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We investigate charge quantization in the Standard Model (SM) through a$\mathbb{CP}^2$ nonlinear sigma model (NLSM), $SU(3)_G/(SU(2)_H \timesU(1)_H)$, and a $\mathbb{CP}^3$ model, $SU(4)_G/(SU(3)_H \times U(1)_H)$. Wealso generalize to any $\mathbb{CP}^k$ model. Charge quantization follows fromthe consistency and dynamics of the NLSM, without a monopole or Grand UnifiedTheory, as shown in our earlier work on the $\mathbb{CP}^1$ model(arXiv:1309.0692). We find that representations of the matter fields under theunbroken non-abelian subgroup dictate their charge quantization under the$U(1)_H$ factor. In the $\mathbb{CP}^2$ model the unbroken group is identifiedwith the weak and hypercharge groups of the SM, and the Nambu-Goldstone boson(NGB) has the quantum numbers of a SM Higgs. There is the intriguingpossibility of a connection with the vanishing of the Higgs self-coupling atthe Planck scale. Interestingly, with some minor assumptions (no vector-likematter and minimal representations) and starting with a single quark doublet,anomaly cancellation requires the matter structure of a generation in the SM.Similar analysis holds in the $\mathbb{CP}^3$ model, with the unbroken groupidentified with QCD and hypercharge, and the NGB having the up quark as apartner in a supersymmetric model. This can motivate solving the strong CPproblem with a vanishing up quark mass.

Charge Quantization and the Standard Model from the $\mathbb{CP}^2$ and $\mathbb{CP}^3$ Nonlinear $\sigma$-Models