BPS Black holes in AdS and a magnetically induced quantum critical point. A. Gnecchi

Transcription

1 BPS Black holes in AdS and a magnetically induced quantum critical point A. Gnecchi June 20, 2017 ERICE ISSP

2 Outline Motivations Supersymmetric Black Holes Thermodynamics and Phase Transition Conclusions and Outlook A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 2

3 Black holes in Supergravity and String/M-Theory Black holes as a Quantum Gravity lab String Theory is a framework that provides a UV completion of a unified theory of GR and QFT AdS/CFT is a tool to investigate quantum gravity Two main areas of investigations 1. Supersymmetric black holes, microscopic derivation 2. Black holes in Anti de Sitter spacetime Fig. from McGreevy, 2009 A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 3

4 Holographic applications Black hole physics can teach about strongly coupled field theories Investigate quantum critical phases of strongly coupled solid state systems Hawking-Page transition for a black hole in anti de Sitter spacetime [ 83] Holographic interpretation as confinement/deconfinement phase transition Witten [ 98] Goal: construct an analytical example from black hole thermodynamics In the dual field theory, states can undergo phase transitions Study the phase space of the black hole solutions A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 4

5 Black holes and Supersymmetry [L. Romans, Nucl.Phys.B383(1992)] Einstein-Maxwell theory Equations of motions e 1 L = 1 4 R F mnf mn Λ G mn = T mn + g mn Λ, m (ef mn ) = 0. The black hole solution is ds 2 = Vdt 2 + dr 2 V + r 2 (dθ 2 + sin θ 2 dφ 2 ) F = Q dt dr + H sin θdθ dφ r 2 with warp factor V (r) = 1 2M r + Z 2 r 2 Λ 3 r 2 A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 5

6 Black holes and Supersymmetry The solution is a supersymmetric solution of N=2 minimal gauged Supergravity e 1 L = 1 4 R+1 2 ψ m γ mnp D n ψ p F mnf mn + Gravitini are minimally coupled SUSY equations + i 8 F mn ψp γ [m γ pq γ n] ψ q 1 2 g ψ m γ mn ψ n 3 2 g 2 D m = m iga m δe a m = Re( ɛγ a ψ m ), δψ m = ˆ m ɛ, δa m = Im( ɛψ m ) Electric solutions preserve 1 2 of the original supersymmetry Magnetic solutions preserve 1 4 of the original supersymmetry, but all have naked singularities at r = 0 A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 6

7 BPS black holes: scalar fields to the rescue Bosonic sector of N = 2 Supergravity Lagrangian S = BPS attractors: - Ungauged Supergravity ( d 4 x R 2 + g i j µz i µ z j + ImN ΛΣ FµνF Λ Λ µν ) 2 g ReN ΛΣɛ µνρσ FµνF Λ ρσ V Σ g - Gauged Supergravity AdS 2 S 2 R 1,3 i Z(q, p, z i, z i ) hor = 0 [Ferrara, Gibbons, Kallosh, Strominger, 95-96] Z Q, V = L Λ q Λ M Λ p Λ V BH = DZ 2 + Z 2 L = G, V = L Λ g Λ M Λ g Λ V g = g i j D i LD j L 3 L 2 Z i L (q, p, zi, z i ) = 0 hor [Dall Agata,A.G., 11] A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 7

8 Thermodynamic ensemble Euclidean path integral formulation of gravity at the semiclassical: [Gibbons, Hawking 76, York, 86] Z = d[g µν ]d[φ] exp{ii e [g µν, φ]}. For a system like black holes and black branes, exhibiting a thermodynamic behaviour, the partition function defines a free energy, which, within a saddle point approximation, corresponds to the Euclidean on-shell action with β = T 1. βf = ln Z = ii e [g, φ ], What are the thermodynamic variables? A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 8

9 Thermodynamic ensemble First law dm = TdS + χ Λ dq Λ + φ Λ dp Λ Electric configuration. For a system with n charges q Λ (Λ = 1,.., n) F (T, χ) = M TS q Λ χ Λ Magnetic configuration. For a system with n charges p Λ (Λ = 1,.., n) F (T, p Λ ) = M TS Adding boundary terms on the action change the boundary conditions: Légendre transformations, change the ensemble. [Hawking, Ross, 95] In Supergravity, that corresponds to an electric-magnetic duality rotation A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 9

10 Black brane in AdS 4 [Klemm, Vaughan, Hristov, Toldo, Vandoren, 10-12] Specify a theory of N = 2 Supergravity with gauging: ( gd L = 4 x R ) 2 µϕ µ ϕ + I ΛΣ FµνF Λ Σ µν V g (φ). with fixed ( ) V g (φ) = 3 2 cosh l 2 AdS 3 ϕ(r), l 2 AdS = 1 g 2. The black brane solutions have metric ( ) ds 2 f (r) dr 2 = H0(r)H1 3(r) dt2 + H 0(r)H 31 (r) f (r) + r 2 (dx 2 + dy 2 ) where H 0(r) = 1 3b, H 1(r) = 1 + b r r the other bosonic fields are F 0 =, f (r) = c1 r + c2 r 2 + r 2 H 0(r)H 1(r) 3, q 2(r 3b) dr dt, F 1 = B 2 2 dx dy, e 8/3ϕ = r + b r 3b. A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 10

11 Good singularity There exists a competing solution in phase space Black brane limit in which the horizon coincides with the singularity: good singularity [Gubser,2000]. g tt(r h ) = 0, as r h r s gxx = g yy = (r 3b)(r + b) 3 Family of black branes with horizon r h = 3b + ɛ For epsilon ɛ 1 g tt(3b + ɛ) = 0 B = 8 2b 2 + (6b2 + χ 2 )ɛ 2b + O(ɛ 2 ). A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 11

12 Phase transition What solution dominates? F = F BB F TG = 27B2 + 32χ B 2. 6 χ There is a second order phase transition at B c = χ2, between a gapless (black brane) and a gapped phase (thermal gas). F DeltaF p p1 No confinement-deconfinement phase transition, however. A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 12

13 Conclusions and Outlook Static supersymmetric black holes in AdS 4 have been constructed and analyzed Black hole attractors for gauged Supergravity have an interesting interpretation in the dual field theory Thermodynamics and electric magnetic duality are interconnected and reveal a nontrivial phase space A quantum critical point emerges from the study of good singularities To do: Attractors for rotating black holes in Anti de Sitter Study phase space at finite temperature Supersymmetry and holographic renormalization Uplift to String/M-theory for generic gaugings to resolve good singularity A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 13

14 Thank you! A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 14

15 BPS black holes in AdS 4 Solutions of N = 2 U(1)-gauged Supergravity, a truncation of SO(8)-gauged N = 8 supergravity [de Wit, Nikolai, 82, Duff & Liu, 99]. Interpolating geometry and scalar fields with nontrivial radial profile ds 2 = e 2U(r) dt 2 + e 2U(r) (dr 2 + e 2ψ(r) dω 2 ) [Cacciatori, Klemm, 09] Two main features wrt asymptotically flat solutions Magnetically charged p Λ g Λ = κ Dirac quantization condition between gravitino and black hole charges Preserve a smaller amount of Supersymmetry: 1 -BPS solutions 4 [A.G., Dall Agata - Hristov, Vandoren 11] They can be obtained as a compactification of M-theory on S 7, and asymptote to AdS 4 S 7. Charges and fluxes are then interpreted as M-branes constituents, they corresponds to BPS states in the dual ABJM theory. [Aharony at al. 08] A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 15

16 BPS black holes in AdS 4 : attractors ungauged Supergravity to gauged Supergravity AdS 2 S 2 R 1,3 i Z(q, p, z i, z i ) hor = 0 AdS 2 S 2 AdS 4 i W hor = 0 i L hor = 0 Translates radial equations in algebraic relations that capture the physics of the (dual) theory A new quantity plays the role of superpotential W = e U Z ie 2(ψ U) L New attractor equations i W h = 0, W h = 0 e 2A = i Z L = R2 S [A.G&G. Dall Agata, 11] Holographically, this quantity corresponds to a field theory partition function in the dual ABJM theory to N=8 Supergravity. Microscopic counting of black hole states in AdS 4 by Zaffaroni, Hristov, Benini by compactifying ABJM on S 2 S 1 A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 16

17 Phase transition Black brane F bb = M bb TS bb + q bb χ bb, M is the mass of the black brane, M = B2 q 2, 4b Thermodynamic potential df bb = S bb dt + q bb dχ bb + m bb db Thermal gas b TG = B. The free energy for the thermal gas is a function of B only, at any temperature F TG = M TG = B2 4b TG = B 3 2, df TG = m TG db The magnetization is qualitatively different for BB and TG m bb = F bb 3 B B = 3 χ,t 2 χ, m TG = B sgn(b). A. Gnecchi June 20, 2017 ERICE ISSP Black holes in AdS and holography 17