Dirac operator on the q-deformed fuzzy sphere and its spectrum

Abstract

The q-deformed fuzzy sphere SqF2(N) is the algebra of (N+1) × (N+1) dim. matrices, covariant with respect to the adjoint action of Uq(su(2)) and in the limit q→1, it reduces to the fuzzy sphere SF2(N). We construct the Dirac operator on the q-deformed fuzzy sphere-SqF2(N) using the spinor modules of Uq(su(2)). We explicitly obtain the zero modes and also calculate the spectrum for this Dirac operator. Using this Dirac operator, we construct the Uq(su(2)) invariant action for the spinor fields on S qF2(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. © SISSA 2006.