Theorem: Relative master equation

On ${\mathfrak{F}}_W$,

exactly in $\kappa$, at ${\hslash }^0$. Consequently $s_{\mathrm{ext}}:=(S_{\mathrm{ext}},\cdot )$ is a nilpotent, odd, ghost-number-one derivation of the local functionals on ${\mathfrak{F}}_W$, and the same holds for every $S^{\prime }\in {\mathfrak{S}}_{\mathrm{kin}}$ whose higher antifield terms satisfy the corresponding degree-$\ge 1$ equations.

Proof. By Theorem thm:breaking, $(S_{\mathrm{ext}},S_{\mathrm{ext}})=\kappa \int c^{\nu }{\mathcal{K}}_{\nu }$. For $c\in {\mathfrak{g}}_W\otimes \Lambda$ the integrand vanishes pointwise: $c^{\nu }(x)=0$ for $x\notin W$ and ${\mathcal{K}}_{\nu }(x)=0$ for $x\in W$ by Lemma lem:Kzero. The functional derivatives of $\mathfrak{K}$ along ${\mathfrak{F}}_W$ vanish as well: the $\gamma$- and $\varphi$-derivatives retain the factor $c^{\nu }$, and the $c$-derivative is $\kappa {\mathcal{K}}_{\nu }$ restricted to $W$, which is zero. Hence $\mathfrak{K}\equiv 0$ on ${\mathfrak{F}}_W$ together with its derivatives, and the graded Jacobi identity gives ${s_{\mathrm{ext}}}^{2}F=\frac{1}{2}((S_{\mathrm{ext}},S_{\mathrm{ext}}),F)=0$. Theorem thm:obstructA1 shows the antifield-degree-zero equation is satisfied by every $S^{\prime }\in {\mathfrak{S}}_{\mathrm{kin}}$ on ${\mathfrak{F}}_W$; the remaining degree-$\ge 1$ equations are the usual conditions on ${S^{\prime }}_{\ge 2}$ and are satisfied, for example, by ${S^{\prime }}_{\ge 2}=0$.