Physlib.Particles.SuperSymmetry.MSSMNu.AnomalyCancellation.LineY3B3

Given two rational numbers $a$ and $b$, this function constructs a charge vector in the space of linear solutions to the MSSM anomaly cancellation conditions ($\text{LinSols}$) by taking the linear combination $a Y_3 + b B_3$. This represents the set of points on the line (or subspace) spanned by the specific solutions $Y_3$ and $B_3$.

For any rational numbers $a$ and $b$, the quadratic anomaly cancellation condition $\text{accQuad}$ vanishes for the charge vector formed by the linear combination $a Y_3 + b B_3$. That is, $\text{accQuad}(a Y_3 + b B_3) = 0$.

In the context of the Minimal Supersymmetric Standard Model (MSSM) anomaly cancellation conditions, let $\text{accCube}$ be the homogeneous cubic map representing the cubic anomaly cancellation condition. For any rational numbers $a, b \in \mathbb{Q}$, the charge vector formed by the linear combination $a Y_3 + b B_3$ satisfies the cubic anomaly equation: $$\text{accCube}(a Y_3 + b B_3) = 0.$$

Given two rational numbers $a, b \in \mathbb{Q}$, this function constructs a full solution to the Minimal Supersymmetric Standard Model (MSSM) anomaly cancellation conditions (ACCs) from the linear combination $a Y_3 + b B_3$. This construction incorporates the linear combination $a Y_3 + b B_3$ (an element of the space of linear solutions) alongside the proofs that it satisfies both the quadratic anomaly condition $\text{accQuad}(a Y_3 + b B_3) = 0$ and the cubic anomaly condition $\text{accCube}(a Y_3 + b B_3) = 0$. The resulting value is an element of the type `Sols`, representing a charge vector that satisfies all linear, quadratic, and cubic ACCs.

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. For any charge vector $R$ that satisfies the linear anomaly cancellation conditions, the trilinear form evaluated at the specific charge vectors $Y_3$, $B_3$, and $R$ vanishes: $$A(Y_3, B_3, R) = 0.$$

In the context of the Minimal Supersymmetric Standard Model (MSSM) anomaly cancellation conditions (ACCs), let $A$ be the symmetric trilinear form (represented by `cubeTriLin`) associated with the cubic anomaly equation. For any rational numbers $a, b \in \mathbb{Q}$ and any charge vector $R$ that satisfies the linear anomaly cancellation conditions, the trilinear form evaluated at the linear combination $a Y_3 + b B_3$ (taken twice) and $R$ vanishes: $$A(a Y_3 + b B_3, a Y_3 + b B_3, R) = 0.$$ This shows that every point on the line through $Y_3$ and $B_3$ is a double point of the cubic anomaly equation with respect to the space of linear solutions.