Physlib.QFT.QED.AnomalyCancellation.Permutations

For a given natural number $n$, the permutation group of $n$ fermions is defined as the symmetric group $S_n$, which consists of all permutations (bijections) of the set $\{0, 1, \dots, n-1\}$.

For any natural number $n$, the set of permutations of $n$ fermions, denoted by $\text{PermGroup } n$ (mathematically the symmetric group $S_n$), is equipped with a group structure where the group operation is the composition of permutations.

For a given natural number $n$ and a permutation $f \in S_n$ (representing an element of `PermGroup n`), this definition provides the $\mathbb{Q}$-linear map from the space of charges $\mathbb{Q}^n$ to itself that acts by permuting the indices. For any charge configuration $S: \{0, 1, \dots, n-1\} \to \mathbb{Q}$, the map sends $S$ to the composition $S \circ f$.

For a given natural number $n$, this definition constructs the group representation of the symmetric group $S_n$ (representing the permutations of $n$ fermions) on the $\mathbb{Q}$-vector space of rational charges $\mathbb{Q}^n$. The representation maps each permutation $f \in S_n$ to a $\mathbb{Q}$-linear automorphism of the charge space, where the action on a charge configuration $S \in \mathbb{Q}^n$ is given by the composition $S \circ f^{-1}$.

For a given natural number $n$, let $f \in S_n$ be a permutation in the permutation group of $n$ fermions, and let $S \in \mathbb{Q}^n$ be a configuration of rational charges. The gravitational anomaly cancellation (ACC) function for a Pure $U(1)$ system, denoted by $\text{accGrav}_n$, is invariant under the permutation of charges: $$\text{accGrav}_n(f \cdot S) = \text{accGrav}_n(S),$$ where $f \cdot S$ denotes the action of the permutation $f$ on the charge vector $S$.

For a pure $U(1)$ anomaly cancellation system with $n$ fermions, let $S \in \mathbb{Q}^n$ be the vector of rational charges and let $f \in S_n$ be a permutation of the fermions (an element of the permutation group $\text{PermGroup } n$). The cubic anomaly cancellation condition, denoted by $\text{accCube}$, is invariant under the permutation of charges, such that $\text{accCube}(f \cdot S) = \text{accCube}(S)$, where $f \cdot S$ represents the action of the permutation on the charge configuration.

For a given natural number $n$, this definition defines a group action of the permutation group $S_n$ on the Pure $U(1)$ anomaly cancellation system (ACC system) consisting of $n$ fermions. The action is defined by the representation of $S_n$ on the space of rational charges $\mathbb{Q}^n$, where a permutation $f \in S_n$ acts on a charge configuration $S$ by permuting its components. This group action preserves the anomaly cancellation conditions of the system, specifically the linear gravitational anomaly cancellation condition $\sum_{i=0}^{n-1} S_i = 0$ and the cubic anomaly cancellation condition $\sum_{i=0}^{n-1} S_i^3 = 0$.

Consider a pure $U(1)$ anomaly cancellation system with $n$ fermions. Let $S \in \mathbb{Q}^n$ be the configuration of rational charges, where $S_i$ denotes the charge of the $i$-th fermion for $i \in \{0, \dots, n-1\}$. Let $f \in S_n$ be an element of the permutation group acting on the system. The group action of $f$ on the charge configuration $S$, denoted by $f \cdot S$, results in a new configuration whose $i$-th component is given by $(f \cdot S)_i = S_{f^{-1}(i)}$.

Let $n$ be a natural number and let $S$ be a solution to the linear anomaly cancellation conditions for the Pure $U(1)$ system with $n$ fermions. For any permutation $f \in S_n$, let $f \cdot S$ denote the action of the permutation on the solution. Then, for any index $i \in \{0, \dots, n-1\}$, the $i$-th component of the permuted solution is equal to the component of $S$ at the index $f^{-1}(i)$. That is, $(f \cdot S)_i = S_{f^{-1}(i)}$.

Given two injections $f, g: \{0, \dots, m-1\} \hookrightarrow \{0, \dots, n-1\}$, which represent two ordered subsets of $m$ fermions within a system of $n$ fermions, this definition constructs a permutation $\sigma$ of the set $\{0, \dots, n-1\}$ (an element of the group $S_n$ associated with `FamilyPermutations n`). This permutation $\sigma$ is defined such that it maps each index $g(i)$ to the index $f(i)$ for all $i \in \{0, \dots, m-1\}$, effectively moving the ordered subset defined by $g$ to the ordered subset defined by $f$.

Given two distinct indices $i, j$ in the set $\{0, \dots, n-1\}$, this definition provides an embedding $f: \{0, 1\} \hookrightarrow \{0, \dots, n-1\}$ such that $f(0) = i$ and $f(1) = j$.

For any distinct indices $i, j \in \{0, \dots, n-1\}$, let $f: \{0, 1\} \hookrightarrow \{0, \dots, n-1\}$ be the embedding (defined as `permTwoInj`) such that $f(0) = i$ and $f(1) = j$. Then $i$ is contained in the range of $f$.

Let $i$ and $j$ be distinct indices in the set $\{0, \dots, n-1\}$. Let $f: \{0, 1\} \hookrightarrow \{0, \dots, n-1\}$ be the embedding (defined as `permTwoInj`) such that $f(0) = i$ and $f(1) = j$. Let $\phi: \{0, 1\} \simeq \text{range}(f)$ be the equivalence mapping each element in the domain $\{0, 1\}$ to its image under $f$. Then the inverse of this equivalence applied to $i$ equals $0$, i.e., $\phi^{-1}(i) = 0$.

Let $i, j$ be distinct indices in the set $\{0, \dots, n-1\}$. Let $f: \{0, 1\} \hookrightarrow \{0, \dots, n-1\}$ be the embedding (defined as `permTwoInj`) that maps $0$ to $i$ and $1$ to $j$. Then $j$ is an element of the range of $f$.

Let $i, j$ be distinct indices in the set $\{0, \dots, n-1\}$. Let $f: \{0, 1\} \hookrightarrow \{0, \dots, n-1\}$ be the embedding (defined as `permTwoInj`) such that $f(0) = i$ and $f(1) = j$. Let $\phi: \{0, 1\} \simeq \text{range}(f)$ be the equivalence mapping each element in the domain to its image under $f$. Then the inverse of this equivalence applied to $j$ equals $1$, i.e., $\phi^{-1}(j) = 1$.

Given two pairs of distinct indices $(i, j)$ and $(i', j')$ in the set $\{0, \dots, n-1\}$, this definition constructs a permutation $\sigma$ in the symmetric group $S_n$ (the group action `FamilyPermutations n` on the Pure $U(1)$ ACC system). This permutation is defined such that it maps $i'$ to $i$ and $j'$ to $j$ (i.e., $\sigma(i') = i$ and $\sigma(j') = j$). It is constructed by composing the injection mapping $\{0, 1\} \to \{i, j\}$ with the inverse of the injection mapping $\{0, 1\} \to \{i', j'\}$ and extending this to a full permutation of $\{0, \dots, n-1\}$.

Let $i, j$ and $i', j'$ be pairs of distinct indices in the set of indices $\{0, \dots, n-1\}$. Let $\sigma$ be the permutation in the symmetric group $S_n$ (associated with the group action on the Pure $U(1)$ anomaly cancellation system) constructed to map the pair $(i', j')$ to $(i, j)$. Then the image of $i'$ under this permutation is $i$, i.e., $\sigma(i') = i$.

Let $i, j$ and $i', j'$ be pairs of distinct indices in the set of indices $\{0, \dots, n-1\}$. Let $\sigma$ be the permutation in the symmetric group $S_n$ defined by `permTwo` that maps the pair $(i', j')$ to $(i, j)$. Then the value of this permutation at $j'$ is $j$, i.e., $\sigma(j') = j$.

Given three distinct indices $i, j, k$ in the set of indices $\{0, \dots, n-1\}$ (represented by $\text{Fin } n$), this definition constructs an embedding (an injective function) $f: \text{Fin } 3 \hookrightarrow \text{Fin } n$. The mapping is defined such that $f(0) = i$, $f(1) = j$, and $f(2) = k$.

Given distinct indices $i, j, k \in \{0, \dots, n-1\}$ and the embedding $f: \{0, 1, 2\} \hookrightarrow \{0, \dots, n-1\}$ (defined as `permThreeInj`) that maps $0 \mapsto i$, $1 \mapsto j$, and $2 \mapsto k$, the index $i$ belongs to the range of $f$.

Let $i, j, k$ be distinct indices in the set $\{0, \dots, n-1\}$. Let $f: \{0, 1, 2\} \hookrightarrow \{0, \dots, n-1\}$ be the embedding (injective function) defined such that $f(0) = i$, $f(1) = j$, and $f(2) = k$. Let $e: \{0, 1, 2\} \cong \text{range}(f)$ be the equivalence (bijection) induced by restricting the codomain of $f$ to its range. Then the inverse of this equivalence, $e^{-1}$, maps the index $i$ to $0$.

Let $i, j, k$ be distinct indices in the set $\{0, \dots, n-1\}$. Let $f: \{0, 1, 2\} \hookrightarrow \{0, \dots, n-1\}$ be the embedding (injective function) defined such that $f(0) = i$, $f(1) = j$, and $f(2) = k$. Then $j$ is contained in the range of $f$.

Let $i, j, k$ be distinct indices in the set $\{0, \dots, n-1\}$. Let $f: \{0, 1, 2\} \hookrightarrow \{0, \dots, n-1\}$ be the embedding (injective function) defined such that $f(0) = i$, $f(1) = j$, and $f(2) = k$. Let $e: \{0, 1, 2\} \cong \text{range}(f)$ be the equivalence (bijection) induced by restricting the codomain of $f$ to its range. Then the inverse of this equivalence, $e^{-1}$, maps the index $j$ to $1$.

Given three distinct indices $i, j, k$ in the set $\{0, \dots, n-1\}$ and the injective mapping $f: \{0, 1, 2\} \hookrightarrow \{0, \dots, n-1\}$ defined such that $f(0) = i$, $f(1) = j$, and $f(2) = k$, then $k$ is an element of the range of $f$.

Given three distinct indices $i, j, k$ in the set of indices $\{0, \dots, n-1\}$ (denoted as $\text{Fin } n$), let $f: \text{Fin } 3 \hookrightarrow \text{Fin } n$ be the injective mapping (embedding) such that $f(0) = i$, $f(1) = j$, and $f(2) = k$. Let $E: \text{Fin } 3 \simeq \text{range}(f)$ be the equivalence between the domain and the range induced by $f$. The theorem states that the inverse of this equivalence, $E^{-1}$, maps the element $k$ (where $k$ is considered as an element of the range of $f$) to the index $2$.

For a system with $n$ fermions, given two sets of three distinct indices $(i, j, k)$ and $(i', j', k')$ in $\{0, \dots, n-1\}$, the definition `permThree` constructs a permutation $\sigma$ in the symmetric group $S_n$. This permutation is defined such that it maps the second triplet to the first triplet: $\sigma(i') = i$, $\sigma(j') = j$, and $\sigma(k') = k$. This permutation belongs to the group of actions defined for the Pure $U(1)$ anomaly cancellation system.

Let $i, j, k$ and $i', j', k'$ be two sets of three distinct indices in $\{0, \dots, n-1\}$. Let $\sigma$ be the permutation in the symmetric group $S_n$ (defined as `permThree`) constructed to map the triplet $(i', j', k')$ to $(i, j, k)$. Then the value of the permutation applied to the first index of the second triplet is the first index of the first triplet, i.e., $\sigma(i') = i$.

Let $n$ be the number of fermions in a Pure $U(1)$ anomaly cancellation system. Given two triplets of distinct indices $(i, j, k)$ and $(i', j', k')$ in the set of fermion indices $\{0, \dots, n-1\}$, let $\sigma$ be the permutation constructed to map the second triplet to the first triplet. Then, the image of the index $j'$ under the permutation $\sigma$ is $j$.

Let $n$ be the number of fermions in a Pure $U(1)$ anomaly cancellation system. Given two triplets of distinct indices $(i, j, k)$ and $(i', j', k')$ in the set of fermion indices $\{0, \dots, n-1\}$, let $\sigma$ be the permutation (defined as `permThree`) constructed to map the second triplet to the first triplet. Then, the image of the index $k'$ under the permutation $\sigma$ is $k$, i.e., $\sigma(k') = k$.

Let $S$ be a solution to the linear anomaly cancellation conditions for the Pure $U(1)$ system with $n$ fermions. Let $P$ be a property (predicate) defined on pairs of rational charges. If there exist two distinct indices $a, b \in \{0, \dots, n-1\}$ such that for every permutation $f$ in the symmetric group $S_n$, the property $P$ holds for the $a$-th and $b$-th components of the permuted solution $f \cdot S$, then $P$ holds for the $i$-th and $j$-th components of $S$ for all pairs of distinct indices $i, j \in \{0, \dots, n-1\}$.

Let $n$ be a natural number, and let $S$ be a solution to the linear anomaly cancellation conditions for the Pure $U(1)$ system with $n$ fermions. Let $P: \mathbb{Q} \times (\mathbb{Q} \times \mathbb{Q}) \to \text{Prop}$ be a property of triplets of rational numbers. Suppose there exist three distinct indices $a, b, c \in \{0, \dots, n-1\}$ such that for every permutation $f$ in the symmetric group $S_n$, the property $P$ holds for the components of the permuted solution $f \cdot S$ at indices $a, b$, and $c$, such that $P((f \cdot S)_a, ((f \cdot S)_b, (f \cdot S)_c))$ is true. Then, for any triplet of distinct indices $i, j, k \in \{0, \dots, n-1\}$, the property $P$ holds for the components of the original solution $S$ at those indices, i.e., $P(S_i, (S_j, S_k))$ is true.