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.

Singularities with symmetries, orbifold Frobenius algebras and mirror symmetry

“…This is, however, expected, since there is the phenomenon of discrete torsion for orbifolds. As we demonstrated in [Ka6] for every G-Frobenius algebra there exists a family of GFrobenius algebras indexed by elements of α ∈ Z 2 (G, k * ) with the same underlying data as mentioned in step 1 (up to a re-scaling of the metrics pairing the twisted sectors). In the last step there is an additional compatibility condition of the pairing, which might force one to again re-scale the pairings between the twisted sectors.…”

“…The constrains can be quite effective, but they define the action at most up to discrete torsion [Ka6]. (4) The G-Frobenius algebra.…”

“…The action of discrete torsion. In [Ka6], we analyzed the phenomenon of discrete torsion and showed that the different choices ϕ can by obtained from a fixed D(k[G]) module by tensoring with the twisted group algebra…”

“…D(k [G]) modules are a special type of G-graded G-modules, namely those, where the G-action acts by conjugation on the G-grading, cf. [Mo,Ka6,JKK].…”

“…Another way to characterize a the G-grading and G-action it to say that it is a D(k[G]) module. This statement is equivalent to saying that A is G-graded and the G-action is such that [Ka6]. We use the nomenclature of D(k[G]) module, rather than G-graded G-module since it includes the condition (*).…”