The Drinfel'd Double and Twisting in Stringy Orbifold Theory

TL;DR

This paper explores the role of the Drinfel'd double and its twists in stringy orbifold theories, providing geometric realizations and demonstrating how to twist orbifold K-theory using co-cycles and gerbes.

Contribution

It introduces a natural framework for G-Frobenius algebras via the Drinfel'd double and offers geometric realizations of the double and its representation ring in orbifold K-theory.

Findings

G-Frobenius algebras are naturally defined as modules over the Drinfel'd double.

The Drinfel'd double is realized as the global orbifold K-theory of the inertia variety.

Twisting orbifold K-theory with co-cycles corresponds to gerbe twistings in the stacky setting.

Abstract

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.