The twisted Drinfeld double of a finite group via gerbes and finite groupoids
This goals of this paper are to understand the twisted Drinfeld double of a finite group, and to understand a finite `toy model' for Freed-Hopkins-Teleman's deep result identifying the Grothendieck ring of the category of highest weight representations of a loop group with the twisted equivariant \(K\)-theory of \(G\). It does this by interpreting both in the context of the categorical description of 3-dimensional quantum field theory (in fact Dijkgraaf-Witten theory) in terms of gerbes and groupoids. The twisted Drinfeld double of a group corresponds to a quantum group for the quantum field theory of a finite group twisted by a 3-cocycle. The formal algebraic definition of the twisted Drinfeld double looks mysterious, but by geometrically representing the 3-cocycle as a gerbe it turns out to have a natural interpretation one step up the categorical ladder. First (and this is perhaps the key insight), the paper identifies an analogue of a `loop group' for a finite group \(G\), or rather for its categorification. Let \(\overline{\mathbb{Z}}\) and \(\overline{G}\) (the author considers also more general groupoids \(\mathcal{G}\)) be categories with one object whose sets of morphisms are isomorphic to the integers \(\mathbb{Z}\) and to the group \(G\) correspondingly. Define \(\Lambda \overline{G}\) to be \(Hom(\overline{Z},\overline{G})\). By the Parmesan Theorem (wonderful name!) \(\Lambda \overline{G}\) indeed behaves as one would expect a loop group for \(G\) to behave, namely \(B\Lambda\overline{G}\equiv \mathcal{L}B\overline{G}\) where \(\mathcal{L}\) means to take the loop space. The Drinfeld double of \(G\) twisted by a 3-cocycle \(\omega\) can now be naturally understood as a twisting of \(\mathbb{C}(\Lambda\overline{G})\) by the 2-cocycle \(\tau(\omega)\), where \(\tau\) is the `transgression map' taking \(\omega\) to a 2-cocycle on \(\Lambda\overline{G}\). The paper points out that indeed, this can be taken as the definition of the twisted Drinfeld double. In this setup the finite analogue to Freed-Hopkins-Teleman's result (that the category of twisted representations of \(\Lambda\overline{G}\), defined as functors from \(\mathbb{C}(\Lambda\overline{G})\) twisted by \(\tau(\omega)\) to vector spaces, is the same as that of equivariant twisted vector bundles over \(G\)) turns out to be trivial. It is a hallmark of good definitions that they may make deep results (at least in toy models) appear trivial. The question of understanding the full Freed-Hopkins-Teleman in these terms (where \(G\) is a Lie group rather than a finite group) seems tantalizing. Connections with ideas of \textit{E. Lupercio} and \textit{B. Uribe} [Q. J. Math. 55, No. 2, 185--201 (2004; Zbl 1066.55006))] are also discussed.
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