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Compact Type IIB D3 tadpole from the sourced five-form Bianchi identity

Physical scope

Conditional compact-current integration of the adopted GKP Type IIB D3-sourced five-form Bianchi identity. It is a Maxwell-flux tadpole identity on the selected cover/quotient, not a local solution, supersymmetry, stability, or quantization theorem.

Statement

Let $X$ be the compact oriented boundaryless six-manifold used consistently for flux and localized-source integrals. Assume the sourced five-form Bianchi identity in the GKP Einstein-frame Type IIB convention is an equality of degree-six currents. Let $T_{f l u x}$ be the specified degree-six current representing $H_{3} \land F_{3}$ , and define $\int_{X} H_{3} \land F_{3} : = ⟨ T_{f l u x}, 1 ⟩$ ; when the field strengths are smooth on $X$ , this is their ordinary top-form integral. Also assume $⟨ ρ_{3}^{l o c}, 1 ⟩ = Q_{3}^{l o c}$ . Then $\frac{1}{2 κ_{0}^{2} μ_{3}} \int_{X} H_{3} \land F_{3} + Q_{3}^{l o c} = 0.$ Using $2 κ_{0}^{2} = (2 π)^{7} α^{' 4}$ and $μ_{3} = (2 π)^{- 3} α^{' - 2}$ , define the dimensionless Maxwell flux charge $Q_{f l u x} = \frac{1}{(2 π)^{4} α^{' 2}} \int_{X} H_{3} \land F_{3};$ then $Q_{f l u x} + Q_{3}^{l o c} = 0$ . This is not a Page-charge equation. The relation does not require flux quantization; it is an integer $N_{f l u x}$ only in a separate model that also assumes the free-period quantization postulate and an integral-cohomology pairing that identifies the normalized de Rham pairing with an integer cup-product evaluation on the oriented fundamental class. The local O3-minus source has charge $- 1 / 4$ per plane in the quotient charge convention. In the cited $T^{6} / Z_{2}$ calculation on the $T^{6}$ cover, the cover source density assigns coefficient $- 1 / 2$ to each of the 64 O3 fixed-point currents; $M$ independent mobile D3-branes contribute $2 M$ on the cover, including their images. For this cited cover specialization, also assume that there is no localized flux on the O3 planes, as explicitly required in Schulz §2.1 after Eq. (2.3) and before Eq. (2.4). This is a separate source-sector premise, not a consequence of the absence of induced D3 charge. With this premise and no additional induced D3 charge, write $N_{f l u x} = Q_{f l u x}$ for the cover integer; the cover tadpole is $N_{f l u x} + 2 M = 32$ , equivalently $\frac{1}{2} N_{f l u x} + M = 16$ . When exotic O3 planes are excluded, the cited $T^{6} / Z_{2}$ model also requires even normalized $F_{3}$ and $H_{3}$ periods on the cover. These are source- and lattice-normalization inputs for that specialization, not consequences of the generic current integration above. Every induced D3 charge from a wrapped D7 curvature or worldvolume field is included only when its source action and normalization are part of the selected model. The conclusion is conditional on the adopted sourced Bianchi identity and does not establish existence, stability, or supersymmetry of a solution.

Statement references

De Rham currents of compact oriented source cycles

Type IIB Calabi–Yau orientifold flux background conventions

An exact top-degree current has zero total charge on a closed manifold

GKP source-coupled five-form Bianchi identity

Local Type IIB D3 Brane Source Maxwell And Page Charge

Type II Dp Brane RR Current And D3 Charge

Giddings–Kachru–Polchinski, §2.1 and §4.1

Graña, §§2.2 and 4.2

Van Riet–Zoccarato, §§4.1 and 4.4

Kachru, Schulz, Tripathy, and Trivedi, §2.1

Schulz, §2.1

Facts & Assumptions

Given: A compact oriented boundaryless $X$ , its declared cover or quotient convention, the degree-six sourced Bianchi identity, and the same normalization for flux periods and source currents.

[F1]

De Rham currents of compact oriented source cycles: a degree- $k$ current acts continuously on smooth $(6 - k)$ -forms; a smooth form $α$ induces $T_{α} (η) = \int_{X} α \land η$ ; for a point $p$ , $⟨ δ_{p}, f ⟩ = f (p)$ .

[F2]

Type IIB Calabi–Yau orientifold flux background conventions: $X$ is compact, oriented, smooth and boundaryless; $α^{'}$ has dimension length squared; $F_{3}$ is the raw RR field and the source and period integrals use the same cover/quotient convention.

[F3]

An exact top-degree current has zero total charge on a closed manifold: for a degree-five current $T$ on this closed six-manifold, $⟨ d T, 1 ⟩ = 0$ .

[A1]

GKP source-coupled five-form Bianchi identity: $d_{X} {\tilde{F}}_{5}^{i n t} = T_{f l u x} + 2 κ_{0}^{2} μ_{3} ρ_{3}^{l o c}$ as degree-six currents, where $T_{f l u x}$ is the specified current representing $H_{3} \land F_{3}$ and $\int_{X} H_{3} \land F_{3} = ⟨ T_{f l u x}, 1 ⟩$ ; also $⟨ ρ_{3}^{l o c}, 1 ⟩ = Q_{3}^{l o c}$ .

[F4]

Local Type IIB D3 Brane Source Maxwell And Page Charge: the localized D3 source current is $J_{6} = μ_{3} δ_{6} (W)$ ; Maxwell charge is gauge-invariant RR flux, while Page charge is a separate potential-corrected local quantity.

[F5]

Type II Dp Brane RR Current And D3 Charge: WZ variation fixes the unit D3 source current, while the self-dual bulk Bianchi coefficient remains a separately adopted physical input.

[A2]

For the cited $T^{6} / Z_{2}$ cover specialization, assume no localized flux on the O3 planes: Schulz §2.1 states this explicitly after Eq. (2.3), before the even cover-period condition (2.4). This condition is retained independently of the assumption of no extra induced D3 charge. In the GKP quotient charge convention an O3-minus plane contributes $- 1 / 4$ . For the cited $T^{6} / Z_{2}$ cover source density, the corresponding coefficient is $- 1 / 2$ at each fixed-point current, as in Schulz §2.1 Eq. (2.6); this source normalization is physical input, not a deduction from the current-pairing theorem. With no exotic O3 planes, Kachru et al. §2.1 footnote 2 require the normalized cover periods to be even.

Proof

1.1F1F3A1F4F5givenalgebra

The right side of [A1] is a degree-six current on $X$ . Since its left side is the differential of a degree-five current, [F3] gives $⟨ d_{X} {\tilde{F}}_{5}^{i n t}, 1 ⟩ = 0$ . By [A1], pairing the Bianchi identity with $1$ gives $⟨ T_{f l u x}, 1 ⟩ + 2 κ_{0}^{2} μ_{3} Q_{3}^{l o c} = 0$ . By the stated pairing convention this is $\int_{X} H_{3} \land F_{3} + 2 κ_{0}^{2} μ_{3} Q_{3}^{l o c} = 0.$ In the smooth sector [F1] identifies $T_{f l u x}$ with the current of the ordinary wedge form. The source coefficient remains the explicit physical input [A1]; [F4]–[F5] confirm its D3 source meaning without deriving its self-dual normalization from WZ variation.

2.1F2step 1.1algebra

The stated constants satisfy $2 κ_{0}^{2} μ_{3} = ((2 π)^{7} α^{' 4}) ((2 π)^{- 3} α^{' - 2}) = (2 π)^{4} α^{' 2} > 0$ . Dividing the equality from step 1.1 by this positive constant yields $\frac{1}{(2 π)^{4} α^{' 2}} \int_{X} H_{3} \land F_{3} + Q_{3}^{l o c} = 0,$ which is exactly $Q_{f l u x} + Q_{3}^{l o c} = 0$ by the stated definition.

3.1A2givenstep 2.1algebra∎

For the specialized cover setup, use the additional no-localized-O3-flux premise [A2], the adopted cover coefficients $- 1 / 2$ at 64 fixed points, $M$ mobile D3-branes and their $M$ images, and the stipulated absence of extra induced D3 charge. The localized cover total is $Q_{3}^{l o c} = 2 M - 64 / 2 = 2 M - 32$ . Substitution into the generic identity in 2.1 gives $N_{f l u x} + 2 M = 32$ and, after division by two, $\frac{1}{2} N_{f l u x} + M = 16$ . The no-localized-flux and even-period restrictions are source-specific physical/lattice inputs; neither follows from integrating the current equation, nor is localized discrete flux inferred to vanish from a smooth de Rham representative alone. No step of the generic current identity uses this additional specialization premise.

Remarks

No flux-integrality input is used in the proof. The conditional integer pairing clause in the Statement is separate from this real-current Stokes deduction. The O3 charge input is used only to interpret the source sum in the stated cover convention. The transitive orientifold-definition carrier and the current R4 self-dual source chain remain unresolved; this item's readiness stays escalated until those current prerequisites are resolved.

Compact Type IIB D3 tadpole from the sourced five-form Bianchi identity

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources