  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.

High Energy Physics - Theory