LemmaStatement: AI-adaptedProof: AI-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.

Nilpotence of the Finite-Jet Non-Abelian BRST Differential

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be nonempty and open, let $g$ be a finite-dimensional real Lie algebra, and let $V = V_{\overset{ˉ}{0}} \oplus V_{\overset{ˉ}{1}}$ be a finite-dimensional graded complex vector space with an even Lie-algebra representation $ρ : g \to {End}_{C} (V)$ . Form the graded-commutative finite-jet algebra over $C^{\infty} (U)$ from an even $g$ -valued local one-form $A = A_{μ}^{a} T_{a} d x^{μ}$ , a ghost $c = c^{a} T_{a}$ with odd components of ghost number one, and matter fields $ψ^{α}$ of fixed component parity and ghost number zero. Let $D_{μ}$ be the total derivatives on coefficient functions and formal field jets, and let $s$ be the left odd derivation of ghost number one specified by $s A_{μ} = \partial_{μ} c + [A_{μ}, c]$ , $s c = - \frac{1}{2} [c, c]$ , $s ψ = - ρ (c) ψ$ , and $s (D_{I} Φ) = D_{I} (s Φ)$ on every multi-index jet. Then $s^{2} = 0$ on the entire finite-jet algebra and $s D_{μ} = D_{μ} s$ for each $μ$ .

Statement references

Barnich, Brandt, and Henneaux, §§2.2–2.3 and 4.1

Gomis, París, and Samuel, §§3.2 and 5.2

Facts & Assumptions

Given: The local fields, finite-jet algebra, total derivatives, and BRST formulas stated above.

[A1]

The structure constants $f^{a}_{b c}$ of $g$ satisfy the ordinary Lie-algebra Jacobi identity.

[A2]

The representation matrices satisfy $[ρ (T_{a}), ρ (T_{b})] = f^{c}_{a b} ρ (T_{c})$ and preserve the declared grading of $V$ .

[F1]

Finite-Jet Non-Abelian BRST Complex: the coefficient bracket is $[X, Y]^{a} = f^{a}_{b c} X^{b} Y^{c}$ , so $[c, c]^{a} = f^{a}_{b c} c^{b} c^{c}$ .

[F2]

Finite-Jet Non-Abelian BRST Complex: $s A_{μ} = \partial_{μ} c + [A_{μ}, c]$ , $s c = - \frac{1}{2} [c, c]$ , and $s ψ = - ρ (c) ψ$ ; $s$ is the left odd derivation and commutes with total derivatives by its jet rule.

Proof

1.1F1F2A1algebra

Since $c$ is odd, the coefficient bracket is graded symmetric on $(c, c)$ and $s [c, c] = 2 [s c, c]$ . Therefore $s^{2} c = - \frac{1}{2} s [c, c] = - [s c, c] = \frac{1}{2} [[c, c], c]$ . Its component is proportional to $f^{a}_{d e} f^{d}_{b c} c^{b} c^{c} c^{e}$ ; because the ghost coefficients anticommute, only the complete antisymmetrization in $b, c, e$ contributes, and the Jacobi identity $f^{a}_{d [b} f^{d}_{c e]} = 0$ makes it vanish.

1.2F1F2A1algebra

Because $A_{μ}$ is even, the left-derivation rule gives $s^{2} A_{μ} = \partial_{μ} (s c) + [s A_{μ}, c] + [A_{μ}, s c] = - [\partial_{μ} c, c] + [\partial_{μ} c + [A_{μ}, c], c] - \frac{1}{2} [A_{μ}, [c, c]] = [[A_{μ}, c], c] - \frac{1}{2} [A_{μ}, [c, c]] = 0$ . Here $\partial_{μ} [c, c] = 2 [\partial_{μ} c, c]$ , and Jacobi with $A_{μ}$ even and $c$ odd gives $[A_{μ}, [c, c]] = 2 [[A_{μ}, c], c]$ .

1.3F1F2A2algebra

For any homogeneous matter component, the odd left-derivation rule gives $s^{2} ψ = - ρ (s c) ψ + ρ (c) s ψ = \frac{1}{2} ρ ([c, c]) ψ - ρ (c)^{2} ψ$ . Writing $c = c^{a} T_{a}$ , graded commutativity yields $ρ (c)^{2} = \frac{1}{2} c^{a} c^{b} [ρ (T_{a}), ρ (T_{b})] = \frac{1}{2} ρ ([c, c])$ by [A2], so $s^{2} ψ = 0$ for either matter parity.

2.1F2step 1.1step 1.2step 1.3∎

On a coefficient function $f (x)$ , both $s D_{μ} f$ and $D_{μ} s f$ vanish. On each jet generator $Φ_{I}$ , the defining rule gives $s D_{μ} Φ_{I} = s Φ_{I + e_{μ}} = D_{μ} D_{I} (s Φ) = D_{μ} s Φ_{I}$ , since the total derivatives commute. Thus $s D_{μ} = D_{μ} s$ on generators, and composition gives $s^{2} Φ_{I} = D_{I} (s^{2} Φ) = 0$ for every multi-index $I$ . The square of an odd derivation is an even derivation because its two cross terms cancel, so steps 1.1–1.3 and this jet calculation make $s^{2}$ vanish on every finite polynomial in the generators.

Depends on

Finite-Jet Non-Abelian BRST Complex