Flux compactifications on projective spaces and the $S$-duality puzzle

Abstract

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 $\mathbb{R}P^3\times\mathbb{R}P^{2k+1}\times S^{6-2k}$ with torsion $H$-flux and to demonstrate the conjectured $T$-duality to $S^3\times S^{2k+1}\times S^{6-2k}$ 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 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.