Mirror symmetry for tori modulo finite groups
Michael Thaddeus, Columbia

Mirror symmetry is a mysterious duality, originating in physics, between pairs of complex algebraic varieties. Many geometric structures on the two varieties, such as their Hodge numbers, are supposed to be interchanged by the duality. Vafa and Witten in 1994 described certain orbifolds, quotients of a complex torus T by a finite group G, for which mirror symmetry had to be slightly modified by introducing a B-field, that is, an element of the group cohomology H^3(G,Z). We will explain how to interpret the B-field as a flat gerbe on the quotient stack [T/G] and generalize the Vafa-Witten construction to a large class of orbifolds. It all turns out to be related to projective representations of crystallographic groups.