Physlib.Particles.BeyondTheStandardModel.Spin10.Basic

The gauge group of the $\operatorname{Spin}(10)$ model is the spin group $\operatorname{Spin}(10)$.

The inclusion homomorphism from the Pati-Salam gauge group $SU(4) \times SU(2) \times SU(2)$ into the $\text{Spin}(10)$ gauge group. This map is constructed by lifting the standard embedding $SO(6) \times SO(4) \hookrightarrow SO(10)$ to their universal covers, resulting in the homomorphism $\text{Spin}(6) \times \text{Spin}(4) \to \text{Spin}(10)$, and precomposing it with the group isomorphism $SU(4) \times SU(2) \times SU(2) \cong \text{Spin}(6) \times \text{Spin}(4)$.

The group homomorphism representing the inclusion of the Standard Model gauge group into the $\operatorname{Spin}(10)$ gauge group. This map is defined as the composition of the inclusion of the Standard Model gauge group into the Pati-Salam group $SU(4) \times SU(2) \times SU(2)$ and the subsequent inclusion of the Pati-Salam group into $\operatorname{Spin}(10)$. This corresponds to the embedding of the Standard Model into the Grand Unified Theory (GUT) group $\operatorname{Spin}(10)$ via the Pati-Salam intermediate symmetry.

The Lie group homomorphism $\text{incl}_{GG} : SU(5) \to \text{Spin}(10)$ that embeds the Georgi-Glashow $SU(5)$ gauge group into the $\text{Spin}(10)$ gauge group. This map represents the specific case for $n=5$ of the general inclusion $SU(n) \hookrightarrow \text{Spin}(2n)$ utilized in Grand Unified Theories (GUTs).

The inclusion map of the Standard Model gauge group into the $Spin(10)$ group, denoted $\text{incl}_{SM \to GG}$, defined as the composition of the inclusion of the Standard Model into the Georgi-Glashow $SU(5)$ group and the inclusion of the Georgi-Glashow $SU(5)$ group into $Spin(10)$.

In the $Spin(10)$ grand unified theory (GUT) model, the inclusion map of the Standard Model gauge group into $Spin(10)$, denoted $\text{incl}_{SM}$, is equal to the inclusion map $\text{incl}_{SM \to GG}$ that factors through the Georgi-Glashow $SU(5)$ subgroup.