Reconstruction and quantization of Riemannian structures

TL;DR

This paper explores reconstructing Riemannian structures from codifferential operators, introduces a new perspective on metrics as cocycle data in differential graded algebras, and extends classical and quantum covariant derivatives.

Contribution

It introduces a novel framework linking metrics and connections to cocycle data in DGAs, and constructs covariant derivatives via quantum analogues of the Koszul formula.

Findings

Reconstruction of Levi-Civita connection from codifferential

Extension of the Hodge Laplacian to metrics

Construction of noncommutative time in DGAs

Abstract

We study how the Riemannian structure on a manifold can be usefully reconstructed from its codifferential $\delta$, including a formula $\nabla_\omega\eta={1\over 2}( \delta(\omega\eta)-(\delta\omega)\eta+\omega(\delta\eta) +L_\omega(\eta)+i_\eta d \omega)$ 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, $\delta$ makes the exterior algebra into a BV algebra. In the invertible case we show that ${\rm Ricci}=-{1\over 2}\Delta g$ where the Hodge Laplacian $\Delta$ 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 $\Omega(M)$ 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 $\Omega(t,d t)$ in one variable, to introduce a noncommutative `time'. Composing these two constructions recovers a previous differential quantisation of $M\times R$.