{\cal A}]}\end{aligned}$$

Here $\gamma_{\mu\nu}$ and ${\cal A}^a_\mu$ are the quantum metric and the quantum gauge field, respectively, and $S$ denotes the bare action. As usual, both of these fields are supposed to transform tensorially with respect to diffeomorphisms, $\delta_{\rm D}$. In addition, ${\cal A}^a_\mu$ defines a connection with respect to Yang-Mills gauge transformations, $\delta_{\rm YM}$. Denoting the vector field that generates the diffeomorphism by $v^\mu$ and the parameter of the Yang-Mills transformation by $\lambda^a$, we have (${\cal L}_v$ denotes the Lie derivative along $v^\mu$): $$\begin{aligned}
\label{full-quantum-metric-diff}\delta_{\rm D}(v) \gamma_{\mu\nu} &=& {\cal L}_v \gamma_{\mu\nu}\\ 
\label{full-quantum-gaugefield-diff}\delta_{\rm D}(v) {\cal A}^a_\mu &=& {\cal L}_v {\cal A}^a_\mu\\
\label{full-quantum-gaugefield-ym}\delta_{\rm YM}(\lambda) {\cal A}^a_\mu &=& - \partial_\mu \lambda^a + f^{abc}\lambda^b {\cal A}^c_\mu\end{aligned}$$

From now on we will assume the Yang-Mills gauge group to be $SU(N)$, so $a$ runs from $1$ to $N^2 - 1$, $f^{abc}$ are the associated structure constants. We demand $S$ to be invariant under both $\delta_{\rm D}$ and $\delta_{\rm YM}$.

Employing the background formalism, the dynamical fields are decomposed according to $$\begin{aligned}
\label{decomp}
\gamma_{\mu\nu} \equiv \bar{g}_{\mu\nu} + h_{\mu\nu} \hspace{0.6cm}\mbox{and}\hspace{0.6cm} {\cal A}^a_\mu \equiv \bar{A}^a_\mu + a^a_\mu\end{aligned}$$

with fixed, but arbitrary background configurations $\bar{g}_{\mu\nu}$ and $\bar{A}^a_\mu$ and fluctuations $h_{\mu\nu}$ and $a^a_\mu$ [@back]. Assuming a translational invariant measure, the 