Extremal black holes from nilpotent orbits

Transcription

1 Extremal black holes from nilpotent orbits Guillaume Bossard AEI, Max-Planck-Institut für Gravitationsphysik Penn State September 2010

2 Outline Time-like dimensional reduction Characteristic equation Fake superpotential Stationary composites Conclusion and outlook [ G. Bossard, H. Nicolai and K. S. Stelle, , ] [ G. Bossard and H. Nicolai, ] [ G. Bossard, , ] [ G. Bossard, Y. Michel and B. Pioline, ]

3 Black holes Four-dimensional black hole solutions Semi-classical string theory Microscopic interpretation of Bekenstein Hawking entropy via microstates counting Non-perturbative symmetries of string theory / M theory SL(2, Z) SO(6, 6)(Z) E 7(7) (Z) E 10 (?)

4 Pure gravity For stationary solutions (space-time M R V ) ds 2 = e 2U( dt + ω µ dx µ) 2 + e 2U γ µν dx µ dx ν Coset representative V = e U e U «σ 0 e SL(2, R)/SO(2) U defined on V dσ = e 4U γ dω For which the equations of motion are R µν (γ) = 1 2 Tr P µp ν d γ VP V 1 = 0 with P 1 2( V 1 dv + (V 1 dv) t).

5 Conserved charges In four dimensions, the Komar mass and its dual are defined from K dg(κ) (with κ being the time-like Killing vector) m 1 s K n 1 s K 8π 8π V where s defines a patch of local sections of M + over an atlas of V. In term of the SL(2, R) Noether charge C 1 ( VP V 1 m n = 4π n m Σ V )

6 Supergravity The vierbein field e a Electromagnetic fields A Λ in l 4 Scalar fields φ A parameterising a symmetric space G 4 /H ε abcde a e b R cd + G AB (φ)dφ A dφ B + N ΛΞ (φ)f Λ F Ξ + M ΛΞ (φ)f Λ F Ξ

7 Time-like dimensional reduction Kaluza Klein Ansatz The metric ds 2 = e 2U( dt + ω µ dx µ) 2 + e 2U γ µν dx µ dx ν where γ is the metric on V and ω µ dx µ the Kaluza Klein vector. And the abelian 1-form fields A Λ = ζ Λ( dt + ω µ dx µ) + ˆζ Λ µ dxµ

8 Duality symmetry The equations of motion permit to dualize ω µ E e 2U + iσ As well as ˆζ Λ ζλ Φ IJ ( ζ Λ, ζ ) Λ and G 4 is enlarged to G σ g = sl(2, R) g 4 2 l 4 = 1 ( 2) l ( 1) 4 ( gl 1 g 4 ) (0) l (1) 4 1(2) h = so(2) h 4 l 4 = 1 ( 2) 1 (2) l ( 1) 4 + l (1) 4 h(0) 4 And ( E, Φ IJ,v ) V G/H

9 Duality symmetry G-invariant equations of motion R µν = 1 k g Tr P µ P ν d VP V 1 = 0 in function of P V 1 dv ( V 1 dv ) h. The Noether charge defined on any 2-cycle Σ of V C 1 VP V 1 4π Σ For Σ V then C g h = ( m, n ) sl(2, R) so(2) ( q Λ, p Λ ) l 4 Σ IJKL g 4 h 4

10 Duality symmetry The Noether charge defined on any 2-cycle Σ of V C 1 VP V 1 4π Σ For Σ V then C g h = ( m, n ) sl(2, R) so(2) ( q Λ, p Λ ) l 4 Σ IJKL g 4 h 4 where Σ IJKL is defined from the G 4 current 3-form Σ IJKL 1 s i κ J IJKL 4π Σ

11 Duality symmetry The Noether charge defined on any 2-cycle Σ of V C 1 4π V 1 0 VP V 1 V 0 For Σ V then C g h = ( m, n ) Σ sl(2, R) so(2) Z ij l 4 Σ ijkl g 4 h 4 where Σ ijkl is defined from the G 4 current 3-form Σ ijkl 1 4π v 1 IJ 0 ij s KL i κ J IJKL v 0 ij Σ

12 Characteristic equation Breitenlohner, Maison and Gibbons theorem: If G is simple, all the non-extremal single-black hole solutions are in the H -orbit of a Kerr solution. Σ ijkl is not a conserved charge, and C is constrained.

13 Characteristic equation Breitenlohner, Maison and Gibbons theorem: If G is simple, all the non-extremal single-black hole solutions are in the H -orbit of a Kerr solution. Σ ijkl is not a conserved charge, and C is constrained. Five-graded decomposition of g with respect with the Schwarzschild Noether charge C = mh g = 1 ( 2) l ( 1) 4 gl 1 g (0) 4 l(1) 4 1(2) Three-graded decomposition of the fundamental representation R R = r ( 1) 4 R (0) 4 r(1) 4

14 Characteristic equation Breitenlohner, Maison and Gibbons theorem: If G is simple, all the non-extremal single-black hole solutions are in the H -orbit of a Kerr solution. Σ ijkl is not a conserved charge, and C is constrained. Generically, one has C 3 = 1 k g Tr C 2 C and for N = 8 supergravity (E 8 ) C 5 = 5 64 Tr C 2 C Tr2 C 2 C

15 Characteristic equation Breitenlohner, Maison and Gibbons theorem: If G is simple, all the non-extremal single-black hole solutions are in the H -orbit of a Kerr solution. Σ ijkl is not a conserved charge, and C is constrained. Generically, one has C 3 = 1 k g Tr C 2 C ( m, n, q Λ, p Λ ) transform all together in a non-linear representation of H.

16 Spherically symmetric black holes C 1 4π V VP V 1 = 1 4π H VP V 1 Tr C 2 only depends on the field U defining the metric. A κ = 4π 1 k g Tr C 2 The Noether charge is nilpotent for extremal spherically symmetric black holes. C 3 = 0 and C 5 = 0 for N = 8

17 H non-semi-simple C 3 = 1 k g Tr C 2 C reduces to a quadratic holomorphic equation in the complex parameters W m + in Z ij q Λ + ip Λ Σ ijkl and can be solved explicitly as Σ ijkl = Z [ijz kl] 2W Σ A ij = Z ijz A W For Pure N 5-extended supergravity, C is a Spin (2N) Cartan pure spinor.

18 H non-semi-simple Then 1 k g Tr C 2 = ( W 2 z 1 2)( W 2 z 2 2) W 2 Extremal non-rotating black holes have Bogomolni saturated electromagnetic charges. For Pure N 5-extended supergravity, all the extremal non-rotating black holes are BPS.

19 H semi-simple C 3 = 1 k g Tr C 2 C is not holomorphic in the complex parameters W m + in Z ij q Λ + ip Λ Σ ijkl and Σ is an irrational function of Z ij and W that can not be written in closed form. For Pure N 6-extended supergravity, there are non-bps extremal non-rotating black holes.

20 Nilpotent orbits Nilpotent adjoint orbits of semi-simple Lie groups C g C n = 0 G C = G/J C have been classified by mathematicians. They admit a symplectic form ω(x, y) C Tr C [x, y] If G/J C g h, the corresponding H -orbit, H /I C C g h C n = 0 H C = H /I C is a Lagrangian submanifold of G/J C with respect with ω.

21 E 8(8) E 7(7) Spin(6,7) SL(2) F 4(4) D oković classification E 8(8) -orbits of nilpotent elements of e 8(8), C = 0 E 6(2) E 6(6) Spin (16)-orbits of nilpotent elements of e 8(8) so (16) Spin (16) SU (8) SU (4) Spin(1,6) SU(2) Sp(3) SU(2) SU(6) Sp(4)

22 D oković classification E 8( 24) -orbits of nilpotent elements of e 8( 24), C = 0 SO(2,11) SL(2) F 4( 52) E 6( 78) E 8( 24) E 7( 25) E 6( 25) SO(3,10) SL(2) F 4( 20) E 6( 14) SL(2, R) E 7( 25) -orbits of nilpotent elements of e 8( 25) ( sl 2 e 7( 25) ). SO(1,10) F 4( 52) E 6( 78) SL(2,R) E 7( 25) E 6( 26) F 4( 52) SO(2, 9) SO(9) SO(10)

23 D oković classification SO(8, 2 + n)-orbits of nilpotent elements of so(8, 2 + n), C S 3 = 0 SO(6, 1+n) SL(2) SO(4, n-1) SO(4, n) SL(2) SO(6, n) Sp(4,R) SO(4, n-2) SO(5, n-1) SO(7, n) SL(2) SO(5, n-2) SO(6, n-2) SO(6, 2) SO(2, n)-orbits of nilpotent elements of so(8, 2 + n) ( so(6, 2) so(2,n) ). SO(5, 1) SO(1, n) SO(4) SO(n-1) SO(4) SO(n) SO(5, 1) SO(1, n-1) SL(2) SO(4) SO(n-2) SO(5) SO(n-1) SO(6, 1) SO(1, n-1) SO(5) SO(n-2) SO(6) SO(n-2)

24 Normal triplets For a nilpotent orbit G e = G/J e of representative e g An sl 2 triplet f, h, e [h,e] = 2e h lies in a Cartan subalgebra of g. For a complex Lie algebra, h determines the orbit. For an orbit H e = H /I e of representative e g h An sl 2 triplet f, h, e [h,e] = 2e h lies in a Cartan subalgebra of h. h determines the orbit.

25 Supersymmetry Dirac equation For N -extended supergravity theories, H = Spin (2N) c H 0, and the Noether charge C transforms as a chiral Weyl spinor of Spin (2N) valued in a representation of H 0. In a harmonic oscillator basis C = ( W + Z ij a i a j + Σ ijkl a i a j a k a l + ) 0

26 Supersymmetry Dirac equation In a harmonic oscillator basis C = ( W + Z ij a i a j + Σ ijkl a i a j a k a l + ) 0 The asymptotic behaviour of the supersymmetry variation of the dilatini translates the BPS condition into the Dirac equation, ( ǫ i α a i + ε αβ ǫ β i ai) C = 0

27 Supersymmetry Dirac equation n N BPS black holes are left invariant by the supersymmetry transformations of parameter satisfying ǫ A α + ε αβ Ω AB ǫ β B = 0 ǫāα = 0 for a symplectic form Ω AB of C 2n satisfying Ω AC Ω BC = δa. B It term of which the Dirac equation reads ( aa Ω AB a B) C = 0 h = 1 ) (Ω AB a A a B Ω AB a A a B n and has as solution C = e 1 2 Ω ABa A a B ( W + Z Ā ) B B aāa 0

28 Supersymmetry Dirac equation For maximal supergravity, this implies that 1 4 BPS black holes have two saturated eigen values of Z ij, z 1 2 = z 2 2 z 3 2 = z 4 2 = W 2 and moreover that the E 7(7) quartic invariant vanishes (W 1 2 Z) = 0 such that the horizon area vanishes. For 1 2-BPS solutions, C is a Majorana Weyl pure spinor Σ ijkl = 1 Z [ijz kl] = 1 2W 48 ε ijklmnpqz mn Z pq W

29 Decomposition of the 1-form P In N = 8 supergravity, the form P reads P = (1 + ) du i 2 e2u dω + e U (v t 1 dφ) ij a i a j + 1 «1 `Dvv 24 ijkl ai a j a k a l 0 It is convenient to consider the dual fields F IJ = da IJ e U v t 1 (dφ) = eu 1 e 4U ωµω µ ( e 4U ω ω Je 2U ω ) v(f) In particular, for static solutions ω µ = 0 and F IJ = dh IJ e U v t 1 (dφ) = e U v(dh) for H IJ Harmonic functions.

30 Fake superpotentiel In the symmetric gauge V = v 0 exp ( C /r ) P V 1 dv ( V 1 dv ) h = C In the parabolic gauge associated to the Kaluza Klein Ansatz P is in the H -orbit of C. ( P = (1+ ) U + e U Z(v) ij a i a j 1 1 ( vv ) ) 24 ijkl ai a j a k a l 0 U = e U W h(z) P = 2 P φijkl = e U G 1 ijkl, mnpq ( ) W mnpq φ

31 1/8 BPS fake superpotential In term of the standard diagonalization of the central charge R k ir l jz kl ˆ= 1 2 eiϕ ( ) ρ ρ ρ ρ 3 the 1/8 BPS fake superpotential is W = ρ 0

32 BPS first order system BPS equations e µ aσ a α β ( ǫ i β a i + ε βγ ǫ γ i ai ) P µ = 0, B ǫ i α = 0 Small anisotropy approximation (ǫ iα a i + ε αβ ǫ βi ai ) P µ = 0 R µν = 0 Cayley first order system (Ω [ij Ω kl] = 0, Ω ij Ω ij = 2) ( Ωij a i a j Ω ij a i a j ) Pµ = 2 P µ, DΩ ij + i 2 e2u dω Ω ij = 0

33 BPS first order system Consistency condition ( Ωij a i a j Ω ij a i a j ) 2 P = 4 P implies (I j i Ω ikω jk ) 8 = 2 6 2I[i k vt 1 (dφ) j]k + Ω ij Ω kl v t 1 (dφ) kl = 0 2I p ( [i Dv v 1 ) jkl]p + 3Ω [ijω pq ( Dv v 1) = 0 kl]pq That is 28 = C = 15 (2 20) R DI j i = 0 dij i = 0

34 1/8 BPS Denef s sum of squares Positive definite quadratic form (e 4U ω µ ω µ < a < 1) ( G,F ) = e 2U 1 e 4U ωµω µ2re ( v(g) ij( e 4U ω ω ie 2U ω ) v(f) ij ) Defining G F 1 2 d( e U v 1 Ω ) 1 2 d( e U Jv t ωω ) the Langragian density reads L = du du 1 4 e4u dω dω + `F, F = `G, G `Dvv 1 ijkl `Dvv 1 `Dvv 1 ijkl`dvv 1 6Ω 1 ijkl ij `Dvv ijkl klpq Ωpq «+ d( )

35 Solution Ω ij u ij IJ = Ω ij v ijij = Φ IJ = 1 ( (H) H IJ 2 4 (H) ( 1 (H) 2 4 (H) 1 2 (H) (H) H IJ H IJ ) + 2H IJ ) 2H IJ e 2U = (H) dω = 2i ( H IJ dh IJ H IJ dh IJ)

36 Regularity For H IJ = a Q IJ a x x a v 1 0 (Ω 0 ) IJ the absence of NUT sources requires b a Q IJ a Q b IJ Q IJ b Q a IJ x a x b = 1 2 ( Ω ij 0 v 0 (Q a ) ij Ω 0 ij v 0 (Q a ) ij )

37 Regularity Transitive action of Spin (12) c E 6(2) v xa (Q a ) ij = e iα a v 0 ( b Q b) ij and for g E 6(2), solution v = h(g,v)vg 1 v x a (gq a ) ij = e iα a v 0( b Q b) ij All embeddings of N = 2 solutions SO (12)/U(6) N = 8 E 7(7) /SU c (8)

38 BPS non-supersymmetric Solutions Z = 0 in exceptional N = 2 supergravity M 4 = E 7( 25) /`U(1) E 6( 78) M 3 = E 8( 24) /`SL(2,R) E 7( 25) Generator h(ω) defined from t abc Ω b Ω c = 0 Ω a Ω a = 2 DΩ a i 2 e2u dω Ω a = 0 Solution of the N = 2 truncation M 4 = SU(1, 1)/U(1) SO(2, 10)/`U(1) SO(10) M 3 = SO(4, 12)/`SO(2, 2) SO(2, 10) ι z(ι(h)) = z(h), ι z I (ι(h)) = z I (H)

39 The 1/2 BPS charge The charge in the 128 is of the form ǫ i α + e iα ε αβ Ω ij ǫ β j = 0 The generator h 1 1 `eiα Ω 2 4 ij a i a j e iα Ω ij a i a j decomposes so (16) = 28 ( 1) gl 1 su (8) (0) 28 (1) such that 16 = 8 ( 1 2 ) 8 (1 2 )

40 The 1/2 BPS charge The charge in the 128 is of the form C = ine 2iα e 1 2 eiα Ω ij a i a j 0 The generator h 1 1 `eiα Ω 2 4 ij a i a j e iα Ω ij a i a j decomposes so (16) = 28 ( 1) gl 1 su (8) (0) 28 (1) such that e 8(8) so (16) = 1 ( 2) 28 ( 1) 70 (0) 28 (1) 1 (2)

41 The 1/2 BPS charge The charge in the 128 is of the form C = ine 2iα e 1 2 eiα Ω ij a i a j 0 The generator h 1 1 `eiα Ω 2 4 ij a i a j e iα Ω ij a i a j decomposes so (16) = 28 ( 1) gl 1 su (8) (0) 28 (1) such that e 8(8) so (16) = 1 ( 2) 28 ( 1) 70 (0) 28 (1) 1 (2)

42 The non-bps charge The charge in the 128 is of the form ( C = (1 + ) ) (e 4 eiα Ω ij a i a j 2iα M + e iα Ξ ij a i a j) 0 The generator h 1 `eiα Ω 2 ij a i a j e iα Ω ij a i a j decomposes so (16) = 28 ( 2) gl 1 su (8) (0) 28 (2) such that e 8(8) so (16) = 1 ( 4) 28 ( 2) 70 (0) 28 (2) 1 (4)

43 The non-bps charge The charge in the 128 is of the form ( C = (1 + ) ) (e 4 eiα Ω ij a i a j 2iα M + e iα Ξ ij a i a j) 0 The generator h 1 `eiα Ω 2 ij a i a j e iα Ω ij a i a j decomposes so (16) = (1 27) ( 2) gl 1 ( sp(4) 27) (0) (1 27 ) (2) such that e 8(8) so (16) = 1 ( 4) ( 1 27 ) ( 2) ( ) (0) (1 27) (2) 1 (4)

44 non-bps fake superpotential In term of the non-standard diagonalization of the central charge R k i R l jz kl ˆ= e iπ ««6 4 e iα + ie iα sin2α +e iα ξ 1 +ξ 2 +ξ ξ ξ ξ 3 13 C7 A5 1 C A the non-bps fake superpotential is W = 2

45 Conclusion The Noether charge satisfies a characteristic equation.

46 Conclusion The Noether charge satisfies a characteristic equation. It is determined in function of the four-dimensional conserved charges.

47 Conclusion The Noether charge satisfies a characteristic equation. It is determined in function of the four-dimensional conserved charges. Extremal solutions are classified by nilpotent H -orbits in g h. which are Lagrangian submanifolds of the corresponding nilpotent G-orbits. are characterised by a Cayley triplet

48 Conclusion For extremal solutions of a given type, the Cayley triplet associates to the coset 1-form P a non-compact generator h of h that determines a first order system of equations [h,p] = 2P In the spherically symmetric case h = h(z) and it determines the fake superpotential. In the static case h determines the mutually local charges. In the stationary case h defines auxiliary functions that permit to render the system first order.

49 Conclusion BPS stationary composites are necessarily 1/2 BPS in an appropriate N = 2 truncation. This is the case for 1/8 BPS composites of maximal supergravity 1/4 BPS and non-bps Z = 0 composites of N = 4 supergravity non-bps Z = 0 composites in N = 2 supergravity with a symmetric moduli space.

50 Outlook Non-BPS stationary composites in maximal supergravity h(ω) P µ + ε µν σ λ ν P σ = 2 P µ Extremal solutions in higher dimensions from higher order nilpotent orbits e 8(8) = 2 ( 3) 27 ( 2) (2 27) ( 1) `gl 1 sl 2 e 6(6) (0) (2 27) (1) 27 (2) 2 (3)

51 Outlook Non-BPS stationary composites in maximal supergravity h(ω) P µ + ε µν σ λ ν P σ = 2 P µ Extremal solutions in higher dimensions from higher order nilpotent orbits e 8(8) = 2 ( 3) 27 ( 2) (2 27) ( 1) `gl 1 sl 2 e 6(6) (0) (2 27) (1) 27 (2) 2 (3) so (16) = 1 ( 3) (2 6) ( 2) (2 6 15) ( 1) `gl 1 gl 1 sp(1) su (6) (0) 128 = 1 ( 3) 15 ( 2) (2 6 15) ( 1) `1 (2 20) R 1 (0)

52

53 Outlook Five-dimensional fake superpotential in terms of E 7(7) nilpotent orbits C = 0 E 7(7) SO(6,6) SL(2,R) Spin(4,5) F 4(4) F 4(4) in SU (8) orbits of the 70 SU (8) SU (4) SU (4) Sp(1) Sp(1) Sp(2) Sp(1) Sp(3) Sp(1) Sp(3)