Dirac operator on the q-deformed Fuzzy sphere and Its spectrum

TL;DR

This paper constructs a Dirac operator on the q-deformed fuzzy sphere, analyzes its spectrum, zero modes, and degeneracies, and explores limits to connect with fuzzy and classical spheres.

Contribution

It introduces a new Dirac operator on the q-deformed fuzzy sphere using $ ext{U}_q( ext{su}_2)$ spinor modules and analyzes its spectral properties.

Findings

Explicit spectrum and zero modes obtained

Novel degeneracy patterns identified for different q values

Connections established with fuzzy and classical spheres in various limits

Abstract

The q-deformed fuzzy sphere $S_{qF}^2(N)$ is the algebra of $(N+1)\times(N+1)$ dim. matrices, covariant with respect to the adjoint action of $\uq$ and in the limit $q\to 1$, it reduces to the fuzzy sphere $S_{F}^2(N)$. We construct the Dirac operator on the q-deformed fuzzy sphere-$S_{qF}^{2}(N)$ using the spinor modules of $\uq$. We explicitly obtain the zero modes and also calculate the spectrum for this Dirac operator. Using this Dirac operator, we construct the $\uq$ invariant action for the spinor fields on $S_{qF}^{2}(N)$ which are regularised and have only finite modes. We analyse the spectrum for both $q$ being root of unity and real, showing interesting features like its novel degeneracy. We also study various limits of the parameter space (q, N) and recover the known spectrum in both fuzzy and commutative sphere.