Physlib.Particles.StandardModel.AnomalyCancellation.FamilyMaps

Given natural numbers $n$ and $m$ representing the number of fermion families, let $\text{SMSpecies}(k) \cong \mathbb{Q}^k$ be the vector space of rational charges for a single fermion species across $k$ families, and let $\text{SMCharges}(k) \cong (\mathbb{Q}^k)^5$ be the total vector space of charges for the five fermion species in the Standard Model. For a $\mathbb{Q}$-linear map $f: \mathbb{Q}^n \to \mathbb{Q}^m$ that acts on the charges of a single species, `SM.chargesMapOfSpeciesMap` defines a $\mathbb{Q}$-linear map $F: \text{SMCharges}(n) \to \text{SMCharges}(m)$. For a total charge configuration $S \in \text{SMCharges}(n)$, the map $F$ is defined such that the charges of each species $i \in \{0, \dots, 4\}$ in the resulting configuration $F(S)$ are obtained by applying $f$ to the charges of the $i$-th species in $S$.

For a single fermion species, given two natural numbers $m$ and $n$ such that $n \leq m$, let $\mathbb{Q}^m$ and $\mathbb{Q}^n$ denote the vector spaces of rational charges across $m$ and $n$ families, respectively. The function `SM.speciesFamilyProj` is the $\mathbb{Q}$-linear projection map $P: \mathbb{Q}^m \to \mathbb{Q}^n$ that maps a charge vector $S = (q_0, q_1, \dots, q_{m-1})$ to its first $n$ components, such that $P(S) = (q_0, q_1, \dots, q_{n-1})$.

For natural numbers $m$ and $n$ such that $n \leq m$, let $\text{SMCharges}(m) \cong (\mathbb{Q}^m)^5$ and $\text{SMCharges}(n) \cong (\mathbb{Q}^n)^5$ be the vector spaces of rational charges for the five fermion species across $m$ and $n$ families, respectively. The function `SM.familyProjection` is the $\mathbb{Q}$-linear projection map $P: \text{SMCharges}(m) \to \text{SMCharges}(n)$. This map takes a total charge configuration and restricts the charges of each of the five fermion species to the first $n$ families, effectively discarding the charges associated with families indexed from $n$ to $m-1$.

For natural numbers $m$ and $n$, this linear map embeds the charges of a single fermion species with $m$ families into the space of charges for $n$ families. Given a charge vector $S \in \mathbb{Q}^m$, the $i$-th component of the resulting vector in $\mathbb{Q}^n$ (where $0 \leq i < n$) is defined as $S_i$ if $i < m$, and $0$ if $i \geq m$. In the case where $n > m$, this corresponds to padding the $m$ charges with zeros; in the case where $n \leq m$, this corresponds to a projection onto the first $n$ families.

For natural numbers $m$ and $n$, this $\mathbb{Q}$-linear map embeds the charge configuration of a Standard Model with $m$ families into one with $n$ families. Given the total charge space $(\text{SMCharges } m).\text{Charges} \cong (\mathbb{Q}^m)^5$, which consists of rational charges for five species of fermions across $m$ families, the map applies the species-level embedding $\text{SM.speciesEmbed } m \ n$ to each species. Specifically, for each species, a charge vector $(q_1, \dots, q_m) \in \mathbb{Q}^m$ is mapped to an $n$-dimensional vector where the $i$-th component is $q_i$ if $i \leq m$ and $0$ otherwise (for $n > m$). This represents the embedding of $m$-family charges onto $n$-family charges with all additional family charges set to zero.

For a given natural number $n$, this is a $\mathbb{Q}$-linear map from the space of charges for a single fermion species with one family, $(SMSpecies \ 1).Charges \cong \mathbb{Q}^1$, to the space of charges for the same species with $n$ families, $(SMSpecies \ n).Charges \cong \mathbb{Q}^n$. The map takes the charge $q$ of the single family and assigns it to each of the $n$ families, effectively mapping $q \mapsto (q, q, \dots, q)$.

For a natural number $n$, this is a $\mathbb{Q}$-linear map from the space of Standard Model charges for a single family, $\text{SMCharges}(1) \cong \mathbb{Q}^5$, to the space of charges for $n$ families, $\text{SMCharges}(n) \cong \mathbb{Q}^{5n}$. The map acts as a universal embedding by taking the rational charge assigned to each of the five fermion species in the 1-family model and replicating it across all $n$ families. Specifically, if the charges for the five species in the single-family case are $(q_1, q_2, q_3, q_4, q_5)$, the map produces a charge configuration where each family $i \in \{1, \dots, n\}$ has the same charges $(q_1, q_2, q_3, q_4, q_5)$.