Dirac Operator on the Quantum Sphere

TL;DR

This paper constructs a Dirac operator on the quantum sphere $S^2_q$ that is covariant under $SU_q(2)$, connecting noncommutative geometry with quantum group symmetries and potential field theory applications.

Contribution

It introduces a new Dirac operator on the quantum sphere $S^2_q$ that generalizes Watamuras' operator and maintains covariance under $SU_q(2)$ actions.

Findings

Dirac operator on $S^2_q$ covariant under $SU_q(2)

Reduces to Watamuras' operator as $q\to 1$

Potential for $SU_q(2)$ invariant field theories

Abstract

We construct a Dirac operator on the quantum sphere $S^2_q$ which is covariant under the action of $SU_q(2)$. It reduces to Watamuras' Dirac operator on the fuzzy sphere when $q\to 1$. We argue that our Dirac operator may be useful in constructing $SU_q(2)$ invariant field theories on $S^2_q$ following the Connes-Lott approach to noncommutative geometry.