  An explicit description of the spectral data of stable U(n) vector bundles on
elliptically fibered Calabi-Yau threefolds is given, extending previous work of
Friedman, Morgan and Witten. The characteristic classes are computed and it is
shown that part of the bundle cohomology vanishes. The stability and the
dimension of the moduli space of the U(n) bundles are discussed. As an
application, it is shown that the U(n) bundles are capable to solve the basic
topological constraints imposed by heterotic string theory. Various explicit
solutions of the Donaldson-Uhlenbeck-Yau equation are given. The heterotic
anomaly cancellation condition is analyzed; as a result an integral change in
the number of fiber wrapping five-branes is found. This gives a definite
prediction for the number of three-branes in a dual F-theory model. The
net-generation number is evaluated, showing more flexibility compared with the
SU(n) case.
