Coupling Higher Form Structures of the EFT of Force Free Electrodynamics to Gravity

We know that the charged Reissner Nordstr$\ddot{o}$m black hole metric is obtained from the Einstein Hilbert gravitational action. This action has the kinetic term $F^2=(da{)}^2$. Motivated by the higher-form symmetry structure of the EFT of Force Free Electrodynamics, we replace the Maxwell field-strength contribution in the Einstein Hilbert action by the gauge-invariant combination $(b-da{)}^2$, where $a_{\mu }$ is the worldsheet gauge field and $b_{\mu \nu }$ is a background two-form field. This ensures that the new action has a higher form symmetry $b\to b+d\mathrm{\Lambda },a\to a+\mathrm{\Lambda }$. Here, unlike in qed, $\mathrm{\Lambda }$ may be any one form (not necessarily a differential one form ${\mathrm{\partial }}_{\mu }\phi$). The higher form symmetry here is one with the conserved current being a two form and the charge integrated on surfaces. Intuitively, it is the number/current of vector field lines that is conserved here, not the current of particles. Thus, integrating over a surface through which the field lines pierce is sufficient to find the number of these lines that are passing through; so the charge is integrated on surfaces, rather than on the volume. After fixing a particular gauge for the fields $a_{\mu }$ and $b_{\mu \nu }$, we obtain a generalized black-hole metric. We find that on hypersurfaces satisfying $(r-t)=$ constant, this metric reduces locally to a Reissner Nordstr$\ddot{o}$m geometry with an effective charge parameter depending on the constant $(r-t)$.