A Geometric Approach to Noncommutative Principal Torus Bundles

A (smooth) dynamical system with transformation group ${\mathbb{T}}^n$ is a triple $(A,{\mathbb{T}}^n,\alpha )$, consisting of a unital locally convex algebra $A$, the $n$-torus ${\mathbb{T}}^n$ and a group homomorphism $\alpha :{\mathbb{T}}^n\to \text{Aut}(A)$, which induces a (smooth) continuous action of ${\mathbb{T}}^n$ on $A$. In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal torus bundles based on such dynamical systems. Our approach is inspired by the classical setting: In fact, after recalling the definition of a trivial noncommutative principal torus bundle, we introduce a convenient (smooth) localization method for noncommutative algebras and say that a dynamical system $(A,{\mathbb{T}}^n,\alpha )$ is called a noncommutative principal ${\mathbb{T}}^n$-bundle, if localization leads to a trivial noncommutative principal ${\mathbb{T}}^n$-bundle. We prove that this approach extends the classical theory of principal torus bundles and present a bunch of (non-trivial) noncommutative examples.