Flux Compactifications on Projective Spaces and The S-Duality Puzzle

We derive a formula for D3-brane charge on a compact spacetime, which includes torsion corrections to the tadpole cancellation condition. We use this to classify D-branes and RR fluxes in type II string theory on RP^3xRP^{2k+1}xS^{6-2k} with torsion H-flux and to demonstrate the conjectured T-duality to S^3xS^{2k+1}xS^{6-2k} with no flux. When k=1, H\neq 0 and so the K-theory that classifies fluxes is twisted. When k=2 the square of the H-flux yields an S-dual Freed-Witten anomaly which is canceled by a D3-brane insertion that ruins the K-theory classification. When k=3 the cube of H is nontrivial and so the D3 insertion may itself be inconsistent and the compactification unphysical. Along the way we provide a physical interpretation for the AHSS in terms of boundaries of branes within branes.