Relating electrodynamics and gravity in two Euclidean dimensions

Two-dimensional electrodynamics coupled to Dirac fermions is mapped onto two-dimensional gravity in the first-order formalism, also including fermions. However, the resulting fermion-gravity coupling deviates from the conventional form, explicitly violating parity and time-reversal symmetries. Additionally, these fermions exhibit an unconventional transformation behavior under $SO(2)$ transformations. Furthermore, we analyze the consistency of this mapping at the quantum level using the path integral formalism and Becchi-Rouet-Stora-Tyutin techniques. Our findings demonstrate that quantum electrodynamics and quantum gravity remain equivalent at the quantum level.