ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pendingjudge 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.

U(1) Fayet–Iliopoulos Cancellation by Charge Trace

Physical scope

A formal perturbative rigid N=1 U(1) Wilsonian example in an unbroken, topologically trivial background at nonzero scale. It asserts no empirical or continuum result.

Example

Take one unbroken compact U(1) gauge factor with two left-chiral superfields of charges $+ 1$ and $- 1$ , one of each and no additional multiplicities. Use the canonical matter kinetic term, a positive field-independent quadratic U(1) gauge-kinetic coefficient, and no nonkinetic holomorphic interaction ( $F_{r e m} = 0$ ). Let the only additional real D-term be the bare field-independent Fayet–Iliopoulos term $ξ \int d^{4} x [V_{1}]_{D}$ , and use the theorem's specified linear Lorenz BRST component graph prescription and a formal perturbative Wilsonian regulator satisfying all its compatibility and infrared inverse conditions at scale $λ > 0$ . Keep the bare FI coefficient $ξ$ fixed as a real number rather than promoting it to an unconstrained background. At zero mass and interaction spurions require the same regulated free momentum kernel and one-vector insertion for both matter components, tensored with their representation identities; only their charge signs differ. Use the theorem's ultraviolet-matched graph prescription with no independent finite FI shifts. Then the perturbative Wilsonian coefficient satisfies $ξ_{λ} = ξ$ at every loop order: the one-loop correction cancels by the charge sum, and the theorem excludes higher-loop corrections. Other real D-term couplings, if added, are not protected by this conclusion.

Example references

Weinberg, the U(1) Fayet–Iliopoulos argument, pp. 7–8

Facts & Assumptions

Given: The $N = 1$ Wilsonian action datum, an unbroken U(1) factor, its two left-chiral representations, and the formal perturbative regulator hypotheses including its local background-spurion extension with independent unconstrained real spurion jets and chirality before constant-background evaluation.

[F1]

Local $N = 1$ Wilsonian Superspace Action Data: for a U(1) factor, $V_{1}$ is a real Abelian prepotential and a constant coefficient $ξ$ defines the Fayet–Iliopoulos D-term; its group is the compact matrix group $U (1)$ with Lie algebra $i R$ and the stated period- $2 π$ charge normalization.

[F2]

$N = 1$ Wilsonian F-Term Nonrenormalization: under its formal cutoff, symmetry, anomaly-free, formal local background-spurion hypotheses, and its fixed-FI promotion exception, common matter-kernel and ultraviolet-matched graph prescription without independent finite FI shifts, the perturbative FI correction is one-loop only, is proportional to the sum of all left-chiral U(1) charges, and vanishes when this sum is zero.

[A1]

The two multiplets have charges $q_{+} = 1$ and $q_{-} = - 1$ , each with multiplicity one.

[A2]

The U(1) background is unbroken, $F_{r e m} = 0$ , the real D-term contains only the bare FI term, and the regulator and measure preserve the gauge and rigid $N = 1$ Ward identities. The bare FI coefficient is not promoted, and both charged components use the common free kernel and one-vector insertion at zero interaction spurions, with ultraviolet matching and no independent finite FI shifts as specified in the Example.

Verification

1.1A1algebra

The total left-chiral charge is $q_{+} + q_{-} = 1 + (- 1) = 0$ .

2.1F1F2A2step 1.1∎

By [F2], the only possible perturbative FI correction is the one-loop tadpole proportional to this total charge, so it is zero; all higher-loop corrections are excluded. Therefore $ξ_{λ} = ξ$ . The theorem makes no corresponding claim about other D terms.

Depends on

Dependency tree · 2 levels

Sources

Steven Weinberg, Non-Renormalization Theorems in Non-Renormalizable Theories