123 lines
5.6 KiB
Text
123 lines
5.6 KiB
Text
/-
|
||
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.Tensors.TensorSpecies.UnitTensor
|
||
/-!
|
||
|
||
## Metrics in tensor trees
|
||
|
||
-/
|
||
|
||
open IndexNotation
|
||
open CategoryTheory
|
||
open MonoidalCategory
|
||
open OverColor
|
||
open HepLean.Fin
|
||
open TensorProduct
|
||
noncomputable section
|
||
|
||
namespace TensorSpecies
|
||
open TensorTree
|
||
|
||
/-- The metric of a tensor species in a `PiTensorProduct`. -/
|
||
def metricTensor (S : TensorSpecies) (c : S.C) : S.F.obj (OverColor.mk ![c, c]) :=
|
||
(OverColor.Discrete.pairIsoSep S.FD).hom.hom ((S.metric.app (Discrete.mk c)).hom (1 : S.k))
|
||
|
||
variable {S : TensorSpecies}
|
||
|
||
lemma metricTensor_congr {c c' : S.C} (h : c = c') : {S.metricTensor c | μ ν}ᵀ.tensor =
|
||
(perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by subst h; fin_cases x <;> rfl))
|
||
{S.metricTensor c' | μ ν}ᵀ).tensor := by
|
||
subst h
|
||
change _ = (S.F.map (𝟙 _)).hom (S.metricTensor c)
|
||
simp
|
||
|
||
lemma pairIsoSep_inv_metricTensor (c : S.C) :
|
||
(Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor c) =
|
||
(S.metric.app (Discrete.mk c)).hom (1 : S.k) := by
|
||
simp only [Action.instMonoidalCategory_tensorObj_V, Nat.succ_eq_add_one, Nat.reduceAdd,
|
||
metricTensor, Monoidal.tensorUnit_obj, Action.instMonoidalCategory_tensorUnit_V]
|
||
erw [Discrete.rep_iso_inv_hom_apply]
|
||
|
||
/-- Contraction of a metric tensor with a metric tensor gives the unit.
|
||
Like `S.contr_metric` but with the braiding appearing on the side of the unit. -/
|
||
lemma contr_metric_braid_unit (c : S.C) : (((S.FD.obj (Discrete.mk c)) ◁
|
||
(λ_ (S.FD.obj (Discrete.mk (S.τ c)))).hom).hom
|
||
(((S.FD.obj (Discrete.mk c)) ◁ ((S.contr.app (Discrete.mk c)) ▷
|
||
(S.FD.obj (Discrete.mk (S.τ c))))).hom
|
||
(((S.FD.obj (Discrete.mk c)) ◁ (α_ (S.FD.obj (Discrete.mk (c)))
|
||
(S.FD.obj (Discrete.mk (S.τ c))) (S.FD.obj (Discrete.mk (S.τ c)))).inv).hom
|
||
((α_ (S.FD.obj (Discrete.mk (c))) (S.FD.obj (Discrete.mk (c)))
|
||
(S.FD.obj (Discrete.mk (S.τ c)) ⊗ S.FD.obj (Discrete.mk (S.τ c)))).hom.hom
|
||
(((OverColor.Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor c) ⊗ₜ
|
||
(OverColor.Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor (S.τ c)))))))) =
|
||
(β_ (S.FD.obj (Discrete.mk (S.τ c))) (S.FD.obj (Discrete.mk c))).hom.hom
|
||
((S.unit.app (Discrete.mk c)).hom (1 : S.k)) := by
|
||
apply (β_ _ _).toLinearEquiv.toEquiv.injective
|
||
rw [pairIsoSep_inv_metricTensor, pairIsoSep_inv_metricTensor]
|
||
erw [S.contr_metric c]
|
||
change _ = (β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).inv.hom
|
||
((β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).hom.hom _)
|
||
rw [Discrete.rep_iso_inv_hom_apply]
|
||
|
||
lemma metricTensor_contr_dual_metricTensor_perm_cond (c : S.C) : ∀ (x : Fin (Nat.succ 0).succ),
|
||
((Sum.elim ![c, c] ![S.τ c, S.τ c] ∘ ⇑finSumFinEquiv.symm) ∘
|
||
Fin.succAbove 1 ∘ Fin.succAbove 1) x =
|
||
(![S.τ c, c] ∘ ⇑(finMapToEquiv ![1, 0] ![1, 0]).symm) x := by
|
||
intro x
|
||
fin_cases x
|
||
· rfl
|
||
· rfl
|
||
|
||
/-- The contraction of a metric tensor with its dual via the inner indices gives the unit. -/
|
||
lemma metricTensor_contr_dual_metricTensor_eq_unit (c : S.C) :
|
||
{S.metricTensor c | μ ν ⊗ S.metricTensor (S.τ c) | ν ρ}ᵀ.tensor = ({S.unitTensor c | μ ρ}ᵀ |>
|
||
perm (OverColor.equivToHomEq (finMapToEquiv ![1, 0] ![1, 0])
|
||
(metricTensor_contr_dual_metricTensor_perm_cond c))).tensor := by
|
||
rw [contr_two_two_inner, contr_metric_braid_unit, Discrete.pairIsoSep_β]
|
||
change (S.F.map _ ≫ S.F.map _).hom _ = _
|
||
rw [← S.F.map_comp]
|
||
rfl
|
||
|
||
/-- The contraction of a metric tensor with its dual via the outer indices gives the unit. -/
|
||
lemma metricTensor_contr_dual_metricTensor_outer_eq_unit (c : S.C) :
|
||
{S.metricTensor c | ν μ ⊗ S.metricTensor (S.τ c) | ρ ν}ᵀ.tensor = ({S.unitTensor c | μ ρ}ᵀ |>
|
||
perm (OverColor.equivToHomEq
|
||
(finMapToEquiv ![1, 0] ![1, 0]) (fun x => by fin_cases x <;> rfl))).tensor := by
|
||
conv_lhs =>
|
||
rw [contr_tensor_eq <| prod_tensor_eq_fst <| metricTensor_congr (S.τ_involution c).symm]
|
||
rw [contr_tensor_eq <| prod_comm _ _ _ _]
|
||
rw [perm_contr_congr 2 1 (by rfl) (by rfl)]
|
||
rw [perm_tensor_eq <| contr_tensor_eq <| prod_perm_right _ _ _ _]
|
||
rw [perm_tensor_eq <| perm_contr_congr 2 1 (by rfl) (by rfl)]
|
||
rw [perm_perm]
|
||
rw [perm_tensor_eq <| contr_swap _ _]
|
||
rw [perm_perm]
|
||
erw [perm_tensor_eq <| metricTensor_contr_dual_metricTensor_eq_unit _]
|
||
rw [perm_perm]
|
||
rw [perm_tensor_eq <| dual_unitTensor_eq_perm _]
|
||
rw [perm_perm]
|
||
apply perm_congr _ rfl
|
||
apply OverColor.Hom.fin_ext
|
||
intro i
|
||
simp only [Functor.id_obj, mk_hom, Function.comp_apply, Equiv.refl_symm, Equiv.coe_refl, id_eq,
|
||
Fin.zero_eta, Matrix.cons_val_zero, List.pmap.eq_1, ContrPair.contrSwapHom,
|
||
extractOne_homToEquiv, Category.assoc, Hom.hom_comp, Over.comp_left, equivToHomEq_hom_left,
|
||
Equiv.toFun_as_coe, types_comp_apply, finMapToEquiv_apply, mkIso_hom_hom_left_apply]
|
||
rw [extractTwo_hom_left_apply]
|
||
simp only [mk_left, braidPerm_toEquiv, Equiv.symm_trans_apply, Equiv.symm_symm,
|
||
Equiv.sumComm_symm, Equiv.sumComm_apply, finExtractOnePerm_symm_apply, Equiv.trans_apply,
|
||
Equiv.symm_apply_apply, Sum.swap_swap, Equiv.apply_symm_apply, finExtractOne_symm_inr_apply,
|
||
Fin.zero_succAbove, List.pmap.eq_2, Fin.mk_one, List.pmap.eq_1, Matrix.cons_val_one,
|
||
Matrix.head_cons, extractTwo_hom_left_apply, permProdRight_toEquiv, equivToHomEq_toEquiv,
|
||
Equiv.sumCongr_refl, Equiv.refl_trans, Equiv.symm_trans_self, Equiv.refl_symm, Equiv.refl_apply,
|
||
predAboveI_succAbove, finExtractOnePerm_apply]
|
||
fin_cases i
|
||
· decide
|
||
· decide
|
||
|
||
end TensorSpecies
|
||
|
||
end
|