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Classical Master Equation and Nilpotence of the BV Differential

Statement

Let $B$ be the graded-commutative canonical field/antifield algebra of the preceding definition: $∣ Φ^{A} ∣ = ϵ_{A}$ , $gh (Φ^{A}) = g_{A}$ , $∣ Φ_{A}^{*} ∣ = ϵ_{A} + 1$ , and $gh (Φ_{A}^{*}) = - 1 - g_{A}$ . Its canonical antibracket is odd, has ghost number $+ 1$ , and satisfies the shifted skew and Jacobi identities stated there. Assume the center of $B$ contains no nonzero element of ghost number one. In the finite-dimensional coordinate algebra this follows because a central element has zero derivative with respect to every field and antifield coordinate and is therefore a numerical constant; for a formal local-functional domain it is an explicit hypothesis. For any even $S_{B V} \in B$ of ghost number zero, define $s F : = (S_{B V}, F)$ . Then $s^{2} = 0$ on $B$ if and only if the classical master equation $(S_{B V}, S_{B V}) = 0$ holds.

In the local-field case of the preceding definition, let $U \subseteq R^{n}$ be nonempty and open, let $g$ be a finite-dimensional real Lie algebra, and let $ρ$ be the stated parity-preserving representation on the graded matter space. Take the minimal fields $A, ψ, c$ and their antifields, with $δ_{λ} A_{μ} = \partial_{μ} λ + [A_{μ}, λ]$ and $δ_{λ} ψ = - ρ (λ) ψ$ for every admitted smooth $λ : U \to g$ . Assume $[δ_{λ}, δ_{μ}] = δ_{[λ, μ]}$ without equations of motion, where $[λ, μ] (x) : = [λ (x), μ (x)]_{g}$ , and that $δ_{λ} Φ \equiv 0$ for every field configuration implies $λ \equiv 0$ . Let $S_{0} [A, ψ]$ be a local functional with well-defined variational derivatives, invariant under these transformations. Choose a functional domain and support or boundary conditions for which the antibracket, integrations by parts, and gauge-variation identity are well-defined and boundary terms vanish. For $s_{0} A_{μ} = \partial_{μ} c + [A_{μ}, c]$ , $s_{0} c = - \frac{1}{2} [c, c]$ , and $s_{0} ψ = - ρ (c) ψ$ , form the minimal action $S_{B V} = S_{0} + \int_{U} d^{n} x \sum_{A} (- 1)^{ϵ_{A}} Φ_{A}^{*} s_{0} Φ^{A}$ in the order specified in the definition. Then this minimal action satisfies the classical master equation $(S_{B V}, S_{B V}) = 0$ .

Statement references

Canonical BV algebra and master-action convention

Gomis, París, and Samuel, §§4.2–4.3 and 4.6

Qiu and Zabzine, §§4.2–4.4

Facts & Assumptions

Given: The canonical BV algebra, its odd antibracket, and an even ghost-number-zero element $S_{B V}$ .

[A1]

The chosen domain has no nonzero central elements of ghost number one; this follows from canonical finite-dimensional coordinates and is separately assumed for any formal local-functional extension. Classical BV Antibracket and Yang–Mills Master-Action Data

[F1]

Classical BV Antibracket and Yang–Mills Master-Action Data: the antibracket has ghost number $+ 1$ , satisfies shifted graded skew-symmetry and shifted Jacobi, and the BV differential convention is $s F = (S_{B V}, F)$ .

[A2]

Classical BV Antibracket and Yang–Mills Master-Action Data: in the minimal local-field setup, the gauge action is irreducible and off-shell closed, $S_{0}$ is invariant, and the functional domain admits the stated integrations by parts with vanishing boundary terms.

[F2]

Classical BV Antibracket and Yang–Mills Master-Action Data: the candidate action is $S_{B V} = S_{0} + S_{1}$ with $S_{1} = \int_{U} d^{n} x \sum_{A} (- 1)^{ϵ_{A}} Φ_{A}^{*} s_{0} Φ^{A}$ in the displayed order.

[F3]

Canonical Antibracket Generates the Declared BRST Field Variations: the antibracket action of $S_{1}$ on each minimal field and ghost coordinate is $(S_{1}, Φ^{A}) = s_{0} Φ^{A}$ .

[F4]

Nilpotence of the Finite-Jet Non-Abelian BRST Differential: the specified $s_{0}$ squares to zero on the minimal fields, ghost, and every finite-jet generator.

Proof

1.1F1algebra

For homogeneous $F$ of parity $∣ F ∣$ , the shifted Jacobi identity with entries $S_{B V}, S_{B V}, F$ gives $((S_{B V}, S_{B V}), F) + 2 (- 1)^{∣ F ∣} ((S_{B V}, F), S_{B V}) = 0$ . Shifted skew-symmetry gives $((S_{B V}, F), S_{B V}) = (- 1)^{∣ F ∣ + 1} (S_{B V}, (S_{B V}, F))$ . Combining these identities yields $s^{2} F = (S_{B V}, (S_{B V}, F)) = \frac{1}{2} ((S_{B V}, S_{B V}), F)$ .

1.2A2F2algebra

Write $S_{1} = \int_{U} d^{n} x \sum_{A} (- 1)^{ϵ_{A}} Φ_{A}^{*} s_{0} Φ^{A}$ . Since $S_{0}$ is antifield-independent, $(S_{0}, S_{0}) = 0$ . The displayed antifield signs and left/right derivative convention give $(S_{0}, S_{1}) = s_{0} S_{0}$ , with $s_{0}$ acting only on $A, ψ$ here. Gauge invariance says the first variation of $S_{0}$ vanishes for every admitted even parameter $λ$ ; its coefficientwise Noether identity therefore remains zero after replacing $λ$ by the formal ghost $c$ , so $(S_{0}, S_{1}) = 0$ . The boundary hypothesis justifies the variational derivatives and integrations by parts in this identity. Both $S_{0}$ and $S_{1}$ are even, so shifted skew-symmetry gives $(S_{1}, S_{0}) = (S_{0}, S_{1}) = 0$ .

2.1step 1.1algebra

If $(S_{B V}, S_{B V}) = 0$ , step 1.1 gives $s^{2} F = 0$ for every $F \in B$ . This is the forward implication.

2.2F1A1step 1.1

Conversely, if $s^{2} F = 0$ for every $F$ , step 1.1 implies $((S_{B V}, S_{B V}), F) = 0$ for every $F$ , so $(S_{B V}, S_{B V})$ is central. Its ghost number is $0 + 0 + 1 = 1$ by [F1], and [A1] therefore forces $(S_{B V}, S_{B V}) = 0$ . This proves the reverse implication.

2.3F1F3F4step 1.1algebra

By [F3], $q F : = (S_{1}, F)$ agrees with $s_{0}$ on each minimal field and ghost, and therefore on every finite-jet expression in them. Applying the identity of step 1.1 with $S_{1}$ in place of $S_{B V}$ gives $((S_{1}, S_{1}), Φ^{A}) = 2 s_{0}^{2} Φ^{A} = 0$ by [F4]. The canonical bracket formula gives $((S_{1}, S_{1}), Φ^{A}) = - δ_{r} (S_{1}, S_{1}) / δ Φ_{A}^{*}$ . Since $(S_{1}, S_{1})$ is linear in antifields, these variational derivatives determine it modulo boundary terms and yield $(S_{1}, S_{1}) = - 2 \int_{U} d^{n} x \sum_{A} Φ_{A}^{*} s_{0}^{2} Φ^{A}$ . It vanishes by [F4].

3.1A2F2step 1.2step 2.3algebra∎

Expanding $(S_{0} + S_{1}, S_{0} + S_{1})$ , the pure $S_{0}$ term and both cross terms vanish by step 1.2, while the pure $S_{1}$ term vanishes by step 2.3. Thus the defined minimal action satisfies the classical master equation under the stated gauge-invariance, off-shell-closure, and domain hypotheses.

Classical Master Equation and Nilpotence of the BV Differential

Statement

Statement references

Facts & Assumptions

Proof

Depends on

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Sources