Conformal Orbifold Theories and Braided Crossed G-Categories

Abstract

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 GLoc A of twisted representations. This category is a braided crossed G-category in the sense of Turaev. 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. We then study the relation between GLoc A 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 = (GLoc 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 GLoc A = Rep A^G \rtimes S, where S \subset Rep A^G is the full subcategory of representations of A^G contained in the vacuum representation of A, and \rtimes refers to the Galois extensions of braided tensor categories of [44,48]. If A is completely rational and G is finite we prove that A has g-twisted representations for every g in G. In the holomorphic case (where Rep A = Vect_C) this allows to classify the possible categories GLoc A and to clarify the role 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.