BRST-invariant FRG flow in quantum Einstein gravity

The thesis concerns a non-standard method of implementing the Functional Renormalization Group (FRG) framework in a quantum field theory of gravity which is capable of explicitly preserving the Becchi-Rouet-Stora-Tyutin (BRST) symmetry of the theory at all stages of the construction. In particular, we firstly review the functional Faddeev-Popov quantization and gauge-fixing of a classical field theory of gravity within the background field method and the role and properties of BRST symmetry. Secondly, we present the standard FRG framework implementation and we discuss how the standard regularization procedure, consisting in manually adding a quadratic regulator term in the metric fluctuation and Faddeev-Popov ghosts to the gauge-fixed action, breaks BRST symmetry and leads to a Wetterich-Morris equation incompatible with the constraint imposed by BRST symmetry, i.e. the Zinn-Justin equation. Finally, we present how a BRST-invariant FRG framework can be introduced by combining the regularization and gauge-fixing procedures in a single step, using a non-standard gauge-fixing choice, which allows to introduce quadratic mass terms in the gauge-fixed action without breaking BRST symmetry, to regularize the theory in an explicitly BRST-invariant manner. Then, we derive the Wetterich-Morris equation stemming from the construction, proving its compatibility with the Zinn-Justin equation and presenting its component form within the Einstein-Hilbert truncation.