The Drinfeld Double and Twisting in Stringy Orbifold Theory

This paper exposes the fundamental role that the Drinfel’d double $\dkg$ of the group ring of a finite group $G$ and its twists $\dbkg$, $\beta \in Z^3(G,\uk)$ 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 $G$–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 $K$–theory of global quotient given by the inertia variety of a point with a $G$ action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full $K$–theory of the stack $[pt/G]$. Finally, we show how one can use the co-cycles $\beta$ above to twist a) the global orbifold $K$–theory of the inertia of a global quotient and more importantly b) the stacky $K$–theory of a global quotient $[X/G]$. This corresponds to twistings with a special type of 2–gerbe.

💡 Research Summary

The paper establishes the Drinfel’d double D(k