Page 527 of Srednicki's QFT book lists the matter content of the Standard Model as
left-handed Weyl fields in three copies of the representation $(1, 2, −1/2)⊕(1, 1, +1)⊕(3, 2, +1/6)⊕(\bar{3̄}, 1, −2/3)⊕(\bar{3}, 1, +1/3)$, and a complex scalar field in the representation $(1, 2, −1/2)$.
But the irreducible representations of $\text{U}(1)$ are just the endomorphisms $e^{i \theta} \to e^{i n \theta}$ indexed by the integer $n$, as described here. What does it mean for the matter fields to lie in non-integer representations of the $\text{U}(1)$ gauge group?
Put more simply: in QED, the compactness of the electromagnetic gauge group $\text{U}(1)$ quantizes the electric charge to integer multiples of $e$. Why doesn't the compactness of the weak hypercharge gauge group $\text{U}(1)$ do the same thing to weak hypercharge in the Standard Model?
