Black holes in N = 8 supergravity

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1 Black holes in N = 8 supergravity Eighth Crete Regional Meeting in String Theory, Nafplion David Chow University of Crete 9 July 2015

2 Introduction 4-dimensional N = 8 (maximal) supergravity: Low energy regime of M-theory compactified on T 7 In perturbation theory, UV finite up to 4 loops,...? Setup for addressing black hole microscopics Aim: Find general (stationary, non-extremal, asymptotically flat) black hole of N = 8 supergravity ST U supergravity: truncation of N = 8 with 4 U(1) gauge fields Classical U-duality (global symmetries) (Cvetič Hull 96) = General ST U solution is generating solution for general N = 8 solution, mixing matter fields, but same 4d metric Equivalent aim: Find general black hole of ST U supergravity 10 parameters: mass, rotation, 4 electric, 4 magnetic (5 electromagnetic charges suffice, but 8 is more symmetric) Based mainly on (with Geoffrey Compère)

3 Outline of solution generating technique ST U supergravity: 4d metric + 4 vectors + 6 scalars reduce on t 3d metric + 5 vectors + 11 scalars Hodge dualize 3d metric + 16 scalars 3d scalars parameterize coset (Breitenlohner Maison Gibbons 88) G/K = SO(4, 4)/SL(2, R) 4 Lagrangian: 3d euclidean gravity + scalars (in SO(4, 4) matrix M) L 3 = R Tr[ 3(M 1 dm) (M 1 dm)] Efficient 1-step generating technique: ds 2 3 = ds 2 3, M = g M 0 g M 0 : seed solution g: SO(4, 4) group element M: new solution : SO(4, 4) generalized transpose (Chong Cvetič Lü Pope 04, Gal tsov Scherbluk 08, Bossard Michel Pioline 09)

4 ST U supergravity 4-dimensional N = 2 supergravity coupled to 3 vector multiplets Bosonic fields: Metric: g ab 4 U(1) gauge fields: A I, I = 1, 2, 3, 4 (or dual fields ÃI) 6 real scalars: ϕ i dilatons, χ i axions, i = 1, 2, 3 (3 complex scalars z i = χ i + ie ϕ i, sometimes called S, T, U) Bosonic Lagrangian of form L 4 = R i ( dϕ i dϕ i + e 2ϕ i dχ i dχ i ) I,J [F IJ(ϕ, χ) F I F J + G IJ (ϕ, χ)f I F J ] 1 2 Scalars parameterize coset G/K = (SL(2, R)/U(1)) 3 Special case: Einstein Maxwell theory (A I all equal, ϕ i = χ i = 0) Field equations invariant under SL(2, R) 3 and triality (permuting SL(2, R) s)

5 ST U supergravity continued Bosonic Lagrangian L 4 = R i=1 ( dϕ i dϕ i + e 2ϕ i dχ i dχ i ) i=1 e2ϕ i ϕ 1 ϕ 2 ϕ 3 ( F i χ i F 4 ) ( F i χ i F 4 ) 1 2 e ϕ 1 ϕ 2 ϕ 3 F 4 F 4 + χ 1 χ 2 χ 3 F 4 F 4 {i,j,k}={1,2,3} (χ i F j F k χ j χ k Fi F 4 ) (Ã1, Ã2, Ã3, A 4 ) duality frame: triality invariance manifest, and prepotential exists SL(2, R) 1 SL(2, R) 2 SL(2, R) 3 global symmetry: ( ) ( ) ( ) ( ) F1 F1 F2 F2 Λ, (Λ T ) 1 F 4 ( ) F4 F 1 F 4 ( F4 Λ F 1 ), ( ) a b Λ = SL(2, R) c d 1, F 3 ( ) F3 F 2 F 3 (Λ T ) 1 ( F3 F 2 z 1 az 1 + b cz 1 + d )

6 Truncations of N = 8 supergravity N = 8 supergravity U-duality STU supergravity A 2 = A 3 ST 2 supergravity (reduction of 6d minimal supergravity) A 2 = A 3 = A 4 S 3 supergravity (reduction of 5d minimal supergravity) A 1 = A 4,A 2 = A 3 ix 0 X 1 supergravity (truncation of N = 4 supergravity) A 2 = A 3 = A 4 =0 Kaluza Klein theory A I = A Einstein Maxwell theory (N = 2 supergravity) A 1 = A 4,A 2 = A 3 =0 Einstein Maxwell dilaton axion theory

7 Basic example: 4d gravity 3d Restrict to stationary solutions = Kaluza Klein reduce on t Reduction ansatz (timelike): ds 2 = e 2U (dt + ω 3 ) 2 + e 2U ds 2 3 Hodge dualize ω 3 to 3d scalar σ: dω 3 = 1 2 e 4U 3 dσ = 3d euclidean gravity + scalars: L 3 = R du du 1 8 e 4U 3 dσ dσ Hidden SL(2, R) Ehlers symmetry revealed in 3d: Scalars (U, σ) parameterize SL(2, R)/U(1)

8 Basic example: 4d gravity 3d: SL(2, R)/U(1) coset sl(2, R) generators: Cartan positive-root negative-root ( ) ( ) ( ) H=, E=, F = Scalars (U, σ) parameterize SL(2, R)/U(1) coset: V = e UH e σe/2 Define M V T V SL(2, R): M is gauge-invariant, i.e. V kv for k SO(2) = U(1) 3d scalars (U, σ) can be extracted from M 3d theory as manifest SL(2, R) coset: L 3 = R du du 1 8 e 4U 3 dσ dσ = R Tr[ 3(M 1 dm) (M 1 dm)] Basic application: generate Taub NUT from Schwarzschild

9 ST U supergravity 3d Reduction ansatz: ds 2 = e 2U (dt + ω 3 ) 2 + e 2U ds 2 3, A I = ζ I (dt + ω 3 ) + A I (3d), Ã I = ζ I (dt + ω 3 ) + ÃI(3d) 16 3d scalars after Hodge dualization: U, σ (dualizing ω 3 ) 8 electromagnetic scalars ζ I, ζ I (dualizing ÃI(3d), A I (3d) ) 3 dilatons ϕ i, 3 axions χ i 3d scalars parameterize SO(4, 4)/SL(2, R) 4 SO(4, 4) generalized transpose : g = ηg T η 1, η = diag( 1, 1, 1, 1, 1, 1, 1, 1) SO(4, 4) matrices satisfy M = M

10 ST U supergravity 3d: SO(4, 4)/SL(2, R) 4 coset 28 so(4, 4) generators: Cartan (4) Positive-root (12) Negative-root (12) H Λ E Λ, E Q I, E P I F Λ, F Q I, F P I Λ = 0: Ehlers SL(2, R) Λ = 1, 2, 3: scalar fields SL(2, R) 3 I = 1, 2, 3, 4: gauge fields 16 scalars parameterize coset in Iwasawa gauge g = g K g H g E : V = exp( UH 0 ) exp( 1 2 i ϕ ih i )exp( i χ ie i ) exp[ I (ζi E Q I + ζ I E P I )] exp( 1 2 σe 0) 3d Lagrangian with manifest SO(4, 4) symmetry: L 3 = R Tr[ 3(M 1 dm) (M 1 dm)], M V V Useful to define 1-form N so(4, 4) (Breitenlohner Maison 00): dn = M 1 3 dm = dω 3 F 0 + I (dai (3d) F P I dãi(3d) F Q I )+...

11 Generating the charged solution Method generalizes Harrison transformation of Einstein Maxwell theory, which uses SU(2, 1)/S(U(1, 1) U(1)) coset 1. Initial seed solution: Kerr Taub NUT (3 parameters m, n, a) 2. Act with group element (adds 8 charge parameters δ I, γ I ) ( g = exp ) ( γ I (E P I F P I ) exp ) δ I (E Q I F Q I ) I I 3. Final 11-parameter solution: 3d scalars from M g Mg (N g 1 N g gives directly 3d 1-forms ω 3, A I (3d), ÃI(3d)) Seed Kerr solution often generates NUT charge = start with more general Kerr Taub NUT seed instead Choice of g motivated by: Symmetry in gauge fields g g = I: preserves asymptotic flatness at spatial infinity

12 Summary of general charged solution 11 physical charges 11 parameters Mass M, NUT charge N m, n Angular momentum J a Electric charges Q I δ I Magnetic charrges P I Set N = 0 (linear constraint on seed m, n) 10-parameter asymptotically flat family Unifies various previous solutions (Demiański Newman 66, Rasheed 95, Cvetič Youm 96, Matos Mora 96, Larsen 99, Lozano-Tellecha Ortín 99, Chong Cvetič Lü Pope 04, Giusto Saxena Ross 07, Compère de Buyl Stotyn Virmani 10) γ I

13 Thermodynamics Thermodynamic quantities satisfy usual relations: First law: de = T ds + Ω dj + I (ΦI dq I + Ψ I dp I ) Smarr relation: E = 2T S + 2ΩJ + I (ΦI Q I + Ψ I P I ) Product of inner and outer horzion areas (Larsen 97, Cvetič Larsen 97, Cvetič Gibbons Pope 10): Cayley hyperdeterminant: S + S = 4π 2 ( J 2 + (Q I, P I ) ) π 2 Z = 1 16 [4( I Q I + I P I ) + 2 I<J Q IQ J P I P J I (Q I) 2 (P I ) 2 ] SL(2, R) 3 -invariant and triality-invariant Special case of E 7(7) quartic invariant (Kallosh Kol 96) Non-extremal entropy formula: S + = 2π ( + F + J 2 + F ) S +, J, E 7(7) -invariant = F (M, Q I, P I, z i ) also invariant

14 Killing tensors and separability Kerr metric admits Killing Stäckel tensor: K ab = K (ab), (a K bc) = 0, K ab P a P b conserved on geodesics General ST U supergravity charged black holes admit Killing Stäckel tensor K ab in string frame d s 2 = Ω 2 ds 2 = Conformal Killing Stäckel tensor Q ab = Q (ab), (a Q bc) = q (a g bc), (q a = 1 6 ( aq b b + 2 b Q b a)) in Einstein frame, components Q ab = K ab Q ab P a P b conserved on null geodesics Separation of: Massive/massless geodesics in string/einstein frame Massless Klein Gordon equation in Einstein frame

15 U(1) 4 gauged supergravity and AdS black holes Maximal N = 8, SO(8)-gauged supergravity truncates to N = 2, U(1) 4 -gauged supergravity (Cvetič et al. 99): L gauged = L ungauged + g 2 i (2 cosh ϕ i + χ 2 i e2ϕ i ) 1 g: gauge-coupling constant/inverse AdS radius Potential breaks 4d global symmetry SL(2, R) 3 U(1) 3 = previous techniques fail, so guess instead Two classes of asymptotically AdS generalizations: Static solutions, 10 parameters: mass, AdS radius, 4 electric charges, 4 magnetic charges, (S 2, R 2, H 2 horizon) Rotating solutions, 8 parameters: mass, NUT charge, rotation, AdS radius, 2 electric charges, 2 magnetic charges (U(1) 2 truncation of N = 4, SO(4)-gauged supergravity) Includes known subcases (Duff Liu 99, Chong Cvetič Lü Pope 05, Lü Pang Pope 13, Lü 13)

16 Summary SO(4, 4) solution generating technique for ST U supergravity Solutions: General rotating black holes of ST U supergravity with 11 parameters: mass, NUT, rotation, 4 electric, 4 magnetic = General black holes in N = 8 supergravity with 59 parameters: mass, NUT, rotation, 28 electric, 28 magnetic Also some AdS black holes of U(1) 4 -gauged supergravity Some properties: E 7(7) -invariant non-extremal entropy formula: Killing tensors, separability Open problems: S = 2π( + F + J 2 + F ) Beyond coset models: inverse scattering, more general N = 2 theories Complete proof of uniqueness

17 This research has been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF), under the grant schemes Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes and the program Thales.

18 Summary of general charged solution: metric Coordinates: (t, r, φ, u), u = n + a cos θ Metric: ds 2 = R U W (dt + ω 3) 2 + W ( dr 2 R + du2 U ) RU dφ2 + a 2 (R U) R = r 2 + a 2 n 2 2mr, U = a 2 (u n) 2 W 2 (r, u) quartic

19 Summary of general charged solution: matter Gauge fields: A I = W δ I ( ) dt + ω3 W Scalar fields: χ i = f i r 2 + u 2 + g i, e ϕ i = W r 2 + u 2 + g i, f i (r, u) = 2(mr + nu)ξ i1 + 2(mu nr)ξ i2 + 4(m 2 + n 2 )ξ i3, g i (r, u) = 2(mr + nu)η i1 + 2(mu nr)η i2 + 4(m 2 + n 2 )η i3