\hat g \: \bar{\Sigma}^a \frac{\partial {\rm G}^a}{\partial a^b_\mu}\delta_{\rm YM}^{\rm G}(\Sigma) a^b_\mu \Big) 
\end{aligned}
\label{Geistwirk-allg}$$

Since we are going to neglect renormalization effects in the ghost sector, the evolution equation for $\Gamma_k$ will contain only the classical ghost action but with the quantum ghost fields replaced by their vacuum expectation values, and the full classical fields $g_{\mu\nu}$ and $A^a_\mu$ identified with their background configurations $\bar{g}_{\mu\nu}$ and $\bar{A}^a_\mu$, respectively; stated differently, in this class of approximations the fluctuations $\bar{h}_{\mu\nu}$ and $\bar{a}^a_\mu$ can be set to zero in $S_{\rm gh}$ even before $S^{(2)}_{\rm gh}$ is computed.

Looking at the $\bar{\Upsilon}$-$\xi$ part of the resulting ghost action we encounter a serious problem: Employing ${\rm G}^a (a; \bar{g}, \bar{A}) \equiv \bar{{\cal D}}^\mu a^a_\mu$ as gauge condition, it is given by $$\label{ups-xi-geist-alt}
 \left(S_{\rm gh}\:[\xi, \bar{\Upsilon}; \bar{g}, \bar{A}]\right)_{\bar{\Upsilon}\xi} = - \int\:{\rm d}^d x\:\sqrt{\bar{g}}\:\big[\bar{\Upsilon}^a \bar{{\cal D}}^{\nu\:ab} \big(\xi^\rho \partial_\rho \bar{A}^b_\nu + (\partial_\nu \xi^\rho) \bar{A}^b_\rho\big)\big]$$

Here the covariant background derivative $\bar{{\cal D}}_\mu$ acts on the ordinary Lie derivative of an $SU(N)$ background connection; therefore, this part of the ghost action is [*not*]{} $\delta^{\rm B}_{\rm YM}$-invariant. In order to resolve this issue, it is useful to look at it from a more abstract point of view.

Ward operators and their algebra
