Line Bundles over Quantum Tori

TL;DR

This paper develops a cohomological framework for defining and classifying line bundles over Quantum Tori, drawing analogies with complex tori and exploring their topological and model-theoretic properties.

Contribution

It introduces a novel cohomological approach to line bundles over Quantum Tori and establishes a structure theorem analogous to the Appel-Humbert Theorem for complex tori.

Findings

Established a cohomological classification of line bundles over Quantum Tori

Defined the Chern class and linked it to an alternating pairing

Connected the theory to model-theoretic studies of Quantum Tori

Abstract

We study the problem of defining line bundles over certain non-Hausdorff spaces known as Quantum Tori, motivated by the proposed theory of Real Multiplication for real quadratic fields. We draw analogies from the theory of Line Bundles over Complex Tori to define a non-trivial cohomological notion of Line Bundles over Quantum Tori. We prove a structure theorem for isomorphism classes of such line bundles analogous to the Appel-Humbert Theorem for Complex Tori. In the second half of this text we consider Lines Bundles over Quantum Tori as topological spaces, and compare this notion with the cohomological definition. We define the Chern class of a line bundle and link this to an alternating pairing on a certain subgroup of Quantum Tori. Finally we investigate how our results are related to the study of Quantum Tori by Zilber using Model theoretic techniques.