PhyslibAlpha.Particles.BeyondTheStandardModel.TwoHDM.GaugeTorus

Given a complex phase $a \in U(1)$ (represented as an element of the unitary group of $\mathbb{C}$), `gaugeCartan a` is the element of the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ defined by the triplet $(I_3, \text{diag}(a, \bar{a}), 1)$. Here, $I_3$ is the $3 \times 3$ identity matrix in $SU(3)$, $\text{diag}(a, \bar{a}) = \begin{pmatrix} a & 0 \\ 0 & \bar{a} \end{pmatrix}$ is a diagonal matrix in $SU(2)$ where $\bar{a}$ denotes the complex conjugate of $a$, and $1$ is the identity element of the $U(1)$ factor.

For any complex phase $a \in U(1)$, let $\text{gaugeCartan}(a)$ be the element of the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ defined as the triplet $(I_3, \text{diag}(a, \bar{a}), 1)$. The projection of this element onto the $U(1)$ factor is equal to the identity element $1$.

For any complex phase $a \in U(1)$, the $SU(2)$ component of the gauge group element $\text{gaugeCartan}(a)$, when represented as a $2 \times 2$ complex matrix, is the diagonal matrix $$\begin{pmatrix} a & 0 \\ 0 & \bar{a} \end{pmatrix}$$ where $\bar{a}$ denotes the complex conjugate of $a$.

For any complex phase $a \in U(1)$ and any Higgs vector $\phi \in \mathbb{C}^2$, the action of the gauge group element $\text{gaugeCartan}(a)$ on $\phi$ is given by the matrix-vector multiplication: $$\text{gaugeCartan}(a) \cdot \phi = \begin{pmatrix} a & 0 \\ 0 & \bar{a} \end{pmatrix} \begin{pmatrix} \phi_1 \\ \phi_2 \end{pmatrix}$$ where $\bar{a}$ denotes the complex conjugate of $a$.