The Drinfel'd Double and Twisting in Stringy Orbifold Theory
arXiv:0708.4006 · doi:10.1142/S0129167X09005431
Abstract
This paper exposes the fundamental role that the Drinfel'd double $\dkg$ of the group ring of a finite group and its twists $\dbkg$, $β\in Z^3(G,\uk)$ as defined by Dijkgraaf--Pasquier--Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that --Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of $\dkg$--modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold --theory of global quotient given by the inertia variety of a point with a action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full --theory of the stack . Finally, we show how one can use the co-cycles above to twist a) the global orbifold --theory of the inertia of a global quotient and more importantly b) the stacky --theory of a global quotient . This corresponds to twistings with a special type of 2--gerbe.
35 pages, no figures