Near-Horizon Extremal Geometries: Coadjoint Orbits and Quantization
IIPM/P-2017/085April 16, 2018
Near-Horizon Extremal Geometries:Coadjoint Orbits and Quantization
R. Javadinezhad † , B. Oblak (cid:91) , M.M. Sheikh-Jabbari ‡ † Physics Department, New York University, New York, USA (cid:91)
Institut f¨ur Theoretische Physik, ETH Z¨urich, CH-8093 Z¨urich, Switzerland ‡ School of Physics, Institute for Research in Fundamental Sciences (IPM),P.O.Box 19395-5531, Tehran, Iran
Abstract
The NHEG algebra is an extension of Virasoro introduced in [arXiv:1503.07861]; it describesthe symplectic symmetries of ( n + 4)-dimensional Near Horizon Extremal Geometries withSL(2, R ) × U(1) n +1 isometry. In this work we construct the NHEG group and classify the (coad-joint) orbits of its action on phase space. As we show, the group consists of maps from an n -torus to the Virasoro group, so its orbits are bundles of standard Virasoro coadjoint orbitsover T n . We also describe the unitary representations that are expected to follow from thequantization of these orbits, and display their characters. Along the way we show that theNHEG algebra can be built from u(1) currents using a twisted Sugawara construction. e-mail:[email protected] e-mail:[email protected] e-mail:[email protected] a r X i v : . [ h e p - t h ] A p r ontents Since the seminal work of Bondi-van der Burg-Metzner, Sachs [1], and Brown-Henneaux [2], it isnow an established fact that in generally covariant field theories there is a certain class of space-time diffeomorphisms (“residual” or “asymptotic” symmetries) to which one can associate conservedsurface charges. Therefore, certain sets of geometries are physically distinct despite being relatedby coordinate transformations. This statement can actually be adapted to any gauge theory [3]and entails a refinement of the na¨ıve notion of general covariance [4]. It is also consistent withholography [5] since the conserved charges associated with asymptotic symmetries are fluxes of variousfield combinations at the space-time boundary [3, 6–9].One elegant application of asymptotic symmetries is the use of universal Cardy-like formulas [10]to reproduce black hole entropy. This was first done in [11] for BTZ black holes in AdS [12] and wassubsequently generalized to extremal Kerr black holes [13], leading to the Kerr/CFT correspondence.In [13] the authors zoomed in on the near-horizon region of a four-dimensional extremal black holewith angular momentum J and looked for asymptotic symmetries of the near-horizon region itself,finding a Virasoro algebra with central charge c = 12 J . Assuming that this reflects the presence of a(chiral) two-dimensional conformal field theory (CFT) and that the Cardy formula is applicable, theyreproduced the entropy of the underlying extremal black hole. In this approach, both extremalityand the near-horizon approximation are crucial. A similar analysis was subsequently applied toextremal black holes in various theories and dimensions, see e.g. [14] and references therein. Laterdevelopments extended these results in two ways:1. It was shown that the Virasoro algebra of Near Horizon Extreme Kerr consists of symplectic ymmetries [16, 17] that do not actually require fall-off conditions for the metric; instead onecan construct a family of mutually diffeomorphic but physically inequivalent solutions thatspan a phase space of “boundary gravitons”.2. The action of symplectic symmetry transformations on space-time can be extended beyond thenear-horizon region, everywhere in the bulk of the extremal black hole [22].In this work we study the symplectic symmetry group of the near-horizon region of a large classof extremal black holes in diverse dimensions, specifically the Near-Horizon Extremal Geometries (NHEGs) of [16, 17, 23]. These are solutions of ( n + 4)-dimensional vacuum Einstein equationswith SL(2 , R ) × U(1) n +1 isometry. The near-horizon geometries of extremal Kerr ( n = 0) and five-dimensional Myers-Perry black holes ( n = 1) fall in this class, while higher-dimensional Myers-Perrysolutions ( n >
1) do not since they lack U(1) n +1 isometries. In that setting, black hole entropyarises as a Noether charge that also happens to be the central charge of the symplectic symmetryalgebra [17, 25]. It is subject to laws of NHEG mechanics that can be obtained from the zerotemperature limit of standard black hole thermodynamics [26].Accordingly, our motivation is to make progress towards the identification of the microstatesresponsible for NHEG entropy by classifying the possible homogeneous phase spaces with NHEGsymmetry. Since the NHEG group extends the Virasoro group familiar from two dimensional CFTs,our first goal will be to define it abstractly, including the central extension that plays a key rolefor entropy-matching. Similarly to the Virasoro group that extends the group Diff( S ) of diffeomor-phisms of the circle, the NHEG group will be based on Diff( T n +1 ), the group of diffeomorphismsof the torus, albeit with an anisotropy vector (cid:126)k related to the n + 1 angular momenta of the back-ground. (For n = 0 the NHEG group reduces to the Virasoro group of Kerr/CFT.) It turns outthat provided (cid:126)k satisfies a natural quantization condition, the NHEG group is a bundle of Virasorogroups over an n -dimensional torus T n . Equipped with these prerequisites we shall classify the or-bits of NHEG backgrounds under the NHEG group, i.e. its coadjoint orbits; these come equippedwith a natural symplectic form left invariant by the action of NHEG transformations. Owing to thesimilarity between NHEG and Virasoro algebras, this classification will be closely related to that ofstandard Virasoro orbits [27–30]. Each NHEG orbit can be seen as a set of physically inequivalentfield configurations dressing a given background, and our goal is effectively to classify all possiblesuch dressings. Note that, within a given orbit, all points correspond to space-time metrics withSL(2 , R ) × U(1) n +1 isometry, having identical n + 1 angular momentum charges and identical entropy.We will also quantize the orbits (promoting Poisson brackets to commutators) and build irreducibleunitary representations of the NHEG algebra. These representations are continuous tensor productsof Verma modules over T n , which will allow us to evaluate their characters.The plan is as follows. We start in section 2 by reviewing the construction of NHEG symmetriesin the gravitational context. This is then used in section 3 to motivate an abstract definition for theNHEG group, its algebra and their central extensions. In section 4 we apply this definition to classify The name “symplectic symmetry” was coined in [15] for cases where the pre-symplectic density ω vanishes.Symplectic (as opposed to asymptotic) symmetries are such that surface charges can be defined on any codimensiontwo compact spacelike surface — not necessarily at infinity [16–19]. Apart from NHEGs, examples of symplecticsymmetries include Brown-Henneaux transformations of Ba˜nados geometries [18, 20, 21] and ADM charges [19]. For n = 0, near-horizon extreme Kerr is the unique metric in this class while for n = 1 there are other solutions, e.g. those obtained in the near horizon limit of extremal black rings or boosted Kerr strings [24]. This procedure will rely on the unproven assumption that coadjoint orbits of the Virasoro group can be quantized,which is a thorny mathematical issue [31]. We make no claims of rigour and confine the discussion to the formal level.
Notation.
The algebra of vector fields on a circle (the Witt algebra) will be denoted as Vect( S ),and its central extension (Virasoro) as (cid:100) Vect( S ). The corresponding groups are Diff( S ) and (cid:100) Diff( S ),respectively. The NHEG algebra, the NHEG group and their central extensions will be respectivelydenoted by Vect (cid:126)k ( T n +1 ), Diff (cid:126)k ( T n +1 ), (cid:100) Vect (cid:126)k ( T n +1 ) and (cid:100) Diff (cid:126)k ( T n +1 ). In this section we review the Near-Horizon Extremal Geometries studied in [16, 17, 23]; they solvethe ( n + 4)-dimensional vacuum Einstein equations and have an SL(2 , R ) × U(1) n +1 isometry group. Metrics.
Following [17], we consider an ( n + 4)-dimensional space-time endowed with a time co-ordinate t , a radial coordinate r , an azimuthal coordinate θ ∈ [0 , π ] and angular coordinates φ i ∈ R where i = 1 , ..., n + 1. These angles are identified as φ i ∼ φ i + 2 π , so they label the points of an( n + 1)-torus T n +1 . We shall think of the coordinates ( t, r, θ, φ , ..., φ n +1 ) as being defined in thenear-horizon region of an extremal black hole; the metric of that region is fixed by SL(2, R ) × U(1) n +1 isometry as ds = Γ( θ ) (cid:20) − r dt + dr r + dθ + γ ij ( θ ) (cid:0) dφ i − k i rdt (cid:1)(cid:0) dφ j − k j rdt (cid:1)(cid:21) , (2.1)where the functions Γ , γ ij are constrained by Einstein’s equations. They generally depend on ( n +2)( n + 1) / n are contained in the components k i , which are arbitrary up toone relation, and the other n ( n + 1) / γ ij [23]. The n = 0 casecorresponds to Near-Horizon Extreme Kerr [13, 32, 33] with k = 1, while n = 1 gives the near-horizon geometry of extremal Myers-Perry black holes or rings with k = 1 /k [24, 32]. Note thatunder SL(2 , R ) isometries which keep the t, r parts of the metric intact, rdt transforms by a closedform, rdt → rdt + dξ , where ξ depends on the details of the transformation. Hence SL(2 , R ) isometriesalso involve translations of φ i along k i , φ i → φ i + k i ξ , so the vector (cid:126)k also affects the generators(Killing vectors) of the SL(2, R ) isometry [25].NHEGs are not black holes (they have no event horizon), but they do have infinitely manybifurcate Killing horizons at constant t, r , all at the same Frolov-Thorne [34] temperature 1 / π [25];see [17, sec. 2.1] for details. The metric on each bifurcation surface is ds H = Γ( θ ) (cid:0) dθ + γ ij ( θ ) dφ i dφ j (cid:1) . (2.2)It is smooth for all values of θ ; although (cid:126)k does not appear here, it is a measurable physical parameteras its components are related to the angular momentum of the black hole — see [17, 25]. Thus, onthe horizon, and of course on the whole NHEG (2.1), one is dealing with an anisotropic torus — atorus with a preferred direction specified by (cid:126)k . In principle, the components of (cid:126)k may take any value;in practice however, the ratios of those components are directly related to ratios of components of3ngular momentum. Assuming that the latter is quantized, this implies that (cid:126)k is proportional to avector with integer entries, i.e. an element of the dual lattice of T n +1 . Throughout this work we willalways assume that this condition is satisfied, as it will be necessary to ensure smoothness of theNHEG group.Let us then consider an anisotropic torus whose (cid:126)k belongs to the dual lattice. In that caseone can use the SL( n + 1 , Z ) volume-preserving symmetry of the torus to bring (cid:126)k , possibly up tonormalization, to the convenient form (cid:126)k = (0 , ..., , . (2.3)However, note that smoothness of the metric (2.2) generally prevents one from finding a global framewhere (cid:126)k has this form for all values of θ . This fact will make the connection between NHEG orbitsand the corresponding space-time metrics somewhat subtle; see section 7. Symplectic symmetries.
Consider a metric g µν solving Einstein’s equations and let χ , χ bevector fields; the corresponding symplectic density is a two-form in field space, ω ( δ χ g µν , δ χ g µν ; g µν ),where δ χ g µν = L χ g µν . This density can be of the Lee-Wald [7] or Barnich-Brandt [3] type, or eitherof them up to a boundary term (see [17] for details). When ω vanishes on-shell for suitable vectorfields, the latter generate symplectic symmetries [17–19]. In such cases, the integrability condition oncharge variations is usually satisfied automatically and surface charges can be defined by integrationover generic compact, space-like, codimension-two surfaces. In contrast to the perhaps more familiarasymptotic symmetries, these surfaces need not be at infinity; for the NHEG (2.1) they can be locatedat arbitrary ( t, r ). Furthermore, one can view the charges as generators of symplectomorphisms ona phase space built by acting on a background metric with finite diffeomorphisms generated by χ ’s.Each such phase space is an orbit of the symplectic symmetry group.For the NHEG (2.1), an interesting family of vector fields generating symplectic symmetries isgiven by [16] χ [ (cid:15) ( (cid:126)φ )] = (cid:15) (cid:126)k · (cid:126)∂ − (cid:126)k · (cid:126)∂(cid:15) (cid:16) r ∂ t + r∂ r (cid:17) , (2.4)where we write (cid:126)φ = ( φ , ..., φ n +1 ) and (cid:15) = (cid:15) ( (cid:126)φ ) is an arbitrary function on T n +1 (it is 2 π -periodicin all φ i ’s); we also let (cid:126)∂ be the gradient operator ( ∂ φ , ..., ∂ φ n +1 ) on T n +1 and write (cid:126)k · (cid:126)∂ = k i ∂ φ i .These vector fields, unlike those of generic asymptotic symmetries, are exact in r . By definition, theirLie brackets span the NHEG algebra ; its structure is most easily described by defining generators χ (cid:126)n = χ [ e i(cid:126)n · (cid:126)φ ], (cid:126)n ∈ Z n +1 , whose brackets read i [ χ (cid:126)m , χ (cid:126)n ] = (cid:126)k · ( (cid:126)m − (cid:126)n ) χ (cid:126)m + (cid:126)n . (2.5)It was shown in [16, 17] that conserved charges associated with the vector fields (2.4) are well-definedand that they close according to the NHEG algebra up to a classical central extension. Indeed, if wedenote by L (cid:126)n the surface charge corresponding to χ (cid:126)n , one has the Poisson brackets i { L (cid:126)m , L (cid:126)n } = (cid:126)k · ( (cid:126)m − (cid:126)n ) L (cid:126)m + (cid:126)n + c
12 ( (cid:126)k · (cid:126)m ) δ (cid:126)m + (cid:126)n, , (2.6)with c
12 = S π = (cid:126)k · (cid:126)J , (2.7)4here S and (cid:126)J are respectively the entropy and angular momenta of the underlying extremal blackhole, or equivalently of the corresponding near-horizon geometry.Starting from a vector field (2.4), one can exponentiate it to get a first glimpse of the (centrallyextended) NHEG group. This exponential is a finite diffeomorphism x (cid:55)→ ¯ x determined by the flowof χ [ (cid:15) ] and it takes the form [17]¯ φ i = φ i + k i F ( (cid:126)φ ) , ¯ r = re − Ψ( (cid:126)φ ) , ¯ t = t − r ( e Ψ( (cid:126)φ ) − , e Ψ ≡ (cid:126)k · (cid:126)∂F, (2.8)which indeed reduces to (2.4) for F = (cid:15) (cid:28)
1. (Note that we are not assuming anything about thenorm of (cid:126)k .) The NHEG phase space is obtained by applying all diffeomorphisms of the form (2.8)to the background metric (2.1); it results in a family of metrics of the form ds = Γ( θ ) (cid:104) − ( σ − d Ψ) + (cid:16) drr − d Ψ (cid:17) + dθ + γ ij ( d ˜ φ i + k i σ )( d ˜ φ j + k j σ ) (cid:105) , (2.9)with σ = e − Ψ rdt + (1 − e − Ψ ) drr , ˜ φ i = φ i + k i ( F − Ψ) . (2.10)Even though each such metric is related to the background (2.1) by a diffeomorphism (2.8), theresulting space-time manifolds should be seen as genuinely distinct configurations of the gravitationalfield because their surface charges differ. Thus, starting from a given NHEG background, one obtainsan entire family of physically distinct metrics labelled by different functions F ( (cid:126)φ ); this family spansan orbit of the (centrally extended) NHEG group. One of the purposes of this paper is precisely toclassify all such orbits and see how the metrics (2.9) fit in that classification. NHEG charges.
On the phase space of metrics (2.9), the surface charges L (cid:126)n generating NHEGtransformations as in (2.6) read [17] L (cid:126)n = (cid:90) H Ω T [Ψ] e − i(cid:126)n · (cid:126)φ , (2.11)where H is the horizon with metric (2.2), Ω is its volume form and T is the “stress tensor” T [Ψ] = 116 πG (cid:16) (Ψ (cid:48) ) − (cid:48)(cid:48) + 2 e (cid:17) (2.12)where primes denote directional derivatives along (cid:126)k , i.e. Ψ (cid:48) = (cid:126)k · (cid:126)∂ Ψ. Now, the transformation lawof Ψ (as defined in (2.8)) under the NHEG algebra is [17] δ (cid:15) Ψ = (cid:15) Ψ (cid:48) + (cid:15) (cid:48) , δ (cid:15) e Ψ = ( (cid:15)e Ψ ) (cid:48) , (2.13)which is to say that e Ψ is a primary field with unit weight under the Virasoro transformationsgenerated by L (cid:126)n ’s whose (cid:126)n is proportional to (cid:126)k . Consistently with this observation, the stress tensor(2.12) can be written in a more inspiring form. Indeed, defining a Schwarzian derivative {F ( (cid:126)φ ); (cid:126)φ } ≡ F (cid:48)(cid:48)(cid:48) F (cid:48) − F (cid:48)(cid:48) F (cid:48) , (2.14)one finds that for F (cid:48) ≡ e Ψ the expression (2.12) can be recast as T [Ψ] = 18 πG (cid:0) F (cid:48) − {F ; (cid:126)φ } (cid:1) . (2.15)Here F is related to the F of (2.8) by F ( (cid:126)φ ) = (cid:126)k · (cid:126)φ/ | k | + F ( (cid:126)φ ). We will use this suggestive rewritingbelow to relate the family of metrics (2.9) to a NHEG coadjoint orbit.5 NHEG group and algebra
In this section we provide an abstract definition of the NHEG group and its algebra, including centralextensions. The main goal is to recover and extend the structures that emerge from the symmetryanalysis summarized in the previous section. In particular we shall assume throughout that theanisotropy vector (cid:126)k takes the simple form (2.3), which entails no loss of generality within the class ofvectors (cid:126)k whose components have rational ratios. The classification of the corresponding coadjointorbits is postponed to section 4.
To define the NHEG group we proceed in two steps: first dealing with its centerless form, then addingcentral extensions.
Centerless version.
Consider an ( n + 1)-torus T n +1 = R n +1 / Z n +1 with coordinates φ , ..., φ n +1 ∈ R , each identified as φ i ∼ φ i + 2 π . The NHEG transformations generated by vector fields of the form(2.4) act on that torus according to (cid:126)φ (cid:55)→ (cid:126)φ + (cid:15) ( (cid:126)φ ) (cid:126)k. (3.1)The set of such infinitesimal transformations is a subalgebra of Vect( T n +1 ) that we denote byVect (cid:126)k ( T n +1 ). For (cid:126)k = (0 , ..., , φ n +1 (cid:55)→ φ n +1 + (cid:15) ( (cid:126)φ ), with φ , ..., φ n leftunchanged; the exponential of this transformation is a diffeomorphism( φ , ..., φ n , φ n +1 ) (cid:55)→ (cid:0) φ , ..., φ n , F ( φ , ..., φ n +1 ) (cid:1) (3.2)where F would have been written as F ( (cid:126)φ ) = φ n +1 + F ( (cid:126)φ ) with the notation of (2.8). It is such that F ( φ , ..., φ n , φ n +1 + 2 π ) = F ( φ , ..., φ n , φ n +1 ) ± π and ∂ F /∂φ n +1 (cid:54) = 0 , F ( φ , ..., φ i + 2 π, ..., φ n +1 ) = F ( φ , ..., φ n , φ n +1 ) + 2 πN i ∀ i = 1 , ..., n (3.3)where N , ..., N n are integers that may take different values for different F ’s (so the last line is justthe requirement that F be 2 π -periodic in φ , ..., φ n modulo 2 π ). Thus the NHEG group isDiff (cid:126)k ( T n +1 ) = C ∞ (cid:0) T n , Diff( S ) (cid:1) . (3.4)It is the set of smooth maps that send a point ( φ , ..., φ n ) on a circle diffeomorphism F ( φ , ..., φ n , · ).In other words, it is a bundle of Diff( S )’s over T n , which already suggests that its central extensionwill be a bundle of Virasoro groups over T n . All our later observations follow from this basic fact.To lighten the notation, from now on we write φ n +1 ≡ ϕ and ( φ , ..., φ n ) ≡ Φ, as well as F ( φ , ..., φ n , φ n +1 ) = F ( ϕ, Φ) ≡ F Φ ( ϕ ) (3.5)so that F ( φ , ..., φ n , · ) = F Φ . We also denote partial derivatives with respect to φ n +1 = ϕ by a prime: ∂ F /∂φ n +1 = F (cid:48) . Derivatives with respect to the remaining coordinates φ , ..., φ n will never appear,as these angles are mere spectators or “parameters” from the point of view of the NHEG group. Inparticular, the group operation is ( F , G ) (cid:55)→ F · G with( F · G )( ϕ, Φ) = F (cid:0) G ( ϕ, Φ) , Φ (cid:1) , i.e. ( F · G ) Φ = F Φ ◦ G Φ ∀ Φ ∈ T n . (3.6)6ote that for n = 0 the “torus” T n = T contains only one point and the NHEG group (3.4) reducesto Diff( S ).A remark: since the fundamental group of the torus T n is Z n , the NHEG group (3.4) has infinitelymany connected components (for n > N i ; any two elements of the NHEG group whose winding numbers differ belong todifferent connected components. Furthermore, the group Diff( S ) has two connected components,corresponding to diffeomorphisms that preserve or break the orientation of the circle. In otherwords, each connected component of the NHEG group can be labelled by (i) the winding number ofits elements, and (ii) the sign plus or minus in the first line of (3.3). In practice however, we will onlydeal with the maximal connected subgroup of the NHEG group, i.e. the component of the identity.It consists of diffeomorphisms F with zero winding number and which preserve the orientation of thecircle so that F (cid:48) >
0. From now on we simply refer to this connected group as “the NHEG group”.
Centrally extended version.
In order to reproduce the centrally extended surface charge algebra(2.6), we need to define a centrally extended version of the NHEG group. Since the coordinates φ , ..., φ n behave as parameters, we can define a “local” central extension such that the correspondingcentral charge is a function of φ , ..., φ n . To see this, consider the set of all pairs ( F , α ) whose entriesare (i) a diffeomorphism F ( φ , ..., φ n , φ n +1 ) = F ( ϕ, Φ) belonging to the NHEG group (3.4), and (ii)a function α ( φ , ..., φ n ) = α (Φ) on the torus T n . Consider then the group operation( F , α ) · ( G , β ) = (cid:0) F · G , α + β + C [ F , G ] (cid:1) (3.7)where the function C [ F , G ] on T n is a simple generalization of the Bott(-Thurston) cocycle [35]: C [ F , G ](Φ) = − π (cid:90) π dϕ log( F (cid:48) Φ ◦ G Φ ) G (cid:48)(cid:48) Φ G (cid:48) Φ . (3.8)The set of such pairs ( F , α ) spans the centrally extended NHEG group, (cid:100) Diff (cid:126)k ( T n +1 ) = C ∞ (cid:0) T n , (cid:100) Diff( S ) (cid:1) ; (3.9)it is an extension of (3.4) by the space C ∞ ( T n ) of smooth, real functions α . This extension is centralsince it commutes with everything, and it implies that the NHEG algebra can have infinitely manycentral charges. The constant, Φ-independent central charge of the surface charge algebra (2.6) isrecovered upon replacing the space C ∞ ( T n ) by R and replacing the group operation (3.7) by( F , α ) · ( G , β ) ≡ (cid:16) F · G , α + β + (cid:90) T n d Φ C [ F , G ] (cid:17) , (3.10)where now α, β ∈ R and d Φ ≡ dφ ...dφ n / (2 π ) n so that (cid:82) T n d Φ = 1. In other words, the centralextension here is the zero-mode projection of the Φ-dependent one in (3.7). As we now verify, theLie algebra that follows from these constructions contains and extends (2.6).
Eq. (3.1) says that the Lie algebra of the NHEG group consists of vector fields (cid:15) = (cid:15) ( (cid:126)φ ) (cid:126)k · (cid:126)∂ where (cid:126)∂ is the gradient on T n +1 . With the convention (cid:126)k = (0 , ..., , = (cid:15) ( ϕ, Φ) ∂ ϕ ≡ (cid:15) Φ ( ϕ ) ∂ ϕ , so from now on we write any element of the NHEG algebra as (cid:15) or (cid:15) Φ . Toobtain the adjoint representation of the NHEG group, we use the general definition (cid:0) Ad F (cid:15) (cid:1) ( ϕ, Φ) ≡ ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:16) F · e t (cid:15) · F − (cid:17) ( ϕ, Φ) (3.11)where the product of group elements is (3.6), so it is a Φ-pointwise multiplication on T n given bycomposition along the S spanned by ϕ . Thus the computation is the same as for the Virasoro group(see e.g. [36, sec. 4.4.2] or [37, sec. 6.1.4]), up to an extra parametric dependence on Φ ∈ T n . Theresult is (cid:0) Ad F (cid:15) (cid:1) Φ ( ϕ ) = (cid:15) Φ ( F − ( ϕ ))( F − ) (cid:48) ( ϕ ) , i.e. (cid:0) Ad F (cid:15) (cid:1) Φ ( F Φ ( ϕ )) = F (cid:48) Φ ( ϕ ) (cid:15) Φ ( ϕ ) . (3.12)We can repeat the same arguments with the centrally extended group (3.7); once more the result isthe same as in the Virasoro case up to a parametric dependence on Φ: (cid:99) Ad ( F ,α ) ( (cid:15) , β ) = (cid:16) Ad F (cid:15) , β − π (cid:90) π dϕ (cid:15) ( ϕ ) {F ; ϕ } (cid:17) , (3.13)where α , β and the second entry on the right-hand side are functions of Φ ∈ T n . For instance, whatwe denote by {F ; ϕ } is the function of Φ and ϕ such that {F ; ϕ } (Φ) ≡ {F Φ ; ϕ } , with {F ; ϕ } = F (cid:48)(cid:48)(cid:48) / F (cid:48) − ( F (cid:48)(cid:48) / F (cid:48) ) the standard Schwarzian derivative.The adjoint representation yields the Lie bracket of the algebra thanks to the definition (cid:2) ( (cid:15) , α ) , ( ζ , β ) (cid:3) ≡ − ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:99) Ad ( e t (cid:15) ,tα ) ( ζ , β ) , (3.14)where the minus sign is a matter of convention. (Note that we are working with the centrally extendedgroup.) Using (3.13) and evaluating the t derivative, one finds (cid:2) ( (cid:15) , α ) , ( ζ , β ) (cid:3) = (cid:16) [ (cid:15) , ζ ] , π (cid:90) π dϕ (cid:15) (cid:48)(cid:48)(cid:48) ( ϕ ) ζ ( ϕ ) (cid:17) , (3.15)where, as before, α , β and the second entry on the right-hand side are functions of Φ. The first entryof the right-hand side involves the standard Lie bracket of vector fields. To make the structure ofthe algebra more explicit, we can expand all functions in Fourier modes on the torus. Thus, we let (cid:126)m, (cid:126)n ∈ Z n +1 and define the NHEG generators L (cid:126)m ≡ (cid:0) e i (cid:126)m · (cid:126)φ (cid:126)k · (cid:126)∂, (cid:1) . (3.16)We also take (cid:126)M ∈ Z n and define the (infinitely many) central charges c ( (cid:126)M, ≡ (0 , e i (cid:126)M · Φ ) . (3.17)Then the Lie bracket (3.15) implies that the central charges commute with everything, while thebrackets of NHEG generators lead to the expected centrally extended NHEG algebra: i (cid:2) L (cid:126)m , L (cid:126)n (cid:3) = (cid:126)k · ( (cid:126)m − (cid:126)n ) L (cid:126)m + (cid:126)n + c (cid:126)m + (cid:126)n
12 ( (cid:126)k · (cid:126)m ) δ (cid:126)k · ( (cid:126)m + (cid:126)n ) , , (3.18)where (cid:126)k ≡ (0 , ..., , (cid:126)k on the dual lattice of T n +1 . Notethat this is the unique maximal central extension of the centerless NHEG algebra (up to redefinitions8f generators, such as shifts in L (cid:126) ); this can be verified directly by requiring the central terms tosatisfy Jacobi identities. In particular, there is no Kac-Moody central extension of the type thatmight be expected for maps from T n to a generic Lie algebra; the reason is that the Virasoro algebrahas no invariant bilinear form.The centerless version of the bracket (3.18) clearly reproduces the symplectic symmetry algebra(2.5), while the restriction of central charges to their zero-mode reproduces the surface charge algebra(2.6). Note that (3.18) has a Virasoro subalgebra generated by L m(cid:126)k , and that the generators L (cid:126)m , (cid:126)m · (cid:126)k = 0 span an Abelian subalgebra that may be viewed as C ∞ ( T n ). An equivalent way to phrase thisis to introduce NHEG generating fields and their Fourier modes along (cid:126)k , L ( ϕ, Φ) = (cid:88) (cid:126)n L (cid:126)n e i(cid:126)n · (cid:126)φ , L n (Φ) ≡ π (cid:90) π dϕ L ( ϕ, Φ) e − inϕ . (3.19)In these terms the bracket (3.18) reads i (cid:2) L m (Φ) , L n (Φ (cid:48) ) (cid:3) = (cid:18) ( m − n ) L n + m (Φ) + c (Φ)12 m δ n + m, (cid:19) δ n (Φ − Φ (cid:48) ) , (3.20)where the central charges of (3.17) are the Fourier modes of c (Φ). Thus the NHEG algebra is a bundleof Virasoro algebras over an n -dimensional torus T n ; this was of course expected from (3.4)-(3.9).The remainder of this paper is devoted to some consequences of this observation. Consider a space-time metric that solves Einstein’s vacuum equations and that takes the NHEG form(2.1). This metric is a point in an infinite-dimensional phase space that consists of gravitational fieldsrepresenting the near-horizon geometry of an extremal black hole. The NHEG symmetry group actson this phase space and relates various physically distinct metrics to one another; metrics that arerelated by NHEG transformations belong, by definition, to the same NHEG orbit. For instance, themetrics (2.9) span one such orbit. As it turns out, the transformation law of NHEG metrics underthe NHEG group is given by the coadjoint representation, so the purpose of this section is to studyand classify NHEG coadjoint orbits. The result will be closely related to the classification of Virasoroorbits [27–29]; see e.g. [30] or [37, chap. 7] for a review.By definition, any Lie group acts on its Lie algebra according to the adjoint representation,and the corresponding dual action is the coadjoint representation. In the present case, assuming (cid:126)k = (0 , ..., , L , c ) where L = L Φ ( ϕ ) dϕ is a Φ-dependent Virasoro coadjoint vector while c = c (Φ) is a Φ-dependent Virasorocentral charge. The pairing between ( L , c ) and the NHEG algebra is (cid:10) ( L , c ) , ( (cid:15) , α ) (cid:11) = 12 π (cid:90) π dϕ (cid:90) T n d Φ L ( ϕ, Φ) (cid:15) ( ϕ, Φ) + (cid:90) T n d Φ c (Φ) α (Φ) . (4.1)For the constant central extension defined in (3.10), the second term of the right-hand side reducesto a product c · α , where both c and α are just real numbers. Strictly speaking, both L and c are the coefficients of volume forms L ⊗ d Φ and c d
Φ on T n , but since NHEGtransformations never affect the Φ coordinates this abuse of notation is harmless. i.e. the coadjoint representation ofthe Virasoro group —, up to a parametric dependence on Φ. Explicitly, this representation is definedby the general formula (cid:10) (cid:99) Ad ∗ ( F ,α ) ( L , c ) , ( (cid:15) , α ) (cid:11) = (cid:10) ( L , c ) , (cid:99) Ad ( F ,α ) − ( (cid:15) , α ) (cid:11) . (4.2)Using the fact that the NHEG adjoint action is (3.13), one finds (cid:99) Ad ∗ ( F ,α ) ( L , c ) = (cid:16) Ad ∗F L − c {F − ; ·} , c (cid:17) , (4.3)where (cid:16) Ad ∗F L − c {F − ; ·} (cid:17) ( ϕ, Φ) = (cid:2) ( F − ) (cid:48) ( ϕ ) (cid:3) L Φ ( F − ( ϕ )) − c (Φ)12 {F − ; ϕ } . (4.4)The same formula follows from the ‘rigid’ central extension of (3.10), except that the correspondingcentral charge is independent of Φ; note the similarity with (2.15).The transformation law (4.4) implies that each coadjoint orbit of the NHEG group is a bundle ofVirasoro orbits over T n ; the fibre at Φ is the Virasoro orbit of the CFT stress tensor L Φ with centralcharge c (Φ) — see fig. 1. Schematically, any NHEG orbit O takes the form of a disjoint union O = (cid:71) Φ ∈ T n O Φ (4.5)where each O Φ is a coadjoint orbit of the Virasoro group. With the Φ-dependent central extension of(3.7), both the coadjoint vectors and the central charges of these Virasoro orbits generally vary withΦ; by contrast, with the rigid central extension of (3.10), the central charge takes the same valuefor all points Φ on T n , but the stress tensors L Φ vary. In principle, this achieves our goal: since theclassification of Virasoro orbits is known, we can describe any NHEG orbit as a bundle of Virasoroorbits over T n . For instance, the orbit (2.9) is obtained when c (Φ) = 6 S/π and L Φ = c/
12 areconstant, with the identification between L and the stress tensor of (2.15) given by A · T = (cid:99) Ad ∗F − L in terms of the horizon area A = 4 GS .Note that, within a NHEG orbit, one can move between different Virasoro orbits by varying Φ,but the orbits that are spanned in the process are not entirely arbitrary. Indeed, in order for theNHEG orbit to be smooth, there must be no singularity in the bundle (4.5); to ensure this, all O Φ ’sshould be specified by the same winding number and have the same type of SL(2 , R ) monodromies,meaning that the monodromy matrices of the associated Hill’s equations should either be all elliptic,or all hyperbolic, or all mutually conjugate and parabolic; see again fig. 1. Thus, for example, if L Φ = − c (Φ ) /
24 at some point Φ , then the corresponding Virasoro orbit O Φ has unit windingnumber and trivial monodromy up to a sign — it is the orbit of a CFT vacuum stress tensorunder conformal transformations —; but smoothness then requires that any other O Φ also be aVirasoro vacuum orbit, which means that there exists a NHEG transformation which turns theNHEG coadjoint vector L into (cid:0) F · L (cid:1) ( ϕ, Φ) = − c (Φ) /
24 for any Φ ∈ T n . In other words, once thecentral charges c (Φ) are fixed, there exists a unique smooth NHEG orbit that contains a fibre whichis a vacuum Virasoro orbit. Amusingly, this means that local data (from the point of view of T n )contains global information within the class of smooth NHEG orbits: knowing O Φ gives information Hill’s equation and the associated monodromies and winding numbers are tools that are routinely used to classifyVirasoro orbits [27, 28]. See e.g. [30] or [37, chap. 7] for a review; see also eq. (4.9) below. T n ; for simplicity we take n = 1 and represent Virasoro orbits by disks, so that NHEG orbits havethe topology of solid tori. The radius of the disk varies with its position Φ, reflecting the fact thatdifferent fibres generally correspond to different Virasoro orbits. In (a) the NHEG orbit is smooth,but in (b) the fibre at Φ ∗ is degenerate (it has zero radius) while its immediate neighbours are not, sothe corresponding NHEG orbit is pathological. This is due to discontinuous jumps between Virasoroorbits of different types; it occurs for instance when a NHEG coadjoint vector L crosses the criticalvalue L Φ ∗ = − c (Φ ∗ ) /
24. We discard such singular NHEG orbits from our classification and onlyinclude smooth orbits such as the one in (a).on the other Virasoro orbits O Φ . Below we provide examples of smooth orbits for which the fibres atdifferent Φ’s are genuinely different Virasoro orbits; but first let us revisit the classification of NHEGorbits from a slightly different point of view. Lie-algebraic data.
Following [29], a quick way to guess the classification of coadjoint orbits isto find the Lie algebras of their stabilizers. The starting point is the Lie algebra representation thatcorresponds to the coadjoint transformation law (4.4). Taking F ( ϕ, Φ) = ϕ + (cid:15) ( ϕ, Φ) in the latterequation and working to first order in (cid:15) , one findsad ∗ (cid:15) L = (cid:15) ( ϕ, Φ) L (cid:48) Φ + 2 (cid:15) ( ϕ, Φ) (cid:48) L Φ − c (Φ)12 L (cid:48)(cid:48)(cid:48) Φ . (4.6)Now, the stabilizer of L consists of those NHEG group elements that leave it fixed; in Lie-algebraicterms this is to say that the algebra of the stabilizer is spanned by (cid:15) ’s for which (4.6) vanishes: (cid:15) ∈ stabilizer algebra ⇐⇒ (cid:15) ( ϕ, Φ) L (cid:48) Φ + 2 (cid:15) ( ϕ, Φ) (cid:48) L Φ − c (Φ)12 L (cid:48)(cid:48)(cid:48) Φ = 0 . (4.7)Of course, this equation and (4.6) coincide with standard Virasoro expressions up to a parametricdependence on the transverse coordinates Φ.Knowing the stabilizer (call it G L ) of a NHEG coadjoint vector L , one infers that the corre-sponding orbit is diffeomorphic to a quotient of the NHEG group by G L . For example, suppose that c (Φ) = c > L Φ = L > − c/
24 are positive constants; then the stabilizer of ( L , c ) is generatedby all (cid:15) ’s such that (cid:15) (cid:48) = 0, so it consists of maps from T n to the group U(1) ∼ = S of rigid rotations.As a result the corresponding NHEG orbit is diffeomorphic to the quotient space O ( L,c ) ∼ = C ∞ (cid:0) T n , Diff( S ) (cid:1)(cid:46) C ∞ (cid:0) T n , S (cid:1) ∼ = C ∞ (cid:0) T n , Diff( S ) /S (cid:1) for constant L (cid:54) = − c/ L Φ ( ϕ ) = − c (Φ) /
24 is the group of smooth maps from T n to PSL(2 , R ).Note once more that smoothness of the orbit restricts the allowed configurations L Φ ( ϕ ): if L Φ = − c (Φ ) /
24 at some point Φ , then G L contains maps that send Φ on PSL(2 , R ); but in order for G L to be smooth, its elements must be able to map any other point Φ on PSL(2 , R ), which impliesas before that L Φ = − c (Φ) /
24 for any Φ.
Parametric Hill’s equations.
The standard classification of Virasoro coadjoint orbits relies onthe monodromies of Hill’s equation [27], so let us comment on the use of this method in the presentcontext. (For a pedagogical review, see [30] or [37, chap. 7].) The starting point is a function ψ ( (cid:126)φ )that we take to be single-valued on T n (so it is 2 π -periodic in the angular coordinates contained inΦ), but not necessarily periodic in ϕ . Given a NHEG coadjoint vector ( L , c ), we require ψ to solvethe parametric Hill’s equation − c (Φ)6 ψ (cid:48)(cid:48) ( (cid:126)φ ) + L Φ ( ϕ ) ψ ( (cid:126)φ ) = 0 . (4.9)The same equation would be relevant to the classification of Virasoro orbits, albeit without parametricdependence on Φ. The fact that L Φ ( ϕ ) is 2 π -periodic in ϕ implies that, for any Φ ∈ T n , any twolinearly independent solutions ψ , ψ of (4.9) behave in a quasi-periodic way: (cid:18) ψ ( ϕ + 2 π, Φ) ψ ( ϕ + 2 π, Φ) (cid:19) = M (Φ) (cid:18) ψ ( ϕ, Φ) ψ ( ϕ, Φ) (cid:19) , (4.10)where the monodromy matrix M (Φ) belongs to SL(2 , R ). Again, the only difference between thissetting and the standard Virasoro case is the dependence on Φ. In particular, similarly to the Virasorocase, the conjugacy class of the monodromy matrix is an invariant label for the orbit of ( L , c ); in thecase at hand, this label is a function on T n since the conjugacy class of M (Φ) depends on Φ. Thus,each NHEG orbit is labelled by Φ-dependent conjugacy classes in SL(2 , R ). In order for the orbitto be smooth, these conjugacy classes must all be of the same type at all Φ’s — elliptic, hyperbolicor parabolic. In addition, Virasoro orbits are labelled by a discrete winding number [30]; smoothNHEG orbits are such that this number is the same for all Φ ∈ T n .We conclude with an example. Consider a NHEG coadjoint vector L Φ ( ϕ ) = L Φ that does notdepend on ϕ (at fixed Φ, it is a constant from the point of view of the coordinate ϕ ). One readilyverifies that a corresponding normalized pair of solutions of Hill’s equation (4.9) is ψ ( ϕ, Φ) = ρ (Φ) (cid:112) K (Φ) e K (Φ) ϕ , ψ ( ϕ, Φ) = 1 ρ (Φ) (cid:112) K (Φ) e − K (Φ) ϕ , L Φ ( ϕ ) = c (Φ)6 K (Φ) , (4.11)where ρ (Φ) is any non-zero function on T n . When the function K (Φ) is real, the correspondingVirasoro orbits are hyperbolic; for K (Φ) = iν (Φ) / ν ∈ (0 , K (Φ) = i/ L Φ = − c (Φ) /
24 and the monodromy matrix is minus the identity.In the first two cases the stabilizer consists of maps from T n to U(1), as in (4.8); in the vacuum caseit is spanned by maps from T n to PSL(2 , R ). 12 NHEG-Kac-Moody algebra
In this section we describe a “free-field” construction of the NHEG algebra in terms of Abelian currentalgebras. This is partly motivated by representation theory, but it is also of interest in computationsof black hole entropy from near-horizon symmetries [21, 22]; we shall briefly return to the latter issuein the last section of this work.Let us consider a set of currents J i ( (cid:126)φ ) , i = 1 , ..., n + 1 and assume that their Fourier modes J i,(cid:126)n = 12 π (cid:90) π dϕ (cid:90) T n d Φ e − i(cid:126)n · (cid:126)φ J i ( (cid:126)φ ) (5.1)satisfy the centrally extended algebra i [ J i, (cid:126)m , J j,(cid:126)n ] = ( (cid:126)k · (cid:126)m ) g ij δ (cid:126)m + (cid:126)n, Z (5.2)where Z is a central charge and g ij is a constant metric on the torus T n +1 . This metric is arbitraryand none of our results will depend on it, so we may choose it to be the same as in (2.1) at somegiven θ . We refer to (5.2) as the NHEG-Kac-Moody algebra.When (cid:126)k is proportional to a vector in the dual lattice of T n +1 (which we assume to be the case),we can take (cid:126)k = (0 , ..., ,
1) and use the notation of section 3. Then eq. (5.2) can be written as J i,m (Φ) = 12 π (cid:90) π dϕ J i ( (cid:126)φ ) e imϕ , i [ J i,m (Φ) , J j,n (Φ (cid:48) )] = m g ij δ m + n, δ n (Φ − Φ (cid:48) ) . (5.3)As we see, J i, (Φ) is central for all Φ ∈ T n . To recover the NHEG algebra, we proceed as in thetwisted Sugawara construction and define L ( ϕ, Φ) ≡ β (Φ) k i J (cid:48) i ( ϕ, Φ) + 12 g ij J i ( ϕ, Φ) J j ( ϕ, Φ) (5.4)where g ij is the inverse of the metric g ij , β (Φ) is an arbitrary function of Φ and as before the primedenotes partial derivative with respect to ϕ . One can readily see that the the modes L n (Φ) of thisquantity close according to the NHEG algebra (3.20) with a Φ-dependent central charge c (Φ) = 12 β (Φ) k i g ij k j Z. (5.5)Note that this is non-zero only if the twisting term βkJ (cid:48) in (5.4) does not vanish. As for the bracketsof NHEG generators with currents, they take the form i [ L (cid:126)m , J i,(cid:126)n ] = − ( (cid:126)k · (cid:126)n ) J i, (cid:126)m + (cid:126)n + iβ ( (cid:126)k · (cid:126)n ) k i δ (cid:126)m + (cid:126)n, Z. (5.6)The central term on the right-hand side implies that the currents J i are not primary fields; thisis of course a standard feature of the twisted Sugawara construction (see e.g. [21, 38]), where thetwist term proportional to J (cid:48) in (5.4) gives rise both to the classical central extension (5.5) and tothe anomaly in (5.6). In the next section we will use this free-field point of view to build NHEGrepresentations. For generic (cid:126)k , we would have L ( (cid:126)φ ) = | k | β ( (cid:126)φ ) (cid:126)k · (cid:126)∂ ( (cid:126)k · (cid:126)J ( (cid:126)φ )) + 12 g ij J i ( (cid:126)φ ) J j ( (cid:126)φ ) with (cid:126)k · (cid:126)∂β ( (cid:126)φ ) = 0. Quantization of NHEG orbits
So far we have analysed NHEG symmetry from a classical perspective involving symplectic manifoldsand Poisson brackets. We now study some aspects of the quantization of these orbits, whereby Poissonbrackets are replaced by commutators according to the canonical prescription i {· , ·} → [ · , · ]. Thisresults in a Hilbert space acted upon by operators L n (Φ) that satisfy the commutator algebra (cid:2) L m (Φ) , L n (Φ (cid:48) ) (cid:3) = (cid:18) ( m − n ) L m + n (Φ) + c (Φ)12 ( m − m ) δ m + n, (cid:19) δ n (Φ − Φ (cid:48) ) . (6.1)Here we have shifted the L (Φ) of (3.20) by − c (Φ) /
24, and we assume that the central charge issome given, strictly positive function c (Φ). The operators L n (Φ) satisfy the Hermiticity conditions L m (Φ) † = L − m (Φ). Similarly, the quantization of the current algebra (5.3) gives commutators[ J i,m (Φ) , J j,n (Φ (cid:48) )] = m g ij δ m + n, δ n (Φ − Φ (cid:48) ) , (6.2)where J i,m (Φ) † = J i, − m (Φ). In the remainder of this section we describe irreducible unitary repre-sentations where these commutator algebras are realized. Since NHEG orbits are bundles of Virasoro orbits over T n , their quantization is in principle straight-forward if one assumes that the quantization of Virasoro orbits goes through. Indeed, if the Hilbertspace obtained by quantizing a Virasoro orbit is the space of a unitary highest-weight representationof the Virasoro algebra, then the Hilbert space obtained by quantizing a NHEG orbit is a continuoustensor product of Virasoro representations — one at each point Φ of T n . In particular, the highest-weight state | h (cid:105) of a NHEG representation is specified by a strictly positive real function h (Φ) on T n . It is such that L (Φ) | h (cid:105) = h (Φ) | h (cid:105) , L m (Φ) | h (cid:105) = 0 ∀ Φ ∈ T n , ∀ m > , (6.3)and we assume it to be normalized: (cid:104) h | h (cid:105) = 1. The NHEG vacuum state | (cid:105) is specified by h (Φ) = 0for all Φ ∈ T n , and it is annihilated by all L m (Φ)’s with m ≥ − N insertion points Φ , ..., Φ N on thetorus, and takes the form L − n (Φ ) L − n (Φ ) ...L − n k (Φ ) ...L − n N (Φ N ) ...L − n NkN (Φ N ) | h (cid:105) (6.4)where 1 ≤ n i ≤ ... ≤ n ik i for all i = 1 , ..., N . Strictly speaking, the norm of any such descendant isinfinite due to the delta function on the right-hand side of (6.1). For instance, (cid:104) h | L m (Φ) L − m (Φ (cid:48) ) | h (cid:105) = (cid:104) mh (Φ) + c (Φ)12 ( m − m ) (cid:105) δ n (Φ − Φ (cid:48) ) , (6.5)so one can think of these descendants as analogues of states with definite position in non-relativisticquantum mechanics. The true elements of the Hilbert space are actually smeared linear combinationsof descendants, such as (cid:90) T n d Φ Ψ(Φ) L − m (Φ) | h (cid:105) (6.6) The quantization of Virasoro orbits is still very much an area of active research; see e.g. [31]. Our standard ofrigour is by no means that of pure mathematics, so we shall bluntly assume that Virasoro quantization does work. T n . More generally, the number of wavefunctionsneeded to smear a descendant (6.4) is the number (cid:80) Ni =1 k i of L m (Φ) operators appearing in itsexpression. In particular, smeared states are generally highly non-local on T n .As in the Verma modules of the Virasoro algebra, the Hilbert space H of the representationis spanned by all linear combinations of the highest-weight state and its descendants; here we areincluding smearing such as (6.6) as a valid way to take linear combinations. Thus, abstractly, theHilbert space is an infinite tensor product of Virasoro Verma modules V c (Φ) ,h (Φ) , one at each pointof the n -torus: H = (cid:79) Φ ∈ T n V c (Φ) ,h (Φ) . (6.7)This product is the quantization of eq. (4.5); it is separable thanks to the fact that Φ is a coordinateon a torus rather than a non-compact manifold. In principle, any irreducible unitary representationof the NHEG algebra takes this form, and similar conclusions apply more generally to any quantumtheory with NHEG symmetry: any such theory is a bundle ( i.e. a tensor product) of chiral two-dimensional CFTs over an n -torus. One might say that T n is a “conformal manifold”, or modulispace, supporting a family of CFTs; the novelty here is that (i) this manifold has a space-timeinterpretation since it is embedded in the NHEG (2.1), and (ii) the symmetry algebra has thismoduli space built in, resulting in wavefunctions such as (6.6) that live on the conformal manifold. Hilbert space from NHEG-Kac-Moody.
Unitary representations of the NHEG algebra canalso be built thanks to the currents J i of section 5. As in the Virasoro case [38], these currents canbe seen as creation/annihilation operators generating the space of the representation when they acton a suitable vacuum state. To define the latter, we start by noting that[ J i, , L (cid:126)m ] = 0 = [ J i, , J j, (cid:126)m ] , ∀ (cid:126)m ∈ Z n +1 , ∀ i, j = 1 , ..., n + 1 , (6.8)implying that the n zero-modes J i, commute with all other generators; in fact, they span the centerof the universal enveloping algebra of NHEG-Kac-Moody. Accordingly, we label the vacuum state | J , ..., J n +1 (cid:105) ≡ | J i (cid:105) by its (local) eigenvalues J i (Φ) under central operators: J i,m (Φ) | J i (cid:105) = 0 and J i, (Φ) | J i (cid:105) = J i (Φ) | J i (cid:105) ∀ m > , ∀ Φ ∈ T n . (6.9)From the point of view of the NHEG algebra obtained by normal-ordering (5.4) and shifting it by c (Φ) /
24, these vacua are primary states of Φ-dependent weight h (Φ) = c (Φ)24 + g ij J i (Φ) J j (Φ): L m (Φ) | J k (cid:105) = 0 ∀ m > , L (Φ) | J k (cid:105) = h (Φ) | J k (cid:105) . (6.10)Each “vacuum” of NHEG-Kac-Moody is thus specified by n + 1 functions J i (Φ) on T n . Once such avacuum has been chosen, its descendants are built by acting with creation operators J i, − n (Φ) , n > T n . This is directly analogous to the construction (6.4)in terms of NHEG generators, and one can indeed show (along the same lines as for the Virasorocase [38]) that the Hilbert space spanned by descendants of | h (cid:105) is isomorphic, as a NHEG-module,to the one spanned by descendants of | J i (cid:105) provided h (Φ) ≥ T n . To conclude this section, let us evaluate the characters of the representations above. First note thatthe Cartan subalgebra of NHEG is infinite-dimensional: as is apparent in (6.1), it is spanned by all15enerators L (Φ) with Φ ∈ T n . This implies that the chemical potential itself is generally a functionof Φ, τ (Φ); as in standard CFT we assume it has positive imaginary part everywhere. The associatedcharacter is χ h,c ( τ ) = Tr H (cid:16) e (cid:82) Tn d Φ 2 πiτ (Φ) L (Φ) (cid:17) (6.7) = (cid:89) Φ ∈ T n Tr V c (Φ) ,h (Φ) (cid:16) e πiτ (Φ) L (Φ) (cid:17) , (6.11)where d Φ ≡ dφ ...dφ n / (2 π ) n as before. For definiteness we assume that c (Φ) > h (Φ) > T n . Then there are no null states in V c (Φ) ,h (Φ) and the counting of L eigenstates atlevel N reduces to the counting of partitions p ( N ) of the integer N . Using this in (6.11), we concludethat χ h,c ( τ ) = (cid:89) Φ ∈ T n e πiτ (Φ) h (Φ) ∞ (cid:89) m =1 − e πimτ (Φ) = exp (cid:20) (cid:90) T n d Φ (cid:18) πiτ (Φ) h (Φ) − ∞ (cid:88) m =1 log (cid:0) − e πimτ (Φ) (cid:1)(cid:19)(cid:21) . (6.12)A similar counting works for the vacuum representation where h (Φ) = 0 for all Φ, but then theproduct and sum over m start at m = 2. In the special case where τ (Φ) is constant, the torusintegral in (6.12) is finite ( (cid:82) d Φ = 1) and the character reduces to χ h,c ( τ ) = q (cid:82) d Φ h (Φ) ∞ (cid:89) m =1 − q m , q ≡ e πiτ . (6.13)This is just the character of a standard unitary Virasoro representation, free of null states, whoseeffective highest weight h eff = (cid:82) d Φ h (Φ) is the average of the h (Φ)’s. (Again, when h = 0 the productwould start at m = 2.) In principle one can perform the same computation when 0 < c (Φ) <
1, inwhich case the local Virasoro representations contain null states and the weights h (Φ) are constrainedby the Kac determinant. Then both c and h are forced to take discrete values, implying that they areboth constant over T n if one assumes continuity. The corresponding NHEG character thus coincideswith the character of a reducible Verma module.Recalling the relation between characters and gravitational partition functions [39], it would beinteresting to see if the NHEG characters (6.12)-(6.13) have anything to do with (one-loop) partitionfunctions of the gravitational field in the near-horizon region of extremal black holes. We will notattempt to address this intriguing issue here. In this work we have built the NHEG group, starting from the NHEG algebra presented in [16, 17].As we have seen, the NHEG group consists of diffeomorphisms of an ( n + 1)-torus that preservethe direction of an anisotropy vector (cid:126)k . When the latter is proportional to an element of the duallattice, the NHEG group effectively becomes a bundle of Virasoro groups over an n -torus, generallywith a position-dependent central charge. This simple observation allowed us to derive, for free,the classification of NHEG coadjoint orbits: each such orbit is a bundle of Virasoro orbits over T n (see fig. 1); the Virasoro orbits at different points generally differ, but they are constrained by therequirement that they form a smooth bundle, leading for instance to a statement of unicity of thevacuum NHEG orbit. In addition we derived the corresponding irreducible unitary representations,which are simply tensor products of Virasoro modules, and we computed their characters. In theremainder of this section we briefly address some open issues, extensions and plausible applicationsof our analysis. 16 rbits versus Metrics. As reviewed in section 2, the NHEG group (with constant central charge)represents the symplectic symmetries of the phase space of metrics (2.9). This space is a homogeneousmanifold with a NHEG-invariant symplectic form, so it is one of the coadjoint orbits described insection 4; in fact it is the (hyperbolic) orbit of the constant coadjoint vector L Φ = c/
12 with c = 6 S/π , provided one identifies (2.15) with (cid:99) Ad ∗F − L up to an overall normalization. This parallelssimilar identifications between AdS metrics and Virasoro coadjoint orbits [18, 40, 41], or their flatspace/BMS analogues [42]. In the same vein, one may ask whether more interesting profiles of NHEGcoadjoint vectors, e.g. those described in (4.11), correspond to well-defined space-time metrics solvingthe vacuum Einstein equations.Unfortunately, a geometric subtlety makes this identification between NHEG coadjoint orbitsand orbits of NHEG metrics somewhat difficult. Indeed, our construction of the NHEG group (andof its orbits) used the fact that the anisotropy vector (cid:126)k can be transformed in such a way that (cid:126)k = (0 , ..., , θ (recall the coordinates usedwhen writing (2.1)). In general, such a θ -dependent transformation is singular, thus preventing adirect identification between NHEG coadjoint vectors L Φ ( ϕ ) and NHEG metrics. A notable exceptionto this general expectation is provided by coadjoint orbits whose representative element is a constant, i.e. those with L Φ ( ϕ ) = const . Intuitively, this is consistent with the na¨ıve expectation that labels ofNHEG orbits should be related to the conserved charges associated with exact symmetries/Killingvectors of the background geometry (see [17–19]). Whether other NHEG orbits correspond to smoothspace-time metrics is a puzzling question, and we hope to return to it in the future. More on NHEG-Kac-Moody.
In section 5 we built the NHEG algebra in terms of u(1) currentsthanks to a twisted Sugawara construction, which we then applied to unitary NHEG representationsalong the same lines as in the Virasoro case [38]. In that context, a natural question is whetherthe currents have a geometric realization in terms of space-time diffeomorphisms, as is the case forinstance in three dimensions [21]. This project has already been carried out for extreme Kerr blackholes [22], but it should be possible to extend it to higher dimensions.A preliminary analysis extending the method of [22] has uncovered a closely related algebra whosegenerators are u(1) currents that depend on individual φ i ’s, from which one can obtain n + 1 copiesof Virasoro algebras [43]; it would be interesting to explore this direction further. NHEG Field Theories.
As mentioned in section 6.1, NHEG-invariant field theories are bundlesof CFTs over a T n moduli space. The point of view adopted in this paper, and the motivation for ourinvestigation, is that such field theories provide putative holographic duals for extremal black holes.It would be interesting to see what constraints are put on such theories by the requirement that theydescribe gravity. An obvious constraint comes from the Poisson bracket algebra (2.6), which saysthat the position-dependent central NHEG charge must in fact be a constant (at least up to quantumcorrections). Optimistically, a better understanding of NHEG field theories might eventually lead toan identification of extremal black hole microstates that does not rely on string theory [44].An illustrative application of the considerations of this paper to the identification of black holemicrostates is provided by the recent “horizon fluff” proposal [38], according to which these mi-crostates can be read off from a subtle relationship between near-horizon and asymptotic symmetryalgebras. This proposal has been successfully worked out for generic BTZ black holes [38] and ex-17remal Kerr black holes [22], and is expected to work for NHEGs too [43]. From this perspective,the orbit analysis presented in this work should be an important technical tool for the explorationof black hole entropy. Acknowledgements
We are especially grateful to Ali Seraj for his contributions in early stages of this project, and wealso thank Geoffrey Comp`ere for comments and discussions. The work of B.O. is supported by theSwiss National Science Foundation, and partly by the NCCR SwissMAP. M.M.Sh-J. would like tothank Abdus Salam ICTP where part of this work was carried out and he acknowledges the ICTPSimons associates program and the ICTP NT-04 network scheme. He also acknowledges supportfrom the Iranian NSF junior research chair in black hole physics.
References [1] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, “Gravitational waves in general rel-ativity. 7. Waves from axisymmetric isolated systems,” Proc. Roy. Soc. Lond. A , 21 (1962) • R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,” Proc. Roy. Soc. Lond. A , 103 (1962); Phys. Rev. , 2851 (1962).[2] J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of AsymptoticSymmetries: An Example from Three-Dimensional Gravity” Commun. Math. Phys. , 207(1986).[3] G. Barnich and F. Brandt, “Covariant theory of asymptotic symmetries, conservation laws andcentral charges,” Nucl. Phys. B , 3 (2002) [hep-th/0111246].[4] M. M. Sheikh-Jabbari, “Residual diffeomorphisms and symplectic soft hairs: The need to re-fine strict statement of equivalence principle,” Int. J. Mod. Phys. D , no. 12, 1644019 (2016)[arXiv:1603.07862 [hep-th]] • “Residual Diffeomorphisms and Symplectic Hair on Black Holes” ,seminar presented in workshop on Recent developments in symmetries and (super)gravity theories ,June 2016, Bogazici Uni. Istanbul.[5] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int.J. Theor. Phys. , 1113 (1999) [Adv. Theor. Math. Phys. , 231 (1998)] [hep-th/9711200] • E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. , 253 (1998) [hep-th/9802150].[6] C. Crnkovic and E. Witten, In *Hawking, S.W. (ed.), Israel, W. (ed.): Three hundred years ofgravitation*, 676-684 and Preprint - Crnkovic, C. (86,rec.Dec.) 13 p • A. Ashtekar, L. Bombelliand O. Reula, PRINT-90-0318 (SYRACUSE).[7] J. Lee and R. M. Wald, “Local symmetries and constraints,” J. Math. Phys. , 725 (1990) • V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamical blackhole entropy,” Phys. Rev. D , 846 (1994) [gr-qc/9403028].188] G. Barnich and C. Troessaert, “Aspects of the BMS/CFT correspondence,” JHEP , 062(2010) [arXiv:1001.1541 [hep-th]].[9] For reviews see e.g. : G. Comp`ere, “Symmetries and conservation laws in Lagrangian gaugetheories with applications to the mechanics of black holes and to gravity in three dimensions,”arXiv:0708.3153 [hep-th] • K. Hajian, “On Thermodynamics and Phase Space of Near HorizonExtremal Geometries,” [arXiv:1508.03494 [gr-qc]] • A. Seraj, “Conserved charges, surface degreesof freedom, and black hole entropy,” [arXiv:1603.02442 [hep-th]].[10] J. L. Cardy, “Operator Content of Two-Dimensional Conformally Invariant Theories,” Nucl.Phys. B , 186 (1986).[11] A. Strominger, “Black hole entropy from near horizon microstates,” JHEP , 009 (1998)[hep-th/9712251].[12] M. Ba˜nados, C. Teitelboim and J. Zanelli, “The Black hole in three-dimensional space-time,”Phys. Rev. Lett. , 1849 (1992), [hep-th/9204099] • M. Ba˜nados, M. Henneaux, C. Teitelboimand J. Zanelli, “Geometry of the (2+1) black hole,” Phys. Rev. D , 1506 (1993) [gr-qc/9302012].[13] M. Guica, T. Hartman, W. Song and A. Strominger, “The Kerr/CFT Correspondence,” Phys.Rev. D , 124008 (2009) [arXiv:0809.4266 [hep-th]].[14] G. Comp`ere, “The Kerr/CFT correspondence and its extensions: a comprehensive review,”Living Rev. Rel. , 11 (2012) [arXiv:1203.3561 [hep-th]].[15] G. Comp`ere, M. Guica and M. J. Rodriguez, “Two Virasoro symmetries in stringy warpedAdS ,” JHEP (2014) 012 [arXiv:1407.7871 [hep-th]].[16] G. Comp`ere, K. Hajian, A. Seraj and M. M. Sheikh-Jabbari, “Extremal Rotating Black Holesin the Near-Horizon Limit: Phase Space and Symmetry Algebra,” Phys. Lett. B , 443 (2015)[arXiv:1503.07861 [hep-th]].[17] G. Comp`ere, K. Hajian, A. Seraj and M. M. Sheikh-Jabbari, “Wiggling Throat of ExtremalBlack Holes,” JHEP , 093 (2015) [arXiv:1506.07181 [hep-th]].[18] G. Comp`ere, P. Mao, A. Seraj and M. M. Sheikh-Jabbari, “Symplectic and Killing symmetriesof AdS gravity: holographic vs boundary gravitons,” JHEP (2016) 080 [arXiv:1511.06079[hep-th]].[19] K. Hajian and M. M. Sheikh-Jabbari, “Solution Phase Space and Conserved Charges: A GeneralFormulation for Charges Associated with Exact Symmetries,” Phys. Rev. D , no. 4, 044074(2016) [arXiv:1512.05584 [hep-th]].[20] M. Ba˜nados, “Three-dimensional quantum geometry and black holes,” AIP Conf.Proc. 484(1999) 147 [hep-th/9901148].[21] H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso, “Soft hairy horizonsin three spacetime dimensions,” Phys. Rev. D (2017) no.10, 106005 [arXiv:1611.09783 [hep-th]] • H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso,“Soft Heisenberg hair on black holes in three dimensions,” Phys. Rev. D (2016) no.10, 101503[arXiv:1603.04824 [hep-th]]. 1922] K. Hajian, M. M. Sheikh-Jabbari and H. Yavartanoo, “Fluffing Extreme Kerr,” arXiv:1708.06378[hep-th].[23] S. Hollands and A. Ishibashi, “All vacuum near horizon geometries in arbitrary dimensions,”Annales Henri Poincar´e (2010) 1537 [arXiv:0909.3462 [gr-qc]].[24] H. Golchin, M. M. Sheikh-Jabbari and A. Ghodsi, “Dual 2d CFT Identification of ExtremalBlack Rings from Holes,” JHEP (2013) 194 [arXiv:1308.1478 [hep-th]] • “More on FiveDimensional EVH Black Rings,” JHEP (2014) 036 [arXiv:1407.7484 [hep-th]].[25] K. Hajian, A. Seraj and M. M. Sheikh-Jabbari, “NHEG Mechanics: Laws of Near Horizon Ex-tremal Geometry (Thermo)Dynamics,” JHEP (2014) 014 [arXiv:1310.3727 [hep-th]] • K. Ha-jian, A. Seraj and M. M. Sheikh-Jabbari, “Near Horizon Extremal Geometry Perturbations: Dy-namical Field Perturbations vs. Parametric Variations,” JHEP (2014) 111 [arXiv:1407.1992[hep-th]].[26] M. Johnstone, M. M. Sheikh-Jabbari, J. Simon and H. Yavartanoo, “Extremal black holes andthe first law of thermodynamics,” Phys. Rev. D (2013) no.10, 101503 [arXiv:1305.3157 [hep-th]].[27] V. F. Lazutkin and T. F. Pankratova, “Normal forms and versal deformations for Hill’s equa-tion,” Funct. Anal. Appl. (October, 1975) 306–311.[28] A. A. Kirillov, “Orbits of the group of diffeomorphisms of a circle and local Lie superalgebras,” Funct. Anal. Appl. (April, 1981) 135–137.[29] E. Witten, “Coadjoint Orbits of the Virasoro Group,” Commun. Math. Phys. , 1 (1988).doi:10.1007/BF01218287[30] J. Balog, L. Feh´er and L. Palla, “Coadjoint orbits of the Virasoro algebra and the global Liouvilleequation,” Int. J. Mod. Phys. A , 315 (1998) [hep-th/9703045].[31] H. Salmasian and K. H. Neeb, “Classification of positive energy representations of the Virasorogroup,” arXiv:1402.6572 [math.RT].[32] J. M. Bardeen and G. T. Horowitz, “The Extreme Kerr throat geometry: A Vacuum analog ofAdS(2) x S**2,” Phys. Rev. D (1999) 104030 [hep-th/9905099].[33] A. J. Amsel, G. T. Horowitz, D. Marolf and M. M. Roberts, “Uniqueness of Extremal Kerr andKerr-Newman Black Holes,” Phys. Rev. D (2010) 024033 [arXiv:0906.2367 [gr-qc]].[34] V. P. Frolov and K. S. Thorne, “Renormalized Stress - Energy Tensor Near the Horizon of aSlowly Evolving, Rotating Black Hole,” Phys. Rev. D (1989) 2125.[35] R. Bott, “On the characteristic classes of groups of diffeomorphisms,” Enseign. Math. (1977),no. 3-4, 209–220.[36] L. Guieu and C. Roger, L’alg`ebre et le groupe de Virasoro . Publications du CRM, Universit´e deMontr´eal, 2007.[37] B. Oblak, “BMS Particles in Three Dimensions,” arXiv:1610.08526 [hep-th]. Published in
Springer Theses , 2017. 2038] H. Afshar, D. Grumiller and M. M. Sheikh-Jabbari, “Near horizon soft hair as microstates ofthree dimensional black holes,” Phys. Rev. D , no. 8, 084032 (2017) [arXiv:1607.00009 [hep-th]] • M. M. Sheikh-Jabbari and H. Yavartanoo, “Horizon Fluffs: Near Horizon Soft Hairs as Microstatesof Generic AdS3 Black Holes,” Phys. Rev. D (2017) no.4, 044007 [arXiv:1608.01293 [hep-th]] • H. Afshar, D. Grumiller, M. M. Sheikh-Jabbari and H. Yavartanoo, “Horizon fluff, semi-classicalblack hole microstates Log-corrections to BTZ entropy and black hole/particle correspondence,”JHEP (2017) 087 [arXiv:1705.06257 [hep-th]].[39] S. Giombi, A. Maloney and X. Yin, “One-loop Partition Functions of 3D Gravity,” JHEP (2008) 007 [arXiv:0804.1773 [hep-th]] • G. Barnich, H. A. Gonzalez, A. Maloney andB. Oblak, “One-loop partition function of three-dimensional flat gravity,” JHEP , 178 (2015)[arXiv:1502.06185 [hep-th]].[40] A. Garbarz and M. Leston, “Classification of Boundary Gravitons in AdS Gravity,” JHEP (2014) 141 [arXiv:1403.3367 [hep-th]] • M. M. Sheikh-Jabbari and H. Yavartanoo, “On 3dbulk geometry of Virasoro coadjoint orbits: orbit invariant charges and Virasoro hair on locallyAdS geometries,” Eur. Phys. J. C , no. 9, 493 (2016) [arXiv:1603.05272 [hep-th]].[41] G. Barnich and B. Oblak, “Holographic positive energy theorems in three-dimensional gravity,”Class. Quant. Grav. (2014) 152001 [arXiv:1403.3835 [hep-th]].[42] G. Barnich and B. Oblak, “Notes on the BMS group in three dimensions: II. Coadjoint repre-sentation,” JHEP , 033 (2015) [arXiv:1502.00010 [hep-th]].[43] K. Hajian, M. M. Sheikh-Jabbari and H. Yavartanoo, Work in progress .[44] A. Strominger and C. Vafa, “Microscopic origin of the Bekenstein-Hawking entropy,” Phys. Lett.B379