Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.PlusU1.QuadSol

In the context of the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a linear solution to the anomaly cancellation conditions and $C$ be a fixed reference charge configuration. For any rational scalars $a, b \in \mathbb{Q}$, the quadratic anomaly cancellation condition $accQuad$ evaluated at the linear combination $a S + b C$ satisfies the identity: $$accQuad(a S + b C) = a(a \cdot accQuad(S) + 2b \cdot quadBiLin(S, C))$$ where $quadBiLin$ is the symmetric bilinear form associated with the quadratic form $accQuad$.

Given a linear solution $S$ of the anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos, this function calculates the rational value $\alpha_1(S)$ defined by: $$\alpha_1(S) = -2 B(S, C)$$ where $B$ is the symmetric bilinear form `quadBiLin` associated with the quadratic anomaly cancellation conditions, and $C$ is a fixed reference charge configuration (typically representing the standard $U(1)$ charges or a specific vector in the charge space).

Given a linear solution $S$ of the anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos, the function $\alpha_2(S)$ calculates the rational value obtained by evaluating the quadratic anomaly cancellation condition $f$ at $S$: $$\alpha_2(S) = f(S)$$ where $f$ is the homogeneous quadratic map `accQuad`. Explicitly, for a charge configuration $S$, this is given by the sum: $$f(S) = \sum_{i=0}^{n-1} \left( Q_i(S)^2 - 2U_i(S)^2 + D_i(S)^2 - L_i(S)^2 + E_i(S)^2 \right)$$ where $Q_i, U_i, D_i, L_i, E_i$ are the charges of the $i$-th generation fermions.

For any solution $S$ to the quadratic anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos, the coefficient $\alpha_2(S)$ is zero. Here, $\alpha_2(S)$ is defined as the value of the homogeneous quadratic map $f$ evaluated at the charge configuration $S$, where $f(S) = \sum_{i=0}^{n-1} (Q_i(S)^2 - 2U_i(S)^2 + D_i(S)^2 - L_i(S)^2 + E_i(S)^2)$.

In the context of the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a linear solution to the anomaly cancellation conditions and $C$ be a fixed reference charge configuration. Let $f$ denote the quadratic anomaly cancellation condition `accQuad` and $B$ denote the associated symmetric bilinear form `quadBiLin`. Define the rational coefficients $\alpha_1(C, S) = -2 B(C, S)$ and $\alpha_2(S) = f(S)$. The quadratic anomaly cancellation condition evaluated at the linear combination of charges $(\alpha_1(C, S)) S + (\alpha_2(S)) C$ is zero: $$f(\alpha_1(C, S) \cdot S + \alpha_2(S) \cdot C) = 0$$

In the context of the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a solution to the linear anomaly cancellation conditions and $C$ be a reference charge configuration. If the coefficients $\alpha_1(C, S)$ and $\alpha_2(S)$ are both zero, then for any rational scalars $a, b \in \mathbb{Q}$, the linear combination $a S + b C$ satisfies the quadratic anomaly cancellation condition: $$accQuad(a S + b C) = 0$$ where $\alpha_1(C, S) = -2 B(S, C)$ is defined using the symmetric bilinear form $B$ associated with the quadratic form, and $\alpha_2(S) = accQuad(S)$ is the evaluation of the quadratic anomaly cancellation condition at $S$.

In the context of the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a solution to the linear anomaly cancellation conditions ($\text{LinSols}$) and $C$ be a fixed reference charge configuration. This function constructs a solution to the quadratic anomaly cancellation conditions ($\text{QuadSols}$) by taking a specific linear combination of $S$ and $C$: $$S' = \alpha_1(C, S) \cdot S + \alpha_2(S) \cdot C$$ where $\alpha_1(C, S) = -2 B(C, S)$ is a rational coefficient derived from the symmetric bilinear form $B$ associated with the quadratic condition, and $\alpha_2(S) = f(S)$ is the evaluation of the quadratic anomaly cancellation condition $f$ at $S$.

In the context of the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a solution to the quadratic anomaly cancellation conditions ($\text{QuadSols}$) and $C$ be a fixed reference charge configuration. The map `genericToQuad` applied to $S$ (viewed as a linear solution) results in the scalar multiplication $\alpha_1(C, S) \cdot S$. This holds because for any quadratic solution $S$, the evaluation of the quadratic anomaly cancellation condition $\alpha_2(S) = f(S)$ is zero, reducing the general definition $\text{genericToQuad}(S) = \alpha_1(C, S) \cdot S + \alpha_2(S) \cdot C$ to its first term. Here, $\alpha_1(C, S) = -2 B(S, C)$ is a rational coefficient derived from the symmetric bilinear form $B$ associated with the quadratic condition.

In the context of the $n$-generation Standard Model with right-handed neutrinos, let $S$ be a solution to the quadratic anomaly cancellation conditions ($\text{QuadSols}$) and $C$ be a fixed reference charge configuration. Let $\alpha_1(C, S) = -2 B(S, C)$ be a rational coefficient, where $B$ is the symmetric bilinear form associated with the quadratic anomaly condition. If $\alpha_1(C, S) \neq 0$, then applying the scalar inverse of $\alpha_1(C, S)$ to the solution generated by the map $\text{genericToQuad}(C, S)$ recovers the original solution $S$: $$(\alpha_1(C, S))^{-1} \cdot \text{genericToQuad}(C, S) = S$$ Here, $\cdot$ denotes the scalar multiplication action of $\mathbb{Q}$ on the space of quadratic solutions.

In the context of the $n$-generation Standard Model with right-handed neutrinos, given a linear solution $S$ to the anomaly cancellation conditions and two rational scalars $a, b \in \mathbb{Q}$, this function constructs a quadratic solution defined by the linear combination $a S + b C$, where $C$ is a fixed reference charge configuration. This construction is valid under the specific conditions that $\alpha_1(C, S) = 0$ and $\alpha_2(S) = 0$. Here, $\alpha_1(C, S) = -2 B(S, C)$ is the coefficient derived from the symmetric bilinear form associated with the quadratic anomaly cancellation condition, and $\alpha_2(S)$ is the evaluation of the quadratic condition at $S$.

Let $S$ be a quadratic solution to the anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos. Suppose that the coefficient $\alpha_1(C, S) = -2 B(S, C)$ is zero, where $B$ is the symmetric bilinear form associated with the quadratic anomaly equations and $C$ is a fixed reference charge configuration. Then, the quadratic solution constructed via the map `specialToQuad` using the linear solution $S$ and rational scalars $a=1$ and $b=0$ is equal to $S$.

In the context of the $n$-generation Standard Model with right-handed neutrinos, given a fixed reference charge configuration $C$, this function constructs a quadratic anomaly cancellation solution from a triple $(S, a, b)$ consisting of a linear solution $S \in \text{LinSols}$ and two rational scalars $a, b \in \mathbb{Q}$. The construction accounts for both generic and special cases: - If $\alpha_1(C, S) = 0$ and $\alpha_2(S) = 0$, the function returns the "special" quadratic solution $a S + b C$. - Otherwise, the function returns the "generic" quadratic solution $\alpha_1(C, S) S + \alpha_2(S) C$ scaled by the rational $a$. Here, $\alpha_1(C, S) = -2 B(S, C)$ is a coefficient derived from the symmetric bilinear form $B$ associated with the quadratic anomaly condition, and $\alpha_2(S) = f(S)$ is the evaluation of the quadratic anomaly condition $f$ at $S$. This map is a surjection from the space of linear solutions and rational pairs to the space of quadratic solutions.

Given a reference charge configuration $C$, this function maps a quadratic solution $S$ (where $S_1$ denotes its underlying linear solution component) of the anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos to a triple $(S_1, q_1, q_2) \in \text{LinSols} \times \mathbb{Q} \times \mathbb{Q}$. The mapping is defined as: $$S \mapsto \begin{cases} (S_1, 1, 0) & \text{if } \alpha_1(C, S_1) = 0 \\ (S_1, \alpha_1(C, S_1)^{-1}, 0) & \text{if } \alpha_1(C, S_1) \neq 0 \end{cases}$$ where $\alpha_1(C, S_1) = -2 B(S_1, C)$ is a coefficient derived from the symmetric bilinear form $B$ associated with the quadratic anomaly equations. This function acts as a right inverse to the map `toQuad`.

For any quadratic solution $S$ of the anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos, let $S_1$ denote its underlying linear solution component. For a given reference charge configuration $C$, the first component of the triple produced by the mapping $\text{toQuadInv}(C, S)$ is equal to $S_1$.

For any quadratic solution $S$ to the anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos, let $S_1$ denote its underlying linear solution component. Given a reference charge configuration $C$, the following equivalence holds: $$\alpha_1(C, S_1) = 0 \iff \alpha_1(C, S'_1) = 0 \land \alpha_2(S'_1) = 0$$ where $S'_1$ is the first component (the linear solution part) of the triple produced by the mapping $\text{toQuadInv}(C, S)$. Here, $\alpha_1(C, X) = -2B(X, C)$ is a coefficient derived from the symmetric bilinear form $B$ associated with the quadratic anomaly equations, and $\alpha_2(X) = f(X)$ is the evaluation of the quadratic anomaly cancellation condition $f$.

Let $S$ be a quadratic solution to the anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos, and let $S_1$ denote its underlying linear solution component. Let $C$ be a fixed reference charge configuration. If the coefficient $\alpha_1(C, S_1) = -2 B(S_1, C)$ is zero, where $B$ is the symmetric bilinear form associated with the quadratic anomaly equations, then the quadratic solution reconstructed via the `specialToQuad` map using the components of $\text{toQuadInv}(C, S)$ is equal to $S$. Specifically, in this case, the inverse map $\text{toQuadInv}(C, S)$ produces the triple $(S_1, 1, 0)$, and it holds that: $$\text{specialToQuad}(C, S_1, 1, 0, \dots) = S$$

For any quadratic solution $S$ to the anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos, let $S_1$ denote its underlying linear solution component, and let $C$ be a fixed reference charge configuration. If the coefficient $\alpha_1(C, S_1) \neq 0$, where $\alpha_1(C, S_1) = -2 B(S_1, C)$ is derived from the symmetric bilinear form $B$ associated with the quadratic anomaly equations, then it holds that: $$(\text{toQuadInv}(C, S))_{2,1} \cdot \text{genericToQuad}(C, (\text{toQuadInv}(C, S))_1) = S$$ Here, $\text{toQuadInv}(C, S)$ maps $S$ to a tuple in $\text{LinSols} \times \mathbb{Q} \times \mathbb{Q}$, with the subscript $1$ extracting the $\text{LinSols}$ component and the subscript $2,1$ extracting the first $\mathbb{Q}$ component. The $\cdot$ denotes the scalar multiplication action of $\mathbb{Q}$ on the space of quadratic solutions.

In the context of the $n$-generation Standard Model with right-handed neutrinos, let $C$ be a fixed reference charge configuration. For any solution $S$ to the quadratic anomaly cancellation conditions (denoted $S \in \text{QuadSols}$), the mapping $\text{toQuad}$ is a right inverse to the mapping $\text{toQuadInv}$. That is, reconstructing a quadratic solution from the components provided by the inverse map returns the original solution: $$\text{toQuad}(C, \text{toQuadInv}(C, S)) = S$$ Here, $\text{toQuadInv}$ maps a quadratic solution $S$ to a triple $(S_1, a, b) \in \text{LinSols} \times \mathbb{Q} \times \mathbb{Q}$, and $\text{toQuad}$ maps such a triple back to the space of quadratic solutions.

In the context of the $n$-generation Standard Model with right-handed neutrinos, let $C$ be a fixed reference charge configuration and $\text{QuadSols}$ be the space of solutions to the quadratic anomaly cancellation conditions. The mapping $\text{toQuad}(C, \cdot): \text{LinSols} \times \mathbb{Q} \times \mathbb{Q} \to \text{QuadSols}$, which constructs a quadratic solution from a linear solution $S \in \text{LinSols}$ and two rational scalars $a, b \in \mathbb{Q}$, is surjective.