Take a classical field theory described by a local Lagrangian depending on a set of fields and their derivatives. Suppose that the action possesses some global symmetry. What conditions have to be satisfied so that this symmetry can be gauged? To give an example, the free Schrödinger field theory is given by the Lagrangian
$$
\mathscr L=\psi^\dagger\biggl(i\partial_t+\frac{\nabla^2}{2m}\biggr)\psi.
$$
Apart from the usual $U(1)$ phase transformation, the action is also invariant under independent shifts, $\psi\to\psi+\theta_1+i\theta_2$, of the real and imaginary parts of $\psi$. It seems that such shift symmetry cannot be gauged, although I cannot prove this claim (correct me if I am wrong). This seems to be related to the fact that the Lie algebra of the symmetry necessarily contains a central charge.
So my questions are: (i) how is the (im)possibility to gauge a global symmetry related to central charges in its Lie algebra?; (ii) can one formulate the criterion for "gaugability" directly in terms of the Lagrangian, without referring to the canonical structure such as Poisson brackets of the generators? (I have in mind Lagrangians with higher field derivatives.)
N.B.: By gauging the symmetry I mean adding a background, not dynamical, gauge field which makes the action gauge invariant.
Best Answer
The guiding principle is: "Anomalous symmetries cannot be gauged". The phenomenon of anomalies is not confined to quantum field theories. Anomalies exist also in classical field theories (I tried to emphasize this point in my answer on this question).
(As already mentioned in the question), in the classical level, a symmetry is anomalous when the Lie algebra of its realization in terms of the fields and their conjugate momenta (i.e., in term of the Poisson algebra of the Lagrangian field theory) develops an extension with respect to its action on the fields. This is exactly the case of the complex field shift on the Schrödinger Lagrangian.
In Galilean (classical) field theories, the existence of anomalies is accompanied by the generation of a total derivative increment to the Lagrangian, which is again manifested in the case of the field shift symmetry of the Schrödinger Lagrangian, but this is not a general requirement. (please see again my answer above referring to the generation of mass as a central extension in Galilean mechanics).
In a more modern terminology, the impossibility of gauging is termed as an obstruction to equivariant extensions of the given Lagrangians. A nontrivial family of classical Lagrangians, exhibiting nontrivial obstructions, are Lagrangians containing Wess-Zumino-Witten terms. Given these terms only anomaly free subgroups of the symmetry groups can be gauged (classically). These subgroups consist exactly of the anomaly free ones. The gauging and the obstruction to it can be obtained using the theory of equivariant cohomology, please see the following article by Compean and Paniagua and its reference list.