Quantum Topological Invariants, Gravitational Instantons and the Topological Embedding.pdf

a r X i v : h e p - t h / 9 6 0 7 2 0 6 v 1 2 7 J u l 1 9 9 6 HUTP-96/A029 hepth/9607206 July, 1996 QUANTUM TOPOLOGICAL INVARIANTS, GRAVITATIONAL INSTANTONS AND THE TOPOLOGICAL EMBEDDING Damiano Anselmi Lyman Laboratory, Harvard University, Cambridge MA 02138, U.S.A. Abstract 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, that 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 instan- tons, 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 ampli- tude. 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. 1 1 Introduction and motivation Given a manifold or, in general, a field configuration, one can define topological quantities like the Pontrjiagin number and the Euler number. In quantum field theory, one mainly deals with spaces of field configurations, rather than single field configurations. Consequently, it can be interesting to study topological invariants of such spaces. These invariants were called quantum in ref. [1], since they involve an integration over the chosen configuration space. The usual topological invariants were called classical. The quantum topological invariants are defined in a way that is originally suggested by topological field theory, if treated with the approach of ref. [2], but that actually live quite independently. No notion of functional integral is strictly necessary, so t