Reconstruction and quantization of Riemannian structures

Abstract

We use algebraic methods to obtain a Cartan-type formula ∇ωη=12(δ(ωη)−(δω)η+ωδη+ω ⊥ dη+dω ⊥ η+d(ω ⊥ η)) for the Levi−Civita connection on a classical Riemannian manifold M in the direction of a 1-form ω (i.e., the usual Levi−Civita connection along the corresponding vector field via the metric). Here, ⊥ denotes a degree −2 bidirectional interior product built from the metric and δ is the divergence or codifferential. We also recover that δ obeys a 7-term relation making the exterior algebra into a Batalin−Vilkovisky algebra. These formulas arise naturally from a novel view of Riemannian structures as cocycles governing the central extension of the classical exterior algebra to a quantum one, motivated by ideas for quantum gravity. The approach also works when the initial exterior algebra is already quantum, allowing us to construct examples of quantum Riemannian structures, including quantum Levi−Civita connections, as cocycle data. Combining with the semidirect product of a differential graded algebra by the quantum differential algebra Ω(t, dt) in one variable, we recover a differential quantization of M×R associated to any conformal Killing vector field on a Riemannian manifold M.