AdS3 black holes and a stringy exclusion principle

“We consider the case X"0, the general case is considered in [276].”

“This equivalence can be interpreted as a modular transformation in the boundary CFT [60].”

“2 This is related to the observation in [19] this non-extremal black string can be understood as the usual CFT on R(2) in the usual Minkowski vacuum but viewed in Rindler space.”

“There are also operators in the D1-D5 system with arbitrarily low dimensions [22,23].”

“s only one asymptotic region and one black hole in a thermal state. The second interpretation of the eternal black hole is that it represents two black holes in disconnected spaces with a common time [7, 8, 9, 10]. We will refer to the disconnected spaces as sheets. The degrees of freedom of the two sheets do not inte…”

“Then the massless field can be an off-diagonal component of the metric on of the four torus [13, 14].”

“s result is to note that at every fixed polar angle θ, the geometry is a quotient of warped AdS 3. The temperature for such quotients is the length of the shift determining the quotient divided by 4π2 [28, 53]. This gives TL = 1 2π for every θ. 13 According to the Cardy formula the entropy for a unitary CFT at large TL…”

“When we unwrap the quotient, the two boundaries of the BTZ Penrose diagram become the two wedges of Rindler space in the CFT [35] (see [36] for a more recent discussion).”

“Finite N truncations in BPS spectra were studied as evidence of a stringy exclusion principle [4]( see also [5]).”

“For n = 2, we thus need the torus partition function, which fortunately is known for a general holographic CFT [9].”

“the partition function was given by Z 0,1( aτ+b cτ+d ). For a given value of τ, one can ask which geometry Mc,d has the largest contribution to Z(τ) in the semiclassical limit k → ∞. As explained in [9,10] and in Sec. 4.2, this question can be answered by maximizing the classical action −4πIm(γτ) as a function of γ. I…”

“Explicit solutions of the equation L+�H = ¯ L+�H = 0 give [ 32 ]:”

“Entropy of BTZ black holes has also been analyzed using the Euclidean action formalism in [117,19, 20, 118].”

“Maximizing this expression over SL(2, Z) images leads to an intricate Euclidean phase diagram with an infinite number of phases tessellating the upper half τ -plane [6, 24, 27].”

“ν2 +3)(r+ −r−) 8πℓ (5.8) TL ≡ (ν2 +3) 8πℓ r+ +r− − p (ν2 +3)r+r− ν ! (5.9) we have ∂θ = πℓ(TLJ2 −TRJ˜2) (5.10) TL and TR are referred to as temperatures in analogy with the BTZ case where it is known [45] that the coefficients of the shifts are the temperatures of the dual 2d CFT. In the coordinates 3.3 the norm is |TLJ…”

“In general at finite temperature, the expression for the energy density will be quite complicated but to leading order in the central charge the energy density does not change until the temperature reaches the high temperature phase RT > (2π)−1 [76].”

“the D1-D5 brane calculations rely only on the IR conformal field theory description of the gauge theory on the branes, and support this duality. Further evidence and results were recently obtained in [38,39]. One of the goalsofthis paper is tocompare in detail the spectrum ofthe conformal field 1There has been a lot of…”

“As shown in [4], the Euclidean black hole background is equivalent to the thermal AdS3 background.”

“ted by an issue in holography. String theory in AdS3 has static solutions describing infinitely-long (p,q) strings, which stretch between two points on the AdS3 boundary [9]. In the dual spacetime CFT [10, 11, 12, 13, 14, 15] the endpoint of a (p,q) string is, as we will explain, an interface separating regions with di…”

“approach which does not lead to a formula for cL,R. Therefore we will simply assume that the conformal symmetry found here connects smoothly to that of the extreme 4 The analysis here follows that of [19] for the BTZ black hole. Although the discussion of [19] was in the context of string theory, the discussion did no…”

“the identifications τE ∼ τE+ 2πτ2 , φ∼ φ+ 2πτ1 . (2.27) This corresponds, from the point of view of the 2–dimensional CFT on the AdSboundary, to considering the CFT on a torus with modular parameter τ[12]. The Euclidean continuation of the regions r>r+ and r<r− give quotients of H3 with modular parameters τout an…”

“o model, as reflected in the (t,u) and (x4,u) submanifolds. havior of other theories compactified on circles with anti-periodic boundary conditions for fermions in the gravity approximation (see, e.g., [2, 25]). In the gravity approximation, in order to study the theory at a finite temperature T, we need to look at Eucli…”

“llows directly from the central charges. Hence, so too does the black hole entropy. 3 Other choices lead to the so-called “SL(2,Z)” family of black holes. 6 Let us illustrate this in more detail; see [12]. An asymptotically AdS3 solution carries an energy H and angular momentum J. In the CFT on the boundary J is the m…”

“he small number of light operators. It is holographically dual to the Hawking-Page phase transition between black holes and a thermal gas in AdS 3, and the Cardy formula equals the black hole entropy [37, 38, 39]. This gives a statistical origin of black hole entropy in many examples. According to the Ryu-Takayanagi f…”

“According to the correspondence, Type IIB superstring theory on AdS3 × S(3) ×M4 is dual to the supersymmetric non-linear sigma model in two dimensions whose target space is the moduli space of Yang-Mills instantons on M4 [13][14].”

“cused on this paper could provide the worldvolume theory for some number of M2-branes.7 For the special case of two M2-branes, the situation is slightly better, due to the stringy exclusion principle [21, 18, 19, 20], which reduces the expected spectrum of operators from the full supergravity result. In this case, the…”

“The partition function of the dual boundary conformal eld theory CFT 2 depends on on ˝. In this framework the modular symmetry is identied with the SL(2;Z) mapping class group of the boundary torus [86], whereas the elliptic symmetry is identied with large gauge transformations of the 3-form eld [44] as we have alread…”

“We note that the semi-classical approximation (57) to the partition function was studied, in the genus one case, in papers [30, 31, 15].”

“As discussed in [9, 28, 29], when using the Cardy formulas”

“One can, for example, take the expression for geodesics in pure AdS3 and then pull it back to the geometry (2.23) (see appendix A).”

“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].”

“e complex coordinate zis defined on a torus, z∼ z+2πn+2πτm, n,m∈ Z. (91) with τits modular parameter. Introducing τis particularly convenient when studying modular invariance on the black hole manifold[25, 39]. We shall not deal with this issue here, so we use (89). 4.4 The quantum space of metrics. State counting We h…”

“The supergravity chiral primary states are given as follows [14][15][16].”

“ienting AdS3 by γ yields thermal AdS3: ds2 = (1+r2/ℓ2)dt2 + dr2 1+r2/ℓ2 + r2dφ2 (2.58) with the identification w ∼= w + 2π ∼= w +2πτ, where w = φ +it/ℓ. To construct an “SL(2,Z) family of black holes” [31,7] we consider the modular transformed version of (2.57) γ = e−iπ aτ+b cτ+d 0 0 eiπ aτ+b cτ+d , (2.59) with (a,b,c,…”

“Related arguments were used in [30, 31].”

“have a BPS string carrying some charge we can take its near horizon limit, which will generically be AdS 3 × S3 × T4. This system has been much studied. A partial list of relevant references includes [28, 29, 30, 31, 32, 33, 34]. A choice of near-horizon limit is a choice of string charge S. We regard the 10 as a symm…”

“ivial cycle ct+dφ in T2 is labeled by a pair of relatively prime integers (c,d). The associated quotient H3/Z will be denoted Mc,d. These geometries were dubbed the “SL(2,Z) family of black holes” by [55]. To describe these manifolds, consider the group of modular transformations γ = a b c d ∈ SL(2,Z) (7.10) which act…”

“From the proof in appendix B it follows that the series in fact converges to the value of the Fourier coefficient of the modular form.”

“ierent black holes are related to each other by a coordinate transformation. To clarify the logic, we rst review the analogous argument for BTZ black holes in the AdS 3/CFT 2 correspondence, made in [53]: The coordinate transformation from Poincare AdS 3 to the BTZ black hole is w ˘e2ˇT (˚ t) near the boundary. The th…”

“For a minimally coupled massive scalar field of mass ms, we could use the dictionary of [8], [22]: ∆ = 1 + √”

“Finite N truncations in BPS spectra were studied as evidence of a stringy exclusion principle [4]( see also [5]).”

“on chiral primaries with the specic generator, (J+ 1 ) ‘ . Actually, the tower of descendant states built using (J+ 1 ) ‘ on a chiral primary have a bulk interpretation as states of a supergraviton [65]. When there are multiple supergravitons, the total state is the tensor product of such supergraviton states. Therefo…”

“In the semiclassical limit, all known saddlepoints of pure 3d gravity that contribute to the partition function have classical action of the form [50, 51]”

“can regard this as a dimensional reduction of AdS3/CFT2 as first pointed out in [7] since the near horizon geometry of 5D near extremal black holes is a rotating BTZ black hole [26, 41] (see also [27]).”

“By using the result (43) on the equivalence between the L−1, L̃−1 action on the state and its differential action on AdS3 in [18] and the following formula for Jacobi polynomials”

“The defect in question in that case is the Q = 12 KK instanton described in the previous section, whose one-defect solution corresponds to an SL(2,Z) black hole [12] in three dimensions.”

“tions on the circle at the boundary. The ground state is AdS3 with no Wilson lines for the U(1) gauge fields. We can consider the spectrum of chiral primaries, i.e. states with L0 = JL, L¯0 = JR as in [11]. ¿From the CFT point of view we can calculate how many of these states we expect. It turns out that there is a sin…”

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“[54] in connection with the stringy exclusion principle[55].”

“There are also an infinite set of saddles [43] which turn out not to contribute at large central charge for a purely imaginary”

“spectrum of AdS excitations at finite “radius”, i.e. at finite N [ 43 ].”

“It should also allow a better understanding of the mapping from the geometry of giant graviton moduli spaces to the space of diagonal gauge-invariant eighth-BPS operators at finite N , generalising the geometrical explanation [3] of the stringy exclusion principle [45].”

“rature T H which appears as the periodicity. There are two other temperature parameters as well, the “left” and “right” temperatures T± which appear in the boundary coordinate transformation (23). In [12], the stress tensor components T±± for the left movers and the right movers were evaluated, and they turned out to…”

“1), one can compute the Euclidean action of this solution and obtain [17] S = − 6 Im 1 τ .”

“we have the bound c1(γ+)+c1(γ−) ≤ N (8.4) We will call this the non-chiral stringy exclusion principle, following the terminology of stringy exclusion principle for the cutoff in the holomorphic case [46]. The bound on the individual branes or anti-branes c1(γ+) ≤ N and c1(γ−) ≤ N can be understood in the semiclassical…”

“gies is bounded by exp(b/G). We will call this bound the topological exclusion principle, similar to how one discusses other bounds on AdS/CFT for two dimensional CFT as a stringy exclusion principle [55]. This is important from many points of view. First, it suggests that topological features have a minimum size in g…”

“There are in fact an infinite number of such geometries, each labelled by an element of the modular group SL(2,Z)1 [9].”

“be time-dependent 4-brane solutions constructed from zero-brane degrees of freedom. Non-commutative spheres were proposed as models of non-commutative space underlying the stringy exclusion principle [7] in [8] and explored further in this context [9][10][11][12]. The fuzzy two-sphere described polarized D0-branes in…”

“Another well-known exact string vacuum is the near-horizon geometry of NS5-branes wrapped on a four-torus (or on a K3 manifold) and fundamental strings [6] [7] [8] [9].”