We present a general method for calculating the moduli spaces of fivebraneswrapped on holomorphic curves in elliptically fibered Calabi-Yau threefolds, inparticular, in the context of heterotic M theory. The cases of fivebraneswrapped purely on a fiber curve, purely on a curve in the base and,generically, on a curve with components both in the fiber and the base are eachdiscussed in detail. The number of irreducible components of the fivebrane andtheir properties, such as their intersections and phase transitions in modulispace, follow from the analysis. Even though generic curves have a large numberof moduli, we show that there are isolated curves that have no moduliassociated with the Calabi-Yau threefold. We present several explicit examples,including cases which correspond to potentially realistic three family modelswith grand unified gauge group SU(5).
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