Physlib.Particles.StandardModel.AnomalyCancellation.Permutations

For a given natural number $n$, representing the number of fermion generations, the permutation group is defined as the product of five symmetric groups $S_n$. Specifically, it is the type of functions mapping each of the five fermion species in the Standard Model (without right-handed neutrinos) to a permutation of its $n$ generations, denoted as $\prod_{i=1}^5 S_n$.

The type $\text{PermGroup } n$, which is the product of five symmetric groups $S_n$ representing the permutations of $n$ fermion generations for each of the five fermion species in the Standard Model (without right-handed neutrinos), is equipped with a group structure. This structure is defined by the pointwise composition of permutations across the product.

For a given natural number $n$ representing the number of fermion generations, let $\text{PermGroup } n \cong \prod_{i=1}^5 S_n$ be the group of permutations of fermion generations for each of the five species in the Standard Model (without right-handed neutrinos). The $\mathbb{Q}$-linear map $\text{chargeMap } f: (\text{SMCharges } n).\text{Charges} \to (\text{SMCharges } n).\text{Charges}$ defines the action of a permutation $f \in \text{PermGroup } n$ on the space of charges. Specifically, if a charge configuration $S$ is represented as a collection of rational charges $q_{i,j}$ for each species $i \in \{0, \dots, 4\}$ and generation $j \in \{0, \dots, n-1\}$, the map transforms $S$ by permuting the generation indices of each species $i$ according to the permutation $f(i)$. Mathematically, the map is defined such that for each species $i$, the projected charges satisfy $(\text{toSpecies } i) (\text{chargeMap } f (S)) = (\text{toSpecies } i) S \circ f(i)$.

For a given number of fermion generations $n \in \mathbb{N}$, the linear representation $\text{repCharges}$ defines the action of the permutation group $\text{PermGroup } n \cong \prod_{i=1}^5 S_n$ on the space of rational charges $(\text{SMCharges } n).\text{Charges} \cong \mathbb{Q}^{5n}$. For any element $f \in \text{PermGroup } n$, the representation acts as a linear map that permutes the generation indices of the charge configuration $S$ according to the inverse permutation $f^{-1}$. Specifically, for each fermion species $j \in \{0, \dots, 4\}$, the resulting charge assignment is the composition of the original charges with $f(j)^{-1}$.

In the Standard Model with $n$ fermion generations, let $f \in \prod_{i=0}^4 S_n$ be an element of the generation permutation group and $S \in \mathbb{Q}^{5n}$ be a configuration of charges. For any fermion species $j \in \{0, \dots, 4\}$, let $\text{toSpecies}_j(S)$ denote the linear projection of the total charges onto the $n$ charges of that specific species. The linear representation of the permutation acting on the charges satisfies: $$\text{toSpecies}_j(\text{repCharges}_f S) = \text{toSpecies}_j S \circ f(j)^{-1}$$ where $f(j)^{-1}$ is the inverse of the permutation associated with the $j$-th species.

In the Standard Model with $n$ fermion generations, let $S$ be the configuration of rational charges and $j \in \{0, 1, 2, 3, 4\}$ represent one of the five fermion species. Let $m$ be a natural number. For any element $f$ in the permutation group $\text{PermGroup } n \cong \prod_{i=1}^5 S_n$, the sum of the $m$-th powers of the charges of the species $j$ is invariant under the group action. That is, $$\sum_{i=0}^{n-1} (\text{toSpecies}_j (f \cdot S)_i)^m = \sum_{i=0}^{n-1} (\text{toSpecies}_j S_i)^m$$ where $f \cdot S$ denotes the representation of the permutation $f$ acting on the charge configuration $S$, and $\text{toSpecies}_j S_i$ is the charge of the $i$-th generation of the $j$-th species.

In the Standard Model with $n$ fermion generations, let $S$ be a configuration of rational charges for 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 $e^c$. Let $f \in \prod_{i=1}^5 S_n$ be an element of the permutation group that acts by permuting the $n$ generation indices of each species independently. The gravitational anomaly is defined by the linear map: $$\text{accGrav}(S) = \sum_{i=0}^{n-1} (6 Q_i + 3 U_i + 3 D_i + 2 L_i + E_i)$$ For any permutation $f$ and charge configuration $S$, the gravitational anomaly is invariant under the action of the group: $$\text{accGrav}(f \cdot S) = \text{accGrav}(S)$$ where $f \cdot S$ denotes the charge configuration after the permutations $f$ have been applied to the generations of $S$.

In the Standard Model with $n$ fermion generations, let $S$ be a configuration of rational charges. Let $f$ be an element of the permutation group $\text{PermGroup } n \cong \prod_{j=1}^5 S_n$, which acts on the charges by permuting the generation indices independently for each of the five fermion species ($Q, u^c, d^c, L, e^c$). The $SU(2)$ gauge anomaly, defined as the sum of charges $\text{accSU2}(S) = \sum_{i=0}^{n-1} (3 Q_i + L_i)$, is invariant under these generation permutations. That is, $$\text{accSU2}(f \cdot S) = \text{accSU2}(S)$$ where $f \cdot S$ denotes the charge configuration resulting from applying the permutations $f$ to the generations of each species in $S$.

In the Standard Model with $n$ fermion generations, let $S$ be a configuration of rational charges for the five fermion species ($Q, u^c, d^c, L, e^c$). The $SU(3)$ gauge anomaly is defined by the linear map $\text{accSU3}(S) = \sum_{i=0}^{n-1} (2 Q_i + u^c_i + d^c_i)$, where $Q_i, u^c_i,$ and $d^c_i$ are the charges of the $i$-th generation left-handed quark doublet, right-handed up-type quark, and right-handed down-type quark, respectively. For any element $f$ in the permutation group $\text{PermGroup } n \cong \prod_{j=1}^5 S_n$ acting on the generations of each species, the $SU(3)$ anomaly is invariant under this action: $$\text{accSU3}(f \cdot S) = \text{accSU3}(S)$$ where $f \cdot S$ denotes the charge configuration after permuting the generation indices according to $f$.

In the Standard Model with $n$ fermion generations, let $S$ be a configuration of rational charges for the five fermion species (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 $e^c$). Let $f \in \text{PermGroup } n \cong \prod_{i=1}^5 S_n$ be a permutation that independently permutes the generation indices for each species. The $Y^2$ anomaly cancellation condition $\text{accYY}(S)$, defined as $$\text{accYY}(S) = \sum_{i=0}^{n-1} (Q_i + 8U_i + 2D_i + 3L_i + 6E_i)$$ is invariant under the action of the permutation $f$, such that: $$\text{accYY}(f \cdot S) = \text{accYY}(S)$$

In the Standard Model with $n$ generations of fermions, let $S$ be a configuration of rational charges and $f \in \text{PermGroup } n$ be a set of permutations that act on the generation indices for each of the five fermion species. The quadratic anomaly cancellation condition $\text{accQuad}(S)$ is invariant under the action of these permutations. That is, $$\text{accQuad}(f \cdot S) = \text{accQuad}(S)$$ where $f \cdot S$ denotes the charge configuration resulting from applying the permutations $f$ to $S$.

In the Standard Model with $n$ fermion generations, let $S$ be a configuration of rational charges. For any element $f$ in the permutation group $\text{PermGroup } n \cong \prod_{i=1}^5 S_n$, which permutes the generation indices of each of the five fermion species independently, the cubic anomaly cancellation condition $\text{accCube}$ is invariant under the group action. That is, $$\text{accCube}(\text{repCharges } f S) = \text{accCube}(S)$$ where $\text{repCharges } f S$ denotes the linear action of the permutation $f$ on the charge configuration $S$.