Conformal Orbifold Theories and Braided Crossed G-Categories

Abstract

The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G − LocA of twisted representations. This category is a braided crossed G-category in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT ❀ modular category ❀ 3-manifold invariant.Secondly, we study the relation between G−LocA and the braided (in the usual sense) representation category Rep A G of the orbifold theory A G . We prove the equivalence Rep A G ≃ (G−LocA) G , which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep A G . In the opposite direction we have G− LocA ≃ Rep A G ⋊ S, where S ⊂ Rep A G is the full subcategory of representations of A G contained in the vacuum representation of A, and ⋊ refers to the Galois extensions of braided tensor categories of [44,48].Under the assumptions that A is completely rational and G is finite we prove that A has g-twisted representations for every g ∈ G and that the sum over the squared dimensions of the simple g-twisted representations for fixed g equals dim Rep A. In the holomorphic case (where Rep A ≃ Vect C ) this allows to classify the possible categories G−LocA and to clarify the rôle of the twisted quantum doubles D ω (G) in this context, as will be done in a sequel. We conclude with some remarks on non-holomorphic orbifolds and surprising counterexamples concerning permutation orbifolds.

The Fusion Algebra of Bimodule Categories

“…The latter equality is seen as follows. Using the defining property of the counit, the unitality of the left A-action ρ l , as well as the functoriality of the braiding and the fact that the left and right A-actions on the bimodule Y commute, one obtains (24) The assertion then follows immediately by the fact [10,18] that the morphism (25) is the idempotent corresponding to the epimorphism that restricts X ⊗ Y to X ⊗ A Y.…”

Galois extensions of braided tensor categories and braided crossed G-categories

“…In the final Subsection 3.4 we show that a subcategory S ⊂ C where S ∼ = Rep G with G finite abelian induces a G-grading on C compatible with the one on C ⋊ S. Similar results are obtained in [19], in particular part II. However, our approach is quite different, more suitable for the application to quantum field theory [28] sketched below, and in places somewhat more satisfactory, e.g. concerning the braiding on C ⋊ S.…”

“…[4, p.238]. For our purposes, in particular the application to conformal field theory [28], the above strict version is sufficient. ✷…”

Galois extensions of braided tensor categories and braided crossed G-categories

“…In the final Section 3.4 we show that a subcategory S ⊂ C where S ∼ = Rep G with G finite abelian induces a G-grading on C compatible with the one on C S. Similar results are obtained in [19], in particular part II. However, our approach is quite different, more suitable for the application to quantum field theory [28] sketched below, and in places somewhat more satisfactory, e.g., concerning the braiding on C S.…”

“…In a companion paper [28] we will show, in the context of algebraic quantum field theory [14], that a chiral conformal field theory A carrying an action of a finite group G gives rise to a braided crossed G-category G-Loc A of 'G-twisted representations.' The full subcategory ∂ −1 (e) ⊂ G-Loc A of grade zero objects is just the ordinary braided representation category Rep A, which does not the G-action into account.…”