The algebra of discrete torsion

Abstract

We analyze the algebraic structures of G-Frobenius algebras which are the algebras associated to global group quotient objects a.k.a. global orbifolds. Here G is any finite group. First, we show that these algebras are modules over the Drinfel'd double of the group ring k [ G ] and are moreover k [ G ] -module algebras and k [ G ] -comodule algebras. We furthermore consider G-Frobenius algebras up to projective equivalence and define universal shifts of the multiplication and the G-action preserving the projective equivalence class. We show that these shifts are parameterized by Z 2 ( G , k * ) and by H 2 ( G , k * ) when considering of G-Frobenius algebras up to isomorphism. We go on to show that these shifts can be realized by the forming of tensor products with twisted group rings, thus providing a group action of Z 2 ( G , k * ) on G-Frobenius algebras acting transitively on the classes G-Frobenius related by universal shifts. The multiplication is changed according to the cocycle, while the G-action transforms with a different cocycle derived from the cocycle defining the multiplication. The values of this second cocycle also appear as a factor in front of the trace which is considered in the trace axiom of G-Frobenius algebras. This allows us to identify the effect of our action by Z 2 ( G , k * ) as so-called discrete torsion, effectively unifying all known approaches to discrete torsion for global orbifolds in one algebraic theory. The new group action of discrete torsion is essentially derived from the multiplicative structure of G-Frobenius algebras. This yields an algebraic realization of discrete torsion defined via the perturbation of the multiplication resulting from a tensor product with a twisted group ring. Additionally we show that this algebraic formulation of discrete torsion allows for a treatment of G-Frobenius algebras analogous to the theory of projective representations of groups, group extensions and twisted group ring modules. Lastly, we identify another set of discrete universal transformations among G-Frobenius algebras pertaining to their super-structure and classified by Hom ( G , Z / 2 Z ) which are essential for the application to mirror symmetry for singularities with symmetries.