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

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Source-free GKP compactifications have vanishing three-form flux and componentwise constant warp factor

Physical scope

Conditional classical source-free two-derivative Type IIB Einstein-frame GKP equation with four-dimensional Poincare invariance and regular compact internal X. It excludes localized sources, boundary terms, higher-derivative/string corrections, and any conclusion about sourced GKP backgrounds, de Sitter space, or constancy of the axio-dilaton.

Statement

Use natural units $ℏ = c = 1$ , length-valued coordinates, and dimensionless metric components. Let $X$ be a nonempty compact oriented smooth six-manifold without boundary, and let $\tilde{g}$ be a smooth Riemannian metric on $X$ . Use $η_{μ ν} = diag (-, +, +, +)$ on four-dimensional Minkowski space and the source-free Einstein-frame GKP metric $d s_{E}^{2} = e^{2 A (y)} η_{μ ν} d x^{μ} d x^{ν} + e^{- 2 A (y)} {\tilde{g}}_{m n} (y) d y^{m} d y^{n} .$ Let $A, α_{4}, C_{0}, ϕ$ be smooth real dimensionless functions on $X$ , with $τ = C_{0} + i e^{- ϕ}$ and $Im τ = e^{- ϕ} > 0$ . Let $F_{3}, H_{3} \in Ω^{3} (X; R)$ be smooth real three-form field strengths, with coordinate components in the GKP normalization of units length $^{- 1}$ , and set $G_{3} = F_{3} - τ H_{3}$ . Define ${\tilde{\nabla}}^{2} u = {div}_{\tilde{g}} (\tilde{\nabla} u)$ , so ${\tilde{\nabla}}^{2} u d {vol}_{\tilde{g}} = d (ι_{\tilde{\nabla} u} d {vol}_{\tilde{g}})$ . In the source-free two-derivative Type IIB Einstein-frame field system, the external Einstein equation in this ansatz is ${\tilde{\nabla}}^{2} e^{4 A} = \frac{e^{8 A}}{12 Im τ} G_{m n p} {\overline{G}}^{m n p} + e^{- 4 A} ({\tilde{g}}^{m n} \partial_{m} α_{4} \partial_{n} α_{4} + {\tilde{g}}^{m n} \partial_{m} e^{4 A} \partial_{n} e^{4 A}),$ where indices on the flux are raised by $\tilde{g}$ . These are unwarped contractions throughout, obtained from the full-metric contractions in GKP equation (2.15) by $g_{E}^{m n} = e^{2 A} {\tilde{g}}^{m n}$ . The equation is conditional model input: it is the source-free specialization of the cited GKP field equations, not a claim about the empirical validity of the truncation.

Under these hypotheses, integrating the displayed equation forces $G_{3} = 0$ , $d α_{4} = 0$ , and $d A = 0$ everywhere on $X$ . $A$ and $α_{4}$ are constant on each connected component; their constants may differ between components. Since $F_{3}, H_{3}$ are real and $Im τ > 0$ , it follows that $F_{3} = H_{3} = 0$ . The equation places no restriction on whether $τ$ is constant.

Statement references

Giddings, Kachru, and Polchinski, §2.1

Graña, §4.1

Cecotti, Chapter 14

Facts & Assumptions

Given: The smooth compact oriented manifold $X$ , the GKP ansatz and conventions in the Statement, $Im τ > 0$ , the source-free field equation displayed there, and the exact smooth-form integration convention.

[A1]

Type IIB Calabi–Yau orientifold flux background conventions: the current definition fixes $G_{3} = F_{3} - τ H_{3}$ , the Einstein/string frame map, orientation, internal metric, warped ansatz, and self-dual five-form profile. Its broader scope is an O3/O7 Calabi–Yau orientifold; this theorem uses the explicitly restated GKP convention subset in the source-free specialization and makes no orientifold-specific conclusion. Its current Step 1 record remains escalated on upstream inputs.

[A2]

Type IIB Two Derivative Supergravity Pseudoaction: the current physical input is the massless source-free leading two-derivative Type IIB pseudoaction; its exact statement is in string frame and excludes localized sources and higher-order terms.

[A3]

Type IIB Five Form Self Duality Constraint: the five-form constraint is imposed separately after variation. This is required by the GKP ansatz and is not supplied by varying the undoubled pseudoaction alone.

[F1]

Smooth forms define currents compatibly with the differential: if $η$ is a smooth $(5)$ -form on $X$ , its smooth-form current satisfies $d T_{η} = T_{d η}$ .

[F2]

An exact top-degree current has zero total charge on a closed manifold: if $T$ is a degree- $5$ current on compact boundaryless $X$ , then $⟨ d T, 1 ⟩ = 0$ .

Proof

1.1F1F2givenalgebra

Put $u = e^{4 A}$ and $η = ι_{\tilde{\nabla} u} d {vol}_{\tilde{g}} \in Ω^{5} (X)$ . The coordinate definition of the divergence gives $d η = ({\tilde{\nabla}}^{2} u) d {vol}_{\tilde{g}}$ . By [F1], this top form defines the exact current $d T_{η}$ ; [F2] therefore gives $\int_{X} {\tilde{\nabla}}^{2} u d {vol}_{\tilde{g}} = ⟨ d T_{η}, 1 ⟩ = 0$ .

2.1A1A2A3step 1.1givenalgebra

In the source equation the full-metric flux contraction is $e^{6 A} G_{m n p} {\overline{G}}_{\tilde{g}}^{m n p}$ , and a full-metric scalar-gradient contraction is $e^{2 A} ∣ d f ∣_{\tilde{g}}^{2}$ . Thus its coefficients $e^{2 A}$ and $e^{- 6 A}$ become respectively $e^{8 A}$ and $e^{- 4 A}$ in the Statement's unwarped notation. Integrate that field equation and use step 1.1. The right-hand side is a sum of continuous nonnegative functions: $Im τ > 0$ , $\tilde{g}$ is positive definite, and $G_{m n p} {\overline{G}}^{m n p} \geq 0$ . Hence its integral is zero. If any summand were positive at a point, continuity would make it positive on an open neighborhood of positive Riemannian volume, contradicting the zero integral. Each summand therefore vanishes everywhere.

3.1step 2.1algebra

Step 2.1 gives $G_{3} = 0$ , $d α_{4} = 0$ , and $d (e^{4 A}) = 0$ . Since $4 e^{4 A} > 0$ , the identity $d (e^{4 A}) = 4 e^{4 A} d A$ gives $d A = 0$ on every component of $X$ . A smooth function with zero differential is locally constant in a coordinate ball by the one-variable fundamental theorem along line segments; connectedness then makes it constant on each connected component. Apply this to both $A$ and $α_{4}$ , without identifying constants on distinct components.

4.1A1step 3.1algebra∎

Write $τ = C_{0} + i τ_{2}$ with $τ_{2} = Im τ > 0$ . The real and imaginary parts of $F_{3} - τ H_{3} = 0$ are $F_{3} - C_{0} H_{3} = 0$ and $- τ_{2} H_{3} = 0$ . The second gives $H_{3} = 0$ , and substitution into the first gives $F_{3} = 0$ . No equation above differentiates $τ$ , so this argument does not imply that it is constant.

Source-free GKP compactifications have vanishing three-form flux and componentwise constant warp factor

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources