# Conformal Orbifold Theories and Braided Crossed G-Categories

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.

open access link
Commun. Math. Phys. 260, 727-762 (2005)

@article{Muger:2005ra,
author = "Muger, M.",
title = "{Conformal orbifold theories and braided crossed
G-categories}",
journal = "Commun. Math. Phys.",
volume = "260",
year = "2005",
pages = "727-762",
doi = "10.1007/s00220-005-1291-z, 10.1007/s00220-005-1422-6",
note = "[Erratum: Commun. Math. Phys.260,763(2005)]",
eprint = "math/0403322",
archivePrefix = "arXiv",
primaryClass = "math.QA",
SLACcitation = "%%CITATION = MATH/0403322;%%"
}

Keywords:

*none*