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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 2 / 28
Overview of the talk
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)
Gravitationspecial tensors
integrable systems
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 2 / 28
Overview of the talk
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)
Gravitationspecial tensors
integrable systems
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 2 / 28
Overview of the talk
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)
Gravitationspecial tensors
integrable systems
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 2 / 28
Overview of the talk
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)
Gravitationspecial tensors
integrable systems
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 2 / 28
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 3 / 28
Higher Dimensional Black Holes
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 4 / 28
Higher Dimensional Black Holes
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 5 / 28
Higher Dimensional Black Holes
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 5 / 28
Higher Dimensional Black Holes
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 5 / 28
Higher Dimensional Black Holes Main properties
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 6 / 28
Higher Dimensional Black Holes Main properties
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!
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 7 / 28
Higher Dimensional Black Holes Main properties
Kerr-NUT(A)dS black holes
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
All properties accounted for by PCKY special tensor!
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 7 / 28
Higher Dimensional Black Holes Main properties
Kerr-NUT(A)dS black holes
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
All properties accounted for by PCKY special tensor!
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 7 / 28
Higher Dimensional Black Holes Main properties
Kerr-NUT(A)dS black holes
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
All properties accounted for by PCKY special tensor!
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 7 / 28
Higher Dimensional Black Holes Main properties
Kerr-NUT(A)dS black holes
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
All properties accounted for by PCKY special tensor!
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 7 / 28
Higher Dimensional Black Holes Main properties
Kerr-NUT(A)dS black holes
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
All properties accounted for by PCKY special tensor!
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 7 / 28
Higher Dimensional Black Holes Main properties
Kerr-NUT(A)dS black holes
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
All properties accounted for by PCKY special tensor!
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 7 / 28
Higher Dimensional Black Holes CKY tensors
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 8 / 28
Higher Dimensional Black Holes CKY tensors
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 9 / 28
Higher Dimensional Black Holes CKY tensors
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) tensor
dω = 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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 9 / 28
Higher Dimensional Black Holes CKY tensors
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 9 / 28
Higher Dimensional Black Holes CKY tensors
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 9 / 28
Higher Dimensional Black Holes CKY tensors
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 9 / 28
Higher Dimensional Black Holes CKY tensors
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 9 / 28
Higher Dimensional Black Holes CKY tensors
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 9 / 28
Higher Dimensional Black Holes CKY tensors
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 9 / 28
Higher Dimensional Black Holes Geometry
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 10 / 28
Higher Dimensional Black Holes Geometry
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 .
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 11 / 28
Higher Dimensional Black Holes Geometry
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 .
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 11 / 28
Higher Dimensional Black Holes Geometry
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 .
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 11 / 28
Higher Dimensional Black Holes Geometry
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 .
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 11 / 28
Higher Dimensional Black Holes Geometry
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 .
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 11 / 28
Higher Dimensional Black Holes Geometry
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 .
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 11 / 28
Higher Dimensional Black Holes Geometry
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 .
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 11 / 28
Higher Dimensional Black Holes Geometry
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 .
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 11 / 28
Recent Results
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 12 / 28
Recent Results Complete set of commuting operators of Dirac equation
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 13 / 28
Recent Results Complete set of commuting operators of Dirac equation
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 14 / 28
Recent Results Complete set of commuting operators of Dirac equation
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 14 / 28
Recent Results Complete set of commuting operators of Dirac equation
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 14 / 28
Recent Results Intrinsic Separability of Dirac Equation
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 15 / 28
Recent Results Intrinsic Separability of Dirac Equation
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 16 / 28
Recent Results Intrinsic Separability of Dirac Equation
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 16 / 28
Recent Results Intrinsic Separability of Dirac Equation
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 16 / 28
Recent Results Intrinsic Separability of Dirac Equation
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 17 / 28
Recent Results Intrinsic Separability of Dirac Equation
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 17 / 28
Recent Results Intrinsic Separability of Dirac Equation
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 17 / 28
Recent Results Intrinsic Separability of Dirac Equation
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 17 / 28
Recent Results Intrinsic Separability of Dirac Equation
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]Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 17 / 28
Recent Results Intrinsic Separability of Dirac Equation
Intrinsic Separability of Dirac Equation 2
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 18 / 28
Recent Results Intrinsic Separability of Dirac Equation
Intrinsic Separability of Dirac Equation 2
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 18 / 28
Recent Results Intrinsic Separability of Dirac Equation
Intrinsic Separability of Dirac Equation 2
Introduce new ‘auxiliary’ operators
Mj ≡ R−1Mj R ,
then:[Mj, Mk] = 0
a solution ofKkξ = 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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 18 / 28
Recent Results Intrinsic Separability of Dirac Equation
Intrinsic Separability of Dirac Equation 2
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 18 / 28
Recent Results Intrinsic Separability of Dirac Equation
Intrinsic Separability of Dirac Equation 2
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 18 / 28
Work in progress and outlook
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 19 / 28
Work in progress and outlook Full integrability of spinning particle motion
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 20 / 28
Work in progress and outlook Full integrability of spinning particle motion
Motivation
Work in progress!
Why interesting?
Genuinely new result for this theoryConnection to Dirac and Papapetrou equationsNew ’miraculous’ cancellations
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 21 / 28
Work in progress and outlook Full integrability of spinning particle motion
Motivation
Work in progress!
Why interesting?
Genuinely new result for this theory
Connection to Dirac and Papapetrou equationsNew ’miraculous’ cancellations
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 21 / 28
Work in progress and outlook Full integrability of spinning particle motion
Motivation
Work in progress!
Why interesting?
Genuinely new result for this theoryConnection to Dirac and Papapetrou equations
New ’miraculous’ cancellations
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 21 / 28
Work in progress and outlook Full integrability of spinning particle motion
Motivation
Work in progress!
Why interesting?
Genuinely new result for this theoryConnection to Dirac and Papapetrou equationsNew ’miraculous’ cancellations
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 21 / 28
Work in progress and outlook Full integrability of spinning particle motion
Meaning real cancellations...
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 22 / 28
Work in progress and outlook Full integrability of spinning particle motion
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 23 / 28
Work in progress and outlook Full integrability of spinning particle motion
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 23 / 28
Work in progress and outlook Full integrability of spinning particle motion
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]
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 23 / 28
Work in progress and outlook Full integrability of spinning particle motion
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]Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 23 / 28
Work in progress and outlook Full integrability of spinning particle motion
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 24 / 28
Work in progress and outlook Full integrability of spinning particle motion
New conserved quantities for the spinning particle theory?
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!
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 24 / 28
Work in progress and outlook Full integrability of spinning particle motion
New conserved quantities for the spinning particle theory?
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!
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 24 / 28
Work in progress and outlook Full integrability of spinning particle motion
New conserved quantities for the spinning particle theory?
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!
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 24 / 28
Work in progress and outlook Full integrability of spinning particle motion
New conserved quantities for the spinning particle theory?
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!
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 24 / 28
Work in progress and outlook Full integrability of spinning particle motion
New conserved quantities for the spinning particle theory?
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!
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 24 / 28
Work in progress and outlook Full integrability of spinning particle motion
New conserved quantities for the spinning particle theory?
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!
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 24 / 28
Work in progress and outlook Full integrability of spinning particle motion
New conserved quantities for the spinning particle theory?
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!
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 24 / 28
Work in progress and outlook Full integrability of spinning particle motion
New conserved quantities for the spinning particle theory?
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!
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 24 / 28
Work in progress and outlook Conclusions
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
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 25 / 28
Work in progress and outlook Conclusions
Conclusions
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)
Open avenuesKilling-Yano bracketElectromagnetic and gravitational perturbations
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 26 / 28
Work in progress and outlook Conclusions
Conclusions
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)
Open avenues
Killing-Yano bracketElectromagnetic and gravitational perturbations
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 26 / 28
Work in progress and outlook Conclusions
Conclusions
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)
Open avenuesKilling-Yano bracket
Electromagnetic and gravitational perturbations
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 26 / 28
Work in progress and outlook Conclusions
Conclusions
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)
Open avenuesKilling-Yano bracketElectromagnetic and gravitational perturbations
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 26 / 28
Work in progress and outlook Conclusions
Being a physics student in Padova...
+ =
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 27 / 28
Work in progress and outlook Conclusions
Being a physics student in Padova...
+ =
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 27 / 28
Work in progress and outlook Conclusions
Being a physics student in Padova...
+
=
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 27 / 28
Work in progress and outlook Conclusions
Being a physics student in Padova...
+ =
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 27 / 28
Work in progress and outlook Conclusions
Thank you!
Marco Cariglia (UFOP) Hidden Symmetries Padova 08-09-2011 28 / 28