The Pati-Salam Model

The Pati-Salam model is a petite unified theory that unifies the Standard Model gauge group into SU(4) x SU(2) x SU(2).

This file currently contains informal-results about the Pati-Salam group.

The Pati-Salam gauge group.

def PatiSalam.GaugeGroupI : InformalDefinition

The gauge group of the Pati-Salam model (unquotiented by ℤ₂), i.e., SU(4) × SU(2) × SU(2).

Equations

• PatiSalam.GaugeGroupI = { deps := [] }

def PatiSalam.inclSM : InformalDefinition

The homomorphism of the Standard Model gauge group into the Pati-Salam gauge group, i.e., the group homomorphism SU(3) × SU(2) × U(1) → SU(4) × SU(2) × SU(2) taking (h, g, α) to (blockdiag (α h, α ^ (-3)), g, diag (α ^ 3, α ^(-3)).

See page 54 of https://math.ucr.edu/home/baez/guts.pdf

Equations

• PatiSalam.inclSM = { deps := [`PatiSalam.GaugeGroupI, `StandardModel.GaugeGroupI] }

def PatiSalam.inclSM_ker : InformalLemma

The kernel of the map inclSM is equal to the subgroup StandardModel.gaugeGroupℤ₃SubGroup.

See footnote 10 of https://arxiv.org/pdf/2201.07245

Equations

• PatiSalam.inclSM_ker = { deps := [`PatiSalam.inclSM, `StandardModel.gaugeGroupℤ₃SubGroup] }

def PatiSalam.embedSMℤ₃ : InformalDefinition

The group embedding from StandardModel.GaugeGroupℤ₃ to GaugeGroupI induced by inclSM by quotienting by the kernel inclSM_ker.

Equations

• PatiSalam.embedSMℤ₃ = { deps := [`PatiSalam.inclSM, `StandardModel.GaugeGroupℤ₃, `PatiSalam.GaugeGroupI, `PatiSalam.inclSM_ker] }

def PatiSalam.gaugeGroupISpinEquiv : InformalDefinition

The equivalence between GaugeGroupI and Spin(6) × Spin(4).

Equations

• PatiSalam.gaugeGroupISpinEquiv = { deps := [`PatiSalam.GaugeGroupI] }

def PatiSalam.gaugeGroupℤ₂SubGroup : InformalDefinition

The ℤ₂-subgroup of the un-quotiented gauge group which acts trivially on all particles in the standard model, i.e., the ℤ₂-subgroup of GaugeGroupI with the non-trivial element (-1, -1, -1).

See https://math.ucr.edu/home/baez/guts.pdf

Equations

• PatiSalam.gaugeGroupℤ₂SubGroup = { deps := [`PatiSalam.GaugeGroupI] }

def PatiSalam.GaugeGroupℤ₂ : InformalDefinition

The gauge group of the Pati-Salam model with a ℤ₂ quotient, i.e., the quotient of GaugeGroupI by the ℤ₂-subgroup gaugeGroupℤ₂SubGroup.

See https://math.ucr.edu/home/baez/guts.pdf

Equations

• PatiSalam.GaugeGroupℤ₂ = { deps := [`PatiSalam.GaugeGroupI, `PatiSalam.gaugeGroupℤ₂SubGroup] }

def PatiSalam.sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup : InformalLemma

The group StandardModel.gaugeGroupℤ₆SubGroup under the homomorphism embedSM factors through the subgroup gaugeGroupℤ₂SubGroup.

Equations

• PatiSalam.sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup = { deps := [`PatiSalam.inclSM, `StandardModel.gaugeGroupℤ₆SubGroup, `PatiSalam.gaugeGroupℤ₂SubGroup] }

def PatiSalam.embedSMℤ₆Toℤ₂ : InformalDefinition

The group homomorphism from StandardModel.GaugeGroupℤ₆ to GaugeGroupℤ₂ induced by embedSM.

Equations

• PatiSalam.embedSMℤ₆Toℤ₂ = { deps := [`PatiSalam.inclSM, `StandardModel.GaugeGroupℤ₆, `PatiSalam.GaugeGroupℤ₂, `PatiSalam.sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup] }