Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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Type II Dp Brane RR Current And D3 Charge

Physical scope

A single positively oriented boundaryless timelike D3 with zero invariant worldvolume field, its local C4 WZ source, and the separately adopted weak source-coupled IIB five-form Bianchi law; no D4/IIB generalization and no derivation of the bulk source coefficient from WZ variation alone.

Statement

For an oriented Lorentzian Type II ten-manifold $X^{10}$ with mostly-plus string-frame metric and a proper smooth oriented timelike D3 embedding $ι : W_{4} ↪ X^{10}$ with $\partial W_{4} = \emptyset$ and invariant worldvolume field $F = 0$ , varying the WZ term $μ_{3} \int_{W_{4}} ι^{*} C_{4}^{W Z}$ with respect to the Hamilton potential $C_{4}^{H}$ while holding $B_{2}, C_{2}^{H}$ fixed gives the localized current $J_{6} = μ_{3} δ_{6} (W)$ , normalized by $\int_{X} η_{4} \land δ_{6} (W) = \int_{W} ι^{*} η_{4}$ . This uses the exact crosswalk $C_{4}^{W Z} = C_{4}^{H} + \frac{1}{2} B_{2} \land C_{2}^{H}$ in Type II Dp Brane DBI WZ Action. Separately, on the oriented source patch of Type IIB D3 Sourced Five Form Bianchi, with $H_{3}$ and ${\tilde{G}}_{3}$ smooth, its adopted equation holds weakly as an equality of degree-six currents: $d {\tilde{G}}_{5} - H_{3} \land {\tilde{G}}_{3} = - 2 κ_{0}^{2} μ_{3} δ_{6} (W) .$ WZ variation fixes the localized current with a positive WZ coefficient; the signed bulk Bianchi law and its self-dual source normalization remain explicit physical inputs. In the wedge-defined Lorentzian convention, the Bianchi source sign is negative relative to the current defined by that WZ pairing. The GKP internal plus-source law is recovered only with the ambient/worldvolume orientation dictionary supplied by the sourced-Bianchi postulate.

Statement References

Polchinski, Chaudhuri, and Johnson, Lecture III §3.3

Kachru, Schulz, and Trivedi, §2.1, Eq. (2.4)

Type II Dp Brane DBI WZ Action

Type IIB D3 Sourced Five Form Bianchi

Facts & Assumptions

Given: An oriented Lorentzian Type II ten-manifold with mostly-plus string-frame metric, a proper smooth oriented timelike D3 embedding $ι : W_{4} ↪ X^{10}$ with $\partial W_{4} = \emptyset$ and $F = 0$ , a compactly supported test variation of $C_{4}^{H}$ with $B_{2}, C_{2}^{H}$ fixed, and the stated source normalization.

[F1]

Type II Dp Brane DBI WZ Action: when $F = 0$ , the D3 WZ term is $μ_{3} \int_{W_{4}} ι^{*} C_{4}^{W Z}$ , where $C_{4}^{W Z} = C_{4}^{H} + \frac{1}{2} B_{2} \land C_{2}^{H}$ . Varying $C_{4}^{H}$ at fixed $B_{2}, C_{2}^{H}$ gives $δ C_{4}^{W Z} = δ C_{4}^{H}$ .

[F2]

Local Type IIB D3 Brane Source Maxwell And Page Charge: source current is normalized by its pairing with test forms; $μ_{3} = (2 π)^{- 3} α^{' - 2}$ . Its current Step 1 record is escalated because the IIB pseudoaction and self-duality inputs are not ready, so the five-form interface remains conditional.

[A1]

Type IIB D3 Sourced Five Form Bianchi: for the proper oriented boundaryless D3 on the oriented source patch, with $H_{3}$ and ${\tilde{G}}_{3}$ smooth, the R4 IIB source equation is the weak equality of degree-six currents $d {\tilde{G}}_{5} - H_{3} \land {\tilde{G}}_{3} = - 2 κ_{0}^{2} μ_{3} δ_{6} (W)$ . This sourced equation is adopted separately from the WZ variation.

[A2]

Type IIB Two Derivative Supergravity Pseudoaction: the current source-free R4 IIB pseudoaction defines $G_{1}, G_{3}, G_{5}, {\tilde{G}}_{3}, {\tilde{G}}_{5}$ and is varied before the separate self-duality condition. Its current Step 1 record is escalated because its upstream readiness inputs are missing, so this framework input remains conditional.

Proof

1.1givenF1F2algebra

For an arbitrary compactly supported variation $δ C_{4}^{H}$ at fixed $B_{2}, C_{2}^{H}$ , [F1] gives $δ C_{4}^{W Z} = δ C_{4}^{H}$ . The first variation is therefore $δ S_{W Z} = μ_{3} \int_{W_{4}} ι^{*} δ C_{4}^{H}$ . By [F2]'s defining pairing this is $\int_{X} δ C_{4}^{H} \land μ_{3} δ_{6} (W)$ , so the localized current is $J_{6} = μ_{3} δ_{6} (W)$ .

2.1givenA1A2step 1.1algebra∎

The weak source-coupled IIB Bianchi equation is exactly [A1], with its $μ_{3}$ coefficient fixed as an adopted field-equation normalization. The deduction in step 1.1 does not determine the coefficient $2 κ_{0}^{2} μ_{3}$ in this bulk equation; that separately adopted self-dual source normalization, its negative sign in the wedge-Hodge/current convention, and the self-dual five-form framework are carried as the explicit physical inputs [A1] and [A2]. Thus the statement holds conditionally and does not claim a derivation from the WZ term alone.

Type II Dp Brane RR Current And D3 Charge

Statement

Statement References

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources