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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Type IIB Einstein Frame Pseudoaction

Physical scope

Conditional pointwise Weyl rewrite of the source-free two-derivative type IIB pseudoaction. The full local divergence from the NS sector is retained; dropping it from the integrated action requires zero boundary flux, while compactly supported bulk variations are unaffected. The five-form constraint remains separate.

Statement

Assume the conventions and action of Type IIB Two Derivative Supergravity Pseudoaction and set $g_{s} = e^{Φ / 2} g_{E}$ on the oriented Lorentzian ten-manifold. For any covariant $p$ -form $F_{p}$ , $∣ F_{p} ∣_{s}^{2} = e^{- p Φ / 2} ∣ F_{p} ∣_{E}^{2}$ , while $\sqrt{- g_{s}} = e^{5 Φ / 2} \sqrt{- g_{E}}$ . Therefore the Einstein-frame action is bulk-equivalent, for compactly supported variations, to

$S_{E} = \frac{1}{2 κ_{0}^{2}} \int {vol}_{g_{E}} (R_{E} - \frac{1}{2} ∣ d Φ ∣_{E}^{2} - \frac{1}{2} e^{- Φ} ∣ H_{3} ∣_{E}^{2}) - \frac{1}{4 κ_{0}^{2}} \int {vol}_{g_{E}} (e^{2 Φ} ∣ G_{1} ∣_{E}^{2} + e^{Φ} ∣ {\tilde{G}}_{3} ∣_{E}^{2} + \frac{1}{2} ∣ {\tilde{G}}_{5} ∣_{E}^{2}) - \frac{1}{4 κ_{0}^{2}} \int C_{4} \land H_{3} \land G_{3} .$

More precisely, the full NS local density includes the additional total-divergence term $- \frac{9}{2} □_{E} Φ$ from String Einstein Frame NS Action Calculation; removing its integral requires the boundary flux to vanish. The middle-degree Hodge star is unchanged by this Weyl rescaling, so the separately imposed constraint is $*_{g_{E}} {\tilde{G}}_{5} = {\tilde{G}}_{5}$ .

Statement references

Supergravity Conventions And Normalizations

Type IIB Two Derivative Supergravity Pseudoaction

String Einstein Frame NS Action Calculation

Lorentzian Hodge Star in Ten Dimensions

Existence and Five-Form Properties of the Lorentzian Hodge Star

Conformal Scaling of the Ten-Dimensional Lorentzian Hodge Star

Type IIB Five Form Self Duality Constraint

Hamilton, §7.3, Theorem 7.11

Sterckx, §3.2.1 and §3.2.3

Facts & Assumptions

Given: A smooth real scalar $Φ$ , the two metrics $g_{s} = e^{Φ / 2} g_{E}$ , the fields and norms in [F1], and the planned string-frame pseudoaction.

[F1]

Type IIB Two Derivative Supergravity Pseudoaction: the RR terms are $- ∣ G_{1} ∣_{s}^{2} - ∣ {\tilde{G}}_{3} ∣_{s}^{2} - \frac{1}{2} ∣ {\tilde{G}}_{5} ∣_{s}^{2}$ with common coefficient $(4 κ_{0}^{2})^{- 1}$ , and the Chern–Simons term is the metric-independent integral $C_{4} \land H_{3} \land G_{3}$ .

[F2]

String Einstein Frame NS Action Calculation: writing $H_{E}^{2} : = H_{M N P} H_{Q R S} G_{E}^{M Q} G_{E}^{N R} G_{E}^{P S}$ , the complete transformed NS density is ${vol}_{g_{E}} [R_{E} - \frac{1}{2} ∣ d Φ ∣_{E}^{2} - \frac{1}{12} e^{- Φ} H_{E}^{2} - \frac{9}{2} □_{E} Φ]$ . By [F5], $∣ H_{3} ∣_{E}^{2} = H_{E}^{2} / 3!$ , so this is ${vol}_{g_{E}} [R_{E} - \frac{1}{2} ∣ d Φ ∣_{E}^{2} - \frac{1}{2} e^{- Φ} ∣ H_{3} ∣_{E}^{2} - \frac{9}{2} □_{E} Φ]$ ; the last summand is a total divergence.

[F3]

Conformal Scaling of the Ten-Dimensional Lorentzian Hodge Star: in dimension ten, $*_{e^{2 ω} g} β = e^{(10 - 2 p) ω} *_{g} β$ for a $p$ -form.

[F4]

Type IIB Five Form Self Duality Constraint: impose the five-form equation only after varying the pseudoaction.

[F5]

Supergravity Conventions And Normalizations: use Lorentzian signature $(-, +, \dots, +)$ , $∣ F_{p} ∣^{2} = F_{M_{1} \dots M_{p}} F^{M_{1} \dots M_{p}} / p!$ , $2 κ_{0}^{2} = (2 π)^{7} α^{' 4}$ , and $g_{E} = e^{- Φ / 2} g_{s}$ ; keep $κ_{0}$ distinct from the asymptotically shifted physical $κ_{10}$ .

Proof

1.1givenF1F5algebra

Write $g_{s} = e^{2 ω} g_{E}$ with $ω = Φ / 4$ , using the frame relation in [F5]. In ten dimensions the determinant contributes $e^{10 ω} = e^{5 Φ / 2}$ , while each inverse metric in the squared norm of a covariant $p$ -form contributes $e^{- 2 ω} = e^{- Φ / 2}$ . Thus $\sqrt{- g_{s}} = e^{5 Φ / 2} \sqrt{- g_{E}}$ and $∣ F_{p} ∣_{s}^{2} = e^{- p Φ / 2} ∣ F_{p} ∣_{E}^{2}$ , as stated.

2.1F1step 1.1algebra

Multiplying the measure and norm factors gives $e^{(5 - p) Φ / 2}$ for a $p$ -form kinetic density: this is $e^{2 Φ}$ for $G_{1}$ , $e^{Φ}$ for ${\tilde{G}}_{3}$ , and $1$ for ${\tilde{G}}_{5}$ . The action coefficient already contains the additional factor $1 / 2$ for the five-form and it is unchanged. The Chern–Simons integral contains only differential forms and orientation, so no metric scaling occurs.

3.1F2F5step 2.1algebra

By [F2], the raw NS three-form term is $- \frac{1}{12} e^{- Φ} H_{E}^{2}$ . Since the convention [F5] defines $∣ H_{3} ∣_{E}^{2} = H_{E}^{2} / 3!$ , this raw contraction equals $- \frac{1}{2} e^{- Φ} ∣ H_{3} ∣_{E}^{2}$ , which is the coefficient in the Statement. The other NS terms give $R_{E} - \frac{1}{2} ∣ d Φ ∣_{E}^{2}$ and the explicitly retained total divergence $- \frac{9}{2} □_{E} Φ$ . Compactly supported variations of its local current integrate to zero by the supplier's stated hypothesis, but its unvaried value is removed only if the boundary flux vanishes. Combining this NS result with step 2.1 proves the bulk-equivalent action formula without discarding an unqualified boundary term.

4.1F3F4givenalgebra∎

Here $ω = Φ / 4$ , so [F3] on $p = 5$ has exponent $(10 - 2 \cdot 5) ω = 0$ ; therefore $*_{g_{s}} {\tilde{G}}_{5} = *_{g_{E}} {\tilde{G}}_{5}$ . Applying [F4] after the Weyl rewrite leaves this same self-duality equation as a separate constraint, not a term varied in either action.

Type IIB Einstein Frame Pseudoaction

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources