Hidden symmetries and the Physics of Higher Dimensional ...Marco Cariglia (UFOP) Hidden Symmetries...

Hidden symmetries and the Physics of HigherDimensional Black Holes

Marco Cariglia1 Pavel Krtouš2 David Kubiznák3

1ICEB, Universidade Federal de Ouro Preto, Minas Gerais, Brazil2Institute of Theoretical physics, Faculty of Mathematics and Physics, Charles University, Czech Republic

3Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada

XVII European Workshop on String Theory, Padova, 8 September 2011

Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 1 / 28

Overview of the talk

Main points to take away

Geometry of higher dimensional black holes is specialRelated to special tensors: Conformal Killing Yano (CKY)Many important physical systems on this background are integrable and displayseparation of variables: HJ, KG, Dirac, spinning particle (preliminary)

Gravitationspecial tensors

integrable systems



Main points to take awayGeometry of higher dimensional black holes is special

Related to special tensors: Conformal Killing Yano (CKY)Many important physical systems on this background are integrable and displayseparation of variables: HJ, KG, Dirac, spinning particle (preliminary)


integrable systems



Main points to take awayGeometry of higher dimensional black holes is specialRelated to special tensors: Conformal Killing Yano (CKY)

Many important physical systems on this background are integrable and displayseparation of variables: HJ, KG, Dirac, spinning particle (preliminary)


integrable systems



Main points to take awayGeometry of higher dimensional black holes is specialRelated to special tensors: Conformal Killing Yano (CKY)Many important physical systems on this background are integrable and displayseparation of variables: HJ, KG, Dirac, spinning particle (preliminary)


integrable systems





integrable systems


Outline

1 Higher Dimensional Black HolesMain propertiesCKY tensorsGeometry

2 Recent ResultsComplete set of commuting operators of Dirac equationIntrinsic Separability of Dirac Equation

3 Work in progress and outlookFull integrability of spinning particle motionConclusions


Higher Dimensional Black Holes

Outline






Physical observations of candidate black holes

Accretion of mass /jetsX-ray binariesActive galactic nuclei

Study:evolution of test particlestest fields: scalar, Dirac, electromagnetic, gravitational

→ Hidden Symmetries














Higher Dimensional Black Holes Main properties

Outline






Kerr-NUT(A)dS black holes

Main propertiesKerr-Schild form

Petrov DSeparability of HJ, KG, DiracTower of isometries and Killing-Stackel tensorsSpinning particle theory has extra worldline SUSYPart of a more general class of metrics (canonical metrics) that admit a PCKYtensor

All properties accounted for by PCKY special tensor!




Main propertiesKerr-Schild formPetrov D

Separability of HJ, KG, DiracTower of isometries and Killing-Stackel tensorsSpinning particle theory has extra worldline SUSYPart of a more general class of metrics (canonical metrics) that admit a PCKYtensor





Main propertiesKerr-Schild formPetrov DSeparability of HJ, KG, Dirac

Tower of isometries and Killing-Stackel tensorsSpinning particle theory has extra worldline SUSYPart of a more general class of metrics (canonical metrics) that admit a PCKYtensor





Main propertiesKerr-Schild formPetrov DSeparability of HJ, KG, DiracTower of isometries and Killing-Stackel tensors

Spinning particle theory has extra worldline SUSYPart of a more general class of metrics (canonical metrics) that admit a PCKYtensor





Main propertiesKerr-Schild formPetrov DSeparability of HJ, KG, DiracTower of isometries and Killing-Stackel tensorsSpinning particle theory has extra worldline SUSY

Part of a more general class of metrics (canonical metrics) that admit a PCKYtensor





Main propertiesKerr-Schild formPetrov DSeparability of HJ, KG, DiracTower of isometries and Killing-Stackel tensorsSpinning particle theory has extra worldline SUSYPart of a more general class of metrics (canonical metrics) that admit a PCKYtensor





Main propertiesKerr-Schild formPetrov DSeparability of HJ, KG, DiracTower of isometries and Killing-Stackel tensorsSpinning particle theory has extra worldline SUSYPart of a more general class of metrics (canonical metrics) that admit a PCKYtensor



Higher Dimensional Black Holes CKY tensors

Outline






CKY tensors

Definition (CKY tensor): a Conformal Killing Yano (CKY) tensor is a p–formsatisfying

∇Xω =1

p + 1X−| dω −

1n− p + 1

X[ ∧ δω = 0 ,

δω = 0→ Killing-Yano (KY) tensordω = 0→ closed conformal Killing-Yano (CCKY) tensor

Hodge duality: KY ↔ CCKY

CCKY tensors closed under exterior product [Krtouš, Kubiznák, Page, Frolov 2007]

Conserved quantities in quantum theory of spin 0 & 12 particles. No anomalies!

Suitable generalization to CKY equation with flux: non-vacuum space-times,black holes of supergravity theories [Kubiznák, Kunduri, Yasui 2009 ; Houri,Kubiznák, Warnick, Yasui 2010; Chow 2010; Krtouš, Kubiznák, Warnick 2011]



CKY tensors


∇Xω =1

p + 1X−| dω −

1n− p + 1

X[ ∧ δω = 0 ,

δω = 0→ Killing-Yano (KY) tensor

dω = 0→ closed conformal Killing-Yano (CCKY) tensor







CKY tensors


∇Xω =1

p + 1X−| dω −

1n− p + 1

X[ ∧ δω = 0 ,








CKY tensors


∇Xω =1

p + 1X−| dω −

1n− p + 1

X[ ∧ δω = 0 ,








CKY tensors


∇Xω =1

p + 1X−| dω −

1n− p + 1

X[ ∧ δω = 0 ,








CKY tensors


∇Xω =1

p + 1X−| dω −

1n− p + 1

X[ ∧ δω = 0 ,




Conserved quantities in quantum theory of spin 0 & 12 particles.

No anomalies!Suitable generalization to CKY equation with flux: non-vacuum space-times,black holes of supergravity theories [Kubiznák, Kunduri, Yasui 2009 ; Houri,Kubiznák, Warnick, Yasui 2010; Chow 2010; Krtouš, Kubiznák, Warnick 2011]



CKY tensors


∇Xω =1

p + 1X−| dω −

1n− p + 1

X[ ∧ δω = 0 ,








CKY tensors


∇Xω =1

p + 1X−| dω −

1n− p + 1

X[ ∧ δω = 0 ,







Higher Dimensional Black Holes Geometry

Outline






Geometry from principal CKY tensor

Principal conformal Killing-Yano (PCKY) tensor = non-degenerate CCKY 2–form

∇Xh = X[ ∧ ξ = 0 , ξa = 1n−1∇bhb

a primary Killing vector.

Implies the canonical metric in 2N + ε dimensions [Houri, Oota, Yasui 2009]

Tower of symmetries:

1) KY tensors: h(j) = h ∧ . . . ∧ h︸︷︷︸total of j factors

, f (j) = ?h(j)

2) Killing tensors: K(j)µν =

1(n− 2j− 1)!(j!)2 f (j)

µλ1...λn−2j−1 f (j)νλ1...λn−2j−1

3) Secondary KVs:ξ(j)µ = K(j)µ

νξν (j = 1, . . . ,N − 1) ,

ξN = f (N) (odd dim.)

They satisfy:[ξ(k), ξ(l)] = 0 , Lξ(k)h(l) = 0 , Lξ(k) f (l) = 0 .





∇Xh = X[ ∧ ξ = 0 , ξa = 1n−1∇bhb





, f (j) = ?h(j)


1(n− 2j− 1)!(j!)2 f (j)

µλ1...λn−2j−1 f (j)νλ1...λn−2j−1


νξν (j = 1, . . . ,N − 1) ,







∇Xh = X[ ∧ ξ = 0 , ξa = 1n−1∇bhb





, f (j) = ?h(j)


1(n− 2j− 1)!(j!)2 f (j)

µλ1...λn−2j−1 f (j)νλ1...λn−2j−1


νξν (j = 1, . . . ,N − 1) ,







∇Xh = X[ ∧ ξ = 0 , ξa = 1n−1∇bhb





, f (j) = ?h(j)


1(n− 2j− 1)!(j!)2 f (j)

µλ1...λn−2j−1 f (j)νλ1...λn−2j−1


νξν (j = 1, . . . ,N − 1) ,







∇Xh = X[ ∧ ξ = 0 , ξa = 1n−1∇bhb





, f (j) = ?h(j)


1(n− 2j− 1)!(j!)2 f (j)

µλ1...λn−2j−1 f (j)νλ1...λn−2j−1


νξν (j = 1, . . . ,N − 1) ,







∇Xh = X[ ∧ ξ = 0 , ξa = 1n−1∇bhb





, f (j) = ?h(j)


1(n− 2j− 1)!(j!)2 f (j)

µλ1...λn−2j−1 f (j)νλ1...λn−2j−1


νξν (j = 1, . . . ,N − 1) ,







∇Xh = X[ ∧ ξ = 0 , ξa = 1n−1∇bhb





, f (j) = ?h(j)


1(n− 2j− 1)!(j!)2 f (j)

µλ1...λn−2j−1 f (j)νλ1...λn−2j−1


νξν (j = 1, . . . ,N − 1) ,







∇Xh = X[ ∧ ξ = 0 , ξa = 1n−1∇bhb





, f (j) = ?h(j)


1(n− 2j− 1)!(j!)2 f (j)

µλ1...λn−2j−1 f (j)νλ1...λn−2j−1


νξν (j = 1, . . . ,N − 1) ,




Recent Results

Outline





Recent Results Complete set of commuting operators of Dirac equation

Outline






Dirac: complete set of commuting operators

Why interesting?

Example of complete set of commuting operators for Dirac equation in curvedspacetime

Proposition (Complete set of commuting operators) The most generalspacetime admitting a PCKY tensor admits a complete set of commutingoperators:

{D,Kξ(0) , . . .Kξ(N−1+ε) ,Mh(1) , . . .Mh(N−1)} .

(Note that the Dirac operator can be written as D = Mh(0) )[Cariglia, Krtouš, Kubiznák 2011]




Why interesting?



{D,Kξ(0) , . . .Kξ(N−1+ε) ,Mh(1) , . . .Mh(N−1)} .





Why interesting?



{D,Kξ(0) , . . .Kξ(N−1+ε) ,Mh(1) , . . .Mh(N−1)} .



Recent Results Intrinsic Separability of Dirac Equation

Outline






Motivation

Why interesting?

Hamilton-Jacobi and Klein Gordon: known theorems for separation of variables

Dirac: theory of separability not well established. However explicit separation ofvariables in Kerr-NUT-(A)dS achieved in [Oota, Yasui 2008].Show separation for Dirac in Kerr-NUT-(A)dS is explained by the complete setof mutually commuting operators



Motivation

Why interesting?

Hamilton-Jacobi and Klein Gordon: known theorems for separation of variablesDirac: theory of separability not well established. However explicit separation ofvariables in Kerr-NUT-(A)dS achieved in [Oota, Yasui 2008].

Show separation for Dirac in Kerr-NUT-(A)dS is explained by the complete setof mutually commuting operators



Motivation

Why interesting?

Hamilton-Jacobi and Klein Gordon: known theorems for separation of variablesDirac: theory of separability not well established. However explicit separation ofvariables in Kerr-NUT-(A)dS achieved in [Oota, Yasui 2008].Show separation for Dirac in Kerr-NUT-(A)dS is explained by the complete setof mutually commuting operators



Intrinsic Separability of Dirac Equation 1

Operators Kk = Kξ(k) and Mj = M 1j! h(j) can be diagonalised

Kkξ = i Ψkξ , Mjξ = Xjξ ,

with eigenfunction ξ in tensorial R-separated form

ξ = R exp(i∑

k Ψkψk) ⊗

ν

χν ,

where {χν} is an N-tuple of 2-dimensional spinors and R is the (Cliffordbundle)-valued prefactor

R =∏κ<λ

(xκ + ι〈κλ〉xλ

)− 12.

Dirac equation reduces to decoupled equations[( ddxν

+X′ν

4Xν+

Ψν

Xνι〈ν〉 +

ε

2xν

)σ〈ν〉 −

(−ι〈ν〉

)N−ν√|Xν |

(ε

i√−c

2x2ν

+ Xν)]

χν = 0 .

Same as solution found in [Oota, Yasui 2008]







ξ = R exp(i∑

k Ψkψk) ⊗

ν

χν ,


R =∏κ<λ


)− 12.


+X′ν

4Xν+

Ψν

Xνι〈ν〉 +

ε

2xν

)σ〈ν〉 −

(−ι〈ν〉

)N−ν√|Xν |

(ε

i√−c

2x2ν

+ Xν)]

χν = 0 .








ξ = R exp(i∑

k Ψkψk) ⊗

ν

χν ,


R =∏κ<λ


)− 12.


+X′ν

4Xν+

Ψν

Xνι〈ν〉 +

ε

2xν

)σ〈ν〉 −

(−ι〈ν〉

)N−ν√|Xν |

(ε

i√−c

2x2ν

+ Xν)]

χν = 0 .








ξ = R exp(i∑

k Ψkψk) ⊗

ν

χν ,


R =∏κ<λ


)− 12.


+X′ν

4Xν+

Ψν

Xνι〈ν〉 +

ε

2xν

)σ〈ν〉 −

(−ι〈ν〉

)N−ν√|Xν |

(ε

i√−c

2x2ν

+ Xν)]

χν = 0 .








ξ = R exp(i∑

k Ψkψk) ⊗

ν

χν ,


R =∏κ<λ


)− 12.


+X′ν

4Xν+

Ψν

Xνι〈ν〉 +

ε

2xν

)σ〈ν〉 −

(−ι〈ν〉

)N−ν√|Xν |

(ε

i√−c

2x2ν

+ Xν)]

χν = 0 .

Same as solution found in [Oota, Yasui 2008]Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 17 / 28



Introduce new ‘auxiliary’ operators

Mj ≡ R−1Mj R ,

then:[Mj, Mk] = 0a solution of

Kkξ = i Ψkξ , Mjξ = Xjξ

can be found in standard separated form (no R factor)operators Mj are operators Mj in the ‘R-representation’

γa = R−1γaR .

[Cariglia, Krtouš, Kubiznák 2011]





Mj ≡ R−1Mj R ,

then:

[Mj, Mk] = 0a solution of



γa = R−1γaR .






Mj ≡ R−1Mj R ,

then:[Mj, Mk] = 0

a solution ofKkξ = i Ψkξ , Mjξ = Xjξ


γa = R−1γaR .






Mj ≡ R−1Mj R ,



can be found in standard separated form (no R factor)

operators Mj are operators Mj in the ‘R-representation’

γa = R−1γaR .






Mj ≡ R−1Mj R ,




γa = R−1γaR .



Work in progress and outlook

Outline





Work in progress and outlook Full integrability of spinning particle motion

Outline






Motivation

Work in progress!

Why interesting?

Genuinely new result for this theoryConnection to Dirac and Papapetrou equationsNew ’miraculous’ cancellations



Motivation

Work in progress!

Why interesting?

Genuinely new result for this theory

Connection to Dirac and Papapetrou equationsNew ’miraculous’ cancellations



Motivation

Work in progress!

Why interesting?

Genuinely new result for this theoryConnection to Dirac and Papapetrou equations

New ’miraculous’ cancellations



Motivation

Work in progress!

Why interesting?

Genuinely new result for this theoryConnection to Dirac and Papapetrou equationsNew ’miraculous’ cancellations



Meaning real cancellations...



Spinning particle theory

Worldline SUSY extension of ordinary classical scalar particle

L =12

gµν

(xµxν + iψµ

Dψν

dτ

).

ψµ grassmannian ’spin’ coordinates.

Equations of motion:

D2xµdτ 2 = xµ + Γµρσ xρxσ = i

2 Rµνκλψκλxν , Dψν

Dτ = ψν + Γνλµxλψµ = 0 .

Supercharge Q = xµψµ = Πµψµ,

{H,Q} = 0 , {Q,Q} = −2iH .

KY tensors ω define extra worldline SUSY

Qω = Πλψµ1 . . . ψµp−1ωλ

µ1...µp−1

− i(p + 1)2ψ

µ1 . . . ψµp+1 dωµ1...µp+1

[Gibbons, Rietdijk, van Holten 1993]





L =12

gµν

(xµxν + iψµ

Dψν

dτ

).

ψµ grassmannian ’spin’ coordinates. Equations of motion:





{H,Q} = 0 , {Q,Q} = −2iH .



µ1...µp−1

− i(p + 1)2ψ

µ1 . . . ψµp+1 dωµ1...µp+1






L =12

gµν

(xµxν + iψµ

Dψν

dτ

).






{H,Q} = 0 , {Q,Q} = −2iH .



µ1...µp−1

− i(p + 1)2ψ

µ1 . . . ψµp+1 dωµ1...µp+1






L =12

gµν

(xµxν + iψµ

Dψν

dτ

).






{H,Q} = 0 , {Q,Q} = −2iH .



µ1...µp−1

− i(p + 1)2ψ

µ1 . . . ψµp+1 dωµ1...µp+1

[Gibbons, Rietdijk, van Holten 1993]Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 23 / 28


New conserved quantities for the spinning particle theory?

Integrability step1: need enough conserved charges

Integrability step 2: want to express xµ in terms of conserved quantities to solvefor motion.Cannot use Qf (j) : ψµ variables not invertible!

Look for new superinvariants K(i),{

Q,K(i)}

= 0, with

K(i) = Kµν(i) ΠµΠν + Lµ(i)α1α2Πµψα1ψα2 +M(i)α1...α4ψ

α1 . . . ψα4

Ansatz: Kµν = f µκ1...κp−1 f νκ1...κp−1 , L ∼ f df , M ∼ ∇L

Finding 1: in general there is an obstruction{

Q,K(i)}∼ R f f 6= 0

Finding 2: obstruction vanishes for Kerr-NUT-(A)dS! (computer simulation)Integrability step 3: check Poisson brackets of conserved quantities:Schouten-Nijenhuis brackets + spin corrections (computer simulation)Integrability step 4: solve for velocities x and integrate the motion




Integrability step1: need enough conserved chargesIntegrability step 2: want to express xµ in terms of conserved quantities to solvefor motion.

Cannot use Qf (j) : ψµ variables not invertible!


Q,K(i)}

= 0, with


α1 . . . ψα4



Q,K(i)}∼ R f f 6= 0





Integrability step1: need enough conserved chargesIntegrability step 2: want to express xµ in terms of conserved quantities to solvefor motion.Cannot use Qf (j) : ψµ variables not invertible!


Q,K(i)}

= 0, with


α1 . . . ψα4



Q,K(i)}∼ R f f 6= 0







Q,K(i)}

= 0, with


α1 . . . ψα4



Q,K(i)}∼ R f f 6= 0







Q,K(i)}

= 0, with


α1 . . . ψα4



Q,K(i)}∼ R f f 6= 0







Q,K(i)}

= 0, with


α1 . . . ψα4



Q,K(i)}∼ R f f 6= 0







Q,K(i)}

= 0, with


α1 . . . ψα4



Q,K(i)}∼ R f f 6= 0

Finding 2: obstruction vanishes for Kerr-NUT-(A)dS! (computer simulation)

Integrability step 3: check Poisson brackets of conserved quantities:Schouten-Nijenhuis brackets + spin corrections (computer simulation)Integrability step 4: solve for velocities x and integrate the motion






Q,K(i)}

= 0, with


α1 . . . ψα4



Q,K(i)}∼ R f f 6= 0

Finding 2: obstruction vanishes for Kerr-NUT-(A)dS! (computer simulation)Integrability step 3: check Poisson brackets of conserved quantities:Schouten-Nijenhuis brackets + spin corrections (computer simulation)

Integrability step 4: solve for velocities x and integrate the motion






Q,K(i)}

= 0, with


α1 . . . ψα4



Q,K(i)}∼ R f f 6= 0



Work in progress and outlook Conclusions

Outline






Conclusions


Open avenuesKilling-Yano bracketElectromagnetic and gravitational perturbations



Conclusions


Open avenues

Killing-Yano bracketElectromagnetic and gravitational perturbations



Conclusions


Open avenuesKilling-Yano bracket

Electromagnetic and gravitational perturbations



Conclusions


Open avenuesKilling-Yano bracketElectromagnetic and gravitational perturbations



Being a physics student in Padova...

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Thank you!


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