  It is known that, for Dirac operators on Riemann surfaces twisted by line
bundles with Hermitian-Einstein connections, it is possible to obtain estimates
for the first eigenvalue in terms of the topology of the twisting bundle
\cite{JL2}. Attempts to generalize topological estimates for higher rank
bundles or higher dimensional manifolds have been so far unsuccessful. In this
work we construct a class of examples which indicates one problem that arises
on such attempts to derive topological estimates.
