Gravity as a four dimensional algebraic quantum field theory

Based on a family of indefinite unitary representations of the diffeomorphism group of an oriented smooth $4$-manifold, a manifestly covariant $4$ dimensional and non-perturbative algebraic quantum field theory formulation of gravity is exhibited. More precisely among the bounded linear operators acting on these representation spaces we identify algebraic curvature tensors hence a net of local quantum observables can be constructed from $C^*$-algebras generated by local curvature tensors and vector fields. This algebraic quantum field theory is extracted from structures provided by an oriented smooth $4$-manifold only hence possesses a diffeomorphism symmetry. In this way classical general relativity exactly in $4$ dimensions naturally embeds into a quantum framework. Several Hilbert space representations of the theory are found. First a "tautological representation" of the limiting global $C^*$-algebra is constructed allowing to associate to any oriented smooth $4$-manifold a von Neumann algebra in a canonical fashion. Secondly, influenced by the Dougan--Mason approach to gravitational quasilocal energy-momentum, we construct certain representations what we call "positive mass representations" with unbroken diffeomorphism symmetry. Thirdly, we also obtain "classical representaions" with spontaneously broken diffeomorphism symmetry corresponding to the classical limit of the theory which turns out to be general relativity. Finally we observe that the whole family of "positive mass representations" comprise a $2$ dimensional conformal field theory in the sense of G. Segal.