Physlib.Particles.SuperSymmetry.MSSMNu.AnomalyCancellation.OrthogY3B3.PlaneWithY3B3

Given an anomaly-free charge vector $R$ that is orthogonal to $Y_3$ and $B_3$, and rational scalars $a, b, c \in \mathbb{Q}$, this function returns the linear combination \[ a Y_3 + b B_3 + c R \] The resulting vector is an element of the space of linear solutions $\text{LinSols}$ for the MSSM anomaly cancellation system.

Let $Y_3$ and $B_3$ be the anomaly-free charge vectors for the MSSM anomaly cancellation system. For any anomaly-free charge vector $R$ that is orthogonal to $Y_3$ and $B_3$, and for any rational scalars $a, b, c \in \mathbb{Q}$, the value of the linear solution $\text{planeY}_3\text{B}_3(R, a, b, c)$ is given by the linear combination $a Y_3 + b B_3 + c R$.

Let $Y_3$ and $B_3$ be anomaly-free charge vectors for the MSSM anomaly cancellation system, and let $R$ be an anomaly-free charge vector orthogonal to $Y_3$ and $B_3$. For any rational scalars $a, b, c, d \in \mathbb{Q}$, the linear combination defined by coefficients $(da, db, dc)$ is equal to the scalar multiple by $d$ of the linear combination defined by $(a, b, c)$: $$\text{planeY}_3\text{B}_3(R, da, db, dc) = d \cdot \text{planeY}_3\text{B}_3(R, a, b, c)$$

Let $R$ be an anomaly-free charge vector orthogonal to $Y_3$ and $B_3$ in the MSSM anomaly cancellation system. For any rational coefficients $a, b, c$ and $a', b', c'$, if $a = a'$, $b = b'$, and $c = c'$, then the linear combinations of $Y_3$, $B_3$, and $R$ are equal: \[ \text{planeY₃B₃}(R, a, b, c) = \text{planeY₃B₃}(R, a', b', c') \]

Let $Y_3$ and $B_3$ be the standard anomaly-free charge vectors for the MSSM anomaly cancellation system. For any anomaly-free charge vector $R$ that is orthogonal to $Y_3$ and $B_3$ such that $R \neq 0$, and for any rational scalars $a, b, c, a', b', c' \in \mathbb{Q}$, if the linear combinations $a Y_3 + b B_3 + c R$ and $a' Y_3 + b' B_3 + c' R$ are equal, then it follows that $a = a'$, $b = b'$, and $c = c'$.

Let $Y_3$ and $B_3$ be anomaly-free charge vectors in the MSSM anomaly cancellation system, and let $R$ be an anomaly-free charge vector that is orthogonal to $Y_3$ and $B_3$ with respect to the linear conditions. For any rational scalars $a, b, c \in \mathbb{Q}$, let $v = a Y_3 + b B_3 + c R$ be the corresponding linear combination of these charges. The value of the quadratic anomaly cancellation condition $\text{accQuad}$ for the vector $v$ is given by: \[ \text{accQuad}(a Y_3 + b B_3 + c R) = c \left( 2a B(Y_3, R) + 2b B(B_3, R) + c B(R, R) \right) \] where $B(\cdot, \cdot)$ denotes the symmetric bilinear form `quadBiLin` associated with the quadratic anomaly condition.

Let $Y_3$ and $B_3$ be the anomaly-free charge vectors for the MSSM, and let $\text{accCube}$ and $\text{cubeTriLin}$ be the associated homogeneous cubic map and symmetric trilinear form, respectively. For any anomaly-free charge vector $R$ orthogonal to $Y_3$ and $B_3$, and for any rational scalars $a, b, c \in \mathbb{Q}$, the value of the cubic anomaly cancellation condition evaluated at the linear combination $a Y_3 + b B_3 + c R$ is given by: $$\text{accCube}(a Y_3 + b B_3 + c R) = c^2 \left( 3 a \cdot \text{cubeTriLin}(R, R, Y_3) + 3 b \cdot \text{cubeTriLin}(R, R, B_3) + c \cdot \text{cubeTriLin}(R, R, R) \right)$$

Given an anomaly-free charge vector $R \in \mathbb{Q}^{20}$ (which is orthogonal to the solutions $Y_3$ and $B_3$) and rational parameters $c_1, c_2, c_3 \in \mathbb{Q}$, this function defines a vector $v$ in the space of linear solutions for the MSSM anomaly cancellation system. The vector $v$ is constructed as a linear combination of the form $v = a Y_3 + b B_3 + c R$, where the coefficients are given by: \[ a = c_2 B(R, R) - 2 c_3 B(B_3, R) \] \[ b = 2 c_3 B(Y_3, R) - c_1 B(R, R) \] \[ c = 2 c_1 B(B_3, R) - 2 c_2 B(Y_3, R) \] Here, $B(\cdot, \cdot)$ denotes the symmetric bilinear form `quadBiLin` associated with the quadratic anomaly cancellation condition. This specific construction ensures that the resulting vector $v$ satisfies the quadratic anomaly condition $\text{accQuad}(v) = 0$.

Let $R$ be an anomaly-free charge vector orthogonal to the solutions $Y_3$ and $B_3$ in the MSSM anomaly cancellation system. For any rational parameters $c_1, c_2, c_3 \in \mathbb{Q}$, the vector $v = \text{lineQuadAFL}(R, c_1, c_2, c_3)$ satisfies the quadratic anomaly cancellation condition, such that $\text{accQuad}(v) = 0$.

Let $R \in \mathbb{Q}^{20}$ be an anomaly-free charge vector that is orthogonal to the solutions $Y_3$ and $B_3$ in the MSSM anomaly cancellation system. For any rational parameters $c_1, c_2, c_3 \in \mathbb{Q}$, $\text{lineQuad}(R, c_1, c_2, c_3)$ defines a quadratic solution (a vector satisfying both linear and quadratic anomaly conditions). The solution is obtained by taking the linear combination $v = a Y_3 + b B_3 + c R$ where the coefficients $a, b, c$ are calculated based on $c_1, c_2, c_3$ and the bilinear form $B(\cdot, \cdot)$ associated with the quadratic anomaly condition, such that $\text{accQuad}(v) = 0$.

Let $R$ be an anomaly-free charge vector orthogonal to the solutions $Y_3$ and $B_3$ in the MSSM anomaly cancellation system. For any rational parameters $c_1, c_2, c_3 \in \mathbb{Q}$, the value of the quadratic solution vector $v = \text{lineQuad}(R, c_1, c_2, c_3)$ is given by the linear combination: \[ v = a Y_3 + b B_3 + c R \] where the coefficients $a, b, c \in \mathbb{Q}$ are defined as: \[ a = c_2 B(R, R) - 2 c_3 B(B_3, R) \] \[ b = 2 c_3 B(Y_3, R) - c_1 B(R, R) \] \[ c = 2 c_1 B(B_3, R) - 2 c_2 B(Y_3, R) \] and $B(\cdot, \cdot)$ denotes the symmetric bilinear form `quadBiLin` associated with the quadratic anomaly cancellation condition.

Let $R$ be an anomaly-free charge vector in the MSSM anomaly cancellation system that is orthogonal to the solutions $Y_3$ and $B_3$. For any rational parameters $a, b, c \in \mathbb{Q}$ and any rational scalar $d \in \mathbb{Q}$, the quadratic solution $\text{lineQuad}$ satisfies the following scaling property: \[ \text{lineQuad}(R, da, db, dc) = d \cdot \text{lineQuad}(R, a, b, c) \] where $\text{lineQuad}(R, a, b, c)$ is the vector in $\text{span}\{Y_3, B_3, R\}$ satisfying the quadratic anomaly condition as defined by the parameters $a, b, c$.

For a given anomaly-free charge assignment $T$, the rational coefficient $\alpha_1(T)$ is defined by the following expression involving the quadratic and cubic anomaly cancellation forms: $$\alpha_1(T) = 3 \cdot f(T, T, B_3) \cdot B(T, T) - 2 \cdot f(T, T, T) \cdot B(B_3, T)$$ where: - $B(\cdot, \cdot)$ is the symmetric bilinear form `quadBiLin` associated with the quadratic anomaly cancellation conditions. - $f(\cdot, \cdot, \cdot)$ is the symmetric trilinear form `cubeTriLin` associated with the cubic anomaly cancellation conditions. - $B_3$ is a fixed reference anomaly-free solution for the MSSM. - $T$ is an element of the subspace `AnomalyFreePerp`.

For a charge vector $T \in \mathbb{Q}^{20}$ (belonging to the subspace `AnomalyFreePerp`), the function $\alpha_2(T)$ is a rational value defined by the symmetric trilinear form $C$ (associated with the cubic anomaly cancellation condition `cubeTriLin`) and the symmetric bilinear form $B$ (associated with the quadratic anomaly cancellation condition `quadBiLin`). The value is given by: $$\alpha_2(T) = 2 C(T, T, T) B(Y_3, T) - 3 C(T, T, Y_3) B(T, T)$$ where $Y_3$ is the specific anomaly-free solution for the MSSM hypercharge assignments. This function serves as a coefficient in the expansion of the cubic anomaly condition on the plane spanned by the solutions.

For a charge vector $T$ in the subspace of anomaly-free perpendicular charges $\text{MSSMACC.AnomalyFreePerp}$, the rational value $\alpha_3(T)$ is defined as: \[ \alpha_3(T) = 6 \left( f(T, T, Y_3) B(B_3, T) - f(T, T, B_3) B(Y_3, T) \right) \] where: - $f$ is the symmetric trilinear form `cubeTriLin` associated with the cubic anomaly cancellation condition. - $B$ is the symmetric bilinear form `quadBiLin` associated with the quadratic anomaly cancellation condition. - $Y_3$ and $B_3$ are the specific anomaly-free charge solutions defined in the MSSM system.

Let $R$ be an anomaly-free charge vector orthogonal to the solutions $Y_3$ and $B_3$ in the MSSM anomaly cancellation system. For any rational parameters $c_1, c_2, c_3 \in \mathbb{Q}$, the value of the cubic anomaly cancellation condition evaluated at the quadratic solution $v = \text{lineQuad}(R, c_1, c_2, c_3)$ is given by the formula: $$\text{accCube}(v) = -4 \left( c_1 B(B_3, R) - c_2 B(Y_3, R) \right)^2 \left( \alpha_1(R) c_1 + \alpha_2(R) c_2 + \alpha_3(R) c_3 \right)$$ where $B(\cdot, \cdot)$ denotes the symmetric bilinear form `quadBiLin` associated with the quadratic anomaly condition, and $\alpha_1(R), \alpha_2(R), \alpha_3(R)$ are the rational coefficients derived from the symmetric trilinear form and bilinear form associated with the MSSM system.

Given a charge vector $R$ in the subspace of anomaly-free charges orthogonal to $Y_3$ and $B_3$, and rational scalars $a_1, a_2, a_3 \in \mathbb{Q}$, the function `lineCube` constructs a specific linear combination of $Y_3, B_3$, and $R$ that satisfies the cubic anomaly cancellation condition. The resulting vector $V$ is defined as: \[ V = (a_2 f(R, R, R) - 3 a_3 f(R, R, B_3)) Y_3 + (3 a_3 f(R, R, Y_3) - a_1 f(R, R, R)) B_3 + 3 (a_1 f(R, R, B_3) - a_2 f(R, R, Y_3)) R \] where $f$ is the symmetric trilinear form `cubeTriLin` associated with the cubic anomaly condition. This vector $V$ is an element of the space of linear solutions for the MSSM anomaly cancellation system and satisfies $\text{accCube}(V) = 0$.

Let $R$ be an anomaly-free charge vector orthogonal to the solutions $Y_3$ and $B_3$ in the MSSM anomaly cancellation system. For any rational scalars $a, b, c, d \in \mathbb{Q}$, the cubic anomaly-free solution constructed with scaled coefficients $(da, db, dc)$ is equal to the scalar multiple by $d$ of the solution constructed with coefficients $(a, b, c)$: $$\text{lineCube}(R, da, db, dc) = d \cdot \text{lineCube}(R, a, b, c)$$

Let $Y_3$ and $B_3$ be anomaly-free charge vectors for the MSSM, and let $R$ be a charge vector in the subspace of anomaly-free charges orthogonal to $Y_3$ and $B_3$. For any rational scalars $a_1, a_2$, and $a_3 \in \mathbb{Q}$, the vector $V$ defined by the linear combination \[ V = (a_2 f(R, R, R) - 3 a_3 f(R, R, B_3)) Y_3 + (3 a_3 f(R, R, Y_3) - a_1 f(R, R, R)) B_3 + 3 (a_1 f(R, R, B_3) - a_2 f(R, R, Y_3)) R \] satisfies the cubic anomaly cancellation condition $\text{accCube}(V) = 0$, where $f$ is the symmetric trilinear form `cubeTriLin`.

Let $R$ be an anomaly-free charge vector in the MSSM anomaly cancellation system that is orthogonal to the solutions $Y_3$ and $B_3$. For any rational scalars $a_1, a_2, a_3 \in \mathbb{Q}$, let $V = \text{lineCube}(R, a_1, a_2, a_3)$ be the corresponding linear combination of $Y_3, B_3$, and $R$ that satisfies the cubic anomaly condition. The value of the quadratic anomaly cancellation condition $\text{accQuad}$ for the vector $V$ is given by: \[ \text{accQuad}(V) = 3 \left( a_1 C(R, R, B_3) - a_2 C(R, R, Y_3) \right) \left( \alpha_1(R) a_1 + \alpha_2(R) a_2 + \alpha_3(R) a_3 \right) \] where $C(\cdot, \cdot, \cdot)$ is the symmetric trilinear form `cubeTriLin` associated with the cubic anomaly, and $\alpha_1(R), \alpha_2(R), \alpha_3(R)$ are rational coefficients depending on $R$ defined via the quadratic and cubic anomaly forms.

In the context of the anomaly cancellation conditions (ACCs) for the MSSM with three generations and right-handed neutrinos, let $f$ be the symmetric trilinear form `cubeTriLin` associated with the cubic anomaly, $B$ be the symmetric bilinear form `quadBiLin` associated with the quadratic anomaly, and $\cdot$ be the dot product on the space of charges. For any anomaly-free solution $T$, let $\text{proj}(T)$ be its projection onto the subspace orthogonal to the basis vectors $Y_3$ and $B_3$. Then the rational coefficient $\alpha_3$ evaluated at $\text{proj}(T)$ satisfies the identity: \[ \alpha_3(\text{proj}(T)) = 6 (Y_3 \cdot B_3)^3 \left( f(T, T, Y_3) B(B_3, T) - f(T, T, B_3) B(Y_3, T) \right). \]

In the context of the anomaly cancellation conditions for the MSSM with three generations and right-handed neutrinos, let $T$ be an anomaly-free solution and $\text{proj}(T)$ be its projection onto the subspace orthogonal to the charge vectors $Y_3$ and $B_3$. Let $\alpha_2$ and $\alpha_3$ be rational coefficients associated with the cubic anomaly cancellation equation, and let $\cdot$ denote the dot product on the space of charges. Then the following identity holds: $$\alpha_2(\text{proj}(T)) = -\alpha_3(\text{proj}(T)) (Y_3 \cdot T - 2(B_3 \cdot T)).$$

Let $T$ be an anomaly-free solution for the MSSM anomaly cancellation system. Let $\text{proj}(T)$ be the projection of the linear component of $T$ onto the subspace orthogonal to the charge vectors $Y_3$ and $B_3$. Let $\alpha_1$ and $\alpha_3$ be rational coefficients defined for the anomaly-free perpendicular subspace, and let $\cdot$ denote the dot product on the space of charges. Then the following identity holds: $$\alpha_1(\text{proj}(T)) = -\alpha_3(\text{proj}(T)) (B_3 \cdot T - Y_3 \cdot T)$$

Let $T$ be an anomaly-free solution for the MSSM anomaly cancellation system, and let $\text{proj}(T)$ be its projection onto the subspace orthogonal to the charge vectors $Y_3$ and $B_3$. If the rational coefficient $\alpha_3(\text{proj}(T))$ is zero, then the rational coefficient $\alpha_1(\text{proj}(T))$ is also zero.

Let $T$ be an anomaly-free solution for the MSSM anomaly cancellation system. Let $\text{proj}(T)$ be the projection of the charge vector $T$ onto the subspace orthogonal to the solutions $Y_3$ and $B_3$. If the rational coefficient $\alpha_3(\text{proj}(T))$ is zero, then the rational coefficient $\alpha_2(\text{proj}(T))$ is also zero.