## 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.