Quantum topological invariants, gravitational instantons and the topological embedding

Certain topological invariants of the moduli space of gravitational instantons are defined and studied. Several amplitudes of two-and four-dimensional topological gravity are computed. A notion of puncture in four dimensions, which is particularly meaningful in the class of Weyl instantons, is introduced. The topological embedding, a theoretical framework for constructing physical amplitudes that are well defined order-by-order in perturbation theory around instantons, is explicitly applied to the computation of the correlation functions of Dirac fermions in a punctured gravitational background, as well as to the most general QED and QCD amplitude. Various alternatives are worked out, discussed and compared. The quantum background affects the propagation by generating a certain effective 'quantum' metric. The topological embedding could represent a new chapter of quantum field theory