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Source Free D3 Page Form Is Closed On A B Field Patch

Physical scope

Differential closure of the local D3 Page five-form on a source-free patch with a chosen B2 potential, dH3=0, and the planned source-free Type IIB component Bianchi identities; smooth-chain flux conservation is conditional on the stated countable-choice premise; no global Page class or quantization is claimed.

Statement

On a source-free smooth patch $U$ with a chosen $B_{2}$ , let $H_{3} = d B_{2}$ , $d H_{3} = 0$ , and use the Type IIB Bianchi identities $d G_{1} = 0$ , $d {\tilde{G}}_{3} = H_{3} \land G_{1}$ , $d {\tilde{G}}_{5} = H_{3} \land {\tilde{G}}_{3}$ . Then the D3 Page five-form $P_{5} = {\tilde{G}}_{5} - B_{2} \land {\tilde{G}}_{3} + \frac{1}{2} B_{2} \land B_{2} \land G_{1}$ is a smooth closed five-form on $U$ . Moreover, assume ${A C}_{ω}$ : for every sequence of nonempty sets $(A_{n})_{n \in N}$ there is a function $f$ with $f (n) \in A_{n}$ for every $n$ . If compact oriented smooth five-manifolds $Σ_{0}, Σ_{1} \subset U$ bound a compact oriented smooth six-manifold $C \subset U$ with $\partial C = Σ_{1} ⊔ (- Σ_{0})$ , then $\int_{Σ_{1}} P_{5} = \int_{Σ_{0}} P_{5}$ . This statement does not assert a global Page class, conservation across patches or large gauge transformations, or quantization.

Statement References

Hamilton, §7.1

Marolf, §§2–4 (terminology illustrated for IIA D4 charge)

Facts & Assumptions

Given: A smooth source-free R4 IIB field configuration on a patch with $B_{2}$ , and the exact component Bianchi identities below.

[F1]

Local Type IIB D3 Brane Source Maxwell And Page Charge: on a patch with the adopted smooth IIB field strengths and chosen $B_{2}$ , the locally defined Page form is $P_{5} = {\tilde{G}}_{5} - B_{2} \land {\tilde{G}}_{3} + \frac{1}{2} B_{2} \land B_{2} \land G_{1}$ ; its closure is conditional on the exact Bianchi identities in [A1].

[F2]

The general Stokes theorem: if ${A C}_{ω}$ holds and $C$ is a compact oriented smooth manifold with boundary, then $\int_{C} d η = \int_{\partial C} j^{*} η$ for each smooth $(\dim C - 1)$ -form $η$ ; its orientation convention is outward-normal-first.

[F3]

The Axiom of Choice states that every family of nonempty sets has a choice function, which selects one member of each set. Only its countable-family restriction ${A C}_{ω}$ , explicitly formulated and assumed in the Statement, is used here, to meet [F2]'s premise in step 3.1. Closure in steps 1.1–2.1 uses no choice assumption.

[A1]

Type IIB Source Free Bianchi Identities: for smooth local potentials with the field-strength definitions in Type IIB Two Derivative Supergravity Pseudoaction, the source-free identities are $d H_{3} = 0$ , $d G_{1} = 0$ , $d {\tilde{G}}_{3} = H_{3} \land G_{1}$ , and $d {\tilde{G}}_{5} = H_{3} \land {\tilde{G}}_{3}$ . This is an adopted local field-basis premise; the displayed closure conclusion is conditional on these exact identities.

Proof

1.1givenF1algebra

Apply the graded Leibniz rule to [F1]. Since $B_{2}$ has degree two, $d (B_{2} \land {\tilde{G}}_{3}) = H_{3} \land {\tilde{G}}_{3} + B_{2} \land d {\tilde{G}}_{3}$ , and $d (B_{2} \land B_{2}) = H_{3} \land B_{2} + B_{2} \land H_{3} = 2 B_{2} \land H_{3}$ .

2.1givenA1step 1.1algebra

Substituting [A1] gives $d P_{5} = H_{3} \land {\tilde{G}}_{3} - H_{3} \land {\tilde{G}}_{3} - B_{2} \land H_{3} \land G_{1} + B_{2} \land H_{3} \land G_{1} + \frac{1}{2} B_{2} \land B_{2} \land d G_{1} = 0$ . The last term vanishes by $d G_{1} = 0$ , so the form is closed locally.

3.1givenF2F3step 2.1algebra∎

On $C$ , the restriction of $P_{5}$ is smooth with compact support because $C$ is compact. Applying [F2] under the countable-choice premise [F3] gives $\int_{Σ_{1}} P_{5} - \int_{Σ_{0}} P_{5} = \int_{\partial C} P_{5} = \int_{C} d P_{5} = 0$ . This conservation statement is conditional on ${A C}_{ω}$ and uses only a smooth bounding chain inside the fixed patch.

Source Free D3 Page Form Is Closed On A B Field Patch

Statement

Statement References

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources