Charged Spinning Black Holes & Aspects Kerr/CFT Correspondence

Transcription

1 Charged Spinning Black Holes & Aspects Kerr/CFT Correspondence

2 I. Black Holes in Supergravities w/ Maximal Supersymmetry (Review) Asymptotically Minkowski (ungauged SG) & anti-desitter space-time (gauged SG) a) Focus on Abelian U(1) charged solutions in D=4 and D=5 & spinning solutions w/ [(D-1)/2]-angular momenta b) Highlight Supersymmetric (BPS) and other Extreme black hole limits II. Aspects of Microscopics: a) Novel microscopic interpretation - ``Kerr/CFT correspondence'' Guica,Hartman,Song,Strominger, arxiv: for entropy of extreme charged spinning black holes in D-dimensions...D.D.K. Chow, M.C., H.Lu & C.N.Pope,arXiv: b) Greybody factors and microscopics:near-& non-extreme charged spinning black holes M.C., Larsen & work in progress w/ F. Larsen; work in progress w/ T. Birkandan

3 Multi-charged black holes in D=4 ungauged supergravities Prototype black hole solutions of N=4 [N=8] supersymmetric ungauged SG in D=4 can be obtained as a toroidal reduction of Heterotic String on T 6 ; D=4 N=4 SG w/ global symmetry O(6,22) x SL(2,R) (T & S-duality) Type IIA String on T 6 ; D=4 N=8 SG w/global symmetry E 7 (U-duality) The relevant subsector generating solution can be obtained from the NS-NS sector of string theory, &T 2 -O(2,2)x SL(2,R)-subsector Four: U(1) 1 (1) x U(1) 1 (2) x U(1) 2 (1) x U(1) 2 (2) [also viewed as D=4 N=2 SG coupled to three vector supermultiplets] Black hole solution w/ all charges turned: obtained by acting w/ generators of global symmetries

4 Generating BPS BLACK HOLE solution of D=4 N=4 SG: w/youm hep-th/ Metric Ansatz: Gauge fields: Supersymmetric (BPS) solution ( = = = 0 ) ; U(1) 4 :

5 ADM Mass: Entropy Finite: Temperature: T H = 0 ( extreme double zero of the horizon) Special cases: All four charges equal: Pairwise equal charges Extreme (BPS) Reissner-Nordstrőm BH (As BPS solution of SG first shown by Gibbons&Hull'84) Dyonic black holes of Kallosh et al.'93 One (a= 3), two (a=1),three (a=1/ 3) equal non-zero charges Dilatonic black holes (c.f.gibbons& Wiltshire'90,Townsend'96; Duff&Rahmfeld96..)

6 w/ Four-charge generating solution: generators of the non-compact N=4 (& N=8) symmetries cannot populate all the charges Need five charge BPS generating solution: w/tseytlin hep-th/ S=

7 Special properties of BPS black-holes: a) Entropy expressed in terms of combination of quartic charge invariant under the non-compact N=4 symmetry O(6,22)x Sl(2,R): w/tseytlin hep-th/ S = = SL(2,R) inv. matrix O(6.22) inv. matrix 28 electirc & 28 magnetic charges N=8 SG The entropy: E 7 quartic invariant-unique: Kol & Kalosh'96 (generalizations to other dimension Ferrara&Kallosh'06, w/hull'96) b) Entropy INDEPENDENT of the values of scalar fields explicitly first demonstrated w/tseytlin hep-th/ [in a general context: attractor mechanism of Ferrara&Kalosh'06...]

8 General Multi-Charged Spinning Black Holes (of D=4 SG) Solution generating technique (c.f., Gibbons, Sen) a) Starting with an uncharged four-dimensional solution that has a timelike Killing vector / t; Kerr black hole solution b) Reducing it to three dimensions on t direction. Yields a three-dimensional theory with an addiitional O(4, 4) global symmetry c) By acting with an O(1, 1)4 subgroup of O(4, 4) on the dimensionally reduced solution generate new solutions involving four parameters δ i characterising O(1, 1)4. d) Upon lifting back to D = 4, we there by arrive at spinning solutions carrying 4 electromagnetic charges, parameterised by the δ i. w/youm hep-th/ also: w/lu& Pope: hep-th/

9 Starting point, D=4 Kerr black hole: m-mass, l-angular momentum

10 Impose 4 O(1,1) boosts on D=3 solution, lift back to D=4: Four charge spinning black hole:

11 D=4 four-charge spinning black hole- Charges, Mass & Entropy Charges and angular momentum: G N = ⅛ ADM mass: Entropy: w/youm hep-th Suggestive of microscopic structure in terms of 2D CFT with S L +S R -more later BPS-limit (one extreme limit): m 0, δ i, J 0 Entropy: S = Other (non-bps) extreme limits when: m=l (analog of extreme Kerr limit) Entropy: S =

12 D=5 black holes in ungauged supergravities Black holes of N=4 (N=8) supersymmetric ungauged SG in D=5 as a toroidal reduction of Heterotic String on T 5 ; D=5 N=4 SG w/ global symmetry O(5,21) xo(1,1) Type IIA String on T 5 ; D=5 N=8 SG w/ global symmetry E 6

13 Toroidal reduction of Heterotic String (Type IIA String) on T (10-D) (D=5). Former D=5, N=4 SG, w/ global symmetry O(5,21) xo(1,1). The relevant subsector to obtain generating solution can also viewed as D=5 N=2 SG coupled to vector super-multiplets: two scalar fields; three U(1)

14 Employing generating technique: a) Reduce D=5 stationary solution-kerr BH with two angular momenta to D=3 on t and one angular direction b) D=3 Largrangian (relevant subsector) has O(3,3) symmetry c) Acting with an O(1, 1)3 subgroup of O(3, 3) transformations on t the dimensionally reduced solution to generate generate new solutions with three parameters δ i d) Upon lifting back to D = 5, arrive at spinning solutions with two angular momenta & three charges parameterised by the three δ i w/youm hep-th/

15 D=5 Kerr Solution: m-mass; l 12 =two angular momenta Myers&Perry'86

16 Metric:

17 Scalar and gauge fields:

18 Somewhat more compact form: w/chong, Lu & Pope: hep-th/ Metric: {l 1,l 2 } {a,b}, r 2 x

19 Solution specified by mass, three charges and two angular momenta: Entropy: w/ Youm hep-th/ Suggestive form in terms of 2D CFT with S L +S R G N =¼ Extreme (BPS) limit: m 0, δ i, J Φ = -J Ψ = J S=π Microscopics: Breckenridge et al.'96 Near-BPS-dilute gas: S= δ 12, but third boost finite: Microscopics:w/Larsen hep-th/ (spinning D=6 string: BTZ (AdS 3 ) x S3) Other Extreme limits: = 0 (analogs of Kerr extreme limit)

20 Black Holes in toroidally compactified string theory in D>5: Generating solution parameterized by 2 charges and [(D-1)/2] angular momenta Employing the same generating technique, these metrics obtained explicitly; w/youm hep-th/ Thermodynamics, i.e. entropy obtained only implicitly In terms of r + - radius of the horizon. Microscopics studied in near-extreme limit: Q 12 >> m

21 Tremendous progress: a) Asymptotically Minkowski Supergravity BH-type Solutions: higher order gravity corrections (Cardoso, et al., Sen,...); multi-centered solution (Denef,..); rings(emparan&reall,...)... N=2 Black Holes (mainly BPS)... c.f., Brossard's talks; Mohaupt's talk; Denef's talk; Warner's talk; de Wit's talk... b) Detailed microscopics ; primary focus on BPS Topological field theory/cft (Dijkgraaf, Ooguri,Vafa..; Sen..;..) c.f,larsen's talk... Microstate geometry (Mathur;...) c.f., de Boer's talk; Denef's talk; Warner's talk, Mathur's talk......

22 (B) Black holes in asymptotically AdS space-time (gauged SG) (general background for addressing AdS/CFT correspondence) Type IIB compactified on five-sphere: a consistent truncation w/ SO(6) gauge symmetry and scalars in symmetric, unimodular rep. of SO(6) w/lu,pope&tran'98 Consistent truncation to Abelian gauge fields U(1) 3 :

23 A consistent truncation of N=8 gauged supergravity to N=2 gauged supergravity coupled to 3 vector supermultiplets For obtaining proto-type: BH's in asymptotic AdS space-time in D=5 3 U(1) gauge fields Potential for scalar fields

24 Static Solution w/ three charges: w/behrndt &Sabra, hep-th/ μ=0 extreme (BPS) limit (singular) Behrndt,Chamseddine&Sabra hep-th/ Global space-time involved, it depends on the size of g-cosmological constant analysis for ``large'' BH's with regular horisons generalization of Hawking-Page phase transitions for charged AdS BH's w/gubser hep-th/ (related, Emparan et al. '99) Extensive further studies of phase diagrams, c.f., Minwalla et al., Microscopics??

25 General D=5 AdS spinning charged black hole solutions Important role as AdS/CFT. Prototype: cosmol. const. ; mass, 3 charges & 2 ang. mom. ; (D=4 boundary CFT ; scaling dim., 3 R charges & spins of dual FT operators) Prototype solution parameterized with: mass, 2-angular momenta, three independent charges not obtained yet! Problem: for gauged supergravities (typically obtained from sphere compactification of effective string theories) - non-compact symmetries (there for toroidal compactifications) are absent No generating techniques to obtain charged solutions from uncharged ones Only special cases obtained with incremental progress with two angular momenta:equal w/lu&pope hep-th/ &educated Ansatz, matches onto ungauged examples & known BPS limits

26 Status of charged AdS BH's w/ two unequal angular momenta : 3-equal charges w/chong, Lu & Pope hep-th/ equal charges hep-th/ charge hep-th/ equal charges 3 rd unequal Mei&Pope General supersymmetric black holes Kunduri, Lucietti, Reall '06 Involved global space-time & thermodynamics: analysis of the global space time, thermodynamics, new BPS BH limits & BPS solitons (regular solutions) w/ Gary Gibbons, H. Lu and C. Pope, hep-th/ [D=4, with 2 pairwise equal charges; D=7, 2-charges, with 3 equal ang. mom. w/z. Chong, H. Lu & C. Pope, hep-th/ ; D= 6, one charge, 2 unequal ang. mom, D. Chow arxiv: ]

27 Metric: ( 2 (co-homegeneity D=5, 3-equal charge, 2 ang. mom. AdS BH: hep-th/ Gauge pot.: Mass: 2 ang. mom.: 3-equal charges:

28 Other extreme limits Thermodynamics: T H 2π Supersymmetric (BPS) limit: Regular Solution (no naked CTC); T H =0

29 Spin off: (digression) New Einstrein-Sasaki from Kerr-deSitter Special case of D=5 neutral rotating AdS BH s & Hunter, Hawking & Taylor 98 finding BPS (scaling) limit ( l 1 = l 2 ) w/gao& Simón hep-th/ Euclidean regime BPS Kerr AdS BH s New Einstein-Sasaki spaces La,b,c ( l 1 l 2 ) local &gobal analysis (regularity), topology /dual FT w/ Lü, Page & Pope hep-th/ ; / (independently Sparks&Martelli,hep-th/ )

30 II. A: Microscopics of extreme spinning black holes (defined by T H =0) in D-dimensions The near horizon metric of extreme spinning solutions as a manifold M fibration over AdS 2: [(D-1)/2] commuting U(1) diffeomorphisms: That generate [(D-1)/2] commuting Virasoro algebra w/ charges = which determine (at Dirac backet level) Virasoro algebra Barnich,Brandt'01 And thus the central charge c i is obtained from: h μν -metric deviation

31 Volume of manifold M Frolov-Thorne Temperature (defined at the horizon in coordinates w/angular velocity Ω H ) Precisely reproduces black hole entropy (via Cardy's formula) For each: i=1,..[(d-1)/2]!

32 First calculated explicitly for a specific case of extreme Kerr BH's in D=4, Guica,Hartman,Song,Strominger and for neutral spinning BH's in D=5,6 Lu,Mei,Pope The general argument of the previous page given in w/chow,lu,pope & shown to be true for most known charged spinning BH's in D 4! ' ' ' '

33 IIB Microscopics of general spinning charged black holes (asympt. Minkowski, only) near- (Kerr)-extreme limit; w/larsen also non-extreme w/larsen work in progress Reminder, tantalizing thermodynamics Chose D=5 three charge two ang. mom. BH (analogous results for D=4 four charge one ang. mom.-also worked out): Description in terms of 2D CFT w/ L and R sector interacting weakly: Not only entropy: Also (inverse) Hawking temperature:

34 Evidence via probes: Scalar Wave Equation w/larsen'97, new insights Equation separable: Radial Equation special structure: Eigenvalue of angular laplacian Denote: =

35 Wave Function: Sequential explicit solution in all near-extreme cases, including In addition, allows for SUPER-RADIANT LIMIT: (c.f. D=4 Kerr, Bredberg,Hartman,Song&Stromginer )

36 Microscopics in terms of 2D CFT: /2 Reproduces emission amplitude from 2D CFT operators: Comparison with SG solution-quantitative & derive 2D CFT length scale:

37 Non-extreme solutions at probe level: Wave equations in the near horizon region SL(2,R) x SL(2,R) symmetry!. w/larsen hep-th/ & work in progress (For d=4 Kerr c.f. Castro, Maloney &Strominger ) The metric w/ manifest SL(2,R) x SL (2,R) ; specific subtractions in ``harmonic functions'': Asymptotic metric: r 2 >>y: Justify a subtraction and analyse the boundary space-time: Expectation related to AdS 3 (BTZ) ---> 2D CFT

38 Microscopics of AdS charged spinning BH's? w/ T. Birkandan, work in progress Testing in probe approximation wave equation separable for all known cases Study of Green's functions complex... Preliminary results for Extreme Spinning AdS Black holes: Explicit retarded Green's Function for D=5 extreme spinning three-equal charge AdS BH in terms of two Frolov Temperatures T 1 and T 2! Interpretation and possible generalizations to near- (non)-extreme cases

39 Black Holes of (gauged) supergravity Rich structure and important implications for Gravity/Field Theory Duality Focus on some aspects of prototype charged spinning BH's & Some intriguing progress on microscopics for Near-extreme (non-bps) and non-extreme Minkowski BH's Hopefully more to come!