PostulateStatement: AI-adaptedProof: Not applicablePipeline-generatedprecheck pass‡ sources checked 2026-11-05

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F-theory flux transversality and supersymmetry conditions

Physical scope

Adopted leading-order supergravity/F-theory consistency conditions on a fixed smooth compact resolved elliptic Calabi–Yau fourfold. Flux backreaction, loop and higher-derivative corrections, singular/non-flat models, and existence of a vacuum are not inferred. The source-free CJS action is not used as a substitute for fermionic supersymmetry equations.

Postulate

Assume Full AC. Fix a smooth compact Kähler resolution $π : {\hat{Y}}_{4} \to B_{3}$ of a flat sectioned elliptic Calabi–Yau fourfold, a zero-section divisor $Z_{0}$ , exceptional Cartan divisors $E_{i}$ for a specified non-abelian gauge algebra, and a Kähler form $J$ ; write $[J] \in H^{2} ({\hat{Y}}_{4}; R)$ for its class. Here $B_{3}$ is the smooth connected complex algebraic threefold of the type-IIB dictionary. For an algebraic line bundle $L$ on $B_{3}$ , let $c_{1} (L)_{R} \in H^{2} (B_{3}; R)$ denote the real image of the integral first Chern class of its underlying topological complex line bundle. Define the space of real divisor classes used here by $V_{B} = {span}_{R} {c_{1} (L)_{R} : L an algebraic line bundle on B_{3}} \subseteq H^{2} (B_{3}; R)$ , and let $D_{a}, D_{b}$ range over $V_{B}$ . All displayed integrals mean evaluation of the indicated degree-eight cup product on the complex-oriented fundamental class $[{\hat{Y}}_{4}]$ ; $[Z_{0}]$ and $[E_{i}]$ are the real Poincaré duals of the specified divisors. For the M2-normalized shifted flux class $[g_{4}] \in H^{4} ({\hat{Y}}_{4}; R)$ , let $g_{4}^{h a r m}$ be its unique harmonic representative. Adopt the following leading-order conditions as distinct requirements: four-dimensional Lorentz invariance requires $\int_{{\hat{Y}}_{4}} g_{4} \land π^{*} D_{a} \land π^{*} D_{b} = 0$ and $\int_{{\hat{Y}}_{4}} g_{4} \land [Z_{0}] \land π^{*} D_{a} = 0$ for every $D_{a}, D_{b}$ ; preserving the chosen non-abelian gauge algebra additionally requires $\int_{{\hat{Y}}_{4}} g_{4} \land [E_{i}] \land π^{*} D_{a} = 0$ for each $E_{i}, D_{a}$ ; and a leading-order supersymmetric flux vacuum requires $[g_{4}]$ of Hodge type $(2, 2)$ and primitive, meaning $[J] ⌣ [g_{4}] = 0$ in $H^{6} ({\hat{Y}}_{4}; R)$ . On the harmonic representative, the pointwise primitivity condition is $J \land g_{4}^{h a r m} = 0$ ; Kähler harmonic theory identifies this with the cohomological condition. These adopted physical conditions do not prove that any such flux or vacuum exists.

Facts & Assumptions

Given: Full AC and The fixed resolved Kähler model, its specified divisors, a shifted-quantized real cohomology class, and the stated leading-order physical regime.

[F1]

Shifted M-theory G4 quantization on a resolved Calabi–Yau fourfold gives the conditional CY4 shift in the local convention. It does not imply Hodge type, primitivity, transversality, or supersymmetry.

[F2]

Type-IIB axiodilaton and elliptic-fiber dictionary supplies the adopted type-IIB/elliptic-fiber setting; its current readiness is escalated on an outstanding Weierstrass dependency, and this item does not treat it as proved.

The source-free classical bosonic CJS action is contextual background only: it does not supply the fermionic supersymmetry variations adopted here.

[F6]

Chern classes from the projective-bundle relation defines the integral first Chern class of a complex line bundle. The smooth base has the required CW homotopy type and its bundles are numerable. The space $V_{B}$ is defined directly by real first Chern classes; no Néron–Severi quotient, projectivity theorem for the base, or comparison modulo algebraic equivalence is required.

[F4]

Compact Kähler and rational Hodge polarization conventions and Compact Kähler Hodge decomposition from elliptic harmonic forms give the compact Kähler conventions and unique harmonic representative of a real de Rham class.

[F5]

Kähler commutators and preservation of harmonic types states that the Kähler Laplacian commutes with Lefschetz multiplication $L = J \land$ ; Kähler identities give Lefschetz sl-two and hard Lefschetz transports that operator to cohomology. These mathematical facts justify the stated harmonic/cohomological primitivity equivalence, not the physical condition itself.

Postulate references

The Axiom of Choice

Chern classes from the projective-bundle relation

Shifted M-theory G4 quantization on a resolved Calabi–Yau fourfold

Type-IIB axiodilaton and elliptic-fiber dictionary

Eleven-Dimensional CJS Supergravity Bosonic Action

Compact Kähler and rational Hodge polarization conventions

Compact Kähler Hodge decomposition from elliptic harmonic forms

Kähler commutators and preservation of harmonic types

Kähler identities give Lefschetz sl-two and hard Lefschetz

F-theory flux transversality and supersymmetry conditions

Postulate

Facts & Assumptions

Postulate references

Depends on

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Sources