=\ - \langle A_\delta, QA_\eta+(F\Psi)^2\rangle
- {\langle\!\langle}\delta\Psi, Y(Q\Psi+X\eta F\Psi){\rangle\!\rangle}\,,
\label{general variation}$$ from which we find the equations of motion, $$QA_\eta + (F\Psi)^2\ =\ 0\,,\qquad
 Q\Psi + X\eta F\Psi\ =\ 0\,.
\label{equations of motion}$$

Before closing this section, we generalize several ingredients for later use. We can define $A_{\mathcal{O}}(t)$ not only for $\mathcal{O}=\partial_t, \eta,$ or $ \delta$, but also for any other derivations of the string product. Although such general $\mathcal{O}$’s are not in general commutative, we assume that they satisfy a closed algebra with respect to the graded commutator of derivations, $\{\mathcal{O}_1,\mathcal{O}_2]\ =\ \mathcal{O}_1\mathcal{O}_2 
-(-1)^{\mathcal{O}_1\mathcal{O}_2}\mathcal{O}_2\mathcal{O}_1$. The generalized $A_{\mathcal{O}}(t)$’s satisfy the equation $$\begin{aligned}
 \mathcal{O}_1A_{\mathcal{O}_2}(t)
 -&(-1)^{\mathcal{O}_1\mathcal{O}_2}\mathcal{O}_2A_{\mathcal{O}_1}(t)
- [\![A_{\mathcal{O}_1}(t)\,, A_{\mathcal{O}_2}(t)]\!]
=\ A_{\{\mathcal{O}_1,\mathcal{O}_2]}(t)\,,\label{gen MC}\end{aligned}$$ which reduces to the Maurer-Cartan-like equation (\[MC\]) when $\{\mathcal{O}_1,\mathcal{O}_2]=0$. Using $A_{\mathcal{O}}(t)$, we can define the covariant derivative $D_{\mathcal{O}}(t)$ on a string field $A$ by $$D_{\mathcal{O}}(t) A\ =\ \mathcal{O} A 
- [\![A_{\mathcal{O}}(t)\,, A]\!]\,.
$$ From (\[gen MC\]), we can show that $$[\![D_{\mathcal{O}_1}(t)\,, D_{\mathcal{O}_2}(t)]\!]\
=\
D_{\{\mathcal{O}_1, \mathcal{O}_2]}(t)\,.
\label{generalized D}$$ As an analog of the linear map $F(t)$ in the Ramond sector, we can also define the linear map $f(t)$ on a general string field $\Phi$ in the NS sector by $$\begin{aligned}
 f(t)\Phi\ =&\ 
 \frac{1}{1+\xi_0(D_\eta(t)-\eta)}\,\Phi
\nonumber\\
=&\ \Phi + \xi_0 [\![A_\eta(t), \Phi]\!]
+ \xi_0 [\![A_\eta(t),\,\xi_0[\![A_\eta(t), \Phi]\!] ]\!]\cdots\,.
\label{f ns}\end{aligned}$$ A homotopy operator for $D_\eta(t)$ in the NS sector is given by the BPZ even opera