On the tensor structure of modules for compact orbifold vertex operator algebras

Abstract

Suppose V G is the fixed-point vertex operator subalgebra of a compact group G acting on a simple abelian intertwining algebra V . We show that if all irreducible V G -modules contained in V live in some braided tensor category of V G -modules, then they generate a tensor subcategory equivalent to the category Rep G of finitedimensional representations of G, with associativity and braiding isomorphisms modified by the abelian 3-cocycle defining the abelian intertwining algebra structure on V . Additionally, we show that if the fusion rules for the irreducible V G -modules contained in V agree with the dimensions of spaces of intertwiners among G-modules, then the irreducibles contained in V already generate a braided tensor category of V G -modules. These results do not require rigidity on any tensor category of V G -modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known; for example they apply when V G is C2-cofinite but not necessarily rational. When V G is both C2-cofinite and rational and V is a vertex operator algebra, we use the equivalence between Rep G and the corresponding subcategory of V G -modules to show that V is also rational. As another application, we show that a certain category of modules for the Virasoro algebra at central charge 1 admits a braided tensor category structure equivalent to Rep SU (2), up to modification by an abelian 3-cocycle.

On rationality for $C_2$-cofinite vertex operator algebras

“…Since W ⊠ W ′ and (W ⊠ W ′ ) ′ are grading-restricted generalized V -modules which are isomorphic as graded vector spaces, it is enough to show that Φ W is injective. We will prove injectivity similar to [McR1,Theorem 4.7].…”

“…Because all irreducible V -modules are projective, in particular V itself, it follows that C is semisimple. So C is a semisimple modular tensor category, and moreover Lemma 3.6 and Proposition 3.7 of [CM] (see also [McR1,Proposition 4.16]) show that V is rational.…”

“…Now U ⊗ V is strongly rational by [McR2,Theorem 1.2(2)]. To show the same for V , we already know or have assumed that V is positive-energy, self-contragredient, and C 2 -cofinite, so by Lemma 3.6 and Proposition 3.7 of [CM] (or [McR1,Proposition 4.16]), it only remains to show that C V is semisimple. Thus suppose f : W 1 → W 2 is a surjection in C V ; we need to show that there exists s :…”

“…Moreover, Φ W is an isomorphism if and only if it is injective, since W ⊠ W ′ and (W ⊠ W ′ ) ′ are isomorphic as graded vector spaces. We then use an argument similar to the proof of [McR1,Theorem 4.7]…”

A general mirror equivalence theorem for coset vertex operator algebras

“…Such results have been achieved in the easier "quasi-classical" setting of commuting actions of a compact group G and its fixed-point vertex operator subalgebra V G on a simple vertex operator algebra V . In this case, the analogue of Theorem 1.1, proved in [McR1], states that the category of finite-dimensional G-modules is tensor equivalent to the semisimple subcategory of V G -modules whose simple objects occur as V G -submodules of V . In [McR1,MY], this theorem was used to transfer rigidity from finite-dimensional SU (2)-modules to many irreducible modules for Virasoro vertex operator algebras at central charges 13 − 6p − 6p −1 , p ∈ Z + .…”