Star Product Geometries

TL;DR

This paper explores noncommutative geometries derived from Drinfeld twists, focusing on noncommutative gravity and quantization, introducing new structural equations and algebraic formulations within this framework.

Contribution

It introduces new structural equations for noncommutative gravity and advances the understanding of scalar field theories using twisted algebraic structures.

Findings

Derived Cartan structural equations for noncommutative torsion and curvature.

Established a noncommutative correspondence principle for quantization.

Analyzed the twisted algebra of classical and quantum observables.

Abstract

We consider noncommutative geometries obtained from a triangular Drinfeld twist. This allows to construct and study a wide class of noncommutative manifolds and their deformed Lie algebras of infinitesimal diffeomorphisms. This way symmetry principles can be implemented. We review two main examples [15]-[18]: a) general covariance in noncommutative spacetime. This leads to a noncommutative gravity theory. b) Symplectomorphims of the algebra of observables associated to a noncommutative configuration space. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e., we establish a noncommutative correspondence principle from *-Poisson brackets to *-commutators. New results concerning noncommutative gravity include the Cartan structural equations for the torsion and curvature tensors, and the associated Bianchi identities. Concerning scalar field theories the deformed algebra of classical and quantum observables has been understood in terms of a twist within the algebra.