Physlib.QFT.QED.AnomalyCancellation.BasisLinear

For a pure $U(1)$ gauge theory with $n+1$ fermions, the function assigns to each index $j \in \{0, 1, \dots, n-1\}$ a charge vector $x \in \mathbb{Q}^{n+1}$. This vector is defined such that its $j$-th component is $1$, its last component (at index $n$) is $-1$, and all other components are $0$. Mathematically, the components $x_i$ of the vector are: \[ x_i = \begin{cases} 1 & \text{if } i = j \\ -1 & \text{if } i = n \\ 0 & \text{otherwise} \end{cases} \] for $i \in \{0, 1, \dots, n\}$. These vectors form a basis for the space of solutions to the linear anomaly cancellation condition $\sum_{i=0}^n x_i = 0$.

For a pure $U(1)$ gauge theory with $n+1$ fermions, let $x^{(j)} \in \mathbb{Q}^{n+1}$ be the $j$-th basis charge vector for the linear anomaly cancellation conditions, where $j \in \{0, \dots, n-1\}$. Then the $j$-th component of the vector $x^{(j)}$ is equal to $1$.

Consider a pure $U(1)$ gauge theory with $n+1$ fermions. Let $x^{(k)} \in \mathbb{Q}^{n+1}$ be the $k$-th basis charge vector for the space of solutions to the linear anomaly cancellation condition, where $k \in \{0, 1, \dots, n-1\}$. For any index $j \in \{0, 1, \dots, n-1\}$ such that $k \neq j$, the $j$-th component of the vector $x^{(k)}$ is $0$.

For a pure $U(1)$ gauge theory with $n+1$ fermions, this function assigns to each index $j \in \{0, 1, \dots, n-1\}$ a charge vector that satisfies the linear anomaly cancellation condition (ACC). This vector is an element of the space $\text{LinSols}$, and its components $x_i \in \mathbb{Q}$ are given by: \[ x_i = \begin{cases} 1 & \text{if } i = j \\ -1 & \text{if } i = n \\ 0 & \text{otherwise} \end{cases} \] for $i \in \{0, 1, \dots, n\}$. These vectors form a basis for the subspace of solutions to the gravitational anomaly condition $\sum_{i=0}^n x_i = 0$.

Consider a pure $U(1)$ gauge theory with $n$ fermions. Let $f_1, \dots, f_k$ be a collection of charge vectors in $\mathbb{Q}^n$ that satisfy the linear anomaly cancellation condition (ACC) $\sum_{l=1}^n (f_i)_l = 0$. For any fermion index $j \in \{1, \dots, n\}$, the $j$-th component of the sum of these solutions is equal to the sum of the $j$-th components of each individual solution: $$\left( \sum_{i=1}^k f_i \right)_j = \sum_{i=1}^k (f_i)_j$$

For a pure $U(1)$ gauge theory with $n+1$ fermions, let $\text{LinSols} \subseteq \mathbb{Q}^{n+1}$ be the vector space of charge vectors $(x_0, x_1, \dots, x_n)$ satisfying the linear anomaly cancellation condition $\sum_{i=0}^n x_i = 0$. This definition establishes a linear isomorphism $\Phi: \text{LinSols} \cong \mathbb{Q}^n$ which serves as the coordinate map for the basis of the solution space. The forward map $\Phi$ sends a solution vector to its first $n$ components: \[ \Phi(x_0, x_1, \dots, x_n) = (x_0, x_1, \dots, x_{n-1}) \] The inverse map $\Phi^{-1}: \mathbb{Q}^n \to \text{LinSols}$ reconstructs the full solution vector using the basis vectors $\mathbf{b}_i \in \text{LinSols}$ (where $\mathbf{b}_i$ has $1$ at index $i$, $-1$ at index $n$, and $0$ elsewhere) as follows: \[ \Phi^{-1}(f) = \sum_{i=0}^{n-1} f_i \mathbf{b}_i \]

For a pure $U(1)$ gauge theory with $n+1$ fermions, let $\text{LinSols}$ be the vector space over $\mathbb{Q}$ consisting of charge vectors $(x_0, x_1, \dots, x_n) \in \mathbb{Q}^{n+1}$ that satisfy the linear anomaly cancellation condition $\sum_{i=0}^n x_i = 0$. This definition constructs a basis for $\text{LinSols}$ indexed by $\text{Fin } n = \{0, 1, \dots, n-1\}$. The basis is defined such that the coordinate representation (the `repr` map) of a solution vector corresponds to the map $\Phi: \text{LinSols} \to \mathbb{Q}^n$ that extracts the first $n$ components of the vector.

For a pure $U(1)$ gauge theory with $n+1$ Weyl fermions, the vector space of linear solutions to the Anomaly Cancellation Conditions (ACCs), denoted by $\text{LinSols}$, is a finite-dimensional module over the field of rational numbers $\mathbb{Q}$. The space $\text{LinSols}$ consists of charge vectors $(x_0, x_1, \dots, x_n) \in \mathbb{Q}^{n+1}$ that satisfy the gravitational anomaly condition $\sum_{i=0}^n x_i = 0$.

For a pure $U(1)$ gauge theory with $n+1$ Weyl fermions, let $\text{LinSols}$ be the vector space over $\mathbb{Q}$ consisting of charge vectors $(x_0, x_1, \dots, x_n) \in \mathbb{Q}^{n+1}$ that satisfy the linear anomaly cancellation condition $\sum_{i=0}^n x_i = 0$. The dimension of this space over the rational numbers $\mathbb{Q}$, denoted by $\operatorname{finrank}_{\mathbb{Q}}(\text{LinSols})$, is equal to $n$.