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docs: Add clusters to informal graph

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jstoobysmith 2024-09-20 17:38:48 -04:00
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21
HepLean/BeyondTheStandardModel/PatiSalam/Basic.lean
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@ -9,7 +9,7 @@ import HepLean.Meta.Informal
# The Pati-Salam Model
The Pati-Salam model is a grand unified theory that unifies the Standard Model gauge group into
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 current contains informal-results about the Pati-Salam group.
@ -27,13 +27,24 @@ informal_definition GaugeGroupI where
math :≈ "The group `SU(4) x SU(2) x SU(2)`."
physics :≈ "The gauge group of the Pati-Salam model (unquotiented by ℤ₂)."
informal_definition embedSM where
physics :≈ "The embedding of the Standard Model gauge group into the Pati-Salam gauge group."
informal_definition inclSM where
physics :≈ "The homomorphism of the Standard Model gauge group into the Pati-Salam gauge group."
math :≈ "The group homomorphism `SU(3) x SU(2) x U(1) -> SU(4) x SU(2) x SU(2)`
taking (h, g, α) to (blockdiag (α h, α ^ (-3)), g, diag(α ^ (3), α ^(-3)))."
ref :≈ "Page 54 of https://math.ucr.edu/home/baez/guts.pdf"
deps :≈ [``GaugeGroupI, ``StandardModel.GaugeGroupI]
informal_lemma inclSM_ker where
math :≈ "The kernel of the map ``inclSM is equal to the subgroup
``StandardModel.gaugeGroupℤ₃SubGroup."
ref :≈ "Footnote 10 of https://arxiv.org/pdf/2201.07245"
deps :≈ [``inclSM, ``StandardModel.gaugeGroupℤ₃SubGroup]
informal_definition embedSMℤ₃ where
math :≈ "The group embedding from ``StandardModel.GaugeGroupℤ₃ to ``GaugeGroupI
induced by ``inclSM by quotienting by the kernal ``inclSM_ker."
deps :≈ [``inclSM, ``StandardModel.GaugeGroupℤ₃, ``GaugeGroupI, ``inclSM_ker]
informal_definition gaugeGroupISpinEquiv where
math :≈ "The equivalence between `GaugeGroupI` and `Spin(6) × Spin(4)`."
deps :≈ [``GaugeGroupI]
@ -54,12 +65,12 @@ informal_definition GaugeGroupℤ₂ where
informal_lemma sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup where
math :≈ "The group ``StandardModel.gaugeGroupℤ₆SubGroup under the homomorphism ``embedSM factors
through the subgroup ``gaugeGroupℤ₂SubGroup."
deps :≈ [``embedSM, ``StandardModel.gaugeGroupℤ₆SubGroup, ``gaugeGroupℤ₂SubGroup]
deps :≈ [``inclSM, ``StandardModel.gaugeGroupℤ₆SubGroup, ``gaugeGroupℤ₂SubGroup]
informal_definition embedSMℤ₆Toℤ₂ where
math :≈ "The group homomorphism from ``StandardModel.GaugeGroupℤ₆ to ``GaugeGroupℤ₂
induced by ``embedSM."
deps :≈ [``embedSM, ``StandardModel.GaugeGroupℤ₆, ``GaugeGroupℤ₂,
deps :≈ [``inclSM, ``StandardModel.GaugeGroupℤ₆, ``GaugeGroupℤ₂,
``sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup]
end PatiSalam