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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?

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  • $\begingroup$ If it helps, the conserved current associated to diffeomorphism invariance is the Einstein tensor : web.mit.edu/edbert/GR/gr5.pdf (chap. 4.2) $\endgroup$
    – Slereah
    Apr 7 '20 at 23:44

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