The Geometry of the SU(2)$\times$ G$_2$-model

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We study elliptic fibrations that geometrically engineer an SU(2)$\times$ G$_2$ gauge theory realized by Weierstrass model for the collision III+$\text{I}_0^{*\text{ns}}$.

• We construct the four possible crepant resolutions of such a Weierstrass model and show that they form a chain of four minimal models connected by flops.
• We compute the generating function for the Euler characteristic of these crepant resolutions.
• In the case of a Calabi-Yau threefold, we consider the compactification of M-theory and F-theory on an SU(2)$\times$ G$_2$-model to a five and six-dimensional supergravity with eight supercharges.