D_\gamma - \frac{1}{2} \bar g^{\alpha\beta}\bar D_\mu\right)\bar h_{\alpha\beta}\\
\textrm{G}^a(\bar a;\bar g,\bar A)&=\hat{g}^{-1}\bar g^{\mu\nu}\bar{\mathcal D}_\mu \bar{a}^a_\nu\end{aligned}$$ The resulting ghost action reads, with $a^a_\mu \neq 0$ and $h_{\mu\nu}\neq 0$ still, $$\begin{gathered}
\label{SGhost}
S_{\textrm{gh}}[h,a,\xi, \bar \xi, \Upsilon, \bar \Upsilon;\bar g,\bar A]=\\
-\int {\textrm{d}^d x\:}\sqrt{\bar{g}}\,\left(\sqrt{2} \bar \xi_\mu\left(\bar g^{\mu\rho}\bar g^{\sigma\lambda}\bar D_\lambda\left(g_{\rho\nu}D_\sigma+g_{\sigma\nu}D_\rho\right)-\bar g^{\rho\sigma}\bar g^{\mu\lambda}\bar D_\lambda g_{\sigma\nu}D_\rho\right)\xi^\nu \right. \\
 + \bar\Upsilon^a \bar g^{\mu\nu} \bar{\mathcal D}_\mu ( \bar F^a_{\rho\nu} \xi^{\rho} + \xi^\rho \partial_\rho a^a_\nu +(\partial_\nu \xi^\rho)a^a_\rho + f^{abc}\bar{A}^b_\rho \xi^\rho a^c_\nu) + \bar\Upsilon^a\left(\bar g^{\mu\rho}\delta^{ab}\bar{\mathcal D}_\mu \nabla_\rho\right)\Upsilon^b\Big)\end{gathered}$$

It can be checked that $S_{\textrm{gh}}$ of eq. (\[SGhost\]) is invariant under background gauge transformations: ${\cal W}^{\rm B}_{\rm YM} S_{\rm gh}=0=\widetilde{\widetilde{{\cal W}^{\rm B}_{\rm D}}} S_{\rm gh}$. While this is true even for non-vanishing fluctuations $h$ and $a$, in the present calculation we shall need $S_{\rm gh}$ only for $h=0=a$.

At this point a remark concerning the expected reliability of this truncation ansatz might be in order. As for its gravitational part, all generalizations of the Einstein-Hilbert truncation explored during the past decade did not change the qualitative picture it gives rise to, at least close to the non-Gaussian fixed point. In the Yang-Mills sector we retained only the first monomial of a systematic derivative expansion. From the analysis in [@nonabavact] without gravity we know 