Field Parametrization Dependence in Asymptotically Safe Quantum Gravity

Motivated by conformal field theory studies we investigate Quantum Einstein Gravity with a new field parametrization where the dynamical metric is basically given by the exponential of a matrix-valued fluctuating field, $g_{\mu \nu }={\bar{g}}_{\mu \rho }(e^h{)}_{\nu }^{\rho }$. In this way, we aim to reproduce the critical value of the central charge when considering $2+ϵ$ dimensional spacetimes. With regard to the Asymptotic Safety program, we take special care of possible fixed points and new structures of the corresponding RG flow in $d=4$ for both single- and bi-metric truncations. Finally, we discuss the issue of restoring background independence in the bi-metric setting.