  Fu and Yau constructed the first smooth family of gauge bundles over a class
of non-Kahler, complex 3-folds that are solutions to Strominger's system, the
heterotic supersymmetry constraints with nonzero H-flux. In this paper, we
begin the study of the massless spectrum arising from compactification using
this construction by counting zero modes of the linearized equations of motion
for the gaugino in the supergravity approximation. We rephrase the question in
terms of a cohomology problem and show that for a trivial gauge bundle, this
cohomology reduces to the Dolbeault cohomology of the 3-fold, which we then
compute.
