Black holes of = 8 supergravity N Geoffrey Compère Université Libre de Bruxelles (ULB) Ascona, July 2014 G. Compère (ULB) 1 / 35 The simplicity of = 8 supergravity N Obtained as the low energy regime of M-theory upon compactification on T7 Inherits E7(7)(R) symmetries broken to E7(7)(Z) at the quantum level Cremmer and Julia (1978), etc In perturbation theory, UV finite up to 4, maybe 8, loops Bern et al. (2009), etc In perturbation theory, amplitudes enjoy remarkable properties : square of N = 4 SYM, etc Kawai, Lewellen, Tye (1986), etc The theory cannot be decoupled from string theory Green, Ooguri, Schwarz (2007) The theory contains BPS extremal black holes with known microscopic counting in M-theory Maldacena, Strominger, Witten (1997). G. Compère (ULB) 2 / 35 Ideal setup to address black hole microscopics All known non-extremal black holes in string theory obey Cvetič and Larsen (1997) 2 S+S− π Z 2 All known extremal black holes have SL(2; R) U(1) × SL(2; R) Virasoro symmetries and obey ! × π2 π2 S = c T = c T + 3 J J 3 Q Q Are there other “IR” universal formulae ? Is there a proof from the “UV" ? In this talk, I will present the most general asymptotically flat non-extremal black hole of = 8 supergravity and N derive some of its properties. G. Compère (ULB) 3 / 35 (Next to simple) Gauged = 8 SO(8) supergravity N Obtained as the low energy regime of M-theory upon compactification on S7 Admits AdS as a = 8 vacuum 4 N When embedded in M-theory, the dual CFT is the ABJM theory at level k = 1. Aharony, Bergman, Jafferis, Maldacena (2008) New sugra theories labelled by a continuous parameter have been recently found Dall’Agata, Inverso, Trigiante (2012) Much less is known on solutions and “IR" relations Known black holes in AdS4 obey Cvetič, Gibbons, Pope (2010) 4 4 S+ S− Si S−i π ` Z 2 In this talk, I will present some black holes of this theory and discuss subtleties with their thermodynamics G. Compère (ULB) 4 / 35 Based on work done with David Chow (ULB) “Seed for general rotating non-extremal black holes of = 8 supergravity”, arXiv :1310.1925 N “Dyonic AdS black holes in maximal gauged supergravity”, arXiv :1311.1204 “Black holes in = 8 supergravity from SO(4; 4) hidden N symmetries”, arXiv :1404.2602 G. Compère (ULB) 5 / 35 Outline 1 STU supergravity and hidden symmetries 2 The general black hole and its properties 3 Microscopic counting for extremal branches 4 Dyonic black holes of maximal gauged supergravity G. Compère (ULB) 6 / 35 I. STU supergravity and hidden symmetries G. Compère (ULB) 7 / 35 From = 8 to STU supergravity N Under U-dualities, the 4d metric will be unchanged. The matter field will be shuffled around. There are 28 abelian gauge fields in = 8 supergravity. If N one starts with a dyonic rotating black hole with 3 additional electric charges (5 charges in total), one can perform U-dualities and generate the general black hole. Cvetič-Hull, 1996 A suitable sector of = 8 supergravity is a = 2 N N supergravity with three vector multiplets known as the STU supergravity Cremmer et al ’85 ; Duff et al. ’96. This is the theory that we will study here. G. Compère (ULB) 8 / 35 STU supergravity In a specific U-duality frame the Lagrangian has the general form Duff et al. ’96 1 = d4xp g R f (z)@ zA@µzB L4 − − 2 AB µ 1 1 k (z)FI FJµν + h (z)µνρσFI FJ −4 IJ µν 4 IJ µν ρσ where zj = xj + i yj, j = 1; 2; 3 are three complex scalar fields AI = (A1; A2; A3; A4) are the four U(1) gauge fields. Triality symmetry : SL(2; R) SL(2; R) SL(2; R) and their × × permutations. G. Compère (ULB) 9 / 35 From = 8 to STU supergravity N =8 N supergravity U-duality STU supergravity A2 = A3 ST2 supergravity (reduction of 6d minimal supergravity) A2 = A3 = A4 A1 = A4,A2 = A3 S3 supergravity iX0X1 supergravity − (reduction of 5d (truncation of minimal supergravity) = 4 supergravity) N A2 = A3 = A4 =0 AI = A A1 = A4,A2 = A3 =0 Einstein– Einstein–Maxwell– Kaluza–Klein theory Maxwell theory dilaton–axion theory ( = 2 supergravity) N G. Compère (ULB) 10 / 35 1 Known solutions =8 N supergravity U-duality Extremal Rotating 4-electric STU supergravity Static Rotating 4-dyonic A2 = A3 ST2 supergravity (reduction of 6d minimal supergravity) A2 = A3 = A4 A1 = A4,A2 = A3 S3 supergravity iX0X1 supergravity − (reduction of 5d (truncation of minimal supergravity) = 4 supergravity) N A2 = A3 = A4 =0 AI = A A1 = A4,A2 = A3 =0 Einstein– Einstein–Maxwell– Kaluza–Klein theory Maxwell theory dilaton–axion theory ( = 2 supergravity) N G. Compère (ULB) 11 / 35 1 Algorithm for solution generation Reduce on time. Get Euclidean 3d Einstein gravity coupled to scalars. SO(4;4) Scalars form the coset model Breitenlohner, Maison, Gibbons, ’88 SL(2;R)4 Take Kerr-Taub-NUT (a; m; n) as a seed Act with SL(2; R)4 SO(4; 4) hidden symmetries ⊂ I Generate all 4 QI + 4 P charges Cancel the total NUT charge at the end by tuning the initial NUT charge n Iterate the loop "Recognize patterns Simplify ! ! Rewrite Recognize patterns . ." ! ! G. Compère (ULB) 12 / 35 II. The general black hole and its properties G. Compère (ULB) 13 / 35 The Kerr metric (1963) R U dr2 du2 RU ds2 = − (dt + ! )2 + W + + dφ2 ; − W 3 R U a2(R U) − where W(r; u) = r2 + u2; 2mrU ! (r; u) = dφ, 3 a(R U) − and R(r) = r2 2mr + a2; − U(u) = a2 u2; − The standard spherical polar angle θ is u = a cos θ. G. Compère (ULB) 14 / 35 The = 8 black hole metric N Input : M; N; m; n; a + 2 harmonic functions L(r), V(u) : R U dr2 du2 RU ds2 = − (dt + ! )2 + W + + dφ2 ; − W 3 R U a2(R U) − where W2(r; u) = (R U)2 + (2Nu + L)2 + 2(R U)(2Mr + V) ; − − 2N(u n)R + U(L + 2Nn) ! (r; u) = − dφ, 3 a(R U) − R(r) = r2 2mr + a2 n2; − − U(u) = a2 (u n)2; − − L(r) = L1r + L2; V(u) = V1u + V2: The standard spherical polar angle θ is u = n + a cos θ. G. Compère (ULB) 15 / 35 Separability and hidden conformal symmetry Define the string frame metric 2 r2 + u2 ds~ = ds2: W It admits an irreducible Killing-Stäckel tensor Kab obeying K = 0: r(a bc) 2 The massive Klein-Gordon equation on ds~ is separable. Therefore, the metric ds2 admits an irreducible conformal Killing-Stäckel tensor with components Qab = Kab obeying Q = q g : r(a bc) (a bc) The massless Klein-Gordon equation on ds2 is separable. [Subcases include : Chow, ’08 ; Keeler, Larsen, ’12] G. Compère (ULB) 16 / 35 Gauge fields : All is known ! We have I @ 1 A = W (dt + !3) : @δI −W The electric and magnetic charges are @M I @N QI = 2 ; P = 2 : @δI − @δI G. Compère (ULB) 17 / 35 Scalar fields The three axions and dilatons are : f W i e−'i χi = 2 2 ; = 2 2 ; r + u + gi r + u + gi where f (r; u) = 2(mr + nu)ξ + 2(mu nr)ξ + 4(m2 + n2)ξ ; i i1 − i2 i3 g (r; u) = 2(mr + nu)η + 2(mu nr)η + 4(m2 + n2)η ; i i1 − i2 i3 are linear functions of r and u. G. Compère (ULB) 18 / 35 Thermodynamics First law and Smarr relation hold I + I δM = T+δS+ + Ω+δJ + Φ+δQI + ΨI δP ; I + I M = 2T+S+ + 2Ω+J + Φ+QI + ΨI P ; Also at the inner horizon, I − I δM = T−δS− + Ω−δJ + Φ−δQI + ΨI δP ; I − I M = 2T−S− + 2Ω−J + Φ−QI + ΨI P : G. Compère (ULB) 19 / 35 Quartic invariant 1 1 2 3 4 ∆ = 16 [4(Q1Q2Q3Q4 + P P P P ) + 2 P Q Q PJPK P (Q )2(PJ)2]: J<K J K − J J The invariant is a Cayley hyperdeterminant, and is manifestly 3 invariant under SL(2; R) upon rewriting as [Duff, ’06] 1 ii0 jj0 kk0 ll0 mm0 nn0 ∆ = 32 aijkai0j0manpk0 an0p0m0 ij [ij] 01 with = , = 1 and components aijk given by (a ; a ) = (Q ; P1); (a ; a ) = (P2; Q ); 000 111 − 1 001 110 2 3 4 (a010; a101) = (P ; Q3); (a011; a100) = (Q4; P ): This invariant is a special case of a more general E7(7) quartic invariant [Kallosh, Kol, ’96]. G. Compère (ULB) 20 / 35 Universal properties of horizons Product of area law : Cvetič, Gibbons, Pope, ’10 A+ A− 2 2 I 2 = 4π J + ∆(QI; P ) π Z 4 4 2 Angular momentum law : 2 Ω+ 2 8π J = (S+ S−) 4π Z T+ − 2 Kinematic relationship : Ω Ω− + = T+ − T− G. Compère (ULB) 21 / 35 A new E7(7)(R) invariant Using these properties, one can prove Cardy’s form : p q S = 2π ∆ + F + J2 + F + − I i Since S+; J; ∆ are E7(7)(R) invariants, then F(M; QI; P ; z1) is invariant as well. Known special cases : For BPS black holes, F = J = 0. In the extremal “fast” rotating limit, F = J2. In the extremal “slow” rotating limit, F = ∆ − For Kerr-Newman : F = M4 M2Q2. 1− In regime Q : F = Q Q Q (M M ) Cvetic-Larsen 1;2;3 ! 1 2 1 2 3 − BPS (2014) G. Compère (ULB) 22 / 35 F is not a rational function of the charges For the Kaluza-Klein black hole (Rasheed-Larsen), 1 1 1 1 F = (M2 P2)(M2 Q2) + (M2 + (P2 + Q2))2H[x] − 4 − 4 3 8 2 2 2 2 where 0 x 54M (P −Q ) 1 and ≤ ≡ (8M2+P2+Q2)3 ≤ p arcsin px arcsin px H[x] = 2 1 x cos + 6px sin 2 − 3 3 − increases monotonically from 0 to 1.