write $D \equiv \partial + \Gamma$ and $\nabla \equiv \partial + A$ for the covariant derivatives that are constructed by means of $\Gamma^\rho_{\mu\nu} = \frac{1}{2}g^{\rho\sigma}\big(\partial_\nu g_{\sigma\mu} + \partial_\mu g_{\sigma\nu} - \partial_\sigma g_{\mu\nu}\big)$ and $A^a_\mu$, respectively; the covariant derivative containing both of these connections is denoted by ${\cal D} \equiv \partial + A +\Gamma$. By adding a bar, we denote their analogues evaluated on the background configurations.

On the Construction of\
Background Gauge Invariant Ghost Actions
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Motivation
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Since we have to fix two gauge invariances by two gauge conditions, $\textrm{F}_\mu(h_{\mu\nu}; \bar{g}_{\mu\nu}, \bar{A}^a_\mu)$ and $\textrm{G}^a(a^a_\mu; \bar{g}_{\mu\nu}, \bar{A}^a_\mu)$ for the diffeomorphisms and the $SU(N)$ transformations, respectively, the associated Faddeev-Popov operator will in general consist of four components. (Here we have already assumed that the diffeomorphism and the $SU(N)$ gauge condition only involve the metric and the gauge field fluctuation separately.) The corresponding classical ghost action will then be of the form $$\begin{aligned}
S_{\rm gh}[h, a, {\cal C}, &\bar{\cal C}, \Sigma, \bar{\Sigma}; \bar{g}, \bar{A}] = - \int\:{\rm d}^d x\:\sqrt{\bar{g}}\:\Big(\kappa^{-1}\:\bar{{\cal C}}_\mu \bar{g}^{\mu\nu}\frac{\partial {\rm F}_\nu}{\partial h_{\rho\sigma}}\delta_{\rm D}^{\rm G}({\cal C}) h_{\rho\sigma}+ \\
& + \kappa^{-1}\:\bar{{\cal C}}_\mu \bar{g}^{\mu\nu}\frac{\partial {\rm F}_\nu}{\partial h_{\rho\sigma}}\delta_{\rm YM}^{\rm G}(\Sigma) h_{\rho\sigma} + \hat{g}\: \bar{\Sigma}^a \frac{\partial {\rm G}^a}{\partial a^b_\mu}\delta_{\rm D}^{\rm G}({\cal C}) a^b_\mu + 