Quantum Principal Fiber Bundles: Topological Aspects

TL;DR

This paper develops a framework for quantum principal fiber bundles, establishing conditions for triviality, a reconstruction theorem, and classification methods, extending classical topological concepts into the quantum realm.

Contribution

It introduces locally trivial quantum principal bundles, proves their triviality criterion, and provides a reconstruction theorem using quantum ch cocycles, advancing quantum topology.

Findings

Quantum bundles admit sections iff they are trivial.

A reconstruction theorem for quantum principal bundles is established.

Classification reduces to classical group problems over quantum spaces.

Abstract

We introduce the notion of locally trivial quantum principal bundles. The base space and total space are compact quantum spaces (unital $C^{\star}$-algebras), the structure group is a compact matrix quantum group. We prove that a quantum bundle admits sections if and only if it is trivial. Using a quantum version of \v{C}ech cocycles, we obtain a reconstruction theorem for quantum principal bundles. The classification of bundles over a given quantum space as a base space is reduced to the corresponding problem, but with an ordinary classical group playing the role of structure group. Some explicit examples are considered.