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

Abstract

The charged Reissner Nordstr$\"{o}$m black hole metric is obtained from the Einstein Hilbert 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 in the EH action by the gauge-invariant combination $(b-da)^2$, where $a_\mu$ is the gauge field and $b_{\mu\nu}$ is a background two-form field which is closed, that is, $db =0$. This ensures that the new action has a higher form symmetry $b \rightarrow b+d\Lambda, a\rightarrow a+\Lambda$. Here, unlike in qed, $\Lambda$ may be any one form (not necessarily a differential one form $\partial_\mu \phi$). The higher form symmetry here is one with the conserved current being a two form and the charge integrated on surfaces. It is the 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. By fixing a gauge for $a_\mu$ and $b_{\mu\nu}$, we obtain the black-hole metric dual to the boundary theory of FFE. There are two different starting points: 1)Consider the $b$ to be exact. The bulk dual to this theory is the RN geometry. 2)Consider the $b$ to have a Dirac Like Monopole and thus to not be exact ($b \neq dc$ for any $c$). This again does not lead to a new bulk dual, it leads us to the Dyonic RN geometry (which arises in the E-M theory when $a$ has the monopole term). Thus, Einstein-FFE leads to the known RN and dyonic solutions of the E-M theory. We note that AdS-CFT applied to this problem states that since the boundary EFT of FFE describes the same theory of a sector of strongly magnetised QED plasma, bulk E-FFE reproduces the same solutions of E-M theory.