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# Advances in Theoretical and Mathematical Physics

## Volume 10 (2006)

### Number 3

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

Pages: 345 – 394

DOI: https://dx.doi.org/10.4310/ATMP.2006.v10.n3.a3

#### Authors

#### 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 $\rpt\times\rp^{2k+1}\times S^{6-2k}$ 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.

Published 1 January 2006