BLACK HOLES IN 3D HIGHER SPIN GRAVITY. Gauge/Gravity Duality 2018, Würzburg

Transcription

1 BLACK HOLES IN 3D HIGHER SPIN GRAVITY Gauge/Gravity Duality 2018, Würzburg

2 What is a Black Hole?

3 What is a Black Hole? In General Relativity (and its cousins): Singularity

4 What is a Black Hole? In General Relativity (and its cousins): Causality: Horizon & Singularity Thermodynamics: Entropy Response: Quasi-normal modes Quantum Information: Excellent Scramblers

5 Not every theory of gravity has a classical geometrical description.

6 What is a Black Hole in Higher Spin Gravity? Causality: Horizon & Singularity Thermodynamics: Entropy Response: Quasi-normal modes Quantum Information: Excellent Scramblers Which of these features implies the rest? Are they always interconnected?

7 What is a Black Hole in Higher Spin Gravity? The goal is to give a definition that does not require a geometric description, and is fully compatible with the gauge symmetries of the theory.

8 What is a Black Hole in Higher Spin Gravity? Along the way, we will challenge the holographic dictionary, and challenge the intuition we exploit from general relativity.

9 1. HS Gravity & Chern-Simons Theory 2. Euclidean Black Holes 3. Extremal Black Holes Outline 4. Eternal Black Holes 5. Outlook

10 3d Higher Spin Gravity Chern-Simons formulation

11 3d Gravity In 2+1 dimensions, we have the luxury of casting general relativity in terms of: [Acucharro & Townsend; Witten] Einstein-Hilbert: Metric, curvature OR Chern-Simons: Gauge connections

12 3d Gravity In 2+1 dimensions, we have the luxury of casting general relativity in terms of: [Acucharro & Townsend; Witten] Einstein-Hilbert: Metric, curvature Local variables. Spacetime is explicit. OR Chern-Simons: Gauge connections Gauge Theory. Topological nature is explicit.

13 3d Gravity In 2+1 dimensions, we have the luxury of casting general relativity in terms of: [Acucharro & Townsend; Witten] Einstein-Hilbert: Metric, curvature Local variables. Spacetime is explicit. OR Chern-Simons: Gauge connections Gauge Theory. Topological nature is explicit. Inclusion of massless higher spin fields is straightforward!

14 I Higher Spin Theories Z Z How to interpret Chern-Simons theory as a theory of gravity? S CS [A] = k Z Tr A ^ da A ^ A ^ A M It is not just a matter of actions and equations of motion. Other important INPUTS are:

15 I Higher Spin Theories Z Z Z S CS [A] = k 4 Z Z How to interpret Chern-Simons theory as a theory of gravity? I Z M Tr A ^ da + 23 A ^ A ^ A It is not just a matter of actions and equations of motion. Z Other important INPUTS are: 1. Gauge Group: Organization of the massless modes A 2 G = G L G R SL(2, R) SL(2, R)

16 I Higher Spin Theories Z Z Z S CS [A] = k 4 Z Z How to interpret Chern-Simons theory as a theory of gravity? I Z M Tr A ^ da + 23 A ^ A ^ A It is not just a matter of actions and equations of motion. Z Other important INPUTS are: Z 1. Gauge Group: Organization of the massless modes A 2 G = G L G R 2. Boundary Conditions: Setup the AdS/CFT dictionary A A AdS = O(1) SL(2, R) SL(2, R)

17 Perturbative Aspects Asymptotic Symmetry Group and Ward Identities SL(2, R) SL(2, R) Vir Vir [Brown & Henneaux] SL(N,R) SL(N,R) W N W N [Campoleoni et al] hs[ ] hs[ ] W 1 [ ] W 1 [ ] [Henneaux & Rey; Gaberdiel & Hartman] With central charge: c = 3` 2G 3 =6k

18 Strategy q Work with a Chern-Simons formulation of higher spin theories. q Emphasis on SL(N), and SUSY cousins, i.e. a finite number of higher spin fields. q Exploit holography: comparison with dual W N theories when possible. q I ll never ever involve a metric in the subsequent definitions.

19 Euclidean Black Holes Thermal Properties

20 Characterizing Solutions Theory Solutions S CS [A] =S CS [A] S CS [Ā] SL(N) SL(N) A(, z, z) =b 1 ( ) a(z, z)+d b( ) Ā(, z, z) =b( ) ā(z, z)+d b 1 ( ) z CFT ρ AdS

21 Physical content in the connection: A(, z, z) =b 1 ( ) a(z, z)+d b( ) a(z, z) =a d + a te dt E z CFT ρ AdS [Gutperle & Kraus; de Boer & Jottar; Bunster et al]

22 Physical content in the connection: A(, z, z) =b 1 ( ) a(z, z)+d b( ) a(z, z) =a d VEV conserved charges + a te dt E Sources z CFT ρ AdS [Gutperle & Kraus; de Boer & Jottar; Bunster et al]

23 Physical content in the connection: A(, z, z) =b 1 ( ) a(z, z)+d b( ) a(z, z) =a d VEV conserved charges + a te dt E Sources From the perspective of CFT dual, I H = H CFT + d X s µ s J s + I d X s µ s Js z CFT ρ AdS [Gutperle & Kraus; de Boer & Jottar; Bunster et al]

24 A(, z, z) =b 1 ( ) a(z, z)+d b( ) a(z, z) =a d + a te dt E Consider N=3, the explicit form of the connection is 0 a L 2W L A Equations of motion + Boundary conditions L : W : vev spin-2 current (energy-momentum tensor) vev spin-3 current

25 A(, z, z) =b 1 ( ) a(z, z)+d b( ) a(z, z) =a d + a te dt E Consider N=3, the explicit form of the connection is [a te,a ]=0 ia te + a = µ L 2W 1 4 L L 2W L 1 A : µ : source spin-2 current (temperature) source spin-3 current z ' z +2 ' z +2

26 A(, z, z) =b 1 ( ) a(z, z)+d b( ) a(z, z) =a d + a te dt E Consider N=3, the explicit form of the connection is ia te + a = µ L 2W 1 4 L L 2W L 1 A z ' z +2 ' z +2 : µ : source spin-2 current (temperature) source spin-3 current Consistency Check: Chern-Simons eoms map to Ward identities of the CFT

27 Smoothness Conditions z ' z +2 ' z +2 z = + it E t E t E Connection should support the topology underneath! A(, z, z) =b 1 ( ) a(z, z)+d b( ) Ā(, z, z) =b( ) ā(z, z)+d b 1 ( ) [Gutperle & Kraus]

28 Smoothness Conditions Hol CE (A) = trivial C E [Gutperle & Kraus]

29 Smoothness Conditions Hol CE (A) = trivial I P exp C E a Use A = b(a + d)b 1 = e 2 ( a z+ a z ) = e 2 il 0 Diagonalize Eigen a z + a z = Eigen il 0 C E Elegant condition that only uses natural variables of Chern-Simons theory. [Gutperle & Kraus]

30 Thermodynamics Holonomy condition leads to thermodynamics! Eigen a z + a z = Eigen il 0 Solve (L,J s ) & µ s (L,J s ) [Gutperle & Kraus; de Boer & Jottar]

31 Thermodynamics Holonomy condition leads to thermodynamics! Eigen a z + a z = Eigen il 0 Solve (L,J s ) & µ s (L,J s ) Combined with the CS variational principle, one can show S = L L + S =2 ktr ( NX (µ s J s µ s Js ) s=3 )L0 Entropy of HSBH Eigen(a ) & Eigen(ā ) [Gutperle & Kraus; de Boer & Jottar]

32 What is an Euclidean Black Hole in Higher Spin Gravity? Causality: Horizon & Singularity Thermodynamics: Entropy Response: Quasi-normal modes Quantum Information: Excellent Scramblers Lesson: Euclidean regularity implies thermodynamic relations

33 Extremal Black Holes And their awkward SUSY features Based on with Bañados, Faraggi and Jottar

34 What defines an extremal black hole? From general relativity, we know Confluence of horizons: inner = outer Zero Hawking temperature Enhancement of symmetries: e.g. AdS 2 Saturation of cosmic censorship: M J BPS conditions (SUSY) Reality observables: e.g. Im(S)=0 Task for HS gravity: Extrapolate one of these conditions in a CS way. Explore how the rest is interconnected.

35 Physical content in the connection: A(, z, z) =b 1 ( ) a(z, z)+d b( ) a(z, z) =a d Charges + a te dt E Sources Simplistic view: Thermodynamics is an eigenvalue problem Eigen a z + a z = Eigen il 0 S =2 ktr ( )L0 : Holonomy condition : Entropy of HSBH Implicit assumption that connections are diagonalizable, which is true of independent values of the charges.

36 Extreme Limit: our proposal a(z, z) =a d Charges + a te dt E Sources Extremal = non-diagonalizable a φ

37 Extreme Limit: our proposal a(z, z) =a d Charges + a te dt E Sources Extremal = non-diagonalizable a φ Via Confluence of eigenvalues Eigen a z + a z = Eigen il 0 Extremal = zero temperature

38 What defines an extremal black hole? From general relativity, we know Confluence of horizons: inner = outer Zero Hawking temperature Enhancement of symmetries: e.g. AdS 2 Saturation of cosmic censorship: M J BPS conditions (SUSY) Reality observables: e.g. Im(S)=0

39 What defines an extremal black hole? For higher spin gravity, we have Confluence of eigenvalues Zero Hawking temperature Enhancement of symmetries: e.g. AdS 2 Saturation of cosmic censorship: M J BPS conditions (SUSY) Reality observables: e.g. Im(S)=0

40 What defines an extremal black hole? From higher spin gravity, we have Confluence of eigenvalues Zero Hawking temperature Enhancement of symmetries: e.g. AdS 2 Saturation of cosmic censorship: M J BPS conditions (SUSY) Reality observables: e.g. Im(S)=0 [Gutperle & Kraus; Henneaux, Perez, Tempo & Troncoso]

41 BPS Conditions Embedded in a supersymmetric version of CS, one can ask when a(z, z) =a d + a te dt E is compatible with SUSY and what are the appropriate BPS bounds (`M=Q ).

42 BPS Conditions Embedded in a supersymmetric version of CS, one can ask when a(z, z) =a d + a te dt E is compatible with SUSY and what are the appropriate BPS bounds (`M=Q ). We derived the bounds on both CFT and Gravity. They match in the large c limit.

43 BPS Conditions Embedded in a supersymmetric version of CS, one can ask when a(z, z) =a d + a te dt E is compatible with SUSY and what are the appropriate BPS bounds (`M=Q ). We derived the bounds on both CFT and Gravity. They match in the large c limit. We found that BPS bounds do not imply extremality. In particular, we can construct SUSY HSBH that are at finite temperature!

44 Eternal Black Holes Exploring bulk locality in HS gravity Based on with Ammon and Iqbal with Iqbal and Llabres with Iqbal and Llabres

45 How to capture casual properties in Higher Spin gravity? How to probe local bulk physics? Singularity Let s quantify this diagram in the CFT language.

46 Eternal HS Black Hole A possible definition: An eternal black is a thermo-field double state in the CFT. Whereas an Euclidean black holes satisfies: Highly entropic state. Hol CE (A) = trivial

47 An eternal black is a thermo-field double state in the CFT. i = 1 p Z X n e 2 (E n+µq n ) Uni L ni R Singularity 1. Signal of a bifurcation point. 2. Symmetric and periodic in Euclidean time. 3. Left-Right correlator should obey half-period relation.

48 Singularity Left Right We want: h O R (t f )O R (t i ) i = h O R (t f )O R (t i i ) i = h O L ( t f )O L ( t i ) i = h O L ( t f i /2)O R (t i ) i = h O R (t f )O L ( t i i /2) i Testing this proposal is difficult: non-trivial to have test particle. Important prior work made use of Vasiliev scalar field. [Kraus & Perlmutter]

49 Singularity Left Right What replaces the notion of distance in HS gravity? How to probe a solution with local fields?

50 Wilson lines in CS Wilson loop encodes the dynamics of a massive particle. Natural replacement of local probes. W R (x i,x f )=hu f P exp Z A P exp Z Ā U i i x i x f [Ammon, AC, & Iqbal; de Boer & Jottar]

51 Highlights W R (x i,x f )=hu f P exp Z A P exp Z Ā U i i q The representation R will dictate the characteristic of the probe: mass and spin. q Until now our observables do not connect A and Ā. Probing local physics requires BOTH connections. q The states U are coherent states in R that combine A and Ā while preserving a diagonal subgroup of SL(N)xSL(N).

52 Features 1. The Wilson line reproduces boundary correlation functions. W R (x i,x f ) =!1 h O(z i )O(z f ) i Connections to entanglement entropy and conformal blocks. 2. For HSBH, the Wilson loop gives thermal entropy. S =2 ktr[( )L0 ] = log(w R (C)) Closed spatial cycle (non-trivial)

53 Singularity Left Right We use W R (x i,x f ) =!1 h O(z i )O(z f ) i And answers depend on the radial function! A(, z, z) =b 1 ( ) a(z, z)+d b( ) Ā(, z, z) =b( ) ā(z, z)+d b 1 ( )

54 KMS conditions h O R (t f )O R (t i ) i = h O R (t f )O R (t i i ) i = h O L ( t f )O L ( t i ) i = h O L ( t f i /2)O R (t i ) i = h O R (t f )O L ( t i i /2) i 1. Wormhole gauge Commonly used by 99.9% of users. No signal of a bifurcation point. No KMS relations. Right side is AAdS. 2. Horizon gauge [Ammon, Gutperle, Kraus, Perlmutter] Radial function adjusted to give a horizon. KMS holds, not AAdS. 3. Kruskal gauge Different choices of radial function KMS holds and AAdS. It works!

55 Singularity Left Right W R (x i,x f ) =!1 h O(z i )O(z f ) i And answers depend on the radial function! A(, z, z) =b 1 ( ) a(z, z)+d b( ) Ā(, z, z) =b( ) ā(z, z)+d b 1 ( ) Lesson: HS Gravity provides a concrete setup where the bulk reconstruction and dual interpretation is not just dictated by obvious symmetries.

56 Outlook There is more to explore

57 Results I didn t discuss but nevertheless important 1. Phase diagram of higher spin black holes [David, Ferlaino & Kumar; Chen, Long & Wang; Bañados, Düring, Faraggi & Reyes] 2. Partition functions in the CFT [Kraus & Perlmutter; Gaberdiel, Hartman & Jin ] 3. Four point correlation functions [Perlmutter; de Boer, AC, Hijano, Jottar, Kraus]

58 1. CFT counterpart A derivation of the universal entropy S =2 ktr formula for W N theories. ( )L0 2. Interplay of Euclidean vs Lorentzian Inner horizons, singularities, wormholes,. Unexplored! IV I W R (x i,x f )=hu f P exp 3. Bulk Locality in Higher Spin Gravity Z A P exp Z Ā U i i

59 THANK YOU!