ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pending

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.

A flat four-torus supplies a geometric Killing-spinor input

Physical scope

A local Euclidean geometric spin/R background example only; no multiplet, gauge-fixed field theory, functional measure, or localization prediction is asserted.

Example

Let $ℓ_{1}, \dots, ℓ_{4} > 0$ be lengths and let $T_{ℓ}^{4} = R^{4} / (ℓ_{1} Z \oplus \dots \oplus ℓ_{4} Z)$ have its quotient flat metric and standard orientation. Use $F_{g}^{+} = T_{ℓ}^{4} \times S O (4)$ and $\tilde{F} = T_{ℓ}^{4} \times Spin (4)$ with covering map $(x, s) \mapsto (x, ρ (s))$ , where $Spin (4) = S U (2)_{+} \times S U (2)_{-}$ and $ρ (u_{+}, u_{-}) q = u_{+} q u_{-}^{- 1}$ on $H ≅ R^{4}$ . Equip this trivial spin lift with its periodic spin structure and zero spin connection. Use the trivial $S U (2)_{R}$ bundle with zero connection and zero supergravity auxiliary fields. For the geometric Killing-spinor equation $\nabla_{μ}^{S} ϵ = 0$ , every constant nonzero section of $S_{+}$ is a solution. These data supply only the geometric spin/R and Killing-spinor input; they do not specify an off-shell multiplet or $Q$ , gauge fixing, an integration contour, a measure, or a regulator, and therefore are not a complete localization package.

Example references

Festuccia and Seiberg, §2, general rigid-background procedure

Facts & Assumptions

Given: The quotient flat metric, the trivial spin and R bundles, the zero connections, zero auxiliary backgrounds, and the equation $\nabla_{μ}^{S} ϵ = 0$ specified in the example.

[F1]

Rigid supersymmetric background and localization data: a full localization package additionally specifies an off-shell multiplet and square, gauge fixing and ghosts, contour, measure or regulator, and boundary conditions; a geometric spinor solution alone does not provide them.

Proof

1.1givenconstructalgebra

The quotient coordinates give the global coframe $d x^{1}, \dots, d x^{4}$ with constant Euclidean metric coefficients. The selected trivial spin lift and zero spin connection identify $S_{+}$ with $T_{ℓ}^{4} \times C^{2}$ and make $\nabla^{S}$ the ordinary derivative. Hence for any fixed $ϵ_{0} \in C^{2} ∖ {0}$ , the section $ϵ (x) = ϵ_{0}$ is smooth, nonzero, and satisfies $\nabla_{μ}^{S} ϵ = 0$ for every $μ$ .

2.1F1givenstep 1.1∎

The specified data contain no field bundles for a matter/gauge multiplet, no off-shell transformation $Q$ , no gauge-fixing or ghost data, and no contour, measure, or regulator. By [F1] they therefore realize only the geometric spin/R and Killing-spinor input and imply no localization statement.

Depends on

Rigid supersymmetric background and localization data