Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.PlusU1.FamilyMaps

For a natural number $n$, the map `familyUniversalLinear` is a $\mathbb{Q}$-linear map from the space of linear solutions for a 1-generation Standard Model with right-handed neutrinos to the space of linear solutions for an $n$-generation model. It takes a charge configuration $S = (Q, u, d, L, e, \nu) \in \mathbb{Q}^6$ that satisfies the four linear anomaly cancellation conditions (ACCs)—the gravitational, $SU(2)$, $SU(3)$, and $Y^2$ mixed anomalies—and embeds it into the $n$-generation charge space by assigning the same charges to every generation. Since the linear ACCs for $n$ generations are the sums of the respective terms over all generations, the resulting configuration satisfies the $n$-generation linear constraints if the original 1-generation configuration satisfies the single-generation constraints.

For a natural number $n$, the map `familyUniversalQuad` embeds the space of 1-generation quadratic solutions for the Standard Model with right-handed neutrinos into the space of $n$-generation quadratic solutions. Specifically, given a 1-generation charge configuration $S = (Q, u, d, L, e, \nu) \in \mathbb{Q}^6$ that satisfies the four linear anomaly cancellation conditions (ACCs)—gravitational, $SU(2)$, $SU(3)$, and $Y^2$ mixed anomalies—as well as the quadratic ACC, this map replicates $S$ across all $n$ generations in $\mathbb{Q}^{6n}$. Since the anomaly equations for $n$ generations are defined as the sums of the single-generation terms, the resulting $n$-generation configuration $(S, S, \dots, S)$ also satisfies the linear and quadratic ACCs.

For a natural number $n$, the map `familyUniversalAF` embeds the set of 1-generation anomaly-free solutions for the Standard Model with right-handed neutrinos into the set of $n$-generation anomaly-free solutions. Given a single-generation charge configuration $S = (Q, u, d, L, e, \nu) \in \mathbb{Q}^6$ that satisfies all six anomaly cancellation conditions (the four linear anomalies, the quadratic anomaly, and the cubic anomaly), this map constructs an $n$-generation configuration by replicating $S$ across all generations. Because the $n$-generation anomaly equations are defined as the sums of the 1-generation terms over the $n$ generations, the resulting configuration in $\mathbb{Q}^{6n}$ (where every generation $i \in \{0, \dots, n-1\}$ has identical charges) also satisfies all the anomaly cancellation conditions.