Reconstruction and quantization of Riemannian structures

Abstract

We study how the Riemannian structure on a manifold can be usefully reconstructed from its codifferential $δ$, including a formula $\nabla_ωη={1\over 2}( δ(ωη)-(δω)η+ω(δη) +L_ω(η)+i_ηd ω)$ for the Levi-Civita covariant derivative in terms of 1-forms, where $L, i$ are respectively the Lie derivative and interior product along the corresponding vector fields. The covariant derivative extends naturally along forms of any degree and to possibly degenerate $(\ ,\ )$. In the nondegenerate case, $δ$ makes the exterior algebra into a BV algebra. In the invertible case we show that ${\rm Ricci}=-{1\over 2}Δg$ where the Hodge Laplacian $Δ$ extends in a natural way to act on the metric. Our results come from a new way of thinking about metrics and connections as a kind of cocycle data for central extensions of differential graded algebras (DGAs), a theory which we introduce. We show that any cleft extension of the exterior algebra $Ω(M)$ on a manifold is associated to a possibly-degenerate metric and covariant derivative. Those for which $d$ is not deformed up to isomorphism correspond to the Levi-Civita case. We provide a construction for such extensions both of classical DGAs and of already non-graded-commutative DGAs, thereby constructing a class of bimodule covariant derivatives via a kind of quantum analogue of the Koszul formula. We also provide a semidirect product of any differential graded algebra by the quantum differential algebra $Ω(t,d t)$ in one variable, to introduce a noncommutative `time'. Composing these two constructions recovers a previous differential quantisation of $M\times R$.