Renormalization group studies of quantum theories of gravity and matter

In this thesis, we study quantum theories of gravity and matter in the Renormalization Group approach. We observe light fermions in our universe which feature a remnant of chiral symmetry. While chiral symmetry appears to remain intact along the Renormalization Group flow of asymptotically safe approaches to quantum gravity, these computations are performed on a flat background. Mean field studies performed on negatively curved backgrounds however indicate chiral symmetry breaking in the form of gravitational catalysis. The study of the mean field RG flow on negatively curved spacetime leads to an upper bound for the ratio of curvature of local patches of spacetime to the RG scale. We extend these calculations to finite temperature and study how thermal fluctuations affect this bound from gravitational catalysis. Applying this thermal extension of the curvature bound to the asymptotic safety scenario of quantum gravity, it translates into an upper bound of numbers of fermion species allowed in our universe. Most approaches towards a UV complete quantum theory of gravity start with General Relativity as its classical theory. Einstein’s formulation of gravity can be summarized as pseudo-Riemannian geometry on a manifold equipped with a metric and a connection. While this connection is restricted to the Levi-Cevita connection in Einstein gravity and therefore is fully determined by the metric, there is a priori no fundamental reason to not use a general connection. This formalism, in which the metric and the general connection are treated as independent degrees of freedom, is referred to as Hilbert-Palatini gravity. In this thesis, we compute the most general solution to the connection for the Einstein-Hilbert-Palatini action and use it in an on-shell reduction scheme to compute the RG flow for the subsequent order in the truncation. We find a UV- attractive fixed point similar to the Reuter fixed point in quantum Einstein gravity.