We study compactifications of F-theory on certain Calabi-Yau threefolds. We find that N = 2 dualities of type II/heterotic strings in 4 dimensions get promoted to N = 1 dualities between heterotic string and F-theory in 6 dimensions. The six dimensional heterotic/heterotic duality becomes a classical geometric symmetry of the Calabi-Yau in

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.

We discuss five-dimensional supersymmetric gauge theories. An anomaly renders some theories inconsistent and others consistent only upon including a Wess-Zumino type Chern-Simons term. We discuss some necessary conditions for existence of nontrivial renormalization group fixed points and find all possible gauge groups and matter content which satisfy them. In some cases, the existence of these fixed points can be inferred from string duality considerations. In other cases, they arise from M-theory on Calabi-Yau threefolds. We explore connections between aspects of the gauge theory and Calabi-Yau geometry. A consequence of our classification of field theories with nontrivial fixed points is a fairly complete classification of a class of singularities of Calabi-Yau threefolds which generalize the "del Pezzo contractions" and occur at higher codimension walls of the Kähler cone. 2/97

We study five-dimensional supersymmetric field theories with one-dimensional Coulomb branch. We extend a previous analysis which led to non-trivial fixed points with E n symmetry (E 8 , E 7 , E 6 , E 5 = Spin(10), E 4 = SU (5), E 3 = SU (3) × SU (2), E 2 = SU (2) × U (1) and E 1 = SU (2)) by finding two new theories: E 1 with U (1) symmetry and E 0 with no symmetry. The latter is a non-trivial theory with no relevant operators preserving the super-Poincaré symmetry. In terms of string theory these new field theories enable us to describe compactifications of the type I ′ theory on S 1 /Z 2 with 16, 17 or 18 background D8-branes. These theories also play a crucial role in compactifications of M-theory on CalabiYau spaces, providing physical models for the contractions of del Pezzo surfaces to points (thereby completing the classification of singularities which can occur at codimension one in Kähler moduli). The structure of the Higgs branch yields a prediction which unifies the known mathematical facts about del Pezzo transitions in a quite remarkable way.

Using "Tate's algorithm," we identify loci in the moduli of F-theory compactifications corresponding to enhanced gauge symmetry. We apply this to test the proposed F-theory/heterotic dualities in six dimensions. We recover the perturbative gauge symmetry enhancements of the heterotic side and the physics of small SO(32) instantons, and discover new mixed perturbative/non-perturbative gauge symmetry enhancements.Upon further toroidal compactification to 4 dimensions, we derive the chain of Calabi-Yau threefolds dual to various Coulomb branches of heterotic strings.