fluctuation fields will replace the full quantum fields as the variables of integration in (\[PI-sketchy\]).

There are now two possibilities to realize the gauge transformations $\delta_{\rm D}$ and $\delta_{\rm YM}$ at the level of the background decomposition:

-   The [*background gauge transformations*]{} $\delta^{\rm B}_{\rm D}$ and $\delta^{\rm B}_{\rm YM}$ are defined such that under diffeomorphisms all the fields transform tensorially, i.e. $$\begin{aligned}
    \label{background-diff}
    \delta^{\rm B}_{\rm D}(v) \Phi &=& {\cal L}_v \Phi\:,\hspace{0.2cm} \Phi \in\{\bar{g}_{\mu\nu}, h_{\mu\nu}, \bar{A}^a_\mu, a^a_\mu\}\end{aligned}$$ With respect to Yang-Mills transformations, the background gauge field transforms as a connection and the fluctuation transforms homogeneously: $$\begin{aligned}
    \label{background-YM-gaugefield} \delta^{\rm B}_{\rm YM}(\lambda) \bar{A}^a_\mu &=& - \partial_\mu \lambda^a + f^{abc}\lambda^b \bar{A}^c_\mu, \hspace{1cm} \delta^{\rm B}_{\rm YM}(\lambda) a^a_\mu = f^{abc}\lambda^b a^c_\mu\end{aligned}$$

-   On the other hand, we can define [*true gauge transformations*]{} $\delta^{\rm G}_{\rm D}$ and $\delta^{\rm G}_{\rm YM}$ by requiring that these shall only affect the fluctuations, not the background fields: $$\begin{aligned}
    \label{true-diff-metric} \delta^{\rm G}_{\rm D}(v) \bar{g}_{\mu\nu} &=& 0, \hspace{0.6cm} \delta^{\rm G}_{\rm D}(v) h_{\mu\nu} = {\cal L}_v \big(\bar{g}_{\mu\nu} + h_{\mu\nu}\big)\\
    \label{true-diff-gaugefield} \delta^{\rm G}_{\rm D}(v) \bar{A}^a_\mu &=& 0, \hspace{0.6cm} \delta^{\rm G}_{\rm D}(v) a^a_\mu = {\cal L}_v \big(\bar{A}^a_\mu + a^a_\mu\big)\\
    \label{true-ym-gaugefield} \delta^{\rm G}_{\rm YM}(\lambda) \bar{A}^a_\mu &=& 0, \hspace{0.6cm} \delta^{\rm G}_{\rm YM}(\lambda) a^a_\mu = - \partial_\mu \lambda^a + 