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.