higher symmetries of the conformal laplacian · higher symmetries of the conformal laplacian j.-p....

Highersymmetries ofthe conformalLaplacian

J.-P. Michel, J.Silhan, R.

Second orderconformalsymmetries of∆YConformal Killingtensors

Natural andconformallyinvariantquantization

Structure of theconformalsymmetries

Examples

DiPirro system

Taub-NUT metric

Application tothe R-separation

Higher symmetries of the conformal Laplacian

J.-P. Michel, J. Silhan, R.

Corfu, May 2013






Examples

DiPirro system

Taub-NUT metric


Introduction

On (R2, g0), we consider the Helmholtz equation

∆φ = Eφ,

where

∆ = ∂2x + ∂2

y , E ∈ R.

Coordinates (u, v) separate this equation⇐⇒ ∃ solutionof the form f (u)g(v)

Coordinates (u, v) orthogonal⇐⇒ g0(∂u, ∂v ) = 0






Examples

DiPirro system

Taub-NUT metric


There exist 4 families of orthogonal separating

coordinates systems :

1 Cartesian coordinates2 Polar coordinates (r , θ) :{

x = r cos(θ)y = r sin(θ)

3 Parabolic coordinates (ξ, η) :{x = ξηy = 1

2(ξ2 − η2)

4 Elliptic coordinates (α, β) :{x =

√d cos(α) cosh(β)

y =√d sin(α) sinh(β)






Examples

DiPirro system

Taub-NUT metric




1 Cartesian coordinates

2 Polar coordinates (r , θ) :{x = r cos(θ)y = r sin(θ)


2(ξ2 − η2)









Examples

DiPirro system

Taub-NUT metric




1 Cartesian coordinates2 Polar coordinates (r , θ) :{

x = r cos(θ)y = r sin(θ)


2(ξ2 − η2)









Examples

DiPirro system

Taub-NUT metric


Figure: Coordinates lines corresponding to the paraboliccoordinates system






Examples

DiPirro system

Taub-NUT metric


Figure: Coordinates lines corresponding to the elliptic coordinatessystem






Examples

DiPirro system

Taub-NUT metric


Bijective correspondence

{Separating coordinates systems}

←→

{Second order symmetries of ∆ : second orderdi�erential operators D such that [∆,D] = 0}

Coordinates system Symmetry

(x , y) ∂2x(r , θ) L2θ(ξ, η) 1

2(∂xLθ + Lθ∂x)

(α, β) L2θ + d∂2x

with Lθ = x∂y − y∂x






Examples

DiPirro system

Taub-NUT metric


Link between the symmetry and the coordinates

system : if the second-order part of D reads as

(∂x ∂y

)A

(∂x∂y

),

the eigenvectors of A are tangent to the coordinates

lines.






Examples

DiPirro system

Taub-NUT metric


On a n-dimensional pseudo-Riemannian manifold (M, g),

∆Y := ∇igij∇j −n − 2

4(n − 1)R,

where R is the scalar curvature of g.

λ0 =n−22n

, µ0 =n+22n

If ∆Y acts between λ0- and µ0-densities, ∆Yconformally invariant : ∆Y (g̃) = ∆Y (g) if g̃ = e

2Υg.

Symmetry of ∆Y : D ∈ Dλ0,λ0(M) such that[∆Y ,D] = 0

Conformal symmetry of ∆Y : D1 ∈ Dλ0,λ0(M) such that∃D2 ∈ Dµ0,µ0(M) such that ∆Y ◦ D1 = D2 ◦∆Y






Examples

DiPirro system

Taub-NUT metric


(M, g) conformally �at : in a coordinates system, g isconformally equivalent to the canonical metric in Rn

Conformal symmetries of ∆Y known (M. Eastwood,J.-P. Michel)

(M, g) Einstein : Ric = f gExistence of a second order symmetry (B. Carter)






Examples

DiPirro system

Taub-NUT metric


1 Second order conformal symmetries of ∆YConformal Killing tensors

Natural and conformally invariant quantization

Structure of the conformal symmetries

2 Examples

DiPirro system

Taub-NUT metric

3 Application to the R-separation






Examples

DiPirro system

Taub-NUT metric


If D ∈ Dkλ0,λ0(M) reads∑|α|6k

Dα∂α1x1 . . . ∂αnxn,

σ(D) =∑|α|=k

Dαpα11. . . pαnn ,

where (x i , pi ) are the canonical coordinates on T∗M

σ(D) can be viewed as a contravariant symmetric tensorof degree k :

σ(D) =∑|α|=k

Dα∂α11∨ . . . ∨ ∂αnn






Examples

DiPirro system

Taub-NUT metric


If D is a conformal symmetry of ∆Y , there exists anoperator D ′ such that ∆Y ◦ D = D ′ ◦∆Y

σ(∆Y ) = H = gijpipj , then {H, σ(D)} ∈ (H), i.e. σ(D)

is a conformal Killing tensor

Conformal Killing tensor K : trace-free part of

∇(i0Ki1···ik) vanishesIf D is a symmetry of ∆Y , [∆Y ,D] = 0, then{H, σ(D)} = 0, i.e. σ(D) is a Killing tensorKilling tensor K : ∇(i0Ki1···ik) = 0The existence of a (conformal) Killing tensor is necessary

to have the existence of a (conformal) symmetry of ∆Y

Is this condition su�cient ?






Examples

DiPirro system

Taub-NUT metric


If D is a conformal symmetry of ∆Y , there exists anoperator D ′ such that ∆Y ◦ D = D ′ ◦∆Yσ(∆Y ) = H = g

ijpipj , then {H, σ(D)} ∈ (H), i.e. σ(D)is a conformal Killing tensor










Examples

DiPirro system

Taub-NUT metric





∇(i0Ki1···ik) vanishes

If D is a symmetry of ∆Y , [∆Y ,D] = 0, then{H, σ(D)} = 0, i.e. σ(D) is a Killing tensorKilling tensor K : ∇(i0Ki1···ik) = 0The existence of a (conformal) Killing tensor is necessary








Examples

DiPirro system

Taub-NUT metric





∇(i0Ki1···ik) vanishesIf D is a symmetry of ∆Y , [∆Y ,D] = 0, then{H, σ(D)} = 0, i.e. σ(D) is a Killing tensor

Killing tensor K : ∇(i0Ki1···ik) = 0The existence of a (conformal) Killing tensor is necessary








Examples

DiPirro system

Taub-NUT metric





∇(i0Ki1···ik) vanishesIf D is a symmetry of ∆Y , [∆Y ,D] = 0, then{H, σ(D)} = 0, i.e. σ(D) is a Killing tensorKilling tensor K : ∇(i0Ki1···ik) = 0

The existence of a (conformal) Killing tensor is necessary








Examples

DiPirro system

Taub-NUT metric













Examples

DiPirro system

Taub-NUT metric


De�nition

A quantization on M is a linear bijection QMλ,λ from the spaceof symbols Pol(T ∗M) to the space of di�erential operatorsDλ,λ(M) such that

σ(QMλ,λ(S)) = S , ∀S ∈ Pol(T ∗M)






Examples

DiPirro system

Taub-NUT metric


De�nition

A natural and conformally invariant quantization is the data

for every manifold M of a quantization QMλ,λ depending on apseudo-Riemannian metric de�ned on M such that

If Φ is a local di�eomorphism from M to a manifold N,then one has

QMλ,λ(Φ∗g)(Φ∗S) = Φ∗(QNλ,λ(g)(S)),

for all pseudo-Riemannian metric g on N and all

S ∈ Pol(T ∗N)QMλ,λ(g) = QMλ,λ(g̃) whenever g and g̃ are conformallyequivalent, i.e. whenever there exists a function Υ suchthat g̃ = e2Υg.






Examples

DiPirro system

Taub-NUT metric


Proof of the existence of QMλ,λ :1 Work by A. Cap, J. Silhan2 Work by P. Mathonet, R.






Examples

DiPirro system

Taub-NUT metric


If K is a conformal Killing tensor of degree 2, there

exists a conformal symmetry of ∆Y with K as principalsymbol i� Obs(K )[ is an exact one-form, where

Obs =2(n − 2)3(n + 1)

pi∂pj∂pl

(C k jl

i∇k − 3Ajl i)

C : Weyl tensor :

Cabcd = Rabcd −2

n − 2(ga[cRicd ]b − gb[cRicd ]a)

+2

(n − 1)(n − 2)Rga[cgd ]b

A : Cotton-York tensor :

Aijk = ∇kRicij−∇jRicik +1

2(n − 1)(∇jRgik −∇kRgij)






Examples

DiPirro system

Taub-NUT metric


If Obs(K )[ = 2df , the conformal symmetries of ∆Ywhose the principal symbol is given by K are of the form

Qλ0,λ0(K )− f + Lλ0X + c,

where X is a conformal Killing vector �eld and where

c ∈ R






Examples

DiPirro system

Taub-NUT metric


If K is a Killing tensor of degree 2, there exists a

symmetry of ∆Y with K as principal symbol i� Obs(K )[

is an exact one-form

If Obs(K )[ = 2df , the symmetries of ∆Y whose theprincipal symbol is given by K are of the form

Qλ0,λ0(K )− f + Lλ0X + c,

where X is a Killing vector �eld and where c ∈ R






Examples

DiPirro system

Taub-NUT metric


Remarks :

1 If (M, g) is conformally �at, no condition on the(conformal) Killing tensor K

2 If Ric = 1nRg and if K is a Killing tensor of degree 2,

then

Obs(K )[ = d

(2− n

2(n + 1)(∇i∇jK ij) +

2− n2n(n − 1)

RgijKij

)and ∇iK ij∇j is a symmetry of ∆Y






Examples

DiPirro system

Taub-NUT metric


On R3, pairs of (local) diagonal metrics and (local)Killing tensors are classi�ed :

Hamiltonian H = gijpipj :

1

2(γ(x1, x2) + c(x3))

(a(x1, x2)p

2

1 + b(x1, x2)p2

2 + p2

3

),

Killing tensor K :

c(x3)a(x1, x2)p2

1+ c(x3)b(x1, x2)p

2

2− γ(x1, x2)p23

γ(x1, x2) + c(x3),

a, b, γ ∈ C∞(R2), c ∈ C∞(R).






Examples

DiPirro system

Taub-NUT metric


If g̃ = 12(γ(x1,x2)+c(x3))

g, then

Obs(K )[ = d(−18

(3Ricij − Rg̃ij)K ij)

Symmetry of ∆Y :

∇iK ij∇j −1

16(∇i∇jK ij)−

1

8RicijK

ij






Examples

DiPirro system

Taub-NUT metric


Four-dimensional �ber bundle M over S2 with

coordinates (ψ, r , θ, φ)

Taub-NUT metric g :(1 +

2m

r

)(dr2 + r2dθ2 + r2 sin2 θdφ2

)+

4m2

1 + 2mr

(dψ + cos θdφ)2

g hyperkähler : there exist three complex structures Jiwhich are covariantly constant and which satisfy the

quaternion relations

J21 = J2

2 = J2

3 = J1J2J3 = −Id.






Examples

DiPirro system

Taub-NUT metric


The skewsymmetric tensor Y of degree 2 is Killing-Yano

i� ∇(λYµ)ν = 0

Killing-Yano tensor Y :

2m2 (dψ + cos θdφ)∧dr + r(r +m)(r +2m) sin θdθ∧dφ

∗Y conformal Killing-Yano tensor :

∇(λ ∗ Yµ)ν =2

3(gλµ∇κ(∗Y κν ) +∇κ(∗Y κ(λ)gµ)ν)

Ji Killing-Yano tensors, hence

Ki = pµpν(∗Y (µλ J

ν)λi

)conformal Killing tensors

Obs(Ki )[ not exact, then there are no conformal

symmetries whose principal symbols are the Ki






Examples

DiPirro system

Taub-NUT metric


The skewsymmetric tensor Y of degree 2 is Killing-Yano

i� ∇(λYµ)ν = 0Killing-Yano tensor Y :

2m2 (dψ + cos θdφ)∧dr + r(r +m)(r +2m) sin θdθ∧dφ

∗Y conformal Killing-Yano tensor :

∇(λ ∗ Yµ)ν =2

3(gλµ∇κ(∗Y κν ) +∇κ(∗Y κ(λ)gµ)ν)

Ji Killing-Yano tensors, hence

Ki = pµpν(∗Y (µλ J

ν)λi

)conformal Killing tensors

Obs(Ki )[ not exact, then there are no conformal

symmetries whose principal symbols are the Ki






Examples

DiPirro system

Taub-NUT metric


Laplace equation : (∆Y + V )ψ = 0, V ∈ C∞(M) is a�xed potential

Helmholtz equation : (∆Y + V )ψ = Eψ, V ∈ C∞(M)is a �xed potential and E ∈ R a free parameterSolving Helmholtz equation : �nding a solution for all E






Examples

DiPirro system

Taub-NUT metric



Helmholtz equation : (∆Y + V )ψ = Eψ, V ∈ C∞(M)is a �xed potential and E ∈ R a free parameter

Solving Helmholtz equation : �nding a solution for all E






Examples

DiPirro system

Taub-NUT metric



Helmholtz equation : (∆Y + V )ψ = Eψ, V ∈ C∞(M)is a �xed potential and E ∈ R a free parameterSolving Helmholtz equation : �nding a solution for all E






Examples

DiPirro system

Taub-NUT metric


Laplace equation R-separable in an orthogonal

coordinates system (x i ) (gij = 0 if i 6= j)

⇐⇒

∃ n + 1 functions R, hi ∈ C∞(M) and n di�erentialoperators Li := ∂

2

i + li (xi )∂i + mi (x

i ) such that

R−1(∆Y + V )R =n∑

i=1

hiLi .






Examples

DiPirro system

Taub-NUT metric


Helmholtz equation R-separable in an orthogonal

coordinates system (x i )

⇐⇒

∀ E ∈ R, ∃ n + 1 functions R, hi ∈ C∞(M) and ndi�erential operators Li := ∂

2

i + li (xi )∂i + mi (x

i ) suchthat

R−1(∆Y + V )R − E =n∑

i=1

hiLi .

R∏n

i=1 φi (xi ) solution of the Laplace or Helmholtz

equation

⇐⇒

Liφi = 0 ∀i






Examples

DiPirro system

Taub-NUT metric


Laplace equation R-separates in an orthogonalcoordinate system if and only if :

(a) ∃ a family of n− 1 conformal Killing tensors (Ki ), wherei = 2, . . . , n, such that, together with K1 = g

ijpipj ,

they Poisson commute : {Ki ,Kj} = 0 for all i , j ,the set {K1, . . . ,Kn} is linearly independent overC∞(M),as endomorphisms of TM, K1, . . . ,Kn admit a basis ofcommon eigenvectors.

(b) ∃ second order conformal symmetries Di , i.e.[∆Y + V ,Di ] ∈ (∆Y + V ), with principal symbolsσ2(Di ) = Ki , for all i = 2, . . . , n.






Examples

DiPirro system

Taub-NUT metric


Laplace equation R-separates in an orthogonalcoordinate system if and only if :(a) ∃ a family of n− 1 conformal Killing tensors (Ki ), where

i = 2, . . . , n, such that, together with K1 = gijpipj ,

they Poisson commute : {Ki ,Kj} = 0 for all i , j ,

the set {K1, . . . ,Kn} is linearly independent overC∞(M),as endomorphisms of TM, K1, . . . ,Kn admit a basis ofcommon eigenvectors.







Examples

DiPirro system

Taub-NUT metric




they Poisson commute : {Ki ,Kj} = 0 for all i , j ,the set {K1, . . . ,Kn} is linearly independent overC∞(M),

as endomorphisms of TM, K1, . . . ,Kn admit a basis ofcommon eigenvectors.







Examples

DiPirro system

Taub-NUT metric











Examples

DiPirro system

Taub-NUT metric


Helmholtz equation R-separates in an orthogonalcoordinate system if and only if :

(a) ∃ a family of n − 1 Killing tensors (Ki ) wherei = 2, . . . , n, such that, together with K1 = g

ijpipj ,


(b) ∃ second order symmetries Di , i.e. [∆Y + V ,Di ] = 0,with principal symbols σ2(Di ) = Ki , for all i = 2, . . . , n.






Examples

DiPirro system

Taub-NUT metric


Helmholtz equation R-separates in an orthogonalcoordinate system if and only if :(a) ∃ a family of n − 1 Killing tensors (Ki ) where


they Poisson commute : {Ki ,Kj} = 0 for all i , j ,

the set {K1, . . . ,Kn} is linearly independent overC∞(M),as endomorphisms of TM, K1, . . . ,Kn admit a basis ofcommon eigenvectors.







Examples

DiPirro system

Taub-NUT metric




they Poisson commute : {Ki ,Kj} = 0 for all i , j ,the set {K1, . . . ,Kn} is linearly independent overC∞(M),

as endomorphisms of TM, K1, . . . ,Kn admit a basis ofcommon eigenvectors.







Examples

DiPirro system

Taub-NUT metric











Examples

DiPirro system

Taub-NUT metric


Link between the (conformal) symmetries and the

R-separating coordinate systems :

Eigenvectors of the Ki ←→ integrable distributionsLeaves of the corresponding foliations←→Coordinateslines of the R-separating coordinate systems






Examples

DiPirro system

Taub-NUT metric




Eigenvectors of the Ki ←→ integrable distributions

Leaves of the corresponding foliations←→Coordinateslines of the R-separating coordinate systems






Examples

DiPirro system

Taub-NUT metric




Eigenvectors of the Ki ←→ integrable distributionsLeaves of the corresponding foliations←→Coordinateslines of the R-separating coordinate systems

Second order conformal symmetries of YConformal Killing tensorsNatural and conformally invariant quantizationStructure of the conformal symmetries

ExamplesDiPirro systemTaub-NUT metric

Application to the R-separation

higher symmetries of the conformal laplacian · higher symmetries of the conformal laplacian j.-p....

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