Moduli Spaces of Fivebranes on Elliptic Calabi-Yau Threefolds

Abstract

We present a general method for calculating the moduli spaces of fivebranes wrapped on holomorphic curves in elliptically fibered Calabi-Yau threefolds, in particular, in the context of heterotic M theory. The cases of fivebranes wrapped 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 each discussed in detail. The number of irreducible components of the fivebrane and their properties, such as their intersections and phase transitions in moduli space, follow from the analysis. Even though generic curves have a large number of moduli, we show that there are isolated curves that have no moduli associated with the Calabi-Yau threefold. We present several explicit examples, including cases which correspond to potentially realistic three family models with grand unified gauge group SU(5).

Figures (9)

• Figure 1: Generic algebraic classes on $X$, where $[C]=\sigma_*\Omega$ and the fiber class $[E_p]=F$ are in $H_2(X,\mathbf{Z})$, and $[\pi^{-1}(C)]=\pi^*\Omega$ and base class $[B]=D$ are in $H_4(X,\mathbf{Z})$.
• Figure 2: The curve $W$ and its image in the base $C$
• Figure 3: The moduli space of lines in the class $l-E_1$. The solid lines represent two examples of the special case where the proper transform of the line splits into two components
• Figure 4: Splitting a single sphere into a pair of spheres
• Figure 5: The moduli space $\mathcal{M}(\sigma_*l-\sigma_*E_1)$.