A Geometric Approach to Noncommutative Principal Torus Bundles
Stefan Wagner
TL;DR
The paper develops a geometrically grounded framework for noncommutative principal torus bundles through smooth dynamical systems and a novel smooth localization toolkit. A dynamical system $(A,\mathbb{T}^n,\alpha)$ yields a noncommutative principal bundle when its localizations around central invariant elements become trivial, thereby extending classical bundle theory to the noncommutative setting. It provides spectral and topological analyses of localized algebras, shows how sections of algebra bundles furnish natural NC bundles, and constructs concrete NC examples including those built from sections, pull-backs, and quantum tori. The work also sketches a path toward classification via noncommutative Čech cohomology and discusses connections to Hopf--Galois theory, signaling a broad program for understanding and organizing NC principal torus bundles.
Abstract
A (smooth) dynamical system with transformation group $\mathbb{T}^n$ is a triple $(A,\mathbb{T}^n,α)$, consisting of a unital locally convex algebra $A$, the $n$-torus $\mathbb{T}^n$ and a group homomorphism $α:\mathbb{T}^n\rightarrow\Aut(A)$, 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 $(A,\mathbb{T}^n,α)$ 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.