A Note on Dirac Operators on the Quantum Punctured Disk

TL;DR

This paper investigates quantum versions of Dirac operators on the punctured disk, constructing a parametrix and analyzing boundary conditions to understand their boundedness properties.

Contribution

It introduces a quantum analog of the Dirac operator with Atiyah-Patodi-Singer boundary conditions and constructs a parametrix for it.

Findings

Constructed a parametrix for the quantum Dirac operator

Showed the operator is bounded outside the zero mode

Extended classical boundary condition analysis to the quantum setting

Abstract

We study quantum analogs of the Dirac type operator $-2\bar{z}\frac{\partial}{\partial\bar{z}}$ on the punctured disk, subject to the Atiyah-Patodi-Singer boundary conditions. We construct a parametrix of the quantum operator and show that it is bounded outside of the zero mode.