Physlib.Particles.BeyondTheStandardModel.RHN.AnomalyCancellation.PlusU1.QuadSolToSol

Given a charge configuration $S \in \mathbb{Q}^{6n}$ that satisfies the quadratic anomaly cancellation conditions (a "quad solution"), the function $\alpha_1$ calculates the rational value: \[ \alpha_1(S) = -3 \cdot f(S, S, v_{B-L}) \] where $f: \mathbb{Q}^{6n} \times \mathbb{Q}^{6n} \times \mathbb{Q}^{6n} \to \mathbb{Q}$ is the symmetric trilinear form associated with the cubic anomaly cancellation condition (ACC) for the Standard Model with right-handed neutrinos, and $v_{B-L} \in \mathbb{Q}^{6n}$ is the vector representing the $B-L$ (Baryon minus Lepton) charges. This coefficient arises in the construction of a full solution to the ACCs from a quadratic solution.

Given a configuration of rational charges $S \in \mathbb{Q}^{6n}$ that satisfies the quadratic anomaly cancellation conditions (referred to as a "quad solution"), the function $\alpha_2$ returns the rational value obtained by evaluating the cubic anomaly cancellation condition (ACC) at $S$. Specifically, $\alpha_2(S) = \mathcal{A}_{\text{cube}}(S)$, where $\mathcal{A}_{\text{cube}}: \mathbb{Q}^{6n} \to \mathbb{Q}$ is the homogeneous cubic map defined by the sum: \[ \mathcal{A}_{\text{cube}}(S) = \sum_{i=0}^{n-1} \left( 6 Q_i(S)^3 + 3 U_i(S)^3 + 3 D_i(S)^3 + 2 L_i(S)^3 + E_i(S)^3 + N_i(S)^3 \right) \] where $Q_i, U_i, D_i, L_i, E_i,$ and $N_i$ are the rational charges assigned to the six fermion species of the $i$-th generation. This value serves as a coefficient in the construction of full solutions to the anomaly cancellation conditions from quadratic ones.

For an $n$-generation Standard Model with right-handed neutrinos, let $S \in \mathbb{Q}^{6n}$ be a configuration of rational charges that satisfies the quadratic anomaly cancellation conditions (a "quad solution"). Let $\mathcal{A}_{\text{cube}}: \mathbb{Q}^{6n} \to \mathbb{Q}$ be the cubic anomaly cancellation condition (ACC) map, and let $f$ be its associated symmetric trilinear form. Define the cubic coefficients $\alpha_1(S) = -3 f(S, S, v_{B-L})$ and $\alpha_2(S) = \mathcal{A}_{\text{cube}}(S)$, where $v_{B-L}$ is the vector of Baryon minus Lepton charges. If $\alpha_1(S) = 0$ and $\alpha_2(S) = 0$, then for any rational scalars $a, b \in \mathbb{Q}$, the charge configuration $a S + b v_{B-L}$ satisfies the cubic anomaly cancellation condition: \[ \mathcal{A}_{\text{cube}}(a S + b v_{B-L}) = 0 \]

For any configuration of rational charges $S$ that is a full solution to the anomaly cancellation conditions for $n$ generations of fermions, the cubic anomaly coefficient $\alpha_2(S)$ (which represents the evaluation of the cubic anomaly cancellation condition $\mathcal{A}_{\text{cube}}$ on $S$) is equal to zero.

For any configuration of rational charges $S \in \mathbb{Q}^{6n}$ that satisfies the quadratic anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos, let $\mathcal{A}_{\text{cube}}: \mathbb{Q}^{6n} \to \mathbb{Q}$ be the cubic anomaly cancellation map and $f$ its associated symmetric trilinear form. Define the rational coefficients $\alpha_1(S) = -3 f(S, S, v_{B-L})$ and $\alpha_2(S) = \mathcal{A}_{\text{cube}}(S)$, where $v_{B-L}$ is the vector of $B-L$ (Baryon minus Lepton) charges. Then the cubic anomaly cancellation condition vanishes for the charge configuration constructed via the $B-L$ mapping with these coefficients: \[ \mathcal{A}_{\text{cube}}(\text{BL.addQuad}(S, \alpha_1(S), \alpha_2(S))) = 0 \]

For any configuration of rational charges $S$ that is a full solution to the anomaly cancellation conditions for $n$ generations of fermions, let $\alpha_1(S) = -3 f(S, S, v_{B-L})$ and $\alpha_2(S) = \mathcal{A}_{\text{cube}}(S)$ be the cubic coefficients, where $f$ is the symmetric trilinear form and $v_{B-L}$ is the $B-L$ charge vector. Then the charge configuration produced by the $B-L$ construction $\text{BL.addQuad}$ satisfies: \[ \text{BL.addQuad}(S, \alpha_1(S), \alpha_2(S)) = \alpha_1(S) \cdot S \] where $\cdot$ denotes the scalar multiplication of the configuration by the rational number $\alpha_1(S)$.

Given a configuration of rational charges $S \in \mathbb{Q}^{6n}$ that satisfies the quadratic anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos (a "quadratic solution"), this function constructs a full solution $S' \in \text{Sols}$ that satisfies both the quadratic and cubic anomaly cancellation conditions. The construction is defined by: \[ S' = \text{BL.addQuad}(S, \alpha_1(S), \alpha_2(S)) \] where $\alpha_1(S) = -3 f(S, S, v_{B-L})$ and $\alpha_2(S) = \mathcal{A}_{\text{cube}}(S)$. Here, $v_{B-L}$ is the vector of $B-L$ (Baryon minus Lepton) charges, $\mathcal{A}_{\text{cube}}$ is the cubic anomaly cancellation map, and $f$ is the symmetric trilinear form associated with $\mathcal{A}_{\text{cube}}$.

Let $S$ be a configuration of rational charges that satisfies the full set of (linear, quadratic, and cubic) anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos. Let $S.1$ denote its underlying charge vector. The theorem states that applying the `generic` construction (which maps quadratic solutions to full solutions) to $S.1$ results in the scalar multiplication of the original solution $S$ by the first cubic coefficient $\alpha_1(S.1)$. That is: \[ \text{generic}(S.1) = \alpha_1(S.1) \cdot S \] where $\alpha_1(S.1) = -3 f(S.1, S.1, v_{B-L})$ is the rational coefficient determined by the symmetric trilinear form $f$ and the $B-L$ charge vector $v_{B-L}$.

Let $S$ be a configuration of rational charges that satisfies the full set of anomaly cancellation conditions (linear, quadratic, and cubic) for the $n$-generation Standard Model with right-handed neutrinos. Let $S.1$ denote its underlying charge vector. If the rational coefficient $\alpha_1(S.1) = -3 f(S.1, S.1, v_{B-L})$ is non-zero, then scaling the solution produced by the generic construction $\text{generic}(S.1)$ by the inverse $(\alpha_1(S.1))^{-1}$ recovers the original solution $S$. That is, \[ (\alpha_1(S.1))^{-1} \cdot \text{generic}(S.1) = S \] where $f$ is the symmetric trilinear form associated with the cubic anomaly cancellation condition and $v_{B-L}$ is the $B-L$ charge vector.

Given a configuration of rational charges $S \in \mathbb{Q}^{6n}$ that satisfies the quadratic anomaly cancellation conditions (a "quad solution"), and rational scalars $a, b \in \mathbb{Q}$, if the cubic coefficients satisfy $\alpha_1(S) = 0$ and $\alpha_2(S) = 0$, then this function constructs a full solution to the anomaly cancellation conditions. The resulting solution is the charge configuration $a S + b v_{B-L}$, where $v_{B-L}$ is the vector of $B-L$ (Baryon minus Lepton) charges. The hypotheses $\alpha_1(S) = 0$ and $\alpha_2(S) = 0$ ensure that this linear combination satisfies the cubic anomaly cancellation condition $\mathcal{A}_{\text{cube}}(a S + b v_{B-L}) = 0$.

For any solution $S$ of the full anomaly cancellation conditions (ACCs) for the Standard Model with $n$ generations of right-handed neutrinos, let $S_{quad}$ be its underlying quadratic solution. If the cubic coefficient $\alpha_1(S_{quad})$ vanishes, then the solution constructed via the "special" mapping using parameters $a=1$ and $b=0$ (corresponding to the charge configuration $1 \cdot S_{quad} + 0 \cdot v_{B-L}$) is equal to $S$. Here, $v_{B-L}$ is the vector of Baryon minus Lepton charges, and the coefficient $\alpha_2(S_{quad})$ is guaranteed to be zero since $S$ is a full solution.

For a system with $n$ generations of the Standard Model with right-handed neutrinos, this function defines a map from the product of the space of quadratic solutions and two rational scalars, $(S, a, b) \in \text{QuadSols} \times \mathbb{Q} \times \mathbb{Q}$, to the space of full solutions $\text{Sols}$. The construction handles two cases based on the cubic coefficients $\alpha_1(S)$ and $\alpha_2(S)$: - If $\alpha_1(S) = 0$ and $\alpha_2(S) = 0$, the result is the "special" solution $a S + b v_{B-L}$, where $v_{B-L}$ is the vector of $B-L$ charges. - Otherwise, the result is the scalar product $a \cdot S_{\text{generic}}$, where $S_{\text{generic}}$ is the full solution constructed from $S$ using the generic method. This map is designed to be a surjection from $\text{QuadSols} \times \mathbb{Q}^2$ onto the set of full anomaly cancellation solutions.

For a system with $n$ generations of right-handed neutrinos, this function maps a solution $S$ of the full anomaly cancellation conditions to a triple $(S_{quad}, a, b) \in \text{QuadSols} \times \mathbb{Q} \times \mathbb{Q}$, where $S_{quad}$ is the underlying quadratic solution. The mapping is defined as follows: If the first cubic coefficient $\alpha_1(S_{quad})$ is zero, the function returns $(S_{quad}, 1, 0)$. If $\alpha_1(S_{quad}) \neq 0$, it returns $(S_{quad}, \alpha_1(S_{quad})^{-1}, 0)$. This map acts as a right-inverse to the `quadSolToSol` function, which constructs full solutions from quadratic ones.

For any solution $S$ to the anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos, the first component of the triple $(S_{quad}, a, b)$ produced by the map $\text{quadSolToSolInv}(S)$ is equal to the underlying quadratic solution component of $S$ (denoted $S.1$).

For any solution $S$ to the anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos, let $S_{quad}$ be its underlying quadratic solution. If the first cubic coefficient of this quadratic solution vanishes, $\alpha_1(S_{quad}) = 0$, then both cubic coefficients $\alpha_1$ and $\alpha_2$ vanish for the quadratic solution component produced by the inverse map $\text{quadSolToSolInv}(S)$. That is, if $S'_{quad} = (\text{quadSolToSolInv}(S))_1$, then $\alpha_1(S'_{quad}) = 0$ and $\alpha_2(S'_{quad}) = 0$.

Let $S$ be a solution to the full anomaly cancellation conditions for the $n$-generation Standard Model with right-handed neutrinos, and let $S.1$ be its underlying quadratic solution. If the first cubic coefficient $\alpha_1(S.1) = -3 \cdot f(S.1, S.1, v_{B-L})$ is non-zero, then it is not the case that both the coefficients $\alpha_1$ and $\alpha_2$ vanish when evaluated at the first component of the triple returned by the inverse mapping $\text{quadSolToSolInv}(S)$.

For any solution $S$ to the full anomaly cancellation conditions (ACCs) for the $n$-generation Standard Model with right-handed neutrinos, let $S.1$ be its underlying quadratic solution. If the cubic coefficient $\alpha_1(S.1)$ vanishes, then applying the `special` reconstruction map to the triple $(S_{quad}, a, b)$ obtained from the inverse mapping $\text{quadSolToSolInv}(S)$—where $S_{quad} = (\text{quadSolToSolInv}(S))_1$, $a = (\text{quadSolToSolInv}(S))_{2,1}$, and $b = (\text{quadSolToSolInv}(S))_{2,2}$—recovers the original solution $S$.

Let $S$ be a solution to the full set of anomaly cancellation conditions (linear, quadratic, and cubic) for the $n$-generation Standard Model with right-handed neutrinos, and let $S.1$ be its underlying quadratic solution component. Suppose that the first cubic coefficient $\alpha_1(S.1) = -3 f(S.1, S.1, v_{B-L})$ is non-zero, where $f$ is the symmetric trilinear form associated with the cubic anomaly cancellation condition and $v_{B-L}$ is the vector of $B-L$ (Baryon minus Lepton) charges. Let $(S_{quad}, a, b) \in \text{QuadSols} \times \mathbb{Q} \times \mathbb{Q}$ be the triple produced by the inverse mapping $\text{quadSolToSolInv}(S)$. Then, the original solution $S$ can be recovered by scaling the solution produced by the generic construction $\text{generic}(S_{quad})$ by the rational scalar $a$: \[ a \cdot \text{generic}(S_{quad}) = S \] where $S_{quad}$ is the first component and $a$ is the second component of the triple.

For the $n$-generation Standard Model with right-handed neutrinos, let $\text{Sols}$ be the space of solutions to the full anomaly cancellation conditions and let $\text{QuadSols}$ be the space of solutions to the quadratic conditions. Let $\text{quadSolToSolInv} : \text{Sols} \to \text{QuadSols} \times \mathbb{Q} \times \mathbb{Q}$ be the mapping that assigns a triple of a quadratic solution and two rational scalars to a full solution, and let $\text{quadSolToSol} : \text{QuadSols} \times \mathbb{Q} \times \mathbb{Q} \to \text{Sols}$ be the mapping that constructs a full solution from such a triple. This theorem states that $\text{quadSolToSolInv}$ is a right-inverse of $\text{quadSolToSol}$, such that for any solution $S \in \text{Sols}$: \[ \text{quadSolToSol}(\text{quadSolToSolInv}(S)) = S \]

For the $n$-generation Standard Model with right-handed neutrinos, the mapping $\text{quadSolToSol} : \text{QuadSols} \times \mathbb{Q} \times \mathbb{Q} \to \text{Sols}$ is surjective.