| import OAI.RepresentationTheory.VertexAlgebra.AlgebraCovariance |
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| namespace OAI |
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| 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) |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| end |
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| end OAI |
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