${\rm U}(n)$ vector bundles on Calabi-Yau three-folds for string theory compactifications

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.

2010 Mathematics Subject Classification

Primary 81T30. Secondary 14J32, 81T13.

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Published 1 January 2005