Silverstein Westphal D4 Charge Scope Distinguishes Two Orientations

Silverstein–Westphal's analyzed geometric ${\varphi }^{2\mathrm{/}3}$ branch uses a D4 wrapped on the diagonal $u_2+{\tilde{u}}_2$ direction and moving along the corresponding diagonal $u_1+{\tilde{u}}_1$. In §3.1.1 the orientifold maps this D4 to an anti-D4, so the pair has zero net D4 source charge. The charged $F_2\wedge H_3$ construction in §3.1.2 instead uses a distinct $u_2-{\tilde{u}}_2$ orientation and gives the ${\tilde{\varphi }}^{2\mathrm{/}5}$ variant. The source's brief §2 discussion of a compensating charge is therefore not a valid blanket premise for the main ${\varphi }^{2\mathrm{/}3}$ branch; the detailed orientation-specific discussion governs.

The local D4 calculation in Wrapped Nil D4 DBI Reduction And Large Field Potential lets the brane move and wrap in the first Nil factor while holding the isomorphic second factor fixed; it derives only the geometric DBI term and does not establish the orientifold projection or any net charge. A flux-charge balance for a source-carrying configuration therefore cannot be attached to the ${\varphi }^{2\mathrm{/}3}$ consumer without resolving this orientation difference.

Silverstein and Westphal, §§2.3.2 and 3.1.1–3.1.2