140 lines
4.3 KiB
Text
140 lines
4.3 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.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.AnomalyCancellation.MSSMNu.Basic
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import Mathlib.Tactic.Polyrith
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import Mathlib.RepresentationTheory.Basic
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/-!
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# Permutations of MSSM charges and solutions
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The three family MSSM charges has a family permutation of S₃⁶. This file defines this group
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and its action on the MSSM.
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-/
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universe v u
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open Nat
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open Finset
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namespace MSSM
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open MSSMCharges
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open MSSMACCs
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open BigOperators
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/-- The group of family permutations is `S₃⁶`-/
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@[simp]
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def PermGroup := Fin 6 → Equiv.Perm (Fin 3)
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@[simp]
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instance : Group PermGroup := Pi.group
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/-- The image of an element of `permGroup` under the representation on charges. -/
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@[simps!]
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def chargeMap (f : PermGroup) : MSSMCharges.Charges →ₗ[ℚ] MSSMCharges.Charges where
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toFun S := toSpecies.symm (fun i => toSMSpecies i S ∘ f i, Prod.snd (toSpecies S))
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map_add' S T := by
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simp only
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rw [charges_eq_toSpecies_eq]
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refine And.intro ?_ $ Prod.mk.inj_iff.mp rfl
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intro i
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rw [(toSMSpecies i).map_add]
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rw [toSMSpecies_toSpecies_inv, toSMSpecies_toSpecies_inv, toSMSpecies_toSpecies_inv]
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rfl
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map_smul' a S := by
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simp only
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rw [charges_eq_toSpecies_eq]
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apply And.intro ?_ $ Prod.mk.inj_iff.mp rfl
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intro i
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rw [(toSMSpecies i).map_smul, toSMSpecies_toSpecies_inv, toSMSpecies_toSpecies_inv]
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rfl
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lemma chargeMap_toSpecies (f : PermGroup) (S : MSSMCharges.Charges) (j : Fin 6) :
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toSMSpecies j (chargeMap f S) = toSMSpecies j S ∘ f j := by
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erw [toSMSpecies_toSpecies_inv]
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/-- The representation of `permGroup` acting on the vector space of charges. -/
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@[simp]
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def repCharges : Representation ℚ PermGroup (MSSMCharges).Charges where
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toFun f := chargeMap f⁻¹
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map_mul' f g :=by
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simp only [PermGroup, mul_inv_rev]
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apply LinearMap.ext
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intro S
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rw [charges_eq_toSpecies_eq]
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refine And.intro ?_ $ Prod.mk.inj_iff.mp rfl
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intro i
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simp only [ Pi.mul_apply, Pi.inv_apply, Equiv.Perm.coe_mul, LinearMap.mul_apply]
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rw [chargeMap_toSpecies, chargeMap_toSpecies]
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simp only [Pi.mul_apply, Pi.inv_apply]
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rw [chargeMap_toSpecies]
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rfl
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map_one' := by
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apply LinearMap.ext
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intro S
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rw [charges_eq_toSpecies_eq]
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refine And.intro ?_ $ Prod.mk.inj_iff.mp rfl
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intro i
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erw [toSMSpecies_toSpecies_inv]
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rfl
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lemma repCharges_toSMSpecies (f : PermGroup) (S : MSSMCharges.Charges) (j : Fin 6) :
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toSMSpecies j (repCharges f S) = toSMSpecies j S ∘ f⁻¹ j := by
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erw [toSMSpecies_toSpecies_inv]
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lemma toSpecies_sum_invariant (m : ℕ) (f : PermGroup) (S : MSSMCharges.Charges) (j : Fin 6) :
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∑ i, ((fun a => a ^ m) ∘ toSMSpecies j (repCharges f S)) i =
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∑ i, ((fun a => a ^ m) ∘ toSMSpecies j S) i := by
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rw [repCharges_toSMSpecies]
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exact Equiv.sum_comp (f⁻¹ j) ((fun a => a ^ m) ∘ (toSMSpecies j) S)
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lemma Hd_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
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Hd (repCharges f S) = Hd S := rfl
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lemma Hu_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
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Hu (repCharges f S) = Hu S := rfl
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lemma accGrav_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
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accGrav (repCharges f S) = accGrav S :=
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accGrav_ext
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(by simpa using toSpecies_sum_invariant 1 f S)
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(Hd_invariant f S)
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(Hu_invariant f S)
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lemma accSU2_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
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accSU2 (repCharges f S) = accSU2 S :=
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accSU2_ext
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(by simpa using toSpecies_sum_invariant 1 f S)
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(Hd_invariant f S)
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(Hu_invariant f S)
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lemma accSU3_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
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accSU3 (repCharges f S) = accSU3 S :=
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accSU3_ext
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(by simpa using toSpecies_sum_invariant 1 f S)
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lemma accYY_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
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accYY (repCharges f S) = accYY S :=
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accYY_ext
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(by simpa using toSpecies_sum_invariant 1 f S)
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(Hd_invariant f S)
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(Hu_invariant f S)
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lemma accQuad_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
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accQuad (repCharges f S) = accQuad S :=
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accQuad_ext
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(toSpecies_sum_invariant 2 f S)
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(Hd_invariant f S)
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(Hu_invariant f S)
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lemma accCube_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
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accCube (repCharges f S) = accCube S :=
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accCube_ext
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(fun j => toSpecies_sum_invariant 3 f S j)
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(Hd_invariant f S)
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(Hu_invariant f S)
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end MSSM
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