Quantization of hypercharge in gauge groups locally isomorphic but globally nonisomorphic to SU(3)_c X SU(2)_L X U(1)_Y

In the Standard Model the hypercharges of quarks and leptons are not determined by the gauge group itself. In a recent paper [C. Hattori et al. Phys. Rev. D83, 015009 (2011)] it is shown that, if the direct product gauge group G_SM is slightly modified to the semidirect product group G'_SM, hypercharges are restricted to quantized values as n/6 mod Z (n = 0,1,3,4). In this brief paper, we examine all of the compact Lie groups locally isomorphic to G_SM, and show that G'_SM (or its isomorphisms) is the unique possibility that yields the correct hypercharge quantization.

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