G}$, has the structure of a semi-direct product of the spacetime diffeomorphisms ${\sf Diff}$ and the local Yang-Mills transformations ${\sf SU(N)}_{\rm loc}$, with the latter forming the invariant subalgebra: ${\bf G} = {\sf Diff} \ltimes {\sf SU(N)}_{\rm loc}$. Whereas the first two relations (\[ward-d-d\]), (\[ward-l-l\]) represent the well-known composition laws of diffeomorphisms and Yang-Mills gauge transformations, the third relation (\[ward-d-l\]) lies at the heart of our problem: diffeomorphisms and local gauge transformations do not commute. Instead, they close on the Lie derivative of the gauge parameter. In particular, this implies that diffeomorphisms do not map $SU(N)$ tensors onto $SU(N)$ tensors.

Modified diffeomorphisms
------------------------

What is called for is an $SU(N)$ covariantization of the ordinary Lie derivative. This is tantamount to a different parametrization of ${\bf G}$ that makes the mixed commutator vanish. This can be achieved by defining [*new diffeomorphisms*]{} which include a ${\sf SU(N)}_{\rm loc}$ transformation with parameter $\lambda^a= \mathcal{A}^a_\mu v^\mu$ [@jackiw]: $$\begin{aligned}
\label{new-diff}
\widetilde{{\cal W}_{\rm D}}(v) \equiv {\cal W}_{\rm D}(v) + {\cal W}_{\rm YM} ({\cal A}\cdot v) \end{aligned}$$

Loosely speaking, this amounts to shifting a certain $v$-dependent part of ${\sf SU(N)}_{\rm loc}$ into the diffeomorphism sector. The notion of an “invariant functional” remains unchanged: $$\begin{aligned}
\label{inv-set-new}
{\cal F}_{\rm 