AdS3 black holes and a stringy exclusion principle

“Early successes include the calculation of black hole entropy [1, 5], thermodynamics [6, 7], and much more.”

“The novel feature of the emerging gravitational theory is a ‘stringy exclusion principle’ ([5]) which follows from the CFT.”

“The CFT corresponding to the former background is expected to be a deformation of the Sym(T 4) orbifold theory, and have small N = (4, 4) superconformal symmetry [5]; the one corresponding to AdS3 × S3 × S3 × S1 has large N = (4, 4) superconformal symmetry but is much less well understood [6].”

“They transform according to the (parabolic) representation of SL(2,R) in the Poincaré coordinates [28, 29, 22, 20, 30].”

“e have also shown that the restriction on the spin (in terms of the level) that is necessary to obtain a ghost-free spectrum corresponds to the stringy exclusion principle of Maldacena and Strominger [5]. Among other things, we regard this is as evidence that the SU(1,1) model with the restriction on the set of allowe…”

“f an underlying conformal symmetry for extreme Kerr. Globally, the θ= θ0 slice ofNHEK is a quotient of AdS3. The reason for this is that the angle φ is periodically identified φ∼ φ+2π. We already know [19] that a quotient of AdS3 space is related to the finite temperature partition function of the dual conformal field th…”

“of all AdS3/CFT2 correspondences since the expectation values of holomorphic and antiholomorphic components of the energy-momentum tensor of the CFT can be directly related to L± [8, 16].”

“Examples include the BTZ black hole [26, 27], analogous handlebody manifolds at higher genus [28], correlators of heavy operators [25,29,30], probes of black hole eigenstates [31–34], collapsing black holes [35], and light correlators [36–38].”

“Topologically, this operation is called ‘Dehn-filling’ and in fact it is possible to write a smooth hyperbolic metric on this manifold [32].”

“CFT [1, 2, 3] allows a translation of many hard questions of quantum gravity to questions in the dual CFT. Intriguing lessons on quantum gravity have taken the form of the stringy exclusion principle [4]. Some aspects of this principle find a geometrical expression in the properties of giant gravitons [5, 6]. The syste…”

“This family can be thought of as related to a different family of black holes in asymptotically AdS3 space with fixed electric potential [71] by a Laplace transform in spacetime.”

“In this respect, the infinite storage capacity of states at the horizon violates the “stringy exclusion principle” [24].”

“To fix this, one can sum over the restricted family of ‘SL(2,Z)’ black holes [46] related by the modular transforms that take θ 7→ θ + 2nπ.”

“The cutoff 1 2N is indeed familiar from the torus T4 [34], which on the level of the vacuum sector can be viewed as the limit of S3× S1 with infinite radius, see [9].”

“One can derive a universal bound for the central charge of a two-dimensional N = 2 SCFT using the infinite-dimensional superconformal algebra, see for example [67].”

“Each term in the PSL(2,Z) sum corresponds to a different AdS3 geometry discussed in [15].”

“[53, 54, 55, 56], the BTZ black hole, when it exists as a proper classical geometry, should be obtained by coarse graining over all microstates of given energy or temperature, consistent with its interpretation as a purely thermal state [43].”

“ncar´e series and sums over asymptotically AdS3 geometries. Also the new regularization will be motivated heuristically. The connection between elements in Γ and AdS3 geometries was suggested in Ref. [21] and refined somewhat in Ref. [5]. It is reviewed for example in Refs. [15, 22] and Ref. [23] from a supergravity pe…”

“regularized Einstein-Hilbert action of this solution is known to be [ 6 ]”

“As in the higher dimensional case, there are two solutions that correspond to either the first or the second circle being contractible in the interior (which correspond to thermal AdS and Euclidean BTZ), but there are now new possibilities obtained by considering a contractible cycle that is a linear combination of th…”

“1 Magnon dispersion relation at strong coupling The near horizon geometry of the D1-D5 system for large λ̃ is AdS3 × S(3) × T 4 described by the following metric [3]: ds(2) = R(2)(− cosh(2) ρdt(2) + dρ(2) + sinh(2) ρdφ(2) + dΩ(2)3) + ds(2)([T (4)]), (4.”

“The bound on the angular momentum parameter is reminiscent of the “stringy exclusion principle” observed in [28] and it would be interesting to explore this in more detail.”

“However, in these cases, when one performs the Euclidean continuation, it is found that the Euclidean BTZ with our boundary conditions is diffeomorphic to thermal AdS as in [40, 41], but with reversed orientation, which is a consequence of the fact that the lapse of AdS spacetime reverses its sign.”

“Applying symmetric orbifold techniques to a free field realisation, the chiral primary operators can be easily constructed [60, 93] and their correlation functions explicitly computed [94–98].”

“involves kcolors as a single row with kboxes. This is illustrated in figure 1. The total number nof boxes in the Young tableaux is bounded by N, and this is related to the stringy exclusion principle [35]. We can use a similar Young tableaux representation for multi-trace states in a gauge theory, associating single-tr…”

“Hamilton theorem which allows us to write tr(XN+1) for an N N matrix in terms of products of lower traces. This has important implications in string theory in the form the stringy exclusion principle [72]. These nite Neects have been studied in a variety of multi-matrix systems [8, 10, 6]. The key lesson is that they…”

“These temperatures can be easily read off from the identifications of the euclidean BTZ solution [38], obtained by the analytic continuation t→ iτ φ̂+ iτ ∼ φ̂+ iτ + 2πm+ in T̂L , m, n ∈ Z (4.”

“4) This geometry has an asymptotically flat extension describing a near-extremal temperature (T+, T−) D1-D5 black string with τ ± asymptotically canonically normalized longitudinal null coordinates [31].”

“Using techniques in the dual conformal field theory, finite N truncations in BPS spectra were studied as evidence of a stringy exclusion principle[11].”

“3) as a sum over topologies with fixed boundary conditions, following [17, 4, 5].”

“(2;R) is actually itself AdS3, and thus we are simply re-asserting the familiar fact that single-particle states on AdS3 transform as highest-weight representations under its isometry group (see e.g. [46]). Now the worldline action describing the system was shown to be S[U;P;;A;A] = Z C ds Tr PU 1DU ds + Tr(P2) c2 ; (…”

“n structures the boundary conditions of the rightmoving SU(2) gauge fields should drop out. This point deserves to be understood more fully. 3 An heuristic version of this sum was first written down in [36]. 16 2.6.3. Spinning in 6-dimensions Actually, we have not yet fully enumerated the distinct types of geometry that…”

“7]. The dyonic string admits the ten-dimensional interpretation [5] of an intersecting NS-NS 1-brane and 5-brane, which in a type II context is in turn related by U-duality to the D1-D5 brane system [8, 9, 10, 11, 12]. This geometry plays a part in recent studies of black holes [14, 15, 16, 17, 18, 19, 10, 11, 12, 20]…”

“The unitary string spectrum emerged from these results agrees with expectations from the AdS/CFT correspondence [3, 4] applied to the case of AdS3 [5, 6, 7].”

“etween the symmetric product of T4 and type IIB string theory on AdS3 × S3 × T4 [2].1 The early checks of this duality included comparison of the moduli spaces [7, 8] and the spectra of both theories [9, 10, 11, 12]. Recently much progress was made in comparing correlation functions. The structure constants of single-…”

“More generally, as we review in appendix A, at fixed temperature and chemical potential there are an infinite number of classical bulk solutions that are related by SL(2,Z) transformations [26].”

“avities in three dimensions are expected to describe e.g. the massless sector of the AdS 3×S3 reduction of D= 6, N= (2,0) 2 supergravity coupled to nT tensor multiplets [5, 6]. Of particular interest [7, 8] have been the theories with nT =5 and nT =21, which correspond to compactifying IIB supergravity on AdS 3×S3×M4…”

“acroscopic entropy, and q is the momentum on the circle S1. In the language of the parent AdS 3, these geometries are the extremal limit1 of the Γ∞\SL(2,Z)/Γ∞ family of AdS 3 black holes discussed in [2,8,9]. The geometry of thermal AdS 3 is a solid torus, and the various solutions correspond to all possible ways of fi…”

“e to start from the supergravity solution with 38 τg= τf, consider all modular transformations of τg, and then sum the exponentials of the Euclidean action of the corresponding supergravity solutions [23,24]. In general all solutions will contribute to the partition function. But we notice that the Euclidean actionsca…”

“The solutions of the equations of motion with a single non-contractible cycle are quotients of the AdS3 manifold by an element (gL, gR) of SL(2,R)×SL(2,R) [30, 31].”

“Following [ 10 ] the chiral primary states can be written”

“S3 [30], i.e. in the AdS3/CFT2 correspondence, with the Virasoro symmetry in the chiral CFT2 dual to the AdS2, i.e. in the Kerr/CFT correspondence. For preceding analysis to relate AdS3 and AdS2, see [11,15,31,32]. We also obtain the correct entropies for this class of the black holes by the Kerr/CFT near horizon geom…”

“y, for fixed gand N there is a critical magnetic field Bcrit ≃ 1/ √ 2πα′N. Notice that the existence of an upper bound on the D4–brane charge is another manifestation of the stringy exclusion principle [39]. Finally, let us note that the map (rH,l,α) → (g,N,B) is generically two–to–one, except at the special points wher…”

“Beyond this point, the stringy exclusion principle [171] is in operation, and cuts off the size of the fuzzy 2-sphere.”

“This class of solutions corresponds to thermal AdS3 and the PSL(2,Z) family of BTZ black holes [23].”

“eem to imply that the closed string bulk variable are simply not good variables for nonperturbative gravity. This reduction of the number of independent variables is the ‘stringy exclusion principle’ [42]. It has been related to the growth of the size of objects at high energy [43]. The information paradox also implie…”

“INTRODUCTION After the formulation of the AdS/CFT correspondence [1], namely the correspondence between gravity in a higher dimensional space and the gauge theory on its boundary, a large amount of research has been devoted in the investigation of AdS black holes [2–9].”

“The change in the leading degeneracy between the non-scaling and scaling regime seems to precisely take into account the stringy exclusion principle [8], which for a chiral primary in the NS sector states that L̃0 ≤ c/12.”

“0 with λ gs = N fixed ? Is there a two-dimensional worldsheet formulation in this double scaling limit which captures the finite N effects such as the cutoffs discussed as the stringy exclusion principle [26, 27, 28] or the enhanced symmetries discussed here, and treats gsN as a deformation parameter that breaks the symme…”

“Various other aspects of this background have been studied in [11, 12, 13, 14, 15].”

“It is familiar from the three-dimensional BTZ black hole that the identifications required to obtain extremal black holes are different than the ones required to obtain non-extremal black holes [31, 211].”

“The twist highest weight ground states are identified with the cohomology of the resolution of diagonals in the symmetric product, and the action of the fermions can be identified with the product in cohomology with the eight even and eight odd cohomology classes of T 4 [62,36,63].”

“The oscillator excitations may be classified according to representations of the symmetric group, and thus matched to the BPS states of the F1-NS5 or D1-D5 system, which are the orbifold cohomology classes of (T (4))/Sn1n5 [45, 6].”

“This is the same structure as the AdS3 black hole Farey Tail [38], and perhaps points to a similar structure of Euclidean saddle-points as in [39].”

“Experiencewith M atrix theory and AdS/CFT suggeststhattheansweris:through thesym m etriesofdeSitter space.”

“Focusing on the left sector of the theory and the corresponding symmetry groups SL(2,R)L × SU(2)L, one can show that the generators of these symmetries can be explicitly realised in the in the NS-NS coordinates as [25, 43]”

“These are explicitly given by the following Killing vectors in AdS3 [22]: L0 = i∂+, L̃0 = i∂−, L±1 = ie ±ix+ [”

“n 2.1 The worldsheet conformal field theory We start with the AdS3 × S3 × K3 background of heterotic string theory. It represents a configuration of wrapped NS5-branes and fundamental strings [14] [15] [16]. The part of – 2 – the worldsheet action which is relevant to discuss the deformation comprises the SL(2,R)k – AdS…”

“The same issue was encountered in the computation of the elliptic genus in supergravity [38], where a suitable notion of degree was introduced to take into account the stringy exclusion principle [54], see also appendix B.”

“s dominated by the classical saddlepoints captured by the classical on-shell action Γ(0). The Euclidean AdS saddle-point in Einstein gravity leads to a classical partition function given by (see e.g. [55]) e −kΓ(0) = (q¯q) k 4 . (30) The 1-loop contribution can be calculated efficiently using heat-kernel methods [56]. T…”

“nite temperature is interpreted as a sum over the modular group. The sum over SL(2;Z) was a sum over all three dimensional geometries which \ll in" a T2 at the conformal boundary of space-time [8, 7, 9, 10]. In the de Sitter case, this sum has a similar interpretation as a sum over ways of lling in a T2 at the Eu…”

“The classical solutions of Euclidean gravity with fixed conformal boundary τ are essentially given by BTZ black holes which correspond to quotients of the AdS vacuum by elements of SL(2,Z) [7].”

“) −2 |N−2i and 1× α (0,0) −3 |N− 3i and 1× α (2,2) −1 |N−1i. (7.14) The fact that the single-particle operators of this form are in one-to-one correspondence with the fixed moduli was already noted in [37]. There are 21+1 of them corresponding to the 21 fixed moduli of supergravity and the size of the 3-sphere. It is ea…”