Pau Figueras et al- Integrability of Five Dimensional Minimal Supergravity and Charged Rotating Black Holes

8/3/2019 Pau Figueras et al- Integrability of Five Dimensional Minimal Supergravity and Charged Rotating Black Holes

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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%22


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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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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)

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(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)

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Pau Figueras et al- Integrability of Five Dimensional Minimal Supergravity and Charged Rotating Black Holes