Discrete torsion, symmetric products and the Hilbert scheme

Recently the understanding of the cohomology of the Hilbert scheme of points on K3 surfaces has been greatly improved by Lehn and Sorger [18]. Their approach uses the connection of the Hilbert scheme to the orbifolds given by the symmetric products of these surfaces. We introduced a general theory replacing cohomology algebras or more generally Frobenius algebras in a setting of global quotients by finite groups [14]. This is our theory of group Frobenius algebras, which are group graded non–commutative algebras whose non–commutativity is controlled by a group action. The action and the grading turn these algebras into modules over the Drinfel’d double of the group ring. The appearance of the Drinfel’d double is natural from the orbifold point of view (see also [17]) and can be translated into the fact that the algebra is a G–graded G–module algebra in the following sense: the G action acts by conjugation on the grading while the algebra structure is compatible with the grading with respect to left multiplication (cf. [16, 20]). In the special case of the symmetric group, we recently proved existence and uniqueness for the structures of symmetric group Frobenius algebras based on a given Frobenius algebra [15], providing explicit formulas for the multiplication in the algebra. This uniqueness has to be understood up to the action of two groups of symmetries on group Frobenius algebras called discrete torsion and super-twisting [16]. The set of G–Frobenius algebras is acted upon by both of these groups. This action only changes some defining structures of a Frobenius algebra in a projective manner while keeping others fixed. Applying this result to the global orbifold cohomology of a symmetric product, where there is a canonical choice for the discrete torsion and super-twists, we obtain its uniqueness. Our latest results on this topic [16] explain the origin of these discrete degrees of freedom. In the special case of the Hilbert scheme as a resolution of a symmetric product the choice of sign for the metric specifies a discrete torsion cocycle that in turn changes the multiplication by a much discussed sign. Assembling our results which we review we obtain: