Some Comments on Kerr/CFT

Transcription

1 Some Comments on Kerr/CFT and beyond Finn Larsen Michigan Center for Theoretical Physics Penn State University, September 10, 2010

2 Outline The Big Picture: extremal limit of general black holes. Microscopics of Kerr/CFT: proposal for precision counting. AdS 2 Quantum Gravity: Kerr/CFT as the theory of (some) diffeomorphisms. Some references: A. Castro and FL, arxiv: A. Castro, C. Keeler, and FL, arxiv: K. Hanaki and FL (in progress). 2

3 Extremal Black Holes: Overview Setting: Black holes in D = 4 SUGRA, single center, asymptotically flat, N = 8, 4, 2 theory. Parameters (M, J, Q I, P I ). Extremal limit: T H = 0. There is an AdS 2 factor in geometry. Distinguish 3 types of extremal Black Holes: i) BPS. ii) non-bps extremal. iii) extremal Kerr. 3

4 Example 1 Theory: M theory on CY S 1 Black hole: M5 on P S 1 (P a divisor) with P 3 0 ( 3 charges ) Momentum = n along S 1 (so 4 charges total) Black hole entropy: S = 2π n P 3 Two extremal limits: BPS: n > 0. The fourth charge break same SUSYs as P. non-bps: n < 0. The fourth charge break opposite SUSYs so none are left. 4

5 Example 2: Kerr-Newman Theory: Einstein-Maxwell, diagonal charges, Q 4 np 3 Black hole: Kerr-Newman Black hole entropy (G 4 = 1): S = 2π [(M 2 12 Q2 ) + ] M 2 (M 2 Q 2 ) J 2 = S L + S R BPS: M 2 = Q 2, J = 0, S = 2π 1 2 Q2 Extremal Kerr: Q 2 = 0, M 4 = J 2, S = 2π J BPS and Kerr both correspond to R in a specific state, L carries entropy 5

6 The Big Picture The general black hole with parameters: (M, J, Q I, P I ) is described by a 2D CFT with L and R chiralities that interact weakly: S = S L + S R, β H = 1 2 (β L + β R ) R-movers have the ability to carry J. BPS: T R 0 (with J = 0). R-movers in ground state, J-carryers not excited. L-movers carry entropy. Extremal Kerr: T R 0 (with J 0). R-movers in a definite state, with J-carryers excited. L-movers carry entropy. General Black Hole: direct product of BPS (L) and non-bps theory (R), with level matching relating the two chiralities. 6

7 Precision Counting: Extremal Kerr Working assumption: the entropy of extremal Kerr comes from the same states (L-movers in our convention) as the BPS entropy. The difference: the R-movers, for Kerr in a state that breaks SUSY spontaneously, instead of the SUSY preserving ground state. Strategy for precision counting: consider the CFT underlying the BPS counting. Keep the dynamical chirality (holomorphic=l-movers) intact, but modify the inert chirality (anti-holomorphic=r-movers) by spectral flow. 7

8 Setting The D1/D5 on K3 S 1, described by the sigma-model on M k /Σ k with M = K3, k = n 1 n The central charge is c = 6k. Excitations at level h = p give asymptotic degeneracy ch S = 2π 6 = 2π kp The 4D version of model involves adding a KK-monopole: basic reasoning remains, but some details change. 8

9 5D Counting Vertex operators: V(z, z) = V L int(z)e if Lϕ L (z)/ 2k V R int( z)e if Rϕ R ( z)/ 2k Bosonized the R-currents: J = 2k ϕ L, J = 2k ϕr Spacetime angular momentum: F R,L = 2j R,L Conformal weights: h R = h int R + 1 4k F R 2 h L = h int L + 1 4k F L 2 Momentum of the state: p = h L h R 9

10 Extremal limit: h int R = 0, h R = 1 4k F 2 R (extremal) Origin of entropy: freedom in Vint L (z) with weight h int L = h L 1 4k F L 2 = p + h R 1 4k F L 2 = p + 1 4k F R 2 1 4k F L 2 The black hole entropy: S = 2π ch 6 = 2π kp + j 2 R j2 L BMPV black hole: special case j R = 0. Extremal 5D Kerr: j L = p = 0 with entropy S = 2π j R 10

11 Comments The 4D version of computation: add KK-monopole SUSY broken in L-sector there is no j L. But j R identified with 4D angular momentum. Uncharged case: n 1 = n 5 = 0 k = 1 a singular limit so computation not justified. Answer analysis (inconclusive): for p = 0 the level k cancels so entropy would work out for Kerr no matter what central charge is claimed. 11

12 Precision Counting The partition function in the RR sector (with ( ) F inserted): Z(τ, z, τ, z) = Tr[( ) F y F L q L 0 c L 24ȳFR q L 0 c R 24 ] The elliptic genus: take z = 0 Z(τ, τ, z, 0) is independent of τ. Combine with spectral flow to map out the dependence on one anti-holomorphic parameter: Z(τ, z, τ, l τ + m) = q 1 k j2 R holomorphic 12

13 Master Partition Function Precision counting is best summarized in terms of an ensemble with arbitrary number (m) strings: a sum over CFTs, with level k = m full (4D) partition function in sector with angular momentum j R : Z jr (τ, z, σ) = (1 q l q m 1 j2 R p m y j ) c b(4lm j 2 ) qy b=0 l,m,j Asymptotic behavior of the entropy S 2π j 2 R + q2 p 2 ( p q) 2 for any charges ( q, p) where the argument of the square root is positive. Clearly: partition function encodes many subleading corrections which we are working out. 13

14 The modular behavior is confusing (and perhaps not right). This work is still in progress 14

15 Extreme Kerr Quantum Geometry Change focus: determine features of the quantum theory from study of the spacetime geometry. (Near) extreme Kerr: warped AdS 2 geometry. Goal: analyze 2D diffeomorphisms with AdS 2 boundary conditions. Result: compute c R = 12J from strictly 2D point of view. Establish consistency with 3D Brown-Henneaux-Strominger treatment of the BTZ black hole gain confidence in results. Interpretation in Kerr setting: AdS 2 quantum gravity is theory is excitations above extremality. 15

16 NHEK Geometry Near extremal limit directly in Kerr geometry (including excitation energy): ds 2 = 1 + cos2 θ [ U 2 ɛ 2 dt 2 l 2 ] + 2 l 2 U 2 ɛ 2dU 2 + dθ 2 +l 2 2 sin2 θ 1 + cos 2 θ (dφ+ U l 2dt)2 This is a warped AdS 2. 2D Black Hole: ds 2 2 = U 2 ɛ 2 l 2 dt 2 + l 2 U 2 ɛ 2dU 2 + dθ 2. B t = U l 2dt In classical theory: diffeomorphic to AdS 2. In quantum theory: an excited sector of AdS 2 quantum gravity. 16

17 2D Effective Theory Effective 2D theory = gravity+u(1) gauge theory S Kerr = l2 4G 4 [R (2) + ] 1l l2 2 2 G µνg µν Notation: appropriate for reduction from 4D Kerr (G 4 = 4πG 2 l 2 is more neutral). The scalar (ψ) is taken to vanish (by asymptotic e.o.m.) Boundary terms guarantee well defined variational principle S bndy = l2 2G 4 dt γ [K 12l + l2 B ab a ] 17

18 Boundary Conditions General 2D spacetime (in convenient gauge) ds 2 2 = dρ 2 + h tt dt 2, Asymptotically AdS 2 boundary conditions B = B t dt h tt = h (0) e 2ρ/l + h (2) + h (4) e 2ρ/l +... B t = B (0) e ρ/l + B (2) + B (4) +... satisfying the constraint: h (0) + l 2 B (0)2 = 0. Diffeomorphism+gauge transformations preserving the gauge conditions ɛ t = ξ(t) l2 2h (0) 2 t ξe 2ρ/l + O(e 4ρ/l ) ɛ ρ = l t ξ(t) Λ = l h (0) 2 t ξ(b (0) e ρ/l B(2) e 2ρ/l ) + O(e 3ρ/l ) 18

19 2D Effective Theory Key computation in standard (higher dim) AdS/CFT: the energy momentum tensor, and other linear response functions. The linear response functions of the 1D boundary theory: T ab = 2 δs l2 = ( 1 γ δγab 4G 4 l γ tt + lbt 2 ) J t = 1 δs γ tt = l3 ( n ν G µt + B t ) γ δb t 2G 4 19

20 Generator of Diffeomorphisms Variation under a diffeomorphism: δ ɛ S = 1 dt γt ab δ ɛ γ ab + 2 dt γj a δ ɛ B a so generator of diffeomorphisms is not just em-tensor: Q ɛ = γγ tt (T tt + J t B t ) Second term not suppressed in 2D (gauge field linearly rising). Expression gives correct notion of local energy T tt + B t J t = l 4G Boundary energy (including measure factors) vanishes in strict extremal limit, as we expect for R-excitations; but scaling limit gives finite notion of excitation energy. ɛ 2 l 2 20

21 Transformation to AdS 2 The AdS 2 black hole is related to AdS 2 in Poincare coordinates ds 2 = l2 y 2(dy2 dw 2 ), through the coordinate transformation y = lɛ U 2 ɛ e ɛt/l2 lu w = U 2 ɛ e ɛt/l2 Λ = 1 ( ) U + ɛ 2 ln U ɛ B = l y dw Comment: coordinate transformation singular at U = ɛ 21

22 More important comment: near U the coordinate change is w = le ɛt/l2, corresponding to Rindler space with T = ɛ 2πl 2. The generator of diffeomorphisms realizes the Virasoso algebra T tt + B t J t = c 12 l{w, t}( wt) 2 = π2 6 clt 2 Comparing with the explicit computation for the AdS 2 black hole, we infer the central charge c = 6l2 G 4 For Kerr G 4 = 2l 2 J c Kerr = 12J. Note: derivation only makes reference to 2D. 22

23 The 3D lift: local part AdS 2 results are consistent with standard results in 3D. Embedding of AdS 2 into AdS 3 : ds 2 3 = l 2 (dθ + B t dt) 2 + dρ 2 + h tt dt 2. The 2D diff+gauge transformations that preserve AdS 2 boundary conditions is a genuine subgroup of diff s that preserve AdS 3 boundary conditions. The 2D Virasoro is identified with the SL(2, R) R of the 3D theory. The central charges match precisely, and the normalizations of energy match. 23

24 The 3D lift: global part The 3D lightcone coordinates w ± = φ ± t are identified as w = r + + r t 8l 2 w + 2l = θ r + r t r + + r 8l 2 Note: boundary conditions are not mapped correctly (e.g. w is periodic, t is not). Remedy: the DLCQ limit w λ 1 w, e 2η λe 2η, r + r = 4λɛ After DLCQ, the boundary conditions work out and the solution is tuned so the (infinitesimal) excitation energy is finite w = t 2l, w+ = θ 2 ɛ 2l 2t 24

25 Isometries The 2π identification on the azimuthal coordinate φ correspond to identifications on the lightcone coordinates w ±. The identification on w + is a hyperbolic holonomy that breaks SL(2) L U(1) L, as it does for BTZ. After the DLCQ limit there is no identifications on w t so SL(2) R is preserved. This group is the isometry group AdS 2. The broken U(1) L SL(2) L is enhanced to a Virasoro algebra, that acts on the excitations. 25

26 Phenomenology of Kerr Entropy Take near extremal limit of general Kerr entropy: S = 2π (M 2 + ) M 4 J 2 2π Central charges: c L = c R = 12J Level (energy) of right movers: h R = ɛ2 λ 2 2l 2 P ( c L 12 +, ɛλ 1 2 (r + r ) cr h R 6 ) Original Kerr/CFT: extremal Kerr = ground state of chiral (L only) CFT Here: focus on the R-sector, c R and excitation energy. 26

27 Kerr/CFT from AdS 2 The central charge is reproduced from AdS 2 quantum gravity. The phenomenological energy has the right form to match a CFT we we need to account for its value E = π2 6 ct 2 HR The conformal weight is finite after the DLCQ limit h = ER λ = c 24 2ɛ2 l 2 First attempt on the coefficient: infer the value R = 2l from our computations. But this value is arbitrary in the CFT (C is for conformal). 27

28 Better: instead we can match the scale invariant ratio: This works out quantitatively! E Rλ = π2 6 ct 2 = T tt + B t J t l After the DLCQ rescaling h c in the black hole regime so Cardy s formula applies. 28

29 Summary Aspects of Kerr/CFT: Relatively firm understanding of origin as a subgroup of diffeomorphism invariance. Beginnings of precision counting. Hints of a structure that generalizes to all black holes. 29