Interacting non-bps black holes

Transcription

1 Interacting non-bps black holes Guillaume Bossard CPhT, Ecole Polytechnique Istanbul, August 2011

2 Outline Time-like Kaluza Klein reduction From solvable algebras to solvable systems Two-centre interacting non-bps solutions Generalisation to N = 8 supergravity Conclusion and outlook [ G. Bossard, C. Ruef, ] and more to come

3 Motivations String theory understanding of BPS black holes Microscopic interpretation of Bekenstein Hawking entropy Wall crossing and black hole bound states Extension to non-bps extremal supersymmetry attractor behaviour Extremal states of the quantum theory?

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) defined U 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 VPV 1 = 0 with P 1 2( V 1 dv + (V 1 dv) t).

5 Supergravity The vierbein field e a Electromagnetic fields A Λ and magnetic duals A Λ in l 4 Scalar fields φ A parametrizing a symmetric space G 4 /K 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

6 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) + wµ Λ dx µ Convenient to parametrize G 4 /K 4 by v(φ) G 4.

7 Duality symmetry The equations of motion permit to dualize d ζ Λ = e 2U N ΛΞ (φ) γ ( dw Ξ + ζ Ξ dω ) + M ΛΞ (φ)dζ Ξ and dσ = e 4U γ dω ( ζλ d ζ Λ ζ Λ dζ Λ ) Heisenberg gauge invariance δζ Λ = C Λ δ ζ Λ = C Λ δσ = c ( C Λ ζ Λ C Λ ζ Λ ) Symmetry G 4 ( l 4 R )

8 Duality symmetry (space-like reduction) Hidden symmetry G g = 1 ( 2) l ( 1) 4 ( gl 1 g 4 ) (0) l (1) 4 1(2) such that U, v(φ), ζ Λ, ζ Λ, σ parametrize G/K in a parabolic gauge as G = ( l 4 R ) ( R + G 4 ) K V = exp [ ζ Λ E Λ + ζ Λ E Λ + σe ] exp [ UH ] v(φ) equations of motion defined with P 1 ( V 1 dv + (V 1 dv) ) 2

9 Duality symmetry (time-like reduction) Hidden symmetry G g = 1 ( 2) l ( 1) 4 ( gl 1 g 4 ) (0) l (1) 4 1(2) such that U, v(φ), ζ Λ, ζ Λ, σ parametrize G/K in a parabolic gauge as G ( l 4 R ) ( R + G 4 ) K V = exp [ ζ Λ E Λ + ζ Λ E Λ + σe ] exp [ UH ] v(φ) equations of motion defined with P 1 ( V 1 dv + (V 1 dv) ) 2

10 Duality symmetry Hidden symmetry SU(2, 1) of Maxwell Einstein su(2, 1) = 1 ( 2) C ( 1) ( gl 1 u(1) ) (0) C (1) 1 (2) such that U, Φ = ζ + i ζ, σ parametrize SU(2, 1)/U(1, 1) in a parabolic gauge SU(2, 1) ( C R ) R + U(1, 1) as e U e U( σ i 2 Φ 2) Φ V = 0 e U 0 0 ie U Φ 1 equations of motion defined with P = du 1 2 e 2U ( dσ + i 2 (ΦdΦ Φ dφ) ) 1 2 e U dφ 1 2 e 2U dσ + i 2 (ΦdΦ Φ dφ) ) du i 2 e U dφ 1 2 e U dφ i 2 e U dφ 0

11 STU truncation Three moduli t i parameterizing SL(2)/SO(2) Four vectors A Λ = (A 0, A i ) ds 2 11 = ( Im[t ]) i 2 3 ( dψ + A 0 )2 ( + i A 3 = i i i ) Im[t i 1 ( 3 ] ds i ( Re[t i ] ( dψ + A 0) + A i) dz i d z i Im[t i ]dz i d z i ) N = 2 supergravity coupled to three vector multiplets. F = X 1 X 2 X 3 X 0

12 Duality symmetry Hidden symmetry SO(4, 4) of the STU model so(4, 4) = 1 ( 2) (2 2 2) ( 1) ( gl 1 sl 2 sl 2 sl 2 ) (0) (2 2 2) (1) 1 (2) such that U, v(φ), ζ Λ, ζ Λ, σ parametrize SO(4, 4)/(SO(2, 2) SO(2, 2)) in a parabolic gauge SO(4, 4) ( R ) ( R + as 3 i=1 ) SL(2, R) SO(2, 2) SO(2, 2) V 81 = exp [ ζ Λ E Λ + ζ Λ E Λ + σe ] exp [ UH ] v(φ) ( ) equations of motion defined with P 1 ( V 1 dv + η(v 1 dv) t η ) 2

13 Duality symmetry Hidden symmetry E 8(8) of the N = 8 supergravity e 8(8) = 1 ( 2) 56 ( 1) ( gl 1 e 7(7) ) (0) 56 (1) 1 (2) such that U, v(φ), ζ Λ, ζ Λ, σ parametrize E 8(8) /(Spin (16)/Z 2 ) in a parabolic gauge as E 8(8) ( 56 R ) ( R + E 7(7) ) Spin (16) V 248 = exp [ ζ Λ E Λ + ζ Λ E Λ + σe ] exp [ UH ] v(φ) equations of motion defined with P 1 ( V 1 dv + (V 1 dv) ) 2

14 Extremal solutions Under-rotating extremal solutions V = R 3 (no ergosphere) This implies either P µ nilpotent γ µν = δ µν R µν = Tr P µ P ν = 0 P µ admits some imaginary eigen values At a horizon U and imaginary eigen values produce exponentially growing oscillating modes [ J. L. Hörnlund] regular extremal solutions: V N G

15 Solvable subalgebra A solvable subalgebra n inside g admits a grading n (p) = ad p 1 n n \ adn p n. which can be defined by h g such that In the symmetric gauge and so we chose h k. [h, n (p) ] = 2p n (p) V = exp( L) for L n (g k )

16 Solvable system of differential equations The function L decomposes into p L(p) such that and d dl (1) = 0 L (p) n (p) (g k ) d dl (2) = 0 d dl (3) = 2 [ dl (1), [L (1), dl (1) ] ] 3 d dl (4) = 2 [ dl (1), [L (1), dl (2) ] ] 2 [ dl (1), [L (2), dl (1) ] ] 2 [ dl (2), [L (1), dl (1) ] ] d dl (5) = 2 [ dl (1), [L (1), [L (1), [L (1), dl (1) ]]] ] + 8 [ [L (1), dl (1) ], [L (1), [L (1), dl (1) ]] ] [ dl (1), [L (2), dl (2) ] ] 2 [ dl (2), [L (1), dl (2) ] ] 2 [ dl (2), [L (2), dl (1) ] ] [ dl (1), [L (1), dl (3) ] ] 2 [ dl (1), [L (3), dl (1) ] ] 2 [ dl (3), [L (1), dl (1) ] ] d dl (6) =...

17 Solvable system of differential equations The explicit solution can then be read from exp( 2L) = VV = e 2U M AB σ e 2U M CB ζ ADE ζ CDE e 2U M AB e 2U M AD ζ DBC and and similarly dω = Tr E VPV 1 = Tr E n 1 k=0 ( 2) k (k + 1)! ad L k dl dw Λ = 1 4 Tr EΛ VPV 1 = 1 4 Tr n 1 ( 2) k EΛ (k + 1)! ad L k dl k=0

18 Papapetrou Majumdar In the case of Maxwell Einstein 2L 0 2L exp 0 2L 2iL = 2L 2iL 0 e 2U( σ i 2 Φ 2) e 2U ie 2U Φ and e 2U = (1 + L) 2 Φ = for any harmonic function L L 2 L = M r gives extremal Reissner Nordström.

19 Nilpotent orbits The nilpotent elements of so(4, 4) e 8(8) lie in E 8(8) /(L e N e ) E 6(2) E 7(7) Spin(6, 7) SL(2) F 4(4) SO(6, 5) SO(4, 4) F 4(4) E 6(6) Regular BPS single-centre solutions: P lies in Spin c (16) /( (SU(2) SU(6)) (C 2 6 R) ) /( E 8(8) E6(2) (C 27 R) ) Regular non-bps single-centre solutions: P lies in Spin (16) /( Sp(4) R 27) /( E c 8(8) E6(6) R )

20 Normal triplets For a nilpotent orbit G C e = G C /J e of representative e g C An sl 2 triplet f, h, e [h, e] = 2e h lies in a Cartan subalgebra of g C. h determines the complex G C -orbit. For an orbit K C e = K C /I e of representative e g C k C An sl 2 triplet f, h, e [h, e] = 2e h lies in a Cartan subalgebra of k C. h determines the complex K C -orbit. Kostant Sekiguchi : N g /G = ( N gc (g C k C ) ) /K C

21 Nilpotent orbits The nilpotent elements of d 4 4a 1 e 8 d 8 lie in D 8 /(L e N e ) A 1 A 5 A 7 A 3 B 3 A 1 C 3 A 3 B 2 [A 1 ] 4 A 1 C 3 C 4 Regular BPS solutions: P lies in /( D 8 (A1 A 5 ) C ) 2 (6+ 6)+1 R 128

22 Nilpotent orbits The nilpotent elements of d 4 4a 1 e 8 d 8 lie in D 8 /(L e N e ) A 1 A 5 A 7 A 3 B 3 A 1 C 3 A 3 B 2 [A 1 ] 4 A 1 C 3 C 4 Regular almost-bps solutions: P lies in /( D 8 (A1 C 3 ) C 2 6) R 128

23 Nilpotent orbits The nilpotent elements of d 4 4a 1 e 8 d 8 lie in D 8 /(L e N e ) A 1 A 5 A 7 A 3 B 3 A 1 C 3 A 3 B 2 [A 1 ] 4 A 1 C 3 C 4 Regular non-bps solutions: P lies in D 8 /( 4 i=1 A (i) 1 C i>j 2i 2j+1) R 128

24 The STU model The supersymmetric orbit h = 2H 0 4 sl 2 = 1 ( 2) ( ) (0) gl 1 sl 2 sl 2 sl 2 1 (2) 16 = ( ) ( 1) D0 3 D2 3 D4 D6 ( D0 3 D2 3 D4 D6 ) (1) The subregular orbit h = 2 i H i 4 sl 2 = (3 1) ( 2) ( gl1 gl 1 gl 1 sl 2 ) (0) (3 1) (2) 16 = (D0 D6) ( 3) ( 3 D2 3 D4 ) ( 1) ( 3 D4 3 D2 ) (1) (D0 + D6) (3) The principal orbit h = 4H i H i 4 sl 2 = 1 ( 4) (3 1) ( 2) ( gl1 gl 1 gl 1 gl 1 ) (0) (3 1) (2) 1 (4) 16 = D0 ( 5) (3 D2) ( 3) (D6 3 D4) ( 1) (D6 3 D4) (1) (3 D2) (3) D0 (5)

25 The STU model The supersymmetric system h = 2H 0 [ F. Denef] ( D0 3 D2 3 D4 D6 ) (1) The almost-bps system h = 4H i H i [ K. Goldstein and S. Katmadas] (D6 3 D4) (1) (3 D2) (3) D0 (5) The composite non-bps system h = 2 i H i (3 D2 3 D4) (1) (D0 D6) (3)

26 ADM mass formula The rotated Noether charge C = 1 4π V 1 0 components decompose into R 3 VPV 1 V 0 C (M + in, Z ij, Π ijkl ) and C n (g k ) determines the ADM mass in function of Z ij and Π ijkl M = W [Z ij, Π ijkl, N = 0]

27 ADM mass formula in the STU model BPS composite : dim[sl(2) 4 /(SO(2) 2 R)] 8 1 = 0 M = e iα Z > 0 = Z almost-bps composite : dim[sl(2) 4 ] 8 1 = 3 M = 1 (3e iα 0 Z e i i α ( i Z e i(α 0 +α i+1 +α i+2 ) Z i + e iα ) ) i Zi > 0 4 non-bps composite: dim[sl(2) 4 /R] 8 1 = 2 M = 1 2 ( e i i αi Z i i e iαi Z i ) > 0

28 non-bps ADM mass formula in the STU model M = 1 2 ( e i i α i Z i e iα i Z i ) > 0 In the single centre case, one has flat directions Im [ e i 2 ( α 0+α i α i+1 α i+2 ) Z i ] = Im [ e i 2 ( α 0+α 1 +α 2 +α 3 ) Z ] the mass formula coincides with the non-standard diagonalization R k i R l j Z kl ˆ= e iπ 4 for α = Λ α Λ. ( ) 1 2 ( e iα + ie iα sin 2α) +e iα M M M M ξ 1+ξ 2 +ξ ξ ξ ξ 3 [ G. Bossard, Y. Michel and B. Pioline]

29 non-bps ADM mass formula in the STU model M = 1 2 ( e i i α i Z i e iα i Z i ) > 0 In general it depends on auxiliary parameters M = W [Z, Z i, β 1, β 2 ] and reproduces the generalised fake superpotential [ A. Ceresole, G. Dall Agata, S. Ferrara and A. Yeranyan] with W [Z, Z i, β 1, β 2 ] β 1 0 W [Z, Z i, β 1, β 2 ] β 2 0

30 Two-centre BPS solution [ F. Denef and B. Bates] The metric depends on 4+4 harmonic functions L Λ, K Λ e 4U = I 4 (L Λ, K Λ ) dω = L Λ dk Λ K Λ dl Λ with L Λ = l i + 2 A q A Λ x x A K Λ = k Λ + 2 A p Λ A x x A

31 Two-centre BPS solution The ADM mass Angular momentum M ADM = Z(q, p) J = p Λ A q B Λ p Λ B q A Λ Horizon area Stability A A = 4π I 4 (q A Λ, p Λ A ) M ADM = Z(q, p) Z(q A, p A ) + Z(q B, p B ) = M ADM A + M ADM B

32 Two-centre BPS solution

33 Two-centre solution The metric reads e 4U = VL 1 L 2 L 3 M 2 dω = dm i L i+1 L i+2 dk i with L i = l i + 2 A q Ai x x A K i = b i l i+1 l i A γ A q Ai x x A and d d M = i d d V = i d ( L i+1 L i+2 dk i ) d ( L i d(k i+1 K i+2 ) K i+1 K i+2 dl i )

34 Two-centre solution The metric reads e 4U = VL 1 L 2 L 3 M 2 dω = dm i L i+1 L i+2 dk i with s L i = l i + 2 A q Ai x x A K i = b i l i+1 l i A γ A q Ai x x A and M = A cos θ A α A x x A V = 1 i b i+1b i+2 l 1l 2l 3 2 A p 0 A +... x x A + 2 A cos θ A γ A α A x x A + 2 i L i K i+1 K i

35 Two-centre solution The metric reads e 4U = VL 1 L 2 L 3 M 2 dω = dm i L i+1 L i+2 dk i with L i = l i + 2 A q Ai x x A K i = b i l i+1 l i A γ A q Ai x x A and ( p i bi+1 A = q A0 = 0 γ A 2 γ A γ B l i l i+2 R ( bi+2 q Bi+1 )q Ai+2 γ A 2 γ A γ B l i l i+1 R q Bi+2 )q Ai+1

36 Two-centre solution The ADM mass M ADM = 1 ( 2 l 1 l 2 l 3 p i b i l i p i + i 1 + b i+1 b i+2 l i q i ) with Angular momentum t i 0 = l i+1 l i+2 b i + i 1 + b 2 i J = α A α B + p Λ A q B Λ p Λ B q A Λ Horizon area A A = 4π I 4 (q A Λ, p Λ A ) α 2 A

37 Two-centre solution

38 Two-centre solution

39 Two-centre solution stability M ADM = W [Z Λ (q Λ, p Λ ), β s (b i )] = W [Z Λ (q A Λ, p Λ A ), β s (b i )] + W [Z Λ (q B Λ, p Λ B ), β s (b i )] < W [Z Λ (q A Λ, p Λ A ), β (Z Λ )] + W [Z Λ (q B Λ, p Λ B ), β (Z Λ )] with W [β] β β=β s 0 W [β] β = 0 β=β therefore M ADM < M ADM A + M ADM B β s = β p i = 1 ( (b i+1 + s) q i+2 + (b i+2 + s) q ) i+1 l i l i+2 l i+1

40 Generalisation to N = 8 supergravity With the decomposition so (16) = 15 ( 2) (2 2 6) ( 1) ( gl 1 sl 2 su(2) su (6) ) (0) (2 2 6) (1) 15 (2) 128 = 2 ( 3) (2 6) ( 2) (2 15) ( 1) (2 20) (0) (2 15) (1) (2 6) (2) 2 (3) One has the Ansatz V = exp ( K ab α e (1)α ab Y ȧ α e (2) α a M α e (3)α) with harmonic functions L ab, K ab, Y ȧ α, and 2 sourced functions d d M = 1 2 ε abcdef d ( L ab L cd dk ef ) d d V = 1 2 ε abcdef d ( L ab d(k cd K ef ) K ab K cd dl ef )

41 Generalisation to N = 8 supergravity Only single-centre non-bps poles with ε αβ q ab α q cd β = 0 Angular momentum vector Scaling factor C A = q ab α e (1)α ab + qȧ αe (2) α a + p α e (3)α dω dm 1 2 ε abcdef L ab L cd dk ef e 4U = 1 6 ε ( abcdef VL ab 3ε α β YαY ȧ ḃ ) β L cd L ef M 2

42 Generalisation to N = 8 supergravity Only single-centre non-bps poles with ε αβ q ab α q cd β = 0 Angular momentum vector Scaling factor C A = q ab α e (1)α ab + qȧ αe (2) α a + p α e (3)α dω dm 1 2 ε abcdef L ab L cd dk ef e 4U = ( V ε α β, L ab, Yα ȧ ) M 2

43 Generalisation to N = 8 supergravity Q 1, Q 2 E 7(7) /E 6(6) and Q 1 + Q 2 of either type C 1, C 2 of Levi E 6(6) and C 1 + C 2 E 8(8) / ( SO(4, 4) R 6 8+6) Everywhere P µ Spin (16) c / ( SO(4) SO(4) R 6 4+1) and only degenerates at the black holes horizons. Orbit of dimension 83 > 57 asymptotic geometry not determined by Q 1 + Q 2 Mass depends on the internal structure, 56 charges + 26 angles. M ADM Ω ij Z ij with Ω SU(8)/(Sp(1) Sp(3)) and constraints Ω(Z ij ).

44 E 8(8) closure diagram e 8(8) so (16)

45 E 8(8) closure diagram e 8(8) so (16) E6 D5 D4 Single centre

46 Almost-BPS in N = 8 supergravity Only single-centre non-bps poles C A = p 0 e (1) + p ab e (1) ab + p a αe (2) α a + q ab e (3) ab + q 0e (5) with either p ab = 0 or p 0 = p a α = 0 Angular momentum vector Scaling factor dω dm ( VZ ab ε αβ Y a αy b β ) dkab e 4U = 1 6 ε ( abcdef VZ ab 3ε α βy αy ȧ ḃ ) β Z cd Z ef M 2

47 E 8(8) closure diagram e 8(8) so (16) E6 D5 D4 Single centre

48 Locally BPS in N = 8 supergravity Angular momentum vector dω L 0 dk 0 + L ab dk ab K 0 dl 0 K ab dl ab Scaling factor e 4U = (L, K) with K 0, K ab, Yα a harmonic, and L 0 = K ab ( ε αβ YαY a β b ) L ab = K 0 ( ε αβ YαY a β b )

49 E 8(8) closure diagram e 8(8) so (16) E6 D5 D4 Single centre

50 Conclusion Solvable subalgebras n g Solvables subsystems Determined by semi-simple elements h [h, P (p) ] = 2p P (p) P (p) = 0 p 0 characterising even nilpotent orbits in g C k C

51 Conclusion D 4 class: STU truncation BPS solutions: 32 harmonic functions, M = Z ij almost-bps solutions: 44 harmonic functions M Z ij + Re[Ω ij Z ij ] with Ω SU(8)/Sp(4) composite non-bps solutions: 44 harmonic functions M Ω ij Z ij with Ω SU(8)/(Sp(1) Sp(3)) + 13 phases fixed

52 Outlook D 4 class: STU truncation locally BPS solutions: 44 harmonic functions almost-bps solutions: 44 harmonic functions M Z ij + Re[Ω ij Z ij ] with Ω SU(8)/Sp(4) composite non-bps solutions: 44 harmonic functions M Ω ij Z ij with Ω SU(8)/(Sp(1) Sp(3)) + 13 phases fixed

53 Outlook D 4 class: STU truncation locally BPS solutions: 44 harmonic functions almost-bps solutions: 44 harmonic functions M Z ij + Re[Ω ij Z ij ] with Ω SU(8)/Sp(4) composite non-bps solutions: 44 harmonic functions M Ω ij Z ij with Ω SU(8)/(Sp(1) Sp(3)) + 13 phases fixed D 5 class: SL(2) SL(4) truncation locally BPS solutions: 52 harmonic functions almost-bps solutions: 52 harmonic functions M Z ij + Re[Ω ij Z ij ] with Ω SU(8)/Sp(2) 2 composite non-bps solutions: 52 harmonic functions M Ω ij Z ij with Ω SU(8)/(Sp(1) 2 Sp(2)) + 5 phases fixed

54 Outlook D 4 class: STU truncation locally BPS solutions: 44 harmonic functions almost-bps solutions: 44 harmonic functions M Z ij + Re[Ω ij Z ij ] with Ω SU(8)/Sp(4) composite non-bps solutions: 44 harmonic functions M Ω ij Z ij with Ω SU(8)/(Sp(1) Sp(3)) + 13 phases fixed E 6 class: SL(6) truncation locally BPS solutions: 56 harmonic functions almost-bps solutions: 56 harmonic functions M Z ij + Re[Ω ij Z ij ] with Ω SU(8)/Sp(1) 4 composite non-bps solutions: 56 harmonic functions M Ω ij Z ij with Ω SU(8)/Sp(1) 4 + phase fixed

55 Outlook N = 2 supergravity in 8 Dimensions Scalars parametrizing SL(2)/SO(2) and SL(3)/SO(3) 2 3 vectors A α i, 3 2-forms B i, a 3-form C α e 7(7) = su(8) (gl1 sl 2 sl 3 sl 4) (0) (2 3 4) (1) ( 3 6) (2) (2 4) (3) 3 (4) 56 = 4 ( 3) (2 3) ( 2) ( 3 4) ( 1) (2 6) (0) (3 4) (1) (2 3) (2) 4 (3)

56 Outlook Truncation in 8 Dimensions Scalars parametrizing SL(2)/SO(2) One 3-form C α sl 6 = so(6) (gl1 sl 2 sl 4 ) (0) (2 4) (3) 20 = 4 ( 3) (2 6) (0) 4 (3) Stationary solutions described with scalars parametrizing E 6(6) /Sp(8, R)