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Shinya Tomizawa KEK
Integrability of Five Dimensional Minimal
Supergravity and Charged Rotating Black Holes
e-Print: arXiv:0912.3199 [hep-th]
Authors : Pau FiguerasElla JamsinJorge V. RochaAmitabh Virmani
http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Figueras%2C%20Pau%22http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Jamsin%2C%20Ella%22http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Rocha%2C%20Jorge%20V%2E%22http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Virmani%2C%20Amitabh%22http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Virmani%2C%20Amitabh%22http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Rocha%2C%20Jorge%20V%2E%22http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Jamsin%2C%20Ella%22http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Figueras%2C%20Pau%228/3/2019 Pau Figueras et al- Integrability of Five Dimensional Minimal Supergravity and Charged Rotating Black Holes
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Outline Introduction
Inverse Scattering Method in Relativity
Non-linear -model in D=5 Minimal SUGRA Inverse Scattering Method in D=5 SUGRA
Results
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Black hole entropy counting
e.g. Fuzzball, Kerr/CFT,
Most proposals are focusing on extreme BHs in SUGRA
For non-extreme BHs, much less developed
All supersymmetric solutions in D=5 minimal SUGRA have been classified
(Gauntlett-Gutowski-Hull-Pakis-Reall 03)
But there is no solution-genenation technique for non-supersymmetric (non-
BPS) solutions even in D=5 minimal SUGRA
EOM : non-linear eq. We need a systematic method
Authors purposes
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Authors purposes
To understand the integrability of D=5 minimal SUGRA and to formulate
systematic solution-generation technique in order to construct not only
BPS solution but also more general non-BPS solutions.
To Generalize the inverse scattering methodwhich was developed by
Belinsky-Zakahlov in Einstein theories to D=5 minimal SUGRA
To construct most generalnon-BPS black rings Supersymemtric black ring (Elvang-Emparan-Mateos-Reall 04)
Non-BPS black ring with 4 parameters (Elavang-Emparan-figures 05)
No supersymmetric limit
More general non-BPS black ring with five parameters may exist
(Elavang-Emparan-figures 05)
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Inverse Scattering Method (ISM)
As solitonic solutions, all known black hole solutions can be derived by ISM( Koikawa 05,Pomerasky 06, Tomizawa- Morisawa-Yasui 06, Tomizawa-Nozawa 06, Pomerasky-Senkov 06,Elvang-Figuras 07,
Izumi 07)
ISM can be applied only to Einstein eqs. with (D-2)-CKVs
ex) in D=5, RU(1)U(1)
In fluid mechanics, ISM was established in order to solve a non-linear KdV eq
in a systematic way (Gardner-Green-Kruskal-Miura 67)
ISM was also extended to other non-linear eqs s.t. Sine Gordon eq & non-linear
Schrodinger eqISM was extend to a special class of D=4 Einstein eq (Belinsky-Zakharov 78, 79)
ISM was extend to a special class of D=4Einstein-Maxwell eq (Alekseev 81)
Recently, ISM has been applied to HD Einstein eq by many authors (Pomerasky
05)
More systematically, ISM can be applied to D=5 minimal SUGRA (09)
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Black di-ring (Iguchi-Mishima 07 )
Black Saturn (Elvang-Figuras07 )
Orthogonal dring (Izumi 07 )
Black holes (Tangherlini 63, Myers-Perry 87 )
Blck rings(Emparan-Reall 02, Mishima-Iguchi 05,
Pomeransky-Senkov 06)
D=5 Asymptotically Flat BH Sols as solitons
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Inverse Scattering Method (ISM)
As solitonic solutions, all known black hole solutions can be derived by ISM( Koikawa 05,Pomerasky 06, Tomizawa- Morisawa-Yasui 06, Tomizawa-Nozawa 06, Pomerasky-Senkov 06,Elvang-Figuras 07,
Izumi 07)
ISM can be applied only to Einstein eqs. with (D-2)-CKVs
ex) in D=5, RU(1)U(1)
In fluid mechanics, ISM was established in order to solve a non-linear KdV eq
in a systematic way (Gardner-Green-Kruskal-Miura 67)
ISM was also extended to other non-linear eqs s.t. Sin Gordon eq & non-linear
Schrodinger eqISM was extend to a special class of D=4 Einstein eq (Belinsky-Zakharov 78, 79)
ISM was extend to a special class of D=4Einstein-Maxwell eq (Alekseev 81)
Recently, ISM has been applied to HD Einstein eq by many authors (Pomerasky
05)
More systematically, ISM can be applied to D=5 minimal SUGRA (09)
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Inverse Scattering Method
in Relativity
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9
(D-2)-metric 2-metric on 2-surface orthogonal to all CKVs
Metric in canonical coordinate
Metric
Pure Einstein theory
Spacetime symmetry:
Existence of (D-2)-CKVs (commuting Killing vectors) is assumed
Assumptions
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Einstein equations
where
; matrixes
Solitonic equations
Constraint condition
For solutions of (1), f is
determined
How to solve
Solve eqs. (1) (4)Normalization For normalized g, solveand
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LAX pair in GR (Belinski and Zakahrov 1979, 1980)
LAX pair
complex parameter
Non-linear solitonic equation can be replaced with a pair oflinear equations:
Comparability
(Solitonic eq)
The metric g can be obtained form by
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Generation technique
Find 0for seed g0
Dressing
New solutionKnown solution
(seed)
Normalization
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13
2N-solitons solutions : new spacetimes
(N-Kerr black holes )(Minkowski spacetimes )
Seed solutions : known spacetimes
Adding
solitons
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Non-linear sigma model in SUGRA
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Einstein-Maxwell-Chern-Simons Theory
Action
EOM
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2-Killing system in D=5 Einstein gravity
(Maison 79)
Assume existence of 2 commuting Killing vectors
Gravitational potentials
Twist potentials
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Assume existence of 2 commuting Killing vectors
Gravitational potentials
Twist potentials
Electromagnetic potentials
2-Killing system in D=5 Minimal SUGRA
(Bouchareb-Clement-Chen-Galtsov-Scherbluk-Wolf 07)
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Action
Invariant under G2(2) transformation (Mizoguchi-Ohta) :
Coupled system of scalar fields and D=3 gravity
Simply
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Introduce 77 coset matrix:
where
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Action
Invariant under G2(2)transformation (Mizoguchi-Ohta) :
Coupled system of scalar fields and D=3 gravity
Simply
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3-Killing system in 5D EMCS
Assume 3rd Killing vector
Metric can be written in canonical coordinares
Gauge potential can be written as
determined by
determined by
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Einstein-Maxwell-Chern-Simons equations
where
; matrixes
Solitonic equations
Constraint condition
For solutions of (1), f is
determined
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L-A pair in D=5 Minimal SUGRA (Figueras-Jamsin-Rocha-Virmani 09)
(Solitonic eq)
M can be obtained from by putting =0
Lax pair
U & V are replaced with
Derivative operators are similarly defined as
Compatibility condition is equal to EOM
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Generation technique
Find 0for seed g0
Dressing
New solutionKnown solution
(seed)
Normalization
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Formula (relation between new solution and seed) is given by
Seed
Main result
Once seeds are given, we can obtain new solutions
with
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Examples
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D=5 Myers-Perry solution (D=5 Kerr solution) can be obtained
from D=5 Schwartzschild solution
D=5 Cvetic-Youm solution (D=5 Kerr-Newman solution) can be obtained
from D=5 Reissner-Nordstom solution
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Results & Open problems
The authors in this paper have developed the integrability of D=5 minimal supergravity with 3
commuting Killing vectors and generalized well-known BZ technique in Einstein gravity to the
supergravity
As examples(tests), in this formalism, some known rotating solutions have been derived from non-
rotaitng solutions in the same theory:D=5 rotating Kerr black hole solution can be reconstructed from D=5 non-rotating Schwarzshild solution
D=5 charged-rotating black hole solution can be reconstructed from D=5 charged non-rotating Reissner-
Nordstrom solution
This result here will be useful for other theory if it can be reduced to a non-linear sigma model
The next step is to understand the physics of dipole charge in this formalism
The authors do not fully understand this formalismCan charged rotating solutions be directly obtained from neutral solutions ?
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EOM
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Dressing dressing matrix
Assume solitonic solutions:
Dressing:
KnownNew
The dressing matrix takes the following form
dressing matrix
77 matrix
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Non-linear -model action
EOMs of the scalar fields are derived from G2 invariant -model action:
Base space: 2D region ={(,z)|0}
Target space:
(Mizoguchi-Ohta 98)
31
(Bouchareb-Clement-Chen-Galtsov-Scherbluk-Wolf 07)
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Non-linear -model approach Under a certain symmetry assumptions, theory can be reduced to 2D non-linear -model
Consider as Boundary value problem of scalar fields
theory target space
D=4 Einstein SU(1,1)
D=4 Einstein-Maxwell SU(1,2)
D=5 Einstein SL(3,R)
D=5 Minimal SUGRA G2 (2)