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–Loc A 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–Loc A and the braided (in the usual sense) representation category Rep A G of the orbifold theory A G . We prove the equivalence RepA G ≃(G–Loc A) 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 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 this allows to classify the possible categories G− Loc A 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.