Physlib.Particles.SuperSymmetry.MSSMNu.AnomalyCancellation.OrthogY3B3.Basic

Given a linear solution $T$ to the MSSM anomaly cancellation conditions, this function maps $T$ to a vector in the subspace of charges orthogonal to the vectors $Y_3$ and $B_3$. The resulting projection is defined by the following linear combination: $$\text{proj}(T) = (B_3 \cdot T - Y_3 \cdot T) Y_3 + (Y_3 \cdot T - 2 (B_3 \cdot T)) B_3 + (Y_3 \cdot B_3) T$$ where $\cdot$ denotes the symmetric bilinear form `MSSMACC.dot` defined on the space of MSSM charges. The construction ensures that the output is perpendicular to both $Y_3$ and $B_3$ with respect to this dot product.

For any linear solution $T$ to the MSSM anomaly cancellation conditions, the underlying charge vector of its projection $\text{proj}(T)$ onto the subspace orthogonal to $Y_3$ and $B_3$ is given by the linear combination: $$(B_3 \cdot T - Y_3 \cdot T) Y_3 + (Y_3 \cdot T - 2 (B_3 \cdot T)) B_3 + (Y_3 \cdot B_3) T$$ where $\cdot$ denotes the dot product on the space of MSSM charges.

Let $T$ be a linear solution to the MSSM anomaly cancellation conditions, and let $a, b, c \in \mathbb{Q}$ be rational scalars. Let $Y_3$ and $B_3$ be the basis vectors of the charge space as defined in the MSSM anomaly cancellation system. The linear combination of $Y_3$, $B_3$, and the projection of $T$ onto the subspace orthogonal to $Y_3$ and $B_3$ can be expressed as: $$a Y_3 + b B_3 + c \cdot \text{proj}(T) = (a + c (B_3 \cdot T - Y_3 \cdot T)) Y_3 + (b + c (Y_3 \cdot T - 2(B_3 \cdot T))) B_3 + (c (Y_3 \cdot B_3)) T$$ where $\cdot$ denotes the dot product (symmetric bilinear form) on the space of MSSM charges.

In the context of the MSSM with three generations and right-handed neutrinos, let $V \cong \mathbb{Q}^{20}$ be the vector space of rational charges. For any linear solution $T \in V$ to the anomaly cancellation conditions, let $\text{proj}(T)$ be the projection of $T$ onto the subspace orthogonal to the vectors $Y_3$ and $B_3$ with respect to the dot product $\cdot$, defined by: $$\text{proj}(T) = (B_3 \cdot T - Y_3 \cdot T) Y_3 + (Y_3 \cdot T - 2 (B_3 \cdot T)) B_3 + (Y_3 \cdot B_3) T.$$ Then, the symmetric bilinear form $B_q$ associated with the quadratic anomaly cancellation condition satisfies the identity: $$B_q(Y_3, \text{proj}(T)) = (Y_3 \cdot B_3) B_q(Y_3, T).$$

For any linear solution $T$ to the MSSM anomaly cancellation conditions, the symmetric bilinear form $B$ (representing the quadratic anomaly condition `quadBiLin`) evaluated on the charge vector $B_3$ and the projection $\text{proj}(T)$ satisfies the identity: $$B(B_3, \text{proj}(T)) = (Y_3 \cdot B_3) B(B_3, T)$$ where $Y_3$ and $B_3$ are fixed charge vectors in the MSSM charge space, $\cdot$ denotes the dot product (`dot`) on the space of rational charges $\mathbb{Q}^{20}$, and $\text{proj}(T)$ is the projection of $T$ onto the subspace orthogonal to $Y_3$ and $B_3$.

Let $T$ be a solution to the anomaly cancellation conditions for the MSSM with three generations and right-handed neutrinos. Let $B$ denote the symmetric bilinear form associated with the quadratic anomaly condition (`quadBiLin`), and let $\cdot$ denote the dot product defined on the space of rational charges $\mathbb{Q}^{20}$. Let $\text{proj}(T)$ be the projection of the linear component of $T$ onto the subspace orthogonal to the vectors $Y_3$ and $B_3$. Then the value of the bilinear form $B$ evaluated on $T$ and its projection $\text{proj}(T)$ satisfies: $$B(T, \text{proj}(T)) = (B_3 \cdot T - Y_3 \cdot T) B(Y_3, T) + (Y_3 \cdot T - 2 (B_3 \cdot T)) B(B_3, T)$$

Let $T$ be a solution to the anomaly cancellation conditions for the MSSM with three generations and right-handed neutrinos. Let $B$ denote the symmetric bilinear form associated with the quadratic anomaly condition (`quadBiLin`), and let $\cdot$ denote the dot product (`dot`) defined on the space of rational charges $\mathbb{Q}^{20}$. Let $\text{proj}(T)$ be the projection of the linear component of $T$ onto the subspace orthogonal to the fixed charge vectors $Y_3$ and $B_3$. Then the quadratic form evaluated at $\text{proj}(T)$ satisfies: $$B(\text{proj}(T), \text{proj}(T)) = 2(Y_3 \cdot B_3) \left[ (B_3 \cdot T - Y_3 \cdot T) B(Y_3, T) + (Y_3 \cdot T - 2(B_3 \cdot T)) B(B_3, T) \right]$$

In the context of the anomaly cancellation conditions for the Minimal Supersymmetric Standard Model (MSSM), let $A$ be the symmetric trilinear form associated with the cubic anomaly equation and let $\cdot$ be the dot product on the space of charges. For any linear solution $T$ to the anomaly cancellation conditions, let $\text{proj}(T)$ be the projection of $T$ onto the subspace orthogonal to the charge vectors $Y_3$ and $B_3$. Then the trilinear form satisfies the following identity: $$A(\text{proj}(T), \text{proj}(T), Y_3) = (Y_3 \cdot B_3)^2 A(T, T, Y_3).$$

In the context of the anomaly cancellation conditions (ACCs) for the Minimal Supersymmetric Standard Model (MSSM), let $A$ be the symmetric trilinear form (represented by `cubeTriLin`) associated with the cubic anomaly equation, and let $\cdot$ denote the dot product on the space of MSSM charges. For any charge vector $T$ that is a linear solution to the ACCs, let $\text{proj}(T)$ be its projection onto the subspace orthogonal to the charge vectors $Y_3$ and $B_3$. Then the trilinear form evaluated at $\text{proj}(T)$, $\text{proj}(T)$, and $B_3$ satisfies the identity: $$A(\text{proj}(T), \text{proj}(T), B_3) = (Y_3 \cdot B_3)^2 A(T, T, B_3).$$

Let $T$ be an anomaly-free solution for the MSSM ACC system. Let $A$ denote the symmetric trilinear form associated with the cubic anomaly cancellation condition, and let $\cdot$ denote the dot product defined on the space of charges. Let $P$ be the projection of the linear components of $T$ onto the subspace orthogonal to $Y_3$ and $B_3$, defined as: $$P = (B_3 \cdot T - Y_3 \cdot T) Y_3 + (Y_3 \cdot T - 2 (B_3 \cdot T)) B_3 + (Y_3 \cdot B_3) T$$ Then the trilinear form evaluated at $(P, P, T)$ satisfies the identity: $$A(P, P, T) = 2 (Y_3 \cdot B_3) \left[ (B_3 \cdot T - Y_3 \cdot T) A(T, T, Y_3) + (Y_3 \cdot T - 2 (B_3 \cdot T)) A(T, T, B_3) \right]$$

Let $T$ be an anomaly-free solution for the MSSM anomaly cancellation system. Let $A$ denote the symmetric trilinear form (`cubeTriLin`) associated with the cubic anomaly condition, and let $\cdot$ denote the dot product defined on the space of charges. Let $\text{proj}(T)$ be the projection of the linear components of $T$ onto the subspace orthogonal to the charge vectors $Y_3$ and $B_3$, defined as: $$\text{proj}(T) = (B_3 \cdot T - Y_3 \cdot T) Y_3 + (Y_3 \cdot T - 2 (B_3 \cdot T)) B_3 + (Y_3 \cdot B_3) T$$ The evaluation of the trilinear form on three copies of this projection satisfies the identity: $$A(\text{proj}(T), \text{proj}(T), \text{proj}(T)) = 3 (Y_3 \cdot B_3)^2 \left[ (B_3 \cdot T - Y_3 \cdot T) A(T, T, Y_3) + (Y_3 \cdot T - 2 (B_3 \cdot T)) A(T, T, B_3) \right]$$