Anomalous D-Brane Charge in F-Theory Compactifications

Tadpole cancellation in F-theory on an elliptic Calabi-Yau fourfold $X\to B_3$ demands some spacetime-filling three-branes (points in $B_3$). If moved to the discriminant surface, which supports the gauge group, and dissolved into a finite size instanton, the second Chern class of the corresponding bundle $E$ is expected to give a compensating contribution. However the dependence of D-brane charge on the geometry of $W$ and on the embedding $i: W\to B_3$ gives a correction to $c_2(E)$. We show how this is reconciled by considering the torsion sheaf $i_*E$ and discuss some integrality issues related to global properties of $X$ as well as the moduli space of this object.