Physlib.QFT.QED.AnomalyCancellation.Odd.Parameterization

For a pure $U(1)$ gauge theory with $2n+1$ fermions, given two vectors of rational coefficients $g, f \in \mathbb{Q}^n$ and a rational scalar $a \in \mathbb{Q}$, this function constructs a specific vector in the space of linear solutions $\text{LinSols} \subset \mathbb{Q}^{2n+1}$. Let $P(g)$ and $P!(f)$ be the charge vectors in $\mathbb{Q}^{2n+1}$ corresponding to the first and second planes of the solution space, and let $P'(g)$ and $P!'(f)$ be their respective representations as elements of the linear solution space. Let $\mathcal{A}(x, y, z) = \sum_{i=1}^{2n+1} x_i y_i z_i$ be the symmetric trilinear form associated with the cubic anomaly. The resulting solution vector is defined as: \[ a \cdot \left( \mathcal{A}(P!(f), P!(f), P(g)) \cdot P'(g) - \mathcal{A}(P(g), P(g), P!(f)) \cdot P!'(f) \right) \] This construction provides a vector that satisfies the linear anomaly cancellation condition $\sum_{i=1}^{2n+1} x_i = 0$, and as noted in the documentation, it serves as a basis for generating solutions that also satisfy the cubic condition $\sum_{i=1}^{2n+1} x_i^3 = 0$.

Consider a pure $U(1)$ gauge theory with $2n+1$ fermions. Let $\mathcal{A}(x, y, z) = \sum_{i=1}^{2n+1} x_i y_i z_i$ be the symmetric trilinear form associated with the cubic anomaly cancellation condition. For any vectors of rational coefficients $g, f \in \mathbb{Q}^n$ and a rational scalar $a \in \mathbb{Q}$, let $P(g)$ and $P!(f)$ be the vectors in the space of charges $\mathbb{Q}^{2n+1}$ defined as the inclusions of the first and second planes of the linear solution space. The charge vector associated with the parameterized linear solution is given by: \[ a \cdot \left( \mathcal{A}(P!(f), P!(f), P(g)) \cdot P(g) - \mathcal{A}(P(g), P(g), P!(f)) \cdot P!(f) \right) \]

Consider a pure $U(1)$ gauge theory with $2n+1$ Weyl fermions. Let $g, f \in \mathbb{Q}^n$ be vectors of rational coefficients and $a \in \mathbb{Q}$ be a rational scalar. Let $x \in \mathbb{Q}^{2n+1}$ be the charge vector defined by the linear parameterization: \[ x = a \cdot \left( \mathcal{A}(P!(f), P!(f), P(g)) \cdot P(g) - \mathcal{A}(P(g), P(g), P!(f)) \cdot P!(f) \right) \] where $P(g)$ and $P!(f)$ are the inclusions of the first and second planes of linear solutions into the space of charges, and $\mathcal{A}(u, v, w) = \sum_{i=1}^{2n+1} u_i v_i w_i$ is the symmetric trilinear form associated with the cubic anomaly. The theorem states that this charge vector satisfies the cubic anomaly cancellation condition: \[ \sum_{i=1}^{2n+1} x_i^3 = 0 \]

For a pure $U(1)$ gauge theory with $2n+1$ Weyl fermions, given two vectors of rational coefficients $g, f \in \mathbb{Q}^n$ and a rational scalar $a \in \mathbb{Q}$, this function constructs a solution vector $x \in \mathbb{Q}^{2n+1}$ that satisfies the Anomaly Cancellation Conditions (ACCs). A solution $x$ must satisfy: 1. The linear condition: $\sum_{i=1}^{2n+1} x_i = 0$. 2. The cubic condition: $\sum_{i=1}^{2n+1} x_i^3 = 0$. The solution is constructed as: \[ x = a \cdot \left( \mathcal{A}(P!(f), P!(f), P(g)) \cdot P(g) - \mathcal{A}(P(g), P(g), P!(f)) \cdot P!(f) \right) \] where $P(g)$ and $P!(f)$ are specific embeddings of the parameter vectors into the space of charges $\mathbb{Q}^{2n+1}$, and $\mathcal{A}(u, v, w) = \sum_{i=1}^{2n+1} u_i v_i w_i$ is the symmetric trilinear form associated with the cubic anomaly.

Consider a pure $U(1)$ gauge theory with $2n+1$ fermions, where $n$ is a natural number. Let $S$ be a solution to the anomaly cancellation conditions (meaning the sum of its components and the sum of the cubes of its components are both zero). Suppose $S$ can be decomposed as $S = P(g) + P!(f)$ for some parameter vectors $g, f \in \mathbb{Q}^n$, where $P(g)$ and $P!(f)$ are the charge vectors in $\mathbb{Q}^{2n+1}$ associated with these parameters. Then the symmetric trilinear form $\mathcal{A}(x, y, z) = \sum_{i=0}^{2n} x_i y_i z_i$ satisfies the following identity: \[ \mathcal{A}(P(g), P(g), P!(f)) = -\mathcal{A}(P!(f), P!(f), P(g)) \]

Let $S$ be a solution to the anomaly cancellation conditions for a pure $U(1)$ gauge theory with $2n+3$ fermions (corresponding to the index $n+1$ in the parameterization). We say that $S$ is in the **Generic Case** if, for every pair of vectors $g, f \in \mathbb{Q}^{n+1}$ such that $S = P(g) + P!(f)$, the symmetric trilinear form $\mathcal{A}$ satisfies: \[ \mathcal{A}(P(g), P(g), P!(f)) \neq 0 \] where $\mathcal{A}(x, y, z) = \sum_{i} x_i y_i z_i$ is the trilinear form associated with the cubic anomaly, and $P(g)$ and $P!(f)$ are the mappings of the parameter vectors into the charge space.

Let $S$ be a solution to the anomaly cancellation conditions for a pure $U(1)$ gauge theory with $2n+3$ fermions. If there exist rational parameter vectors $g, f \in \mathbb{Q}^{n+1}$ such that the charge vector $S$ can be decomposed as $S = P(g) + P!(f)$ and the symmetric trilinear form $\mathcal{A}(x, y, z) = \sum_i x_i y_i z_i$ satisfies $\mathcal{A}(P(g), P(g), P!(f)) \neq 0$, then $S$ satisfies the condition for the Generic Case.

Let $S$ be a solution to the anomaly cancellation conditions (ACC) for a pure $U(1)$ gauge theory with $2n+3$ Weyl fermions (where $n+1$ is the dimension of the parameterizing subspaces). The solution $S$ is said to be in the **special case** if, for every pair of rational vectors $g, f \in \mathbb{Q}^{n+1}$ such that the charge vector of $S$ can be decomposed as $S = P(g) + P!(f)$, the symmetric trilinear form $\mathcal{A}$ associated with the cubic anomaly satisfies: \[ \mathcal{A}(P(g), P(g), P!(f)) = 0 \] where $\mathcal{A}(x, y, z) = \sum_i x_i y_i z_i$, and $P, P!: \mathbb{Q}^{n+1} \to \mathbb{Q}^{2n+3}$ are the linear maps embedding the parameter vectors into the space of charges.

Let $S$ be a solution to the anomaly cancellation conditions for a pure $U(1)$ gauge theory with $2n+3$ fermions, where $S$ is represented by a vector of rational charges in $\mathbb{Q}^{2n+3}$ satisfying $\sum x_i = 0$ and $\sum x_i^3 = 0$. Suppose there exist coefficient vectors $g, f \in \mathbb{Q}^{n+1}$ such that the charge vector of $S$ can be expressed as the sum of points in the two linear solution planes, $S = P(g) + P!(f)$, and the symmetric trilinear form $\mathcal{A}(x, y, z) = \sum_i x_i y_i z_i$ satisfies $\mathcal{A}(P(g), P(g), P!(f)) = 0$. Then $S$ satisfies the condition for the **Special Case** in the classification of odd-dimensional anomaly solutions.

Let $S$ be a solution to the anomaly cancellation conditions for a pure $U(1)$ gauge theory with $2n+3$ Weyl fermions ($n \in \mathbb{N}$), where the solution consists of a vector of rational charges $x \in \mathbb{Q}^{2n+3}$ satisfying $\sum x_i = 0$ and $\sum x_i^3 = 0$. Then $S$ satisfies either the **Generic Case** condition or the **Special Case** condition. Specifically, for any decomposition of the charge vector as $x = P(g) + P!(f)$ with $g, f \in \mathbb{Q}^{n+1}$, it must hold that either the symmetric trilinear form $\mathcal{A}(P(g), P(g), P!(f)) \neq 0$ (Generic Case) or $\mathcal{A}(P(g), P(g), P!(f)) = 0$ (Special Case), where $\mathcal{A}(u, v, w) = \sum_i u_i v_i w_i$.

Consider a pure $U(1)$ gauge theory with $2n+3$ Weyl fermions ($n \in \mathbb{N}$). Let $S$ be a solution to the anomaly cancellation conditions, which consist of a vector of rational charges $x \in \mathbb{Q}^{2n+3}$ satisfying $\sum_i x_i = 0$ (gravitational anomaly) and $\sum_i x_i^3 = 0$ (gauge anomaly). If $S$ is in the **Generic Case**—meaning that for any decomposition of its charge vector as $x = P(g) + P!(f)$ with $g, f \in \mathbb{Q}^{n+1}$, the symmetric trilinear form satisfies $\mathcal{A}(P(g), P(g), P!(f)) \neq 0$—then $S$ can be represented by the parameterization. Specifically, there exists a rational scalar $a \in \mathbb{Q}$ and parameter vectors $g, f \in \mathbb{Q}^{n+1}$ such that \[ S = a \cdot \left( \mathcal{A}(P!(f), P!(f), P(g)) \cdot P(g) - \mathcal{A}(P(g), P(g), P!(f)) \cdot P!(f) \right) \] where $P(g)$ and $P!(f)$ are the standard embeddings of the parameter vectors into the charge space and $\mathcal{A}(u, v, w) = \sum_i u_i v_i w_i$.

Consider a pure $U(1)$ gauge theory with $2n+3$ Weyl fermions ($n \in \mathbb{N}$). Let $S$ be a solution to the anomaly cancellation conditions (ACC), which consist of a charge vector $x \in \mathbb{Q}^{2n+3}$ such that $\sum_{i=1}^{2n+3} x_i = 0$ and $\sum_{i=1}^{2n+3} x_i^3 = 0$. If $S$ satisfies the "special case" condition—meaning that for any decomposition of its charge vector as $x = P(g) + P!(f)$ (where $g, f \in \mathbb{Q}^{n+1}$ are parameter vectors), the symmetric trilinear form $\mathcal{A}(x, y, z) = \sum_i x_i y_i z_i$ satisfies $\mathcal{A}(P(g), P(g), P!(f)) = 0$—then $S$ also satisfies the "line in cubic" property. This property states that every linear combination of the vectors $P(g)$ and $P!(f)$ is a solution to the cubic anomaly cancellation condition: \[ \sum_{i=1}^{2n+3} (a P(g)_i + b P!(f)_i)^3 = 0 \] for all rational coefficients $a, b \in \mathbb{Q}$.

Consider a pure $U(1)$ gauge theory with $2n+3$ Weyl fermions ($n \in \mathbb{N}$). Let $S$ be a solution to the anomaly cancellation conditions (ACC), consisting of a charge vector $x \in \mathbb{Q}^{2n+3}$ such that $\sum_{i=1}^{2n+3} x_i = 0$ and $\sum_{i=1}^{2n+3} x_i^3 = 0$. If every permutation $\sigma$ of the charge vector $x$ satisfies the "special case" property—meaning that for every decomposition of the permuted vector $\sigma(x) = P(g) + P!(f)$, the symmetric trilinear form satisfies $\sum_{i=1}^{2n+3} P(g)_i^2 P!(f)_i = 0$—then the solution $S$ satisfies the `LineInCubicPerm` property. This implies that for every permutation $\sigma \in S_{2n+3}$, any linear combination of the parameter vectors $P(g)$ and $P!(f)$ associated with $\sigma(x)$ is itself a solution to the cubic anomaly equation: \[ \sum_{i=1}^{2n+3} (a P(g)_i + b P!(f)_i)^3 = 0 \] for all $a, b \in \mathbb{Q}$.

Consider a pure $U(1)$ gauge theory with an odd number of fermions $N = 2(n+2)+1$ for $n \in \mathbb{N}$ (specifically $N \ge 5$). Let $S$ be a solution to the anomaly cancellation conditions, consisting of a charge vector $x \in \mathbb{Q}^N$ such that $\sum_{i=1}^N x_i = 0$ and $\sum_{i=1}^N x_i^3 = 0$. If every permutation $\sigma \in S_N$ of the charge vector $x$ satisfies the "special case" property—meaning that for every decomposition of the permuted vector $\sigma(x) = P(g) + P!(f)$, the symmetric trilinear form satisfies $\sum_{i=1}^N P(g)_i^2 P!(f)_i = 0$—then the solution $S$ must be the zero vector, i.e., $x = (0, \dots, 0)$.