Massive IIA D6/O6 tadpole on a compact Calabi–Yau threefold

Let $E$ be an oriented smooth four-dimensional external spacetime patch, choose $\chi \in {\mathrm{\Omega }}_c^4(E)$ with ${\int }_E\chi =1$, and let $X$ be a smooth compact oriented Calabi–Yau threefold without boundary in the broad convention of Compact Calabi–Yau manifold conventions: a connected compact Kähler complex threefold with a specified holomorphic isomorphism $K_X\cong {\mathcal{O}}_X$. Use the chosen orientation on $X$, with product spacetime $M:=E\times X$ oriented external factor first. Let finitely many D6/O6 sources wrap product worldvolumes $E\times {\tilde{W}}_a$, where each ${\tilde{W}}_a\subseteq X$ is a compact oriented embedded three-manifold without boundary and the product orientations are used. Put $S:={\bigcup }_a{\tilde{W}}_a$. Let ${\mathrm{\Sigma }}_3\subseteq X$ be a compact oriented embedded three-manifold without boundary transverse to every ${\tilde{W}}_a$. For each source, take $q_a=+1$ for a D6 and $q_a=-2$ for an O6-minus plane in the cited source convention.

Write ${\pi }_X:M\to X$ for projection and use ${\pi }_{X\ast }$ for integration over the oriented four-dimensional $E$ fibre, with the convention of A space-filling source current reduces to the negative internal integration current.

Use the natural-unit convention $\mathrm{\hslash }=c=1$ and ${\mathrm{\ell }}_s:=\sqrt{{\alpha }^{\mathrm{\prime }}}$. For a smooth manifold $N$ of dimension $n$, a degree-$k$ de Rham current is a continuous real-linear functional on compactly supported smooth $(n-k)$-forms, with differential $⟨dT,\zeta ⟩=(-1{)}^{k+1}⟨T,d\zeta ⟩$; a smooth form ${\alpha }_k$ determines $T_{{\alpha }_k}(\zeta )={\int }_N{\alpha }_k\wedge \zeta$. Let $T_2$ be an internal degree-two de Rham current on $X$, represented by a smooth two-form on $X\setminus S$, with no smoothness imposed on $S$. Its external-independent extension $F_2$ to $M$ is the degree-two current defined by $⟨F_2,{\zeta }_8⟩:=⟨T_2,{\pi }_{X\ast }{\zeta }_8⟩$ for ${\zeta }_8\in {\mathrm{\Omega }}_c^8(M)$, where ${\pi }_{X\ast }$ is integration over the oriented four-dimensional fibre $E$. Let $F_0\in \mathbb{R}$ be constant and let $H_3\in {\mathrm{\Omega }}^3(X)$ be a smooth closed internal form, pulled back to $M$; the condition $d_X{H}_3=0$ means there is no NS5 magnetic current in this model. Thus $F_0$ and the smooth coordinate components of $F_2,H_3$ have units of inverse length, and an $H_3$ period has units of length squared.

Assume the localized Bianchi identity holds distributionally in the Graña source convention, $d{F}_2-F_0{H}_3=2{\kappa }_0^2{\mu }_6{J}_3=2\pi {\mathrm{\ell }}_s{J}_3,2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4},{\mu }_6=(2\pi {)}^{-6}{\alpha }^{\mathrm{\prime }-7\mathrm{/}2}\mathrm{.}$ Here $H_3$ in the equation denotes its smooth pullback current $T_{{\pi }_X^{\ast }{H}_3}$. The source current is the test-first current from Type Iia D6 O6 Localized F2 Current and Cover Charge: for every compactly supported test seven-form ${\zeta }_7$, ${\int }_M{\zeta }_7\wedge {\delta }_3(E\times {\tilde{W}}_a)={\int }_{E\times {\tilde{W}}_a}{\zeta }_7$, and $J_3={\sum }_a{q}_a{\delta }_3(E\times {\tilde{W}}_a)$. Precisely, in the mathematical de Rham-current functional convention $T_{\alpha }(\zeta )={\int }_M\alpha \wedge \zeta$, the source equation means $-⟨d{F}_2-F_0{T}_{H_3},{\zeta }_7⟩=2\pi {\mathrm{\ell }}_s{\sum }_a{q}_a{\int }_{E\times {\tilde{W}}_a}{\zeta }_7\mathrm{.}$ The minus sign is $(-1{)}^{3\cdot 7}$ from converting the source's test-first wedge to the functional-current pairing; its reduction is proved by A space-filling source current reduces to the negative internal integration current. The current coefficient $2\pi {\mathrm{\ell }}_s$ has units of length and the localized source has coordinate density of units length${}^{-3}$, so both sides have coordinate units of length${}^{-2}$.

For each pair define the source-first signed intersection where ${\sigma }_x=+1$ when a positive source tangent basis followed by a positive test-cycle tangent basis orients $T_{x}X$. Define

Under the directly adopted degree-three current equation of Type Iia D6 O6 Localized F2 Current and Cover Charge, reduction on the external patch and pairing with the compact dual cycle gives $N_{D6}({\mathrm{\Sigma }}_3)-2N_{O{6}^{-}}({\mathrm{\Sigma }}_3)+\frac{F_0}{2\pi {\mathrm{\ell }}_s}{\int }_{{\mathrm{\Sigma }}_3}{H}_3=0.$ The sign conversion from the source's test-first wedge current to the mathematical de Rham current convention is proved in A space-filling source current reduces to the negative internal integration current; it is essential because the current-compatible regularized intersection of two three-cycles is $-1$ times the source-first sign.

For the separate $T^3\mathrm{/}{\mathbb{Z}}_2$ orientifold covering convention cited by Polchinski, Chaudhuri, and Johnson, there are eight O6-minus fixed planes and their total charge is $-16$ D6 units. This count applies only to that stated cover. Under the separate free-period postulate Massive IIA Romans-mass and NS three-form flux quantization, the flux term is an integer product; the current identity itself is a real charge equation without that postulate.

Compact Calabi–Yau manifold conventions

A closed dual test form pairs a compact three-cycle with flux and source currents

A space-filling source current reduces to the negative internal integration current

A source-cycle current pairs with a transverse dual cycle by intersection number