83 lines
2.9 KiB
Text
83 lines
2.9 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.StandardModel.Basic
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/-!
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# The Pati-Salam Model
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The Pati-Salam model is a petite unified theory that unifies the Standard Model gauge group into
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`SU(4) x SU(2) x SU(2)`.
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This file currently contains informal-results about the Pati-Salam group.
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-/
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namespace PatiSalam
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/-!
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## The Pati-Salam gauge group.
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-/
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/-- The gauge group of the Pati-Salam model (unquotiented by ℤ₂), i.e., `SU(4) × SU(2) × SU(2)`. -/
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informal_definition GaugeGroupI where
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deps := []
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/-- The homomorphism of the Standard Model gauge group into the Pati-Salam gauge group, i.e., the
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group homomorphism `SU(3) × SU(2) × U(1) → SU(4) × SU(2) × SU(2)` taking `(h, g, α)` to
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`(blockdiag (α h, α ^ (-3)), g, diag (α ^ 3, α ^(-3))`.
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See page 54 of https://math.ucr.edu/home/baez/guts.pdf
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-/
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informal_definition inclSM where
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deps := [``GaugeGroupI, ``StandardModel.GaugeGroupI]
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/-- The kernel of the map `inclSM` is equal to the subgroup `StandardModel.gaugeGroupℤ₃SubGroup`.
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See footnote 10 of https://arxiv.org/pdf/2201.07245
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-/
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informal_lemma inclSM_ker where
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deps := [``inclSM, ``StandardModel.gaugeGroupℤ₃SubGroup]
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/-- The group embedding from `StandardModel.GaugeGroupℤ₃` to `GaugeGroupI` induced by `inclSM` by
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quotienting by the kernel `inclSM_ker`.
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-/
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informal_definition embedSMℤ₃ where
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deps := [``inclSM, ``StandardModel.GaugeGroupℤ₃, ``GaugeGroupI, ``inclSM_ker]
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/-- The equivalence between `GaugeGroupI` and `Spin(6) × Spin(4)`. -/
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informal_definition gaugeGroupISpinEquiv where
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deps := [``GaugeGroupI]
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/-- The ℤ₂-subgroup of the un-quotiented gauge group which acts trivially on all particles in the
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standard model, i.e., the ℤ₂-subgroup of `GaugeGroupI` with the non-trivial element `(-1, -1, -1)`.
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See https://math.ucr.edu/home/baez/guts.pdf
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-/
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informal_definition gaugeGroupℤ₂SubGroup where
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deps := [``GaugeGroupI]
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/-- The gauge group of the Pati-Salam model with a ℤ₂ quotient, i.e., the quotient of `GaugeGroupI`
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by the ℤ₂-subgroup `gaugeGroupℤ₂SubGroup`.
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See https://math.ucr.edu/home/baez/guts.pdf
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-/
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informal_definition GaugeGroupℤ₂ where
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deps := [``GaugeGroupI, ``gaugeGroupℤ₂SubGroup]
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/-- The group `StandardModel.gaugeGroupℤ₆SubGroup` under the homomorphism `embedSM` factors through
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the subgroup `gaugeGroupℤ₂SubGroup`.
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-/
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informal_lemma sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup where
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deps := [``inclSM, ``StandardModel.gaugeGroupℤ₆SubGroup, ``gaugeGroupℤ₂SubGroup]
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/-- The group homomorphism from `StandardModel.GaugeGroupℤ₆` to `GaugeGroupℤ₂` induced by `embedSM`.
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-/
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informal_definition embedSMℤ₆Toℤ₂ where
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deps := [``inclSM, ``StandardModel.GaugeGroupℤ₆, ``GaugeGroupℤ₂,
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``sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup]
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end PatiSalam
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