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 Ramond-Ramond fluxes in type II string theory on R{double-struck}P³ × R{double-struck}P^2k+1 × S^6-2k with torsion H-flux and to demonstrate the conjectured T-duality to S³ × S^2k+1 × S^6-2k with no flux. When k = 1, H = 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 dual K-theory flux 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 Atiyah-Hirzebruch spectral sequence in terms of the boundaries of branes within branes.