feat: Informal Pati-Salam

HepLean/BeyondTheStandardModel/PatiSalam/Basic.lean

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+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.StandardModel.Basic
+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
+`SU(4) x SU(2) x SU(2)`.
+
+This file current contains informal-results about the Pati-Salam group.
+
+-/
+
+namespace PatiSalam
+/-!
+
+## The Pati-Salam gauge group.
+
+-/
+
+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."
+  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_definition gaugeGroupISpinEquiv where
+  math :≈ "The equivalence between `GaugeGroupI` and `Spin(6) × Spin(4)`."
+  deps :≈ [``GaugeGroupI]
+
+informal_definition gaugeGroupℤ₂SubGroup where
+  physics :≈ "The ℤ₂-subgroup of the un-quotiented gauge group which acts trivially on
+    all particles in the standard model."
+  math :≈ "The ℤ₂-subgroup of ``GaugeGroupI with the non-trivial element (-1, -1, -1)."
+  ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
+  deps :≈ [``GaugeGroupI]
+
+informal_definition GaugeGroupℤ₂ where
+  physics :≈ "The gauge group of the Pati-Salam model with a ℤ₂ quotient."
+  math :≈ "The quotient of ``GaugeGroupI by the ℤ₂-subgroup `gaugeGroupℤ₂SubGroup`."
+  ref :≈ "https://math.ucr.edu/home/baez/guts.pdf"
+  deps :≈ [``GaugeGroupI, ``gaugeGroupℤ₂SubGroup]
+
+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]
+
+informal_definition embedSMℤ₆Toℤ₂ where
+  math :≈ "The group homomorphism from ``StandardModel.GaugeGroupℤ₆ to ``GaugeGroupℤ₂
+    induced by ``embedSM."
+  deps :≈ [``embedSM, ``StandardModel.GaugeGroupℤ₆, ``GaugeGroupℤ₂,
+    ``sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup]
+
+end PatiSalam