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We study crepant resolutions of Weierstrass models of$\text{SU}(2)\!\times\!\text{SU}(3)$-models, whose gauge group describes thenon-abelian sector of the Standard Model. The$\text{SU}(2)\!\times\!\text{SU}(3)$-models are elliptic fibrationscharacterized by the collision of two Kodaira fibers with dual graphs that areaffine Dynkin diagrams of type $\widetilde{\text{A}}_1$ and$\widetilde{\text{A}}_2$. Once we eliminate those collisions that do not havecrepant resolutions, we are left with six distinct collisions that are relatedto each other by deformations. Each of these six collisions has eight distinctcrepant resolutions whose flop diagram is a hexagon with two legs attached totwo adjacent nodes. Hence, we consider 48 distinct resolutions that areconnected to each other by deformations and flops. We determine topologicalinvariants---such as Euler characteristics, Hodge numbers, and tripleintersections of fibral divisors---for each of the crepant resolutions. Weanalyze the physics of these fibrations when used as compactifications ofM-theory and F-theory on Calabi--Yau threefolds yielding 5d ${\mathcal N}=1$and 6d ${\mathcal N}=(1,0)$ supergravity theories respectively. We study the 5dprepotential in the Coulomb branch of the theory and check that thesix-dimensional theory is anomaly-free and compatible with a 6d uplift from a5d theory.

48 Crepant Paths to $\text{SU}(2)\!\times\!\text{SU}(3)$