This article explains how to construct an action for a model of quantum gauge fields with gauged group (G1×···× GK)/Γ, where each Gk is a compact Lie group – either U(1) or a simple 1-connected Lie group – and Γ is a discrete subgroup of the center of G1×···× GK. The action may involve several independent coupling constants, one for each factor Gk and one for each pairing of U(1) factors (if the numerator has more than one U(1) factor). The factor Γ affects the set of possible interaction terms, but it doesn't affect the Lie algebra.
One of the motives for this article is the fact that the gauged group of the Standard Model(s) of particle physics has the form (SU(3)× SU(2)× U(1))/Γ. More than one choice for Γ is consistent with everything that is currently known about the real world, but the most likely choice is a group with six elements for which the resulting gauged group is not a product.
