Black Holes and Five-Brane Thermodynamics Emil Martinec University of Chicago

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8-1-1999 Black Holes and Five-brane Thermodynamics Emil Martinec University of Chicago

Vatche Sahakian Harvey Mudd College

Recommended Citation Emil Martinec and Vatche Sahakian. "Black holes and five-brane thermodynamics." Phys. Rev. D 60, 064002 (1999). doi: 10.1103/ PhysRevD.60.064002

This Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. PHYSICAL REVIEW D, VOLUME 60, 064002

Black holes and ﬁve-brane thermodynamics

Emil Martinec* and Vatche Sahakian† Enrico Fermi Institute and Department of Physics, University of Chicago, 5640 S. Ellis Avenue, Chicago, Illinois 60637 ͑Received 24 February 1999; published 2 August 1999͒

4 4 5 6 The phase diagram for Dp-branes in M theory compactified on T ,T/Z2 ,T, and T is constructed. As for the lower-dimensional tori considered in our previous work ͓E. Martinec and V. Sahakian, Phys. Rev. D 59, 124005 ͑1999͔͒, the black brane phase at high entropy connects onto matrix theory at low entropy; we thus recover all known instances of matrix theory as consequences of the Maldacena conjecture. The difficulties that arise for T6 are reviewed. We also analyze the D1-D5 system on T5; we discuss its relation to matrix models of M5-branes, and use spectral flow as a tool to investigate the dependence of the phase structure on angular momentum. ͓S0556-2821͑99͒03316-0͔

PACS number͑s͒: 04.70.Dy

I. SUMMARY OF RESULTS AND DISCUSSION coupling, depend on the radial position in the associated low- A. Introductory remarks energy black supergravity solution. Since the horizon radius decreases with decreasing entropy, and only the horizon ge- Black hole thermodynamics has played an important role ometry is relevant to the thermodynamics, the entropy pa- ͑ ͓ ͔ in elucidating the structure of M theory see 1–3 for re- rametrizes a path through the moduli space of the low-energy ͒ ͓ ͔ views . In the context of the Maldacena conjecture 4–6 , supergravity. Along this path, it may be necessary to perform black hole thermodynamics generates predictions for the U-duality transformations to achieve a valid low-energy de- thermodynamics of gauge theory in various strong-coupling scription of the horizon geometry. This is why, at high en- regimes. This conjecture posits ͑in its extended form͒ that all tropy, the charge carried by the system is the Dp-brane num- of M theory in spacetimes with particular asymptotic bound- ber, while at low entropy, it is interpreted as momentum. The ary conditions is equivalent ͑dual͒ to a theory without gravity. Recently ͓7͔, the present authors constructed a phase two lie on an orbit of the U-duality group E p(Z). Further- diagram for maximally supersymmetric Yang-Mills ͑SYM͒ more, phase transitions may occur in the geometrical region ͑ ͒ theory on tori Tp, pϭ1,2,3, by systematically exploiting due to de localization of the horizon on cycles across which ͑ ͒ ͓ ͔ these ideas. It was seen that a number of different geometri- it is initially un smeared 15 . Such transitions are involved cal phases ͑i.e. those with a valid low-energy supergravity in the passage from black Dp-branes to matrix theory black description as black objects͒ arise as the entropy and cou- holes ͓16–19͔. pling are varied.1 The boundaries of the region of geometri- Thermodynamics is one of many probes of matrix- cal phases are correspondence curves ͓8͔, where the curva- Maldacena duality. It is a particularly useful one in that it ture of the geometry becomes string scale at the horizon of canonically associates an energy scale ͑that of a typical the object. Hawking quantum͒ with a particular place in the geometry Generically, the thermodynamics at high entropy contains ͑the horizon͒. The fact that the geometry appropriate to the a phase of black Dp-branes, while at low entropy one finds description of this scale undergoes a sequence of duality 11-dimensional black holes in the light-cone ͑LC͒ frame. transformations as we go from IR ͑matrix theory regime͒ to The reason is quite simple ͓7͔: The scaling limit specified by UV ͑Maldacena regime͒ means that the interpretation of Maldacena and the limit prescribed by Sen and Seiberg for probes as scattering states in discrete LC quantization compactifications of matrix theory [9,10] are one and the ͑DLCQ͒ M theory is only valid up to some scale, beyond same.2 One and the same gauge theory describes both; for which one should pass to a description in terms of scattering example, on Tp, black Dp-branes characterize the density of off of black p-branes in a dual geometry. Using the relation states in the regime of SYM entropies SտN2, whereas ma- between the energy and the radial scale probed ͓20͔, this trix theory ͓13,14͔ describes the regime SϽN. implies that matrix theory is only valid ͑in the sense of ac- The point is that the scales of various features of the ge- curately describing flat-space supergravity͒ up to some dis- ometry, for instance proper size of the torus and the string tance from the source.3 The precise relation between the Maldacena or near- horizon limit of N Dp-branes on Tp and matrix theory on Tp *Email address: [email protected] †Email address: [email protected] 1 The entropy is most useful in parametrizing the behavior of the 3This conclusion was independently reached from a somewhat theory since it is directly tied to the horizon area of the low-energy different perspective in ͓21͔. The analysis of supersymmetric quan- supergravity solution. The energy can then be read from the equa- tum mechanics ͑SQM͒ in this latter work is equivalent to the large tion of state of the relevant black hole. V limit of the phase diagrams here and in ͓7͔. In Sec. II D of ͓7͔ it 2It was shown that these two limits are related in ͓11,12͔. Dem- was observed that the D0 geometry breaks down at the correspon- onstrating their complete equivalence requires further specifying dence point, where the temperature of the system is T ϳ 1/3 2 ͓ ͔ the dimensionless quantities to be held fixed, in particular the scale N R111/lpl . Using the energy-distance relations of 20 , the re- ϳ 1/3 of the energy. sult rmax N lpl follows.

0556-2821/99/60͑6͒/064002͑23͒/$15.0060 064002-1 ©1999 The American Physical Society EMIL MARTINEC AND VATCHE SAHAKIAN PHYSICAL REVIEW D 60 064002 with N units of longitudinal momentum goes as follows ͓7͔: It is related to the DLCQ description of five-brane dynam- ␣Јϭ 2 ics ͓33–36͔. The Maldacena limit is lstr˜0, with the gauge coupling 2 ϭ (pϪ3)/2 ⌺ gY gstrlstr and the coordinate size i of the torus The analysis will clarify the relation of the D-brane de- cycles held fixed. This limit isolates the gauge theory dy- scription of the system to one in terms of NS five-branes and namics on the Dp-brane while decoupling gravity ͑for p fundamental strings ͓31͔, as low-energy descriptions of dif- Ͻ6). Natural energy scales in the gauge theory are measured ferent regions of the phase diagram ͑for earlier work, see with respect to the torus size.4 On the other hand, the ͓37͔͒. Finally, we will explore the use of spectral flow in the Seiberg-Sen prescription for matrix theory on Tp ͓9,10͔ in- superconformal theory to determine the spectral density of volves type IIA string theory with N D0-branes or, equiva- the theory as a function of angular momentum on S3. lently, M theory with N units of momentum on a circle of 2 radius R11 ; then one takes the limit lpl˜0, with R11 /lpl and ͑ ͒ 4 5 6 the transverse torus cycle sizes Ri /lpl held fixed. The rela- B. Phase diagrams for T ,T, and T ͑ tion between the two sets of parameters is simply cf. As in ͓7͔, the phase diagrams for Dp-branes on tori, p ͓ ͔͒ p 11,12,7 the T duality on all cycles of T that maps ϭ4,5,6, have a number of common features. The vertical Dp-branes to D0-branes and vice versa: axis of the diagrams will be entropy; for the horizontal axis p l3 we take the size V of cycles on the torus T in 11- 2 ϭ pl dimensional Planck units, as measured in the LC M theory lstr R11 appearing in the lower right corner ͑the phase of boosted 11D black holes͒. N is the charge carried by the system: the l3 ⌺ ϭ pl brane number in the high entropy regimes and longitudinal i R11Ri momentum in the low-energy, LC M-theory phase. Through- out the various phases, the corresponding gravitational cou- ͑ Ϫ ͒ p ͑ ϭ l p 3 /2 l plings vanish in the Maldacena limit except for p 6, where ϭͩ pl ͪ pl gstr ͟ the limit keeps the Planck scale of the high-entropy phase R ϭ R 11 i 1 i held fixed͒, implying the decoupling of gravity for the dual

Ϫ dynamics. Solid lines in the diagrams denote thermodynamic l2 p 3 p l 2 ϭͩ pl ͪ pl ͑ ͒ transitions separating distinct phases, while dotted lines rep- gY ͟ . 1 R11 iϭ1 Ri resent symmetry transformations which change the appropriate low-energy description. We do not expect sharp phase Thus, the two limits are clearly identical. transitions along these dotted curves since the scaling of the 5 In this work, we extend our analysis of such compactifi- equations of state is unchanged across them. cations to pϭ4,5, where the relevant theories involve the The structure of the phase diagram for VϾ1 is identical to dynamics of five-branes ͓22,23,9,10͔, and pϭ6, where the the cases encountered in ͓7͔͑see, for example, Fig. 1͒.At definition of matrix theory is problematic ͓9,10,24–26͔.In high entropies and large M-theory Tp, we have a perturba- ϩ the process of generating the phase diagram, we will redis- tive (p 1)D SYM gas phase. Its Yang-Mills coupling gY cover all the remaining prescriptions for generating matrix increases toward the left, cf. Eq. ͑1͒. The effective dimen- theory compactifications; we will also comment on the diffi- sionless coupling is of order 1 on the double lines bounding culties encountered for pϭ6 ͑and a proposal by Kachru this phase, which are Horowitz-Polchinski correspondence et al. ͓27͔ for overcoming them͒. For pϭ5, we map out the curves. As the entropy decreases at large V, there is a D0- phase diagram of the six-dimensional ‘‘little string theories’’ brane phase arising on the right and middle of the diagrams. compactified on a five-torus T5. Its description as a thermodynamic state within SYM theory In addition, we will analyze the phase diagram of the would be highly interesting. It has a Horowitz-Polchinski D1-D5 system, which arises in diverse contexts: correspondence point at SϳN2, where a zero specific heat It has played a central role in our understanding of black transition is to occur ͓38͔, and localizes into a LC 11D black hole thermodynamics ͓28͔. hole phase for entropies SϽN. The line SϳN separates the It is a prime example of the Maldacena conjecture, due to 11D phases that are localized on the M-theory circle ͑whose the rich algebraic structure of (1ϩ1)D superconformal coordinate size is R11) from those that are delocalized, uni- theories which are proposed duals to string theory on formly across the diagram ͓16–19͔. The 11D black hole ϫ 3ϫM ͓ ͔ p AdS3 S 4 4,29–31 . phase at small entropy becomes smeared across the T when It describes the ‘‘little string’’ theory of five-branes the horizon size becomes smaller than the torus scale V;we ͓23,32͔, where the little strings carry both winding and mo- denote generally such smeared phases by an overline ͑in this mentum charges. case 11d). This ͑de͒localization transition of the horizon on the compact space extends above the SϳN transition, separating the black Dp-brane phase from the black D0-brane 4For pÞ3, the Yang-Mills coupling is dimensionful, and should be referred to the torus scale as well. When we say that a dimensionful quantity is held fixed in the decoupling limit, we mean the 5This does not in principle exclude the possibility of smoother energy in the system relative to that scale. ͑i.e. higher order͒ transitions.

064002-2 BLACK HOLES AND FIVE-BRANE THERMODYNAMICS PHYSICAL REVIEW D 60 064002 phase.6 Susskind ͓39͔ has argued that, on the SYM side, one should regard this localization transition as an analogue of the Gross-Witten large N transition ͓40͔. The localization transition line runs into the correspondence curve separating the SYM gas phase from the geometrical phases at SϳN2. Thus as we move to the left ͑decreasing V, i.e. increasing the bare SYM coupling͒ at high entropy SϾN2, the SYM gas phase reaches a correspondence point; on the other side of the transition is the phase of black Dp-branes. A further common feature of the diagrams is a ‘‘self-duality’’ point at Vϳ1 and SϳN(8Ϫp)/(7Ϫp), where a number of U-duality curves meet. In contrast, the structure of the phase diagrams for VϽ1 depends very much on the specific case at hand. Compacti- fications on Tp, pϭ1,2,3, were analyzed in ͓7͔; we now describe the specifics of this region for pϭ4,5,6. Figure 1 is the phase diagram of T4 compactification. There are six different phases, several of which—the 11D and 11d black hole, black D0- and Dp-branes, and SYM gas ͑ ͒ phases—were discussed above. In a slight shift of emphasis, FIG. 1. Phase diagram of the six-dimensional 2,0 theory on 4ϫ 1 ϭ 4 we have relabeled the black D4-brane phase as the black T S . S is entropy, V R/lpl is the size of a cycle on the T of M5-brane phase, since its description in terms of the latter light-cone M theory, and N is longitudinal momentum quantum. object extends to the region VϽ1 ͑in fact, even for a patch of The dotted lines denote symmetry transformations: M, M lift or reduction; T, T duality; S, S duality. The solid lines are phase tran- VϾ1 the D4-brane becomes strongly coupled and must be ͒ sition curves. Double solid lines denote correspondence curves. The lifted to M theory . The appropriate dual non-gravitational ϭ ͑ ͒ dashed line is the extension of the axis V 1, and is merely in- description involves the six-dimensional 2,0 field theory on cluded to help guide the eye. The labels are defined as follows: D0, 4ϫ 1 T S , where the last factor is the M-theory circle; the scale black D0 geometry; W11, black 11D wave geometry; 11DBH, 11D of Kaluza-Klein excitations given by the size of this circle LC black hole; D0, black smeared D0 geometry; W11, black ͑times the number of branes͒ sets the transition point be- ͑ ͒ smeared 11D wave geometry; 11DBH, 11D smeared LC black tween the 2,0 and SYM descriptions. This M5 phase con- hole; D4, black D4 geometry; M5, black M5 geometry; F1, black sists of six patches that we cycle through via duality trans- smeared fundamental string geometry; WB, black smeared type IIB formations required to maintain a valid low-energy wave geometry; 10DBH, type IIB boosted black hole. The phase description. The energy per entropy increases toward the left ͑ ͒ 4 diagram can also be considered that of the 2,0 theory on T /Z2 and toward higher entropies; this is to be contrasted with the ϫ 1 ͓ ͔ S by reinterpreting the F1, WB, 10D phases, and the matrix cases analyzed in 7 where the IR limit appears toward the string phase as those of a Heterotic theory. left of the diagrams. This behavior is a consequence of the reversal of the direction of renormalization group ͑RG͒ flow equivalently boosted type IIB holes͒. between pϽ3 and pϾ3. As we continue to the left and/or Figure 1 is trivially modified to give the phase diagram of Ͻ 4 ͑ ͒ 4 ϫ 1 down on the figure at small volume V 1, the T is small the 2,0 theory on T /Z2 S . The additional structure does while the M-theory circle remains large; eventually one re- not affect the critical behavior of the diagram. The change duces string theory along the cycles of the T4, and the M5- appears in the chain of dualities we perform on the dotted brane dualizes into a string. Somewhat further in this direc- lines of the diagram. The orbifold quotient metamorphoses tion, we encounter a Horowitz-Polchinski correspondence into world-sheet parity, and the fundamental string patch ͑la- curve, and a transition to a phase consisting of a matrix beled F1) becomes that of the heterotic string. The emerging string ͓41–43͔ with the effective string tension set by the matrix string phase at the correspondence point is then that adjacent geometries. Using Maldacena’s conjecture, we thus of a heterotic theory. We thus confirm the suggestion ͓44,45͔ validate earlier suggestions to describe matrix strings using to describe heterotic matrix strings via the ͑2,0͒ theory on the ͑2,0͒ theory ͓22,23,9͔. This matrix string phase has a 4 ϫ 1 T /Z2 S . One can also propose to extend the dual theory correspondence curve also for low entropies, now with re- of an intermediate state obtained in the chain of dualities ͑ spect to a phase of smeared LC M-theory black holes or between the M5 and the F1 patches into the matrix string regime; we then have heterotic matrix strings encoded in the O(N) theory of type-I D-strings, as suggested in ͓46–48͔. 6Initially, the D0-brane phase becomes smeared to D0; as the Similar statements can be made about matrix theory orbi- entropy increases, the effective geometry of the latter patch be- folds or orientifolds in other dimensions. comes substringy at the horizon, and one should T dualize into the The thermodynamic phase diagram of five-branes ͑some- black Dp-brane patch. Both the D0 and Dp patches have the same times called the theory of ‘‘little strings’’ ͓49,23,32͔͒ on T5 equation of state, since they are related by a symmetry transforma- is shown in Fig. 2. We have a total of seven distinct phases. tion of the theory; they are different patches of the same phase. We again shift the notation somewhat, relabeling the black

064002-3 EMIL MARTINEC AND VATCHE SAHAKIAN PHYSICAL REVIEW D 60 064002

FIG. 3. Phase diagram of the D6 system. S is entropy, V ϭ 6 5 R/lpl is the size of a cycle on the T of the LC M theory, and N FIG. 2. Phase diagram of ‘‘little string’’ theory on T . The la- is longitudinal momentum. The dotted lines are symmetry transfor- beling is as in Fig. 1. D0, black D0 geometry; W11, black 11D mations: M, M lift or reduction; T, T duality; S, S duality. The wave geometry; 11DBH, 11D LC black hole; D0, black smeared solid lines are phase transition curves. Double solid lines denote D0 geometry; W11, black smeared 11D wave geometry; 11DBH, correspondence curves. The labels are defined as follows: M¯ TN, 11D smeared LC black hole; D5, black D5 geometry; NS5B, black Mˆ TN, black Taub-NUT geometry; D6,g D6, black D6 geometry; five branes in type IIB theory; NS5A, black five-branes in type IIA D0,g D0, black D0 geometry; W11,g W11, black 11D wave geom- theory; M5, black M5-brane geometry;g M5, black smeared M5- etry; 11DBH,g 11DBH, 11D LC black hole; D0,¯˜ D0, black smeared brane geometry; Mˆ W11, black smeared wave geometry in Mˆ D0 geometry; W11,¯˜ W11, black smeared 11D wave geometry; theory; Mˆ W11, black smeared wave geometry in Mˆ theory; 11DBH,¯˜ 11DBH, 11D smeared LC black hole. d11DBH, smeared boosted black holes in Mˆ theory.

g smaller V to a phase whose equation of state is that of a (5 D5 phase as a black M5 phase, since the latter extends the ϩ1)D gas. It is interesting that the Hagedorn transition is 7 Ͻ validity of the description to V 1. The equation of state of seen here as a localization-delocalization transition in the this high-entropy regime is black geometry. Yet further in this direction, the system lo- 2 calizes at NϳS to a dual LC Mˆ theory on a T4ϫS1ϫS1; here lpl SϳEN1/2ͩ ͪ VϪ5/2, ͑2͒ the horizon is smeared along the square T4, localized along R 11 both S1 factors, and carrying momentum along the last S1. characteristic of a string in its Hagedorn phase. The tempera- This Mˆ phase on the lower left is U dual to the LC M theory ture determines the tension of the effective string. We have a on the lower right. patch of black NS5-branes in the middle of the diagram. The D6 phase diagram has two important features ͑see They appear near the Vϳ1 line, at which point a T-duality Fig. 3͒. First of all, the Maldacena limit keeps fixed the ˜ ϳ 2 Ϫ2 transformation exchanges five-branes in type IIA and IIB Planck scale l pl (lpl/R11)V of the high-entropy black theories. The type IIB Neveu-Schwarz five-brane ͑NS5͒ Taub-Newman-Unti-Tamburino ͑Taub-NUT͒ geometry8 patch connects to a D5-brane patch via S duality. The type ͓50͔. Thus, gravity does not decouple, and the limit does not IIA NS5 patch lifts to a patch of M5-branes on T5ϫS1 at lead to a non-gravitational dual system that would serve as strong coupling on the left. The extra circle is the M circle the definition of M theory in such a spacetime. A symptom transverse to the wrapped M5-branes; the horizon undergoes of this lack of decoupling of gravity is the negative specific a localization transition on this circle at lower entropy and/or heat SϰE3/2 of the high-entropy equation of state. This property is related to the breakdown of the usual UV-IR corre-

7The tilde is meant to distinguish this 11-dimensional phase ͑where the M circle is transverse to the ﬁve-branes͒ from the 11- 8In the Maldacena limit, the near horizon geometry is that of an ͑ ͒ dimensional LC phase on the lower right, whose M circle has a asymptotically locally Euclidean ALE space with ANϪ1 singular- different origin. ity.

064002-4 BLACK HOLES AND FIVE-BRANE THERMODYNAMICS PHYSICAL REVIEW D 60 064002 spondence of Maldacena duality ͓51,20͔. The energy-radius tropy increases at fixed but not large Calabi-Yau coordinate relation of ͓20͔ determined by an analysis of the scalar wave size V, one finds the horizon smears over the Calabi-Yau equation in the relevant supergravity background is in fact space and eventually one reaches the D0 patch of smeared the relation between the horizon radius and the Hawking black D0-branes. The proper size of the Calabi-Yau space at temperature of the associated black geometry; thus, for p the horizon in string units is decreasing along this path; ϭ6 decreasing energy of the Hawking quanta is correlated eventually one reaches the curve along which one should to increasing radius of the horizon, as a consequence of the perform the duality transformation, in this case mirror sym- negative specific heat. This is to be contrasted with the situ- metry. Naively, in the mirror, as the entropy increases fur- ation for pϽ5, where the positive specific heat means in- ther, the T3 wrapped by the D3-branes is increasing in size, creasing horizon radius correlates to increasing temperature, while the base S3 continues to shrink; the high-entropy phase and pϭ5, where the Hawking temperature is independent of would seem to be described by D3-branes on the special the horizon radius in the high-entropy regime. Now, the tem- Lagrangian cycle of the mirror Calabi-Yau space near a co- perature in any dual description must be the same as in the nifold singularity. However, the duality transformation will supergravity description. For pϽ5, the dual is a field theory; not change the equation of state, since the D0 patch and high temperature means UV physics dominates the typical everything above it are related by symmetries of the theory. interactions, leading to the UV-IR correspondence. For p The only thing that could change this conclusion is a further ϭ5, the dual is a ‘‘little string’’ theory; the temperature is phase transition, but there is no candidate. We conclude that unrelated to the horizon radius ͑and thus the total energy͒ on the high-entropy phase is again one with negative specific the gravity side, and unrelated to short-distance physics in heat, and thus cannot be that of a field theory.9 the dual ‘‘little string’’ theory ͑since high-energy collisions of strings do not probe short distances͒. Hence the UV-IR C. D1-D5 system correspondence already breaks down at this point. For p ϭ6, there is nothing to say—large radius ͑large total energy͒ As a further example of our methods, we have examined 4ϫ 1 corresponds to low temperature of probes ͑Hawking quanta͒, the D1-D5 system on T S , which as we mentioned above can be considered as the ‘‘little string’’ theory of Q5 five- and any dual description could not have high energy or tem- 1 perature related to short distance physics, since it is a theory branes, with Q1 units of string winding along the S . Figure that contains gravity ͑so high energy makes big black holes͒. 4 shows the thermodynamic phase diagram. In the Mal- A second key feature is the duality symmetry ͑cf. ͓52͔͒ dacena limit, this theory is a representation of the algebra of Ϫ Nϭ ϩ V V 1 of the diagram relating the VϽ1 structure to that (4,4) superconformal transformations in (1 1)D ˜ ͓ ͔ ϵ discussed above for VϾ1. Note that this duality symmetry 4,55,56,29–31 . We have defined k Q1Q5 and q ϵ inverts the T6 volume as measured in Planck units rather Q1 /Q5 . We keep k fixed, but q may be viewed as a vari- Ͻ Ͻ than string units. The duality interchanges momentum modes able in the range 1 q k, thus moving some of the dotted with five-brane wrapping modes, while leaving membrane curves of duality transformation, but not altering phase tran- ϳ wrapping modes fixed; in other words, the dual space is that sition curves. For q 1, we can exchange the roles of Q1 and seen by the M5-brane. It is possible that this symmetry ex- Q5 via duality transformations across the diagram; the struc- ϭ ϭ tends to any Calabi-Yau compactification of M theory, since ture is unchanged. The other limit, q k, is the Q5 1 the volume of the Calabi-Yau sits in a universal hypermul- bound. The vertical axis on the diagram is again the entropy, tiplet whose moduli space appears to be SU(2,1)/U(2) ͓53͔; while the horizontal axis is the six-dimensional string cou- ϵ ͱ ϭ ␣Ј2 if the discrete identifications involve the appropriate element pling g6 gs / v of the D1-D5 patch, where v V4 / is 4 ͓ Ϫ2 of SU(2,1;Z), there will be a dual Calabi-Yau compactifi- the volume of the T in string units equivalently g6 is the cation of roughly the inverse size seen by M-theory five- volume of the T4 in appropriate string units of the NS5 five- branes wrapping the original Calabi-Yau compactification. brane ͑NS5FB͒ phase͔. The phase diagram has a symmetry ͑ ͒ The thermodynamic perspective also sheds light on a pro- g6˜1/g6 inversion of the torus in the NS5FB phase ; this is posal of Kachru, Lawrence, and Silverstein ͓27͔ for a defi- the T-duality symmetry of the little string theory. From the nition of matrix theory compactifications on Calabi-Yau perspective of the D1-D5 patch, we can consider the entire spaces. Generically, string theory on a Calabi-Yau space phase diagram as that of the 1ϩ1D conformal theory that does not have a T duality that inverts its volume in string arises in the IR of this gauge theory, which is conjectured to ϫ 3ϫ 4 units. Rather, these authors suggest that the appropriate du- be dual to the near-horizon geometry AdS3 S T of the ality to consider, analogous to the T-duality transformation D1-D5 system. In this patch, the D-strings are wrapped on a used by Sen-Seiberg for torus compactifications, is the mir- cycle of size R5 . This parameter is absent from the scaling ror symmetry transformation. This transformation relates relations of all curves because of conformal symmetry. D0-branes in type IIA theory on a given Calabi-Yau space to D3-branes wrapping a special Lagrangian submanifold of the ͓ ͔ type IIB mirror 54 ; locally, the Calabi-Yau space looks like 9Note that one could also imagine performing the same duality 3 3 aT fibered over an S base, and mirror symmetry is T sequence to describe matrix theory on K3 in terms of two-branes on duality on the fiber. Thus, it is proposed that some sort of the torus fiber of a near-degenerate mirror K3. In this case one 3ϩ1 gauge dynamics might yield an appropriate underlying knows that this description is related to the five-brane description description. Consider the phase diagram that should arise. At given above by duality, and hence indeed has a (5ϩ1)D equation low entropy, one has the 11D black hole phase. As the en- of state at high entropy, rather than a (2ϩ1)D equation of state.

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boost, thus connecting with the proposal of ͓33͔ for a matrix theory of this system. The lower boundary of this phase occurs at entropies of order Sϳͱk, where a Bogomol’nyi- Prasad-Sommerfield ͑BPS͒ matrix string phase emerges and the diagram is truncated at finite entropy. We find agreement with Vafa’s argument ͓57͔ that the BPS spectrum in the Ra- mond ͑R͒ sector of the D1-D5 system is that of fundamental type IIB strings carrying winding and momentum ͑some- times called Dabholkar-Harvey states ͓58͔͒. Similarly, chas- ing through the sequence of dualities for the D1-D5 system on K3ϫS1, one finds that the BPS spectrum consists of Dabholkar-Harvey states of the heterotic string. The character of the phase diagram is different at the ex- ϳ ͑ ϳ ϳ ͑ Ӷ treme limits q 1 i.e. Q1 Q5) and q k i.e. Q5 Q1). The location of the transition curves bounding the NS5WA patch ͑type IIA NS five-branes with a wave as the low- energy description͒ depend on the ratio xϭln q/ln k. For roughly equal charges qϳ1, xϳ0, this patch disappears, as do the related M5W and M5W patches and the NS5FB patch of fundamental strings and type IIB NS5-branes. The D-brane description predominates the phase diagram, except at low energies where there is a large patch describing fundamental strings with winding and momentum. The opposite ϳ regime, say fixed Q5 and large Q1 so that x 1, is the regime discussed by ͓31͔; it is also relevant to the ‘‘DLCQ’’ descrip- FIG. 4. Thermodynamic phase diagram of ‘‘little strings’’ tion of the five-brane ͓33͔. Indeed, the high-entropy region 1 4ϫ 1 wound on the S of T S , with Q1 units of winding and Q5 SϾk is taken over by the NS5FB patch up to the correspon- ϵ Ͻ ϵ Ͻ five-branes. k Q1Q5 and 1 q Q1 /Q5 k. g6 is the six dimen- dence curve, while in the low-entropy domain SϽk, the sional string coupling of the D1-D5 phase. The labels are defined as follows: D1D5, black D1-D5 geometry; NS5FB, black NS5 geom- NS5WA patch expands to squeeze out the D0D4, M5W, and etry with fundamental strings in type IIB theory; D0D4, black F1WB patches, and the localized phase is covered by the D0-D4 geometry; D0D4, black smeared D0-D4 geometry; M5W, M5W patch—longitudinal M-theory five-branes with a large black boosted M5-brane geometry; M5W, black smeared boosted boost, just what one needs for an infinite momentum frame M5-brane geometry; NS5WA, black boosted NS5 branes in type or DLCQ description. We discuss the DLCQ limit in detail 11 IIA theory; F1WB, black boosted fundamental strings in type IIB in Sec. II E below. theory; F1WB, black smeared and boosted fundamental strings in For simplicity, we have restricted the set of parameters we type IIB theory; L, localization transitions. have considered in the phase diagram to the entropy and the coupling g6 . It is straightforward to see what will happen as Analogous to the singly charged brane systems we have been other moduli of the near-horizon geometry are varied. Con- 4 discussing, at high entropies there is a ‘‘(1ϩ1)D gas’’ phase sider for instance decreasing one of the T radii, keeping the ͑ total volume fixed. At some point, the appropriate low en- at small g6 large V4), which passes across a correspondence curve to the black brane phase as the coupling increases. ergy description will require T duality on this circle, shifting Being determined by conformal symmetry and quantization from D1-branes dissolved into D5-branes, to D2-branes, end- of the central charge, the equation of state does not change ing on D4-branes. One can then chase this duality around the across this ‘‘phase transition.’’ Starting in the ‘‘(1ϩ1)D diagram: The NS5FB phase becomes M2-branes ending on gas’’ phase and decreasing the entropy, Sϳk corresponds to M5-branes; the NS5WA, D0D4, and D0D4 phases become the point where the thermal wavelength in the (1ϩ1)D con- D1-branes, ending on D3-branes; and the M5W and M5W formal theory becomes of the order of the size of the box R5 . phases become those of fundamental strings ending on D3- This is again a Horowitz-Polchinski correspondence curve branes. The near-extremal F1WB phase is unaffected. One from the side of lower entropies, analogous to the SYM theo- can also imagine replacing the T4 by K3. Moving around the 10 ϳ 2 ries at S N . There is a localization transition on the R5 K3 moduli space, when a two-cycle becomes small, a D3- cycle cutting obliquely across the diagram. The localized brane wrapping the vanishing cycle becomes light; one phase can be interpreted as that of M5-branes with a large should consider making a duality transformation that turns Q1 or Q5 into the wrapping number on this cycle.

10There is similarly a hidden phases of zero speciﬁc heat between the gas phase and the lower, localized phase, as can be seen by the 11The relation between the Maldacena conjecture and matrix mod- discontinuity in temperatures that occurs between SϾk and SϽk. els of M5-branes has also been considered recently in ͓59͔.

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Thus the D1-D5 system appears to have a remarkably essential in order to regularize the singularities in the effec- varied life. On the one hand, it can describe low-energy su- tive description. The (1ϩ1)D CFT on ͑symmetric products ϫ 3 ͒ pergravity on a 6D space, namely AdS3 S ; the common of K3 is simply singular, and does not contain the objects coordinate of the branes is the angle coordinate on AdS3. which are needed. These objects are carried, however, as This space parametrizes physics of the Coulomb branch of fluxes on the five-brane one starts with; the energy cost of the gauge theory. On the other hand, the same system de- these excitations simply becomes small at the relevant points ϩ scribes the ‘‘decoupled’’ dynamics of the five-brane, another in moduli space, suggesting that the (5 1)D string-theoretic 6D system12—except that the spatial coordinates are now character of the dynamics does not fully decouple in the T4ϫS1, with the T4 apparently related to the physics of the Maldacena limit. Similarly, one might expect that lower di- ͑ Higgs branch of the gauge theory, and the S1 the dimension mensional examples of the Maldacena conjecture e.g. those involving AdS or AdS ) are not fully captured by quantum common to the branes. In the Maldacena limit, the theory is 2 3 mechanics or more elaborate (1ϩ1)D field theories. As a representation of the (1ϩ1)D superconformal group; in mentioned above, it is known that the background fields of the DLCQ limit, it describes light-cone M5-branes. the near-horizon limit of the D1-D5 system correspond in the The careful reader will have noted that we have refrained symmetric orbifold CFT to turning off the CFT resolution of from characterizing the nongravitational dual of the D1-D5 the Z singularities of Symk(T4). It may be that branes wrap- ϩ 13 2 geometry as a (1 1)D field theory. The standard candi- ping these vanishing cycles are again the needed ingredient ϩ date for this dual is the (1 1)D conformal field theory for a well-defined description at these points of moduli k 4 ͑CFT͒ on Sym (T ) ͑or K3͒. This CFT is supposed to pro- space. vide a description of nonperturbative string theory on ϫ 3ϫ 4 ͑ ͒ AdS3 S T or K3 . Indeed, it captures the high-entropy D. Spectral flow and angular momentum thermodynamics ͓28͔ as well as the BPS spectrum ͓57,29,60͔. However, the near horizon geometry appears to The (1ϩ1)D Nϭ(4,4) superconformal algebra has two put the CFT at a singular point in its moduli space ͓61,62͔; canonical realizations, depending on whether one chooses also, there appears to be a mismatch in the level of the U(1)4 antiperiodic ͑NS͒ or periodic ͑R͒ boundary conditions on the affine algebra of Noether charges acting on the T4 ͓63͔.A fermionic generators. The spacetime geometry in the Mal- ϫ 3ϫM basic problem also arises in the phase diagram of Fig. 4. In dacena limit of the D1-D5 system is AdS3 S 4 .2 ϩ the high-entropy phase, S duality connects the D1-D5 patch 1D supergravity on asymptotically locally AdS3 space- to the NS5-F1B patch as one moves to stronger coupling. It times carries a realization of this superconformal algebra ͓56,55͔; being a subgroup of the ͑super͒diffeomorphism is straightforward to check that, crossing the boundary g6 Ϫ ͓ ͔ ϳq 1/2, the energy scale of a D1-brane wrapping the torus group, the symmetry extends to the full string theory 31 . T4 becomes less than that of a fundamental string; the appro- AdS3 itself is the vacuum state, and resides in the NS sector priate effective description is the S-dual one. In fact there has since the Killing spinors are antiperiodic; hence low-energy to be an entire decuplet of strings transforming under the supergravity about this vacuum is described by NS sector O(5,5;Z) U-duality group; the proper low-energy descrip- representation theory. The R sector is what one naively dis- tion favors one pair of these, electrically and magnetically covers as the near-horizon limit of D1-D5 bound states on M ϫ 1 1 charged under one of the five six-dimensional B-fields ͓the 4 S , since the supercharges are periodic on S . subgroup of U-duality fixing the description is14 O(5,4;Z)]. A similar situation occurs, for example, in D3-brane 3 The problem is that the objects carrying these charges, which gauge theory. The gauge theory on S describes supergravity ϫ 5 ͓ ͔ are the lightest objects in the theory at intermediate coupling, on AdS5 S in ‘‘global coordinates’’ 65 , where time trans- are not apparent in the Symk(T4) symmetric orbifold any- lation is generated by the dilation operator in the conformal 3 ͑ 3 where on its moduli space. Similarly, in the D1-D5 system group. The gauge theory on R or T ) describes supergrav- ϫ 5 ͑ on K3 there should be a full O(5,21) 26-plet of strings; in ity on a slice of AdS5 S in ‘‘Poincare´ coordinates’’ with 3 this case, tensionless strings corresponding to wrapped D3- periodic identifications for T ), where time translation is branes arise when a 2-cycle on the K3 degenerates, and are generated by a conformal boost operator. The Poincare´ slice is obtained as the limiting near-horizon geometry of black D3-branes in the full string theory. There is no map between 3 3 12 gauge theory on S and gauge theory on T . Seven-dimensional, if we include the circle transverse to the A major difference in the D1-D5 system is that, since the M5-brane. 13 one-dimensional sphere and torus are the same, the NS and R The following remarks reflect ongoing discussions of the first sectors can be related by a continuous twist of boundary author with D. Kutasov and F. Larsen. In particular, it was D. conditions known as spectral flow. This operation shifts con- Kutasov that raised the question of whether the dual object is a field formal dimensions (h ,h ) and J charges ( j , j )by͓66͔ theory. L R 3 L R 14There are BPS charges corresponding to these objects wrapping (␩) ϭ (0) Ϫ ␩ (0) ϩ␩2 hL,R hL,R 2 jL,R k T5, which are central charges in the 10D supersymmetry algebra. Just as in the case of the transverse five-brane in matrix theory ͓64͔, (␩) ϭ (0) Ϫ␩ ͑ ͒ jL,R jL,R k. 3 these charges decouple from the supersymmetry algebra in the Mal- ͑ ͒ dacena limit; nevertheless the objects remain as finite energy exci- Here, Ja are the SU 2 chiral R-symmetry currents of the Nϭ ϭ 1 ϩ tations carrying conserved charges. (4,4) algebra, E 2 (hL hR) is the energy, and P

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FIG. 5. Allowed region for states belonging to unitary repre- sentations of the ͑NS͒ superconformal algebra. The dashed curve represents the continuous spectral 2 flow h␩ϭ j␩/k of the point hϭ j ϭ0. Spectral flow slides the boundary polygon along the pa- rabola; a half unit of flow gives the Ramond sector ͑inset͒.

ϭ 1 Ϫ ␩ϭ 1 2(hL hR) is the momentum along x5 . We will restrict our horizon region; 2 maps the R sector of the wrapped brane attention to the Pϭ0 sector. The mode expansions of the system to the NS sector, with the R ground state of maximal ͑ ϭϮ 1 Ϯ␩ supercurrents which have j 2) are shifted by . Thus charge mapping to the NS vacuum. ␩ϭ ϩ 1 ෈ spectral flow by n 2, n Z, relates NS sector states to Unitarity implies that any allowed highest weight repre- R sector states. Moreover, spectral flow by integral amounts sentation of the superconformal group must have hу͉j͉ ͓71͔. ␩෈Z maps a given sector onto itself; the spectrum maps to Spectral flow then implies that allowed states must lie inside itself, but individual states are not preserved. This, combined the shaded region of the (h, j) plane in Fig. 5.16 In particular, Ϫ spectral flow forces a cutoff on the spectrum of BPS super- with the charge conjugation symmetry j˜ j, means that the full spectrum of states in the theory ͑for both NS and R gravity states ͑regardless of whether they are single- or ͒ ϭ ϭ ͒ ϭ sectors with jL jR j is determined by, e.g., NS sector multi-particle configurations at j k; as is easily seen from states with 0р jрk/2. This relation implies a relation be- the figure, states on the line hϭ j beyond this point lie out- ϭ ͑ tween standard conventions in the literature: h(Ram) h(NS) side the allowed region since they would have to flow from Ϫ ϭ ϭ ͒ k/4, and j(Ram) 0 corresponds to j(NS) k/2. states that violate the BPS bound . This feature was termed In fact, there is a simple operation on the full string theory the ‘‘stringy exclusion principle’’ in ͓29͔; we see that it de- that reduces to spectral flow in the near-horizon limit of the pends only on some rather mild assumptions about the quan- D1-D5 bound state: It is the orbifold described by Rohm tization of Chern-Simons supergravity ͑i.e. the global struc- ͓ ͔ ϫ ͒ 67 . The SU(2)L SU(2)R R symmetry of the near-horizon ture of the class of geometries under consideration . All such ϱ supersymmetry of the D1-D5 system is inherited from the restrictions disappear in the classical k˜ limit. Lorentz group of the asymptotically flat spacetime in which Spectral flow determines the density of states—at high it is embedded in the original string theory. Thus the entropy and far from the boundary of the allowed region—in R-symmetry twist is nothing but the imposition of the terms of the Cardy formula ͓73,74͔ for zero charge, twisted boundary condition ϭ ␲ͱ ͑ Ϫ 1 ͒Ϫ 2 ϩ ␲ͱ ͑ Ϫ 1 ͒Ϫ 2 ͑ ͒ S 2 k hL 4k jL 2 k hR 4k jR, 5 ⌽͑ ϭ ͒ϭ ͓ ␲␩͑ 3 ϩ 3 ͔͒⌽͑ ϭ ͒ ͑ ͒ x5 R5 exp i4 JL JR x5 0 . 4 which is precisely the density of states for D1-D5 black holes In the near-horizon region, the geometry is asymptotically with angular momentum ͑remembering the shift in conven- ϫ 3ϫM AdS3 S 4 , and the spectral flow operation can be un- tions͒. The expression must be invariant under spectral flow, derstood ͓68͔ in the effective Chern-Simons supergravity when the thermal wavelength is much smaller than the size theory that arises ͓69͔. There, spectral flow is implemented of the system, because the fermion boundary conditions are by coupling the U(1)ϫU(1) Cartan R-symmetry currents to irrelevant. Near the boundary of the allowed region, the den- a source; a shift in the energy arises due to the usual relation sity of states will differ from this expression. between regularization ͑framing͒ of Wilson line sources and A qualitative sketch of the phase diagram as a function of conformal spin in Chern-Simons theory ͓70͔.15 It is interest- energy and angular momentum is given in Fig. 6. The loca- ing that, although this twist breaks supersymmetry in the full theory, anti–de Sitter supersymmetry is restored in the near- 16These curves are slightly different from the unitarity boundaries of ͓71,72͔ since we are only asking that a state be the spectral flow 15Thus, very little of the quantum structure of gravity is being of some state in an allowed representation, rather than that it be an used here. allowed superconformal highest weight.

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FIG. 6. Qualitative phase diagrams for the D1-D5 system as a function of energy and angu- ͑ ͒ ϵ 2 Ͼ lar momentum a for coupling geff g6k 1, where g6 is the six dimensional string coupling; ͑ ͒ Ͻ b for coupling geff 1. SQM stands for super quantum mechanics ͓38͔, a phase corresponding to energy independent entropy Sϳk.

tions of the phase boundaries are not precisely determined,17 Teitelboim-Zanelli ͑BTZ͒ black hole phase with equation of since we only accurately know the phase structure in the state ͑5͒ takes over, as the (5ϩ1)D black hole delocalizes on vicinity of the NS ( jϭ0)andR(jϭk/2) sectors. The R S3. As a function of angular momentum, there are then phase sector structure is that of Sec. II C, and outlined in the pre- boundaries where the NS and R structures abut one another. vious section; the NS phase structure was discussed in ͓75͔: More details may be found in Sec. II F. There is a ‘‘supergravity gas’’ phase ͑i.e. the predominant states are dressed Fock space states of low-energy supergrav- II. DETAILS FOR THE PHASE DIAGRAMS ity͒ about the AdS vacuum; at somewhat higher energy the entropy is dominated by a long string phase; then the string The details of our results can be found in the coming undergoes a correspondence transition to a (5ϩ1)D sections. The D4, the D4 on an orbifold, D5, D6 and D1-D5 ͑ ϫ 3 systems are analyzed in detail in Secs. II A, II B, II C, II D, Schwarzschild black hole i.e. localized on AdS3 S and M and II E, respectively. The discussion about spectral ﬂow and smeared on 4); and ﬁnally, at high energy the Banados- angular momentum can be found in Sec. II F.

A. 2,0 theory on T4؋S1 17Since we are now considering ﬁnite k, the boundaries between „ … phases are not sharp anyway; they are crossover transitions rather The M5 phase. Our starting point will be D4-branes than singularities in derivatives of the free energy. wrapped on the T4 which is T-dual to the matrix theory

064002-9 EMIL MARTINEC AND VATCHE SAHAKIAN PHYSICAL REVIEW D 60 064002 description. This phase consists of six geometrical patches cases studied in ͓7͔. We will therefore be brief in the descrip- and is described by the equation of state tion of the right half of the phase diagram; a complete discussion can be found in the cited paper. We sketch quickly R11 the scaling of the various transition curves encountered along Eϳ V8/5S6/5NϪ3/5, ͑6͒ l2 this chain in the M5 phase. pl The smeared black D0 geometry (D0) localizes on the T4 obtained from the geometry of N D4-branes. The geometries for are parametrized by the harmonic functions SϽV9/2N1/2, ͑14͒ 3 3 q r0 Hϭ1ϩ , hϭ1Ϫ , ͑7͒ r3 r3 into a phase of localized black D0-branes, and gets M lifted to smeared M-theory black waves (W11) at with SϳN4/3VϪ4/3. ͑15͒ 2 5 S N lpl r5ϳ l5 VϪ4, q3ϳ . ͑8͒ 0 N pl 4 2 At V R11 We next describe the six patches of this phase. Vϳ1, ͑16͒ The black D4 brane geometry ͑D4͒ is given by the metric and dilaton it is seen to be necessary to reduce this latter geometry along one of the cycles of the T4 to the geometry of type IIA 2 ϭ Ϫ1/2͑Ϫ 2ϩ 2 ͒ϩ 1/2͑ Ϫ1 2ϩ 2 ⍀2͒ 3 ds10 H hdt dy(4) H h dr r d 4 , waves, then to T dualize on the remaining T to a type IIB theory, and ﬁnally to S dualize to the geometry of black type ␾ Ϫ1/4 e ϭH . ͑9͒ IIB waves, to be discussed below. The W11 geometry furthermore collapses at We are using the convention that the asymptotic values of the dilaton are absorbed into the gravitational coupling. The SϳN ͑17͒ parameters of this geometry are related to the moduli of the light cone M theory introduced above as follows: into the phase described by the geometry of light cone M Ϫ 4 l 1/2 l3 l2 V 1 theory black holes smeared on the T . ϭͩ pl ͪ Ϫ4 ␣Јϭ pl Ϸ pl ͑ ͒ The black M5 geometry ͑M5͒ is obtained from the D4 gstr V , , y , 10 R11 R11 R11 geometry we started with by lifting it, at strong couplings, to where in the last equation, we use the notation Ϸ to denote an M˜ theory. It is described by the metric the compactiﬁcation scale for the four y coordinates, all as- 2 ϭ Ϫ1/3͑ 2 ϩ 2Ϫ 2͒ϩ 2/3͑ Ϫ1 2ϩ 2 ⍀2͒ sumed equal in size. This geometry is subject to the follow- ds11 H dx11 dy4 hdt H h dr r d 4 , ing restrictions: ͑18͒ The Horowitz-Polchinski correspondence principle requires and the M˜ theory is parametrized: Ͼ 12 Ϫ2 ͑ ͒ S V N . 11 l2 2 l2 l2 ˜3 ϭ ͩ pl ͪ Ϫ4 ˜ ϭ pl Ϫ4 Ϸ pl Ϫ1 l pl lpl V , R11 V , y 4 V . Otherwise, we connect to a phase described by perturba- R11 R11 R11 tive (4ϩ1)D SYM. ͑19͒ Requiring that the y’s be bigger than the string scale yields This geometry is subject to the following restrictions: Requiring that the curvature at the horizon be greater than Ͼ 2 4/3 ͑ ͒ ˜ S V N . 12 the Planck scale l pl yields

Otherwise, we T dualize into the geometry of N smeared NϾ1; ͑20͒ black D0-branes. Requiring that small coupling at the horizon yield i.e., there is no dual geometrical description for Nϳ1. What- SϽV12N4/3. ͑13͒ ever the string theoretical description of a few M5-branes is to be, it will take over the phase description beyond this Beyond this point, we describe the vacuum via the geom- point. etry of black M5-branes. Requiring that the T4, as measured at the horizon, be big- The T-duality transformation yielding the geometry of ger than the Planck scale yields the condition smeared D0-branes beyond Eq. ͑12͒ leads us for VϾ1 onto a phase structure identical to the ones encountered in the three SϾVϪ3N4/3. ͑21͒

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We must otherwise reduce to the geometry ofg D4 in some SϾV2N4/3. ͑26͒ type IIAg theory residing on T3ϫS1 ͑where we have isolated We otherwise T dualize on x , along the string, and an arbitrary one of the four circles to be that of M reduction 11 obtain the geometry of smeared type IIB black waves. to type IIA͒. Localization on x is of no concern, since the symmetry Reducing tog D4-branes wrapped on T3ϫS1, we find that 11 3 structure of the metric does not allow the Gregory- the size of the T as measured at the horizon is smaller than LaFlamme localization ͑25͒͑i.e. the brane is stretched along the string scale set by ˜␣Ј for entropies satisfying the reverse this cycle͒. of Eq. ͑21͒. We then T dualize the D4g-branes to D1g-branes wrapped on S1. We find than the type IIB string coupling The type IIB smeared black wave geometry (WB) is the T ͓ ͑ ͔͒ measured at the horizon is bigger than one for the reverse of dual on x11 of F1 Eq. 22 , Eq. ͑21͒. We then S dualize to the geometry of type IIB black Ϫ ds2 ϭ͑HϪ1͒͑dx Ϫdt͒2ϩdx2 Ϫdt2ϩH 1͑1Ϫh͒dt2 fundamental strings smeared on the T3 (F1): 10 11 11 ϩdy2ϩhϪ1dr2ϩr2d⍀2 2 ϭ Ϫ1͑ 2 Ϫ 2͒ϩ 2ϩ Ϫ1 2ϩ 2 ⍀2 3 4 ds10 H dx11 hdt dy3 h dr r d 4 e␾ϭ1, ͑27͒ e␾ϭHϪ1/2. ͑22͒ and the type IIB theory is parametrized by The type IIB theory is parametrized by 3 2 Ϫ4 Ϫ2 2 ˜g ϭV , ˜␣Јϭl V , ˜R ϷR , y Ϸl V . ͑28͒ l V l s pl 11 11 3 pl ϭ pl ␣Јϭ 2 Ϫ4 Ϸ Ϫ2 ˜ Ϸ pl Ϫ4 ˜gs , ˜ lplV , y 3 lplV , R11 V . R11 R11 The relevant restrictions are: ͑23͒ Localization on x11 occurs at This geometry is the correct dual in this phase provided that: SϳN. ͑29͒ The curvature at the horizon is smaller than the string scale ˜␣Ј: The system collapses into a new phase described by the geometry of a boosted type IIB black hole smeared on T3. Ͼ Ϫ3 1/2 ͑ ͒ S V N . 24 The string coupling at the horizon becomes bigger than 1 at Beyond this point, the stringy description is that of a highly excited matrix string, as we will see shortly. Vϳ1. ͑30͒ The T3 as measured at the horizon is smaller than the transverse size of the object ͑set by the angular part of the We then are instructed to perform the chain of dualities ͒ ͑ ͒ metric ; this yields again Eq. 24 . As the box size becomes S,T(3) ,M, bringing us back to the geometry of light cone M bigger than the size of the object, the system localizes on the theory black waves W11. T3. Taking into account the changes to the geometry and thermodynamics as in ͓7͔, We thus conclude the analysis of the M5 phase. The dual theory can be inferred from the M5 patch; it is the six- 2 ϩ Ϫ1 2ϩ 2 ⍀2 Ϫ1 2ϩ 2 ⍀2 ͑ ͒ 4ϫ 1 dz(p) f dr r d d˜ f dr r d dϩp dimensional 2,0 theory wrapped on T S . Extending the validity of this theory throughout the phase diagram, we con- 3 6ϳ 6 Ϫ15/2 Ϫ3/4 clude that we can interpret it as the phase diagram of the r0˜r0 lplSV N ͑2,0͒ theory. We now move onto the other phases of the ͑2,0͒ 8 theory; we will be brief in the discussion of the right half of lpl q3 q6ϳ NVϪ10, ͑25͒ the diagram, since it overlaps in content with the lower di- ˜ 2 R11 mensional SYM cases ͓7͔. The smeared type IIB black hole (10DBH). This phase is we find that the localized fundamental string has its described by the equation of state Horowitz-Polchinski point again at Eq. ͑24͒. Furthermore, as needed for consistency with this statement, we find that the R11 1 change in the equation of state for this localized phase does Eϳͩ ͪ V8/5S8/5. ͑31͒ N 2 not affect the analysis regarding the matrix string phase we lpl will perform later. Other restrictions on the localized F1 geometry are all seen to be satisfied in the region of the param- and consists of the type IIB hole obtained from the type IIB eter space of interest. wave geometry WB at SϳN, and the smeared 11D LC hole The T3 as measured at the horizon must not be substringy. obtained from the 11D wave geometry W11. Its correspon- We find than the size of the torus as measured at the horizon dence point can be found by minimizing the Gibbs energies is at the self-dual point. between the equation of state ͑31͒ and that of the matrix The size of x11 as measured at the horizon is greater than string, which we perform below. The smeared hole geometry the string scale ˜␣Ј: localizes on the T4 at

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؋ 1 9 SϳV , ͑32͒ B. „2,0… theory on T4 /Z2 S Inspired by the previous discussion of the phase structure where the localized 11D LC black hole emerges. of the ͑2,0͒ on T4ϫS1, we further consider the phase struc- The black D0 phase ͑D0͒. This phase consists of the ge- ture of this theory on T4/Z ϫS1. This corresponds to a cor- ͑ ͒ 2 ometries of localized black D0 branes D0 and its M lift ner in the moduli space of K3ϫS1; particularly, in addition ͑ ͒ light cone M theory waves W11 ; the two patches meet at to considering a square T4, we will be ignoring phase dynamics associated with the 16ϫ4 moduli that blow up the ϳ 8/7 ͑ ͒ S N . 33 ﬁxed points ͓77͔. Our parameter space is again two dimensional, entropy S, and the volume of the T4. There are only The equation of state is two novelties that arise, both leaving the global structure of the phase diagram unchanged, modifying only the interpre- R11 1 tation of the various patches of geometry. Eϳͩ ͪ S14/9N2/9, ͑34͒ N l2 The ﬁrst change arises from the effect of the orbifold on pl the duality transformations; we will obviously be driven into obtained from the D0 geometry. The W11 patch collapses the other branch of the web of dualities that converge onto M ͑ ͓ ͔͒ into a light cone M theory black hole phase at Eq. ͑17͒. The theory cf. 78 . We proceed from the 11D phase of the black D0-brane patch has its Horowitz-Polchinski correspon- previous discussion, upward and counterclockwise on the ϳ 2 phase diagram. We have M theory on a light-cone circle dence point at S N . This is an interesting transition dis- 4 cussed in greater detail in ͓7͔;ontheS-V phase diagram, the times T /Z2 . We reduce on R11 to D0-branes in type IIA 4 ͑ ͒ (4ϩ1)D perturbative SYM phase emerges beyond this residing on the T /Z2 at Eq. 15 . Under this orbifold, the point. massless spectrum has positive parity eigenvalue. We T du- 4 ͑ ͒ The 11D black hole phase ͑11D BH͒. The equation of alize on T at Eq. 12 , getting to the patch of D4-branes in 4 state is given by type IIA wrapped on T /Z2. We remind the reader of the transformation

Ϫ R11 ␤ 1ϭ␤ ͑ ͒ Eϳ NϪ1S16/9. ͑35͒ T(4) (4)T(4) (4) , 39 l2 pl where we have used the properties of the reﬂection operator More details about this phase can be found in ͓7,76,18͔. on the spinors

The matrix string phase. The F1 geometry encountered ␤ ϭ⌫⌫i ␤2ϭ Ϫ ͒FL ␤ ␤ ϭ ͑ ͒ i , ͑ 1 , ͕ i , j͖ 0, 40 above breaks down via the Horowitz-Polchinski principle of i correspondence at Eq. ͑24͒. The emerging phase is that of a with the T-duality operation reﬂecting the left moving matrix string. This can be veriﬁed as follows: using the string spinors only. Here, (Ϫ1)FL is the left moving fermion op- scale given in this geometry ͑23͒, we can write down the ˜ 4 erator. We then M lift to M5-branes in M theory on T /Z2 equation of state of the matrix string phase ϫS1 at Eq. ͑13͒. Next, we have to apply the chain of duali- ͑ ͒ g ties M,T(3) ,S near 21 . From the M reduction we obtain D4 R 3 F ϳ 11 Ϫ1 2 4 ͑ ͒ branes on T /(Ϫ1) L⍀. This is because the M reduction E 2 N S V . 36 lpl along an orbifold direction yields the twist eigenvalues, for the massless spectrum, ͑ ͒͑ Matching this energy with that of the M5 phase 6 or that ϩ ␾ϩ Ϫ (1)Ϫ (3)ϩ ͑ ͒ of the localized F1 geometry͒ yields Eq. ͑24͒. Similarly, we g␮␯ , , B␮␯ , C , C , 41 ͑ ͒ can match the equation of states 36 and that of the type IIB while the world-sheet parity operator ⍀ acts on this spectrum ͑ ͒ hole 31 , yielding the matrix string-boosted type IIB hole as transition curve at (1),(2),(5),(6) (0),(3),(4),(7),(8) g␮␯ϩ, ␾ϩ, B␮␯Ϫ, C ϩ, C Ϫ, Ϫ SϳV 6. ͑37͒ ͑42͒

Ϫ FL Perturbative (4ϩ1)D SYM phase. The scaling of the and the action of ( 1) yields equation of state is ﬁxed by dimensional analysis and yields NSNSϩ,RRϪ. ͑43͒

R 1 3 11 1/2 5/4 The T duality on T brings us to D1-branes in type IIB Eϳͩ ͪ VN S . ͑38͒ 1ϫ 3 ⍀ 1ϫ 3 N l2 theory on S T / , which is type I theory on S T . This pl is because

Ϫ This phase borders that of the D4-branes and D0-branes. ␤ Ϫ ͒FL⍀ 1ϭ Ϫ ͒FL⍀ ͑ ͒ T(3) (3)͑ 1 T ͑ 1 . 44 The ﬁnal phase diagram is that of the ͑2,0͒ on T4ϫS1 or, (3) as we see from the LC black hole phase, that of LC M theory Finally, the S duality culminates in the geometry of N black on T4. heterotic strings smeared on the T3. The Horowitz-

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Polchinski correspondence curve ͑24͒ patches this phase with onto that of the heterotic matrix string phase, whose equation of state obeys the scaling ͑36͒. We thus verify the following l4 N S2 2ϳ pl 4ϳ 4 Ϫ5 ͑ ͒ previous suggestions ͓44–46,79͔ from the perspective of q , r0 lplV . 49 R2 V5 N Maldacena’s conjecture: 11 The phase is described by the equation of state18 Heterotic matrix string theory emerges in the UV of the ͑2,0͒ theory. 1 R11 Heterotic matrix strings can be described via the O(N) Eϭ SNϪ1/2V5/2, ͑50͒ 2␲ 2 SYM of type-I D-strings lpl

The structure of the phase diagram has not changed, but the characteristic of a string in a Hagedorn phase. Our starting labelling of some of the phases has. The additional symmetry point is the black D5 geometry ͑D5͒ given by structure of the orbifold background entered our discussion 2 ϭ Ϫ1/2͑Ϫ 2ϩ 2 ͒ϩ 1/2͑ Ϫ1 2ϩ 2 ⍀2͒ trivially; the critical behaviors are unaffected. ds10 H hdt dy(5) H h dr r d 3 To complete the discussion, we need to address a second e␾ϭHϪ1/2. ͑51͒ change to the T4 compactification. The localization transi- ͑ ͒ tions, say the one occurring at Eq. 14 , are of a somewhat The patch is parametrized by different nature than the ones encountered earlier. Localized 3 2 black geometries on orbifold backgrounds are unstable to- lpl lpl lpl g ϭ VϪ5, ␣Јϭ , y Ϸ VϪ1. ͑52͒ ward collapse toward the nearest fixed point; by virtue of str R R (5) R being above extremality, there are static forces, and by virtue 11 11 11 of the symmetry structure of the orbifold, there is no balance The relevant restrictions are: of forces as in the toroidal case. It is then most probable that the localized D0-branes sit at the orbifold points, with their The Horowitz-Polchinski correspondence principle be sat- black horizons surrounding the singularity. The most natural isfied for geometry is the one corresponding to 16 black D0 geom- SϾV15/2NϪ1/2. ͑53͒ etries distributed among the singularities, yielding a non- singular geometry outside the horizons. Beyond this point, we sew onto the perturbative (5 ϩ1)D SYM phase whose equation of state is given by Eq. C. Little strings and five-branes on T5 ͑47͒. In this section, we study the thermodynamics of five- Requiring that the coupling at the horizon be small yields 5 branes wrapped on a square T . The notation is as before; we SϽV15/2N3/2. ͑54͒ express all equations in terms of the parameters of a LC M 5 theory on T . The structure of the phase diagram for VϾ1is We then S dualize to the geometry of black NS5 branes in similar to the one already encountered. We will therefore not the type IIB theory. discuss the D0, D0, W11, W11, 11DBH, 11DBH, and per- The condition of large T5 cycles at the horizon requires turbative (5ϩ1)D phases except for noting that the only 3/2 3/2 changes to our previous discussion are to Eqs. ͑15͒, ͑31͒ and SϾV N . ͑55͒ ͑38͒, which become, respectively, Otherwise, we T dualize on the T5 and obtain the geom- SϳVϪ5/2N3/2 ͑45͒ etry of smeared D0-branes, D0. The black NS5 geometry ͑NS5B͒ is the S dual of Eq. ͑51͒, R11 1 Eϳͩ ͪ V5/2S3/2 ͑46͒ 2 2 ϭϪ 2ϩ 2ϩ ͑ Ϫ1 2ϩ 2 ⍀2͒ N lpl ds10 hdt dy5 H h dr r d 3

␾ϭ 1/2 ͑ ͒ R11 1 e H , 56 Eϳͩ ͪ VN3/5S6/5. ͑47͒ N 2 lpl and the new asymptotic moduli are

We start from the D5 geometry and move counterclockwise 4 2 R11 lpl lpl on the phase diagram. g ϭ V5, ␣Јϭ VϪ5, y Ϸ VϪ1. ͑57͒ str l 2 5 R The M5 phase (M5g). This phase consists of seven geo- pl R11 11 metrical patches. For two of these, the D0 and W11, we refer The relevant restrictions are: the reader to ͓7͔. The relevant harmonic functions are

2 2 q r0 Hϭ1ϩ , hϭ1Ϫ , ͑48͒ 18Note that we have kept track of the exact numerical coefﬁcient r2 r2 for this equation of state for later use.

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Requiring that the cycle size of the ﬁve y’s be greater than Otherwise, we localize in the manner of Gregory- the string scale yields the condition LaFlamme on ˜x11 to the geometry of localized M5-branes. The correspondence condition yields VϾ1. ͑58͒ SϾVϪ15/2NϪ3/2, ͑67͒ Beyond this point, we need to T dualize on the T5 and we obtain the geometry of black NS5-branes in a type IIA which is rendered irrelevant by the previous condition. theory. Requiring that the cycle size of the y’s at the horizon be The correspondence point is at bigger than the Planck scale yields

Nϳ1. ͑59͒ SϾV3/2N3/2. ͑68͒

We note that the dual theory is the non-local ͑1,1͒ theory Beyond this point, we reduce on one of the cycles of T5 to of type IIB NS5-branes. At low energy, it is described by a type IIA theory. We find that we need to further T dualize (5ϩ1)D perturbative SYM. on the remaining T4. The resulting geometry of black D0- branes is found strongly coupled at the horizon; we therefore The geometry of the black NS5 branes in type IIA theory lift to another Mˆ theory, and we have the geometry of black ͑NS5A͒ is the T dual of NS5B ͓Eq. ͑56͔͒: Mˆ waves smeared on the T4. ds2 ϭϪhdt2ϩdy2 ϩH͑hϪ1dr2ϩr2d⍀2͒ 10 (5) 3 The geometry of smeared waves in the Mˆ theory (Mˆ W11) is e␾ϭH1/2. ͑60͒ given by ds2 ϭ͑HϪ1͒͑dxˆ Ϫdt͒2ϩdxˆ 2 Ϫdt2ϩHϪ1͑1Ϫh͒dt2 The parameters of the type IIA theory are 11 11 11 ϩ 2 ϩ 2 ϩ Ϫ1 2ϩ 2 ⍀2 ͑ ͒ 4 2 dy(4) dx˜ 11 h dr r d 3 . 69 R11 lpl lpl g ϭ VϪ5/2, ␣Јϭ VϪ5, y Ϸ VϪ4. ͑61͒ str l 2 (5) R The parameters of the Mˆ theory are pl R11 11 ˆ ϭ ˆ ϭ Ϫ2 Ϸ Ϫ2 ˜ Ϸ Ϫ5 The new restrictions are: R11 R11 , l pl lplV , y (4) lplV , R11 lplV . ͑70͒ The correspondence point occurs for The relevant restrictions are: Nϳ1. ͑62͒ Requiring that the cycle size of ˜x11 at the horizon be big- ˆ The dual theory is the non-local ͑2,0͒ theory of type IIA ger than the Planck scale l pl yields NS5-branes, related to the ͑1,1͒ theory we encountered above Ͻ ͑ ͒ via a T duality on the T5. V 1. 71 Requiring small coupling at the horizon yields Otherwise, we reduce along ˜x11 to a type IIA theory, T 4 SϾVϪ15/2N3/2. ͑63͒ dualize on the T , and M lift back to the original LC M theory with Planck scale lpl and five torus moduli Vlpl . Otherwise, we lift to an M˜ theory and obtain smeared The system would localize on ˜x11 unless g Ϫ M5-branes. SϾV 15/2N1/2. ͑72͒ g The smeared black M5 geometry (M5) is described by the We then have localized waves in Mˆ theory which are still metric smeared on the remaining T4. We find that the cycle sizes of the four y’s as measured at ds2 ϭH2/3͑dx˜ 2 ϩhϪ1dr2ϩr2d⍀2͒ϩHϪ1/3͑dy2 Ϫhdt2͒. 11 11 3 (5) the horizon are of order the Planck scale ˆl . ͑64͒ pl The system would localize on the T4 unless ˜ Ϫ The parameters of the M theory are SϾV 3/2N1/2. ͑73͒

5 2 lpl lpl This condition is never realized because of the other re- ˜R ϭl VϪ5, ˜l 3 ϭ VϪ10, y Ϸ VϪ4. ͑65͒ 11 pl pl 2 (5) R strictions. R11 11 The system can localize on xˆ 11 unless The new restrictions are: SϾN. ͑74͒

Requiring that the size of ˜x11 as measured at the horizon be smaller than the size of the object gives Otherwise, we collapse to the geometry of an 11D black hole in LC Mˆ theory; this black hole is still smeared on the Ϫ15/2 1/2 4 SϾV N . ͑66͒ T and on ˜x11 .

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We thus conclude the discussion of this phase comprised of SϳVϪ15. ͑83͒ seven patches. We have two non-local theories sitting on top of the phase, the ͑2,0͒ theory and the ͑1,1͒ theory, related by Localization on the T4 occurs at Sϳ1, and therefore is not a T duality, and bounded by three curves due to finite size seen on our phase diagrams. This can be seen by matching effects, and one curve due to the correspondence principle. Eq. ͑82͒ with The black M5 phase ͑M5͒. This phase consists of two patches. The M5 patch ͑M5͒ is the localized version of Eq. R11 Eϳ NϪ1V4S16/9, ͑84͒ ͑64͒, 2 lpl ds2 ϭH2/3͑hϪ1dr2ϩr2d⍀2͒ϩHϪ1/3͑dy2 Ϫhdt2͒, ͑75͒ 11 4 (5) i.e. the equation of state of the totally localized hole in the Mˆ with the changes theory. This completes the phase diagram obtained from the D5 l5 system. The structure can be verified by using the various 3ϳ pl Ϫ10 5ϳ 5 Ϫ1 2 Ϫ10 ͑ ͒ equations of state. We conclude by noting that there are sev- q 2 NV , r0 lplN S V . 76 R11 eral different interpretation of this diagram. It is that of the ͑2,0͒ theory; it is that of the ͑1,1͒ related to the latter by T The equation of state becomes duality, but it also encompasses the phase structure of LC M theory on T5. Various previous observations regarding ma- R ϳ 11 6/5 4 Ϫ3/5 ͑ ͒ trix theory on T5 are thus confirmed ͓23,22͔ via the Mal- E 2 S V N . 77 lpl dacena conjecture. In other words, SϳN1/2(y E)5/6 in the parameters ͑65͒ of (5) D. D6 system this patch; this equation of state is characteristic of a (5 ϩ1)D gas, as one expects for the theory on the M5-brane at The Taub-NUT ͑Newman-Unti-Tamburino͒ phase. This large volume and sufficiently low energy. The new restric- phase consists of 8 patches. The harmonic functions are tions are: q r0 The correspondence point is now at Hϭ1ϩ , hϭ1Ϫ , ͑85͒ r r Nϳ1. ͑78͒ with Reduction on the y’s along the discussion for the smeared ϳ 2/3 Ϫ1/3 Ϫ2 M5g branes encountered above occurs at r0 S N V lpl

4/3 3 SϳN . ͑79͒ lpl qϳ NVϪ6. ͑86͒ R2 We then emerge into the phase of Mˆ W11 black waves. 11 The equation of state is The geometry of 11D black waves (Mˆ W11) is obtained via ˆ R localization on ˜x11 of the smeared geometry MW11. The ϳ 11 2/3 Ϫ1/3 4 ͑ ͒ resulting phase is still smeared on the T4. It can however E 2 S N V . 87 lpl further localize at Our starting point is the black D6 geometry ͑D6͒, given SϳN ͑80͒ by d 2 ϭ Ϫ1/2͑Ϫ 2ϩ 2 ͒ϩ 1/2͑ Ϫ1 2ϩ 2 ⍀2͒ along xˆ 11 into a smeared 11D LC black hole 11DBH. The ds10 H hdt dy(6) H h dr r d 2 condition of localization on the T4 is, however, e␾ϭHϪ3/4 SϽN1/2, ͑81͒ ͒ ͑ Ϫ1 ץϭ Frty y rH . 88 and therefore never arises due to Eq. ͑80͒. 1¯ 6 The smeared 11D LC black hole phase (11DdBH). This The parameters of this type IIA theory are phase is described by the equation of state l3 l 3/2 l2 ␣Јϭ pl ϭͩ pl ͪ Ϫ6 Ϸ pl Ϫ1 ͑ ͒ , gs V , y V . 89 R11 R R R Eϳ NϪ1V4S8/5. ͑82͒ 11 11 11 l2 pl The various restrictions are: 4 It is smeared on the T but localized on ˜x11 . Minimizing its Weak coupling at the horizon requires Gibbs energy as given by Eq. ͑82͒ with respect to that of the ͑ ͒ 2 6 hole smeared on ˜x11 , Eq. 46 yields the transition curve SϽN V . ͑90͒

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Otherwise, we lift to a Taub-NUT geometry in 11 dimen- pling at the horizon is bigger than one, so we S dualize, and sions. find that the T5 is substringy. We T dualize again and find The correspondence point is at that the resulting type IIA wave geometry is strongly coupled at the horizon. We therefore, and finally, lift to a black wave SϾV6. ͑91͒ geometry in an M˜ theory. The chain of dualities is then ˜ ˜ Perturbative 6ϩ1D SYM emerges beyond this point. M,T5 ,S,T5 ,M. The new M wave geometry (MW11) is pa- T duality on the y (6) must be applied unless rametrized by Ͼ 2 ͑ ͒ Ϸ Ϫ5 ˜ ϭ Ϫ4 Ϸ ͑ ͒ S N . 92 z(6) lplV , l pl lplV , x11 R11 . 100 ˜ Otherwise, we have the geometry of smeared D0-branes. The M circle is one of the z(6) , and the wave is along x11 . This geometry localizes on the T6 for The black Taub-NUT patch in M¯ theory (M¯ TN) is given by the geometry SϽVϪ9/2N1/2. ͑101͒

2 Ϫ 2 Ϫ ds ϭH͑h 1dr2ϩr2d⍀ ͒ϩH 1͑dx ϪAd␾͒2 6 ˜ Ͻ 11 2 11 The T at the horizon is bigger than l pl for V 1, and x11 at ˜ Ϫ 2ϩ 2 ͑ ͒ the horizon is bigger than l pl for hdt dy(6) . 93 Ͻ 6 2 ͑ ͒ We have introduced a gauge potential Aϭ(1Ϫcos ␪)N/2 lo- S V N . 102 ␪Þ cally ( 0) for the magnetically charged 2-form dual to Eq. g ͑88͒. In the Maldacena limit, this is an 11-dimensional ALE Otherwise, we reduce to a type IIA theory along x11 to the ¯ ¯˜ space with ANϪ1 singularity. The parameters of this M geometry of smeared D0-branes, D0. The curvature at the theory are horizon is small with respect to the Planck scale for Ϫ 3 6 2 SϾV 3N1/2, ͑103͒ lpl lpl lpl ˜R ϭ VϪ6, ˜l 3 ϭ VϪ6, y Ϸ VϪ1. ͑94͒ 11 2 pl 3 (6) R R11 R11 11 which is rendered irrelevant by the other considerations. ¯˜ The relevant restrictions are: The smeared D0 geometry D0 is parametrized by 3/2 3 The correspondence point R11 lpl g ϭͩ ͪ V6, ␣Јϭ VϪ12, z Ϸl VϪ5. ͑104͒ str l R (6) pl SϾNϪ1V6. ͑95͒ pl 11 6 g This is seen to be irrelevant. A T duality on the T takes us to the D6 geometry for 6 The T must be bigger than the Planck scale: SϾN2, ͑105͒ Ͼ ͑ ͒ V 1. 96 and localization on the T6 occurs for Eq. ͑101͒. g Otherwise, we have to reduce along one of the cycles to a The D6 geometry is parametrized by type IIA theory, and T dualize along the other five cycles to l 3/2 l3 l2 a type IIB Taub-NUT geometry. We then need to S dualize, ϭͩ pl ͪ ␣Јϭ pl Ϫ12 Ϸ pl Ϫ7 ͑ ͒ gstr , V , z(6) V . 106 and T dualize again on the five torus; finally, we lift to a R11 R11 R11 Taub-NUT geometry in an M˜ . Instead of following this path, This has a correspondence point at we will map the VϽ1 region from the D0 geometry. SϳVϪ6, ͑107͒ As mentioned above, we now pick up the trail from it the D0 patch. This patch localizes on the T6 to the D0 geometry for and lifts to a Taub-NUT geometry in some Mˆ theory for SϾV9/2N1/2, ͑97͒ SϾVϪ6N2. ͑108͒ and lifts to an M theory wave W11 for The Taub-NUT geometry Mˆ TN obtained from theg D6 patch SϽVϪ6N2. ͑98͒ is parametrized by

6 ͑ ͒ 3 2 2 The latter localizes on the T at Eq. 97 . For lpl lpl lpl Rˆ ϭ VϪ6, ˆl ϭ VϪ6, z Ϸ VϪ7. ͑109͒ 11 2 pl R (6) R VϽ1 ͑99͒ R11 11 11 we need to reduce the W11 geometry along one of the cycles It patches onto the M¯ TN geometry at Vϳ1 via a chain of 6 of the T , and T dualize on the other five. We find the cou- five dualities M,T5 ,S,T5 ,M discussed above.

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ϳ ␣ We note the symmetry of the diagram about V 1. The This type IIB theory is parametrized by gstr , Ј, and resides 4ϫ 1 4 remaining phases were encountered in the previous SYM on T S ; the T is square with volume V4 , and we define examples; there is a phase of localized black D0-branes, a ͓29͔ LC black hole, a smeared LC black hole, and a (6ϩ1)D perturbative SYM phase. In the D6 system, each of these V g ϵ 4 ϵ s ͑ ͒ ϳ v , g6 . 114 phases has a mirror phase about the V 1. The structure can ␣Ј2 v1/2 be further verified by matching the energies, at fixed entropy, 1 of the various phases. This completes the phase diagram for The S is compact with radius R5 . The Maldacena limit cor- the D6 system, shown in Fig. 3. We note that: responds to The gravitational coupling does not vanish in the M¯ TN and Mˆ TN patches, whereas it does for all the other patches of r R5 ␣Ј 0 with , , g and v held fixed. the diagram. ˜ ␣Ј ␣Ј1/2 6 For pϭ5,6 diagrams involving (pϩ1)D SYM, the energy ͑115͒ decreases for higher entropies, unlike the pϽ5 cases; i.e., ϫ 3ϫ 4 the specific heat is negative. This reduces the geometry above to AdS3 S T . The (1 ϩ1)D boundary theory is conformal with central charge c ϭ E. Little strings with winding charge 6k. The gravitational coupling in our conventions is ϭ͑ ␲͒7 2 ␣Ј4 ͑ ͒ We will map here the thermodynamic phase diagram of G10 2 gstr . 116 Q5 five-branes and Q1 strings. Our starting point is the D1-D5 geometry. From the area law, we have Black five-branes and strings. This phase consists of 12 ͑2␲͒4 patches. We start with the D1D5 geometry ͑D1D5͒ given by ϭ ͑ ͒ S r1r5r0R5V4 117 G10 2 ϭ͑ ͒Ϫ1/2͑Ϫ 2ϩ 2͒ϩ 1/2 Ϫ1/2 2 ds10 H1H5 hdt dx5 H1 H5 dx(4) or ϩ͑ ͒1/2͑ Ϫ1 2ϩ 2 ⍀2͒ H1H5 h dr r d 3 ϳ ␣Ј Ϫ1/2 Ϫ1/2 Ϫ1 ͑ ͒ r0 Sgstr k v R5 . 118 Ϫ e␾ϭH1/2H 1/2 1 5 The Arnowitt-Deser-Misner ͑ADM͒ mass is Ϫ1 ␳2 3 1 ͑2␲͒ R V ϩ ͪ ϭ ␳2͑␧ ͒ 5 4 2 2 2 ͩ ץϭ F 1 , F␪ ␪ ␾ 2 ␪ ␪ ␾ . ϭ ͓ ϩ ͑ ϩ ͔͒ ͑ ͒ rtx5 r 2 1 2 5 3 1 2 M 3r0 2 r1 r5 , 119 r 2 G10 ͑110͒ yielding the equation of state The harmonic functions are given by S2 r2 r2 Eϭ ͑120͒ ϭ ϩ i ϭ ϭ Ϫ 0 ͑ ͒ 8␲2kR Hi 1 i 1,5, h 1 ; 111 5 r2 r2 characteristic of a (1ϩ1)D conformal field theory ͓28͔. ␳ ϭ the charge radii of the branes i , i 1,5, are related to the The various restrictions on the D1-D5 geometry are: parameters r by i The Horowitz-Polchinski correspondence principle dic- ͑kq͒1/2 tates ␳2ϭ͑2␲͒4g ␣Ј3 , ␳2ϭg k1/2qϪ1/2␣Ј, 1 str V 5 str Ͼ Ϫ1/2 ͑ ͒ 4 g6 k . 121

␳2ϭ ͱ 2ϩ 2 ͑ ͒ Beyond this point, the (1ϩ1)D conformal theory takes i 2ri r0 ri . 112 over. Its equation of state is fixed by conformal symmetry; Here we make a distinction between the antisymmetric ten- using Cardy’s formula ͓73,74͔ and the central charge 6k,we sor field strength’s harmonic functions and those of the met- find precisely Eq. ͑120͒, as expected. ric, since we will be interested below in the numerical coef- Requiring that the coupling at the horizon be small yields ficients of some of the equations of state; the extremal limit Ͻ Ϫ1/2 ͑ ͒ ␳ g6 q . 122 corresponds to r0˜0 with the i held fixed. For scaling purposes, we can write ␳ ϭr in the Maldacena limit. We i i Otherwise, we S dualize to the geometry of NS5-branes also have traded the two integers Q and Q for the new 1 5 and fundamental strings. variables k and q: Requiring the T4 as measured at the horizon be big with respect to the string scale gives k Q ϵͱkq, Q ϵͱ . ͑113͒ 1 5 q qϾ1. ͑123͒

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Otherwise, we apply a T duality on the T4, and exchange ϭ ␣Ј Ϫ1 3 ϭ ␣Ј2 Ϫ1 R11 gstr R5 , lpl gstr R5 , the roles of Q1 and Q5 . Without loss of generality, we re- Ϫ strict our attention to qϾ1 only. We also note that qϽk; the x Ϸ␣Ј1/2v1/4, x Ϸ␣ЈR 1 . ͑132͒ ϭ (4) 5 5 upper bound corresponds to Q5 1. We therefore have The restrictions are: 1ϽqϽk. ͑124͒ The correspondence principle requires Requiring that the size of x as measured at the horizon be 5 Ͼ Ϫ1 1/2 ͑ ͒ bigger than the string scale gives S g6 q . 133

Ϫ SϾg 1/2k3/4. ͑125͒ This condition is rendered irrelevant by the others for q 6 Ͻk.Atqϳk, it coincides with the localization condition on Otherwise, we T dualize to the geometry of smeared x5 we will ﬁnd shortly. D0-D4 branes. Requiring that the size of x5 as measured at the horizon be bigger than the Planck scale yields The smeared D0D4 geometry (D0D4) is given by Ͻ Ϫ1 3/4 Ϫ1/4 ͑ ͒ S g6 k q . 134 2 ϭϪ͑ ͒Ϫ1/2 2ϩ 1/2 Ϫ1/2 2 ds10 H1H5 fdt H1 H5 dx(4) Otherwise, we reduce along x5 to a type IIA theory and to ϩ͑ ͒1/2͑ 2ϩ Ϫ1 2ϩ 2 ⍀2͒ H1H5 dx5 f dr r d 3 the geometry of boosted NS5-branes. Requiring that the size of the T4 as measured at the hori- ␾ϭ 3/4 Ϫ1/4 ͑ ͒ e H1 H5 . 126 zon be bigger than the Planck length gives Ͼ 1/2 3/4 Ϫ1/4 ͑ ͒ The parameters of this type IIA theory become S g6 k q . 135 ϭ ␣Ј1/2 Ϫ1 ␣Јϭ␣Ј ˜gs gstr R5 , ˜ , Otherwise, we reduce to a type IIA theory along one of the cycles of the T4. We ﬁnd as always that the other three Ϸ␣Ј1/2 1/4 Ϸ␣Ј Ϫ1 ͑ ͒ x(4) v , x5 R5 . 127 cycles are substringy and T dualize along them. Finally, the resulting boosted D1 geometry is seen to be strongly coupled The restrictions are: at the horizon, and we S dualize to the geometry of boosted type IIB fundamental strings smeared on x . Small curvature at the horizon yields the condition 5 The localization condition on x5 is as for the D0-D4 phase Ͼ Ϫ1/2 ͑ ͒ ͑ ͒ g6 k . 128 130 . This will be rendered irrelevant by the subsequent condi- The geometry of NS5 branes and fundamental strings ͑ ͒ tions. NS5FB is obtained from the D1-D5 geometry via S duality: Small coupling at the horizon requires 2 ϭ Ϫ1͑Ϫ 2ϩ 2 ͒ϩ Ϫ1 2 ds10 H1 fdt H1dx(4) H1 dx5 Ͼ 1/2 3/4 1/2 ͑ ͒ Ϫ S g6 k q . 129 ϩ ͑ 1 2ϩ 2 ⍀2͒ H5 f dr r d 3

Otherwise, we lift to the geometry of smeared boosted ␾ Ϫ e ϭH 1/2H1/2 . ͑136͒ M5-branes. 1 5 Requiring that the size of x5 as measured at the horizon be The parameters of the type IIB theory are smaller than the transverse size of the object yields Ϫ ˜g ϭg 1 , ˜␣Јϭg ␣Ј, x Ϸ␣Ј1/2v1/4, x ϷR . ͑137͒ Ͼ Ϫ1 1/2 ͑ ͒ s str str (4) 5 5 S g6 k . 130 The restrictions are: Beyond this point, the system localizes in the manner of Small curvature at the horizon requires Gregory and LaFlamme along x5 , and we have the geometry of localized D0-D4-branes. kϾq, ͑138͒ Finally, a large T4 is associated with the condition ͑123͒. which is trivially satisﬁed. The smeared boosted M5 geometry (M5W) is the M lift of Large x at the horizon requires the D0-D4 geometry ͓Eq. ͑126͔͒ at Eq. ͑129͒: 5 SϾk3/4q1/4. ͑139͒ 2 ϭ Ϫ1 Ϫ1/3͑Ϫ 2ϩ 2 ͒ ds11 H1 H5 fdt H1dx(4) Ϫ Otherwise, we T dualize along x and emerge into the ϩH2/3͑dx2ϩ f 1dr2ϩr2d⍀2͒ 5 5 5 3 geometry of boosted NS5-branes in type IIA theory; the lat- ϩ Ϫ1/3͓ Ϫ͑ Ϫ1Ϫ ͒ ͔2 ͑ ͒ ter was encountered from the M5W phase via an M reduction H1H5 dx11 H1 1 dt . 131 along x5 . The parameters of this M theory are Large T4 at the horizon requires

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Ͻ ͑ ͒ g6 1. 140 theory, which is T duality of the little string. As we scan through 1ϽqϽk, at the lower bound the phase structure is 4 Otherwise, we T dualize along the T , yielding to a simi- such that, via dualities exchanging Q1 and Q5 , a mirrored lar system with altered asymptotic moduli. phase diagram for qϽ1 emerges; for the upper bound, the geometrical vacua across the diagram break down via the The boosted black type IIB string geometry (F1WB) is ob- correspondence principle. These comments carry over to the tained from the M5W patch by a chain of three dualities as other phases, which we describe next. ͑ ͒ described after Eq. 135 ; this gives the metric The black localized boosted M5 phase ͑M5W͒. The local- 2 ϭϪ͑ ͒Ϫ1 2ϩ Ϫ1 2 ϩ 2 ization transition along x5 yields the change in the harmonic ds10 H1H5 fdt H1H5 dxˆ 11 dx(3) functions ϩ Ϫ1 2ϩ 2 ⍀2ϩ 2 f dr r d 3 dx5 r3 r3 ␾ϭ Ϫ1/2 ͑ ͒ Ϫ 0 Ϫ i ͑ ͒ e H5 . 141 f 1 , Hi 1 , 146 ˜ r3 ˜ r3 The parameters of the type IIB theory are with ϭ Ϫ1␣ЈϪ1/2 3/4 ␣Јϭ 2 ␣Ј2 Ϫ1 Ϫ2 gˆ str gstr v R5 , ˆ gstr v R5 2 Ϫ Ϫ Ϫ ␣Ј g x Ϸg ␣ЈvϪ1/2R 1 , x Ϸ␣ЈR 1 , xˆ Ϸg ␣ЈR 1 . 3ϳ str 1/2 1/2 (3) str 5 5 5 11 str 5 r1 k q ͑142͒ vR5

The restrictions are: ␣Ј2g 3ϳ str 1/2 Ϫ1/2 ͑ ͒ r5 k q . 147 Small curvature at the horizon requires R5

SϾk1/2. ͑143͒ The expression ͑118͒ for the entropy is unaffected by this transition, unlike all other cases encountered here and in ͓7͔. This will be rendered irrelevant by other restrictions. The equation of state of the localized phase becomes Large x5 as measured at the horizon requires 3 g6 S g Ͻ1. ͑144͒ Eϳ ͩ ͪ . ͑148͒ 6 1/2 R5 k Otherwise, we T dualize along x5 , and obtain a similar geometry. There are three patches. The localized boosted fundamen- ͑ ͒ ͑ ͒ Localization on x5 occurs unless condition 130 is satis- tal string F1WB is obtained from the F1WB patch by lo- fied. calization on x5 . The relevant restrictions are: ͑ ͒ Localization on x(3) occurs unless condition 143 is satisfied. This is irrelevant in view of Eq. ͑130͒. Small curvature at the horizon requires Requiring that xˆ as measured at the horizon be bigger 11 Ͼ 1/2 ͑ ͒ than the string scale yields Eq. ͑123͒. S k . 149 Small coupling at the horizon requires the reverse of Eq. ͑135͒. At this point, we emerge in a matrix string phase carrying And, finally, we note that the geometry is at the self-dual two charges. More on this later. ͑ ͒ Localization on x(3) occurs unless Eq. 149 is satisfied. point for the three cycles x(3) . This is similar to what we saw in the (4ϩ1)D SYM case. ͑ ͒ The geometry of boosted NS5 branes of the type IIA theory Large x11 at the horizon requires Eq. 123 . ͑NS5WA͒ is obtained from the NS5FB patch via T duality or Small coupling at the horizon necessitates the M5W via M reduction. The only relevant restriction is Ϫ that of large T4 at the horizon. This occurs for SϽk2/3q 1/6. ͑150͒ Ͻ ͑ ͒ g6 1. 145 Beyond this point, we apply the chain S,T(3) ,M to patch onto the localized boosted M5 geometry. The reverse of this Otherwise, we have the T dual, and identical, geometry with chain was described in the smeared case above. different asymptotic moduli. Finally, the geometry is at the self-dual point for the x(3) We have completed the boosted M5 phase up to the con- cycles. dition ͑145͒. We note that all duality transformations along ϳ g6 1 leave the geometries unchanged, and change the The localized boosted black M5 brane geometry ͑M5W͒ has asymptotic moduli. It is easy then to check that venturing the same parameters as Eq. ͑132͒. It is subject to one addi- Ͼ into domains with g6 1 yields a mirrored picture of the tional non-trivial condition, that of M reduction along x11 ϳ phase diagram about g6 1. Our six patches have six other unless ϳ mirror geometries across the g6 1 line. We therefore see a Ͻ 2/3 1/3 ͑ ͒ signature of a strong-weak symmetry g6˜1/g6 in the dual S k q . 151

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We then emerge into the geometry of localized D0-D4- which is indeed the equation of state of a Hagedorn string branes. with tension proportional to L/Q5 , as has been seen previ- The localized D0-D4 brane geometry ͑D0-D4͒ is subject ously from several related points of view ͓80,81͔.Itisa to the following additional condition; its curvature at the nontrivial check that this equation of state agrees precisely horizon is small when with Eq. ͑50͒ when we use the parameters ͑65͒ of the M5g phase.19 SϽk. ͑152͒ The same exercise can be repeated for the localized ͑M5W͒ phase. The equation of state ͑148͒ in DLCQ param- ϳ 2 Otherwise, the dual geometrical description breaks down. eters, again assuming light-cone kinematics ELC M /2P, Comparing the equations of state ͑148͒ and ͑120͒,wefind takes the form that we do not have a match at Sϳk. This is identical to the situation encountered in all the SYM cases at SϳN2. There ϳ 1/2 1/6 ␯1/4 ͒2/3 ͑ ͒ S Q Q ͑ lplM . 156 is a non-trivial transition at this point through a phase with 5 1 4 zero specific heat. On the (1ϩ1)D gas side, Sϳk is where In terms of scaling, this equation of state is the energy- the thermal wavelength becomes the size of the box, R5 ; the ϩ ͑ dynamics is then frozen into a quantum mechanics. entropy relation of a (2 1)D gas extensive in the box size ␯1/4 The BPS matrix string. At Sϳk1/2, the emerging phase is 4 ), although it is difficult to explain the dependence on Q1 that of the Ramond ground states of the conformal theory, and Q5 . A natural candidate for the object being observed which are those of a BPS matrix string. The situation can be here is an excited M2-brane embedded in the M5-brane ͑ ͒ compared to the matrix string transition of the M5-brane on which is indeed one of the bound states of M theory . The T4ϫS1 discussed in Sec. II A. There, we found a correspon- Q1 dependence appears to violate Lorentz invariance; it ϳ Ϫ3 1/2 would be interesting to understand why light-cone kinemat- dence curve at S V Q5 , beyond which a perturbative string description should be valid. At the transition, the ratio ics does not work in the low-energy, low-entropy regime; of the cycle sizes at the horizon of the T4 and the S1 was and why the low-entropy phase is not a boosted version of the (5ϩ1)D gas found for the M5-brane in Eq. ͑77͒. ˜ 2ϳ 6ϳ 2 Ӷ (y 4 /R11) V Q5 /S ; in particular, since V 1, the dynamics is effectively one dimensional. In the localized ͑ ͒ 2 M5W phase of the D1-D5 system, we have (y 4 /R11) F. Spectral flow and rotating black holes ϳ Ϫ1 ␣Ј 1/2 2 2␣Ј2 ϳ Ϫ1 ϳ 1/2 H1 ( v R5/gs ) Q1 at the transition S k , We now turn to a discussion of angular momentum in the 2 which is again of order Q5 /S . We conclude that the two D1-D5 system. As pointed out in Sec. I D, spectral flow is an transitions are the same. In the present case, the emerging adiabatic twisting of boundary conditions in the full string phase is BPS; a perturbative string carrying both winding Q5 theory before the Maldacena limit; the near-horizon limit and momentum Q1 quanta obeys the Virasoro constraints maps the twist onto the spectral flow operation in the superconformal algebra. On the geometry side, a point on the 2ϭ͑ ͒2ϩ͑ ͒2ϩ ϩ E Q1lstr /R Q5R/2lstr 2NL 2NR unitarity diagram ͑Fig. 5͒ in the NS sector, far from the boundary and at high conformal weight, is described by the ϭ ϭ Ϫ ͑ ͒ ͑ k Q1Q5 NL NR ; 153 BTZ black hole geometry independent of the fermion boundary conditions͓͒82,29,30,83͔, in a space which is Ӷ 3ϫ 3ϫM when e.g. the left oscillator level NL NR , there are of order asymptotically locally AdS S 4 ; in the R sector, such 1/2 k states, and the system becomes BPS saturated at NR a point represents the near horizon geometry of a rotating ϭ0. D1-D5 system ͓84͔͑due to the shift in conventions between Comments on DLCQ of the M5-brane. As mentioned in canonical definitions of NS and R sector quantum numbers͒. ӷ ϭ ϫ the introductory summary, the limit Q5 fixed, Q1 Q5 ,is The isometry SO(4) SU(2)L SU(2)R of the transverse relevant to the DLCQ description of the M5-brane ͓33–36͔. S3 combines with the ͑4,4͒ supersymmetry generators and ϫ In terms of the D1-D5 parameters, the DLCQ parameters are the SL(2,R) SL(2,R) symmetry of AdS3 to yield two copies of the Nϭ4 superconformal algebra; a gauge transforma- 2 1/3 lpl R5 tion in SU(2) can be used to shift the boundary conditions ϭl ͩ ͪ R str l g on the supercharges, yielding an isomorphism between the 11 str str NS and R sectors ͓66͔. The charges under the Cartan sub- groups of each of the two SU(2)’s parametrize the angular x R 1/3 (4) ϭ 1/4ͩ 5 ͪ ϵ␯1/4 momenta of the rotating D1-D5 or BTZ hole geometries. The v 4 lpl lstrgstr subalgebra of concern is then two copies of the Nϭ2 superconformal algebra with two U(1) R-symmetry generators x l R 1/3 J3 ϭ 1 qU(1)ϵ j that implement the spectral flow. We re- 5 ϭ str 5 ϵ ͑ ͒ L,R 2 L,R ͩ ͪ L. 154 strict our attention to equal left and right U(1) charges. lpl R5 lstrgstr

Converting Eq. ͑120͒ to DLCQ parameters, we ﬁnd 19The light-cone scaling was determined in ͓81͔; our contribution ϭ ␲ ͒1/2 ͑ ͒ S 2 ͑Q5 /L lplM, 155 is a check that the precise numerical coefﬁcient agrees.

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Consider first the NS sector. The qualitative features of where, in the last step, we have taken the non-relativistic the density of states about jϭ0 were discussed in ͓75͔. limit hЈӷ jЈ; we will see that this is justified. In the relativ- There are several phases. Consider the regime of sufficiently istic limit the hole approaches extremality; one obtains a 2 ϵ 2 Ͼ 3 Јϳ Ј large effective coupling geff g6k 1; in the present notation, gravitational wave on S with h j , thus matching onto the ϭ տ ͓ ͔ for ERAdS 2h k one is in the BTZ black hole phase 75 BPS spectrum of supergravity states. This regime occurs 20 ϳ 1/2 Ϫ3/2Շ Շ near the boundary of the unitarity plot, where the Hagedorn with S (kh) . For kgeff h k, there is a phase of (5 ϩ1)D Schwarzschild black holes because the horizon local- or gas phase takes over the (5ϩ1)D black hole. The local- 3 Ϫ1/3 4/3 3 Ͻ izes on S ; the entropy is of order Sϳk h . The lower ization on the S will then occur if SBTZ S6D at a given 3/2Շ energy, i.e. bound is the correspondence point; thus, for geff h Ϫ Ϫ Շkg 3/2 , there is a Hagedorn phase, with Sϳhg 1/2 . Fi- eff eff hЈ jЈ 2 hЈ 8/3 nally, for hՇg3/2 , the system is in a supergravity gas phase, Ͻͩ ͪ ϩͩ ͪ ͑ ͒ eff . 159 ϳ 5/6 Ͻ ϩ k k k with S h . At weak coupling geff 1, the (5 1)D Schwarzschild phase and the supergravity gas phase disap- For hЈϽk, i.e., for the horizon size smaller than the size of pear. the S3, we can ignore the last term, and we have the condi- Consider next the R sector ͑i.e. jϷk/2), and define hЈ ϵ Ϫ 1 ϭ Ϫ 1 tion h 4k and jЈ j 2k as the energy and angular momentum in R sector conventions. We first focus on the regime Ј2 Ͼ j geff 1, i.e. the middle part of the diagram in Fig. 4. For hЈϽ . ͑160͒ Ϫ2 Շ Ј Ͻ Ј k geff k h , we have the black D1-D5 system. For 0 h Շ Ϫ2 geff k, we have the M5W or D0-D4 phase localized on x5 . Ј Јϳ Note that for j near zero, the corresponding localization Finally, at h 0, the BPS matrix string cuts the diagram at condition cannot be met ͓85͔. For jЈϳk, however, this equa- ϳ 1/2 Ͻ finite and large entropy S k . For geff 1, we have an tion can be satisfied: The direct analysis in the NS sector ϳ Շ ЈՇ additional phase with entropy S k for geffk h k squeezed jЈϭ 1 k shows that a localized phase exists at large enough between the D1-D5 and D0-D4 phases. In the phase diagram 2 geff . We conclude that the D1-D5 system indeed localizes on of Fig. 4, this corresponded to the horizontal line segment at the S3 at a critical value of the angular momentum. Note that Sϳk. As argued in ͓85͔, we see that the D1-D5 system with- 3 our uncertainty in the location of the transition is due to the out angular momentum does not localize on the S at low fact that it is sensitive to the numerical accuracy of the equa- energies, whereas the stationary BTZ hole in the NS sector ͓ ͔ tions, not just the scalings of the thermodynamic parameters does undergo such a localization 75 . The spectral flow map ͑and therefore lies beyond the scope of our geometrical adiabatically relates the states of these two sectors; the dif- analysis͒. Continuing to lower conformal weights in the R fering phase structures obtained at zero charge in the NS and sector, the equation of state of the rotating D0-D4 phase is ͑ ϭ ͒ R sectors the latter flowing to e.g. j k/2 in the NS sector given by ͓86͔ then implies that the spinning D1-D5 system must undergo a 3 localization transition on the S at a critical value of the hЈ 2/3 angular momentum. We next analyze the possibility for such S2ϳkͩ ͪ Ϫ jЈ2. ͑161͒ g a transition. 6 The equation of state of the rotating D1-D5 phase can be In the NS sector, the rotating BTZ hole localizes on S3 as the extracted from the corresponding geometry ͓84͔, and is given ͑ ͒ ͑ ͒ energy is lowered; the equation of state is roughly Eq. 158 by Eq. 5 ͑without the primes͒. As extremality is approached, the spin- S2 ϳkhЈϪ jЈ2. ͑157͒ ning black hole reaches the correspondence point and be- BTZ comes a large fundamental string carrying angular momen- This phase should collapse at a critical value of jЈ toa(5 tum ϩ1)D black hole localized and spinning on the S3. Angular ϳ͓ Ϫ1/2 Ϫ 2͔ ͑ ͒ momentum is introduced in this phase by spinning up the SHag geff h j . 162 black hole along an orbit on the equator of S3 with momen- ϳ Below this, there should be a transition where a supergravity tum p jЈ/RAdS ; kinematic relations and the Schwarzschild equation of state then imply gas extremizes the free energy. The phase structure about the NS sector jϳ0 should sew onto the phase structure about the ϳ Ϫ1/3͑ Ј2Ϫ Ј2͒2/3 Ϫ1/3 Ј4/3 ͑ ͒ ϳ S6D k h j ˜k h , 158 R sector j k/2 in some intermediate regime. Most of the above formulas are not invariant under spectral flow; they are determined in an analysis about zero angular momentum 20The standard conventions for the BTZ metric, where length and relative to the NS or R sectors, and may be corrected by ϳ time scales are referred to the AdS curvature radius, differ from large gravitational back reaction when j k. We leave an those of the D1-D5 geometry encountered in the R sector, where analysis of these effects to future work. For g Ͻ1, the picture is slightly different ͑see Fig. 6͒. scales are often referred to the scale R5 . Matching the asymptotics eff ϳ ϳ 4 There is no (5ϩ1)D black hole or supergravity gas phase in of the metrics yields the relation ENSRAdS ERR5 h, where RAdS ϳ G6k. We write subsequent equations in terms of the invariant the NS sector. The phase labeled SQM has an entropy which conformal weight h to avoid confusion. is energy independent Sϳk. As mentioned above, it corre-

064002-21 EMIL MARTINEC AND VATCHE SAHAKIAN PHYSICAL REVIEW D 60 064002 sponds to the horizontal line at Sϳk in Fig. 4. We again inspired definitions of quantum gravity ͑i.e. using finite N defer a detailed analysis to future work. dual nongravitational systems͒ where the dual theory is in It is a curious fact that, for finite k, the spectral flow finite volume ͑e.g. a torus͒; the finite density of states due to relation between the NS and R sectors implies an IR cutoff in the IR cutoff in the gauge theory imposes an effective cutoff the spectrum of particle states in the latter. In the NS sector, at large radius in the supergravity—even though the classical the eigenmodes of the free scalar wave equation have a natu- wave equations in such geometries can have continuous ϳ 2 ␣Ј Ϫ1/2 ral gap in the spectrum of order 1/RAdS (g6k ) ; how- spectra. It would be interesting to understand this phenom- ever, in the R sector, the free spectrum is continuous. Nev- enon better ͑it is not obviously related to the UV-IR corre- ertheless, in the full quantum theory, spectral flow from one spondence of ͓51͔͒. sector to the other implies that the finite density of states in the NS sector gives a finite R sector density of states; in ACKNOWLEDGMENTS other words, finite k generates an effective IR cutoff. This ϱ cutoff disappears in the classical k˜ limit, as one sees for We wish to thank J. Harvey, D. Kutasov, F. Larsen, and example in the fact that the number of BPS states in the R A. Lawrence for helpful discussions. This work was sup- sector is O(ͱk). This feature is a property of all Maldacena- ported by DOE grant DE-FG02-90ER-40560.

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