TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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A classical master solution defines a square-zero differential

Statement

Let $(A, Δ)$ be as in BV algebra with a degree-one generator, and let $S \in A^{0}$ satisfy the classical master equation in Classical and formal quantum master equations. Then $d_{S} : A \to A$ , $d_{S} (a) = {S, a}$ , has degree $+ 1$ and satisfies $d_{S}^{2} = 0$ .

Facts & Assumptions

Given: A BV algebra over a characteristic-zero field, an element $S \in A^{0}$ , and the CME ${S, S} = 0$ .

[F1]

Classical and formal quantum master equations: the classical master equation for a degree-zero element $S$ is ${S, S} = 0$ .

[F2]

The derived BV bracket is an odd Poisson bracket: the bracket has degree $+ 1$ and satisfies ${a, {b, c}} = {{a, b}, c} + (- 1)^{(∣ a ∣ + 1) (∣ b ∣ + 1)} {b, {a, c}}$ .

Proof

1.1givenF2algebra

Since the bracket has degree $+ 1$ by [F2] and $∣ S ∣ = 0$ , the map $d_{S} (a) = {S, a}$ sends $A^{r}$ into $A^{r + 1}$ and has degree $+ 1$ .

1.2givenF2algebra

Apply the Jacobi identity [F2] with $a = b = S$ and third entry $x$ ; its sign is $(- 1)^{(0 + 1) (0 + 1)} = - 1$ , so ${S, {S, x}} = {{S, S}, x} - {S, {S, x}}$ and therefore $2 d_{S}^{2} (x) = {{S, S}, x}$ .

2.1givenF1step 1.2algebra∎

The CME [F1] makes the right side zero. Characteristic zero makes $2$ invertible, hence $d_{S}^{2} (x) = 0$ for every $x \in A$ .

Depends on

Used by

A one-variable BV algebra and a classical master solution Example

Dependency tree · one level

3 results within one dependency step of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ezra Getzler, Batalin–Vilkovisky algebras and two-dimensional topological field theories, arXiv:hep-th/9212043v3 (standard reference, not scraped)
Yvette Kosmann-Schwarzbach and Juan Monterde, Divergence operators and odd Poisson brackets, Annales de l’Institut Fourier 52 (2002) (standard reference, not scraped)