An explicit description of the spectral data of stable ${\rm U}(n)$ vector bundles
on elliptically fibered Calabi–Yau three-folds 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 ${\rm U}(n)$
bundles are discussed. As an application, it is shown that the ${\rm U}(n)$
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 5-branes is found. This gives a definite prediction for
the number of 3-branes in a dual F-theory model. The net-generation
number is evaluated, showing more flexibility compared with the ${\rm SU}(n)$
case.