FC Sets and Twisters: The Basics of Orbifold Deconstruction
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Communications in
Mathematical Physics
FC Sets and Twisters: The Basics of Orbifold Deconstruction Peter Bantay Institute for Theoretical Physics, Eötvös Loránd University, Pázmány Péter s. 1/A, 1117 Budapest, Hungary. E-mail: [email protected] Received: 10 December 2019 / Accepted: 21 July 2020 Published online: 15 September 2020 – © The Author(s) 2020
Abstract: We present a detailed account of the properties of twisters and their generalizations, FC sets, which are essential ingredients of the orbifold deconstruction procedure aimed at recognizing whether a given conformal model may be obtained as an orbifold of another one, and if so, to identify the twist group and the original model. The close analogy with the character theory of finite groups is discussed, and its origin explained. 1. Introduction Orbifold deconstruction, i.e. the procedure aimed at recognizing whether a given 2D conformal model is an orbifold [12,15] of another one, and if so, to identify (up to isomorphism) the relevant twist group and the original model, is an effective tool to better understand both the general properties of conformal models and the precise structure of their orbifolds. The basic ideas have been described in [3,6], focusing on conceptual issues without going into the mathematical details. The purpose of the present paper is to fill this gap by giving a formal treatment of the concepts underlying the deconstruction procedure. The starting point of orbifold deconstruction is the observation [3,6] that every orbifold has a distinguished set of primaries, the so-called vacuum block, consisting of the descendants of the vacuum, and that this vacuum block has quite special properties: it is closed under the fusion product, and all its elements have integral conformal weight and quantum dimension. Such sets of primaries were termed ’twisters’ because of their relation to twist groups and twisted boundary conditions. Twisters provide the input for the deconstruction procedure: to each different twister corresponds a different deconstruction, with possibly different twist groups and/or deconstructed models. It turns out that most properties of twisters can be understood in the more general context of FC sets, which are those sets of primaries that are closed under the fusion product. As we shall see, these show deep analogies with character rings of finite groups,
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especially the so-called integral FC sets, all of whose elements have integral quantum dimension. In case of twisters, this analogy with character theory is of course far from being accidental, for it stems from their relation with the twist group of the corresponding orbifold, and it allows the generalization of several important group theoretic notions (like nilpotency, solubility, etc.) to general FC sets. In this respect, a most interesting question is: to what extent do classical results about groups generalize to selected classes of FC sets? We shall encounter several such conjectural generalizations on the way, e.g. of Lagrange’s and Ito’s cele
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