Heat kernel methods in perturbative quantum gravity

The heat kernel method is a powerful technique in mathematical physics, with applications ranging from black hole entropy to mathematical finance. It consists in a variety of perturbation methods applied to elliptic second-order differential operators on manifolds, which allow to study asymptotic expansions and singularities of Green functions. Perturbative quantum gravity is grounded on the background field method, where the metric tensor is split into a fixed background and quantum perturbations. By moving to euclidean time, the kinetic operators of BRST-quantised gravity become elliptic operators of second-order in partial derivatives, which can be studied via heat kernel techniques. The heat kernel coefficients obtained in this expansion correspond to the counterterms needed to renormalise the one-loop effective action; once computed on-shell, i.e. by using Einstein equations, they become gauge invariant. Up to now, only the first three coefficients for perturbative quantum gravity were known, so the main goal of this thesis is to compute the fourth one, which allows to study renormalisation theory for D=6 gravity at one-loop, behaving similarly to D=4 gravity at two-loops. Both theories are known to be non-renormalisable, and our calculation shows the precise coefficient of the one-loop term that gives the logarithmic divergences in D=6 extended to arbitrary dimensions (for D>6 the divergences are not anymore logarithmic). Our result is in accordance with the one-loop calculation performed independently through the N=4 spinning particle in the worldline formalism. The computations are then extended to the case of matter fields coupled to the graviton, in the vacuum approximation; a suitable extension of these results to the general matter case might have a role in evaluating quantum corrections to the entropy of Kerr-Newmann black holes.