DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Formal differential gauge equivalence of star products

Definition

Let $⋆$ and $⋆^{'}$ be differential star products on the same real Poisson manifold $(M, π)$ , with $A = C^{\infty} (M, R)$ . They are formally gauge equivalent if there is an invertible $R ⟦ ℏ ⟧$ -linear map $T : A ⟦ ℏ ⟧ \to A ⟦ ℏ ⟧$ of the form $T = id + \sum_{r \geq 1} ℏ^{r} D_{r},$ where each $D_{r} : A \to A$ is a differential operator and $D_{r} (1) = 0$ , such that $f ⋆^{'} g = T^{- 1} (T f ⋆ T g) .$ The condition $D_{r} (1) = 0$ makes $T (1) = 1$ , so this change of variables preserves the unit. The map is formal and coefficientwise; no analytic convergence or classification of equivalence classes is asserted.

Depends on

Formal differential star products and first-order normalization

Used by

Formal equivalence changes the first-order cochain by a Hochschild coboundary Proposition

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Maxim Kontsevich, Deformation Quantization of Poisson Manifolds, I (standard reference, not scraped)
Nima Moshayedi, Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theory (standard reference, not scraped)