LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge 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.

Gauge invariance of the adjoint cubic superpotential

Physical scope

Pure matrix algebra over the even chiral coefficient ring with a specified conjugation action and trace form; no physical premise is used.

Statement

Let $R$ be the commutative ring of even smooth chiral superfields on the specified flat superspace, with the supercommutative complex coefficient algebra of the superspace definition. Let $Φ_{1}, Φ_{2}, Φ_{3}$ be traceless $2 \times 2$ matrices over $R$ . Use $[X, Y] = X Y - Y X$ and the complex-bilinear trace form $κ_{C} (X, Y) = - 2 {Tr}_{2} (X Y)$ , where ${Tr}_{2} A = \sum_{j = 1}^{2} A_{j j}$ . Suppose a supplied invertible even chiral matrix $U \in G L_{2} (R)$ acts simultaneously by $Φ_{i}^{'} = U Φ_{i} U^{- 1}$ . Then the even chiral cubic polynomial $κ_{C} (Φ_{1}, [Φ_{2}, Φ_{3}])$ and every constant multiple are invariant. This includes the substitution $U = e^{i Λ}$ whenever the given even chiral parameter defines that invertible chiral matrix. The trace form restricts to the real form $κ (X, Y) = - 2 {Tr}_{2} (X Y)$ on the explicitly defined real Lie algebra of anti-Hermitian traceless complex $2 \times 2$ matrices. No physical model, gauge measure or extended-supersymmetry premise is used.

Statement references

Four-dimensional $N = 1$ chiral superfield on flat superspace

Four-dimensional $N = 1$ superspace Grassmann algebra and Berezin integration

Facts & Assumptions

Given: The even chiral matrices, inverse and simultaneous conjugation action in the Lemma, together with the displayed definitions of bracket and trace form.

[F1]

Four-dimensional $N = 1$ chiral superfield on flat superspace: even smooth chiral superfields obey ${\overset{ˉ}{D}}_{\dot{α}} f = 0$ on the classical coefficient-function domain.

[F2]

Four-dimensional $N = 1$ superspace Grassmann algebra and Berezin integration: the coefficients are supercommutative and the odd derivatives obey the graded Leibniz rule; even coefficients commute without signs.

Proof

1.1givenF1F2algebra

Products and sums of even chiral entries remain even and chiral by [F1] and the Leibniz rule in [F2]. Differentiating $U U^{- 1} = I$ and using $\overset{ˉ}{D} U = 0$ gives $\overset{ˉ}{D} (U^{- 1}) = 0$ . Thus the substitutions and all polynomial products are defined in the commutative even chiral ring. For matrices $A, B$ over that ring, index expansion gives ${Tr}_{2} (A B) = \sum_{i, j} A_{i j} B_{j i} = \sum_{j, i} B_{j i} A_{i j} = {Tr}_{2} (B A)$ . In particular ${Tr}_{2} (U A U^{- 1}) = {Tr}_{2} (A)$ , so tracelessness is preserved.

2.1givenstep 1.1algebra∎

Associativity and inverse cancellation give $[U Φ_{2} U^{- 1}, U Φ_{3} U^{- 1}] = U [Φ_{2}, Φ_{3}] U^{- 1}$ . Hence the transformed cubic is $- 2 {Tr}_{2} (U Φ_{1} [Φ_{2}, Φ_{3}] U^{- 1}) = - 2 {Tr}_{2} (Φ_{1} [Φ_{2}, Φ_{3}])$ by step 1.1. Every constant multiple is likewise invariant. Restricting the same formula to anti-Hermitian traceless ordinary complex matrices gives exactly the stated real trace form; no physical supplier is needed for this restriction.

Gauge invariance of the adjoint cubic superpotential

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources