Twisted submanifolds of R^n

Abstract

We propose a general procedure to construct noncommutative deformations of an embedded submanifold $M$ of ${\mathbb{R}}^n$ determined by a set of smooth equations $f^a(x)=0$. We use the framework of Drinfel'd twist deformation of differential geometry of [Aschieri et al., Class. Quantum Gravity 23 (2006), 1883]; the commutative pointwise product is replaced by a (generally noncommutative) $\star$-product determined by a Drinfel'd twist. The twists we employ are based on the Lie algebra ${\Xi }_t$ of vector fields that are tangent to all the submanifolds that are level sets of the $f^a$; the twisted Cartan calculus is automatically equivariant under twisted tangent infinitesimal diffeomorphisms. We can consistently project a connection from the twisted ${\mathbb{R}}^n$ to the twisted $M$ if the twist is based on a suitable Lie subalgebra $\mathfrak{e}\subset {\Xi }_t$. If we endow ${\mathbb{R}}^n$ with a metric then twisting and projecting to the normal and tangent vector fields commute, and we can project the Levi-Civita connection consistently to the twisted $M$, provided the twist is based on the Lie subalgebra $\mathfrak{k}\subset \mathfrak{e}$ of the Killing vector fields of the metric; a twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can be characterized in terms of generators and polynomial relations. We present in some detail twisted cylinders embedded in twisted Euclidean ${\mathbb{R}}^3$ and twisted hyperboloids embedded in twisted Minkowski ${\mathbb{R}}^3$ [these are twisted (anti-)de Sitter spaces $d{S}_2,Ad{S}_2$].

arXiv commentLatex file, 41 pages, 1 figure with 4 images. This new version has been shortened avoiding material that can be found elsewhere. To appear in Lett. Math. Phys

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• On the Notion of Noncommutative Submanifold
• Twisted Quadrics and Algebraic Submanifolds in ${\mathbb{R}}^n$
• Energy cutoff, effective theories, noncommutativity, fuzzyness: the case of O(D)-covariant fuzzy spheres