  We show that the complex cohomologies of Bott, Chern, and Aeppli and the
symplectic cohomologies of Tseng and Yau arise in the context of type II string
theory. Specifically, they can be used to count a subset of scalar moduli
fields in Minkowski compactification with RR fluxes in the presence of either
O5/D5 or O6/D6 brane sources, respectively. Further, we introduce a new set of
cohomologies within the generalized complex geometry framework which
interpolate between these known complex and symplectic cohomologies. The
generalized complex cohomologies play the analogous role for counting massless
fields for a general supersymmetric Minkowski type II compactification with
Ramond-Ramond flux.
