An operator algebraic characterization of the Riemannian vacuum Einstein equation in four dimensions

In this paper, using connected compact oriented smooth 4-manifolds, some representations of the hyperfinite II_1-type factor von Neumann algebra are constructed. The Murray--von Neumann coupling constant of these representations gives rise to a new smooth 4-manifold invariant whose very first properties are investigated.
Moreover as a part of this construction, a connected compact oriented smooth 4-manifold admits an embedding into the hyperfinite II_1 factor. This embedding, on the one hand, induces a Riemannian metric on the manifold such that its Riemannian curvature tensor belongs to the von Neumann algebra; on the other hand the metric induces a periodic dynamics on the von Neumann algebra, what we call the Hodge dynamics on the hyperfinite II_1 factor. It is observed that the metric is Einstein i.e., satisfies the (Riemannian) vacuum Einstein equation with a possibly non-zero cosmological constant, if and only if its Riemannian curvature tensor belongs to the fixed-point-subalgebra of the Hodge dynamics.
Finally, we make a comprehensive enumeration of all representations of the hyperfinite II_1 factor constructed here, from the viewpoint of thermal equilibrium states and phase transitions in algebraic quantum field theory.