  We analyze the ground states of a two-dimensional sigma-model whose target
space is an elliptically fibered K3, with the sigma-model compactified on a
circle with boundary conditions twisted by a duality symmetry. We show that the
Witten index receives contributions from two kinds of states: (i) those that
can be mapped to cohomology with coefficients in a certain line bundle over the
target space, and (ii) states whose wave-functions are localized at singular
fibers. We also discuss the orbifold limit and possible connections with
geometric quantization of the target space.
