Conformal Orbifold Theories and Braided Crossed G-Categories

Michael Müger
March 19, 2004
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 $\mathrm{GLoc} A$ 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 $\mathrm{Rep} A$. Combining this with [29], where $\mathrm{Rep} A$ 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 $\mathrm{GLoc} A$ and the braided (in the usual sense) representation category $\mathrm{Rep} A^G$ of the orbifold theory $A^G$. We prove the equivalence $\mathrm{Rep} A^G = (\mathrm{GLoc} A)^G$, which is a rigorous implementation of the insight that one needs to take the twisted representations of $A$ into account in order to determine $\mathrm{Rep} A^G$. In the opposite direction we have $\mathrm{GLoc} A = \mathrm{Rep} A^G \rtimes S$, where $S \subset \mathrm{Rep} A^G$ 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 $\mathrm{Rep} A = \mathrm{Vect}_C$) this allows to classify the possible categories $\mathrm{GLoc} A$ and to clarify the role of the twisted quantum doubles $D^\omega(G)$ 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.