Reference:

Chapter 11.3.1 of Freedman and Van Proeyen's [Supergravity][1] textbook.

\begin{eqnarray} 
\notag
\delta(a,\lambda) \phi(x) &=& (a^\mu(x) P_\mu -\frac{1}{2}\lambda^{\mu\nu}(x)M_{\mu\nu}) \phi(x) \\\notag
&=& (a^\mu(x) \partial_\mu +\lambda^{\mu\nu}(x) x_\nu \partial_\mu) \phi(x)\\\notag
&=& (a^\mu(x) +\lambda^{\mu\nu}(x) x_\nu) \partial_\mu \phi(x) \\\notag
&=& (\xi^\mu(x)) \partial_\mu \phi(x) \\\notag
&=& L_\xi \phi(x) \\
&=& \delta_{gct} \phi(x) 
\end{eqnarray}


where we generalized the old spacetime translation vector $a^\mu(x)$ to curved spacetime with $\xi^\mu(x)= a^\mu(x)+ \lambda^{\mu\nu}(x) x_\nu$. So we will have general coordinate transformations (GCTs) parametrized by $\xi^\mu(x)$ and local Lorentz transformations (LLTs) parametrized by $\lambda^{ab}(x)$.


I am trying to understand the introduction of "covariant GCTs" (CGCTs) in the context of gauged spacetime translations. CGCTs are defined by equation 11.61 in the referenced above

\begin{equation}
\delta_{cgct} (\xi) = \delta_{gct}(\xi) - \delta(\xi^\mu B_\mu)
\end{equation}

This is motivated by the following:

Consider the standard transformation of scalar fields given by equations 11.1 and 11.2 in the reference above

\begin{equation}
\delta(\epsilon) \phi^i(x) = - \epsilon^A(x) t_A{}^i{}_j \phi^j
\end{equation}

Now, we showed above what the transformation of the scalar field under GCTs is, so lets say that the symmetry ($T_A = - (t_A)^i{}_j$) is GCT, i.e. $\partial_\mu$, and the parameter ($\epsilon$) is $\xi$.

Then we have, as before,

\begin{equation}
\delta(\xi) \phi^i(x) = \xi^\mu(x) \partial_\mu \phi^i(x)
\end{equation}

The authors then state on page 228, 


"This is correct, *but it has the undesirable property that it does not transform covariantly under internal symmetry.* We fix this by adding a field dependent gauge transformation and thus define

\begin{equation}
\delta_{cgct}(\xi) \phi^i = \xi^\mu \partial_\mu \phi^i(x)+(\xi^\mu A_\mu{}^A)t_{A}{}^i{}_j \phi^j"
\end{equation}

where $\phi^i$ and $\xi$ are still functions of spacetime, the $(x)$ has just been neglected for brevity.

My confusion lies in "**but it has the undesirable property that it does not transform covariantly under internal symmetry.**"

Can anyone expound on this?


  [1]: https://www.cambridge.org/gb/academic/subjects/physics/theoretical-physics-and-mathematical-physics/supergravity?localeText=United%20Kingdom&locale=en_GB&remember_me=on