Non-commutative Zariski geometries and their classical limit

TL;DR

This paper explores non-commutative Zariski geometries, demonstrating their realization via $C^*$-algebras and establishing a limit process that yields classical gauge fields on Riemann surfaces.

Contribution

It introduces a limit construction connecting non-commutative geometries to classical gauge theories, providing new insights into their structure and classical correspondence.

Findings

Non-commutative Zariski geometries can be represented by $C^*$-algebras.

A natural limit process produces classical U(1)-gauge fields from non-commutative geometries.

The approach bridges non-commutative and classical geometric frameworks.

Abstract

We undertake a case study of two series of nonclassical Zariski geometries. We show that these geometries can be realised as representations of certain noncommutative $C^*$-algebras and introduce a natural limit construction which for each of the two series produces a classical U(1)-gauge field over a 2-dimensional Riemann surface.