henriette elvang et al- supersymmetric black rings and three-charge supertubes

8/3/2019 Henriette Elvang et al- Supersymmetric black rings and three-charge supertubes

http://slidepdf.com/reader/full/henriette-elvang-et-al-supersymmetric-black-rings-and-three-charge-supertubes 1/30

Supersymmetric black rings and three-charge supertubes

Henriette Elvang, 1 Roberto Emparan, 2 David Mateos, 3 and Harvey S. Reall 4

1 Department of Physics, University of California, Santa Barbara, California 93106-9530, USA2 Institucio Catalana de Recerca i Estudis Avancats (ICREA), Departament de Fı ´ sica Fonamental,

and C.E.R. en Astrofı ´ sica, Fı´ sica de Partı ´ cules i Cosmologia,Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain

3 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2W9, Canada4

Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA(Received 11 November 2004; published 26 January 2005)

We present supergravity solutions for 1=8-supersymmetric black supertubes with three charges andthree dipoles. Their reduction to ve dimensions yields supersymmetric black rings with regular horizonsand two independent angular momenta. The general solution contains seven independent parameters andprovides the rst example of nonuniqueness of supersymmetric black holes. In ten dimensions, thesolutions can be realized as D1-D5-P black supertubes. We also present a worldvolume construction of asupertube that exhibits three dipoles explicitly. This description allows an arbitrary cross section butcaptures only one of the angular momenta.

DOI: 10.1103/PhysRevD.71.024033 PACS numbers: 04.70.Dy, 04.65.+e, 11.25.–w, 11.25.Uv

I. INTRODUCTIONThe black hole uniqueness theorems establish that, in

four spacetime dimensions, an equilibrium black hole hasspherical topology and is uniquely determined by its con-served charges. It was realized a few years ago that theseresults do not extend to ve dimensions. The D 5 vac-uum Einstein equations admit a solution describing a sta-tionary, asymptotically at black hole with an eventhorizon of topology S1 S2: a rotating black ring [1].The solution is not uniquely determined by its conservedcharges (mass and angular momentum) and, moreover,these charges do not even distinguish black rings fromblack holes of spherical topology. Charged black ringsolutions with similar properties were constructed in [2,3].

The black rings of [1–3] entail a nite violation of black hole uniqueness, since there are nitely many solutionswith the same conserved charges. It has been suggestedthat black rings exhibiting a continuously innite violationof black hole uniqueness might also exist [4], and suchsolutions were recently constructed [5]. In their simplestguise, these are described by solutions of ve-dimensionalEinstein-Maxwell theory. Physically, they describe rotat-ing loops of magnetically charged black string. Since theloop is contractible, these solutions carry no net magneticcharge. They do carry, however, a nonzero magnetic dipolemoment. This is a nonconserved quantity, often referred toas ‘‘dipole charge.’’ The black rings of [5] are character-ized by their mass, angular momentum and dipole charge,hence there is a continuous innity of solutions for xedconserved charges.

The dipole charge has a simple microscopic interpreta-tion [5]. The black rings of [5] can be obtained fromdimensional reduction of an 11-dimensional solution de-scribing M5-branes with four worldvolume directionswrapped on an internal six-torus and one worldvolume

direction forming the S1 of the black ring in the noncom-pact dimensions. The dipole charge of the black ring is justthe number of M5-branes present. The most general solu-tion of [5] has three independent dipole charges, since itarises from the orthogonal intersection of three stacks of M5-branes wrapped on T 6, with the common string of theintersection forming the S1 of the ring. Classically, thedipole charges are continuous parameters, whereas in thequantum theory they are quantized in terms of the numberof branes in each stack.

Dipole moments and angular momentum play an im-portant role in another class of solutions of recent interest:the supertubes of [6–8]. Black rings become black tubeswhen lifted to higher dimensions, and Ref. [3] identiedcertain charged nonsupersymmetric black tubes as ther-mally excited states of two-charge supertubes carrying D1-brane and D5-brane charges and a dipole charge associatedto a Kaluza-Klein monopole (KKM). Supertubes have alsobeen the subject of interest from a different directionfollowing the realization that the nonsingular, horizon-free supergravity solutions describing these objects are inone-to-one correspondence with the Ramond-sectorground states of the supersymmetric D1-D5 string inter-section [9–11]. It has been conjectured that supergravitysolutions for three-charge supertubes might similarly ac-count for the microstates of supersymmetric ve-dimensional black holes [12]. This proposal has motivateda number of interesting studies on the D1-D5 system andsupertubes [13–17] including the rst examples of non-singular three-charge supergravity supertubes without ho-rizons [13,14].

Investigations of the relationship between black ringsand supertubes have previously been done in the frame-work of the supergravity solutions found in [1–3].However, these do not admit a supersymmetric limit withan event horizon, and this complicates understanding the

PHYSICAL REVIEW D 71, 024033 (2005)

1550-7998 =2005 =71(2)=024033(30)$23.00 024033-1 2005 The American Physical Society



microscopic origin of their entropy. 1 It has been conjec-tured, though, that supersymmetric black rings should exist[15,16]. The additional ingredient of supersymmetry of theblack ring is important for two reasons. First, many of thesolutions of [1–3,5] are believed to be classically unstable,whereas a supersymmetric black ring should be stable.Second, it should facilitate a precise quantitative compari-son between black rings, worldvolume supertubes, and the

microscopic conformal eld theory (CFT) of the D1-D5system.

Recently, we found the rst example of a supersymmet-ric black ring [18]. It is a three-parameter solution of minimal D 5 supergravity. We shall see that, upon oxi-dation to ten dimensions, this solution describes a black supertube carrying equal D1-brane, D5-brane and momen-tum (P) charges, and equal D1, D5 and KKM dipole mo-ments. One purpose of the present paper is to generalizethis solution to allow for unequal charges and unequaldipole moments. We shall present a seven-parameter black supertube solution labeled by three charges, three-dipolemoments and the radius of the ring.

Our solution contains several previously known familiesof solutions as special cases. First, it reduces to the solutionof [18] in the special case of three equal charges and threeequal dipoles. Second, it reduces to the two-charge super-gravity supertubes of [7] when one of the charges and twoof the dipoles vanish. Third, in the zero-radius limit, thesolution reduces to the four-parameter solution describingsupersymmetric black holes of spherical topology [19].Finally, in the innite-radius limit the dipole momentsbecome conserved charges and the solution reduces tothe six-charge black string of [16].

In ve dimensions, our solution describes a supersym-

metric black ring. Although it is determined by sevenparameters, it carries only ve independent conservedcharges, namely, the D1, D5 and momentum charges(which determine the mass through the saturated BPSbound), and two independent angular momenta. Henceclassically, the continuous violation of black hole unique-ness discovered for the dipole black rings of [5] alsoextends to supersymmetric black holes. In string/M theory,the net charges and dipole charges must be integer-quantized, since they represent the number of branes andunits of momenta. As a consequence of the charge quan-tization, the violation of uniqueness is nite.

One might wonder whether this lack of uniqueness couldbe a problem for a string theory calculation of the entropyof black rings. After all, the original entropy calculations[20] simply counted all microstates with the same con-served charges as the black hole, which clearly will notwork here. But note that there is no conict with thecomputation of the entropy of black holes of spherical

topology, as performed by Breckenridge, Myers, Peet,and Vafa (BMPV) [19], since the supersymmetric BMPVblack hole has two equal angular momenta, whereas ourblack rings always have unequal angular momenta. For therings themselves, the proposal of [3] is essentially that weshould resolve the nonuniqueness by counting only micro-states belonging to specic sectors of the D1-D5 CFT, withthe precise sector being determined by the values of the

dipole charges. It will be interesting to see whether this canbe done at the orbifold point of the CFT. We will make afew more comments on the issue of nonuniqueness in theconclusions of the paper.

Two-charge supertubes were originally discovered assolutions of the Dirac-Born-Infeld (DBI) effective actionof a D-brane in a Minkowski vacuum [6]. In this worldvo-lume picture the branes associated to net charges arerepresented by uxes on the worldvolume of a tubular,higher-dimensional brane; the latter carries no net chargeitself but only a dipole charge. In this description the back-reaction on spacetime of the branes is neglected. Thesupergravity solution for a two-charge supertube [7,8]describes this back-reaction.

The worldvolume description has proven extremely il-luminating for the physics of two-charge supertubes.For example, it has led to a new way of counting theentropy of the D1-D5 system that does not use its CFTdescription [17]. It is therefore desirable to have an analo-gous description for three-charge supertubes. A rst step inthis direction was given in [15], where a worldvolumedescription based on the DBI action of a D6-brane thatexhibits explicitly three charges and two dipoles wasfound. However, generic three-charge supertubes carrythree dipoles, as can be understood from the fact each

pair of charges expands to a higher-dimensional brane.A worldvolume description based on D-branes thatincorporates the third dipole seems problematic, since thelatter necessarily corresponds to an object that cannot becaptured by an open string description, such as NS5-branesor KKMs [15]. This difculty can be circumvented bygoing to M-theory, where the three branes with netcharges can be taken to be three orthogonal M2-branes,whereas the three dipoles are associated to three M5-branes. (This is also the most symmetric realization of the three-charge supertube.) We will show that there existsupersymmetric solutions of the effective action of asingle M5-brane in the M-theory Minkowski vacuum thatcarry up to four M2-brane charges and six M5-brane di-poles. We call these ‘‘calibrated supertubes’’ because theworldspace of the M5-brane takes the form S C, where Sis a calibrated surface and C is an arbitrary curve. Whilethe three-charge calibrated supertube captures all dipolesand shows that an arbitrary cross section is possible, italso suffers from limitations. We will discuss these indetail in the corresponding section. Sufce it to say herethat the calibrated supertube only captures one of the

1 Ref. [5] made some progress in this direction by studying anonsupersymmetric extremal ring with a horizon.

ELVANG, EMPARAN, MATEOS, AND REALL PHYSICAL REVIEW D 71, 024033 (2005)

024033-2



angular momenta, as opposed to the two present in thesupergravity description.

The paper is organized as follows. In Sec. II we presentthe black supertube solution as an M-theory congurationwith three orthogonal M2-brane charges and three M5-branes intersecting over a ring. Then in Sec. III we describeother useful coordinates for the solution, calculate itsphysical parameters, and study its causal structure and

horizon geometry. In Sec. IV we dualize the solution to aD1-D5-P black supertube, which is shown to possess aremarkably rich structure. Section V discusses how ourblack rings contain two independent continuous parame-ters which are not xed by the asymptotic charges, andtherefore realize innite nonuniqueness of supersymmetricblack holes. In Sec. VI we analyze some particular casescontained within our general solution, and study the ‘‘de-coupling limit,’’ relevant to AdS/CFT duality. Section VIIis devoted to the construction of worldvolume supertubeswith three charges and three dipoles. In Sec. VI we give apreliminary comparison of the supergravity black tubeswith worldvolume supertubes. We conclude in Sec. IX.

The derivation and analysis of the solutions entail manytechnical details that, for the sake of readability, we havefound convenient to move out of the main body of thepaper into a number of extended appendices. These are thederivation of the supersymmetric rings in minimal super-gravity and in U 1 N supergravity theories (Appendices Aand B), the conditions for the absence of causal anomalies(Appendix C), and the proof of regularity of the horizon(Appendix D).

II. THREE-CHARGE BLACK SUPERTUBEIN M-THEORY

The most symmetric realization of a supertube withthree charges and three dipoles is an M-theory congura-tion consisting of three M2-branes and three M5-branesoriented as indicated by the array 2

Q1 M2: 1 2 – – – – – ;

Q2 M2: – – 3 4 – – – ;

Q3 M2: – – – – 5 6 – ;

q1 m5: – – 3 4 5 6 ;

q2 m5: 1 2 – – 5 6 ;

q3 m5: 1 2 3 4 – – :

(2.1)

We will denote by zi the coordinates along the 123456-directions, which we take to span a six-torus. The threeM5-branes wrap a common circular direction, parame-trized by , in the four-dimensional space transverse to

the three M2-branes. Since this circle is contractible, theM5-branes do not carry conserved charges but are insteadcharacterized, as we will see, by their dipoles q i. The M2-branes do carry conserved charges Qi .

The D 11 supergravity solution describing this sys-tem takes the form 3

ds 211 ds 2

5 X 1 dz21 dz2

2 X 2 dz23 dz2

4

X 3 dz25 dz26 ;A A1 ^dz1 ^dz2 A2 ^dz3 ^dz4

A3 ^dz5 ^dz6 ;

(2.2)

where A is the three-form potential with four-form eldstrength F dA . The solution is specied by a metricds 2

5, three scalars X i , and three one-forms Ai , with eldstrengths F i dAi , which are dened on a ve-dimensional spacetime by

ds 25 ÿH 1H 2H 3 ÿ2=3 dt ! 2 H 1H 2H 3 1=3dx 2

4 ;

Ai H ÿ1i dt !

ÿqi

21 y d 1 x d ;

X i H ÿ1i H 1H 2H 3 1=3 ; (2.3)

where

dx 24

R2

xÿ y 2dy2

y2 ÿ 1y2 ÿ 1 d 2

dx2

1 ÿ x2 1 ÿ x2 d 2 ; (2.4)

H 1 1Q1 ÿ q2q3

2R2 xÿ y ÿq2q3

4R2 x2 ÿ y2 ;

H 2 1Q

2 ÿq

3q

12R2 xÿ y ÿq

3q

14R2 x2 ÿ y2 ;

H 3 1Q3 ÿ q1q2

2R2 xÿ y ÿq1q2

4R2 x2 ÿ y2 ;

(2.5)

and ! ! d ! d with

! ÿ1

8R2 1 ÿ x2 q1Q1 q2Q2 q3Q3

ÿ q1q2q3 3 x y ;

! 12

q1 q2 q3 1 y ÿ1

8R2 y2 ÿ 1

q1Q1 q2Q2 q3Q3

ÿq1q2q3 3 x y :

(2.6)

Note that the six-torus in (2.2) has constant volume, since

X 1 X 2 X 3 1: (2.7)

This constraint implies that the ve-dimensional metric2 In such arrays, we shall reserve capital letters (M2) for branescarrying conserved charges and lower case letters (m5) forbranes carrying dipole charges. 3 The action of D 11 supergravity is given in Eq. (B8).

SUPERSYMMETRIC BLACK RINGS AND THREE-CHARGE . . . PHYSICAL REVIEW D 71, 024033 (2005)

024033-3



ds 25 is the same as the Einstein-frame metric arising from

reduction of the above solution on T 6. Note as well that,although the functions H i are not harmonic, they appear inthe metric (2.2) as would be expected on the basis of the‘‘harmonic superposition rule’’ for the three M2-branes.

The metric dx 24 (which we shall sometimes refer to as

the ‘‘base space’’) is just the at metric on E 4 written in‘‘ring coordinates’’ [1–3,5,21]. These foliate E 4 by sur-

faces of constant y with topology S1 S2, which areequipotential surfaces of the eld created by a ringlikesource. They are illustrated in Fig. 1. The coordinatestake values in the ranges ÿ1 x 1 and ÿ1 < y

ÿ1; ; are polar angles in two orthogonal planes in E 4

and have period 2 . Asymptotic innity lies at x ! y !ÿ1. Note that the apparent singularities at y ÿ1 and x

1 are merely coordinate singularities, and that x;parametrize (topologically) a two-sphere. The locus y

ÿ1 in the four-dimensional geometry (2.4) is a circle of radius R > 0 parametrized by . We will show that in thefull geometry (2.3) this circle is blown up into a nite-area,regular horizon. Note also that the function x

ÿy is har-

monic in (2.4), with Dirac-delta sources on the circle aty ÿ1.

The angular momentum one-form ! is globally welldened, since ! x 1 ! y ÿ1 0, i.e.,there are no Dirac-Misner strings. If these had been presentthen removing them would have required a periodic iden-tication of the time coordinate, rendering the solutionunphysical [3]. In contrast, the potentials Ai are not glob-ally well dened since there are Dirac strings at x 1(but not at x ÿ1 or y ÿ1). This poses no problem,

however, because their gauge-invariant eld strengths arewell dened.

As mentioned above, Q i and q i are constants that mea-sure the charges and the dipole moments of the congura-tion. We assume that

Q1 q2q3 ; Q2 q1q3 ; Q3 q1q2 ; (2.8)

so that H i 0 (this assumption will be justied below).

For later convenience, we deneQ 1 Q1 ÿ q2q3 ; Q 2 Q2 ÿ q3q1 ;

Q 3 Q3 ÿ q1q2 ;(2.9)

which obviously satisfy 0 Q i Qi , and

q q1q2q31=3: (2.10)

Integer powers of q such as q2 and q3 should not beconfused with individual dipole moments, which we al-ways label by a subindex, i.e., as q2 and q3.

It is shown in Appendices A and B that the elds (2.2),(2.2) and (2.3) provide a supersymmetric solution of D11 supergravity. This is done as follows. First D 11supergravity is reduced on T 6 using the ansatz (2.2) andthe constraint (2.7). This gives a N 1 D 5 supergrav-ity theory with gauge group U 1 3 consisting of D 5minimal supergravity coupled to two U 1 vector multip-lets. The bosonic elds of this theory are the metric, thethree Abelian gauge elds Ai and the three scalars X i ,which obey the constraint (2.7). This is a special case of a more general U 1 N theory obtained by coupling minimalsupergravity to N ÿ 1 vector multiplets. A general form forsupersymmetric solutions of the latter theory was obtainedin [22,23], generalizing the results of [24] for the minimaltheory. Using these results, it is a simple task to extend ourconstruction of the supersymmetric black ring solutionfrom the minimal theory to this more general theory. Thegeneral N -charge supersymmetric black ring solution isgiven in Appendix B. For the special case of the U 1 3

theory, it reduces to the solution (2.3). The analysis of [22,23] reveals that all supersymmetric solutions of theD 5 theory preserve either four or eight supersymme-tries, and a complete list of the latter was given in [22]. Itfollows that our solution preserves four supersymmetriesand hence gives a 1=8 BPS solution of D 11supergravity.

In the special case of three equal charges Qi Q andthree equal dipoles q

iq, the D 5 solution (2.3) re-

duces to the supersymmetric black ring solution of mini-mal supergravity constructed in [18]. We shall show thatthe general seven-parameter solution (2.3) also describessupersymmetric black rings.

III. PHYSICAL PROPERTIES

In this section we shall compute the physical quantitiesthat characterize the black supertube solution, determine

x = − 1 x = + 1

x

ψ y

y = − 1

x = const

FIG. 1. Coordinate system for black ring metrics (from [5,21]).The diagram sketches a section at constant t and . Surfaces of constant y are ring-shaped, while x is a polar coordinate on theS2 (roughly x cos ). x 1 and y ÿ1 are xed-point sets(i.e., axes) of @ and @ , respectively. Asymptotic in nity lies at x y ÿ1.


024033-4



the necessary and suf cient conditions to avoid causalpathologies and demonstrate that it has a regular horizon.We rst introduce some new coordinate systems that areuseful for different aspects of the analysis.

A. Coordinate systems

The coordinates employed in the previous section dis-play the solution in a form that involves simple functions of x and y and indeed provide the easiest way to derive it. Toobtain the charges measured at in nity, however, it isconvenient to introduce coordinates in which the asymp-totic atness of the solution becomes manifest.Speci cally, we change x;y ! ; through

sinR y2 ÿ 1p xÿ y

; cosR 1 ÿ x2p

xÿ y; (3.1)

with 0 < 1, 0 =2. De ne also

2R2

xÿ y2 ÿ R2 2 4R2 2cos2q : (3.2)

In these coordinates the at base space metric is

dx 24 d 2 2 d 2 sin2 d 2 cos2 d 2 ; (3.3)

The functions entering the solution are

H 1 1Q1 ÿ q2q3 q2q3

2

2 ;

H 2 1Q2 ÿ q3q1 q3q1

2

2 ;

H 3 1Q3 ÿ q1q2 q1q2

2

2 ;

(3.4)

! ÿ2cos2

2 2 q1Q1 q2Q2 q3Q3 ÿq3 3ÿ2 2 ;

(3.5)

! ÿq1 q2 q32R2 2sin2

2 R2 ÿ2sin2

2 2

q1Q1 q2Q2 q3Q3 ÿ q3 3 ÿ2 2

;

Ai H ÿ1i dt !

q i

22 R2 ÿ d

2 ÿ R2 ÿ d :

Note that ÿ1 is a harmonic function in (3.3) with Dirac-delta sources on a ring at R, =2 (see Fig. 2). As

! 1 the ve-dimensional metric (2.3) is manifestlyasymptotically at, and the 11-dimensional metric (2.2)is asymptotically at in the directions transverse to all of the M2-branes.

This coordinate system foliates E4 in a familiar manner,but is quite unwieldy for studying the structure of thesolution near the ring. There is yet a third system of

coordinates that proves useful for later applications, inparticular, for describing the decoupling limit in

Sec. VI D. These coordinates are de

ned by changing x;y ! r; through

r 2 R2 1 ÿ x xÿ y

; cos2 1 x xÿ y

; (3.7)

where 0 r < 1, 0 =2. The at base space met-ric is

dx 24

dr 2

r 2 R2 d 2 r 2 R2 sin2 d 2

r 2cos2 d 2 ; (3.8)

where now the function de ned in (3.2) takes the form

r 2 R2cos2 ; (3.9)

and does not involve any surds. Surfaces at constant r aretopologically S3’s that enclose the ring. The inner disk of

Θ = 0

Θ = π/ 2Θ = π/ 2ρ = R

FIG. 2. Coordinates ; , in a section at constant t, , (thefour quadrants are obtained by including also constant and ). Solid lines are surfaces of constant , dashed linesare at constant . The ring lies at R, =2.

θ = 0

θ = π/ 2θ = π/ 2 r = 0

r = const

FIG. 3. Coordinates r; , in a section at constant t, , (and , ). Solid lines are surfaces of constant r , dashed

lines are at constant . The axis of consists of the segmentsr 0 and =2.


024033-5



the ring, f x 1g, corresponds now to fr 0 ;0 < <=2g, and the outer annulus, f x ÿ1g, is f0 < r < 1 ;=2g(see Fig. 3). The horizon of the ring lies at r 0,=2. The full solution becomes manifestly at as r ! 1 ,

where r and come to coincide with the previous and .The functions de ning the solution are now

H 1 1Q1

ÿ

q2q3R2 cos22 ; (3.10)

with the obvious permutations of (123) giving H 2 and H 3,and

! ÿr 2cos2

2 2 q1Q1 q2Q2 q3Q3

ÿ q3 12R2 cos2

;

! ÿq1 q2 q3R2sin2

ÿr2 R2 sin2

2 2

q1Q1 q2Q2 q3Q3 ÿ q3 12R2 cos2

:

(3.11)

For convenience, we also give the gauge potentials

Ai H ÿ1i dt !

q iR2sin2 d ÿ cos2 d :

(3.12)

B. Physical parameters

If we assume that the zi directions are all compact withlength 2 ‘ , then the ve-dimensional Newton ’s constant

G5 is related to the 11D coupling constant through 28 G 5 2 ‘ 6. The mass and angular momenta in ve di-mensions can be read off from the asymptotic form of theabove metric,

M

4G5Q1 Q2 Q3 ;

J

8G5q1Q1 q2Q2 q3Q3 ÿ q1q2q3 ;

J

8G52R2 q1 q2 q3 q1Q1 q2Q2

q3Q3 ÿ q1q2q3 :

(3.13)

The M2-brane charges carried by the solution are given by

Q i2 ‘ 2

2 2 Z S3 T 4? 11F

116 G 5 Z S3

X i ÿ2 ? 5 F i

4G5Q i ; (3.14)

where ? 11 and ? 5 are the 11-dimensional and ve-dimensional Hodge dual operators with respect to themetrics (2.2) and (2.3), respectively. The S3 is the sphere

at in nity in the ve-dimensional spacetime, and T 4 de-notes the 3456, 1256 and 1234 four-torus for i 1 ;2 ;3,respectively. The solution saturates the BPS bound

M Q 1 Q 2 Q 3: (3.15)

As we have explained, the M5-branes do not carry any netcharges. However, their presence can be characterized byappropriate uxes, to which we will refer as ‘‘ dipole

charges, ’’ through surfaces that encircle the ring once,namely, by

D i2 ‘ 4

2 2 Z S2 T 2F 1

16 G 5 Z S2F i

q i

8G5 ;

(3.16)

where the S2 is a surface of constant t, y, and in themetric (2.2), and T 2 is a two-torus in the 12-, 34- and 56-directions for i 1 ;2 ;3, respectively. In computing theseintegrals it is useful to observe that the rst summand in Ai

does not contribute, since, because ! is globally wellde ned, it leads to a total derivative.

These dipole constituents generate a dipole

eld com-ponent of F near in nity. Asymptotically, the magneticcomponents of the three-form potential are given by

Ai ! ÿ4G5J

qiR2 cos2

2 ; (3.17)

Ai ! ÿ

4G5J

ÿ q iR2 sin2

2 : (3.18)

The presence of nonzero -components is easily under-stood. Consider for example A1 (the interpretation for A2

and A3 is analogous). This corresponds to a nonzero A 12

that, upon Hodge dualization, leads to a nonzero compo-nent ~A 03456 , as expected for the potential sourced by M5-branes along the 3456 -directions, as in the array (2.1).The magnitude of this dipole moment is set by the coef -cient in brackets in (3.17). The second contribution, q1R2,is exactly as would be expected for a one-dipole supertubesource [7,8]. The interpretation of the contribution propor-tional to J is more subtle, and its origin can presumablybe understood in the same way as that of J itself, whichwill be discussed in Sec. VIII.

Hodge dualization of the Ai components would seem-

ingly suggest the presence of M5-brane sources that wrapthe -direction. However, examination of the details of thesupergravity solution reveals that there are no such sources.Instead, the correct interpretation of these components isthat they are sourced by the M2-branes in the presence of the M5-branes. Consider, for example, a two-charge/one-dipole supertube consisting of the M2-branes along the 12-and 34-directions and the M5-brane along the1234 -directions. The M2-branes can be represented by uxes H 012 and H 034 of the M5-brane worldvolume three-form. These couple minimally to, and hence act as sources


024033-6



of, the A 12 and A 34 components of the supergravitypotential through the Wess-Zumino term of the M5-braneaction, SWZ RA 3 ^H 3.

We can infer from the expressions (3.17) and (3.18) thatthe gyromagnetic ratio of the supersymmetric ring is g3, as for the BMPV black hole [25].

In the quantum theory the charges will be integer-quantized, with [26]

N i

4G5

2=3Q i ; (3.19)

n i

4G5

1=3q i ; (3.20)

corresponding to the numbers of M2- and M5-branes in thesystem, respectively.

C. Causal structure and horizon geometry

The results of [24] reveal that many rotating supersym-metric solutions exhibit closed causal curves (CCCs). Inthis section, we shall derive a simple criterion for theabsence of such pathologies in a general ve-dimensionalsupersymmetric solution and then examine when our so-lution (2.3) satis es this criterion.

Any supersymmetric solution of D 5 supergravitytheory admits a nonspacelike Killing vector eld V [22,27], which de nes a preferred time orientation. In aregion where V is timelike, the metric can be written as(see Appendix A for details)

ds 2 ÿ f 2 dt ! 2 f ÿ1hmndxmdxn ; (3.21)

where V @=@tand hmn is a Riemannian metric on a four-

dimensional space with coordinates xm

. The metric hmn ,scalar f and 1-form ! ! mdxm are all independent of t.For our solution, f ÿ1 H 1H 2H 3 1=3, xm f ;y; ;xg,and hmn is at.

Consider a smooth, future-directed, causal curve in sucha region. Let U denote the tangent to the curve and aparameter along the curve. Then, using a dot to denote aderivative with respect to , we have

0 ÿV U f 2 _t ! m _ xm ; (3.22)

because the curve is future-directed. Furthermore,

f 2 _t _t 2! m _ xm ÿU 2 gmn _ xm _ xn ; (3.23)

where

gmn f ÿ1hmn ÿ f 2! m! n : (3.24)

Let us now assume that gmn is positive-de nite. Then theright-hand side of Eq. (3.23) is non-negative because U 20. We shall show that this implies _t > 0. Consider rst thespecial case in which the right-hand side of (3.23) vanishes.This implies that _ xm 0. Equation (3.22) then gives _t > 0(we cannot have _t 0 as that would imply U 0). Now

consider the general case in which the right-hand side of (3.23) is positive. Then either (i) _t > 0 and _t 2! m _ xm > 0or (ii) _t < 0 and _t 2! m _ xm < 0. However, it is easy to seethat (ii) is inconsistent with (3.22). Hence we must have (i)so _t > 0. Therefore t must increase along any causal curve,so if t is globally de ned then such a curve cannot intersectitself.

In summary, the condition that gmn be positive-de nite

is suf cient to ensure that there are no closed causal curvescontained entirely within a region in which V is timelikeand t is globally de ned.

For our solution, the coordinates t;x;y; ; coversuch a region. Hence to show that our solution has noCCCs at nite y, i t is suf cient to show that gmn ispositive-de nite for ÿ1 < y < ÿ1. This reduces to show-ing that gij is positive-de nite, where i; j are the ; directions. In Appendix C we show that a necessary andsuf cient condition for gij to be positive-de nite is 4

2Xi<jQ iq iQ j q j ÿXi

Q 2i q2

i 4R2q3Xiqi ; (3.25)

where we use the Q i de ned in (2.9). It follows from theabove argument that our solution is free of CCCs at nite yif the inequality (3.25) is satis ed. This might be regardedas providing an upper bound on the radius R for a given setof charges Qi and qi . However, as we shall see, R is not thephysical radius of the ring. An equivalent expression thatinvolves only physical quantities is

4q1q2 Q 1Q 2 ÿ q34G5 J ÿ J

Q 1q1 Q 2q2 ÿ Q 3q3

2: (3.26)

This expression yields now an upper bound on J

ÿJ for

given charges. Other equivalent forms are obtained bypermutations of (123).

As y ! ÿ1 we nd

g L2 q2

41 ÿ x2 O 1

y ; (3.27)

where

L1

2q2 2Xi<jQ iq iQ j q j ÿXi

Q 2i q2

i ÿ 4R2q3Xiq i

1=2

1q2 q1q2 Q 1Q 2 ÿ q3

4G5 J ÿ J

ÿ14

Q 1q1 Q 2q2 ÿ Q 3q32

1=2 ; (3.28)

4 We assume q i > 0 so the inequality (3.25) requires Q i to liein the region interior to one of the sheets of a two-sheetedhyperboloid in R 3. One sheet lies entirely in the positive octantof R 3 (i.e., Q i > 0) and the other entirely in the negative octant.We have assumed that we are dealing with the positive octant.This justi es our earlier restriction (2.8).


024033-7



which is real and non-negative as a consequence of (3.25).If (3.25) were violated then @=@ would become timelikeas y ! ÿ1 in a neighborhood of x 1 so some orbits of @=@ would be closed timelike curves.

In [18] we showed that the BPS ring solution of minimalsupergravity can be analytically extended through an eventhorizon at y ÿ1when the inequality (3.25) is strict (i.e.,when L > 0). The same is true of the general solution

presented above. The method of extending the solution isthe same as in the minimal theory — the details are pre-sented in Appendix D. There we show that the geometry of a spacelike section of the horizon is the product of a circleof radius L and a round 2-sphere of radius q=2,

ds 2H L2d 02 q2

4d 2 sin2 d 2 ; (3.29)

The ring circle is parametrized by the coordinate 0, whichis a good coordinate on the horizon, while itself is not.These coordinates differ by a function of y [see (D1) fordetails] and hence have the same period 2 . The S2 coor-dinates are

ÿ and cos x, which are well

behaved at the horizon. The horizon area is

A H 2 2Lq2: (3.30)

Observe that the proper circumferential length of the ringhorizon is 2 L , not 2 R , and can be arbitrarily large orsmall for xed R. In a form more symmetric in the threeconstituents,

L1q2 Q1Q2Q3 ÿ Q 1Q 2Q 3 ÿ

4G5J

2

ÿ q3 4G5 J

ÿJ

1=2

: (3.31)

When L 0, it is shown in Appendix D 3 that the solutionhas a null orbifold singularity instead of a regular eventhorizon.

Finally we should mention that the restriction (3.25)guarantees only that CCCs are absent in the region exteriorto the event horizon of the black ring. There will certainlybe CCCs present behind the event horizon.

IV. THE DOUBLE HELIX: D1-D5-P BLACKSUPERTUBE

The solution of 11-dimensional supergravity given inSec. II can be Kaluza-Klein reduced to a solution of typeIIA supergravity and then dualized to a type IIB solutionwith net charges D1, D5, and momentum (P). We studyhere the properties of this black supertube solution.

A. The IIB solution

Perform a KK reduction of (2.2) along z6, and T-dualizeon z5, z4, z3 (using [28]) to get a IIB supergravity solution.

The solution has D1-D5-P charges and D1, D5 and Kaluza-Klein monopole (kkm) dipoles. The D1-D5-P supergravitysolution describes a three-charge black supertube. In stringtheory, a D1-D5-P supertube is actually a double D1-D5helix that carries momentum in the direction parallel to itsaxis, along z z5, and which coils around the direction of the ring . The D1 and D5 branes are bound to a tube madeof KK monopoles spanning the ring circle and z1 ; z2 ; z3 ; z4,

with the direction z being the U 1 ber of the KK mono-poles. In array form

Q1 D5: z 1 2 3 4 –

Q2 D1: z – – – – –

Q3 P: z – – – – –

q1 d1: – – – – –

q2 d5: – 1 2 3 4

q3 kkm: z 1 2 3 4 :

(4.1)

Dualizing the supergravity solution as described above, we nd that the string frame metric of the D1-D5-P black supertube is

ds 2 ÿ X 3 1=2ds 25 X 3 ÿ3=2 dz A3 2

X 1 X 3 1=2dz 24

ÿ1

H 3 H 1H 2p dt ! 2 H 3H 1H 2p dz A3 2

H 1H 2p dx 24

H 2H 1s dz 2

4 ; (4.2)

where ds 25, X i , and Ai are given in (2.3). 5 The other non-

vanishing elds are the dilaton and RR 3-form eldstrength:

e2 H 2H 1

F 3 X 1 ÿ2 ? 5 F 1 F 2 ^ dz A3 :

(4.3)

The Bianchi identity and equation of motion of F 3 aresatis ed as a consequence of the Bianchi identities andequations of motion of the D 5 gauge elds F 1 and F 2.

This solution can be S-dualized to give a purely NS-NSsolution of type II supergravity, and it then describes an F1-NS5-P supertube. A trivial T-duality along any of the atdirections z 4 of the NS5 brane maps the F1-NS5-P super-tube to a solution of IIA supergravity, which when upliftedto 11 dimensions along a direction ~z provides an embed-ding in D 11 supergravity different than the one in (2.2).This new embedding describes M2 and M5 branes that

5 Any supersymmetric solution of IIB supergravity must admita globally de ned null Killing vector eld [29]. For this solutionit is easy to see that @=@tis globally null.


024033-8



intersect over a helical string, and which are bound to atube of KK monopoles. Reducing it along z yields a super-tube with D0-F1-D4 charges bound to D2-D6-NS5 tubularbranes. T-dualizing this along ~z gives back a D1-D5-Psupertube with the charges Qi shuf ed compared to the rst con guration (4.1) and (4.2).

Because of its particular relevance to the microscopicCFT description of black holes, in the following we will

mostly focus on the D1-D5-P version of the solution. Weassume that the directions z4 are compact with length 6

2 ‘ , while the length along z is 2 R z. The numbers of D5 and D1 branes and momentum units are then

N D51

gs‘ 2s

Q1 ; N D11

gs‘ 2s

‘‘ s

4Q2 ;

N P1

g2s ‘ 2

s

Rz

‘ s

2 ‘‘ s

4Q3 ;

(4.4)

and the dipole components

nD11

gs‘ s

Rz

‘ s

‘‘ s

4

q1 ; nD51

gs‘ s

Rz

‘ s q2 ; (4.5)

where gs and ‘ s are the string coupling constant and stringlength. The quantization condition on the KKM dipole willbe rederived below.

B. Structure of the D1-D5-P supertube

1. KK dipole quantization

The solution (4.2) possesses nontrivial structure alongthe sixth direction z, so it is more appropriately viewedfrom a six-dimensional perspective. The quantization of

the KK dipole charge follows then from purely geometricconsiderations [3]. The metric ds 25 is clearly regular at x

ÿ1 and y ÿ1. However, A3 is not regular at x 1unless we perform a gauge transformation. This gaugetransformation is a shift in the coordinate z:

z ! z z ÿ q3 (4.6)

under which the dangerous terms transform as

dz ÿq3

21 x d dz

q3

21 ÿ x d ; (4.7)

which is now regular at x 1. However, z parametrizes acompact Kaluza-Klein direction, so z z 2 R z. This

implies that the coordinate transformation (4.6) is globallywell de ned only if

q3 nKKRz ; (4.8)

for some positive integer nKK (as we know q3 > 0). Hence

the dipole charge q3 is quantized in units of the radius of the KK circle, as expected for a KK monopole charge. Thisis dual to the quantization conditions (4.5).

2. Horizon geometry

The D 5 no-CCC condition (3.25) is suf cient toensure that the IIB solution is also free of naked CCCsbecause the extra terms in the metric (4.2) are manifestlypositive. 7 Subject to the quantization condition (4.8), theIIB solution is regular at nite y. As y ! ÿ1 , the confor-mal factors multiplying the three terms in the rst line of (4.2) remain nite and nonzero (since they just involvepowers of the D 5 scalar elds). We know that ds 2

5 isregular at y ÿ1when L > 0 so it remains only to showthat dz A3 is also regular there. The gauge transforma-tion that achieves this is described in Appendix D. It is thenapparent that y ÿ1 is an event horizon of the IIBsolution.

Following Appendix D, the geometry of a spatial slicethrough the event horizon is (in string frame)

ds 2 qL 2

q1q2p d 02 q3 q1q2p 4

d 2 sin2 d 2

q1q2p q3

dz0ÿq3

21 cos d

ÿq1Q 1 q2Q 2 ÿ q3Q 3

2q1q2d 0

2 q1

q2s dz24 ; (4.9)

where, recall, q q1q2q31=3 and cos x. The coordi-

nate z0differs from z only by a function of y, and hence hasthe same period 2 R z. The coordinate transformation

z00 z0ÿq1Q 1 q2Q 2 ÿ q3Q 3

2q1q2 0 (4.10)

reveals that this is locally a product of S1, parametrized by 0, with a locally S3 geometry parametrized by z00 ; ;

(and T 4). The locally S3 part is only globally S3 in thespecial case nKK 1. For nKK > 1 it is a homogeneouslens space S3=Z nKK . Note, however, that the coordinatetransformation (4.10) is not globally well de ned unless

q1Q 1 q2Q 2 ÿ q3Q 3 2q1q2mRz ; (4.11)

6 For simplicity we use the same letter ‘ to denote the compactradii in the IIB solution as in the D 11 solution, even if theyare not invariant under the dualities that relate them. We hopethat this does not cause any confusion.

7 We shall not determine whether this condition is also neces-sary for CCCs to be absent in the IIB solution since the D 5description seems to be more relevant (and more stringent) forthe purpose of analyzing causal anomalies. In the case of theBMPV solution it is known that CCCs can be removed from theIIB solution by working in the universal covering space, andonly appear when the z direction is compacti ed [25]. This is notthe case here. For instance, the component of the IIB metricis not automatically positive near the horizon without imposingsome condition — and (3.25) is suf cient for this.


024033-9



with m an integer. In this special case the horizon geometryis the product S1 S3=Z nKK . If this equation is not sat-is ed then the horizon geometry is given by a regularnonproduct metric on S1 S3=Z nKK .8 To avoid confu-sion, we emphasize that Eq. (4.11) does not have to besatis ed in general but, when it is satis ed, the geometry of the horizon factorizes. It is worth observing that in this case

the no-CCC bound (3.26), and also the expression for theentropy, simplify considerably.

The near-horizon limit of the IIB solution is obtained byde ning y ÿR2= L~r , t ~t= and ! 0. In terms of the coordinates regular at the horizon introduced inAppendix D, we take r L~r=R, v ~v= . In this limitwe obtain a locally AdS3 S3 T 4 spacetime 9 :

ds 2 2qq1q2p d ~vd ~r

4Lq1q2p ~rd ~vd 0 qL 2

q1q2p d 02 q3 q1q2p 4

d ~2 sin2 ~d 2 q1q2p q3

dz0ÿq3

21 cos~ d ÿ

q1Q1 q2Q2 ÿ q3Q3 q3

2q1q2d 0

2 q1

q2s dz24: (4.12)

The near-horizon metric (4.12) is locally the same as that of (oxidized) BMPV. Note however that the roles of somecoordinates, like 0and z0, are exchanged relative to BMPV. We will revisit this issue in Sec. VI D.

The area of the horizon is interpreted as usual as associated to an entropy. Expressed in terms of the brane numbers, theentropy of the D1-D5-P black supertube is

SA H

4G5

2 N D1N D5N P

ÿN D1N D5N P

ÿJ 2

ÿnD1nD5nKK J

ÿJ

q ; (4.13)

where we have de ned, in analogy to (2.9),

N D1 N D1 ÿ nD1nKK ; N D5 N D5 ÿ nD5nKK ; N P N P ÿ nD1nD5: (4.14)

An alternative form is

S 2 N D1N D5 ÿ N D1N D5 N P N D1N D5 ÿ nKK J ÿ J nD1nD5 ÿ J 2q ; (4.15)

which suggests the interpretation that the system decom-poses into two sectors, with central charges c06 N D1N D5 ÿ N D1N D5 and c00 6N D1N D5.

It is also worth noting that, in terms of integer branenumbers Eq. (4.11) is

nD1N D5 nD5N D1 ÿ nKKN P 2mnD1nD5: (4.16)

Note that all dependences on the moduli gs , Rz=‘ s and ‘=‘ sdrop out from this equation. It would be interesting tounderstand its microscopic origin.

V. NONUNIQUENESS

A supersymmetric black ring solution is completelyspeci ed by the seven dimensionful parameters Qi , qi ,and R. Such a solution carries only ve independent con-served charges: three gauge charges proportional to the Qi

and two angular momenta J and J . The mass of thesolution is not an independent charge since it is determinedby the saturated BPS bound in terms of the gauge charges.

The three-dipole charges of the solution, proportional tothe qi , are not conserved charges. Eqs. (3.13) can be used toeliminate R and one combination of the dipoles in favor of J and J . The remaining two dipoles can still be variedcontinuously while keeping the conserved charges xed.Supersymmetric black rings are therefore not uniquelydetermined by the latter, but exhibit in nite nonuniquenessin the classical theory. In this section we will examineseveral aspects of this nonuniqueness.

To simplify the analysis, let us take all gauge charges tobe equal, Q Qi . The BPS bound then xes the ADMmass to be M 3 Q= 4G5 . Since we wish to compareproperties of different rings with the same mass and gaugecharges, we de ne the dimensionless angular momenta,horizon area and dipole charges as

j ; 27 32G5s

J ;

M 3=2 ; aH A H

G5M 3=2 ; (5.1)

and

i3 p 2

qi

G5M 1=2qi

Qp : (5.2)8 Formally, this is the same as the oxidation of BMPV: see

Eq. (6.8) of [30] with u ! 0, 0! 2z00=q3, ! . (Lensspaces do not arise from an asymptotically at BMPV black hole but they do arise from obvious quotients of BMPV. Theycan also arise in the near-horizon geometry of oxidized D 4black holes [31].)

9 This is the string frame metric. For consistency with notationto be used later, we have changed ! ~.


024033-10



Note from (3.13) that j > j . By (2.8), the i ’s mustsatisfy i j 1 for i Þ j . As discussed above, we caneliminate one of the parameters i . Solving for 3 gives

34 2p j ÿ 2 ÿ 1

1 ÿ 2 1: (5.3)

For a regular horizon we need i > 0 and hence 4 2p j >

1 2.It is illustrative to specialize to the case 1 2 ,i.e., equal pitches for the D1 and D5 helices. Substituting in

3 as given above, we nd

a H 16 1=2

33=22 2p j ÿ

1 ÿ 2 3 2

ÿ 2 2p j 1 2 ÿ4 2p 2 j 1=2

: (5.4)

Since 4 2p j 1 2 2 and 1, we require

j < j max

1

4 2p 23 2

ÿ2 2p j 1 2 :

(5.5)

This is just the condition (3.25) needed to avoid nakedCCCs. For the BMPV black hole, it is well-known thatrequiring the spacetime to be free of naked CCCs imposesan upper bound on the angular momenta. However, in thecase of BMPV the angular momenta must be equal inmagnitude and there is no equivalent of the nonuniquenessparameter (dipole moment) , so the bound comes outmuch simpler than (5.5).

In [18], we studied the supersymmetric black ringsobtained from the general solutions of Sec. II by takingall charges Qi equal and all dipole moments q i equal. Thatspecialized system does not exhibit nonuniqueness, sincethe three-parameter solution is speci ed uniquely by theconserved charges (the net charge and the two angularmomenta). In particular, we plotted in [18] the horizonarea a H as a function of j and j . For the more generalcase at hand, we can make such a plot for each value of .As an example, Fig. 4 shows a H vs j and j for xed0:4. Note that for nonzero dipole moments, there are upperand lower bounds on both angular momenta (a featurepresent also in nonsupersymmetric rings [5]). It would beinteresting to understand the precise microscopic origin of these bounds and how they depend on the dipole moments.

The expression (5.4) for the horizon area a H illustratesthe nonuniqueness: the net charges are xed and even whenboth the angular momenta are speci ed, we can still vary

. In particular, we can x j and j , and plot the entropya H as a function of . More generally, we can include both

1 and 2. Then we can use one parameter to x thehorizon area a H and still have another parameter to vary.We conclude that for given net charges Qi and angularmomenta j ; , there are in nitely many supersymmetric

black rings with the same horizon area, except for the black ring that maximizes the area, which is unique. This might

suggest to recover a notion of uniqueness, at least amongsupersymmetric black rings, by adding the condition thatthe solution have maximum entropy for given conservedasymptotic charges. We emphasize, however, that thisadditional requirement is absent from the traditional notionof black hole uniqueness, and is also known to be insuf -cient to distinguish between nonsupersymmetric black rings and black holes of spherical topology [1].

Figure 5 illustrates the nonuniqueness of supersymmet-ric black rings. It shows for xed values of j and j thehorizon area a H as a function of the dipole parameters 1and 2. The bounds of the covered region are set by therequirement that there be no naked CCCs.

Now consider what happens when we uplift the super-symmetric black ring to a D 10 D1-D5-P supertube, asdescribed in Sec. IV. In the quantum theory, the net chargesand dipoles are quantized in terms of the number of branesin the D1-D5-P con guration. Using (4.4) and (4.5), we nd that the restrictions (2.8) on the net charges Q i anddipole moments qi become

N D5 nD5nKK ; N D1 nD1nKK ; N P nD1nD5: (5.6)

j φ

j ψ 5

4 √21

5 √2

15 √2

79185 √2

FIG. 4. The dimensionless area a H as a function of j and jfor xed 0:4. Note that the third dipole charge is not heldconstant, but is determined by the other parameters. Darkerregions correspond to smaller area. On the left, the triangularregion is bounded by the line j j . The lower bound is aconsequence of j being bounded from below for nonvanishingdipole moment: 2 2p j . The region is bounded on the rightby the line determined by j max

in (5.5). This is a consequence of requiring that there are no naked CCCs. The area a H vanishes atthe bottom and right boundaries of the triangular region. For xed j , a H is maximized when j ! j .


024033-11



So the number of D1- and D5-branes and units of momentarestrict the number of dipole branes and KK monopoles.This shows that upon quantization of the charges, thenonuniqueness becomes nite (but still very large).

VI. PARTICULAR CASES AND LIMITS

In this section we study various limits of the supersym-metric black ring solution. First in Sec. VI A, we considerthe limit R ! 0 where the solution reduces to the BMPVblack hole. In the in nite-radius limit R ! 1 , the black ring becomes a black string in ve dimensions. We show inSec. VI B that in this limit our solution reproduces theblack string metric found in [16]. Sec. VI C contains spe-cial cases of the general black ring solution: one is theoriginal two-charge supertube solution [6,7], the other thethree-charge solution with only two nonzero dipole mo-ments. Finally, in Sec. VI D, we study the decoupling limitrelevant for the AdS/CFT correspondence.

A. BMPV black hole

Consider the solution in the ; coordinates of (3.1),(3.2), and (3.3) 10 . If we set R 0 we nd H i 1 Qi= 2

and

! ÿ4G5J

cos2

2 ; ! ÿ4G5J

sin2

2 ;

Ai H ÿ1i dt ! ;

(6.1)

where

J

8G5

q1Q1 q2Q2 q3Q3

ÿq1q2q3 : (6.2)

This is the BMPV black hole with three independentcharges Qi and angular momenta J J J [19].Note that the parameters q i have become redundant sincethey enter the solution only through the angular momen-tum J . In particular, they no longer appear in Ai and

therefore no longer have the interpretation of dipolecharges. The horizon is located at ! 0. Topologicallythe horizon is a three-sphere. The horizon area is

A BMPV 2 2 Q1Q2Q3 ÿ4G5J

2s : (6.3)

This is not the R ! 0 limit of the horizon area of thesupersymmetric black ring (3.30),

limR! 0

A ring 2 2 Q1Q2Q3 ÿ Q 1Q 2Q 3 ÿ4G5J

2s ;

(6.4)

which is always smaller than that of the BMPV black holewith the same asymptotic charges, except possibly whenboth areas vanish. The areas are compared in Fig. 6 for theparticular case of equal charges Q i Q and dipoles qiq, i.e., for the solutions of minimal supergravity.

A clue to a (macroscopic) understanding of this effectfollows by considering an analogy to a two-center extremalReissner-Nordstrom (RN) solution. Observe that if we take

2 / 27 1 / 8 j 2

a H

163 √π

3

169 √π

3

FIG. 6. Area of the BMPV black hole (dashed) and limit R !0 of the area of the supersymmetric black ring (solid), vs spin2,for xed mass [in terms of the variables a H and j in (5.1)]. Forsimplicity the three charges and dipoles are set equal, Q i Q,qi q so these are solutions of minimal D 5 supergravity.

0.45 η2

η1

0.45

0.63 η2

η1

0.63

0.99η2

η1

0.99

FIG. 5. Contour plots of the dimensionless area a H versus 1 and 2, for xed values of j and j . For all three plots 4 2p j 2:1,but j varies for each case: from left to right 4 2p j 0:7, 1:; 1:7. The plots are symmetric in 1 and 2, and the darker regionscorrespond to smaller area. The regions for which the black rings exist are bounded by the condition that there be no naked CCCs. Atthis boundary, the area a H vanishes.

10 The limit can equally well be taken in the r; coordinates(3.7): when R 0 one has r , .


024033-12



a diameter section of the black ring solution, at constant and , then the resulting four-dimensional geometrycontains two in nite throats and therefore is similar to thegeometry of two extremal black holes. The limit in whichthe inner radius of the ring shrinks to zero is in this senseanalogous to the process in which the two centers of theextremal black holes are taken to coincide. If two RNextremal black holes, each of charge Q, are separate, the

total mass is 2Q and the total area is 2 4 Q2

. When theircenters coincide the mass is still 2Q but the entropy jumpsto 4 2Q 2. So the limit of zero separation is discontinu-ous, and indeed the topology of the solution changes. Thissame effect occurs for the black ring: the area of thesolution with R 0, i.e., the BMPV black hole, is largerthan the limit R ! 0 of the ring area, and the topology of the two solutions are different.

B. Innite-radius limit

In this limit we take R ! 1 , but keep the charges perunit length along the tube ( 2 R ) nite by de ning nite

linear densities (the factors n account for different di-mensionalities of spheres for integration)

Q iQ i

2 R2 3

2

Qi

2R: (6.5)

We keep qi xed and de ne nite coordinates r , , by 11

r ÿR=y; cos x; R : (6.6)

Then we get

H 1 ! 1Q1

rq2q3

4r 2 ; (6.7)

! d ! ÿ q1 q2 q3

2rq1Q1 q2Q2 q3Q3

4r 2

q1q2q3

8r3 d ; (6.8)

! ! 0 ; (6.9)

with H 2 and H 3 given by permutations of (123). With thiswe reproduce the metric for the ‘‘ at supertube ’’ in [16].

C. Simpler supertubes

1. Two charges and one dipole

The original supergravity supertube solutions in D 6[7,32,33] correspond to setting Q3 0 and therefore q1q2 0. In this case J 0 and J =4G5 R2q3. Thebound (3.25) is trivially saturated, but this inequality isonly suf cient to eliminate CCCs when qi > 0. In thepresent case, it is easy to use the results of Appendix C

to see that the necessary and suf cient condition for theabsence of CCCs is 12 q2

3 Q1Q2=R2, which agrees pre-cisely with the worldvolume analysis of these supertubes in[6,7].

2. Three charges and two dipoles

The solution with q3 0 and Q1 ;2 ;3 Þ 0, q1 ;2 Þ 0, ismore complicated than the previous one in that the BPSequations that it solves are nonlinear and therefore thefunctions H i are not harmonic. There are also two inde-pendent angular momenta.

The solution can be interpreted as a helical D1-D5 stringcarrying momentum in the direction of the axis of thehelix, along which it is smeared, but this time the KKmonopole tube is absent. It is different than the D1-D5-Pgyrating strings of [34], since it has two independentangular momenta and the area is always zero. In fact,now there is a naked curvature singularity at y ÿ1where R R diverges.

Absence of causal anomalies imposes again constraints

on the parameters. Equation (3.25) reduces to ÿ q1Q1 ÿq2Q22 0 so we require

q1Q1 q2Q2: (6.10)

In terms of quantized brane numbers (4.4) and (4.5), thisequation becomes

N D1

nD1

N D5

nD5 ; (6.11)

which means that the D1 and D5 helices have the samepitch and can therefore bind to form a D1-D5 helix.

From Appendix C, it is now easy to see that the neces-sary and suf cient condition for absence of CCCs is thatthe radius R be bounded above like 13

R2 Q3 ÿ q1q2 Q1Q2

q1q2 Q1 Q2: (6.12)

Equation (6.11) is the precise T-dual (in the z direction)to a constraint found for supertubes with D0-D4-F1charges and D2-D6 dipoles using the worldvolume theoryof D6-branes [15] 14 . Equation (6.12) is very similar to, butnot exactly the same as, another equation derived in [15]for the same system. We will return to this point inSec. VIII.

11 These r , are the same as introduced in the near-horizonstudy of Appendix D.

12 In this case, the rst two lines of Eq. (C3) vanish, as does thesecond term on the third line, so the leading order (cubic)behavior as y ! ÿ1 comes from Y . Demanding Y > 0 givesthis inequality.

13 The rst line of Eq. (C3) vanishes, as does the second term onthe second line, so the leading (quartic) behavior as y ! ÿ1comes from X . Demanding X > 0 gives this inequality. It is theneasy to see that Y > 0 so the remaining terms in (C3) arepositive.

14 Equation (6.10) was also recovered in the in nite-radius limitin [16].


024033-13



It is perhaps surprising that despite having the three D1-D5-P charges, the area of this solution always vanishes.The apparent reason is that near the core at y ! ÿ1 thesolution is mostly dominated by terms involving the dipolemoments q i . There is a curvature singularity at the core andthe moduli also blow up there. However, it will be shown in[35] that this solution admits thermal deformations, i.e.,there exists a family of nonextremal solutions with regular

horizons, which in the extremal limit reduce to the super-symmetric solution with three charges and two dipoles.

One can easily check that the solutions with threecharges and one dipole, as well as the ones with twocharges and two dipoles, always possess CTCs near y !ÿ1. This parallels the fact that these supertubes do nothave a sensible Born-Infeld description [15,16].

D. Decoupling limit

In the decoupling limit of the D1-D5-P solution we send0 ‘ 2

s ! 0 and keep the string coupling gs xed, in sucha way that the geometry near the core decouples from the

asymptotically

at region. Our solution contains severalmore parameters than previous D1-D5-P con gurations, sowe shall describe this limit in some detail.

We work with the coordinates r; de ned in (3.7).Since we want to keep xed the energies (in string units) of the excitations that live near the core, then r= 0and R= 0must remain nite. We are also interested in a regimewhere the size of the z direction, Rz, is large compared to‘ s , so that winding modes can be ignored and the momen-tum modes are the lowest excitations. So when we take

0! 0, Rz will be xed, and then also q3 as a result of (4.8). Further, we take the T 4 length scale ‘ ‘ s so that theenergy scale of both momentum and winding modes on the

T 4

is large. Finally, we keep the string coupling xed andalso keep the number of branes and units of momentum xed. Using (4.4) and (4.5) the limiting solution is obtainedtaking 0! 0 while

r= 0 ; Q1 ;2= 0 ; Q3= 02 ; R=0 ;q1 ;2= 0 ; q3

(6.13)

are held xed. Then the length scales in the supergravitysolution are arranged as

r R Q3p q1q2p Q1Q21=4 q3p q1q2

1=4

(6.14)

with Q1 Q2 and q1 q2. Recall from (4.1) that Q1, Q2,and Q3 label D5, D1, and momentum charge, respectively,and q1, q2, and q3 correspond to the dipole charges of,respectively, d1- and d5-branes and the Kaluza-Kleinmonopoles making up the supertube. Observe also thatQ i scales like Qi .

After rescaling the metric and the gauge elds A1 and A2

by an overall factor of 0, we obtain a new solution of IIBsupergravity. This decoupled solution has the same form as

(4.2), (4.3), and (2.3) with, in the coordinates of (3.8),

H 1 ;2Q1 ;2 ÿ

q2 ;1q3R2 cos22 ;

H 3 1Q3 ÿ

q1q2R2 cos22 ;

(6.15)

! ÿq3

R2sin2

ÿr 2 R2 sin2

2

4G5J ÿ q3 R2 cos2

:

while ! and Ai remain as in (3.11) and (3.12), and J is asin (3.13). We can gauge-transform A3 ! A3 ÿ dt , i.e., z !z ÿ t, so that A3

t H ÿ13 ÿ 1 vanishes at r ! 1 . I t i s

apparent that this decoupling limit amounts to the familiarprocedure of ‘‘ removing the 1 ’s’’ from the functions H 1 ;2associated to the D5 and D1 branes, with the rst term of ! modi ed so the result remains a solution of the eldequations.

The decoupling limit is not in general the same as thenear-horizon limit (4.12) analyzed earlier. In the near-horizon limit r is taken to be much smaller than R andindeed than any other scale in the system, so one coversonly a small region of the decoupled solution. Also, thenear-horizon limit in (4.12) exists for any black ring,whereas here we are restricting the parameters to theranges in (6.14). These differences between the two limitsare in fact also present for BMPV [36]. The two limitsnevertheless commute, so the new solution has a regularhorizon of nite-area. After an appropriate rescaling, L isthe same as (3.28), when expressed in terms of physicalquantities only. A slight difference is hidden, though, in the

fact that in the decoupling limit J ÿ J changes to

J ÿ J decoupled

4G5R2q3: (6.16)

This is simply a consequence of the restrictions (6.13) onthe relative values of the parameters, which imply, inparticular, that q3 q1 ;2. As a consequence, rings withq3 q1 ;2 are not expected to be fully captured by the dualCFT description of D1-D5 systems.

At asymptotic in nity, r ! 1 , the metric becomes(omitting the T 4 factor)

ds 2

!r 2

Q1Q2p ÿdt2 dz2 Q

1Q

2

p dr 2

r 2

Q1Q2p d 2 sin2 d 2 cos2 d 2 ; (6.17)

so we recover the asymptotic geometry of global (as z isperiodic) AdS3 times S3, both with equal radius

‘ 1 Q1Q21=4: (6.18)

For certain particular values of the parameters we getsolutions which are everywhere locally AdS3 S3. This


024033-14



happens when R 0, i.e., BMPV in the decoupling limit[36], and also for two-charge D1-D5 supertubes that satu-rate the CTC-bound q2

3R2 Q1Q2. The decoupling limitof the latter is global AdS3 times a rotating S3, with aconical singularity if nKK > 1 [32].

However, in general our solution is (locally) the productspace AdS3 S3 only at asymptotic in nity and near thehorizon. In the latter region this occurs in a rather unusual

manner. In order to nd the near-horizon limit of thedecoupling solution in r; coordinates, near r 0,

=2, we take the gauge in which A3t H ÿ1

3 so we are in aframe that corotates with the horizon, and de ne

r2 ~rL cos2~2

; R2cos2 ~rL sin2~2

; t ~t= :

(6.19)

Sending ! 0, the geometry that results is the same as(4.12), but now in coordinates that cover only the regionoutside the horizon,

ds 22Lq1q2p ~rd ~td

qL2

q1q2p d 2q3 q1q2p

4d~r2

~r 2

q1q2p q3

dz ÿq3

21 cos~ d

2

q3 q1q2p 4

d ~2 sin2 ~d 2 ; (6.20)

where we use ÿ and, for simplicity, we assume(4.11) is satis ed so that we can use a shift z ! z mRz to bring the geometry to a product of two factors. The rstfactor is locally isometric to AdS3 with radius

‘ nh q3p q1q21=4: (6.21)

This factor is globally the same as the near-horizon limit of an extremal BTZ black hole with mass and spin

M BTZ 2L2=q2 ; J BTZ M BTZ‘ nh ; (6.22)

and horizon at ~r 0. The second factor is the quotientspace S3=Z nKK , with the same radius ‘ nh.

The appearance of AdS3 S3 near the horizon is verydifferent from the factorization into AdS3 S3 in theasymptotic region (6.17). The AdS3 near the horizon spansthe coordinates ~t; ~r; whereas near the asymptoticboundary it spans t;r;z — the relation between t and ~t just amounts to the redshift near the horizon, but the othercoordinates are not simply related.

The direction in which the near-horizon geometry ro-tates is . In contrast, in the decoupling limit of BMPV, thenear-horizon geometry rotates in the z-direction and arisesfrom the linear momentum in this direction. Here therotation of the near-horizon geometry arises, in a sense,from the rotation of the ring, but there is no simple rela-tionship between J and J BTZ. Furthermore, the radii of thetwo AdS3’s are different

‘ 1 > ‘ nh: (6.23)

The curvature of the solution near the horizon is notcontrolled by the net D1 and D5 charges but instead bythe dipole charges q i . As a consequence, the simple argu-ment for the statistical calculation of the entropy of theBTZ black hole, from the charges under the Virasoroalgebra of diffeomorphisms at the boundary of AdS3

[37], does not seem to apply easily to the computation of the entropy of the ring.

So the full decoupling solution interpolates in a highlynontrivial way between two different factorizations of thesix-dimensional solution, both of which are locally of theform AdS3 S3. Between these two limiting regions, thesolution has nonvanishing Weyl curvature and is generi-cally quite complicated. Indeed, already when the rstsubleading terms near r ! 1 are considered, the geometrydoes not factorize.

This decoupled solution must admit a dual description interms of an ensemble of supersymmetric states of the dualCFT. It should be very interesting to identify and count thedegeneracy of these states to reproduce the entropy of theblack ring. Given the two limiting AdS geometries, itmight be useful to view the solution as dual to a renormal-ization group ow, with Eq. (6.23) implying that the centralcharge is greater in the UV than in the IR, as expected fromthe c-theorem.

VII. WORLDVOLUME SUPERTUBES ANDKA ¨ HLER CALIBRATIONS

Consider three M5-branes intersecting as in the array(2.1), with the -circle replaced by a curve C in the E 4

space transverse to the M2-branes. If C is a straight line,then the worldspace of the three M5-branes may be de-scribed as that of a single M5-brane with (in general)nonsingular worldspace S C, where S i s a K ahler-calibrated surface of degree-four (that is, of complex di-mension two) embedded in the C 3 E 6 space spanned bythe z1 ; . . . ; z6 coordinates. This con guration preserves 1=8of the 32 supersymmetries of the M-theory Minkowskivacuum. Here we will show that an M5-brane with world-space S C and appropriate uxes of the worldvolumethree-form H also preserves 1=8 of the supersymmetries(albeit a different set) for any arbitrary curve C in E4. If Cis closed then this con guration carries no net M5-branecharges, but only three M2-brane charges and three M5-brane dipoles, and thus provides the rst worldvolumedescription of a three-charge supertube in which the threedipoles are visible. We call this a calibrated supertube . Infact, we will show that any Kahler calibration (of appro-priate degree to be interpreted in terms of M5-branes) givesrise to a calibrated supertube in a similar manner. It wouldbe interesting to investigate the relationship between super-tubes and other types of calibrations (SLAG calibrations,exceptional calibrations, etc.), with calibrations in more


024033-15



general backgrounds, and with calibrations correspondingto nonstatic branes.

To avoid any confusion, it is worth mentioning from thestart a limitation of the worldvolume description we areabout to present. This is the fact that, although the con- gurations we will construct do carry multiple charges anddipoles globally, they may be locally regarded as standardtwo-charge/one-dipole supertubes. Globally, therefore,

they may be viewed as resolved junctions of standardsupertubes, in the same sense that certain calibrations canbe viewed as resolved junctions of M5-branes. One veryconcrete manifestation of this limitation is that the world-volume description does not capture the second angularmomentum visible in the supergravity description.

Despite this limitation, it is remarkable that such non-singular junctions of supertubes can preserve supersym-metry, 15 and we regard the construction below as a rststep towards a more sophisticated worldvolume descrip-tion of three-charge black supertubes.

A. Ka ¨hler calibrations and intersecting M5-branes

We begin by reviewing a few facts about Ka ¨ hler cali-brations. We follow closely the discussion in [39,40]. Letu j z2 jÿ1 iz2 j , with j 1 ; . . . ; n, be complex coordi-nates on C n E 2n, with metric

ds 2 Xn

j 1du j du j X

2n

j 1dz2

j ; (7.1)

and

H i2 X

n

j 1du j ^du j X

n

j 1dz2 jÿ1 ^dz2 j (7.2)

the associated Ka ¨ hler two-form. Then1p !

H p (7.3)

is a calibration of degree 2p in C n (p n) associated tothe group SU n [41]. This means that the volume of any(hyper)surface S E2n of dimension 2p is bounded frombelow as

Z Sd2p detgp Z S

; (7.4)

where are coordinates on S, g is the induced metric on S,and a pullback of onto S is understood. If this bound issaturated, the surface S is said to be calibrated by , orKahler-calibrated (since is constructed from the Ka ¨ hlerform). All complex surfaces in C n are Ka hler-calibrated[39]. Since S minimizes its volume within its homologyclass, a static M5-brane with worldspace S E 5ÿ2p is asolution of the M5-brane equations of motion. Moreover,

any two tangent (hyper)planes to S are related by anSU n SO 2n rotation, from which it follows that afraction 1=2n of supersymmetry is preserved.

The Ka hler calibrations of interest here are those thathave an interpretation in terms of intersecting M5-branes,namely, those with n 2 ;3 ;4. For n 2 there is only anSU 2 calibration of degree-two, corresponding to aRiemann surface S embedded in C 2. An M5-brane with

worldspace S E3

can be interpreted as the intersection of two M5-branes

M5: 1 2 – – 5 6 7

M5: – – 3 4 5 6 7 ;(7.5)

where the C 2 space corresponds to the 1234-directions.For n 3 there are two relevant calibrations, of degrees

two and four. The rst one corresponds to a Riemannsurface S in C 3. An M5-brane with worldspace S E 3

preserves 1=8-supersymmetry and can be interpreted asdescribing the triple intersection

M5: 1 2 – – – – 7 8 9

M5: – – 3 4 – – 7 8 9

M5: – – – – 5 6 7 8 9:(7.6)

The SU 3 calibration of degree-four corresponds to asurface S of complex dimension two embedded in C 3.

An M5-brane with worldspace S E can be interpretedas the triple intersection

M5: – – 3 4 5 6 7

M5: 1 2 – – 5 6 7

M5: 1 2 3 4 – – 7:(7.7)

In both SU 3 cases, the C 3 space corresponds to the123456-directions. As mentioned above, the second casecorresponds to the M5 intersection in (2.1) with the circlereplaced by a line.

Finally, the only SU 4 calibration that has an interpre-tation in terms of M5-branes is that of degree-four, whichcorresponds to a surface S of complex dimension twoembedded in C 4.16 An M5-brane with worldspace S E

15 Non-supersymmetric supertube junctions have been previ-ously studied in [38].

16 An SU 4 calibration of degree-two has an interpretation asan intersection of four M2-branes, and one of degree six as anintersection of a number of D6-branes.


024033-16



can be interpreted as the sixtuple intersection

M5: 1 2 3 4 – – – – 9

M5: 1 2 – – 5 6 – – 9

M5: 1 2 – – – – 7 8 9

M5: – – 3 4 5 6 – – 9

M5: – – 3 4 – – 7 8 9

M5: – – – – 5 6 7 8 9:

(7.8)

Although in all arrays above we have displayed the M5-branes as being orthogonal, this need not be the case; theycan intersect at arbitrary SU n angles, which are encodedin S. Let us illustrate this for the SU 3 calibration of degree-four, represented by (7.7). The surface S can gen-erally be speci ed as the locus F u1 ; u2 ; u3 0, where F is a holomorphic function. The choice of F determines theSU 3 angles between the three M5-branes, which arise atasymptotic regions of S. As an example, consider F f 1 f 2 f 3

ÿc, where f i

P3 j 1 a j

i u j are linear functions

and a ji and c are constants. The induced metric on S hasthree asymptotically at regions that can be identi ed withthe three M5-branes. A simple way to determine these is toset c 0. In this case the locus F 0 consists of threecomplex planes, f 1 0, f 2 0 and f 3 0, the anglesbetween them being determined by the constants a j

i . Thiscorresponds to a singular intersection of three M5-branesthat extend along these three planes. Setting now c Þ 0smooths out the intersection and hence allows the entirecomplex two-surface to be interpreted as a single M5-brane, but does not alter the orientations of the asymptoticregions, which therefore can still be identi ed with threedistinct M5-branes.

Despite the fact that the arrays above do not necessarilyspecify the SU n angles between the intersecting branes,each array is useful in summarizing the number of partic-ipating branes and the set of supersymmetries preserved byeach intersection. For example, the con guration repre-sented by the array (7.5) preserves 1=4-supersymmetry,corresponding to the Killing spinors subject to the con-straints

ÿ 012567 ; ÿ 034567 ; (7.9)

each of them being associated to one of the M5-branes.Similarly, the con guration represented by (7.8) preserves

1=16-supersymmetry, corresponding to any four of the sixM5-branes (the two projectors associated to any two of theM5-branes are implied by the those of the other four).

B. Calibrated supertubes

We are now in a position to show that each of the Ka ¨ hlercalibrations above gives rise, through turning on appropri-ate worldvolume uxes, to a calibrated supertube.Choosing S E2 S0, with S0a Riemann surface, in the

SU 3 and SU 4 calibrations of degree-four, representedby the arrays (7.8) and (7.7), we recover the SU 2 andSU 3 calibrations of degree-two, represented by the arrays(7.5) and (7.6), respectively. The two degree-two calibra-tions can therefore be regarded as ‘‘ degenerate ’’ cases of the 2 degree-four calibrations, and for this reason we willonly discuss the latter two. These give rise to the three-charge/three-dipole SU 3 calibrated supertube

Q1 M2: 1 2 – – – – –

Q2 M2: – – 3 4 – – –

Q3 M2: – – – – 5 6 –

q1 m5: – – 3 4 5 6

q2 m5: 1 2 – – 5 6

q3 m5: 1 2 3 4 – –

(7.10)

and to the four-charge/six-dipole SU 4 calibrated super-tube

Q1 M2: 1 2– – – – – – –

Q2 M2: – – 3 4 – – – – –

Q3 M2: – – – – 5 6 – – –

Q4 M2: – – – – – – 7 8 –

q1 m5: 1 2 3 4 – – – –

q2 m5: 1 2 – – 5 6 – –

q3 m5: 1 2 – – – – 7 8

q4 m5: – – 3 4 5 6 – –

q5 m5: – – 3 4 – – 7 8

q6 m5: – – – – 5 6 7 8 :

(7.11)

In the two arrays above, we have denoted by the coor-dinate along the curve C. Note that each M5-brane can bethought of as originating from the expansion of a pair of M2-branes, as in a ‘‘ standard ’’ two-charge supertube.

Consider therefore an M5-brane with worldspace S C,where S is a complex two-surface in C 3 or C 4 and C is anarbitrary curve in E 4 or E 2, with a worldvolume three-form ux H dB2 given by

H d ^H dx0 ^H 0 ; (7.12)

where again H is understood to be pulled-back onto theM5-brane worldvolume. H 0is a two-form determined interms of H by the (generalized) self-duality conditionsatis ed by H , but whose explicit expression will not beneeded. Closure of H follows from that of H . As we willsee below, this H -flux induces M2-brane charges on theM5-brane as in the arrays (7.10) or (7.11). We claim thatthis con guration preserves 1=8 or 1=16 of the supersym-metries of the M-theory Minkowski vacuum, generated byKilling spinors subject to the constraints associated to


024033-17



the M2-branes, that is,

ÿ 012 ; ÿ 034 ; ÿ 056 (7.13)

for the three-charge supertube, and

ÿ 012 ; ÿ 034 ;

ÿ 056 ; ÿ 078

(7.14)

for the four-charge supertube. Note that there is no trace of a condition associated to the M5-branes, as expected fromthe absence of M5-brane net charges. 17

To prove this, it is convenient to adopt a (static) gauge inwhich x0, xa x1 ; . . . ; x4 , and are worldvolume coor-dinates on the M5-brane, where parametrizes the crosssection C, speci ed as x x . For the three-chargesupertube x x7 ; . . . ; x10 , whereas for the four-chargesupertube x x9 ; x10 . Without loss of generality wechoose to be the af ne parameter along C, that is,

@ x @ x 1. The complex surface S is speci edas xm xm xa , where xm x5 ; x6 for the three-chargesupertube and xm x5 ; . . . ; x8 for the four-charge super-tube. In both cases, xm satisfy the appropriate Cauchy-Riemann equations,

@1 x5 @2 x6 ; @2 x5 ÿ@1 x6 ; . . . (7.15)

where the dots stand for the same expression with f1 ;2gand/or f5 ;6greplaced by f3 ;4gand/or f7 ;8g. Note that inthese coordinates the only nonzero components of H are

H ab 1 a b 2 3 a b 4 @a x5@b x6 (7.16)


H ab 1 a b 2 3 a b 4 @a x5@b x6 @a x7@b x8

(7.17)for the four-charge supertube. In both cases, the inducedmetric on the M5-brane worldvolume takes the form

ds 2 ÿdx20 d 2 gab dxa dxb ; (7.18)

where

gab ab @a xm@b xnmn : (7.19)

The number of supersymmetries of the Minkowski vac-uum preserved by the M5-brane is the number of Killingspinors that satisfy the condition ÿ M5 [42], whereÿ M5 is the matrix appearing in the kappa-symmetry trans-

formations of the M5-brane worldvolume fermions. Wework with a unit-tension M5-brane and the covariant for-mulation of [43], which contains an auxiliary scalar eld athat we eliminate by the gauge choice a x0. Under these

circumstances the kappa-symmetry matrix takes the sameform for the three- and four-charge supertubes, namely,

ÿ M5ÿ 0

det gab habp 14

abcd H ab cd ÿ P 1234 ; (7.20)

where 1234 1,

a ÿ a @a xmÿ m ; @ x ÿ (7.21)

are the worldvolume Dirac matrices induced by the space-time, constant Dirac ÿ -matrices, i1...in i1 i2 in ,

P 18

abcd H ab H cd (7.22)

is the momentum density along @ , and

hab12

gac gbd

cdef

detgp H ef : (7.23)

Making use of (7.13), (7.14), (7.15), (7.16), and (7.17), atedious but straightforward calculation reveals that

1234 ÿP ; (7.24)

where

P 1 ab @a x6@b x6 (7.25)


P 1 ab @a x6@b x6 ab @a x8@b x8

abcd @a x5@b x6@c x7@d x8 (7.26)

for the four-charge supertube. We note that the Cauchy-Riemann equations imply ab @a x5@b x5 ab @a x6@b x6,and analogously with f5 ;6greplaced by f7 ;8g.It follows that the two terms inside the round brackets in(7.20) cancel each other, and hence all information aboutthe cross section C, which is entirely encoded in , dropsfrom the supersymmetry equation. This now reduces to

14

abcd H ab ÿ 0 cd det gab habq : (7.27)

Using again (7.13), (7.14), (7.15), (7.16), and (7.17), onecan verify that this equation is identically satis ed, with thedeterminant given by

det gab habq 2 detgabp 2P : (7.28)

Essentially the same arguments given above can be usedto show that these supertube con gurations saturate thebounds found in [44,45]. It is worth remarking, though,that the con gurations studied in detail in those referencesall have zero momentum density, i.e., P P a 0.

17 If the curve C is not closed but instead extends to in nity,then there are net M5-brane charges, and there is also a net linearmomentum. The preserved supersymmetries are still the onesabove because the linear momentum cancels exactly the M5-brane charges in the supersymmetry algebra [8].


024033-18



C. Physical properties

In this section we will show that a calibrated supertubecan be regarded, at a given point, as a standard supertubewith one M5 dipole and two M2 charges. For the standardsupertube, supersymmetry xes both these densities andthe shape of the tube in the directions transverse to thecross section (i.e., it xes S) [6]. 18 For a calibrated super-tube, stability is instead achieved locally in such a way thatthe only restriction on this shape is that S be a calibratedsurface; the charge densities may then also vary, as they aregiven by the pullback onto S of the Ka hler form. Thisallows the calibrated supertube to carry, globally, morethan two net charges and one dipole, as we have seenabove. In this sense, the worldvolume three-charge super-tube constructed in this paper may be regarded as a smooth junction of three two-charge supertubes associated to thethree asymptotic regions of S.

The cross section of the calibrated supertube, like that of the standard one, is supported against collapse by the‘‘ centrifugal force ’’ associated to the Poynting momentumdensity generated by the product of the worldvolumecharge densities. To see this, we note that the momentumdensity P is, at each point, the product of the two M2-brane charge densities carried by the M5-brane at thatpoint. Recall that the M2-brane charge density in theab -directions tangent to a given point on the M5-brane isgiven by the momentum density ab conjugate to theworldvolume two-form potential Bab . This is because theM5-brane action depends only on the background-covariant combination H A , where H dB2 and apullback onto the M5 worldvolume of the supergravitythree-form potential A is understood. It follows from thisthat the M2-brane charge density carried by the M5 is

@L M5

@A 0ab A 0

@L M5

@_Bab A 0

ab : (7.29)

This momentum is determined in terms of the worldspacecomponents of H by the constraint associated to the self-duality condition of H [46]. In the present case it takes theform 19

ab 12

abcd H cd ; (7.30)

so P may be rewritten as

P 18 abcd

ab cd : (7.31)

Now, at each point on S an orthonormal basis of its tangentspace may be chosen such that the antisymmetric tensorab is skew-diagonal, with skew-eigenvalues and 0.

These measure the magnitude of the two independent M2-brane charge densities at the given point, whereas the

orthonormal basis determines their orientations. In termsof these densities we have

P 0 ; (7.32)

as anticipated.Equation (7.32) is completely analogous to that for a

two-charge standard supertube [6,7]. We now show that therest of the relations between the charge densities, theangular momentum and the size of the cross section alsoare, at a given point, as those of the two-charge supertube.For simplicity, we assume that C is a circle of radius R insome plane, so we set R . It is important to rememberthat the densities that enter these relations are densities perunit area of S, obtained by normalizing by detgabp andintegrating over C. The normalization is most easily ac-counted for by working in the orthonormal basis used tode ne and 0, so that detgab 1. By virtue of thesecond equality in (7.28), this implies

P 0 1 ; (7.33)

as for the for the standard supertube [8]. The C-integratedM2-brane densities are

%1

2 Z Cd R ;

%0 12 Z C

d 0 R 0 ;(7.34)

where we have used the fact that is the af ne parameteralong C and that ; 0are -independent. It then followsfrom (7.33) that

R %%0

q : (7.35)

Similarly, the angular momentum is

J R

2 Z Cd P R2 ; (7.36)

and hence

J %%0: (7.37)

We thus see that R, J , %, and %0obey the same relations asfor a standard supertube with unit dipole, i.e., constructedfrom a single M5-brane. If instead n M5-branes are super-posed, then these relations become

R %%0q =n; J %%

0=n: (7.38)

We conclude by showing that the energy of the cali-brated supertube may be written as the sum of the corre-sponding M2-brane charges, as expected fromsupersymmetry. The M5-brane energy density E can beextracted from [47]. In our case it takes the form

E 2 2P 2 det g h 6P 2 : (7.39)

Integrating over the M5-brane worldspace and using the

18 In the simplest case of a D2-supertube, it forces the chargedensities and shape to be either constant or those of a D2-BIon.

19 For ease of notation we are absorbing a factor of 4 in ab

with respect to the de nition of [46].


024033-19



de nition (7.22) of P we obtain the total energy

E Z C Sd d 4 xE 6p Z C S

d d 4 xP

6p 2 Z C S

d ^H ^H : (7.40)

Employing now the de nition (7.2) of H , this becomes

E 6p 2 Xn

j 1Z C Sd ^dx2 jÿ1 ^dx2 j ^H 6p 2 X

n

j 1Q j ;

(7.41)

as we wanted to see. In the above expression appropriatepull-backs onto the M5-brane are understood, as always,and we have used the fact that the total charge Q1 asso-ciated to an M2-brane in the 12-directions is

Q1 Z C Sd d 4 x 12 Z C S

d d 4 xH 34

Z C S

d ^dx1 ^dx2 ^H ; (7.42)

and similarly for the rest of the Q j . In this last equationH 34 stands for the 34-component of the pullback of H , asopposed to the 34-component of H itself.

VIII. SUPERGRAVITY VS WORLDVOLUMEDESCRIPTION OF THREE-CHARGE

SUPERTUBES

In the previous sections we have provided a detaileddescription of three-charge supertubes within supergravity.We have also developed a worldvolume construction of supertubes with three charges and three dipoles in Sec. VII.A third framework for describing these systems, in terms of the microscopic CFT of D1-D5 systems, is currently underinvestigation. For two-charge supertubes, the agreementbetween the supergravity and the worldvolume descrip-tions is perfect [6 – 8]. Here we offer some preliminaryobservations aimed at exploring whether a similar connec-tion for three-charge supertubes may exist.

In order to investigate this, let us take the two-chargesupertube as the basic ‘‘ building block ’’ . For the sake of generality and simplicity, we phrase the discussion in termsof supergravity charges Qi , q i instead of quantized branenumbers which would require singling out a speci c U-duality frame.

For the two-charge supertube, both the worldvolume andsupergravity descriptions (and the CFT too) yield the samerelations between the parameters,

4G5

J R2q3

Q1Q2

q3: (8.1)

It is convenient to assume that the supergravity no-CCCbound is saturated, since then the correspondence withworldvolume supertubes is particularly simple [7,8]. All

the Q1 and Q2 branes are ‘‘ dissolved ’’ in the supertube,thus contributing to the angular momentum, and the pro leof the supertube is uniquely xed to be circular.

Consider now three-charge/two-dipole supertubes.Using either the worldvolume analysis of [15] or ourresults from supergravity one obtains (6.10) [also easilyinterpreted within the microscopic D1-D5 view, see(6.11)].

Let us now regard this supertube, within the worldvo-lume view, as the superposition of two two-charge super-tubes of equal radius R, one with parameters Q1 ; Q03 ; q2 ,the other with Q2 ; Q003 ; q1 , and total charge Q3 Q03Q003 . In terms of the worldvolume construction of Sec. VII,this simply means that the intersection between the tubes isnot resolved but remains singular. Assume that each super-tube separately satis es corresponding relations (8.1).Then it is easy to derive [15]

Q3 R2q1q2Q1 Q2

Q1Q2: (8.2)

The supergravity analysis, which does not allow for simplesuperpositions of the two supertubes but includes insteadnonlinear interactions, yields, in contrast, the CTC-bound(6.12) which, when saturated, can be written as

Q 3 R2q1q2Q 1 Q 2

Q 1Q 2: (8.3)

(In this case Q 1 Q1, Q 2 Q2 but Q 3 < Q 3.) Thissuggests that in order to recover the supergravity expres-sions from the worldvolume we must replace

Qi ! Q i: (8.4)

Q i is then seen to play the role of an ‘‘ effective charge. ’’

The origin of this ‘‘ replacement rule ’’ is unclear. Q3 andQ 3 coincide when Q3 q1q2. This is the case if we takethe limit of very large radius while keeping nite the lineardensity of Q3, so Q3 R, while q i R0 (see Sec. VI B).However, at nite R the worldvolume and supergravityresults differ: the supergravity radius is smaller, for givencharges. This suggests that the discrepancy might be due toclosed-string self-attraction of the ring, which would causethe tube radius to shrink, and which would not be at work in the worldvolume description. However, if this were thecase then one might expect that, when expressed in termsof quantized brane numbers, the string coupling should beinvolved in the differences between (8.2) and (8.3), but it isnot. Another possibility is that the replacement (8.4) ariseswhen the system actually forms a single supertube (orblowing up the intersection), instead of a simple superpo-sition. However, the analysis of such single-brane super-tubes in Sec. VII shows no evidence for this effect.Ref. [15] did also consider proper supertubes with threecharges and two dipoles, and obtained essentially (8.2)instead of (8.3).


024033-20



The spin is the sum of the spins of each supertube, so

4G5

J R2 q1 q2 (8.5)

[independently of whether (8.4) is applied or not]. This isprecisely the leading value of the angular momentum atlarge R. In fact, it seems more appropriate to consider thatthe spin (8.5) in this worldvolume approach accounts ex-

actly for the value of J ÿ J , but not for the self-dualcontribution to the angular momentum that includes J . Inthe limit R ! 1 the latter vanishes, so the discrepancydisappears.

One might remark that in the supergravity solutions R isnot the proper radius of the ring (which, for q3 0,actually diverges at y ! ÿ1 ). However, R, as the radiusin the base space, does play the role of the supertube radiusin the two-charge supergravity supertubes (8.1). In thesupergravity expressions for physical quantities R can al-ways be eliminated in favor of the physical charges, di-poles and angular momenta (in particular, usingJ

ÿJ

/R2). So we can employ it as, at least, a useful

auxiliary quantity that can be related to the worldvolumesupertube radius.

Now regard the three-charge/three-dipole supertube asthe superposition of three two-charge supertubes (see also[15]). Again, this corresponds to considering that the in-tersection of the three M5-branes in Sec. VII is not re-solved. The supertubes have parameters Q01 ; Q02 ; q3 ,Q001 ; Q03 ; q2 , Q002 ; Q003 ; q1 . The total charge of the i-th

constituent is Qi Q0i Q00i . The radii of the three super-tubes must be the same, so

R2 Q01Q02q2

3

Q001Q03q2

2

Q002Q003

q2

1

: (8.6)

Furthermore, applying (6.10) to pairwise combinations of the tubes we obtain the constraints

q1Q01 q3Q003 ; q2Q002 q1Q001 ; q2Q02 q3Q03:(8.7)

After some algebra one can write the equation for theradius in the form

2Xi<jQ iqiQ j q j ÿXi

Q2i q2

i 4R2q3Xiq i: (8.8)

If we now perform the substitution (8.4) we recover exactly

the condition for saturation of the no-CCC bound fromsupergravity, Eq. (3.25).

The angular momentum is the sum

4G5

J

Q01Q02q3

Q001 Q03q2

Q002Q003q1

R2 q1 q2 q3 ;

(8.9)

and the same comments apply as in (8.5). If we take thislast equation as giving J ÿ J instead of just J , then it

can be combined with (8.8) and the substitution (8.4) toreproduce the condition for saturation of (3.26).

We see that one key feature of the supergravity descrip-tion that the worldvolume construction does not seem tocapture is the second angular momentum J . We mustremember that the calibrated supertube presented in theprevious section is a solution of the Abelian theory on asingle M5-brane. One may therefore speculate that incor-

porating non-Abelian effects, namely, working with thetheory on more than one M5-brane, is necessary to repro-duce J . Although this possibility cannot be discardedwithout further investigation, it is hard to see how thiswould explain that the solutions with a linear cross section(i.e., those obtained as the in nite-radius limit of the ring)carry zero J . Here we would like to speculate that theexplanation may instead be that J is not carried by theworldvolume supertube source itself, but that it is insteadgenerated as a Poynting momentum by crossed electric andmagnetic supergravity gauge elds. Although the argu-ment we present is somewhat heuristic, it is based ongeneral grounds and may describe the correct physicalorigin of J . In particular, it explains why J 0 for alinear cross section.

The idea is that J is given by the integral over aspacelike hyper-surface of the T 0 component of theenergy-momentum tensor that appears on the right-hadside of Einstein equations. Since the only bosonic eld of D 11 supergravity other than the metric is the three-form potential A , this has a unique contribution,

T 0 F 0mnp F mnp ; (8.10)

which for our solution takes the form

T 0 F 012mF 12m F 034mF 34m F 056mF 56m: (8.11)

This is indeed a product of electric and magnetic compo-nents of F . We wish to argue on general grounds that thismust be nonzero for a brane array as (2.1) except if the

-direction is a straight line.Indeed, we know the rst M2-brane must generate a

nonzero component A 012, whose magnitude must be pro-portional to Q1. Similarly, the rst M5-brane must source anonzero component ~A 03456 of the six-form potential dualto A , whose magnitude must be proportional to q1.Analogous statements apply to the other two M2/M5 pairs,so let us concentrate on the rst pair.

The key difference between a linear and circular (or,more generally, any nonlinear) cross section is that, in thelinear case, all elds may depend on a single radial coor-dinate in E 4 [see (3.3)], because of SO 3 rotationalsymmetry around the string. This means that the nonzerocomponents of the gauge potentials above lead to thenonzero components F 012 , ~F 03456 of the corresponding eld strengths. Hodge-dualizing ~F we see that the onlymagnetic component of F is F 12 . It follows that the


024033-21



contractions in (8.11) vanish and hence J 0. This is infact the only result compatible with SO 3 symmetry, sinceany nonzero angular momentum in the E3 space transverseto the string would break this symmetry down to U 1 . Inthe case of a circular cross section this breaking is alreadypresent from the beginning by the choice of plane in whichthe ring lies.

Indeed, for a circular cross section (in fact, for any

nonlinear cross section) the contractions (8.11) do notvanish. This is because now the elds depend on twocoordinates

1 cos ; 2 sin ; (8.12)

so ~F has nonzero components ~F 03456 1 and ~F 03456 2 .Dualizing we nd that F has nonzero magnetic compo-nents F 12 1 and F 12 2 and therefore that the contrac-tions in (8.11) do not vanish in general. Moreover, theelectric components are proportional to Q i , whereas themagnetic ones are proportional to qi , so naively T 0 is

proportional to Q1q1 Q2q2 Q3q3. This gives the rstterm in J , and therefore it satis es the requirement that itbe zero for a standard two-charge/one-dipole supertube.However, it misses the second term in J , proportional toq1q2q3. This is because our argument ignored the fact thatthe electric component A 012 is not just proportional to Q1but actually contains a term proportional to q2q3 (see theexpression (2.5) for H 1). Analogously, the electric compo-nents A 034 and A 056 contain terms proportional to q1q3and q1q2, respectively. Each of these electric components,when multiplied by the corresponding magnetic compo-nent, gives a term proportional to q1q2q3.

The argument above suggests that, although the precisevalue of J depends on some details of the solution, thefact that it is nonvanishing follows on general groundsfrom the presence of brane sources oriented as in the(2.1). From a mechanical viewpoint, one may say that inorder to bend the rst M5-brane to close one of its direc-tions into a circle, in the presence of the rst M2-brane, anangular momentum must be generated by the crossedelectric and magnetic elds they source.

Although tentative, the observations in this section pointto nontrivial connections between the worldvolume andsupergravity descriptions of supertubes with three charges.The justi cation of (8.4) remains an important open issue.It is presumably signi cant that it makes appearance onlywhen the BPS equations solved by the supergravity solu-tion are nonlinear. The perfect agreement observed be-tween worldvolume and supergravity for two-chargesupertubes would then seem to be a chance effect of thelinearity of the system. In fact there does not seem to beany a priori reason to expect perfect agreement.Supersymmetry, in particular, does not provide any clearreason for this. The physical origin of J needs furtherinvestigation too.

Finally, one might note that the supergravity constraints,derived by requiring absence of causal anomalies, do notactually x the angular momentum, for given charges, butinstead impose an upper bound on it. The cases where thebound is not saturated include the black supertubes withnonzero area. From the worldvolume perspective, it hasbeen argued that two-charge supertubes with pro les otherthan circular, which do not saturate the bound on the

angular momentum [8], are degenerate. Upon quantization,their degeneracy is equal to the degeneracy of the Ramondground states of the supersymmetric D1-D5 string [9,17].Thus it would seem natural to conjecture that the degen-eracy of three-charge worldvolume supertubes obtained byquantizing their moduli space of arbitrary pro les, cansimilarly reproduce the entropy of three-charge black supertubes, possibly after making the substitution (8.4). If we consider such a supertube as made of three superposedtwo-charge supertubes as in the construction above, it iseasy to see that the no-CCC bound comes out correctly —

after making the replacement (8.4) — but the entropy is toosmall. Hence supertubes with L > 0 are, not surprisingly,quite more complicated than these simple composites.Even if one considered the worldvolume three-charge re-solved supertubes constructed in the previous section, itmight still be that a calculation of their entropy fails toreproduce exactly the Bekenstein-Hawking entropy of three-charge supertubes. Instead, our analysis suggeststhat the worldvolume description might only reproducethe supergravity results up to the replacement (8.4), andthen accounting only for the non-self-dual part of theangular momentum, J ÿ J .

IX. CONCLUDING REMARKS

Supersymmetric ve-dimensional black holes are of considerable interest in string theory because they admita simple microscopic description in terms of D-branes[20]. Until now, the largest known family of such black holes was the four-parameter BMPV family [19]. Our work has shown that this is a limiting case of a larger seven-parameter family of supersymmetric black rings. Clearlythe challenge now is to obtain a microscopic description of these solutions that correctly accounts for their entropy.

It is natural to ask whether we have now exhausted thecatalogue of supersymmetric D 5 black holes. Onemight be tempted to speculate that there are many furthersurprises to be discovered. These are constrained, however,by the analysis of possible near-horizon geometries of supersymmetric black holes in Refs. [4,48], which allowsfor only three possibilities (at least for the class of D 5supergravity theories considered in Appendix B): (i) atspace, (ii) AdS3 S2, and (iii) near-horizon BMPV, withcorresponding horizon geometry (i) T 3, (ii) S1 S2 or(iii) (possibly a quotient of) a homogeneously squashedS3. It was also shown that the only asymptotically atsolution of type (iii) is the BMPV black hole. Hence if


024033-22



there exist any further supersymmetric black hole solutionsthen they must either have a at near-horizon geometry orthey must be black rings distinct from the ones presentedhere.

Could there exist supersymmetric black rings distinctfrom the ones presented here? Our worldvolume analysisof three-charge supertubes suggests that there might existcorresponding supergravity solutions with pro les other

than circular. If such solutions had horizons then the resultsof [4,48] prove that the near-horizon geometry must be thesame as for the circular black rings presented here, sodeviations from circularity would only be apparent awayfrom the horizon. On the other hand, Refs. [12 – 14] haveconstructed regular horizon-free solutions with threecharges and suggest the existence of a larger class of them. So maybe solutions with noncircular cross sectionswould belong to this class instead. Clearly, the space of physically relevant D1-D5-P solutions is far from beingcompletely mapped out.

The fact that three-charge supersymmetric black ringsexhibit nonuniqueness might seem at rst to be a dif culty

for a microscopic description. However, what appears to bean obstacle may actually be a very useful ingredient to-wards a more complete understanding of the D1-D5-Psystem. It has been argued in [3,5], in the context of near-extremal solutions, that string theory could containthe necessary states to account for black rings. One needsto appropriately identify the dipole constituents, or thephase in which the strings are. From a thermodynamicalviewpoint, solutions that are characterized by the sameasymptotic charges should be regarded as being only lo-cally stable in general. If we maximize the entropy byvarying the two independent dipoles we nd a uniquesolution, which should be the only globally thermodynami-cally stable con guration. Still, it would seem that all localequilibrium states, and not only the global maxima, shouldadmit a microscopic description. It would indeed be verysurprising if string theory could not account for solutionsof its low energy supergravity limit that seem completelypathology-free.

A remarkable aspect of supergravity solutions is the wayin which they capture highly nontrivial constraints betweenparameters that arise in a microscopic description. It wasalready known for the BMPV black hole and for the two-charge supertube that the condition that CCCs be absentyields the correct upper bounds on angular momentumrequired by the microscopic CFT or worldvolume theory. 20

For supertubes with three charges we have observed simi-lar nontrivial results in Sec. VIII. However, the worldvo-lume description falls just short of perfect agreement withsupergravity: a basis has to be found for the simple (partial) x of (8.4). The origin of J also needs to be better

understood. Although we have presented a speculativeexplanation for this origin, further investigation is clearlyneeded to establish a connection, at the same level as thatfor two-charge supertubes, between the worldvolume andsupergravity constructions of three-charge supertubesstudied in this paper.

ACKNOWLEDGMENTS

We thank J. Gauntlett, G. Horowitz, D. Marolf, and R.Myers for useful discussions. This work was presented byH. E. and H.S. R. at the GR-17 Conference in Dublin, July18 – 23, 2004. We would like to thank the audience, inparticular, V. Hubeny, M. Rangamani, and S. Ross, forpositive feedback. H. E. was supported by the DanishResearch Agency and NSF Grant No. PHY-0070895. REwas supported in part by UPV00172.310-14497, FPA2001-3598, DURSI 2001-SGR-00188, HPRN-CT-2000-00131.D.M. is supported in part by funds from NSERC of Canada. H. S. R. was supported in part by the NationalScience Foundation under Grant No. PHY99-07949.

Note added. —Following our publication of the minimalsupersymmetric black ring in [18], the solutions describingthree charge supersymmetric black rings have been foundindependently by two other groups [51,52]. Solutions de-scribing concentric black rings have also been constructed[50,52].

APPENDIX A: DERIVATION OF THE RING INMINIMAL 5D SUPERGRAVITY

1. Supersymmetric solutions of minimal supergravity

Minimal D 5 supergravity is a theory with eightsupercharges with bosonic action

I 116 G 5 Z R ? 5 1 ÿ 2F ^? 5F ÿ 8

3 3p F ^F A ;

(A1)

where F dA. Any supersymmetric solution of this the-ory admits a globally de ned nonspacelike Killing vector eld V [27] that cannot vanish [4]. In a region where V istimelike, coordinates t; xm can be introduced so that V @=@tand the line element can be written

ds 2 ÿ f 2 dt ! 2 f ÿ1hmndxmdxn ; (A2)

where hmn is a Riemannian metric on a four-dimensionalspace referred to as the ‘‘ base space ’’ B . The metric hmn ,scalar f and 1-form ! ! mdxm are all independent of t.Supersymmetry implies that hmn is a hyper-Ka¨ hler metricon B and that the Maxwell eld strength is given by [24]

F 3p

2d f dt ! ÿ

13p G ; (A3)

with

G 12 f d! ? 4d! ; (A4)

20 Such constraints from supergravity do also arise, alterna-tively, from the requirement that the solution admits a thermaldeformation.


024033-23



where ? 4 denotes the Hodge dual on B with respect to themetric hmn with orientation de ned so that the complexstructures are anti-self dual. These conditions are neces-sary for supersymmetry; it turns out that they are alsosuf cient. In the orthonormal basis e0 f dt ! , ei

f ÿ1=2ei with ei an orthonormal basis for hmn , the Killingspinor equation is solved by [24]

t; x f 1=2

x ; (A5)where is any chiral spinor on B that is covariantlyconstant with respect to the Levi-Civita connection of hmn . This implies that any supersymmetric backgroundpreserves at least 1=2 supersymmetry. In fact the onlyallowed fractions of supersymmetry are 0 ;1=2 and 1.This is easy to understand by noting the isomorphismSpin 1 ;4 Sp 1 ;1 under which the irreducible spinorrepresentation becomes a quaternion doublet [49]. TheKilling spinor equation is linear so any solution can bemultiplied by a constant quaternion hence the generalsolution must have a multiple of four real degrees of

freedom.We are interested in supersymmetric solutions so wemust also impose the equations of motion for this theory.The Bianchi identity for F gives

dG 0 ; (A6)

and the Maxwell equation reduces to [24]

r 2 f ÿ1 49 G 2 2

9GmnG mn ; (A7)

where r 2 is the Laplacian on B with respect to h. TheEinstein equation is automatically satis ed as a conse-quence of the above equations [24].

2. Minimal supersymmetric black rings

We start by choosing the base space to be at spacewritten in the form of Eq. (2.4) with orientation y x

1 (note that at space admits anti-self-dual hyper-Ka ¨ hlerstructures of either orientation). We make the Ansatz

! ! x;y d ! x;y d : (A8)

Equation (A6) gives

@ x f ! ;y ! ;x @y f ! ;x ÿy2 ÿ 11 ÿ x2 ! ;y ;

@y f ! ;y ! ;x @ x f ! ;y ÿ 1 ÿ x2

y2 ÿ 1! ;x :

(A9)

We assume

! ;xy2 ÿ 11 ÿ x2 ! ;y ; (A10)

so (A9) reduces to

f ! ;y ! ;x32q (A11)

where q is a constant. This determines

G 34q dx ^d dy ^d (A12)

and

G 2 9q2 x ÿ y 4

8R4 : (A13)

Now we seek a solution to Eq. (A7). We obtain it as aharmonic piece from solutions to the homogeneousLaplace equation r 2 f ÿ1 0, plus a solution to the inho-mogeneous Poisson equation sourced by (A13). It seemsreasonable to take the harmonic piece to contain a term / xÿ y, since this is the solution to the Laplace equation inR 4 with delta-function sources on a circle of radius R,which appears in two-charge supertube solutions [2,3]. Inorder to solve the Poisson equation, one looks for a simpleenough function of x and y, which is antisymmetric under x $ y, vanishes at in nity ( x;y ! ÿ 1), and is singular onthe circle at y

! ÿ1. A quick survey leads to the solution

ÿq2 x2 ÿ y2 = 4R2 . Then we take

f ÿ1 1Q ÿ q2

2R2 xÿ y ÿq2

4R2 x2 ÿ y2 ; (A14)

where we have normalized so that f ! 1 as x; y ! ÿ 1.Higher harmonics in f ÿ1, such as xy xÿ y , would lead tomore singular behavior on the ring and so are discarded.

It remains to solve Eqs. (A10) and (A11) to determine ! .Equation (A10) is equivalent to

! 1 ÿ x2 @ xW; ! y2 ÿ 1 @yW; (A15)

for some function W x;y . Substituting this into Eq. (A11)

gives@ x 1 ÿ x2 @ xW @y y2 ÿ 1 @yW 3

2qf ÿ1: (A16)

We demand that ! vanish at x 1 and ! vanish aty ÿ1, hence W must be nite at x 1 and at y ÿ1.Looking for a solution of the form W X 1 x Y 1 yleads to

X 01 x ÿq

8R2 3Q ÿ q2 3 x ;

Y 01 y ÿ3q

2 1 ÿ y ÿq

8R2 3Q ÿ q2 3 y :(A17)

We are free to add a solution of the homogeneous equation[i.e., Eq. (A16) with q 0]. If we look for solutions of theform X 2 x Y 2 y subject to the above regularity conditionsthen we are led to X 2 x Y 2 y /P l x P l y where P l areLegendre polynomials. In general, we can add an in nitesum of such terms to W . However, in order to avoid theorbits of @=@ and @=@ being closed timelike curves asy ! ÿ1 , ! and ! can diverge no faster than y2, whichrestricts us to l 2. More careful inspection reveals thatthe norm of @=@ diverges as y ! ÿ1 if a l 2 term is


024033-24



present so we need l 1, corresponding to the solution

! ÿq

8R2 1 ÿ x2 3Q ÿ q2 3 x ky

! 3q2

1 y ÿq

8R2 y2 ÿ 1 3Q ÿ q2 3 y kx ;

(A18)

where k is a constant. We then nd, near y

ÿ1,

g 12 k q2 xy O y0 ; (A19)

so we must choose k ÿq2 to prevent some orbits of @=@ from being closed timelike curves (CTCs). Thiscompletes the derivation of the solution given in [18].

APPENDIX B: SUPERSYMMETRIC BLACK RINGSIN U 1 N THEORIES

1. The theory

The method of [24] has been generalized to the case of minimal supergravity coupled to N

ÿ1 Abelian vector

multiplets with scalars taking values in a symmetric space[22,23]. The action for such a theory is

I 1

16 G 5 Z R ? 1 ÿ GIJ dX I ^?dX J ÿ GIJ F I ^?F J

ÿ16

CIJK F I ^F K AK ; (B1)

where I ; J ;K 1 ; . . . ; N . The constants CIJK are symmet-ric in IJK and obey

CIJK CJ 0LM CPQ K 0JJ 0 KK 0 4

3 I LCMPQ : (B2)

The N

ÿ1 dimensional scalar manifold is conveniently

parametrized by the N scalars X I , which obey the con-straint 21

16 CIJK X I X J X K 1: (B3)

It is then convenient to de ne

X I 16CIJK X J X K ; (B4)

so X I X I 1. The matrix GIJ is de ned by

GIJ 92 X I X J ÿ 1

2CIJK X K ; (B5)

with inverse

GIJ 2 X I X J ÿ 6CIJK X K ; (B6)

where CIJK CIJK . We also have

X I 92C

IJK X J X K : (B7)

Reducing the D 11 supergravity action,

I 1

16 G 11 Z R11 ? 11 1 ÿ12

F ^? 11F

ÿ16

F ^F ^A ; (B8)

to D 5 on T 6 using the Ansatz (2.2) with the constraint(2.7) yields precisely the action (B1) with N 3, CIJK 1 if IJK is a permutation of 123 and CIJK 0 other-

wise, andGIJ

12diag X 1 ÿ2 ; X 2 ÿ2 ; X 3 ÿ2 : (B9)

2. Supersymmetric solutions

Anysupersymmetric solution of this theory must admit anonspacelike Killing vector eld V [22] so, in a regionwhere V is timelike, we can introduce coordinates just as inthe minimal theory described in Appendix A.Supersymmetry again implies that B ; h is a hyper-Kahler manifold and that the Maxwell elds can be written[22] 22

F I d fX I dt ! I ; (B10)

where I are self-dual 2-forms on B satisfying

X I I ÿ23G : (B11)

The above conditions are both necessary and suf cient forthe existence of a supercovariantly constant spinor of thesame form (A5) as in the minimal theory [23].

We also need to satisfy the equations of motion. TheBianchi identity for F I is

d I 0 ; (B12)

and the Maxwell equation is [22]

r 2 f ÿ1 X I 16CIJK

J K ; (B13)

where, r 2 is the Laplacian on B and, for 2-forms andon B , 1=2 mn

mn , raising indices with hmn . Theremaining equations of motion are satis ed automatically[23].

3. Supersymmetric black rings

We now want to generalize our black ring solution of theminimal theory to a solution of the theory (B1). We pro-ceed by analogy with the minimal theory. First we choose

B to be at space written in the form (2.4). Next we need to nd some closed, self-dual 2-forms I on B . We alreadyknow one example of such a 2-form from the minimaltheory, namely G . This suggests the Ansatz

I ÿ12q

I dy ^d dx ^d ; (B14)

for some constants qI . Equation (B13) reduces to21 Given this constraint, the scalars should be written in terms of

unconstrained variables before the action is varied. 22 Set 0 in [22] to obtain these results.


024033-25



r 2 f ÿ1 X I 1

12R4 CIJK qJ qK xÿ y 4: (B15)

Comparison with the corresponding equation of the mini-mal theory immediately provides a solution:

13

H I f ÿ1 X I

X I 1

6R2 QI ÿ12 CIJK q

J

qK

x ÿ y

ÿ1

24R2 CIJK qJ qK x2 ÿ y2 ; (B16)

where the constants QI are arbitrary but demanding f ! 1at in nity implies that the constants X I must obey the samealgebraic restrictions as X I . These restrictions also imply

f ÿ3 16C

IJK H I H J H K : (B17)

Next, from Eq. (B11) we have

G ÿ32 X I I ; (B18)

which we have to solve to determine ! . A natural Ansatz is

! qI ! I (B19)

where ! I obeys12 d! I ? 4d! I

14H I dy ^d dx ^d : (B20)

This is exactly the same as the equation we had to solve inthe minimal theory and can be solved in the same way —

we have

! I 1 ÿ x2 @ xW I ; ! I y2 ÿ 1 @yW I (B21)

where W I is regular at x 1 and y ÿ1 but can divergelogarithmically at y 1, and must obey

@ x 1 ÿ x2 @ xW I @y y2 ÿ 1 @yW I 12H I : (B22)

We can just read off a solution by carrying over resultsfrom the minimal theory:

! I ÿ1

8R2 1 ÿ x2 QI ÿ1

6R2 CIJK qJ qK 3 x y ;

(B23)

! I 32

1 y X I ÿ1

8R2 y2 ÿ 1

QI ÿ1

6R2 CIJK qJ qK 3 x y : (B24)

So nally we have

! ÿ1

8R2 1 ÿ x2 qI QI ÿ q3 3 x y ; (B25)

! 32

1 y qI X I ÿ1

8R2 y2 ÿ1 qI QI ÿq3 3 x y ;

(B26)

where

q3 16CIJK qI qJ qK : (B27)

The electric charges are given by

Q I 1

8 G 5 Z GIJ ? F J 4G5

QI : (B28)

The mass and angular momenta can be read off by compar-ing the asymptotics of the above solution with the minimalring. We nd

M X I Q I

4G5 X I QI : (B29)

J

8G5qI QI ÿ q3 ;

J

8G56R2qI X I qI QI ÿ q3 :

(B30)

The solutions of Sec. II are obtained by replacing indicesI ; J ;K with i ; j ;k , choosing X i 1=3, i.e., X i 1, andde ning qi q i .

APPENDIX C: POSITIVITY OF THE - METRIC

The determinant of the – part of the ve-dimensional metric (2.3) is

f ÿ2h h ÿ f! 2 h ÿ f! 2 h ; (C1)

where hmn is the base space metric (2.4). If is positivethen the – metric is either positive-de nite or negativede nite. To see which, note that

f ÿ1h g ÿ f! 2 h ; (C2)

so > 0 implies g > 0. Hence the - metric ispositive-de nite when > 0.

We nd that

xÿ y 4

1 ÿ x2 ÿ1 ÿ y fR 4y2 ÿ x2 1 ÿ y xÿ y 2

64R6 2Xi<jQ iqiQ j q j ÿXi

Q 2i q2

i ÿ 4R2q3Xiq i

x ÿ y 2 1 ÿ y8R4 X ÿ x y q3Xi

q i 1 ÿ x XiÞ jQ iq iq j

x ÿ y 2

4R2 Y 1 ÿ y4R2 y2 ÿ x2 Xi<j

q iq j 2 xÿ y XiQ i 1 ÿ y (C3)


024033-26



where Q i is de ned in Eq. (2.9) and

X Q 1Q 2Q 3

R2 xÿ y 1 y XiQ iq2

i ;

Y 1 ÿ y

R2 Xi<jQ iQ j 1 y Xi

q i2:

(C4)

We derived this expression by rst grouping together terms involving the same power of R. For to be positive as y ! ÿ1we need2Xi<j

Q iq iQ j q j ÿXiQ 2

i q2i 4R2q3

Xiq i: (C5)

For qi > 0, this inequality implies that the Q i must lie in the region interior to one sheet of a double sheeted hyperboloid. Itis easy to see that it cannot be satis ed if any of the Q i vanishes hence the allowed region lies entirely within the positiveoctant 23 of R3, i.e., Q i > 0.All of the remaining terms in are manifestly positive except for X and Y . These can be seen to be positive as follows:

Y > ÿ 1 y1

R2Xi<jQ iQ j ÿ Xi

qi2

ÿ1 y4R2q3 4q3Xi<j

Q iQ j ÿ 4R2q3 Xiq i

2

ÿ1 y4R2q3 4q3Xi<j

Q iQ j ÿ Xiq i 2Xi<j

Q iqiQ j q j ÿXiQ 2

i q2i

ÿ 1 y4R2q3 q1 Q 2q2 Q 3q3 ÿ Q 1q1

2 q2 Q 3q3 Q 1q1 ÿ Q 2q22

q3 Q 1q1 Q 2q2 ÿ Q 3q32

0:Here the rst inequality is just 1 ÿ y > ÿ1 ÿ y and the second follows from (C5). For X we have

X ÿ1 yQ 1Q 2Q 3

R2 ÿXiQ iq2

i ÿ1 y

4R2q3Pi qi4Q 1Q 2Q 3q3Xi

qi ÿ4R2q3 Xiq i X j Q j q2

j

ÿ1 y

4R2q3

Pi

q i4Q 1Q 2Q 3q3

Xi

q i ÿ

X j

Q j q2 j 2

Xi<j

Q iqiQ j q j ÿ

Xi

Q 2i q2

i

ÿ1 y

4R2q3Pi q iQ 1q2

1 Q 2q2 Q 3q3 ÿQ 1q12 Q 2q2

2 Q 3q3 Q 1q1 ÿQ 2q22 Q 3q2

3 Q 1q1 Q 2q2 ÿQ 3q32

0:

The rst inequality is just x ÿ y ÿ1 ÿ y and the secondfollows from (C5). In summary, the only condition re-quired for to be positive is (C5).

APPENDIX D: EXTENSION THROUGH THEHORIZON

1. Five-dimensional solutionThe solution (2.3) can be extended through y ÿ1in

the same way as the minimal ring [18]: let r ÿR=y and

dt dv ÿ A r dr; d d 0ÿ B r dr;

d d 0ÿ B r dr;(D1)

where

A r A2

r 2 A1

r A0 ; B r

B1

rB0 ; (D2)

and Ai , Bi are determined by requiring that the solution inthe coordinates v; r;x; 0 ; 0 is analytic at r 0. Firstnote that the scalars X I are already analytic at r 0. The

electromagnetic potentials Ai

are given by

Ai 4qir 2

q3 1 O r dv ÿq i

21 x O r d 0

ÿqi

21ÿ xÿ

2Q iq i ÿP j Q j q j

q3 O r d 0

ÿb i

0B1

r4qi A2

q3 bi0B0 b i

ÿ1B1 O r dr; (D3)23 The other sheet lies in the negative octant but shall wedisregard this region.


024033-27



where b i0 and b i

ÿ1 are certain constants and O r denotesterms that can be expanded as a series in positive powers of r . This is analytic up to a term that can be removed by agauge transformation. For the metric, we nd that gr 0diverges as 1=r unless we choose A2

ÿL2 q1q2q31=3B1= 2R , where L was introduced in

(3.28). We then nd that gr r has a 1=r2 divergence unlesswe choose 24

B1 ÿq

2L: (D4)

This implies

A2Lq2

4R: (D5)

Now gr r diverges as 1=r unless we choose

A11

4R2Lq 2 Q 1Q 2Q 3 ÿ R2XiQ iq2

i

q3

q1 q2 q3 R2

: (D6)

The metric is now analytic at r 0. However we still havethe freedom to choose the nite part of the coordinatetransformation. After the above transformation, gr r is alinear function of x at r 0 and we can choose A0, B0 tocancel this function so that gr r 0 at r 0. The expres-sions for A0 and B0 are lengthy and unilluminating so weshall not present them here. The metric nally takes theform

ds 25 ÿ

16r 4

q4 dv 2 2RL

dvd r4r3sin2

Rqdvd 0

4rRq

dvd 0 1L

q1 q2 q3 rsin2 drd 0

2qL2R

cos ÿ c drd 0 L2d 02

q2

4d 2 sin2 d 0ÿ d 02 . . . (D7)

We have set x cos . The ellipsis denotes terms in gr rstarting at O r , as well as subleading (integer) powers of rin all of the metric components explicitly written above.The constant c is given by

c1

2LRq1q2q3Q 1Q 2Q 3 ÿ R2Xi<j

Q i Q j qiq j

ÿ q3 q1 q2 q3 R2 : (D8)

The above metric is analytic in r hence so is its determi-nant. At r 0, the determinant vanishes if, and only if,sin2 0, which is just a coordinate singularity. Hence theinverse metric is also analytic in r so the above coordinatesde ne an analytic extension of our solution through thesurface r 0.

The supersymmetric Killing vector eld V @=@visnull at r 0. Furthermore, V dx R=L dr at r 0 so

V is normal to the surface r 0. Hence r 0 is a nullhyper-surface and a Killing horizon of V , i.e., our solutionhas an event horizon at r 0.

The metric of a spatial cross section of the horizon canbe written

ds 2horizon L2d 02 q2

4d 2 sin2 d 2 (D9)

with 0ÿ 0. The near-horizon geometry is locallyAdS3 S2 where the AdS3 has radius q1q2q3

1=3 and theS2 has radius q1q2q3

1=3=2.

2. The IIB solutionConsider now the IIB solution (4.2). As y ! ÿ1 , the

conformal factors multiplying the three terms in (4.2)remain nite and nonzero. Hence, after transforming tothe coordinates v; r; ; 0 ; 0, the only part of the metricthat is not manifestly regular at r 0 is the part involving A3. To make this regular we need a gauge transformation,i.e., a shift in z. Using Eq. (D3), the required shift is

dz dz0 b30B1

r4 A2

q1q2b3

0B0 b3ÿ1B1 dr; (D10)

which gives

dz A3 dz0 4r2

q1q21 O r dv

ÿq3

21 cos O r d 0

ÿq3

21ÿcos ÿ

2Q 3q3 ÿP j Q j q j

q3 O r d 0

O r dr: (D11)

This is manifestly analytic at r 0. We still have V dx /dr at r 0 where V @=@vso r 0 is a Killing horizon

of V .

3. Null orbifold singularity of rings with L 0and q i Þ 0

Consider now solutions where all the dipoles qi arenonzero but L 0. The analysis is similar to the case L >0 so we shall simply sketch it. Let y ÿR2=r 2 and changeas in (D1). Take A A2=r2 and B B2=r 2. A2 and B2 canbe chosen to cancel the divergent term in gr r . There are no

24 The overall sign here is arbitrary; making the opposite choicewould lead to an extension of the metric through the past horizonrather than the future horizon.


024033-28



other divergences in gr r nor gr 0. The determinant of themetric vanishes at r 0, but this comes from the coef -cient r 2 in g 0 0and is a coordinate singularity analogous tothe one in the Poincare patch of AdS3. There remains a r 0

piece in gr r which is x dependent (but does not vanish atany x), and could be eliminated by adding an r0 term to Bwhich is also a linear function of x, but this is not actuallynecessary. So there is no curvature singularity at y

ÿ1.

The near-horizon (or, more appropriately, ‘‘ near-core ’’ )geometry is more simply expressed in coordinates whichonly cover the outer region. Changing y ÿR2= ~r 2 andt ~t= , and sending ! 0 we nd

ds 25

4~r 2

qd~td q2 d~r 2

~r2q2

4d ~2 sin2 ~d 2 : (D12)

This is, like in the cases with L > 0, locally AdS3 S2.The AdS3 part is written in double-null form, and since theorbits of @ are closed we nd a null orbifold singularity at~r 0 instead of a regular horizon. The near-core limit for

the corresponding IIB solutions is easily obtained by com-paring to (4.12).

The solutions where some of the q i vanish, and possiblyalso some Q i , are studied in Sec. VI.

[1] R. Emparan and H. S. Reall, Phys. Rev. Lett. 88 , 101101(2002).

[2] H. Elvang, Phys. Rev. D 68 , 124016 (2003).[3] H. Elvang and R. Emparan, J. High Energy Phys. 11

(2003) 035.[4] H. S. Reall, Phys. Rev. D 68 , 024024 (2003).[5] R. Emparan, J. High Energy Phys. 03 (2004) 064.[6] D. Mateos and P. K. Townsend, Phys. Rev. Lett. 87 , (2001)

011602.[7] R. Emparan, D. Mateos, and P.K. Townsend, J. High

Energy Phys. 07 (2001) 011.[8] D. Mateos, S. Ng, and P.K. Townsend, J. High Energy

Phys. 03 (2002) 016.[9] O. Lunin and S. D. Mathur, Phys. Rev. Lett. 88 , 211303

(2002).[10] O. Lunin, S. D. Mathur, and A. Saxena, Nucl. Phys. B655 ,

185 (2003); O. Lunin and S. D. Mathur, Nucl. Phys. B623 ,342 (2002);

[11] O. Lunin, J. Maldacena, and L. Maoz, hep-th/0212210.[12] S. D. Mathur, A. Saxena, and Y. K. Srivastava, Nucl. Phys.

B680 , 415 (2004); S. D. Mathur, hep-th/0401115.[13] S. Giusto, S. D. Mathur, and A. Saxena, hep-th/0406103;

S. Giusto, S. D. Mathur, and A. Saxena, Nucl. Phys. B701 ,357 (2004).

[14] O. Lunin, J. High Energy Phys. 04 (2004) 054.[15] I. Bena and P. Kraus, Phys. Rev. D 70 , 046003 (2004).[16] I. Bena, Phys. Rev. D 70 , 105018 (2004).[17] B. C. Palmer and D. Marolf, J. High Energy Phys. 06

(2004) 028; D. Bak, Y. Hyakutake, and N. Ohta, Nucl.Phys. B696 , 251 (2004); D. Bak, Y. Hyakutake, S. Kim,and N. Ohta, hep-th/0407253.

[18] H. Elvang, R. Emparan, D. Mateos, and H.S. Reall, Phys.Rev. Lett. 93 , 211302 (2004).

[19] J. C. Breckenridge, R. C. Myers, A. W. Peet, and C. Vafa,Phys. Lett. B 391 , 93 (1997); A. A. Tseytlin, Mod. Phys.Lett. A 11 , 689 (1996); J. C. Breckenridge, D.A. Lowe,R. C. Myers, A. W. Peet, A. Strominger, and C. Vafa, Phys.Lett. B 381 , 423 (1996);

[20] A. Strominger and C. Vafa, Phys. Lett. B 379 , 99 (1996).[21] R. Emparan and H. S. Reall, Phys. Rev. D 65 , 084025

(2002).[22] J. B. Gutowski and H.S. Reall, J. High Energy Phys. 04

(2004) 048.[23] J. B. Gutowski, hep-th/0404079.

[24] J. P. Gauntlett, J. B. Gutowski, C. M. Hull, S. Pakis, andH.S. Reall, Classical Quantum Gravity 20 , 4587(2003).

[25] C. A. R. Herdeiro, Nucl. Phys. B582 , 363 (2000).[26] I. R. Klebanov and A. A. Tseytlin, Nucl. Phys. B475 , 179

(1996).[27] G.W. Gibbons, D. Kastor, L. A. J. London, P. K.

Townsend, and J. H. Traschen, Nucl. Phys. B416 , 850(1994).

[28] E. Bergshoeff, C. M. Hull, and T. Ort ın, Nucl. Phys. B451 ,547 (1995).

[29] E. J. Hackett-Jones and D. J. Smith, hep-th/0405098.[30] J. B. Gutowski, D. Martelli, and H. S. Reall, Classical

Quantum Gravity 20 , 5049 (2003).[31] M. Cvetic, H. Lu, and C.N. Pope, Nucl. Phys. B549 , 194

(1999).[32] V. Balasubramanian, J. de Boer, E. Keski-Vakkuri, and

S.F. Ross, Phys. Rev. D 64 , 064011 (2001); J.M.Maldacena and L. Maoz, J. High Energy Phys. 12,(2002) 055.

[33] O. Lunin and S.D. Mathur, Nucl. Phys. B610 , 49(2001).

[34] G.T. Horowitz and D. Marolf, Phys. Rev. D 55 , 835(1997); G. T. Horowitz and D. Marolf, Phys. Rev. D 55 ,846 (1997).

[35] H. Elvang, R. Emparan, and P. Figueras (to be published).[36] M. Cveticˇ and F. Larsen, Nucl. Phys. B531 , 239

(1998).[37] A. Strominger, J. High Energy Phys. 02 (1998) 009.[38] D. Bak and S.W. Kim, Nucl. Phys. B622 (2002) 95.[39] G.W. Gibbons and G. Papadopoulos, Commun. Math.

Phys. 202 , 593 (1999).[40] J. P. Gauntlett, N. D. Lambert, and P.C. West, Commun.

Math. Phys. 202 , 571 (1999).[41] R. Harvey and H. B. Lawson, Acta Math. 148 , 47 (1982);

F. R. Harvey, Spinors and Calibrations (Academic, NewYork, 1990).

[42] E. Bergshoeff, R. Kallosh, T. Ortin, and G. Papadopoulos,Nucl. Phys. B502 , 149 (1997).

[43] I. A. Bandos, K. Lechner, A. Nurmagambetov, P. Pasti,D. P. Sorokin, and M. Tonin, Phys. Rev. Lett. 78 , 4332(1997).

[44] O. Barwald, N. D. Lambert, and P. C. West, Phys. Lett. B463 , 33 (1999).


024033-29



[45] J. P. Gauntlett, N. D. Lambert, and P.C. West, Adv. Theor.Math. Phys. 3, 91 (1999).

[46] E. Bergshoeff, D. P. Sorokin, and P.K. Townsend, Nucl.Phys. B533 , 303 (1998).

[47] J. P. Gauntlett, J. Gomis, and P. K. Townsend, J. HighEnergy Phys. 01 (1998) 003.

[48] J. B. Gutowski, J. High Energy Phys. 08 (2004) 049.

[49] R.L. Bryant, Pseudo-Riemannian Metrics with ParallelSpinor Fields and Vanishing Ricci Tensor , Semin. Congr.,Vol. 4 (Soc. Math. France, Paris, 2000), pp. 53 – 94.

[50] J. P. Gauntlett and J. B. Gutowski, hep-th/0408010.[51] I. Bena and N. Warner, hep-th/0408106.[52] J. P. Gauntlett and J. B. Gutowski, hep-th/0408122.


henriette elvang et al- supersymmetric black rings and three-charge supertubes

Documents