where $\varepsilon$ is an infinitesimal real parameter, and $Y^{\alpha}(\phi(x),x)$ is a generator, such that the transformation (1) is a quasi-symmetry$^2$ of
the Lagrangian density
whenever $\varepsilon$ an $x$-independent global parameter, such that the last term on the rhs. of eq. (2) vanishes. Here $j^{\mu}$ and $J^{\mu}:=j^{\mu}-f^{\mu}$
$$ \tag{3} J^{\mu}~:=~j^{\mu}-f^{\mu}$$
are the bare and the full Noether currents, respectively. The corresponding on-shell conservation law reads$^3$
$$ \tag{3} d_{\mu}J^{\mu}~\approx~ 0, $$$$ \tag{4} d_{\mu}J^{\mu}~\approx~ 0, $$
cf. Noether's first Theorem.
Here $f^{\mu}$ are so-called improvement terms, which are not uniquely defined from eq. (2). Under mild assumptions, it is possible to partially fix this ambiguity by assuming the following technical condition
$$ \tag{5}\sum_{\alpha}\frac{\partial f^{\mu}}{\partial(\partial_{\nu}\phi^{\alpha})}Y^{\alpha}~=~(\mu \leftrightarrow \nu), $$
which will be important for the Theorem 1 below. We may assume without loss of generality that the original Lagrangian density
$$ \tag{4} {\cal L}~=~{\cal L}(\phi(x), \partial \phi(x); A(x),F(x);x). $$$$ \tag{6} {\cal L}~=~{\cal L}(\phi(x), \partial \phi(x); A(x),F(x);x). $$
$$\tag{5} F_{\mu\nu}~:=~\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$$$$\tag{7} F_{\mu\nu}~:=~\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$$
$$ \tag{6} \delta A_{\mu}~=~d_{\mu}\varepsilon. $$$$ \tag{8} \delta A_{\mu}~=~d_{\mu}\varepsilon. $$
$$ \tag{7} D_{\mu}\phi^{\alpha}~=~\partial_{\mu}\phi^{\alpha} - A_{\mu}Y^{\alpha}, $$$$ \tag{9} D_{\mu}\phi^{\alpha}~=~\partial_{\mu}\phi^{\alpha} - A_{\mu}Y^{\alpha}, $$
which is invariant $\delta (D\phi)=0$transforms covariantly
$$ \tag{10} \delta (D_{\mu}\phi)^{\alpha}~=~\varepsilon (D_{\mu}Y)^{\alpha}$$
under gauge transformations (1) and (68).
One One may then prove under mild assumptions the following theoremTheorem 1.
Theorem 1. The gauge transformations (1) and (68) are a quasi-symmetry for the following so-called gauged Lagrangian density
$$ \tag{8}\widetilde{\cal L}~:=~ \left.{\cal L}\right|_{\partial\phi\to D\phi}+\left. A_{\mu} f^{\mu}\right|_{\partial\phi\to D\phi}. $$$$ \tag{11} \widetilde{\cal L}~:=~ \left.{\cal L}\right|_{\partial\phi\to D\phi}+\left. A_{\mu} f^{\mu}\right|_{\partial\phi\to D\phi}. $$
$$ \tag{9} {\cal L}
~=~ \frac{i}{2}(\phi^*\partial_{0}\phi - \phi \partial_{0}\phi^*)
- \frac{1}{2m} \sum_{k=1}^3(\partial_{k}\phi)^*\partial^{k}\phi. $$$$ \tag{12} {\cal L}
~=~ \frac{i}{2}(\phi^*\partial_0\phi - \phi \partial_0\phi^*)
- \frac{1}{2m} \sum_{k=1}^3(\partial_k\phi)^*\partial^k\phi. $$
$$ \tag{10} i\partial_{0}\phi~\approx~-\frac{1}{2m}\partial_{k}\partial^{k}\phi
\qquad \Leftrightarrow \qquad
i\partial_{0}\phi^*~\approx~\frac{1}{2m}\partial_{k}\partial^{k}\phi^*. $$$$ 0~\approx~\frac{\delta S}{\delta\phi^*}
~=~ i\partial_0\phi~+\frac{1}{2m}\partial_k\partial^k\phi $$
$$ \tag{13} \qquad \Leftrightarrow \qquad
0~\approx~\frac{\delta S}{\delta\phi}
~=~ -i\partial_0\phi^*~+\frac{1}{2m}\partial_k\partial^k\phi^*. $$
$$ \tag{11} \delta \phi~=~\varepsilon
\qquad \Leftrightarrow \qquad
\delta \phi^*~=~\varepsilon^*. $$$$ \tag{14} \delta \phi~=~Y\varepsilon
\qquad \Leftrightarrow \qquad
\delta \phi^*~=~Y^*\varepsilon^*, $$
where $Y\in\mathbb{C}\backslash\{0\}$ is a fixed non-zero complex number. Bear in mind that the above Theorem 1 is only applicable to a single real transformation (1). Here we are trying to apply Theorem 1 to a complex transformation, so we may not succeed, but let's see how far we get. The complexified Noether currents are
$$ \tag{12} j^0~=~ \frac{i}{2}\phi^*, \qquad
j^k~=~-\frac{1}{2m}\partial_{k}\phi^*, \qquad k~\in~\{1,2,3\},$$$$ \tag{15} j^0~=~ \frac{i}{2}Y\phi^*, \qquad
j^k~=~-\frac{1}{2m}Y\partial^k\phi^*, \qquad k~\in~\{1,2,3\},$$
$$ \tag{13} f^0~=~ -\frac{i}{2}\phi^*, \qquad f^k~=~0, $$$$ \tag{16} f^0~=~ -\frac{i}{2}Y\phi^*, \qquad f^k~=~0, $$
$$ \tag{14} J^0~=~ i\phi^*, \qquad
J^k~=~-\frac{1}{2m}\partial_{k}\phi^*, $$$$ \tag{17} J^0~=~ iY\phi^*, \qquad
J^k~=~-\frac{1}{2m}Y\partial^k\phi^*, $$
and the corresponding complex conjugate relations (15)-(17). The gaugedinfinitesimal complex gauge symmetry is defined to be
$$ \tag{18} \delta A_{\mu}~=~d_{\mu}\varepsilon
\qquad \Leftrightarrow \qquad
\delta A_{\mu}^*~=~d_{\mu}\varepsilon^*. $$
The Lagrangian density (811) reads
$$ \widetilde{\cal L}
~:=~\frac{i}{2}(\phi^*D_{0}\phi - \phi D_{0}\phi^*)
- \frac{1}{2m} \sum_{k=1}^3(D_{k}\phi)^*D^{k}\phi
+\frac{i}{2}(\phi A_0^* - \phi^* A_0) $$$$ \widetilde{\cal L}
~=~\frac{i}{2}(\phi^*D_0\phi - \phi D_0\phi^*)
- \frac{1}{2m} \sum_{k=1}^3(D_k\phi)^*D^k\phi
+\frac{i}{2}(\phi Y^* A_0^* - \phi^*Y A_0) $$
$$ \tag{15}
~=~\frac{i}{2}\left(\phi^*(\partial_{0}\phi-2A_0) - \phi (\partial_{0}-2A_0\phi)^*\right)
- \frac{1}{2m} \sum_{k=1}^3(D_{k}\phi)^*D^{k}\phi . $$$$ \tag{19} ~=~\frac{i}{2}\left(\phi^*(\partial_0\phi-2Y A_0)
- \phi (\partial_0\phi-2YA_0)^*\right)
- \frac{1}{2m} \sum_{k=1}^3(D_k\phi)^*D^k\phi . $$
We emphasize that the gauged Lagrangian density $\widetilde{\cal L}$ is not just the minimally coupled original Lagrangian density $\left.{\cal L}\right|_{\partial\phi\to D\phi}$. The last term on the rhs. of eq. (811) is important too. An infinitesimal gauge transformation of the gauged Lagrangian density is a total time derivative
$$ \tag{16} \delta\widetilde{\cal L}~=~\frac{i}{2}\partial_0(\varepsilon^*\partial_0\phi -\varepsilon\partial_0\phi^*) $$$$ \tag{20} \delta\widetilde{\cal L}~=~\frac{i}{2}d_0(\varepsilon^*Y^*\phi -\varepsilon Y\phi^*) + i|Y|^2(\varepsilon A_0^* - \varepsilon^* A_0) $$
for arbitrary infinitesimal $x$-dependent local gauge parameter $\varepsilon=\varepsilon(x)$. SoNote that the the local complex transformations (14) and (18) is not a (quasi) gauge symmetry of the Lagrangian density (19). The obstruction is the second term on the rhs. of eq. (20). Only the first term on the rhs. of eq. (20) is a total time derivative. However, let us restrict the gauge parameter $\varepsilon$ and the gauge field $A_{\mu}$ to belong to a fixed complex direction in the complex plane,
$$ \tag{21}\varepsilon,A_{\mu}~\in ~ e^{i\theta}\mathbb{R}.$$
Here $e^{i\theta}$ is some fixed phase factor, i.e. we leave only a single real gauge d.o.f. Then the second term on the rhs. of eq. (20) vanishes, so the gauged Lagrangian density (1519) hashas a real (quasi)gauge gauge symmetry in accordance with Theorem 1. Note that the field $\phi$ is still a fully complex variable even with the restriction (21). Also note that the Lagrangian density (19) can handle both the real and the imaginary local shift transformations (14) as (quasi) gauge symmetries via the restriction construction (21), although not simultaneously.
III) An incomplete list offor further readingstudies:
Peter West, Introduction to Supersymmetry and SupergrativitySupergravity, 1990, Chap. 7.
Henning Samtleben, Lectures on Gauged Supergravity and Flux Compactifications, Class. Quant. Grav. 25 (2008) 214002, arXiv:0808.4076.
