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

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Dp Brane Leading Curvature WZ Coupling

Physical scope

The source-normalized leading disk-level order-alpha'^2 Dp curvature coupling with vanishing B and worldvolume gauge field; does not establish its worldsheet amplitude locally, convert tension normalization to the R4 charge normalization, or assert an all-orders formula.

Statement

In the democratic WZ convention and under the hypotheses of [A1], for $p \geq 3$ the leading curvature correction to the Dp WZ coupling is $μ_{p} \int_{W} (ι^{*} C_{p + 1} + \frac{π^{2} α^{' 2}}{24} ι^{*} C_{p - 3} \land (tr (R_{T} \land R_{T}) - tr (R_{N} \land R_{N}))),$ through order $α^{' 2}$ , with the source coefficient $μ_{p} = ((2 π)^{p} α^{' (p + 1) / 2})^{- 1}$ equal to the bare WZ coefficient in Type II Dp Brane DBI WZ Action. This is a source-reported disk result, not a local derivation of the worldsheet amplitude.

Facts & Assumptions

Given: A smooth Dp embedding, constant dilaton, $B_{2} = F_{A} = 0$ , the leading WZ coupling, and the separately adopted truncated disk curvature factor.

[F1]

Type II Dp Brane DBI WZ Action: the leading WZ coupling is $μ_{p} \int_{W} [ι^{*} C \land e^{F}]_{p + 1}$ in the democratic component basis, with $μ_{p}$ the RR charge normalization and $T_{p} = μ_{p} / g_{s t r}$ the constant-dilaton DBI tension.

[A1]

Disk Level Truncated Dp Curvature WZ Factor: for an allowed Type II dimension $3 \leq p \leq 9$ and a smooth oriented timelike embedding with nondegenerate induced metric, use the projected tangent and normal connections and the ambient democratic RR potentials specified there. At disk level, with constant dilaton and $B_{2} = F_{A} = 0$ , its order- $α^{' 2}$ expression has coefficient $μ_{p}$ and tangent-minus-normal curvature traces. The traces are four-forms on $W$ ; only the ambient potentials are pulled back. In its explicit BBG crosswalk $C_{r}^{B B G} = g_{s t r} C_{r}$ and $T_{(p)}^{B B G} = μ_{p} / g_{s t r}$ , so the WZ product is $μ_{p} C_{r}$ , not an identification of BBG tension with $μ_{p}$ . The supplier retains the massless D8 restriction and separate D9 supplement.

Proof

1.1givenA1algebra

The adopted source expression [A1] has degree- $p + 1$ term $ι^{*} C_{p + 1}$ and order- $α^{' 2}$ term $(π^{2} α^{' 2} / 24) ι^{*} C_{p - 3} \land (tr (R_{T} \land R_{T}) - tr (R_{N} \land R_{N}))$ , with the stated tangent-minus-normal sign and coefficient $μ_{p}$ . These are exactly the terms in the theorem statement.

2.1givenF1A1step 1.1algebra∎

Restricting [A1] to terms through order $α^{' 2}$ leaves precisely the stated formula. The crosswalk in Type II Dp Brane DBI WZ Action uses the same WZ polyform and charge normalization; the physical constant-dilaton tension $T_{p} = μ_{p} / g_{s t r}$ belongs to the DBI clause and is not substituted into the WZ coefficient. No higher orders are asserted.

Dp Brane Leading Curvature WZ Coupling

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources