Quantum Groups on Fibre Bundles

TL;DR

This paper develops a sheaf-theoretic framework connecting locality and noncommutative geometry, introducing quantum spaces and extending classical fibre bundle concepts to the quantum setting, including $q$-deformed instanton models.

Contribution

It introduces a sheaf-theoretic approach to quantum spaces, extending principal and vector bundles to the noncommutative setting with new quantum geometric constructions.

Findings

Quantum spaces are formally defined within the sheaf framework.

Classical bundle constructions are successfully extended to quantum bundles.

$q$-deformed instanton models are constructed for all integer indices.

Abstract

It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of quantum spaces noncommutative principal and vector bundles are defined and their properties are studied. Important constructions in the classical theory of principal fibre bundles like associated bundles and differential calculi are carried over to the quantum case. At the end $q$-deformed instanton models are introduced for every integral index.