas (29) and (30). Various ingredients needed in the proof are supplied in section 1. In section 3 we outline the particularities of the case that C comes from endomorphisms of a factor [BEK], and describe further relations between Theorem O and structures arising in quantum field theory; this latter part is, necessarily, not self-contained.