A}^a_\mu (x)\frac{\delta}{\delta {\cal A}^a_\mu(x)}\\
& \hspace{2.1cm} + \delta_{\rm D}(v){\cal C}^\mu (x)\frac{\delta}{\delta {\cal C}^\mu(x)} + \delta_{\rm D}(v)\bar{\cal C}_\mu (x)\frac{\delta}{\delta \bar{\cal C}_\mu(x)}\\
&\hspace{2.1cm} + \delta_{\rm D}(v)\Sigma^a (x)\frac{\delta}{\delta \Sigma^a(x)} + \delta_{\rm D}(v)\bar{\Sigma}^a (x)\frac{\delta}{\delta \bar{\Sigma}^a(x)}\bigg)
\end{aligned}\label{ward-diff}$$

and Yang-Mills gauge transformations: $$\begin{aligned}
{\cal W}_{\rm YM}(\lambda) &\equiv -\int\:{\rm d}^d x\:\Big(\delta_{\rm YM}(\lambda)\gamma_{\mu\nu} (x)\frac{\delta}{\delta \gamma_{\mu\nu}(x)} + \delta_{\rm YM}(\lambda){\cal A}^a_\mu (x)\frac{\delta}{\delta {\cal A}^a_\mu(x)}\\
& \hspace{2.0cm} + \delta_{\rm YM}(\lambda){\cal C}^\mu (x)\frac{\delta}{\delta {\cal C}^\mu(x)} + \delta_{\rm YM}(\lambda)\bar{\cal C}_\mu (x)\frac{\delta}{\delta \bar{\cal C}_\mu(x)}\\
& \hspace{2.0cm} + \delta_{\rm YM}(\lambda)\Sigma^a (x)\frac{\delta}{\delta \Sigma^a(x)} + \delta_{\rm YM}(\lambda)\bar{\Sigma}^a (x)\frac{\delta}{\delta \bar{\Sigma}^a(x)}\Big)
\end{aligned}\label{ward-su}$$

In these integrals the measure factor $\sqrt{{\rm det}(\gamma_{\mu\nu})}$ cancels against a similar one which would render the functional derivatives tensorial. We define the set of all invariant functionals by $$\begin{aligned}
\label{inv-set}
{\cal F}_{\rm inv} \equiv \{F\:|\:{\cal W}_{\rm D}(v) F = 0 \:\wedge\: {\cal W}_{\rm YM}(\lambda) F = 0 ~ \forall ~ v^\mu,\:\lambda^a\}\end{aligned}$$

Computing the algebra of these operators leads to $$\begin{aligned}
\label{ward-d-d}[{\cal W}_{\rm D}(v_1), {\cal W}_{\rm D}(v_2)] &=& {\cal W}_{\rm D}([v_1, v_2])\\
\label{ward-l-l}[{\cal W}_{\rm YM}(\lambda_1), {\cal W}_{\rm YM}(\lambda_2)] &=& {\cal W}_{\textrm{YM}}(f \lambda_1\lambda_2)\\
\label{ward-d-l}[{\cal W}_{\rm D}(v), {\cal W}_{\rm YM}(\lambda)] &=& {\cal W}_{\rm YM}({\cal L}_v \lambda)\end{aligned}$$

where $[v_1, v_2]$ denotes the Lie bracket and $(f \lambda_1\lambda_2)^a \equiv f^{abc}\lambda^b_1\lambda^c_2$. This algebra implies that the total group of gauge transformations, ${\bf 