Twisted Quadrics and Algebraic Submanifolds in ${\mathbb{R}}^n$

Abstract

We propose a general procedure to construct noncommutative deformations of an algebraic submanifold $M$ of ${\mathbb{R}}^n$, specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of ${\mathbb{R}}^n$, arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883-1911, 2006), whereby the commutative pointwise product is replaced by the $\star$-product determined by a Drinfel'd twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds $M_c$ that are level sets of the $f^a(x)$, where $f^a(x)=0$ are the polynomial equations solved by the points of $M$, employing twists based on the Lie algebra ${\Xi }_t$ of vector fields that are tangent to all the $M_c$. The twisted Cartan calculus is automatically equivariant under twisted ${\Xi }_t$. If we endow ${\mathbb{R}}^n$ with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted $M$ is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and $\star$-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in ${\mathbb{R}}^3$ except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean ${\mathbb{R}}^3$ and twisted hyperboloids embedded in twisted Minkowski ${\mathbb{R}}^3$ [the latter are twisted (anti-)de Sitter spaces $d{S}_2$, $Ad{S}_2$].

arXiv commentLatex file, 50 pages, 2 figures. arXiv admin note: text overlap with arXiv:2003.03854

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Cited by in corpus (3)

• Twisted submanifolds of R^n
• Fuzzy hyperspheres via confining potentials and energy cutoffs
• BMS algebras in 4 and 3 dimensions, their quantum deformations and duals