Spectrum of Spin 1 Dirac Operators on the Fuzzy 2-Sphere

TL;DR

This paper investigates the spectrum of spin 1 Dirac operators on the fuzzy 2-sphere, providing numerical and analytical insights into their properties, inequivalence, zero modes, and eigenvalue relations, with implications for higher spins.

Contribution

It introduces a method to analyze the spectrum of spin 1 Dirac operators on the fuzzy sphere and extends zero mode counting to higher spins, revealing universal eigenvalue relations.

Findings

Operators are unitarily inequivalent.

Spectrum resembles fermionic positive and negative eigenvalues with zero modes.

Universal relation between eigenvalues and multiplicities.

Abstract

We numerically find out the spectrum of the $3$ spin $1$ Dirac operators found in~\cite{ApbPP}. We give an analytic and numerical proof that they are unitarily inequivalent. Since these operators come paired with an anticommuting chirality operator, we find their spectrums to resemble those of fermions with positive and negative eigenvalues along with a number of zero modes. We give a method to count the number of zero modes which can be extended to higher spins on $S_F^2$. An universal relation between the energy eigenvalues of the spin 1 Dirac operator and their multiplicities is found. This helps us predict the energy eigenvalues for an arbitrarily large cut-off $L$, a problem which is computationally difficult to handle.