Dirac type operators on the quantum solid torus with global boundary conditions

TL;DR

This paper introduces a noncommutative space called the quantum solid torus, studies quantum Dirac operators with boundary conditions, and proves their ellipticity through compact inverses, extending classical boundary value problem theory.

Contribution

It defines a new noncommutative manifold with boundary and analyzes Dirac operators with boundary conditions, establishing their elliptic properties.

Findings

Quantum solid torus as a noncommutative manifold with boundary

Existence of compact inverse for quantum Dirac operators

Ellipticity of boundary value problems in noncommutative setting

Abstract

We define a noncommutative space we call the quantum solid torus. It is an example of a noncommutative manifold with a noncommutative boundary. We study quantum Dirac type operators subject to Atiyah-Patodi-Singer like boundary conditions on the quantum solid torus. We show that such operators have compact inverse, which means that the corresponding boundary value problem is elliptic.