Index Theorem for the $q$-Deformed Fuzzy Sphere

Abstract

We calculate the index of the Dirac operator defined on the q-deformed fuzzy sphere. The index of the Dirac operator is related to its net chiral zero modes and thus to the trace of the chirality operator. We show that for the q-deformed fuzzy sphere, a $\uq$ invariant trace of the chirality operator gives the q-dimension of the eigenspace of the zero modes of the Dirac operator. We also show that this q-dimension is related to the topological index of the spinorial field. We then introduce a q-deformed chirality operator and show that its $\uq$ invariant trace gives the topological invariant index of the Dirac operator. We also explain the construction and important role of the trace operation which is invariant under the $\uq$, which is the symmetry algebra of the q-deformed fuzzy sphere. We briefly discuss chiral symmetry of the spinorial action on q-deformed fuzzy sphere and the possible role of this deformed chiral operator in its evaluation using path integral methods.