D2-branes in B fields

TL;DR

The note resolves an apparent puzzle where a D2-brane in a background $B$ field seems to acquire a non-quantized D0-brane charge from $-\int_\Sigma B$. By analyzing both the M-theory momentum cancellation and the Type IIA bulk contributions, it shows that for D2-branes wrapping homotopically trivial cycles the $\int B$ contribution is exactly cancelled, leaving the total D0-charge determined solely by the quantized flux $\frac{1}{2\pi} \int F$. This cancellation is interpreted as momentum conservation in M-theory and remains valid upon dimensional reduction to IIA, where a bulk $G^{(4)}$-coupling cancels the $\int_\Sigma B$ piece. The result clarifies D0-charge quantization in D2-brane systems with flux, resolves the SU(2) spherical D2-brane puzzle posed by Bachas, Douglas and Schweigert, and generalizes to higher D$p$-branes and potential connections to noncommutative geometry and Myers dielectric effects.

Abstract

This note focuses on the coupling of a type IIA D2-brane to a background B field. It is shown that the D0-brane charge arising from the integral over the D2-brane of the pullback of the B field is cancelled by bulk contributions, for a compact D2-brane wrapping a homotopically trivial cycle in space-time. In M-theory this cancellation is a straightforward consequence of momentum conservation. This result resolves a puzzle recently posed by Bachas, Douglas and Schweigert related to the quantization of R-R charges on stable spherical D2-branes on the group manifold SU(2).