Compactifications of F-Theory on Calabi--Yau Threefolds -- II

We continue our study of compactifications of F-theory on Calabi--Yau threefolds. We gain more insight into F-theory duals of heterotic strings and provide a recipe for building F-theory duals for arbitrary heterotic compactifications on elliptically fibered manifolds. As a byproduct we find that string/string duality in six dimensions gets mapped to fiber/base exchange in F-theory. We also construct a number of new $N=1$, $d=6$ examples of F-theory vacua and study transitions among them. We find that some of these transition points correspond upon further compactification to 4 dimensions to transitions through analogues of Argyres--Douglas points of $N=2$ moduli. A key idea in these transitions is the notion of classifying $(0,4)$ fivebranes of heterotic strings.