There exists "folk-theorem" about the impossibility to have global symmetries in a consistent theory of quantum gravity. For example, see Global symmetries in quantum gravity .
Typical argument sounds like this:
Quantum gravity may break global symmetries because the global charge can be eaten by virtual black holes or wormholes.
But nonetheless, one can construct conserved quantities in pure gravity: $$ J^{\mu\nu\rho}=\varepsilon^{\mu\nu\rho\sigma}\partial_\sigma\sqrt{g} $$ $$ J^{\mu\nu\rho}=\varepsilon^{\mu\nu\rho\sigma}\partial_\sigma R $$
Or in gravity with matter: $$ J^{\mu\nu\rho}=\varepsilon^{\mu\nu\rho\sigma}\partial_\sigma\phi $$ $$ J^{\mu\nu} =\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma} $$ $$ J^{\mu\nu}=\varepsilon^{\mu\nu\rho\sigma}\partial_\rho\sqrt{g}\partial_\sigma\phi $$
Such currents trivially conserved, and doesn't act on fields, but acts on monopole-like operators.
Does this current correspond to global symmetry?
Are some applications of such currents?
