Ground states of duality-twisted sigma-models with K3 target space

We analyze the ground states of a two-dimensional sigma-model whose target space is an elliptically fibered K-3, with the sigma-model compactified on S-1 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.