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e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import OAI.RepresentationTheory.VertexAlgebra.AlgebraCovariance
namespace OAI
noncomputable section
open UniformSpace Set
open scoped Topology
namespace StronglyRationalVOA.CFTTypeVOA.UnitaryStructure
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
{A : CFTTypeVOA V} (U : A.UnitaryStructure)
lemma stressPropagator_commutes_of_graph (hc : A.toVertexAlgebra.C2Cofinite)
(f : StressPath) (Q : Completion V →L[ℂ] Completion V)
(hQ : ∀ t x y, (x,y) ∈ (U.closedSmearedOperator hc A.conformalVector 2
(f.coefficients t) (f.rapid t)).graph →
(Q x,Q y) ∈ (U.closedSmearedOperator hc A.conformalVector 2
(f.coefficients t) (f.rapid t)).graph) (t : ℝ) :
Q*U.stressPropagator hc f t=U.stressPropagator hc f t*Q := by
have hp (x y : U.smoothEnergySpace) :
inner ℂ (U.stressPropagator hc f t x) (Q (U.stressPropagator hc f t y))=
inner ℂ (x:Completion V) (Q y) := by
have hd (s : ℝ) : HasDerivAt (fun s =>
inner ℂ (U.stressPropagator hc f s x) (Q (U.stressPropagator hc f s y))) 0 s := by
have H := (U.stressPropagator_derivative hc f s x).inner ℂ
((Q.restrictScalars ℝ).hasFDerivAt.comp_hasDerivAt s (U.stressPropagator_derivative hc f s y))
have he := SymmetricOperator.formalAdjoint_graph_pairing
(SymmetricOperator.selfAdjoint_formal
(U.stress_closedSmearedOperator_selfAdjoint hc (f.rapid s) (f.real s)))
(U.smoothSmear_graph hc A.conformal_mem (f.rapid s) (U.stressPropagatorSmooth hc f s x))
(hQ s _ _ (U.smoothSmear_graph hc A.conformal_mem (f.rapid s)
(U.stressPropagatorSmooth hc f s y)))
have hre (v : Completion V) : (Q.restrictScalars ℝ) v=Q v := rfl
simp only [Function.comp_apply,hre] at H
change inner ℂ
(U.smoothSmear hc A.conformal_mem (f.rapid s) (U.stressPropagatorSmooth hc f s x) : Completion V)
(Q (U.stressPropagator hc f s y))=
inner ℂ (U.stressPropagator hc f s x)
(Q (U.smoothSmear hc A.conformal_mem (f.rapid s) (U.stressPropagatorSmooth hc f s y) : Completion V)) at he
rw [U.stressPropagatorDerivative_smoothSmear hc f s x, U.stressPropagatorDerivative_smoothSmear hc f s y, map_smul,inner_smul_left,inner_smul_right,Complex.conj_I,he] at H
convert H using 1
first | rfl | ring
have he := is_const_of_deriv_eq_zero (fun s => (hd s).differentiableAt)
(fun s => (hd s).deriv) t 0
simpa only [U.stressPropagator_zero,one_apply_eq_self] using he
have hall (x y : Completion V) :
inner ℂ (U.stressPropagator hc f t x) (Q (U.stressPropagator hc f t y))=
inner ℂ x (Q y) := by
refine U.smoothEnergySpace_dense.denseRange_val.induction_on x ?_ ?_
· exact isClosed_eq ((U.stressPropagator hc f t).continuous.inner continuous_const)
(continuous_id.inner continuous_const)
· intro x
refine U.smoothEnergySpace_dense.denseRange_val.induction_on y ?_ ?_
· exact isClosed_eq (continuous_const.inner (Q.continuous.comp (U.stressPropagator hc f t).continuous))
(continuous_const.inner Q.continuous)
· intro y; exact hp x y
ext y
apply ext_inner_left ℂ
intro z
obtain ⟨x,rfl⟩ := (U.stressPropagatorEquiv hc f t).surjective z
exact (hall x y).trans ((U.stressPropagatorEquiv hc f t).inner_map_map x (Q y)).symm
lemma angularStressPath_local_action (hc : A.toVertexAlgebra.C2Cofinite)
(φ : PositiveCircleGeometry.SmoothLift)
{O : Set (AddCircle (2*Real.pi))} (hO : IsOpen O)
(hv : ∀ t, tsupport (φ.angularVelocity t).onCircle ⊆ Oᶜ)
{Q : Completion V →L[ℂ] Completion V} (hQ : Q ∈ U.localAlgebra hc O)
(t : ℝ) : Q*U.stressPropagator hc φ.angularStressPath t=
U.stressPropagator hc φ.angularStressPath t*Q := by
apply U.stressPropagator_commutes_of_graph hc φ.angularStressPath Q
intro s x y hxy
let T : U.PrimaryTestField :=
⟨A.conformalVector,2,A.conformal_mem,A.conformal_quasiPrimary,U.stress_hermitian,
φ.angularVelocity s,φ.angularVelocity_real s,Or.inr (Or.inl ⟨rfl,rfl⟩)⟩
apply T.commutant_preserves_graph hc (hv s) _ hxy
apply U.primaryAlgebra_locality hc disjoint_compl_right
rwa [U.localAlgebra_eq_primaryAlgebra hc hO] at hQ
end StronglyRationalVOA.CFTTypeVOA.UnitaryStructure
namespace StronglyRationalVOA.PositiveCircleGeometry.SmoothLift
variable (φ : SmoothLift)
lemma velocity_zero_of_fixed (t x : ℝ) (hx : φ x=x) : φ.velocity t x=0 := by
have hi : φ.interpolate t x=x := by dsimp [interpolate]; rw [hx]; ring
have hinv : φ.inverse t x=x := by simpa only [hi] using φ.inverse_interpolate t x
simp only [velocity,hinv,hx,sub_self,mul_zero]
end StronglyRationalVOA.PositiveCircleGeometry.SmoothLift
namespace StronglyRationalVOA.SmoothCircleLift
lemma angularVelocity_zero_of_fixed (F : SmoothCircleLift) (t x : ℝ) (hx : F x=x) :
F.unitLift.angularVelocity t x=0 := by
have hu : F.unitLift (x/(2*Real.pi))=x/(2*Real.pi) := by
change F (2*Real.pi*(x/(2*Real.pi)))/(2*Real.pi)=_
rw [mul_div_cancel₀ _ (by positivity : (2*Real.pi:ℝ)≠0),hx]
change ((2*Real.pi*F.unitLift.velocity t (x/(2*Real.pi)) : ℝ):ℂ)=0
rw [F.unitLift.velocity_zero_of_fixed t _ hu,mul_zero,Complex.ofReal_zero]
lemma angularVelocity_support_of_fixed (F : SmoothCircleLift)
{O : Set (AddCircle (2*Real.pi))} (hO : IsOpen O)
(hfixed : ∀ x : ℝ, (x : AddCircle (2*Real.pi)) ∈ O → F x=x) (t : ℝ) :
tsupport (F.unitLift.angularVelocity t).onCircle ⊆ Oᶜ := by
apply closure_minimal _ hO.isClosed_compl
intro z hz hmem
obtain ⟨x,rfl⟩ := QuotientAddGroup.mk_surjective z
exact hz (F.angularVelocity_zero_of_fixed t x (hfixed x hmem))
end StronglyRationalVOA.SmoothCircleLift
namespace StronglyRationalVOA.CFTTypeVOA.UnitaryStructure
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
{A : CFTTypeVOA V} (U : A.UnitaryStructure)
lemma diffeomorphism_local_action_of_fixed_lift (hc : A.toVertexAlgebra.C2Cofinite)
(F : SmoothCircleLift) {O : Set (AddCircle (2*Real.pi))} (hO : IsOpen O)
(hfixed : ∀ x : ℝ, (x : AddCircle (2*Real.pi)) ∈ O → F x=x)
{Q : Completion V →L[ℂ] Completion V} (hQ : Q ∈ U.localAlgebra hc O) :
(U.diffeomorphismImplementer hc F).conjStarAlgEquiv Q=Q := by
have he := U.angularStressPath_local_action hc F.unitLift hO
(F.angularVelocity_support_of_fixed hO hfixed) hQ 1
have hx (x : Completion V) := congrArg (fun B : Completion V →L[ℂ] Completion V => B x) he
ext x
exact ((hx ((U.diffeomorphismImplementer hc F).symm x)).symm.trans
(congrArg Q ((U.diffeomorphismImplementer hc F).apply_symm_apply x)))
end StronglyRationalVOA.CFTTypeVOA.UnitaryStructure
end
end OAI
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