Twisted submanifolds of $${\mathbb {R}}^n$$

Abstract

We propose a general procedure to construct noncommutative deformations of an embedded submanifold M of $${\mathbb {R}}^n$$ R n determined by a set of smooth equations $$f^a(x)=0$$ f a ( x ) = 0 . We use the framework of Drinfel’d twist deformation of differential geometry of Aschieri et al. (Class Quantum Gravity 23:1883, 2006); 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 $$\Xi _t$$ Ξ t of vector fields that are tangent to all the submanifolds that are level sets of the $$f^a$$ f a (tangent infinitesimal diffeomorphisms); the twisted Cartan calculus is automatically equivariant under twisted $$\Xi _t$$ Ξ t . We can consistently project a connection from the twisted $${\mathbb {R}}^n$$ R n to the twisted M if the twist is based on a suitable Lie subalgebra $${\mathfrak {e}}\subset \Xi _t$$ e ⊂ Ξ t . If we endow $${\mathbb {R}}^n$$ 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 $${\mathfrak {k}}\subset {\mathfrak {e}}$$ k ⊂ 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 $$\star $$ ⋆ -polynomial relations. We present in some detail twisted cylinders embedded in twisted Euclidean $${\mathbb {R}}^3$$ R 3 and twisted hyperboloids embedded in twisted Minkowski $${\mathbb {R}}^3$$ R 3 [these are twisted (anti-)de Sitter spaces $$dS_2,AdS_2$$ d S 2 , A d S 2 ].

Twisted geometry for submanifolds of $\mathbb{R}^n$

“…A unitary or real twist F on induces a twisted commutative left F -module * -algebra X ★ and a twisted symmetric F -equivariant X ★ - * -bimodule T ★ according to the previous section. In more detail, T ★ = , ∈N 0 T , ★ is defined by (20) and the F -action is given by (19), where we replace Δ by Δ F . Above, Ω ★ denotes the X ★bimodule of twisted 1-forms, i.e.…”

“…So far these questions have been answered by making sense of many special examples of noncommutative (NC) submanifolds , but have not received sufficient general treatment, except in few articles (see e.g. [18,19,21,27,30]). This proceeding summarizes the contributions to the topic of Ref.…”

“…This proceeding summarizes the contributions to the topic of Ref. [18,19], which address the above questions systematically within the framework of deformation quantization [6], in the particular approach based on Drinfel'd twisting [12] of Hopf algebras, for embedded submanifolds of R consisting of points of fulfilling a set of equations…”

“…Here ≡ ( 1 , ..., ) : R → R are smooth functions such that the Jacobian matrix = / is of rank on all R ; or, more generally, where is well-defined and is of rank on an open subset D ⊂ R , and consists of the points of D fulfilling (1). In fact, in [18,19] one…”

“…In words, we induce a quantization of a submanifold via a quantization of the manifold R , given the commutativity of (3). As said, in [18,19] we are interested in the situation when is a submanifold given in terms of generators ( 1 , . .…”