Interacting non-bps black holes

Transcription

1 Interacting non-bps black holes Guillaume Bossard CPhT, Ecole Polytechnique IPhT Saclay, November 2011

2 Outline Time-like Kaluza Klein reduction From solvable algebras to solvable systems Interacting non-bps solutions Outlook Mainly on [ G. Bossard, C. Ruef, ] and more to come.

3 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.

4 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 )

5 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

6 Beyond STU truncation 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 Non-supersymmetric truncation in 8 dimensions Scalars parametrizing SL(2)/SO(2) One 3-form C α In four dimensions one gets moduli in SL(6)/SO(6) and 20 electromagnetic fields.

7 Duality symmetry Hidden symmetry E 6(6) of the pertinent truncation e 6(6) = 1 ( 2) 20 ( 1) ( gl 1 sl 6 ) (0) 20 (1) 1 (2) such that U, v(φ), ζ Λ, ζ Λ, σ parametrize E 6(6) /(Sp(8, R)/Z 2 ) in a parabolic gauge as E 6(6) ( 20 R ) ( R + SL(6) ) Sp(8, R) V 27 = 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

8 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

9 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 )

10 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) =...

11 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

12 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)

13 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)

14 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

15 Supersymmetric solutions The metric reads [ B. Bates and F. Denef] and moduli such that e 4U = I 4 (H) t i = dω = H Λ dh Λ H Λ dh Λ I 4 H i ih i I 4 H 0 ih 0 H Λ = h Λ + A p Λ x x A H Λ = h Λ + A q Λ x x A

16 Almost BPS solutions [ I. Bena, G. Dall Agata, S. Giusto, C. Ruef and N. P. Warner] The metric reads e 4U = V 1 6 cijk L i L j L k M 2 dω = dm VL i dk i and moduli t i = K i + such that V and K i are harmonic and [ K. Goldstein and S. Katmadas] 3cijk L j L k ( M + ie 2U ) Vc pqr L p L q L r d dl i = 1 2 c ijkd ( Vd(K j K k ) K j K k dv ) d dm = d ( VL i dk i)

17 Non-BPS composite solutions The metric reads e 4U = V 1 6 c ijkl i L j L k M 2 dω = dm 1 2 c ijkl i L j dk k and moduli t i = K i + such that L i and K i are harmonic and 6L i ( M + ie 2U ) c jkl L j L k L l d d M = 1 2 c ijkd ( L i L j dk k) d d V = 1 2 c ijkd ( L i d(k j K k ) K j K k dl i)

18 Non-BPS composite solutions The metric reads e 4U = V 1 6 c ijkl i L j L k M 2 dω = dm 1 2 c ijkl i L j dk k with L i = l i + 2 A p i A x x A K i = k i + 2 A γ A p i A x x A and M = A cos θ A α A x x A V = 6 c ijk l i l j l 1 k 2 c ijk l i k j k k 2 A p 0 A +... x x A + 2 A cos θ A γ A α A x x A c ijk L i K j K k +...

19 Non-BPS composite solutions The metric reads e 4U = V 1 6 c ijkl i L j L k M 2 dω = dm 1 2 c ijkl i L j dk k with L i = l i + 2 A p i A x x A K i = k i + 2 A γ A p i A x x A and ( q A i = c ijk p j γ A A l k k k + p k ) B B γ A ) x B A(γ A x B p 0 A = 0

20 Non-BPS composite solutions Let us define d i such that q i = c ijk p j d k then for p 0 = 0 I 4 (q, p) = 2 3 c ijkp i p j p k( q c lpqp l d p d q) using a Jordan algebra identity.

21 Non-BPS composite solutions The ADM mass M ADM = 1 2 ( ) Z + (t i t i )D i Z 0 identical to [ E. Gimon, F. Larsen and J. Simón] for single centre Angular momentum J = A α A + A>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

22 Two-centre solution

23 Two-centre solution stability [ A. Ceresole, G. Dall Agata, S. Ferrara and A. Yeranyan] M ADM = W [Z Λ (q Λ, p Λ ), β s ] = W [Z Λ (q A Λ, p Λ A ), β s ] + W [Z Λ (q B Λ, p Λ B ), β s ] < 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 q i = c ijk p j Re[(1 is)t k 0 ]

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

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

26 Outlooks It is very appealing that the most general systems have 56 harmonic functions for N = 8 and generalise the three, BPS, almost BPS and non-bps systems. However, the same Ansatz of harmonic functions does not permit to find regular solutions in higher degree orbits Generalisations including non-physical poles or multipole harmonics? Would help to have the explicit solution f ABC = 1 x x A 1 x x B 1 x x C Need to study the general solution to find the duality invariant equations for the distances in terms of the charges, and domain of stability. Generalisation to arbitrary (non-symmetric) cubic pre-potential.