DefinitionDefinition: Literature-sourcedProof: 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.

Calabi–Yau threefold data used by string compactifications

Physical scope

Geometric input data for compactifications in this pair. Broad canonical-bundle triviality makes no claim of exact SU(3) holonomy, an N=2 spectrum, or a supergravity solution.

Definition

A geometric Calabi–Yau threefold compactification datum consists of a specified smooth connected compact complex threefold $X$ , a Kähler class $[J]$ , a chosen trivialization $Ω$ of $K_{X}$ , orientation and integral-cohomology conventions, and any additional bundle, involution, flux, brane, or source data stated for the physical model. It uses the broad Calabi–Yau convention: $K_{X} ≅ O_{X}$ does not imply exact $S U (3)$ holonomy, simple connectivity, or $h^{p, 0} = 0$ for $0 < p < 3$ . Claims of precisely four-dimensional $N = 2$ supersymmetry require the corresponding holonomy and worldsheet conditions in addition to this broad definition.

Definition references

Compact Calabi–Yau manifold conventions

Facts & Assumptions

Given: A smooth connected compact complex threefold $X$ , a Kähler form $J$ on $X$ , a specified holomorphic trivialization $Ω$ of $K_{X}$ , an orientation, and declared integral-cohomology conventions. Any other physical data are supplied separately.

[F1]

Compact Calabi–Yau manifold conventions defines the broad Calabi–Yau convention as connected, compact, Kähler, with a specified holomorphic trivialization of the canonical line; it explicitly does not assert exact $S U (n)$ holonomy or intermediate $h^{p, 0}$ vanishing.

[F2]

Kähler class, canonical bundle, and Ricci form defines a Kähler form as a smooth real positive closed $(1, 1)$ -form and its cohomology class as a real de Rham class.

Proof

technique · direct interface verification

Given: The smooth connected compact complex threefold, Kähler form, nowhere-zero holomorphic canonical form, orientation, integral convention, and separately supplied model data in the Facts & Assumptions section.

1.1F1F2given

By [F1] and [F2], the first three data in the Given clause are exactly the broad Calabi–Yau geometry interface: $X$ is connected, compact, and Kähler, and $Ω$ is the chosen nowhere-zero holomorphic section supplied by the trivialization of $K_{X}$ . The orientation and integral-cohomology conventions are additional choices attached to this compactification datum; they are not consequences of broad CY status.

2.1F1F2givenstep 1.1∎

The listed bundles, involutions, fluxes, branes, and sources are explicitly supplied model data. Nothing in [F1] or [F2] supplies them, and [F1] expressly permits broad CY manifolds without exact $S U (3)$ holonomy. Therefore no strict-holonomy or $N = 2$ conclusion follows from this definition alone.

Calabi–Yau threefold data used by string compactifications

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources