Supersymmetric Black Holes in String Theory Napoli, October 2006 Thomas Mohaupt University of Liverpool Supersymmetric Black Holes in String Theory – p.1 Outline 1. Introduction: the laws of black mechanics, the string–black hole correspondence and BPS states. 2. BPS black holes in = 2 compactifications: special geometry, attractor mechanism, N variational principle, and higher derivative corrections. With comments on nonsupersymmetric black holes. 3. OSV conjecture, large and small black holes. 4. Concluding remarks. Supersymmetric Black Holes in String Theory – p.2 The Laws of Black Hole Mechanics (0) κS = const. κS (1) δM = 8π δA + !δJ + φδq. (2) δA 0. ≥ (3) κS = 0 cannot be reached in finite time by any physical process. ks = surface gravity, M = mass, A = horizon area, ! = horizon angular velocity, J = angular momentum, φ = chemical (= electrostatic) potential, q = electric charge. J.M. Bardeen, B. Carter and S.W. Hawking (1973) Suggests: κ T ; A S S ∼ ∼ where T = temperature and S = Entropy. Assumptions: (0), (1): stationary black hole, (2): ‘predictable’ space-time, Einstein’s field equations satisfied, (dominant/weak) energy condition. However, (0), (1) hold irrespective of the details of the field equations, if appropriate symmetry conditions are imposed! Supersymmetric Black Holes in String Theory – p.3 The Laws of Black Hole Mechanics (2) Modified assumptions: generally covariant Lagrangian with black hole solution such that (i) black hole is static or stationary, axisymmetric, t φ reflection symmetric, − (ii) event horizon is a Killing horizon, (iii) a Cauchy hypersurface exists. Then (0) κS = const. R. Wald and Racz (1995) κS (1) δM = 2π δS + !δJ + φδq. R. Wald (1993), ... provided that M; J; S; q are defined as appropriate surface charges. Entropy: S = Q[ξ] ZH ξ = ‘horizontal’ Killing vector field, Q = Noether two-form. Supersymmetric Black Holes in String Theory – p.4 The Laws of Black Hole Mechanics (2) Modified assumptions: generally covariant Lagrangian with black hole solution such that (i) black hole is static or stationary, axisymmetric, t φ reflection symmetric, − (ii) event horizon is a Killing horizon, (iii) a Cauchy hypersurface exists. Then (0) κS = const. R. Wald and Racz (1995) κS (1) δM = 2π δS + !δJ + φδq. R. Wald (1993), ... provided that M; J; S; q are defined as appropriate surface charges. Entropy: δ S = 2π L "µν "ρσphd2Ω δR ZH µνρσ "µν = normal bivector of horizon, phd2Ω = induced volume form. Supersymmetric Black Holes in String Theory – p.4 Black Hole Thermodynamics Hawking radiation: κS THawking = (GN = c = ~ = 1) : 2π First law: κ δM = S δA + 8π · · · A S = : ) 4 Supersymmetric Black Holes in String Theory – p.5 Black Hole Thermodynamics Hawking radiation: κS THawking = (GN = c = ~ = 1) : 2π First law (generalized version): κ δM = S δS + 2π · · · A S = + corrections from higher derivative terms 4 Supersymmetric Black Holes in String Theory – p.5 Black Hole Thermodynamics Hawking radiation: κS THawking = (GN = c = ~ = 1) : 2π First law (generalized version): κ δM = S δS + 2π · · · A S = + corrections from higher derivative terms 4 S = Smacro = thermodynamical or macroscopic entropy. Unterlying microscopic theory (=quantum gravity) should specify the microscopic states of the black hole. Microscopic entropy: Smicro = log d(M; J; q) ; d = #microstates : Expect: Smacro = Smicro : Benchmark for theories of quantum gravity! Supersymmetric Black Holes in String Theory – p.5 Black Hole Thermodynamics Hawking radiation: κS THawking = (GN = c = ~ = 1) : 2π First law (generalized version): κ δM = S δS + 2π · · · A S = + corrections from higher derivative terms 4 More ambitiously: derive black hole thermodynamics starting from a microscopic partition function. Example: OSV conjecture for supersymmetric black holes. Supersymmetric Black Holes in String Theory – p.5 The String – Black Hole Correspondence Idea: black hole microstates = string states at large mass or large coupling. Perturbative string regime Semiclassical gravity regime. $ pα0 r pα0 r . S S L. Susskind (1993), G.T. Horowitz and J. Polchinski (1997). Supersymmetric Black Holes in String Theory – p.6 The String – Black Hole Correspondence Idea: black hole microstates = string states at large mass or large coupling. Perturbative string regime Semiclassical gravity regime. $ pα0 r pα0 r . S S Compare 4d Schwarzschild black hole to open bosonic string (truncated). 2 0 Relation of gravitational and string scale: GN = gS α : String mass formula: α0M 2 n (for large excitation number n) : ≈ String entropy: S = log d(n) pn : String ∼ Schwarzschild radius of string state: r g2 pn pα0 : S ≈ S Supersymmetric Black Holes in String Theory – p.6 The String – Black Hole Correspondence Idea: black hole microstates = string states at large mass or large coupling. Perturbative string regime Semiclassical gravity regime. $ pα0 r pα0 r . S S Perturbative string in flat space: g2 pn 1 r pα0 ; S S S ) S BH String Semiclassical black hole: g2 pn 1 r pα0 ; S S : S ) S BH String Transition(?): g2 pn 1 r pα0 ; S S : S ≈ ) S ≈ BH ≈ String Supersymmetric Black Holes in String Theory – p.6 The String – Black Hole Correspondence Idea: black hole microstates = string states at large mass or large coupling. Perturbative string regime Semiclassical gravity regime. $ pα0 r pα0 r . S S Need to interpolate between two accessible regimes. Intermediate regime not under control. Matching of entropies up to (1): O S S : BH ∼ String Consider supersymmetric (BPS) states. A. Strominger and C. Vafa (1996), ... Interpolation more plausible. Can compute Smacro S and S S to high precision in their respective ≡ BH micro ≡ String regimes. Find ‘exact’ matching S S ; BH ≈ String including subleading corrections (in large mass = semiclassical expansion). Supersymmetric Black Holes in String Theory – p.6 The String – Black Hole Correspondence Idea: black hole microstates = string states at large mass or large coupling. Perturbative string regime Semiclassical gravity regime. $ pα0 r pα0 r . S S Besides fundamental strings, also solitonic p-branes are associated with microscopic degrees of freedom. E.g. the pioneering work of A. Strominger and C. Vafa involved D-branes, rather than fundamental strings. We will discuss examples involving fundamental (and also solitonic) strings later. Supersymmetric Black Holes in String Theory – p.6 The String – Black Hole Correspondence Idea: black hole microstates = string states at large mass or large coupling. Perturbative string regime Semiclassical gravity regime. $ pα0 r pα0 r . S S Beyond matching numbers: OSV conjecture 2 Z Ztop : BH ≈ j j ZBH = black hole partition function, Ztop = partition function of the topological string. H. Ooguri, A. Strominger and C. Vafa (2004) Supersymmetric Black Holes in String Theory – p.6 BPS states Supersymmetry algebra (4d, Weyl spinors): + µ Qα; Q = 2σ f β g αβ Supersymmetric Black Holes in String Theory – p.7 BPS states -extended supersymmetry algebra (4d, Weyl spinors): N QA; Q+B = 2σµ δAB f α β g αβ QA; QB = ZAB f α β g αβ A; B; : : : = 1; : : : ; N. Central charges = skew eigenvalues of ZAB: 2 2 2 M Z1 Z2 0 : ≥ j j ≥ j j ≥ · · · ≥ Saturation of inequalities shortened (BPS) multiplets. ) Supersymmetric Black Holes in String Theory – p.7 BPS states -extended supersymmetry algebra (4d, Weyl spinors): N QA; Q+B = 2σµ δAB f α β g αβ QA; QB = ZAB f α β g αβ A; B; : : : = 1; : : : ; N. Examples: = 2: N 1. M > Z : generic massive multiplet. j j 2. M = Z : short or 1 -BPS multiplet. j j 2 = 4: N 1. M > Z1 > Z2 : generic massive multiplet. j j j j 1 2. M = Z1 > Z2 : -BPS multiplet. j j j j 4 1 3. M = Z1 = Z2 : -BPS multiplet. j j j j 2 1 Supersymmetric vacua = ‘ 1 -BPS’ (fully supersymmetric). Supersymmetric Black Holes in String Theory – p.7 BPS solitons BPS states can be realized as finite energy solutions Φ0 of the field equations (asymptotic to vacuum). Killing spinors " residual (rigid) supersymmetry of Φ0: $ δ" Φ = 0 : jΦ0 Example: the extreme Reissner-Nordstrom black hole regarded as a solution of = 2 N Supergravity = Einstein-Maxwell + 2 Gravitini. Has 4 Killing spinors and interpolates between two supersymmetric vacua (8 Killing spinors): Minkowski space at infinity and AdS2 S2 (with covariantly constant gauge fields) at × horizon. G. Gibbons (1982) Supersymmetric Black Holes in String Theory – p.8 Embedding into string theory String compactifications give supergravity plus matter. We consider: 2 Het=(K3 T ) and type-II/CY3 × = 2 Supergravity + n vector multiplets ( + n hypermultiplets + n tensor −! N V H T multiplets ). Het=T 6 and type-II/(K3 T 2) × = 4 Supergravity + n vector multiplets. −! N V Main tool: specical geometry of = 2 vector multiplets. N B. de Wit and A. Van Proeyen (1984). All vector multiplet couplings are encoded in a holomorphic prepotential. Field equations are invariant under Sp(2nV + 2; ) rotations which generalize the electric-magnetic duality of Maxwell theory, and include stringy symmetries such as T-duality and S-duality. Supersymmetric Black Holes in String Theory – p.9 Special geometry (1) Multiplets: A i Gravity multiplet: e ; ; µ . f µ µ A g i A Vector multiplet: µ; λ ; z . fA g i = 1; 2, A = 1; : : : ; nV . Bosonic Lagrangian: −1 R A µ B i −I −Ijµν i +I +Ijµν 8πe bos = g (z; z)@µz @ z + IJ (z; z)F F IJ (z; z)F F L − 2 − AB 4 N µν − 4 N µν I = 0; 1; : : : ; nV . I Fµν = (anti-)selfdual part of field strength. To make electric-magnetic duality manifest, define dual field strength: δ G L : Ijµν ' δF Ijµν Supersymmetric Black Holes in String Theory – p.10 Special Geometry (2) Field equations (not: action) are invariant under Sp(2nV + 2; ) rotations. Symplectic vectors: I I I Fµν p F Gauge fields and charges: ; = 0 1 0 1 0 1 GIjµν qI H GI @ A @ A @ H A XI Scalars: 0 FI 1 @ A A where ‘scalars’ I are related to the physical scalars A by A X and X z z = X0 @F FI = : @XI Prepotential F (X) is holomorphic and homogenous of degree 2: F (λXI ) = λ2F (X) : Supersymmetric Black Holes in String Theory – p.11 Special Geometry (3) Poincaré Supergravity Conformal Supergravity ! nV vector multiplets (nV + 1) vector multiplets Φ 2nV +2 MV M ! CMV M −! XI zA XI 0 FI 1 @ A is a complex cone over .