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Conformal Orbifold Theories and Braided Crossed G-Categories

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An Erratum to this article was published on 11 October 2005

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 AG of the orbifold theory AG. We prove the equivalence RepAG≃(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 AG. In the opposite direction we have is the full subcategory of representations of AG 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 Dω(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.

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Authors and Affiliations

  1. Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands

    Michael Müger

  2. Department of Mathematics, Radboud University, Nijmegen, The Netherlands

    Michael Müger

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  1. Michael Müger
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Correspondence to Michael Müger.

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Communicated by Y. Kawahigashi

Supported by NWO through the “pioneer” project no. 616.062.384 of N. P. Landsman.

An erratum to this article can be found at http://dx.doi.org/10.1007/s00220-005-1422-6

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Müger, M. Conformal Orbifold Theories and Braided Crossed G-Categories. Commun. Math. Phys. 260, 727–762 (2005). https://doi.org/10.1007/s00220-005-1291-z

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