Type Iia D6 O6 Localized F2 Current and Cover Charge

Adopt directly the degree-three sourced identity of Graña, §4.2, Eq. (4.18), with constant $F_0\in \mathbb{R}$ and smooth closed $H_3\in {\mathrm{\Omega }}^3(M^{10})$ on an oriented smooth ten-dimensional manifold $M^{10}$ without boundary. No higher-degree RR equation or truncated-polyform identity is adopted here.

For $0\le k\le 10$, a degree-$k$ current $T$ is a continuous real-linear functional on ${\mathrm{\Omega }}_c^{10-k}(M^{10})$, the smooth compactly supported test forms with their usual test-function topology (on each fixed compact support, all coordinate derivatives converge uniformly). Use the test-first convention: a smooth $k$-form $A$ represents $T_A(\eta )={\int }_{M^{10}}\eta \wedge A$, and define $(dT)(\eta )=(-1{)}^{10-k}T(d\eta )$ for $\eta \in {\mathrm{\Omega }}_c^{9-k}(M^{10})$. This sign agrees with exterior differentiation of smooth forms by integration by parts.

Let each ${\iota }_a:W_a\hookrightarrow M^{10}$ be a proper smooth embedding of an oriented seven-dimensional submanifold without boundary; the chosen orientation determines the sign of the integration current below. Proper means that inverse images of compact sets are compact. Require local finiteness: each compact subset of $M^{10}$ meets only finitely many $W_a$. The source set $S={\bigcup }_a{\iota }_a(W_a)$ is then closed. Define the degree-three integration current by ${\delta }_3(W_a)({\omega }_7):={\int }_{W_a}{\iota }_a^{\ast }{\omega }_7,{\omega }_7\in {\mathrm{\Omega }}_c^7(M^{10}),$ and put $J_3={\sum }_a{q}_a{\delta }_3(W_a)$. Properness makes every displayed integral finite; local finiteness makes the sum finite on each compact test support. In submanifold charts integration over a compact support is bounded by finitely many sup norms, so these are continuous functionals.

Take $F_2$ to be a degree-two current on all of $M^{10}$ whose restriction to $M^{10}\setminus S$ is represented by a smooth two-form: $F_2(\eta )=\int \eta \wedge F_2^{\mathrm{r}\mathrm{e}\mathrm{g}}$ for tests supported there. Adopt the current equation $d{F}_2-F_0{T}_{H_3}=2\pi \sqrt{{\alpha }^{\mathrm{\prime }}}{J}_3=2{\kappa }_0^2{\mu }_6{J}_3,2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4},{\mu }_6=(2\pi {)}^{-6}{\alpha }^{\mathrm{\prime }-7\mathrm{/}2}\mathrm{.}$ Explicitly, for every ${\omega }_7\in {\mathrm{\Omega }}_c^7(M^{10})$, it means $F_2(d{\omega }_7)-F_0{\int }_{M^{10}}{\omega }_7\wedge H_3=2\pi \sqrt{{\alpha }^{\mathrm{\prime }}}{\sum }_a{q}_a{\int }_{W_a}{\iota }_a^{\ast }{\omega }_7\mathrm{.}$ Its source-free restriction is $d{F}_2^{\mathrm{r}\mathrm{e}\mathrm{g}}=F_0{H}_3$. Existence of a global $F_2$ satisfying this equation is an adopted condition, not a theorem for arbitrary source and flux data. Set $q_a=+1$ for a D6 and $q_a=-2$ for an O6-minus plane in the cited source convention. In the specific $T^3\mathrm{/}{\mathbb{Z}}_2$ covering-space example of Polchinski, Chaudhuri, and Johnson, there are $2^{9-p}=8$ fixed planes for $p=6$, giving total covering-space charge $-16$ D6 units.

In the natural-unit convention, target coordinates have length dimension, $F_0,H_3,F_2^{\mathrm{r}\mathrm{e}\mathrm{g}}$ have coordinate components of dimension length${}^{-1}$, the D6 tension ${\mu }_6$ has dimension length${}^{-7}$, and the transverse current has coordinate components of dimension length${}^{-3}$. Thus $2\pi \sqrt{{\alpha }^{\mathrm{\prime }}}{J}_3$ and $d{F}_2-F_0{H}_3$ both have components of dimension length${}^{-2}$. The stated O6 count is only in the cited covering-space convention.

Graña, §4.2, Eqs. (4.18)–(4.19)

Polchinski, Chaudhuri, and Johnson, Lecture III §3.3

The stated units and normalization use the conventions definition. The degree-three source equation is adopted directly from Graña Eq. (4.18), independently of any claim about a truncated RR polyform. No global tadpole or other quotient count is inferred.