  We analyze zero mode counting problems for Dirac operators that find their
origin in string theory backgrounds. A first class of quantum mechanical models
for which we compute the number of ground states arises from a string winding
an isometric direction in a geometry, taking into account its energy due to
tension. Alternatively, the models arise from deforming marginal bound states
of a string winding a circle, and moving in an orthogonal geometry. After
deformation, the number of bound states is again counted by the zero modes of a
Dirac operator. We count these bound states in even dimensional asymptotically
linear dilaton backgrounds as well as in Euclidean Taub-NUT. We show multiple
pole behavior in the fugacities keeping track of a U(1) charge. We also discuss
a second class of counting problems that arises when these backgrounds are
deformed via the application of a heterotic duality transformation. We discuss
applications of our results to Appell-Lerch sums and the counting of domain
wall bound states.
