Classical and formal quantum master equations

Let $(A,\mathrm{\Delta })$ be a BV algebra in the degree-$+1$ convention of BV algebra with a degree-one generator, with derived bracket $\{-,-\}$. For an element $S\in A^0$, the classical master equation (CME) is $\{S,S\}=0.$

For a formal version, take $A$ over $\mathbb{C}$, let $\mathrm{\hslash }$ be a central formal variable of degree $0$, and write $A[\text{\hspace{0.17em}⁣}[\mathrm{\hslash }]\text{\hspace{0.17em}⁣}]$ for formal power series with the Cauchy product. Extend $\mathrm{\Delta }$ and $\{-,-\}$ coefficientwise. For $S_{\mathrm{\hslash }}\in A^0[\text{\hspace{0.17em}⁣}[\mathrm{\hslash }]\text{\hspace{0.17em}⁣}]$, the formal quantum master equation (QME) in the convention used here is $\frac{1}{2}\{S_{\mathrm{\hslash }},S_{\mathrm{\hslash }}\}-i\mathrm{\hslash }\mathrm{\Delta }{S}_{\mathrm{\hslash }}=0$ as an equality in $A^1[\text{\hspace{0.17em}⁣}[\mathrm{\hslash }]\text{\hspace{0.17em}⁣}]$, checked coefficientwise. Each coefficient uses only a finite Cauchy sum, so this is a formal algebraic condition and requires no analytic convergence.

Kosmann-Schwarzbach–Monterde write a corresponding Maurer–Cartan equation as $\mathrm{\Delta }w+\frac{1}{2}\{w,w\}=0$. Where $w=i{S}_{\mathrm{\hslash }}\mathrm{/}\mathrm{\hslash }$ is formed in the Laurent-series extension, multiplying that equation by $-{\mathrm{\hslash }}^2$ gives the QME displayed here. This scalar substitution is only a normalization conversion. Neither equation defines a measure, proves a renormalization claim, or by itself establishes anomaly cancellation. The CME differential consequence is proved in A classical master solution defines a square-zero differential.