On hidden symmetry of higher-dimensional black holes

On hidden symmetry of higher-dimensional black holes

Summer In symmetry of stitute 2011, Fuji, August 07, 2011 Hidden symmetry of symmetry of supergravity black holes holes Tsuyoshi Houri (OCAMI) In symmetry of collaboration symmetry of with D. Kubizn symmetry of ak holes, C. M. Warn symmetry of ick holes (DAMTP) an symmetry of d Y. Yasui (OCU) Ref. Yasui and TH, Prog. Theor. Phys. Suppl. 189 (2011) 126-164. Motivation symmetry of of this work holes We wan symmetry of t to solve the higherdimen symmetry of sion symmetry of al gravitation symmetry of al theories an symmetry of d explicitly con symmetry of struct the solution symmetry of s that have physically an symmetry of d

mathematically in symmetry of terestin symmetry of g properties. But, it is difficult to solve them in symmetry of gen symmetry of eral. In symmetry of vestigatin symmetry of g k holesn symmetry of own symmetry of solution symmetry of s, we con symmetry of sider a gen symmetry of eralization symmetry of of them. We f ocus on symmetry of hidden symmetry of symmetries of black holes holes. Exact solution symmetry of s vacuum black holes holes n with S horizon symmetry of topology vacuum Ein symmetry of stein symmetry of s Eq. Four dimensions

Schwarzschild (1916) Kerr (1963) Carter (1968) mass, NUT, rotation symmetry of , Higher dimensions

rotation symmetry of s, i (1916) Tan symmetry of gherlin symmetry of Myers-Perry (1986) Gibbon symmetry of s-Lu-Page-Pope (2004) mass, NUTs, [(D-1)/2] [(D-1)/2] 5-dim. Hawk holesin symmetry of g, et al. (1998) Chen symmetry of -Lu-Pope (2006)

[D/2-1] [(D-1)/2] Rotatin symmetry of g black holes hole solution symmetry of In symmetry of 1963, Kerr discovered a solution symmetry of describin symmetry of g rotatin symmetry of g black holes holes in symmetry of a vacuum. wher e Geometry of Kerr spacetime Kerrs metric where - Two parameters

mass M an symmetry of gular momen symmetry of tum J=Ma - Two isometries time tran symmetry of slation symmetry of /t axial symmetry / - Rin symmetry of g sin symmetry of gularity at =0, i.e., r=0, =/2 - Two horizon symmetry of s at r=r s.t. (r )=0 Geodesics in symmetry of the Kerr spacetime In symmetry of 1968, Cater demon symmetry of strated that the Hamilton symmetry of - Jacobi equation symmetry of f or geodesics f or the Kerrs metric can symmetry of be separated f or a solution symmetry of an symmetry of d then symmetry of the f un symmetry of ction symmetry of s R(r) an symmetry of d () f ollow where

Scalar fields in symmetry of the Kerr spacetime He also demon symmetry of strated that the massive Klein symmetry of Gordon symmetry of equation symmetry of f or the Kerrs metric can symmetry of be separated f or a solution symmetry of an symmetry of d then symmetry of the f un symmetry of ction symmetry of s R(r) an symmetry of d () f ollow where Separation symmetry of of variables in symmetry of various equation symmetry of s f orgeodesics the Kerrs metric - Hamilton symmetry of - Jacobi equation symmetry of f or - Klein symmetry of - Gordon symmetry of equation symmetry of Carter (1968)

- Maxwell equation symmetry of - Lin symmetry of earized Ein symmetry of stein symmetry of s equation symmetry of Teuk holesolsk holesy (1972) - Neutrin symmetry of o equation symmetry of - Dirac equation symmetry of Teuk holesolsk holesy (1973), Un symmetry of ruh (1973) Chan symmetry of drasek holeshar (1976), Page (1976) Hidden symmetry of symmetries In symmetry of order to give an symmetry of accoun symmetry of t of such in symmetry of tegrabilities an symmetry of d separabilities, a

gen symmetry of eralization symmetry of of Killin symmetry of g symmetry has been symmetry of studied sin symmetry of ce 1970s. vector Killing vector symmetri c Killing-Stackel (KS) Stackel (1895) conformal Killing vector conformal Killing-Stackel (CKS)

antisymmetric Killing-Yano (KY) Yano conformal Killing-Yano (CKY) Tachibana (1969), Kashiwada (1952) (1968) Plan symmetry of of this talk holes 0. Introduction

1. Review Hidden symmetry of Kerr black holes 2. On spacetimes admitting CKY symmetry 3. A generalization of CKY symmetry 4. Summary & Outlook 1. In symmetry of troduction symmetry of Hidden symmetry of symmetry of Kerr black holes holes Complete in symmetry of tegrable system Liouville

in symmetry of tegrability Geodesic equation symmetry of f or F(x, p) : a con symmetry of stan symmetry of t of motion symmetry of Poisson symmetry of s brack holeset Liouville in symmetry of tegrability mean symmetry of s that there exists a maximal set of Poisson symmetry of commutin symmetry of g in symmetry of varian symmetry of ts. Con symmetry of stan symmetry of ts of motion symmetry of an symmetry of d Killin symmetry of g ten symmetry of sors Assume

= 0 ; Killin symmetry of g equation symmetry of Def. Killing-Stackel tensor (KS) is a rank-n symmetric tensor K obeying the Killing Stackel (1895) equation Hamilton symmetry of - Jacobi approach For a D-dimen symmetry of sion symmetry of al man symmetry of if old (MD, g), a local coordin symmetry of ate system xa is called a separable coordin symmetry of ate system if a Hamilton symmetry of - Jacobi equation symmetry of in symmetry of these coordin symmetry of ates where 0 is a con symmetry of stan symmetry of t, is completely in symmetry of tegrable by (additive) separation symmetry of of variables, i.e., where Sa(xa, c) depen symmetry of ds on symmetry of ly on symmetry of the correspon symmetry of din symmetry of g coordin symmetry of ate xa an symmetry of d in symmetry of cludes D

con symmetry of stan symmetry of ts c=(c1, , cD). rr-Separability structure Theor. A D-dimensional manifold (MD, g) admits separability of H-J equation for geodesics if and only if 1. There exist r indep. commuting Killing vectors X(i) : 2. There exist D-r indep. rank-2 Killing tensors K()), which satisfy 3. The Killing tensors K()) have in common D-r eigenvectors X()) s.t. Ben symmetry of en symmetry of ti-Fran symmetry of caviglia (1979) Commen symmetry of ts :- Some examples which are n symmetry of ot separable but in symmetry of tegrable are k holesn symmetry of own symmetry of cf .) . Gibbon symmetry of s-TH-Kubizn symmetry of ak holes-Warn symmetry of ick holes

Hidden symmetry of symmetry of Kerr spacetime I Kerr spacetime admits a ran symmetry of k holes-2 irreducible Walk holeser-Pen symmetry of rose (1970) Killin symmetry of g ten symmetry of sor . Commen symmetry of ts :- Kerr spacetime has 4 in symmetry of depen symmetry of den symmetry of t an symmetry of d mutually commutin symmetry of g con symmetry of stan symmetry of ts of geodesic motion symmetry of , which are correspon symmetry of din symmetry of g to 2 Killin symmetry of g vectors an symmetry of d 2 ran symmetry of k holes-2 Killin symmetry of g ten symmetry of sors. : : : : - On symmetry of e also fin symmetry of ds that this Killin symmetry of g ten symmetry of sor admits the r2separability structure of the H-J equation symmetry of f or

geodesics. Hidden symmetry of symmetry of Kerr spacetime II The Killin symmetry of g ten symmetry of sor K can symmetry of be written symmetry of as the square of a ran symmetry of k holes-2 Killin symmetry of g-Yan symmetry of Pen symmetry of o ten symmetry of sor f. (1973) rose-Floyd rank-2 KY equation Commen symmetry of ts :- Killin symmetry of g-Yan symmetry of o ten symmetry of sor (KY) is a ran symmetry of k holes-p an symmetry of tiYan symmetry of o symmetric ten symmetry of sor f obeyin symmetry of g (1952) . - Havin symmetry of g a Killin symmetry of g-Yan symmetry of o ten symmetry of sor, on symmetry of e can symmetry of always

con symmetry of struct the correspon symmetry of din symmetry of g Killin symmetry of g ten symmetry of sor. On symmetry of the other han symmetry of d, n symmetry of ot every Killin symmetry of g ten symmetry of sor can symmetry of be decomposed in symmetry of terms of Collin symmetry of a Killin symmetry of -Yan symmetry of o ten symmetry of sor. i son symmetry of g(1976), Stephan symmetry of Hidden symmetry of symmetry of Kerr spacetime III Moreover, the Killin symmetry of g-Yan symmetry of o ten symmetry of sor f gen symmetry of erates two Killin symmetry of g vectors. Hughston symmetry of -Sommers (1973) In symmetry of the en symmetry of d, all the symmetries n symmetry of ecessary f or complete in symmetry of tegrability an symmetry of d separability of the H-J equation symmetry of f or geodesics can symmetry of be gen symmetry of erated

by a sin symmetry of gle ran symmetry of k holes-2 Killin symmetry of g-Yan symmetry of o ten symmetry of sor. Hidden symmetry of symmetry of Kerr spacetime IV The Killin symmetry of g-Yan symmetry of o ten symmetry of sor is derived f rom a 1f orm poten symmetry of tial b, Carter (1987) Commen symmetry of ts : - Obviously, is closed 2-f orm. - On symmetry of e fin symmetry of ds that h is a con symmetry of f ormal Killin symmetry of g-Yan symmetry of o ten symmetry of sor (CKY) of ran symmetry of k holes-2, i.e., it f ollows where Tachiban symmetry of a (1969) Hidden symmetry of symmetry of Kerr spacetime V Kerrs

metric where - KY 2-form - CCKY 2-form - rank-2 Killing tensor Symmetry operators a sym. op. f or D f or a diff. op. D Klein symmetry of - Gordon symmetry of equation symmetry of For the scalar Laplacian symmetry of , are symmetry operators, i.e.,

Carter (1977) Dirac equation symmetry of For the Dirac operator D, the operator is symmetry operator when symmetry of ever f is a Killin symmetry of g-Yan symmetry of o ten symmetry of sor. Carter-McLen symmetry of aghan symmetry of (1979) Separability structures f or Kerr black holes hole Algebraic type of curvature is type-D. Geodesic motion symmetry of is completely in symmetry of tegrable. Carter (1968) Hamilton symmetry of - Jacobi equation symmetry of is separable. Klein symmetry of - Gordon symmetry of equation symmetry of is separable.

Carter (1968) K-G symmetry operators exist. Carter (1977) Chan symmetry of drasek holeshar (1976) Dirac equation symmetry of is separable. Dirac symmetry operators exist. Carter-McLen symmetry of aghan symmetry of (1979) A closed CKY 2-f orm exists. Carter (1987) Carters metric Kerrs metric

where coord. trasf . (Boyers coordin symmetry of ates) where Carter (1968) The off-shell metric with Q an symmetry of d P replaced by arbirary f un symmetry of ction symmetry of s Q(r) an symmetry of d P(p) is said to Spacetimes admittin symmetry of g a Killin symmetry of g-Yan symmetry of o ten symmetry of sor Theor. Let (M4, g) be a vacuum type-D space-time. The following

conditions are equivalent: 1. (M4, g) is without acceleration. 2. (M4, g) is one of Carters class. 3. (M4, g) admits a r2-separability structure. 4. (M4, g) admits a Killing-Yano Demian symmetry of sk holesitensor. -Fran symmetry of caviglia (1980) Theor. A spacetime (M4, g) admits a rank-2 KillingYano tensor if and only if the metric is of Carters class, i.e., Dietz-Rudiger (1982), Taxiarchis (1985) Carters metric in symmetry of Ein symmetry of stein symmetry of -Maxwell theory The Carters metric obeys the Ein symmetry of stein symmetry of -Maxwell equation symmetry of s when symmetry of provided

that the f un symmetry of ction symmetry of s tak holese the f orm an symmetry of d the vector poten symmetry of tial reads This metric has six in symmetry of depen symmetry of den symmetry of t parameters. Pleban symmetry of sk holesi-Demian symmetry of sk holesi metric The importan symmetry of t f amily of type D in symmetry of f our dimen symmetry of sion symmetry of s can symmetry of be represen symmetry of ted by the seven symmetry of -parameter metric. Pleban symmetry of sk holesi-Demian symmetry of sk holesi (1976) This metric obeys the Ein symmetry of stein symmetry of -Maxwell equation symmetry of s provided that the f un symmetry of ction symmetry of s tak holese the f orm an symmetry of d the vector poten symmetry of tial reads Relation symmetry of ship b/w Carters metric an symmetry of d P-D

metric Pleban symmetry of sk holesi-Demian symmetry of sk holesi metric where rescale relabel where Set =0 an symmetry of d =1. Then symmetry of we recover the Carters f amily. 197 5 196 8 197 3

197 3 191 8 197 5 196 6 196 196 5 196 9

8 191 6 196 6 196 196 8 9 196 3 196 8 196 9 196

4 196 195 3 1 196 8 191 196 8 196 1 196 2 2 197

2 197 3 195 195 9 9 191 191 7 6 TABLE I in symmetry of Pleban symmetry of sk holesi, Demian symmetry of sk holesi, An symmetry of n symmetry of al. Phys. 98 (1976) 98-127 2. On symmetry of Spacetimes admittin symmetry of g con symmetry of f ormal Killin symmetry of g-Yan symmetry of o (CKY) symmetr

y Exact solution symmetry of s vacuum black holes holes vacuum Ein symmetry of stein symmetry of s with Sn horizon symmetry of topology Eq. Four dimensions Schwarzschild (1916) Kerr (1963) Carter (1968) mass,

NUT, rotation symmetry of , Higher dimensions mass, rotation symmetry of , i (1916) Tan symmetry of gherlin symmetry of Myers-Perry (1986)

Gibbon symmetry of s-Lu-Page-Pope (2004) 5-dim. Hawk holesin symmetry of g, et al. (1998) Chen symmetry of -Lu-Pope (2006) The most general known solution = higher-dimensional Kerr-NUT-(A)dS NUT, D-dimen symmetry of sion symmetry of al Kerr-NUT-(A)dS metric D = 2n +

Chen symmetry of -Lu-Pope (2006) where D=2 n D=2n +1 This metric satisfies Ein symmetry of stein symmetry of Eq. Four-dimen symmetry of sion symmetry of al Kerr-NUT-(A)dS metric where

Five-dimen symmetry of sion symmetry of al Kerr-NUT-(A)dS metric where Six-dimen symmetry of sion symmetry of al Kerr-NUT-(A)dS metric where Seven symmetry of -dimen symmetry of sion symmetry of al Kerr-NUT-(A)dS metric where D-dimen symmetry of sion symmetry of al Kerr-NUT-(A)dS metric D = 2n +

Chen symmetry of -Lu-Pope (2006) where D=2 n D=2n +1 This metric satisfies Ein symmetry of stein symmetry of Eq. How about higher dimen symmetry of sion symmetry of s? Higher dim. Kerr-NUT-(A)dS Kubizn symmetry of ak holes-Frolov (2007) A closed CKY 2-f orm exists.

Geodesic motion symmetry of is completely in symmetry of tegrable. Page-Kubizn symmetry of ak holes-Vasudevan symmetry of -Krtous (2007) Algebraic type of curvature is type-D. Hamamoto-TH-Oota-Yasui (2007) Hamilton symmetry of - Jacobi equation symmetry of is separable. Klein symmetry of - Gordon symmetry of equation symmetry of is separable. Frolov-Krtous-Kubizn symmetry of ak holes (2007) K-G symmetry operatorsSergyeyev, exist. Krtous (2008) Dirac equation symmetry of is separable. Oota-Yasui (2008)

Dirac symmetry operators exist. Ben symmetry of n symmetry of -Charlton symmetry of (1996), Wu (2009) Hidden symmetry of symmetries There exist two n symmetry of atural (symmetric an symmetry of d an symmetry of ti-symmetric) gen symmetry of eralization symmetry of s of (con symmetry of f ormal) Killin symmetry of g vector. vector Killing vector symmetri c Killing-Stackel (KS)

Stackel (1895) conformal Killing vector conformal Killing-Stackel (CKS) antisymmetric Killing-Yano (KY) Yano conformal Killing-Yano (CKY) Tachibana (1969), Kashiwada

(1952) (1968) Gen symmetry of eralization symmetry of s of Killin symmetry of g vector Def. Killing-Stackel tensor (KS) is a rank-p symmetric tensor K obeying Stackel (1895) Def. Killing-Yano tensor (KY) is a rank-p anti-symmetric tensor f obeying Yano (1952) Properties of KY ten symmetry of sors an symmetry of d KS ten symmetry of sors

Prop. When f is a rank-n Killing-Yano (KY) tensor, then rank-2 symmetric tensor K defined by Kab = fa fb is a Killing-Stackel (KS) tensor. Prop. Let K be a rank-n Killing-Stackel tensor field and be a geodesic with tangent p. Then Kabc papbpc is constant along . Con symmetry of f ormal Killin symmetry of g-Yan symmetry of o ten symmetry of sor Def. Conformal Killing-Yano tensor

(CKY) is a rank-p anti-symmetric tensor k obeying wher e Tachiban symmetry of a (1969), Kashiwada (1968) Prop. Let k be a CKY p-form for a metric g. Then, ~k = p+1 k is a CKY pform for the metric ~g = 2 g. Subclasses of CKY ten symmetry of sors Def. Equivalently, CKY is a p-form k obeying for an arbitrary vector X. covariantly constant ; h is a closed CKY

; f is a KY ; is a special Tachibana-Yu KY (1970) form Basic properties of CKY ten symmetry of sors Prop. The Hodge star * maps CKY pforms into CKY (D-p)-forms. In particular, the Hodge star of a closed CKY p-form is a KY (D-p)-form versa. Prop.

When h1and andvice h2 is a closed CKY p-form and q-form, respectively, then h3 = h1 h2 is a closed CKY (p+q)form. Basic properties of hidden symmetry of symmetries CKY KY CCKY

rank-2 KS rank-2 CKS Tower of hidden symmetry of symmetries CCKY (2) CCKY(0) CCKY(2)CCKY(4) volu KY(Dfor me 2) m

metr KS(2 ic ) KY(D4) CCKY(2n2) KY(D KS(2) 2n+1)

KS(2) Geodesic in symmetry of tegrability in symmetry of higher dimen symmetry of sion symmetry of s closed CKY 2form closed CKY 2j-form KY (D-2j)-formrank-2 KS tensor const. of motion Killing vector

const. of motion *nontrivia l dimen symmetry of sion symmetry of even symmetry of (D=2n symmetry of ) odd (D=2n symmetry of +1) Killing vector # Killin symmetry of g vector n symmetry of n symmetry of +1

# KS ten symmetry of sor n symmetry of n symmetry of Krtous-Kubizn symmetry of ak holes-Page-Frolov TH-Oota-Yasui (2007) (2006) On symmetry of e f urther fin symmetry of ds that such a spacetime admits r(n+)separability structure, that is, separability of H-J TH-Oota-Yasui (2007) equation symmetry of f or geodesics. Man symmetry of if olds admittin symmetry of g a closed CKY 2-f orm Theor. Suppose a Riemannian manifold (MD, g) admits a non-degenerate closed CKY 2-form h. Then the metric takes the form where

TH-Oota-Yasui (2007), Krtous-Frolov-Kubizn symmetry of ak holes (2008) Ein symmetry of stein symmetry of metrics with a n symmetry of on symmetry of -degen symmetry of erate CKY 2-f orm whe n symmetry of in symmetry of 2n symmetry of dimen symmetry of sion symmetry of in symmetry of 2n symmetry of +1 dimen symmetry of sion symmetry of , This metric satisfies Ein symmetry of stein symmetry of Eq. Then symmetry of , the metric coin symmetry of cides with that of KerrNUT-(A)dS metric. In symmetry of this mean symmetry of , on symmetry of ly vacuum spacetime admittin symmetry of g a n symmetry of on symmetry of -degen symmetry of erate CKY 2f orm is the Kerr-NUT-(A)dS spacetime.

In symmetry of the case of degen symmetry of erate CCKY ten symmetry of sors It is con symmetry of ven symmetry of ien symmetry of t to see the eigen symmetry of values of a ran symmetry of k holes-2 closed CKY by . TH-Oota-Yasui (2008) The D-dim. gen symmetry of eralized Kerr-NUT-(A)dS offshell metric is Where is arbitrary K-dim metric an symmetry of d Kahler metric with the Kahler f orm . is 2m j-dim We cant determine

them any more without Ein symmetry of stein symmetry of metrics with a degen symmetry of erate CKY 2-f orm When symmetry of is K-dim Ein symmetry of stein symmetry of metric, is 2mjdim Ein symmetry of stein symmetry of -Kahler metric with the Kahler f orm an symmetry of d where ( This metric satisfies Ein symmetry of stein symmetry of Eq. Man symmetry of if olds admittin symmetry of g a special KY Theor. Let (Mn, g) be a compact, simply connected manifold admitting a special KY.

Then M is either isometric to Sn or M is a Semmelmann (2002) Sasakian, 3-Sasakian, nearly oraweak 2n+1 2Example Let (M , g,Kahler , ) be Sasak holesG ian symmetry of manifold. man symmetry of if old with Killin symmetry of g vector field . Then symmetry of is a ran symmetry of k holes-(2k+1) special KY f or k = 0, , n, which satisfies f or an symmetry of y vector field X an symmetry of d an symmetry of y k

3. A gen symmetry of eralization symmetry of of CKY symmetry Hidden symmetry of symmetry of charged BH in symmetry of 5-dim. min symmetry of imal SUGRA - Charged rotatin symmetry of g BH Chon symmetry of g-Cvetic-Lu-Pope (2005) Kn symmetry of own symmetry of f acts : Existen symmetry of ce of a ran symmetry of k holes-2 Killin symmetry of g ten symmetry of sor Davis-Kun symmetry of duri-Lucietti (2005) Existen symmetry of ce of a GCCKYKubizn symmetry of 2- ak holes-Kun symmetry of dri-Yasui (2009)

Gen symmetry of eralized con symmetry of f ormal Killin symmetry of g-Yan symmetry of o ten symmetry of sor Def. Generalized CKY is a p-form k if a 3-form T exists obeying for an arbitrary vector X. Note: This con symmetry of n symmetry of ection symmetry of gives . Subclasses of GCKY ten symmetry of sors ; h is a generalized closed CKY ; f is a GKY

Basic Properties of GCKY symmetry 1) A GCKY 1-f orm is equal to a con symmetry of f ormal Killin symmetry of g 1-f orm. 2) The Hodge star maps GCKY p-f orms in symmetry of to GCKY (D-p)f orms. In symmetry of particular, the Hodge star of a closed GCKY p-f orm is a GKY (D-p)-f orm d dvice 3) When symmetry of h1an symmetry of an symmetry of h2 versa. is a closed GCKY p-f orm an symmetry of d q-f orm, respectively, then symmetry of h3 = h1 h2 is a closed GCKY (p+q)-f orm. 4) When symmetry of f is a G(C)KY p-f orm, then symmetry of ran symmetry of k holes-2 symmetric ten symmetry of sor K defin symmetry of ed by Kab=fa fb is a (con symmetry of f ormal) Killin symmetry of g ten symmetry of sor. GCKY GKY

rank-2 KS GCCKY rank-2 KS Tower of hidden symmetry of symmetries GCCK Y(2) GCCKYGCCKYGCCKY (0)

volu for me m metr ic (2) (4) GKY(D GKY(D-2)

KS(2 ) 4) GCCKY(2 n-2) GKY(D KS(2) 2n+1) KS(2)

Geodesic in symmetry of tegrability GCCKY 2form GCCKY 2j-form GKY (D-2j)-formrank-2 Killing tensor const. of motion Killin symmetry of g vector con symmetry of st. of motion symmetry of Killin symmetry of g vector

commen symmetry of ts : - Con symmetry of stan symmetry of ts of motion symmetry of gen symmetry of erated f rom a GCCKY 2-f orm are in symmetry of in symmetry of volution symmetry of , i.e., - on symmetry of e doesn symmetry of t have Killin symmetry of g vectors. Dirac symmetry operator Ben symmetry of n symmetry of -Charlton symmetry of , Class.Quan symmetry of t.Grav.14 (1997) TH-Kubizn symmetry of ak holes-Warn symmetry of ick holes-Yasui, arXiv:1002.3616 Th. Let be a gen symmetry of eralized con symmetry of f ormal Killin symmetry of g-Yan symmetry of o (GCKY) p-f orm obeyin symmetry of g Then symmetry of the operator satisfies Massless Dirac symmetry operators In symmetry of the case A van symmetry of ishes, is an symmetry of symmetry

operator f or massless Dirac equation symmetry of , i.e., ( on symmetry of -shell ) An symmetry of omaly The last term A = A(p+2) + A(p-2) is written symmetry of explicitly as Massive Dirac symmetry operators Col. Let be a gen symmetry of eralized Killin symmetry of g-Yan symmetry of o (GKY) p-f orm such that an symmetry of an symmetry of omaly A van symmetry of ishes. Then symmetry of there exists an symmetry of operator such that ( off-shell ) Col. Let be a gen symmetry of eralized closed con symmetry of f ormal Killin symmetry of g-Yan symmetry of o (GCCKY) p-f orm such that an symmetry of an symmetry of omaly A van symmetry of ishes. Then symmetry of there exists an symmetry of operator M such that

( off-shell ) The symmetry operators in symmetry of terms of gamma matrices Hidden symmetry of symmetry of CCLP black holes hole - GCCKY 2-form Kubizn symmetry of ak holes-Kun symmetry of duri-Yasui (2009) with It was shown symmetry of that this 2-f orm produces a ran symmetry of k holes-2 Killin symmetry of g ten symmetry of sor discovered by DavisKun symmetry of duri-Lucietti.

- Separation of variables H-J, K-G an symmetry of d Dirac equation symmetry of s are separable. Davis-Kun symmetry of duri-Lucietti (2005), Wu (2009) 4-dim. heterotic SUGRA We con symmetry of sider the f ollowin symmetry of g theory where This action symmetry of gives an symmetry of boson symmetry of ic part of the lowen symmetry of ergy effective action symmetry of of heterotic strin symmetry of g theory. Kerr-Sen symmetry of black holes holes where

Sen symmetry of (1992) Hidden symmetry of symmetry of Kerr-Sen symmetry of black holes holes Kn symmetry of own symmetry of f acts : Algebraic properties of curvature Burin symmetry of sk holesii (1995) Separability of the Hamilton symmetry of - Jacobi equation symmetry of Blaga-Blaga (2001) Separability of the Klein symmetry of - Gordon symmetry of equation symmetry of Wu-Cai (2003)

Existen symmetry of ce of a ran symmetry of k holes-2 Killin symmetry of g ten symmetry of sor (strin symmetry of g Hiok holesi-Miyamoto (2008) f rame) Question symmetry of s : Separability of the Dirac equation symmetry of ? Why does such a separation symmetry of occur? D-dimen symmetry of sion symmetry of al heterotic SUGRA We con symmetry of sider the n symmetry of ave gen symmetry of eralization symmetry of of heterotic supergravity where This k holesin symmetry of d of action symmetry of gives a boson symmetry of ic part of supergravity such as heterotic supergravity compactified on symmetry of a torus in symmetry of each dimen symmetry of sion symmetry of .

Higher-dimen symmetry of sion symmetry of al Kerr-Sen symmetry of black holes holes where Cvetic-Youm (1996), Chow (2008) Hidden symmetry of symmetry of Kerr-Sen symmetry of black holes holes Kn symmetry of own symmetry of f acts : Chow (2008) Hamilton symmetry of - Jacobi equation symmetry of is separable. Ran symmetry of k holes-2 Killin symmetry of g ten symmetry of sors exist. Question symmetry of s :

Does the separation symmetry of of the K-G equation symmetry of occurs? How about the Dirac equation symmetry of ? If separable, where does such a structure come f rom? Hidden symmetry of symmetry of Kerr-Sen symmetry of black holes holes - GCCKY 2-form with TH-Kubizn symmetry of ak holes-Warn symmetry of ick holes-Yasui (2010) - Separation of Ok holesai (1994), Blaga, et al. (2001), Wu-Cai (2003), Hiok holesi-Miyamoto(2008)

variables Chow (2008), HKWY (2010) f rame Ein symmetry of stein symmetry of strin symmetry of g H-J K-G Dirac* separable separable separable separable

TH-Kubizn symmetry of ak holes-Warn symmetry of ick holes-Yasui (2010) - Symmetry For the torsion symmetry of T=H, on symmetry of e can symmetry of produce the symmetry operators operators f or the Laplacian symmetry of an symmetry of d the modified Dirac T/3 4. Summary & Outlook Outlook holes Summary We have studied properties of spacetimes admittin symmetry of g a con symmetry of f ormal Killin symmetry of g-Yan symmetry of o symmetry an symmetry of d its gen symmetry of eralization symmetry of . Especially, a ran symmetry of k holes-2 CCKY an symmetry of d GCCKY 2-f orm. If the torsion symmetry of is absen symmetry of t, we have shown symmetry of that

such symmetry characterizes vacuum black holes hole solution symmetry of s with spherical horizon symmetry of topology. If the torsion symmetry of is persen symmetry of t, we have shown symmetry of that such symmetry are seen symmetry of in symmetry of the solution symmetry of s of supergravities such as 5-dim. min symmetry of imal SUGRA an symmetry of d heterotic supergravity. Exact solution symmetry of s of 5-dim. U(1)3 SUGRA Cvetic-Youm (1996) Galtsov-Sherbluk holes Cvetic-Lu-Pope(2008) (2004) min symmetry of imal SUGRA

theumost eral nkngen symmetry of ow solution symmetry of Chon symmetry of g-Cvetic-Lu(2005) Pope susy limit Gaun symmetry of tlett(2003) Gutowsk holes i Mei-Pope (2007)

n susy limit Chon symmetry of g-Cvetic-LuPope Chow (2007) (2005) Kun symmetry of duri-LuciettiReall (2006) Man symmetry of if olds with special holon symmetry of omy Type-IIB supergravity on symmetry of correspon symmetry of den symmetry of c e

AdS5X5 (Examples of Sasak holesi-Ein symmetry of stein symmetry of ) S5 T1,1 SCF T It is k holesn symmetry of own symmetry of that Sasak holesi-Ein symmetry of stein symmetry of an symmetry of d Calabi-Yau metrics are derived f rom vacuum rotatin symmetry of g BH by tak holesin symmetry of g a limit. eve Calabilimit vacuum rotatin symmetry of g n symmetry of Yau od Sasak holesiBH d

Ein symmetry of stein symmetry of Ex) 5-dim. Kerr(A)dS 6-dim. Kerr-NUT(A)dS Sasak holesi-Ein symmetry of stein symmetry of Yp,q La,b,c resolved Calabi-Yau con symmetry of e Even symmetry of Dimen symmetry of sion symmetry of s Space admittin symmetry of g a CCKY 2-f orm vacuum rotatin symmetry of g BH

Apostolov et al (2002) Space admittin symmetry of g a GCCKY 2-f orm charged rotatin symmetry of g BH TH, Oota, Yasui (2008) Krtous, Frolov, Kubiznak (2008)

Kahler man symmetry of if old admittin symmetry of g a Hamilton symmetry of ian symmetry of 2-f orm Calabi-Yau limit man symmetry of if old limi HKWY (2010) t KT man symmetry of if old admittin symmetry of g a Hamilton symmetry of ian symmetry of 2-f orm with torsion symmetry of Calabi-Yau with torsion symmetry of

Spacetimes admittin symmetry of g a GCCKY 2-f orm assumption symmetry of i.e. D-dim metric GCCKY 2f orm in symmetry of troduce can symmetry of on symmetry of ical basis Orthon symmetry of oramal f rame s.t.

n symmetry of on symmetry of degen symmetry of erate: the f orm of where Q is an symmetry of arbitrary f n symmetry of . the on symmetry of ly compon symmetry of en symmetry of ts van symmetry of ishin symmetry of g. are n symmetry of on symmetry of - local multi-Hermitian symmetry of structure s.t. (1) For each , J is complex

structure. (2) g is Hermitian symmetry of : Bismut torsion symmetry of where s.t. M, g, J, , B is called Kahler with torsion symmetry of (KT) man symmetry of if on symmetry of d. When symmetry of B=0, then symmetry of it becomes Kahler man symmetry of if old. relation symmetry of ship b/w the torsion symmetry of T of GCCKY 2-f orm an symmetry of d the Bismut torsion symmetry of B 3 types of solution symmetry of s: Mixed These doesn symmetry of t exist when symmetry of T=0

(1) type: special solution symmetry of where an symmetry of d (2) we have gen symmetry of eral solution symmetry of where (3) Mixed f or simplicity, in symmetry of 4 dimen symmetry of sion symmetry of s con symmetry of struction symmetry of of metrics (1) : where

This in symmetry of cludes Kerr-Sen symmetry of black holes holes (2) where (3) Mixed n symmetry of ot

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    "Exploiting the WWW: Lessons from a UK Research Project on a Health Record Broker " Professor Michael Rigby Mark Turner Keele University, United Kingdom
  • The Thomas-Killman Conflict Mode Instrument

    The Thomas-Killman Conflict Mode Instrument

    Gets to heart of issue. Moves people beyond polarized positions. Sets stage for mutual understanding. Leads to group cooperation. Sets stage for issue re-framing. Sets stage for generating creative options. Examples of two positions: The Thomas-Killman Conflict Mode Instrument No...
  • Challenging Sectors for Mitigation Market Cost of Renewable

    Challenging Sectors for Mitigation Market Cost of Renewable

    Cross-laminated timber (CLT) is a wood panel product made from solid-sawn lumber. CLT makes it possible to build mid-rise wooden structures. 33 m high, 121 unit apartment block in London (Dalston Lane) 12-story CLT building planned for Portland, Oregon (Framework)
  • GEOFFREY CHAUCER The Canterbury Tales EARLY LIFE  1342-1400

    GEOFFREY CHAUCER The Canterbury Tales EARLY LIFE 1342-1400

    Lack of alliteration Best known for writing The Canterbury Tales, but also produced several other works CHAUCER'S MAIN WORKS During the French period he wrote poems modelled on French romance styles and subjects like: The Romaunt of the Rose (before...
  • Alberta Business Beat Volume 14, July 2016 1

    Alberta Business Beat Volume 14, July 2016 1

    Alberta SMEs are slightly more optimistic about the future. Although optimism in the future of their own business has increased slightly versus last quarter, a full 16% more SMEs believe that the Alberta economy will be better off or the...
  • Presentación de PowerPoint - WordPress.com

    Presentación de PowerPoint - WordPress.com

    [email protected] …la rutina de aconsejar la práctica de ejercicio sin más es poco útil, al menos en Atención Primaria… Por lo que la prescripción debe ser explícita e incluir información sobre: tipo de ejercicico modo, intensidad, duración, frecuencia ritmo de...
  • New Vector Resonance as an Alternative to Higgs Boson

    New Vector Resonance as an Alternative to Higgs Boson

    Klein-Nishina limit Thomson Klein-Nishina limit (α=1/137) σCompHEP= 2.0899 nb σLEP=1.9993+- 0.0026 nb Tevatron LEP = 0.01627 CompHEP t H g g g t u u d u u d b b p p pp ttH +X tt bb + X...
  • Back To School Second Grade - Bushy Park Elementary School

    Back To School Second Grade - Bushy Park Elementary School

    Poem of the Month . Every month each student needs to memorize a Poem of the Month. Teachers will give students a copy to glue in their agenda book to practice at home. Each child will receive a monthly certificate...