( ) 1. (I) Hidden Symmetry and Exact Solutions in Einstein Gravity Houri-Y.Y: Progress Supplement (2011) (II) Generalized Hidden Symmetries and Kerr-Sen Black Hole Houri-Kubiznak-Warnick-Y.Y: JHEP (2010) (III) Einstein, -Einstein - 2011 4
CKY CKY CKY 4 Kerr CKY
4-dim. Kerr metric g = r ρ 2 ( dt a sin 2 θdϕ ) 2 + ρ 2 r dr 2 + ρ 2 dθ 2 + sin2 θ ρ 2 ( adt (r 2 + a 2 )dϕ ) 2 r = r 2 + a 2 2mr, ρ 2 = r 2 + a 2 cos 2 θ m:, a: :, Klein-Gordon [Carter 1968] Maxwell, [Teukolsky 1972] Dirac [Chandrasekhar 1976]
t, ϕ. f = (f ab ) : f = f ab = f ba, a f bc + b f ac = 0 0 a cos θ ar sin θ 0 a cos θ 0 0 a 2 cos θ sin 2 θ ar sin θ 0 0 r(r 2 + a 2 ) sin θ 0 a 2 cos θ sin 2 θ r(r 2 + a 2 ) sin θ 0 t r θ ϕ
(1) Carter (Walker-Penrose 1970, Floyd 1973) C = K ab p a p b, K ab f ac f c b ( ) (2) Klein-Gordon (Carter 1977): K = a K ab b (3) Dirac (Carter-McLenaghan 1979): f = iγ 5 γ a (f b a b (1/6)γ b γ c c f ab )
(CKY) rank-2 CKY ( 1969): a h bc + b h ac = 2g ab ξ c g cb ξ a g ca ξ b, ξ c = 1 n 1 a h ac rank-p CKY ( 1968): p 1 a h bc1 c p 1 + b h ac1 c p 1 = 2g ab ξ c1 c p 1 + ( 1) i g ci (aξ b)c1 ĉ i c p 1 1 ξ c1 c p 1 = D p + 1 a h ac1 c p 1 ξ c1 c p 1 0 CKY. i=1
h a1 a 2 a p h = 1 p! h a 1 a 2 a p dx a 1 dx a p D CKY p-form X h = 1 p + 1 i(x)dh 1 D p + 1 X δh δh = 0 dh = 0 closed CKY closed CKY (p-form) (D-p form)
1 S n, E n, H n (Riemann) ds n, E 1,n 1, AdS n (Lorentz). Special CKY [ -Yu 1970] : X (dh) = cx h c
[Semmelmann 2002] Special CKY Einstein S (a) S n (b) -Einstein (c) 3- (d) Nearly Kähler (e) Weak G 2 C(S) ( ) (a) Euclid (b) Calabi-Yau (c) Hyperkähler (d) G 2 (e) Spin(7)
( ) SUSY k T k {1} 1 6 Calabi-Yau SU(3) 1/2 7 G 2 G 2 1/8 8 Hyper-Kähler Sp(2) 1/4 8 Calabi-Yau SU(4) 1/8 8 Spin(7) Spin(7) 1/16
CKY ds n, AdS n, 4 Kerr (Lorentz) S n, H n, (Riemann) (rank-2 closed CKY), Klein-Gordon, Dirac rank-2 closed CKY
1. 2. CKY 3. (CKY) 4. CKY (GCKY) 5. Einstein
2. CKY 1970 Einstein Solutions of Einstein s Equation Lecture Notes in Physics 205 (1984) Gravitational Solitons V. Belinski (2001) (5 [Pomeransky-Sen kov] [Elvang-Figueras, - ] Kerr-Schild (D ) Minkowski (AdS) g µν = η µν + h µν Ansatz: h µν k µ k ν (
D ( 4) R µν = Λg µν mass rotation NUT Λ parameter Myers-Perry (1986) 0 1+[(D-1)/2] Gibbons-Lü-Page-Pope (2004) non-zero 2+[(D-1)/2] Chen-Lü-Pope (2006) non-zero D Kerr-NUT-de Sitter Chen-Lü-Pope )
Kerr-NUT-de Sitter metric [Chen-Lü-Pope 2006] (a) D = 2n (b) D = 2n + 1 g (2n+1) = g (2n) = n µ=1 n µ=1 dx 2 µ Q µ (x) + dx 2 µ Q µ (x) + ( n n ) 2 Q µ (x) σ k 1 (ˆx µ )dψ k µ=1 ( n n Q µ (x) µ=1 k=1 k=1 σ k 1 (ˆx µ )dψ k ) 2 + c σ n ( n ) 2 σ k dψ k k=0 σ k σ 1 = x 2 1 + x2 2 + + x2 n, σ 2 = x 2 1 x2 2 + + x2 n 1 x2 n,
Q µ (x) = X µ /U µ, U µ = n ν=1,ν µ (x 2 µ x2 ν ) X µ = X µ (x µ ) x µ. Einstein (a) X µ = (b) X µ = n k=0 n k=0 c k x 2k µ + b µx µ (D = 2n) c k x 2k µ + b µ + ( 1)n c x 2 µ c k, b µ, c,, NUT,. (D = 2n + 1)
D = 4 Kerr-NUT de Sitter metric: where g (4) = x2 y 2 X(x) dx2 + y2 x 2 Y (y) dy2 + X(x) x 2 y 2(dt + y2 dψ) 2 + Y (y) y 2 x 2(dt + x2 dψ) 2, X(x) = (a 2 x 2 )(1 + λx 2 ) + 2Mx Y (y) = (a 2 y 2 )(1 + λy 2 ) + 2Ly Ric (4) µν = 3λg(4) µν a : angular momentum, M : mass, L : NUT Bolyer-Lindquist coordinate (x, y, ψ, t) (r, θ, ϕ, τ ) x = 1r, y = a cos θ, ψ = ϕ/a, t = τ aϕ
D = 5 Kerr-NUT-de Sitter metric where g (5) = x2 y 2 X(x) dx2 + y2 x 2 Y (y) dy2 + X(x) x 2 y 2(dt + y2 dψ 1 ) 2 + Y (y) y 2 x 2(dt + x2 dψ 1 ) 2 a2 b 2 x 2 y 2 ( dt + (x 2 + y 2 )dψ 1 + x 2 y 2 dψ 2 ) 2 X(x) = 1 x 2(a2 x 2 )(b 2 x 2 )(1 + λx 2 ) 2M Y (y) = 1 y 2(a2 y 2 )(b 2 y 2 )(1 + λy 2 ) 2L R (5) µν = 4λg(5) µν a, b : angular momenta, M : mass, L : NUT
D = 6 Kerr-NUT-de Sitter metric g (6) = (x2 y 2 )(x 2 z 2 ) dx 2 + (y2 x 2 )(y 2 z 2 ) dy 2 + (z2 x 2 )(z 2 y 2 ) dz 2 X(x) Y (y) Z(z) X(x) + (x 2 y 2 )(x 2 z 2 ) (dt + (y2 + z 2 )dψ 1 + y 2 z 2 dψ 2 ) 2 Y (y) + (y 2 x 2 )(y 2 z 2 ) (dt + (z2 + x 2 )dψ 1 + z 2 x 2 dψ 2 ) 2 Z(z) + (y 2 x 2 )(y 2 z 2 ) (dt + (x2 + y 2 )dψ 1 + x 2 y 2 dψ 2 ) 2 where X(x) = (a 2 x 2 )(b 2 x 2 )(1 + λx 2 ) 2Mx Y (y) = (a 2 y 2 )(b 2 y 2 )(1 + λy 2 ) 2L 1 y Z(z) = (a 2 z 2 )(b 2 z 2 )(1 + λz 2 ) 2L 2 z R (6) µν = 5λg(6) µν a, b : angular momenta, M : mass, L 1, L 2 : NUT s
CKY 2-form [Kubiznak-Frolov (2007)] h = n dψ k dσ k = k=1 n x µ e µ e µ+n µ=1 σ k = σ k (x 1,, x n ) : M D R n Killing K (j) ( Carter ): h = h (j) h h = f (j) h (j) = K (j) ab f a (j) f (j) b (j = 0,, n 1) Killing η (j) ξ a (δh) a = η (j) a K (j) ab ξb (j = 0,, n 1 + ϵ) C (j) = K (j) ab pa p b, C (j) = η (j) a p a
D Kerr-NUT-de Sitter [Frolov-Krtous-Kubiznak 2007] Klein-Gordon [Frolov-Krtous-Kubiznak 2007] Dirac [Oota-Y.Y. 2008] [Kunduri-Lucietti-Reall 2006, Murata-Soda 2008, Oota-Y.Y. 2009]
Kerr-NUT-de Sitter Klein-Gordon [Sergyeyev-Krtous 2008] O K (i) = a K (i)ab b, O η (j) = η (j) Dirac [Benn-Kress 2004, Cariglia-Krtous-Kubiznak 2011] O h (i) = γ a h (i) a + αδh (i), O η (j) = η (j)a a + βdη (j)
3. (CKY) [Houri-Oota-Y.Y 2007, Krtous-Frolov-Kubiznak 2008] CKY 2-form h dh = 0, Einstein Kerr-NUT-de Sitter. Einstein : Kerr-NUT-de Sitter : Einstein-Kähler M 1 M 2 M N where N = #{CKY }, dimm i =.
Einstein [Houri-Oota-Y.Y 2009] g = n µ=1 dx 2 µ P µ (x) + n P µ (x) µ=1 ( n 1 k=0 σ k (ˆx µ )θ k ) 2 + N n (x 2 µ ξ2 i )g(i) i=1 µ=1 g (i) m i Kähler. P µ (x) = X µ N i=1 (x2 µ ξ2 i )m i Uµ, U µ = n ν=1 (ν µ) (x 2 µ x2 ν ) X µ = b µ x µ + x µ xµ θ k 1-form 0 n N a k y 2(k 1) (y 2 ξ 2 i )m i dy. k=1 i=1 dθ k + 2 N i=1 ( 1) k+n ξ 2n 2k 1 i ω (i) = 0, k = 0, 1,, n 1.
2 (a) (b) CKY CKY ( Einstein ( CKY
3-2 CKY (GCKY) 11 10 (typei, typeii, hetero...) (a) Ricci, Calabi-Yau (b) Special CKY Einstein, -Einstein (c) Flux)
SUGRA (a) R 1,p (, AdS p+2 (Special CKY ) (b) CKY GCKY)
GCKY? SUGRA GCKY SUSY ϕ (BPS
(A) D=5 Minimal Gauged Supergravity S = d 5 x g (R + 12g 2 14 ) F µνf µν M 5 + 1 3 F F A 3 M 5 Chong-Cvetic-Lu-Pope ( ) GCKY=CKY with T = F ( 3-form) Klein-Gordon Dirac
(B) S = d D x g e (R ϕ + µ ϕ µ ϕ 14 F µνf µν 16 ) H µνλh µνλ M D [D=4: Sen 1992, : Chow 2010] GCKY=CKY with T = H( 3-form) Klein-Gordon Dirac (C) 4,5,6,7 [Chow 2008-2010]
GCKY Yano-Bochner(1953) Curvature and Betti Number Kubiznak-Kundri-Y.Y (2009) Rank-2 GCKY: T X h = 1 p + 1 i(x)dt h 1 D p + 1 X δ T h (d T e a T e a, δ T = d T ) T a h bc + T b h ac = 2g ab ξ c g cb ξ a g ca ξ b, ξ a = 1 T 3-form T = (T abc ) D 1 T b hb a (Γ T ) a bc = Γa bc + 1 2 T c ab
GCKY d T closed GCKY δ T closed GCKY Killing-Yano with torsion Hodge d T closed 2-form h h = h (j) h h = f (j) h (j) = K (j) ab f a (j) f (j) b [K (i), K (j) ] = 0, ( Klein-Gordon, Dirac [Houri-Kubiznak-Warnick- Y.Y 2010]
Rank-2 closed GCKY D. (1) CKY (T = 0): {Kerr-NUT-de Sitter } ( N i=1 M i) (2) GCKY (T = T 0 0): Kahler [Apostolov et.al, 2002-2005] Sasakian, Calabi-Yau [Houri-Kubiznak-Warnick-Y.Y 2011] D = 2n Hermitian, D = 2n + 1 CR. D = 2n T = T 0 + T B. T B Hermitian (M, g, J) B g = 0, B J = 0 Bismut (3-form).
g = g ab dx a dx b = {e a } ( ) = [e a, e b ] = f c ab e c ( ) = a and T a ( ) = R abcd and R T abcd ( ) (1) GCKY {e µ, eˆµ } h = n µ=1 x µe µ eˆµ GCKY T a h bc = g ab ξ c g ac ξ b R T abcd h de R T abed h dc = g ae T b ξ c + g bc T a ξ e + ξ e T abc {e c}
(2) [e µ, e ν ]= x ν Qν x 2 µ x2 ν e µ x µ Qµ x 2 µ x2 ν e ν, [e µ, eˆµ ]=K µ e µ + L µ eˆµ + ν µ ( 2xµ Qν x 2 µ x2 ν T µˆµˆν ) eˆν, [e µ, eˆν ]= x µ Qµ x 2 µ eˆν, [eˆµ, eˆν ] = 0 x2 ν Q µ = X µ(x µ ) n, U µ = (x 2 µ U x2 ν ) µ ν=1(ν µ) K µ, L µ, T µˆµˆν ( : T µˆµˆν = K µ = 0, L µ = µ Qµ = Kerr-NUT-de Sitter
Hermitian (g, J, Ω) Bismut g 2 Ω: g = n (e µ e µ + eˆµ eˆµ ), Ω = µ=1 n mu=1 e µ eˆµ J: J(e µ ) = eˆµ, J(eˆµ ) = e µ [J(e a ), J(e b )] [e a, e b ] J([e a, J(e b )]) J([J(e a ), e b ]) = 0 Bismut T B = (T B abc ) T B abc = dω(j(e a), J(e b ), J(e c ))
e µ = dxµ Qµ, eˆµ = ( n R µ σ k 1 (ˆx µ )dψ k 1 Φ n S ν n σ k 1 (ˆx ν )dψ k ) k=1 ν=1 k=1 3-form T = T 0 + T B Qν T 0 = 2x µ x 2 µ e µ eˆµ eˆν, x2 ν ( Q µ R ν T B = µ log Φ + 2x ) µ Qν R µ x 2 µ e µ eˆµ eˆν x2 ν 3n 1 : X µ = X µ (x µ ), Y µ = Y µ (x µ ), Z µ = Z µ (x µ ) Q µ = X µ U µ, R µ = Y µ U µ, S µ = Z µ U µ
D (g ab, ϕ, H abc, F ab ) H =db A da (3-form) F =da (2-form) : R ab a b ϕ F c a F bc 1 4 H cd a H bcd = 0 d(e ϕ F ) = e ϕ H F, d(e ϕ H) = 0 ( ϕ) 2 + 2 2 ϕ + 1 2 F abf ab + 1 12 H abch abc R = 0
H=T ( (a) Kerr-Sen BH X µ = Y µ = n 1 k=0 n 1 c k x 2k µ + b µx µ, Z µ = qx µ + k=0 d k x 2k µ (b) CYT (Calabi-Yau with torsion) X µ = 1 4 Hol( T )=SU(n) n 1 k=0 c k x 2k 2 µ, Y µ = n 1 k=0 c k x 2k µ, Z µ = HKT (Hyper Kähler with torsion) n 1 k=0 d k x 2k µ
1. 2. 3. (CKY) 4. CKY (GCKY) CKY (with torsion) SUGRA Hermitian CKY 2-form. SUGRA. GCKY.
5. Einstein Page (1978): 4 Kerr-de Sitter CP(1) S 2 Einstein Page S 2 bundle over Einstein-Kähler [Berard-Bergery 1986, Page-Pope 1987] S 3 bundle over CP(1) [Hashimoto-Sakaguchi-Y.Y. 2004] S n bundle over CP(1) [Gibbons-Lü-Page-Pope 2004] S n bundle over Einstein-Kähler [Lü-Page-Pope 2004] CKY
D Kerr-NUT-de Sitter CKY : {α A }={, NUT, } CKY : {α A } {ξ i } ξ i CKY M = {Kerr-NUT-de Sitter} N M i N = #{ξ i }, M i : Kähler-Einstein, dim M i = ξ i i=1
Einstein g = n µ=1 dx 2 µ P µ (x) + n P µ (x) µ=1 ( n 1 k=0 σ k (ˆx µ )θ k ) 2 + N n (x 2 µ ξ2 i )g(i) i=1 µ=1 g (i) m i Kähler. P µ (x) = X µ N i=1 (x2 µ ξ2 i )m i Uµ, U µ = n ν=1 (ν µ) (x 2 µ x2 ν ) X µ = b µ x µ + x µ xµ 0 n N a k y 2(k 1) (y 2 ξ 2 i )m i dy. k=1 i=1
M {a k } = {α A } {ξ i }. : a 1 x 1 a 2 x 2 : ( q k (a 1,, a n ) pa k m + 1 A k log f m+1 A k (1 t) δ n l=1 (1 ta l) = f i (A 1,, A n )t i, A k = 1/(1 a 2 k ) i=0 M = {Kerr-NUT-de Sitter} N N M i = M = ( M i ) i=1 i=1
7-dim. Einstein metrics on S 5 -bundle over CP(1) {a 1, a 2, a 3 } (q 1, q 2, q 3 ) Z Z Z q 1 =a 1 (a 2 2 1)(a2 3 1)(3 a2 1 2a2 2 2a2 3 + a2 2 a2 3 + a2 1 a2 2 a2 3 )/ q 2 =a 2 (a 2 1 1)(a2 3 1)(3 a2 2 2a2 1 2a2 3 + a2 1 a2 3 + a2 1 a2 2 a2 3 )/ q 3 =a 3 (a 2 1 1)(a2 2 1)(3 a2 3 2a2 1 2a2 2 + a2 1 a2 2 + a2 1 a2 2 a2 3 )/ =3 3a 2 1 + a4 1 3a2 2 + a2 1 a2 2 + a4 2 3a2 3 + a2 1 a2 3 + a2 2 a2 3 + 6a2 1 a2 2 a2 3 3a 4 1 a2 2 a2 3 3a2 1 a4 2 a2 3 + a4 1 a4 2 a2 3 + a4 3 3a2 1 a2 2 a4 3 + a4 1 a2 2 a4 3 + a2 1 a4 2 a4 3
7-dim. Einstein metrics on S 5 -bundle over CP(1) 150 100 200 50 0 200 150 100 50 0 0 50 100 150 200
9-dim. Einstein metrics on S 5 -bundle over CP(2) 60 40 20 60 40 20 20 40 60
BPS [Hashimoto-Skaguchi-Yasui, 2004] = CKY : x µ 1 + ϵy µ, ϵ 0 Odd dim. Kerr-NUT-de Sitter = Sasaki-Einstein metrics Even dim. Kerr-NUT-de Sitter = Calabi-Yau metrics
CKY 2-form Einstein [Oota-Y.Y 2010] B: Einstein-Kähler manifold with c 1 (B) = pα H 2 (B, Z), p Z >0. P k1,k 2,,k n : T n bundle over B classified by integers (k 1,, k n ) M (ϵ) k 1,k 2,,k n (ϵ = 0, 1): S 2n ϵ bundle over B associated with P k1,k 2,,k n THEOREM 1. If k α are positive integers satisfying 0 < k 1 + k 2 + + k n < p, then M (0) k 1,k 2,,k n admits an Einstein metric with Λ > 0. THEOREM 2. If k α are positive integers, then M (1) k 1,k 2,,k n admits an Einstein metric with Λ > 0. Especially, if k 1 + k 2 + + k n = p, then M (1) k 1,k 2,,k n admits a Sasaki-Einstein metric.