Physlib.Particles.StandardModel.AnomalyCancellation.Basic

For a natural number $n$ representing the number of fermion generations (families), `SMCharges n` is the system of charges for the Standard Model without right-handed neutrinos. It is defined as an anomaly cancellation condition (ACC) system with $5n$ charges, representing the 5 species of fermions (typically the left-handed quark doublet $Q$, right-handed up-type quark $u^c$, right-handed down-type quark $d^c$, left-handed lepton doublet $L$, and right-handed charged lepton $e^c$) across each of the $n$ families.

For a natural number $n$, representing the number of fermion generations (families) in the Standard Model, `SMSpecies n` is the system of charges associated with a single species of fermions. This defines a vector space of dimension $n$ (typically over the rational numbers $\mathbb{Q}$) where each component corresponds to the charge of that specific species in one of the $n$ families.

For a Standard Model with $n$ families, this equivalence $\simeq$ identifies the set of charges $(SMCharges\, n).Charges$ with the set of functions $f: \{0, \dots, 4\} \to (\{0, \dots, n-1\} \to \mathbb{Q})$. This maps the total collection of $5n$ rational charges to a representation organized by the five fermion species (the index $i \in \{0, \dots, 4\}$ representing $Q, u^c, d^c, L, e^c$) and the $n$ families (the index $j \in \{0, \dots, n-1\}$).

For a given index $i \in \{0, 1, 2, 3, 4\}$ representing one of the five fermion species in the $n$-family Standard Model (typically $Q, u^c, d^c, L, e^c$), this definition is a $\mathbb{Q}$-linear map that projects the total collection of $5n$ charges, $S \in (SMCharges\, n).Charges$, onto the $n$ charges associated with the $i$-th species, $(SMSpecies\, n).Charges \cong \mathbb{Q}^n$.

For two charge assignments $S, T \in (\text{SMCharges } n).\text{Charges}$ in the $n$-family Standard Model, $S$ is equal to $T$ if and only if for every fermion species index $i \in \{0, 1, 2, 3, 4\}$, the projection of $S$ onto the $i$-th species is equal to the projection of $T$ onto the $i$-th species. That is, $S = T \iff \forall i, \text{toSpecies}_i(S) = \text{toSpecies}_i(T)$.

In the Standard Model with $n$ fermion families, let $f: \{0, 1, 2, 3, 4\} \to (\text{Fin } n \to \mathbb{Q})$ be a function that assigns rational charges to each of the five fermion species ($Q, u^c, d^c, L, e^c$) across the $n$ families. If we reconstruct the total charge vector $S \in (\text{SMCharges } n).\text{Charges}$ from $f$ using the inverse of the species-family equivalence ($\text{toSpeciesEquiv}^{-1}$), then the linear projection of $S$ onto the $i$-th species ($\text{toSpecies}_i$) is identical to the $i$-th component of the original configuration $f$. That is, for any $i \in \{0, 1, 2, 3, 4\}$: $$(\text{toSpecies}_i)(\text{toSpeciesEquiv}^{-1}(f)) = f(i)$$

For a Standard Model with $n$ fermion families, `SMCharges.Q` is the $\mathbb{Q}$-linear projection that extracts the charges of the $Q$ species (left-handed quark doublets) from the total collection of $5n$ fermion charges. It maps a charge configuration $S \in (SMCharges\, n).Charges$ to a vector in $\mathbb{Q}^n$, where each component $Q_i$ for $i \in \{0, \dots, n-1\}$ represents the rational charge assigned to the $Q$ field in the $i$-th generation.

For the $n$-family Standard Model, `SMCharges.U` is the $\mathbb{Q}$-linear map that projects the total configuration of $5n$ charges onto the $n$ rational charges associated with the right-handed up-type quark species ($u^c$). Specifically, given a total charge assignment, it extracts the vector in $\mathbb{Q}^n$ where each component $q_j$ corresponds to the charge of the $u^c$ fermion in the $j$-th generation for $j \in \{0, \dots, n-1\}$.

For an $n$-family Standard Model, `SMCharges.D` is the $\mathbb{Q}$-linear map that projects the total collection of $5n$ fermion charges $S \in (\text{SMCharges } n).\text{Charges}$ onto the $n$ charges associated with the right-handed down-type quark species ($d^c$). The result is an element in $(\text{SMSpecies } n).\text{Charges}$, effectively a vector $(d^c_1, d^c_2, \dots, d^c_n) \in \mathbb{Q}^n$ representing the charges of the down-type quarks across all generations.

For a given set of $5n$ charges representing the fermion species in the $n$-family Standard Model, this definition is the $\mathbb{Q}$-linear projection that extracts the charges associated specifically with the left-handed lepton doublet $L$. It maps the total charge vector in $(\text{SMCharges } n).\text{Charges} \cong \mathbb{Q}^{5n}$ to the specific charges $(l_1, l_2, \dots, l_n) \in \mathbb{Q}^n$ for each of the $n$ fermion families.

For an $n$-family Standard Model, `SMCharges.E` is the $\mathbb{Q}$-linear map that projects the total system of $5n$ fermion charges onto the $n$ charges associated with the fifth fermion species, which corresponds to the right-handed charged leptons $E$ (or $e^c$). Given a total charge vector $S \in \mathbb{Q}^{5n}$, the map returns a vector in $\mathbb{Q}^n$ representing the charges assigned to the $E$ species across all $n$ generations.

For a Standard Model with $n$ fermion families, let $S$ be the collection of charges for all fermion species. Let $Q_i, U_i, D_i, L_i,$ and $E_i$ denote the charges of the left-handed quark doublet, right-handed up-type quark, right-handed down-type quark, left-handed lepton doublet, and right-handed charged lepton for the $i$-th generation, respectively, where $i \in \{0, \dots, n-1\}$. The gravitational anomaly is defined as the $\mathbb{Q}$-linear map: $$\text{accGrav}(S) = \sum_{i=0}^{n-1} (6 Q_i + 3 U_i + 3 D_i + 2 L_i + E_i)$$ This expression represents the linear gravitational anomaly cancellation condition, where the coefficients $(6, 3, 3, 2, 1)$ correspond to the number of degrees of freedom (color and weak isospin) for each fermion species.

Consider an $n$-family Standard Model. Let $S$ and $T$ be two sets of rational charges for the $5n$ fermion representations in the model. Let $j \in \{0, \dots, 4\}$ index the five fermion species ($Q, u^c, d^c, L, e^c$) and let $(\text{toSpecies } j) S_i$ denote the charge of the $i$-th generation of species $j$ under the charge assignment $S$. If for every species $j$, the sum of charges over all $n$ families is equal for $S$ and $T$: $$\sum_{i=0}^{n-1} (\text{toSpecies } j) S_i = \sum_{i=0}^{n-1} (\text{toSpecies } j) T_i$$ then the gravitational anomaly $\text{accGrav}(S)$ is equal to $\text{accGrav}(T)$, where the gravitational anomaly is defined as: $$\text{accGrav}(S) = \sum_{i=0}^{n-1} (6 Q_i + 3 U_i + 3 D_i + 2 L_i + E_i)$$

For a Standard Model with $n$ fermion families, let $S$ be the collection of charges for all species. Let $Q_i$ denote the charge of the left-handed quark doublet and $L_i$ denote the charge of the left-handed lepton doublet for the $i$-th generation, where $i \in \{0, \dots, n-1\}$. The $SU(2)$ anomaly is defined as the $\mathbb{Q}$-linear map: $$\text{accSU2}(S) = \sum_{i=0}^{n-1} (3 Q_i + L_i)$$ This expression represents the gauge anomaly cancellation condition associated with the $SU(2)_L$ gauge group, where the factor of 3 accounts for the number of color degrees of freedom for the quarks.

In the $n$-family Standard Model, let $S$ and $T$ be two charge configurations. If for each of the five fermion species $j \in \{0, \dots, 4\}$, the sum of the charges across all $n$ families is the same for $S$ and $T$, such that $$\sum_{i=0}^{n-1} (\text{toSpecies } j)(S)_i = \sum_{i=0}^{n-1} (\text{toSpecies } j)(T)_i,$$ then the $SU(2)$ gauge anomalies of the two configurations are equal: $$\text{accSU2}(S) = \text{accSU2}(T).$$ Here, $\text{accSU2}(S)$ is defined as $\sum_{i=0}^{n-1} (3 Q_i + L_i)$, where $Q_i$ and $L_i$ are the charges of the $i$-th family left-handed quark and lepton doublets, respectively.

For a Standard Model with $n$ fermion families, let $S$ be the collection of charges for all fermion species. Let $Q_i$, $U_i$, and $D_i$ denote the charges of the left-handed quark doublet, the right-handed up-type quark, and the right-handed down-type quark for the $i$-th generation ($i \in \{0, \dots, n-1\}$), respectively. The $SU(3)$ anomaly is defined as the $\mathbb{Q}$-linear map: $$\text{accSU3}(S) = \sum_{i=0}^{n-1} (2 Q_i + U_i + D_i)$$ This expression represents the gauge anomaly cancellation condition associated with the $SU(3)_C$ gauge group. The factor of 2 for the $Q_i$ charges accounts for the $SU(2)_L$ doublet nature of the left-handed quarks.

In the $n$-family Standard Model, let $S$ and $T$ be two charge configurations. If for each of the five fermion species $j \in \{0, \dots, 4\}$, the sum of the charges across all $n$ families is the same for $S$ and $T$, such that $$\sum_{i=0}^{n-1} (\text{toSpecies } j)(S)_i = \sum_{i=0}^{n-1} (\text{toSpecies } j)(T)_i,$$ then the $SU(3)$ gauge anomalies of the two configurations are equal: $$\text{accSU3}(S) = \text{accSU3}(T).$$ Here, $\text{accSU3}(S)$ is defined as the sum $\sum_{i=0}^{n-1} (2 Q_i + U_i + D_i)$, where $Q_i, U_i,$ and $D_i$ are the rational charges of the $i$-th generation left-handed quark doublet, right-handed up-type quark, and right-handed down-type quark, respectively.

This linear map from the space of charges to the rational numbers $\mathbb{Q}$ represents the $Y^2$ anomaly cancellation condition for a Standard Model with $n$ fermion families. For a given charge assignment $S$, the map is defined by summing the charges across all $n$ generations as follows: $$\text{accYY}(S) = \sum_{i=0}^{n-1} (Q_i + 8U_i + 2D_i + 3L_i + 6E_i)$$ where $Q_i, U_i, D_i, L_i,$ and $E_i$ are the rational charges assigned to the $i$-th generation of the five fermion species: the left-handed quark doublet ($Q$), the right-handed up-type quark ($u^c$), the right-handed down-type quark ($d^c$), the left-handed lepton doublet ($L$), and the right-handed charged lepton ($e^c$), respectively.

Let $S$ and $T$ be two sets of charges for the $n$-family Standard Model. If for each of the five fermion species $j \in \{Q, u^c, d^c, L, e^c\}$, the sum of charges across all $n$ families is the same for $S$ and $T$, i.e., $$\sum_{i=0}^{n-1} (toSpecies_j(S))_i = \sum_{i=0}^{n-1} (toSpecies_j(T))_i$$ then the $Y^2$ anomaly cancellation condition values are equal: $$\text{accYY}(S) = \text{accYY}(T)$$ where $\text{accYY}$ is defined by the linear map $\text{accYY}(S) = \sum_{i=0}^{n-1} (Q_i + 8U_i + 2D_i + 3L_i + 6E_i)$.

This definition defines a symmetric bilinear map on the space of charges for the $n$-family Standard Model without right-handed neutrinos. Given two charge configurations $S$ and $T$ in $\mathbb{Q}^{5n}$, the map evaluates to the rational sum over the $n$ generations: $$\sum_{i=0}^{n-1} \left( Q_S^{(i)} Q_T^{(i)} - 2 U_S^{(i)} U_T^{(i)} + D_S^{(i)} D_T^{(i)} - L_S^{(i)} L_T^{(i)} + E_S^{(i)} E_T^{(i)} \right)$$ where $Q^{(i)}, U^{(i)}, D^{(i)}, L^{(i)}$, and $E^{(i)}$ denote the rational charges in the $i$-th generation for the left-handed quark doublet, right-handed up-type quark, right-handed down-type quark, left-handed lepton doublet, and right-handed charged lepton, respectively.

For the $n$-family Standard Model without right-handed neutrinos, `SMACCs.accQuad` is the homogeneous quadratic map $f: \mathbb{Q}^{5n} \to \mathbb{Q}$ that represents the quadratic anomaly cancellation condition. For a charge configuration $S$, it is defined as the sum over the $n$ generations: $$\sum_{i=0}^{n-1} \left( (Q_i)^2 - 2(U_i)^2 + (D_i)^2 - (L_i)^2 + (E_i)^2 \right)$$ where $Q_i, U_i, D_i, L_i$, and $E_i$ denote the rational charges of the $i$-th generation for the left-handed quark doublet, right-handed up-type quark, right-handed down-type quark, left-handed lepton doublet, and right-handed charged lepton, respectively.

Let $S$ and $T$ be two charge configurations for the $n$-family Standard Model. Let $q_{j,i}^{(S)}$ denote the charge of fermion species $j \in \{0, 1, 2, 3, 4\}$ (corresponding to the species $Q, u^c, d^c, L, e^c$) in the $i$-th generation for configuration $S$. If for every species $j$, the sum of the squares of the charges across all $n$ generations is the same for $S$ and $T$, i.e., $$\forall j \in \{0, \dots, 4\}, \quad \sum_{i=0}^{n-1} (q_{j,i}^{(S)})^2 = \sum_{i=0}^{n-1} (q_{j,i}^{(T)})^2$$ then the quadratic anomaly cancellation condition evaluates to the same value for both configurations: $$\text{accQuad}(S) = \text{accQuad}(T)$$ This is an extensionality lemma for the quadratic anomaly cancellation condition.

Let $V$ be the space of charges for the $n$-family Standard Model, where an element $S \in V$ represents a collection of rational charges for the five fermion species across $n$ generations: $Q$ (left-handed quark doublet), $U$ (right-handed up-type quark), $D$ (right-handed down-type quark), $L$ (left-handed lepton doublet), and $E$ (right-handed charged lepton). `SMACCs.cubeTriLin` is the symmetric trilinear form $f: V \times V \times V \to \mathbb{Q}$ defined by summing the products of the charges of each species over the $n$ families, weighted by their respective gauge representation dimensions: $$f(S^{(1)}, S^{(2)}, S^{(3)}) = \sum_{i=0}^{n-1} \left( 6 Q_i^{(1)} Q_i^{(2)} Q_i^{(3)} + 3 U_i^{(1)} U_i^{(2)} U_i^{(3)} + 3 D_i^{(1)} D_i^{(2)} D_i^{(3)} + 2 L_i^{(1)} L_i^{(2)} L_i^{(3)} + E_i^{(1)} E_i^{(2)} E_i^{(3)} \right)$$ where $Q_i^{(j)}, U_i^{(j)}, D_i^{(j)}, L_i^{(j)}, E_i^{(j)}$ denote the charges of the $i$-th generation of the $j$-th charge configuration.

Let $V$ be the module of charges for the $n$-family Standard Model, where a charge configuration $S \in V$ assigns rational charges $Q_i, U_i, D_i, L_i, E_i$ to the five species of fermions for each generation $i \in \{0, \dots, n-1\}$. The cubic anomaly cancellation condition (ACC) is defined as the homogeneous cubic map $f: V \to \mathbb{Q}$ given by evaluating the symmetric trilinear form $\text{cubeTriLin}$ on the diagonal $f(S) = \tau(S, S, S)$: $$f(S) = \sum_{i=0}^{n-1} \left( 6 Q_i^3 + 3 U_i^3 + 3 D_i^3 + 2 L_i^3 + E_i^3 \right)$$ This sum represents the total cubic gauge anomaly, where the coefficients $(6, 3, 3, 2, 1)$ correspond to the dimensions of the representations of the respective fermion species under the Standard Model gauge group.

Let $S$ and $T$ be two charge configurations for the $n$-family Standard Model, assigning rational charges $Q_i, U_i, D_i, L_i, E_i$ to the five species of fermions for each generation $i \in \{0, \dots, n-1\}$. If for each species $X \in \{Q, U, D, L, E\}$, the sum of the cubes of the charges across all generations is the same for both $S$ and $T$, i.e., $$\sum_{i=0}^{n-1} (X_{S, i})^3 = \sum_{i=0}^{n-1} (X_{T, i})^3$$ then the values of the cubic anomaly cancellation condition for $S$ and $T$ are equal: $\text{accCube}(S) = \text{accCube}(T)$.