Extending general covariance: Moyal-type noncommutative manifolds

TL;DR

This paper explores how noncommutative geometry modifies the symmetry structures in general relativity, distinguishing between twisted and deformed diffeomorphisms and their implications for noncommutative manifolds.

Contribution

It introduces a framework to compare twisted and deformed diffeomorphisms in noncommutative gravity, highlighting the relevance of deformed symmetries for noncommutative manifolds.

Findings

Deformed diffeomorphisms modify the hypersurface-deformation algebra.

Twisted diffeomorphisms retain the classical algebra structure.

Non-locality effects are evident in the modified brackets.

Abstract

In the Hamiltonian formulation of general relativity, Einstein's equation is replaced by a set of four constraints. Classically, the constraints can be identified with the generators of the hypersurface-deformation Lie algebroid (HDA) that belongs to the groupoid of finite evolutions in space-time. Taken over to deformed general relativity, this connection allows one to study possible Drinfeld twists of space-time diffeomorphisms with Hopf-algebra techniques. After a review of noncommutative differential structures, two cases --- twisted diffeomorphisms with standard action and deformed (or $\star$-) diffeomorphisms with deformed action --- are considered in this paper. The HDA of twisted diffeomorphisms agrees with the classical one, while the HDA obtained from deformed diffeomorphisms is modified due to the explicit presence of $\star$-products in the brackets. The results allow one to distinguish between twisted and deformed symmetries, and they indicate that the latter should be regarded as the relevant symmetry transformations for noncommutative manifolds. The algebroid brackets maintain the same general structure regardless of space-time noncommutativity, but they still show important consequences of non-locality.