What is a chiral 2d CFT? And what does it have to do with extremal black holes?
Vijay Balasubramanian, Jan de Boer, M.M. Sheikh-Jabbari, Joan Simon
aa r X i v : . [ h e p - t h ] D ec arXiv:0906.3272 IPM/P-2009-022UPR-T-1209
What is a chiral 2d CFT?
And what does it have to do with extremal black holes?Vijay Balasubramanian ,a , Jan de Boer ,b , M.M. Sheikh-Jabbari ,c,d and Joan Sim´on ,e,f a David Rittenhouse Laboratory, University of Pennsylvania, Philadelphia, PA 19104, USA b Instituut voor Theoretische Fysica, Valckenierstraat 65,1018XE Amsterdam, The Netherlands c School of Physics, Institute for Research in Fundamental Sciences (IPM) ,P.O.Box 19395-5531, Tehran, Iran d The Abdus Salam ICTP, Strada Costiera 11, 34014, Trieste, ITALY e School of Mathematics and Maxwell Institute for Mathematical Sciences,King’s Buildings, Edinburgh EH9 3JZ, United Kingdom f Kavli Institute for Theoretical Physics,University of California, Santa Barbara CA 93106-4030, USA
Abstract
The near horizon limit of the extremal BTZ black hole is a “self-dual orbifold” of AdS .This geometry has a null circle on its boundary, and thus the dual field theory is aDiscrete Light Cone Quantized (DLCQ) two dimensional CFT. The same geometrycan be compactified to two dimensions giving AdS with a constant electric field. Thekinematics of the DLCQ show that in a consistent quantum theory of gravity in thesebackgrounds there can be no dynamics in AdS , which is consistent with older ideasabout instabilities in this space. We show how the necessary boundary conditionseliminating AdS fluctuations can be implemented, leaving one copy of a Virasoroalgebra as the asymptotic symmetry group. Our considerations clarify some aspects ofthe chiral CFTs appearing in proposed dual descriptions of the near-horizon degreesof freedom of extremal black holes. e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] ontents d CFT 23 DLCQ of a d CFT is a chiral CFT 64 Asymptotic symmetries and the chiral Virasoro algebra 85
AdS quantum gravity and dual chiral CFTs 106 Extremal Kerr black hole and its dual chiral CFT 137 Discussion 16 In the vicinity of their horizons, extremal black holes in many dimensions, both in flat andanti-de Sitter spaces, contain an AdS component with a constant electric field . Proposeddualities between AdS space and a conformal quantum mechanics [2, 3, 4, 5, 6] or a chiral1+1 dimensional conformal field theory (CFT) [5, 7] have been used to explain the statisticaldegeneracy of extremal black holes. In [7, 8] it was shown the AdS geometry with a constantelectric field can be understood as the compactification of an orbifold of AdS with a nullboundary. Systematically applying the rules of the AdS/CFT correspondence then suggeststhat the dual theory on the 1+1 dimensional boundary is a Discrete Light Cone QuantizedCFT [7, 8, 9, 6]. Because of the highly boosted kinematics of a DLCQ theory, only one chiralsector of the 2 d CFT survives. Such chiral theories thus seem to appear universally in thedual descriptions of extremal black holes.In this paper, we develop aspects of this DLCQ - extremal black hole correspondence.The essential features can be understood by considering the extremal BTZ geometry, whichitself appears in the near-horizon geometry of many asymptotically flat or AdS black holes. Itis well known that the BTZ black holes are dual to thermal ensembles in a 1+1 dimensionalCFT. Thermal ensembles in a single chiral sector of this CFT are dual to the extremalblack holes and explain their statistical degeneracy. Taking a limit which focuses on thevicinity of the BTZ horizon gives a locally AdS geometry that is a circle fibration over anAdS base. From the three dimensional perspective this is precisely the self-dual orbifoldof [8, 10]. Dimensionally reducing over the circle fibre gives an AdS geometry with anelectric flux – precisely the spacetime appearing in [7]. As we will show in sections 2 and This statement is an actual theorem in four and five dimensions, under certain isometry assumptionsand for extremal black holes with finite area horizons [1].
1, the same focusing limit applied to the CFT dual to BTZ effectively applies a DLCQprocedure that isolates the chiral sector carrying the extremal black hole entropy. Thus,one chiral set of Virasoro generators of the CFT is frozen in this limit, in the sense thatthere are no physical states charged under them. It turns out that the same chiral sectoralso contains the SL(2 , R ) isometries of the AdS geometry, while the surviving SL(2 , R )in the limiting chiral CFT appears as an enhancement of the U(1) symmetry of the circlefibration. Specifically, we show that there exists a consistent set of boundary conditionson the fluctuations of the near horizon extremal BTZ metric, as in the Brown-Henneauxanalysis [11], that enhances the U(1) isometry to an asymptotic chiral Virasoro algebra.This is consistent with recent proposals that the description of extremal black holes in termsof an AdS throat requires asymptotic boundary conditions eliminating AdS excitationsand enhancing a U(1) appearing in the geometry to a Virasoro symmetry [12, 13].Usually in the AdS/CFT duality, the isometries of spacetime are realized in the dual asglobal symmetries which then organize the representations of physical states. The surprisehere is that the SL(2 , R ) symmetry inherited in the CFT from the spacetime isometries actstrivially on the space of physical states. This has two implications. First, the chiral duals tothe near-horizon geometry of extremal black holes are incapable of describing non-extremalexcitations. Second, even after the addition of an electric field to AdS , 2 d quantum gravitywith this asymptotics has no dynamics. This is consistent with the idea that finite energyexcitations in AdS destroy its asymptotic structure [14]. These two points are related to thefact that non-extremal black holes do not have AdS throats. Similarly, in the classic settingof the D1-D5-string, extremal black holes arise from chiral excitations, and non-extremalityrequires excitations of both left and right movers.The self-dual orbifold and AdS with a flux also appear in the near horizon limit of theextremal Kerr black hole in four dimensions [12] suggesting the appearance of a chiral CFTdual. However, in this setting (as in [15, 16, 17]) the near-horizon AdS geometries appearin a “warped” way, with their metric multiplied by a function of another angular directionin the overall spacetime. We suggest that reduction over this additional direction can giverise to an effective three dimensional gravity with a negative cosmological constant with theself-dual orbifold as a solution. The dual description of this space as a chiral 2 d CFT thenexplains the statistical degeneracy of Kerr.
Note added:
In the last stage of preparation of this article two papers appeared on thearXiv [18] arguing that there is no dynamics in the chiral 2 d CFT proposed to be dual tothe near horizon extremal Kerr geometry, in agreement with our results. d CFT
BTZ black holes are three dimensional, asymptotically AdS spacetimes with metric [19] ds = − ( r − r )( r − r − ) r ℓ dt + ℓ r ( r − r )( r − r − ) dr + r ( dφ − r + r − ℓr dt ) . (2.1)2hey have ADM angular momentum and mass J r + r − ℓ , M = r + r − ℓ (2.2)given in terms of two parameters: the inner and outer horizons r ± . These are locally AdS spacetimes, differing from global AdS by a quotient under a discrete identification. This isthe origin of the periodicity in φ in (2.1), i.e. φ ∼ φ + 2 π . Regularity of the metric requires | J | ≤ M . The BTZ black holes also appear as components in the near-horizon geometryof black holes in many dimensions with both vanishing and negative cosmological constants(e.g. see [20]). The extremal BTZ black holes ( M = J ) have coincident inner and outerhorizons M = J = ⇒ r + = r − ≡ r h . (2.3)Globally, the generator of the discrete quotient of AdS giving rise to the extremal black holelies in a different conjugacy class from the generator giving rise to the non-extremal blackhole [21].According to the AdS/CFT correspondence, quantum gravity in AdS is dual to a 2 d conformal field theory (CFT) with equal left and right central charges c [11] c = 3 ℓ G , (2.4)where G is Newton’s constant in three dimensions. The BTZ black holes are thermal statesin this CFT having left and right-moving temperatures T R = 14 π r + − r − ℓ , T L = 14 π r + + r − ℓ , (2.5)with energy and angular momentum: L − c
24 = M − J, ¯ L − c
24 = M + J. (2.6)(In our conventions M and J are both dimensionless; their natural units are given by theAdS radius ℓ .) In the extremal ( M = J ) black hole the right-movers are in the groundstate L = c
24 ; T R = 0 (2.7)while the left moving temperature T L = π r h ℓ and ¯ L are arbitrary. The extremal BTZentropy (and that of higher dimensional black holes of which it is the near horizon limit) is In theories with supersymmetry these are indeed the obvious ground states in the RR sector. Thiscondition can even correspond to ground states in the NS sector, because in many examples the quantumnumbers L and ¯ L are not exactly identical to the standard CFT quantum numbers but can e.g. receivecontributions from gauge fields which make them spectral flow invariant, in which case this condition reallyimplies that the states have to be chiral primary. Although we have no proof that L = c/
24 always impliesthat the states have to be ground states of some sort, we will continue to refer to these states as groundstates and hope that this will not cause any confusion. L − c/
24 = 2 M ,at least when ¯ L ≫ c/ u = t/ℓ − φ , ˆ v = t/ℓ + φ, r − r = ℓ e ρ , (2.8)in which the metric takes the form ds = r d ˆ u + ℓ dρ − ℓ e ρ d ˆ u d ˆ v . (2.10)The variables ˆ u, ˆ v have a periodicity { ˆ u, ˆ v } ∼ { ˆ u − π, ˆ v + 2 π } . (2.11)On the cylindrical boundary of AdS ( ρ → ∞ ), d ˆ u and d ˆ v become null directions. Thus thetwo chiral Virasoro algebras of the dual conformal field theory are associated to asymptoticreparameterizations ˆ u → f (ˆ u ) and ˆ v → g (ˆ v ).Since the horizon is located at ρ → −∞ , we take the near horizon limit ρ = ρ + r, u = ˆ u r + ℓ , v = e ρ ℓr + ˆ v, { u, v } ∼ { u − π r + ℓ , v + 2 π ℓr + e ρ } ( ρ → −∞ )(2.12)while keeping r, u, v and r + fixed. (See [7] for the first discussion of this limit.) Theresulting metric, which describes the geometry in the vicinity of the extremal horizon, ds = ℓ ( du + dr − e r du dv ) (2.13)is identical in form to (2.10) but there is a crucial difference. In the ρ → −∞ limit, theidentification (2.11) becomes { u, v } ∼ { u − π r + ℓ , v } . (2.14)Thus, the boundary of (2.13) ( r → ∞ ) is a “null cylinder” – it has a metric conformal to du dv , the standard lightcone metric on a cylinder, but has a compact null direction ( u ).The periodicity of u encodes the temperature of the left-moving thermal state that gave theoriginal extremal BTZ black hole its statistical degeneracy.Rewriting the radial coordinate as y = e r gives ds = ℓ (cid:18) − y dv + dy y (cid:19) + ℓ (cid:18) du − y dv (cid:19) . (2.15) For later use note that a generic BTZ metric in the ˆ u, ˆ v, ρ coordinate system takes the form [22] ds = ℓ (cid:2) L + d ˆ u + L − d ˆ v + dρ − ( e ρ + L + L − e − ρ ) d ˆ ud ˆ v (cid:3) . (2.9)where L ± = ℓ ( r + ± r − ) . Recalling (2.5), L + = (2 πT L ) , L − = (2 πT R ) . Despite the resemblance of the limit (2.12) and the coordinate changes one makes in taking the Penroselimit, (2.12) is not a Penrose limit, as the geometry we obtain after the limit is not a plane-wave. S fibration over AdS which arises as a discrete identification of AdS . Thegenerator of this discrete group sits inside the SL(2 , R ) L subgroup of the initial SL(2 , R ) L × SL(2 , R ) R isometry group of AdS [7, 8]. To be precise, the parametrization of SL(2 , R ) (i.e.AdS ) that is relevant for the metric (2.15) is G = (cid:18) v (cid:19) (cid:18) √ y √ y − √ y √ y (cid:19) (cid:18) e u e − u (cid:19) , (2.16)in terms of which the metric (2.15) is ds = ℓ G − d G ) (2.17)Under u → u − πr + /ℓ , G is identified by the right action of (cid:18) e − πr + /ℓ e πr + /ℓ (cid:19) . (2.18)The isometry group is SL(2 , R ) R × U(1) L , the first factor corresponding to the isometries ofthe AdS base. On the boundary of the spacetime these isometries act to reparameterizethe non-compact coordinate v . In fact, this geometry is precisely the self-dual orbifold ofCoussaert and Henneaux [10]. The present coordinate system covers only part of the globalspacetime described in [8, 10].Since (2.13) is asymptotically locally AdS , we expect the dual field theory to still be a twodimensional conformal field theory, but defined on a boundary null cylinder. To understandwhat that means, we can follow [8] and regulate the CFT by cutting off the self-dual orbifoldat a fixed, large radius. Following the usual AdS/CFT reasoning, this implements a UVcutoff in the field theory. We will remove the cutoff by sending r → ∞ . At any fixed r , themetric (2.13) is conformal to ds = du − e r du dv (2.19)Now consider a standard cylinder with its usual Cartesian metric ds = − dt + dφ and { φ , t } ∼ { φ − β, t } . We will use coordinates u = t − φ ; t = 2 t = ⇒ ds = du − du dt ; { u , t } ∼ { u + β, t } . (2.20)We now boost the cylinder with a rapidity 2 γ (˜ u = e γ u ) and then reparameterize theboosted cylinder so that the identification is still occurring at fixed t . The metric thenbecomes ds = e − γ (cid:0) d ˜ u − e γ d ˜ u dt (cid:1) ; { ˜ u , t } ∼ { ˜ u + βe γ , t } (2.21)Rescaling the coordinates as ˜ u → e − γ ˜ u and t → e − γ t gives the metric ds = d ˜ u − e γ d ˜ u dt ; { ˜ u , t } ∼ { ˜ u + β, t } . (2.22) Strictly speaking, these SL(2 , R ) transformations include U(1) gauge transformations compensating thetransformation of the gauge field on AdS . r surfaces of the near-horizon BTZ metric (2.19) are conformal to aboosted cylinder. As r → ∞ the boost becomes infinite, precisely realizing the proceduredefined by Seiberg [23] for realizing the Discrete Light Cone Quantization (DLCQ) of a fieldtheory. In Sec. 3 we will show that following the usual kinematics of DLCQ, only one chiralsector of the CFT dual to AdS will survive at finite energies.We can also see the latter by directly examining the near-horizon limit (2.12). Acting inthe CFT dual to AdS , the near horizon limit of the extremal BTZ black hole focuses in onenergies so low that they lie below the black hole mass gap, thus eliminating all non-extremaldynamics [7] (also see [6]). This will isolate one chiral sector (the left-movers), since non-extremal, finite energy excitations necessarily involve excitations of the right-movers also.Explicitly, the infinite rescaling in the coordinate ˆ v relates translations as ∂ v ∼ e − ρ ∂ ˆ v . (2.23)Thus, recalling the ∂ ˆ v is the right-moving Hamiltonian in the CFT dual to AdS , any finite-energy right-moving excitation, i.e. any excitation | s i with ∂ ˆ v | s i = ( L − c/ | s i 6 = 0, willbe infinitely blue shifted in the Hamiltonian ∂ v that is well defined in the ρ → −∞ limit.In other words, we should only be keeping the states satisfying ∂ ˆ v | s i = ( L − c/ | s i = 0 (2.24)which are the ground states in the right-moving sector.We can also directly follow how the near-horizon limit (2.12) acts on the left and rightmoving Virasoro generators of the CFT dual to AdS . These generators are L n − c δ n, = e in ˆ v ∂∂ ˆ v , ¯ L n − ¯ c δ n, = e in ˆ u ∂∂ ˆ u . (2.25)As ρ → −∞ in the near-horizon limit (2.12), it is evident that ¯ L n are essentially unchangedwhile the L n annihilate all the finite energy states because of the condition (2.24). d CFT is a chiral CFT
In the previous section we reviewed how the near-horizon geometry of extremal BTZ is dualto the DLCQ of a 2 d CFT. We now examine how such theories are quantized. Consider a2 d CFT on a cylinder ds = − dt + dφ = − du ′ dv ′ ; u ′ = t − φ, v ′ = t + φ (3.1)where φ is a circle with radius R . Here { φ, t } ∼ { φ + 2 πR, t } ; { u ′ , v ′ } ∼ { u ′ − πR, v ′ + 2 πR } (3.2)Let P u ′ and P v ′ denote momentum operators in the v ′ and u ′ directions respectively. Theireigenvalues P v ′ = (cid:16) h + n − c (cid:17) R , P u ′ = (cid:16) h − c (cid:17) R , n ∈ Z (3.3)6re given in terms of the quantized momentum n along the S , the 2 d central charge c andan arbitrary value of h with h ≥ h + n ≥
0. These are related to the eigenvalues ofthe standard operators L , ¯ L used in radial quantization on the plane by ¯ L = h + n and L = h . We will assume that the 2 d CFT is non-singular, and therefore that the spectrumis discrete.Following Seiberg [23], consider a boost with rapidity γu ′ → e γ u ′ , v ′ → e − γ v ′ . (3.4)The boost leaves metric (3.1) invariant. However the identifications are now { u ′ , v ′ } ∼ { u ′ − πRe γ , v ′ + 2 πRe − γ } . (3.5)We want to match the boundary structure appearing in the boundary of the near horizongeometry with the DLCQ of the starting boundary cylinder. To do so, consider the limit γ →∞ with R e γ fixed. This describes a null cylinder geometry with metric ds = − du ′ dv ′ and u ′ a compact null direction. The same infinite boost was presented in different coordinatesin (2.20) – (2.22). However, notice that since v ′ → e − γ v ′ = e − γ ( t + φ ) and 0 ≤ φ ≤ π R ,as γ → ∞ any finite changes in v ′ come from changes in t . Thus, in the limit, dv ′ ∝ dt and ds = − du ′ dv ′ ≈ − e − γ du ′ dt which is conformal to the dominant piece of the metric in(2.22).More explicitly, the periodicities of the boundary coordinates under the limit γ → ∞ with R − ≡ R e γ fixed are (cid:18) φt (cid:19) ∼ (cid:18) φt (cid:19) + (cid:18) πR (cid:19) − infinite boost → (cid:18) u ′ v ′ (cid:19) ∼ (cid:18) u ′ v ′ (cid:19) + (cid:18) πR − πR − e − γ (cid:19) (3.6)We can now identify { u ′ , v ′ } with the lightcone boundary coordinates of AdS in (2.12) via u ′ = u ( ℓ/r + ) R − and v ′ = v ( r + /ℓ ) R − . Then, comparing (2.12) and (3.6), it is evident thatthe action of the near horizon limit on u, v precisely reproduces the identifications inducedby the infinite boost in DLCQ. Thus, from this perspective also, the dual to the near-horizongeometry of the extremal BTZ black hole should be the DLCQ of the 1+1 dimensional CFTdual to AdS .Because of the kinematics of the DLCQ boosts, P v ′ = (cid:16) h + n − c (cid:17) e − γ R , P u ′ = (cid:16) h − c (cid:17) e γ R . (3.7)Keeping P u ′ (momentum along v ′ ) finite in the γ → ∞ limit requires h = c/
24. This leadsto P v ′ = n · e − γ R = nR − . (3.8)Thus the DLCQ limit (3.7) freezes the right moving sector. Equivalently, it generates aninfinite energy gap in this sector, while the gap in the left-moving sector (whose energy is In our previous analysis the cylinder in coordinates (2.20) was boosted but also reparameterized – thisis why the metric transformed to (2.21). P v ′ ) is kept finite. All physical finite energy states in this limit only carrymomentum along the compact null direction u ′ . Therefore, the DLCQ γ → ∞ limit definesa Hilbert space H H = {| anything i L ⊗ | c/ i R } . (3.9)It is worth noting that the extremal D1-D5-p black hole (whose near horizon limit isthe BTZ black hole) is precisely dual to states of this form with the right movers in the RRground state, and the left movers in a highly excited state the statistical degeneracy of whichexplains the black hole entropy [24].Since the spectrum of the DLCQ theory is chiral we might wonder what remains in thislimit of the Virasoro algebra of the CFT we started with. Denoting the right moving Virasorogenerators by L m , all states L m | c/ i ( m <
0) have infinite energy in the DLCQ limit, sincetheir action always changes the right-moving energy. Explicitly, consider the generators L q ∼ e iqv ′ ∂∂v ′ , ¯ L p ∼ e ipu ′ ∂∂u ′ , (3.10)with L − c/
24, ¯ L − c/
24 being generators of translations along v ′ and u ′ respectively. Afterthe boost (3.4) the quantization conditions for p, q become: q = kRe − γ = kR − e γ , p = m · Re γ = mR − , k, m ∈ Z . (3.11)Thus, there is a single copy of the Virasoro algebra, generated by ¯ L p , which survives thelimit. This is acting on the left movers, as expected from the spectrum defining the Hilbertspace of the theory. Notice the generators of this algebra are acting on the compact directionof the DLCQ null cylinder. Summary:
The DLCQ of a non-singular 2 d CFT freezes the right moving sector to itsground states | c/ i while keeping the full left moving sector. Hence, the DLCQ limit givesa chiral 2 d CFT with the same central charge as the original one. Applied to the BTZblack hole (Sec. 2), we learn that the near-horizon geometry of extremal BTZ is dual to onechiral sector of the 2 d CFT with central charge c = 3 ℓ/ G that is dual to AdS gravity. Thesurviving chiral sector is in the state in which it was placed to realize the dual to an extremalblack hole, namely a thermal state at a temperature T L = T DLCQ = R − / (2 π ), correspondingto the left-moving thermal state | c/ i ⊗ | T = R − / π i in the Hilbert space of the CFT dualto AdS . In the AdS/CFT correspondence, the isometries of spacetime manifest themselves as globalsymmetries of the dual field theory, and physical states are organized in representations ofthe isometry group. For this reason, various authors [8, 25, 26] have considered how thephysical states of fields in the near-horizon BTZ geometry (2.13) or (2.15) transform underthe SL(2 , R ) × U(1) isometry group. Now recall that the DLCQ analysis of the dual field8heory in the previous section showed that the physical states of this theory must live ina chiral CFT. It would have been natural to expect that the SL(2 , R ) isometries providethe global part of the associated Virasoro algebra. The surprise is that this is not thecase. Specifically, the SL(2 , R ) isometries are associated to reparameterizations of the non-compact coordinate v on the boundary, while the physical states only carry momentum alongthe compact null direction u on which only the U(1) part of the isometry group acts. Thus,AdS/CFT is telling us that physical states cannot be charged under the SL(2 , R ) isometrygroup associated to the AdS base in (2.15).Why would a consistent quantum theory of gravity around the near-horizon BTZ back-ground (2.15) require the absence of excitations in the AdS base of this geometry? Perhapsbecause any such fluctuations would cause the space to “fragment” leading to the appearanceof multiple boundaries to the spacetime [14]. In the next section we will compactify (2.15)and examine its stability to excitations in the AdS base. Below we will simply accept thelesson from the analysis of the dual DLCQ field theory and implement boundary conditionsfor the spacetime that preserve only the predicted spectrum. Boundary conditions:
To this end, we will follow the asymptotic symmetry group analysisof Brown and Henneaux [11] by identifying the boundary conditions for “allowed” metricfluctuations close to the spacetime boundary. First recall the Brown-Henneaux boundaryconditions for AdS . In the ˆ u, ˆ v, r coordinates [22], where the background AdS metric takesthe form ds = ℓ ( dr r − r d ˆ ud ˆ v ) these boundary conditions at large r are [11] δg ˆ u ˆ u ∼ δg ˆ v ˆ v ∼ δg ˆ u ˆ v ∼ O (1) , δg rr ∼ O ( 1 r ) , δg r ˆ u ∼ δg r ˆ v ∼ O ( 1 r ) . (4.1)Order one fluctuations in δg ˆ u ˆ u , δg ˆ v ˆ v correspond to normalizable modes in the dual 2 d CFTand these may be chosen arbitrarily. For example, writing a generic BTZ black hole in theˆ u, ˆ v coordinates, the constant parts of g ˆ u ˆ u and g ˆ v ˆ v determine the ADM mass and angularmomentum of the black hole (2.9). Thus, order O (1) fluctuations in δg ˆ u ˆ u , δg ˆ v ˆ v correspondto changing the mass and angular momentum in the dual 2 d CFT. A general deformation of δg ˆ u ˆ u , δg ˆ v ˆ v would be non-extremal and would thus excite both chiral sectors of the dual CFT.By contrast, we want to restrict to extremal excitations. Recalling the form of BTZ metric(2.9), one may easily observe that imposing the extremality condition L = c/
24 requires amore stringent boundary condition on the variations in g ˆ v ˆ v . The arguments of Sec. 2 and 3for taking the DLCQ limit and in particular (3.7) then suggest that we should replace theboundary condition on g ˆ v ˆ v by δg ˆ v ˆ v ∼ O ( 1 r ) . (4.2)The remainder of the Brown-Henneaux boundary conditions in (4.1) can be kept intact.Further analysis shows that these are forming a set of consistent boundary conditions. Infact this set is equivalent to choosing a subset of (4.1) that preserve the null nature of thenon-compact coordinate v (up to transformations which are trivial at large r ). Asymptotic Symmetry Group:
The asymptotic symmetry group (ASG) of a space-time is the set of symmetry transformations (diffeomorphisms) which preserve the boundaryconditions modulo the set of diffeomorphisms the generators of which vanish (reduce to aboundary integral) after implementation of the boundary conditions. Equipped with the9bove boundary conditions we can compute the ASG for the case of the near horizon ex-tremal BTZ or the self-dual orbifold of AdS . We seek diffeomorphisms (vector fields ζ )whose action on the metric (Lie derivative L ζ g ) generates metric fluctuations compatiblewith the above boundary conditions. More mathematically, if g αβ = g αβ + δg αβ , where g αβ stands for the asymptotic metric, then one is looking for vector fields ζ satisfying( L ζ g ) αβ ∼ δg αβ , (4.3)where the symbol ∼ stands for same order of magnitude in the large r expansion sense.Since our boundary conditions are closely related but more restrictive than those ofBrown-Henneaux [11], we can use their explicit analysis of the generators of the asymptoticsymmetry group and simply impose the additional constraint on δg vv (4.2) on them. Theallowed diffeomorphisms are ζ u = 2 f ( u ) + 12 r g ′′ ( v ) + O ( r − ) (4.4a) ζ v = 2 g ( v ) + 12 r f ′′ ( u ) + O ( r − ) , (4.4b) ζ r = − r ( f ′ ( u ) + g ′ ( v )) + O ( r − ) (4.4c) g ′′′ ( v ) = 0 = ⇒ g = A + B v + C v . (4.5)Here, the connection to the Brown-Henneaux diffeomorphisms is made explicit: the diffeo-morphisms generated by ζ = ζ α ∂ α of (4.4) are exactly those of Brown-Henneaux [11] andthe constraint δg vv = O ( r ) is implemented by (4.5). One set of allowed diffeomorphismsis specified by a periodic function f ( u ) = f ( u + 2 π ). The analysis of generators of thesediffeomorphisms follows directly from those of Brown and Henneaux and they lead to a chi-ral Virasoro algebra at central charge c = 3 ℓ/ G (2.4). The remaining three parameterfamily of diffeomorphisms in (4.5) describes the SL(2 , R ) isometries of the self-dual orbifold.The isometries of the original extremal black holes were just a U(1) × U(1). In that casea Brown-Henneaux analysis with the extremal constraint would have also yielded (4.4) withthe constraint g ′′′ = 0. However, in the original geometry g has to be a periodic functionwhich restricts the solutions to the constraint to g = A only. The process of taking the nearhorizon limit led to an identification in u alone, and thus, g need not be periodic, allowingthe three parameter solution above. The isometry generators that appear in this way, arenot simply related to the SL (2 , R ) generated by L , L ± (2.25). AdS quantum gravity and dual chiral CFTs Consider the two-dimensional Einstein-Maxwell-Dilaton theory with a negative cosmologicalconstant: S = ℓ G Z d x √− g (cid:20) e ψ ( R + 2 ℓ ) − ℓ e ψ F µν F µν (cid:21) (5.1) Our analysis here suggests that the Left and Right CFT’s introduced in [27] may be identical. Thispoint deserves further investigation. F µν is the U (1) field strength. This action has an AdS solution with curvature R = − ℓ , constant ψ and constant electric flux : ds = − ℓ r ( − dt + dr ) , F tr = 2 Qr , e − ψ = Q . (5.2)This action may be obtained from the dimensional reduction of the 3 d Einstein-Hilbertaction with 3 d Newton constant G and cosmological constant − /ℓ via restriction to themassless sector of the Kaluza-Klein tower. Likewise the reduction of the near-horizon BTZgeometry (2.15) to two dimensions is precisely (5.2). The radius of the extremal BTZ horizonbecomes ℓQ . The action (5.1) has another two parameter family of solutions in which ψ isnot a constant [7] – these lift to generic BTZ black holes.Because of this connection between two and three dimensions, we expect that quantumgravity around the background (5.2) is dual to a subsector of the DLCQ chiral CFT thatis developed in Sec. 2 and 3, and is only fully consistent when embedded in string theorywith all the resulting additional degrees of freedom. The electric field strength Q is relatedto the DLCQ compactification scale R − in (3.6) while the central charge is related to the 2 d Newton constant: c = 3 ℓ/ (2 G ) = 3 / (4 πG ).Quantum gravity in the AdS background (5.2) was explored in [25] from the perspectiveof the spacetime conformal field theory, and in [26] from the perspective of the boundarystress tensor. Both of the papers consider spectra including states charged under the SL(2 , R )isometry group of AdS , and analyze a Virasoro algebra which includes this SL(2 , R ). How-ever, as shown in previous sections, a consistent quantum theory of gravity in this backgroundshould not have any states charged under the isometry group. The reason for this is thatexcitations supported in AdS back-react strongly and can modify the asymptotic structureof the spacetime [14].To see this, let us write the two dimensional metric in a gauge in which the metric isconformally flat ds = e φ ( σ + ,σ − ) dσ + dσ − , ≤ σ ± ≤ π , (5.3)and consider the variation of the action (5.1) with respect to the 2 d metric. We find ∇ + ∇ + e ψ = 8 πG T ++ (5.4)and similarly for the −− component. If we regard (5.1) as arising from compactification of athree dimensional theory, besides the contributions from ψ and the gauge field, we can alsoinclude all contributions of massive Kaluza-Klein modes in T ++ . We may now follow thediscussion in section 2.2 of [14] (see Eqs. (2.16) and (2.17) there). Integrating (5.4) against e − φ dσ + , we obtain e − φ ∂ + e ψ | σ + =0 − e − φ ∂ + e ψ | σ + = π = − πG Z dσ + e − φ T ++ (5.5) An analysis of Schwinger pair creation of charged particles in
AdS in the presence of a constant electricfield was performed in [28]. A bound between the mass of particle excitations and the background electricfield was derived to ensure the stability of these backgrounds. This bound is satisfied in supersymmetric AdS × S spacetimes and is also saturated for the two dimensional vacuum solution discussed here. T −− . Assuming a null energy condition ( T ++ ≥ T ++ requires at least one of the two terms on the left hand side of this equationto be non-zero. Since e − φ vanishes quadratically near the boundary of AdS , this impliesthat e ψ must diverge at one of the AdS boundaries. This is inconsistent with the constantvalue e − ψ = Q in (5.2), which is related from the three dimensional point of view to the com-pactification radius. This shows that preserving the boundary conditions requires T ++ = 0and a similar argument requires T −− = 0. Thus perturbations cannot have any dependenceon σ + , σ − , as their back-reaction would destroy the boundary of the geometry. The back-ground geometry (5.2) has an SL(2 , R ) isometry, and if perturbations do not depend on σ + , σ − , then the perturbation cannot break the SL(2 , R ) symmetry either. In other words,all degrees of freedom transform trivially under SL(2 , R ), in agreement with the analysis inprevious section. This argument used the fact that AdS has two disconnected boundaries. In Sec. 2the analysis of the CFT dual was carried out in coordinates that only intersected a singleboundary, but it was shown in [8] that, globally, the self-dual orbifold geometry has twoboundaries, each of which is a null cylinder carrying a DLCQ of a CFT. To see this, transformthe coordinates in (2.15) as y = cos τ cosh z + sinh z ; v = sin τ cosh z cos τ cosh z + sinh z , (5.6)so that the self-dual orbifold metric becomes ds = ℓ (cid:0) − cosh z dτ + dz (cid:1) + ℓ ( du + A ′ ) (5.7)where A ′ is a gauge field with constant field strength in global AdS . This is the globalself-dual orbifold of [8]. The entire range of v is covered by a finite range of global time τ .Thus each patch of the form (2.15) intersects one boundary of the global spacetime at either z = ±∞ .In view of this, both the near-horizon limit of extremal BTZ (2.13) and the 3 d uplift of(5.2) can be regarded globally as dual to two DLCQ CFTs, each giving rise to one chiral theory(see [8, 6] for discussion). From this perspective we can presumably view the description ofthe self-dual orbifold as a thermal state in a single CFT as emerging from tracing over theHilbert space living in one of the boundaries. This is in analogy with the usual treatmentof the eternal BTZ black hole as either an entangled state in two CFTs defined on the twoboundaries of the geodesically complete spacetime, or as a thermal state in a single CFT[31]. The statistical degeneracy of the thermal state in the chiral CFT dual to the spacetime(2.15) then measures the area of the familiar Poincare horizon of this coordinate patch (see[3] for a similar perspective). One difference between BTZ and the self-dual orbifold is thatwhile the BTZ boundaries are causally disconnected, a light ray can travel between the The fact that AdS “fragments” in this way has led to the suggestion [14] that the dual of a one-dimensional conformal field theory should involve a sum over tree-like geometries with many different asymp-totic AdS boundaries. While some partial progress has been made in developing this picture [29, 30], itis still unclear whether this is the right way to think about AdS , or whether it eventually will lead toconnection with the fuzzball proposal, and we will not further pursue this possibility in this paper. base of the self-dual orbifold cannot be consistently excited. It was shown in [26] that the most generalsolution of the dimensional reduction of 3 d gravity with a negative cosmological constant ina particular gauge can be put in the form g µν dx µ dx ν = dη −
14 ( h ( t ) e η/L + h ( t ) e − η/L ) dt . (5.8)At the boundary η → ∞ , the boundary metric is determined by h ( t ). One can choosea coordinate t such that h = 1. The subleading behavior is determined by h ( t ). Thediffeomorphisms that preserve this gauge and leave h unchanged were determined by [26].However it turns out that while these are normalizable deformations of the boundary at η → + ∞ , they are not normalizable deformations at the other boundary η → −∞ – i.e.they change h . In fact, there are no deformations at all which both preserve the gauge andare normalizable at both boundaries, except the isometries. This again suggests that it isnot possible to deform AdS without disrupting the spacetime boundary. The extremal 4 d Kerr black hole is given by ds = − ∆ R (cid:16) d ˆ t − a sin θd ˆ φ (cid:17) + sin θR (cid:16) (ˆ r + a ) d ˆ φ − ad ˆ t (cid:17) + R ∆ d ˆ r + R dθ , (6.1)where R = ˆ r + a cos θ, ∆ = (ˆ r − a ) . (6.2)Its ADM mass and angular momentum are function of the horizon size aM = a, J = a G . (6.3)In the quantum theory, J is quantized (to half integers) in units of ~ . This black hole haszero Hawking temperature and its Bekenstein-Hawking entropy is S BH = 2 πM ~ G = 2 π ~ J . (6.4)In the near horizon ǫ → r = a + ǫ r, ˆ t = 2 atǫ , ˆ φ = φ + tǫ , (6.5)while keeping the un-hatted parameters and coordinates fixed, we obtain the near horizonextremal Kerr (NHEK) geometry [32, 12] ds = 2 G J Ω( θ ) (cid:20) − r dt + dr r + dθ + Λ( θ ) ( dϕ + rdt ) (cid:21) , (6.6)13here ϕ ∈ [0 , π ] , ≤ θ ≤ π andΩ( θ ) = 1 + cos θ , Λ( θ ) = 2 sin θ θ . (6.7)This metric at a given θ has the form of a warped circle fibration over AdS in which thefiber radius depends on the angle θ . If Λ and Ω were constants this would be precisely theself-dual orbifold of (2.15) times a circle. Indeed, as emphasized in [12], constant θ slices looklike squashed self-dual orbifolds. The coordinates in (6.6) cover only part of the spacetime,with a boundary at r → ∞ – globally, like the self-dual orbifold, there are two boundaries.One sees similar squashed geometries with AdS and AdS factors in decoupling limits ofnear-extremal black holes in anti-de Sitter space [15, 16, 17]. (Also see [33, 34, 35].)The Kerr black hole is invariant under time and angular ˆ φ translations. This isometrygroup is enhanced to SL(2 , R ) × U(1) in the near horizon, just as in the self-dual orbifold.The U(1) is generated by ∂ ϕ , whereas the SL(2 , R ) acts both on the AdS subspace andalong the fiber to preserve the form of dϕ + rdt [12].In [12], the asymptotic symmetry group preserving certain boundary conditions for thefluctuations of the NHEK was calculated. The corresponding diffeomorphisms they foundwere of the form ζ λ = λ ( ϕ ) ∂ ϕ − rλ ( ϕ ) ′ ∂ r . (6.8)These generate a chiral Virasoro algebra. In [12] it was proposed that this Virasoro algebrashould be understood as the symmetry group of a chiral 2 d CFT dual to quantum gravityaround the near horizon Kerr geometry. The central charge of this chiral CFT was computedto be c Ext. Kerr = 12
J . (6.9)The NHEK is then associated with a thermal state of the chiral 2 d CFT at tempera-ture T NHEK = 1 / π . Upon applying the Cardy formula for the entropy of 2 d CFTs, theBekenstein-Hawking entropy of the extremal Kerr black hole (6.4) is reproduced. The con-sistency of the boundary conditions proposed in [12] required the vanishing of the charge ofthe U (1) τ ∈ SL (2 , R ), i.e. E R = 0 (6.10)in the notation used in [12], for all physical states. Thus, like for the self-dual orbifold, thereare no physical excitations of the AdS factor in the geometry. The E R = 0 condition actslike the restriction to extremality in the BTZ black hole that we studied in Sec. 2.The analogies between the Kerr-CFT construction [12] and the analysis of the self-dualorbifold in previous sections suggests that chiral CFT of [12] is the DLCQ of an ordinary twodimensional conformal field theory. Ideally, we would like to find a consistent Kaluza-Kleinreduction of gravity in the NHEK geometry to the three-dimensional self-dual orbifold. As afirst step, we make a connection between the NHEK geometry and 3 d gravity with a negativecosmological constant. For the NHEK geometry we consider then the four dimensional metricreduction ansatz: ds = L Ω (cid:2) − ∂ σ β ( t, σ ) (cid:0) − dt + dσ (cid:1) + dθ + Λ ( dϕ + β ( t, σ ) dt ) (cid:3) , (6.11)14here Ω = (1 + cos θ ) / θ/ (1 + cos θ ). The equation of motion derivedfor β using this ansatz and the four dimensional Einstein equation without a cosmologicalconstant is identical to the equation of motion obtained from the three-dimensional ansatz ds = ℓ (cid:2) − ∂ σ β ( t, σ ) (cid:0) − dt + dσ (cid:1) + ( dϕ + β ( t, σ ) dt ) (cid:3) . (6.12)and Einstein’s equation with a cosmological constant R µν + 2 ℓ g µν = 0 . (6.13)Here R is the Ricci tensor computed for the 3 d metric. Although this obviously does notshow that there should exist a Kaluza-Klein reduction from four to three dimensions whichreduces the NHEK geometry to the self-dual orbifold of AdS , it does show that the twotheories share some dynamics.We can also derive the central charge derived in [12] from the 4 d NHEK geometry, bymatching parameters with the three-dimensional reduction ansatz. To do this, note first thatthe above 3 d equation of motion can be obtained from the Lagrangian L = p − det g ( R + 2 ℓ ) , (6.14)which describes 3 d gravity in the presence of a negative cosmological constant. The 3 d New-ton constant is then computed by integrating over the compact direction θ in our reductionansatz 1 G = 2 L R π dθ Ω Λ G ℓ = 4 L G ℓ . (6.15)Thus the 3 d action is S = 116 πG Z d x L , (6.16)Note that its vacuum solution is an AdS with radius R AdS = ℓ . Since L = 2 G J , using theBrown-Henneaux formula for the central charge, we have c = 3 R AdS G = 12 J . (6.17)This matches (6.9). We earlier showed that the AdS central charge also matches the centralcharge of the chiral CFT that is dual to self-dual orbifold.This suggests the proposed chiral 2 d CFT dual to extremal Kerr [12] is the DLCQ of a2 d CFT with the following identifications: (a) The DLCQ compactification radius R − is anarbitrary physical scale and has been set equal to one in the Kerr/CFT analysis [12], (b)The E R = 0 condition in [12] is mapped to L = c/
24 DLCQ condition, (c) The extremalKerr ADM angular momentum J is equal to the light-cone momentum P + of the DLCQdescription.One should note that identifying the chiral 2 d CFT duals proposed for extremal blackholes [12, 13] as the DLCQ of a 2 d CFT also explains why we can use Cardy’s formula to15ount the number of states. If we only knew that the states had to form representations of asingle Virasoro algebra, we would not be able to use modular invariance, and unitarity alonedoes not determine the asymptotic growth of the number of states. Still, there are to ourknowledge no general statements about the asymptotic growth of the number of states ofthe form | c/ i R ⊗ | anything i in an arbitrary CFT. If the left-movers are Ramond groundstates, and it is a theory with supersymmetry, one can estimate the number of states of thisform using the elliptic genus and its modular properties [36], and it would be interesting toestablish similar results for more general CFT’s.While our results have provided some evidence that DLCQ of a CFT is dual to thenear-horizon extremal Kerr, it would have been more satisfactory to have a consistent andcomplete reduction of 4 d gravity with NHEK boundary conditions [12] to 3 d gravity with acosmological constant. In a similar setting where squashed AdS factors appear in a decou-pling limit of R-charged black holes in AdS and AdS , progress towards such a reductionhas been made [15, 16, 17]. In this paper we have shown that the near-horizon limit of the extremal BTZ black hole,which leads to the so-called self-dual orbifold geometry, is dual to the DLCQ of a non-chiral 2dCFT, which is a chiral 2d CFT with the same central charge. We have also provided evidencethat various “chiral CFTs” that have appeared in the literature as dual CFTs to extremalblack holes should really be thought of as DLCQ of ordinary two-dimensional CFTs. This,among other things, justifies the use of Cardy formula to account for the extremal black holeentropy using this chiral CFT duals. It would be desirable to develop this picture in moredetail. In particular, it would be interesting to study correlation functions in the DLCQtheory and the corresponding bulk-boundary dictionary. Another outstanding problem isto establish more rigorously that generic extremal black holes, upon taking a near-horizonlimit, are indeed dual (once suitable boundary conditions are imposed) to the DLCQ of aconformal field theory. If this is indeed the case, one would expect that the parent 2 d CFTof the DLCQ theory might also have a string theoretic realization, e.g. in the form of awarped AdS solution of string theory. In other words, one might seek some sort of mapfrom extremal black hole solutions to AdS solutions. We have seen hints of such a map in[15, 16, 17], but whether it exists in the general case is unclear.One curiosity about the self-dual orbifold geometry is that it is dual to thermal state ina DLCQ CFT. The ground state of the DLCQ theory does not appear to have a bona fidegeometric dual. This is unlike AdS gravity with a standard cylindrical boundary wherethe ground state describes empty AdS and thermal states describe black holes. This seemsto be a general feature of gauge-gravity duality for DLCQ field theories [37]. Specifically, it is not dual to a very near horizon limit of the M = 0 BTZ black hole as one can explicitlycheck. cknowledgements We would like to thank John McGreevy, Bindusar Sahoo, Gary Horowitz and MatthewRoberts for useful discussions. J.S. would like to thank the organizers of the KITP pro-gramme“Fundamentals of String Theory” and the Departmento de F´ısica de Part´ıculas inSantiago de Compostela for hospitality during different stages of this project. The workof J.S. was partially supported by the Engineering and Physical Sciences Research Council[grant number EP/G007985/1]. This research was supported in part by the National ScienceFoundation under Grant No. NSF PHY05-51164. VB was partly supported by DOE grantDE-FG02-95ER40893 and partly by NSF grant NSF OISE-0443607. JdB was supported inpart by the FOM foundation.
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