The Geometry of the SU(2) × G2-model

A bstract We study elliptic fibrations that geometrically engineer an SU(2) ×G 2 gauge theory realized by a Weierstrass model for the collision III + I 0 * ns . We find all the distinct crepant resolutions of such a model and the flops connecting them. We compute the generating function for the Euler characteristic of the SU(2) ×G 2 -model. In the case of a Calabi-Yau threefold, we consider the compactification of M-theory and F-theory on an SU(2) × G 2 -model to a five and six-dimensional supergravity theory with eight super-charges. By matching each crepant resolution with each Coulomb chamber of the five-dimensional theory, we determine the number of multiplets and compute the prepotential in each Coulomb chamber. In particular, we discuss the counting numbers of hypermultiplets in the presence of singularities. We discuss in detail the cancellation of anomalies of the six-dimensional theory.