Topological mirror symmetry with fluxes

Motivated by SU(3) structure compactifications, we show explicitly how to construct half-flat topological mirrors to Calabi-Yau manifolds with NS fluxes. Units of flux are exchanged with torsion factors in the cohomology of the mirror; this is the topological complement of previous differential-geometric mirror rules. The construction modifies explicit SYZ fibrations for compact Calabi-Yaus. The results are of independent interest for SU(3) compactifications. For example one can exhibit explicitly which massive forms should be used for Kaluza-Klein reduction, proving previous conjectures. Formality shows that these forms carry no topological information; this is also confirmed by infrared limits and old classification theorems