Point mass geometries, spectral flow and AdS3–CFT2 correspondence

Nuclear Physics B 564 Ž2000. 128–141 www.elsevier.nlrlocaternpe

Point mass geometries, spectral flow and AdS 3 –CFT2 correspondence Justin R. David a

a,1

, Gautam Mandal a , Sachindeo Vaidya a , Spenta R. Wadia a,b,2

Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India b Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Trieste 34014, Italy Received 20 July 1999; accepted 22 September 1999

Abstract We discuss, in terms of the AdS 3 –CFT2 correspondence, a one-parameter family of Žasymptotically AdS 3 . conical geometries which are generated by point masses and interpolate between AdS 3 and BTZ space-times. We show that these correspond to spectral flow in N s Ž4,4. SCFT2 which interpolate between NS and R sectors. Our method involves representing the conical spaces as solutions of three-dimensional supergravity based on the supergroup SUŽ1,1 <2. = SUŽ1,1 <2.. The boundary SCFT we use is based on the D1rD5 system. The correspondence includes comparing the Euclidean free energies between supergravity and SCFT for the family of conical spaces including BTZ black holes, and a concrete proposal for representing the point mass by a vertex operator in the SCFT. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction and summary The recently conjectured AdS dq 1rCFTd correspondence has led to a number of remarkable predictions for N s 4 Yang–Mills theory in four dimensions. Considering the fact that such a correspondence potentially defines quantum gravity in these backgrounds, one should be able to describe various interesting dynamical phenomena in gravity as well, including, e.g. black hole formation. The case of d s 3 is particularly attractive because both AdS 3 and CFT2 are more tractable than their higher dimensional counterparts and has nevertheless a rich physics content. A particularly interesting class of solutions of three-dimensional gravity with L - 0 are conical space-times w1x which are generated by a point mass at the origin Žsee Eq. 1

Address after August 31, 1999: Department of Physics, University of California, Santa Barbara, CA 93106, USA. 2 Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore 560012, India. 0550-3213r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 6 2 1 - 5

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Ž5... If the mass exceeds a certain critical value, the conical space-time becomes a BTZ black hole. Indeed, it was observed by w4x Peleg and Steif that in the context of a gravitational collapse of a shell of variable rest mass, the resulting space-time is conical unless the rest mass exceeds a critical value. At this value there is a critical phase transition and a BTZ black hole is formed. It was pointed out by Brown and Henneaux w2x that the conical space-times are ‘‘asymptotically AdS 3’’ and they belong to the class of geometries whose ‘‘asymptotic isometry group’’ is Virasoro = Virasoro. Therefore it seems reasonable to expect, in the light of AdSrCFT correspondence, that these spaces should correspond to boundary conformal field theories. It is already known w3x that the ‘‘end-points’’ of the one-parameter family of metrics Žg s 0,1., namely the pure AdS 3 and the zero-mass BTZ black hole Žsee below Eq. Ž8.., correspond to the NS and R sectors of a N s Ž1,1. superconformal field theory ŽSCFT.. If we go back to the origin of the AdS 3 –CFT2 correspondence w7x, namely to the near-horizon limit of the D1rD5 system, the natural supersymmetry of the SCFT is N s Ž4,4.. ŽThis corresponds to a supergravity based on the group G s SUŽ1,1 <2. = SUŽ1,1 <2. w10x.. In terms of such a SCFT, there is a natural interpolation between the NS and R sectors called spectral flow w11x. It was conjectured some time back w13x that the conical spaces should correspond to spectral flow, the defect angle being related to the parameter of the flow Žsee Eq. Ž21... One evidence was that the ADM mass in supergravity matched exactly with L0 q L0 in the SCFT Žsee Eq. Ž26.. for all values of the interpolating parameter. This connection has also been mentioned recently in w14x. In incorporating conical spaces in the ambit of AdS 3 –CFT2 correspondence, the first step is to realize the conical spaces as supersymmetric solutions of three-dimensional gravity, since the boundary theory mentioned above is supersymmetric. If one tries to generalize to conical spaces the work of w3x which embeds BTZ and AdS 3 in the framework of Ž1,1.-type 3 AdS 3 , the conical spaces turn out to be non-supersymmetric: the Killing spinors, in particular, are quasiperiodic and hence not globally defined. If, however, one tries to realize these as solutions of SUŽ1,1 <2. = SUŽ1,1 <2. supergravity, which is natural from the viewpoint of the boundary theory, the Killing spinors become globally defined. This construction also allows us to establish the main point of the bulk-to-boundary correspondence in this case: since the same supergroup is represented in the bulk theory and in the boundary theory, we can find the operator in the supergroup which deforms the value of the spectral parameter in the SCFT and the operator which changes the value of the defect angle in the supergravity. We find that the same operator causes the one-parameter flow in both cases, thus establishing the correspondence between the spectral flow and conical spaces. As additional evidence for this correspondence we compute the Euclidean free energy of the family of space-times and compare with the corresponding quantity in SCFT. The BTZ free energy, which has already been discussed in w17,18x, is shown to be reproduced by SCFT based on the symmetric product SQ 1 Q 5ŽT 4 . at high temperatures. The free energies of the conics are reproduced at low temperatures.

3 For the definition of Ž p,q .-type AdS 3 supergravity, please see w22x. Note the difference with the notation N s Ž p,q . which represents the number of supersymmetries in the left- and right-moving sectors respectively of the SCFT.

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As a part of our construction we find a vertex operator in the SCFT which corresponds to creation of a point mass geometry in an asymptotically AdS 3 space. The scattering of such point masses Žincluding the cases when they form a black hole w5,6x. is naturally represented in terms of correlation function of such vertex operators. This paper is organized as follows. In Section 2 we discuss the conical spaces as solutions of three-dimensional supergravity Žwith L - 0. based on G s SUŽ1,1 <2. = SUŽ1,1 <2.. In Section 3 we make the identification with spectral flow in N s Ž4,4. SCFT. In Section 4 we compute the Euclidean free energies of these spaces from supergravity. In Section 5 we compute the free energies from the SCFT viewpoint and compare with the results of Section 4. In Section 6 we discuss the issue of scattering of the point masses and black hole formation in terms of correlation functions in the SCFT.

2. Conical spaces as solutions of supergravity based on G The action for three-dimensional supergravity based on G ' SUŽ1,1 <2. = SUŽ1,1 <2. is as follows 1

Ss

16p GNŽ3.

Hd

3

x eR q

y 8l e mnr Ami En Ari y

ž

2 l2

Žwith cosmological constant L - 0. w15,16x:

e y e mnrcm Dn cr

4 i e i jk 3

Ami Anj Ark

ye mnrcmX DnX crX q 8l e mnr AXmi En AXri y

ž

/ 4 i e i jk 3

AXmi AXnj AXrk

/

,

Ž 1.

where l 2 s y1rL, GNŽ3. is the three-dimensional Newton’s constant, and Dn s En q v a bn g a br4 y e a n g arŽ2 l . y 2 Ani s i and DnX s En q v a bn g a br4 q e a n g arŽ2 l . y 2 AXni s i. The basic fields appearing in the Lagrangian are the vierbein ema , cr , Ami , cmX and AXmi . The same three-dimensional supergravity can be obtained from type IIB string theory compactified on K s T 4 = S 3 Žwith constant flux on S 3 .. Recall that the near-horizon geometry of the D1rD5 system involves K = AdS 3 whose Žsuper.isometries are G . Similarly the near-horizon geometry of the five-dimensional black hole is K = BTZ w17x. This implies in an obvious fashion that AdS 3 and BTZ are solutions of Ž1.. Furthermore, the three-dimensional Newton’s constant GNŽ3. and the AdS radius l are given by the following string theoretic expressions: GNŽ3. s 4

l s

4p 4 g s2 V4 l 3

,

16p 4 g s2 Q1 Q5 V4

,

Ž 2.

where V4 is the volume of T 4 and g s is the string coupling Žwe are working in the units a X s 1..

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Conical spaces As mentioned in Section 1 there is a one-parameter family of classical solutions of three-dimensional gravity Žwith L - 0. discovered by Deser and Jackiw which are asymptotically AdS 3 and interpolate between the AdS 3 and zero-mass BTZ solutions. These represent geometries around a point mass m, 0 - m - 1rŽ4GNŽ3. . which create a conical singularity at the origin. The metric is given by

ž

ds 2 s ydt 2 g q

r2

/ ž

q dr 2 g q

l2

r2 l2

y1

/

q r 2 df 2 .

Ž 3.

It is easy to see that there is a conical singularity at the origin r s 0 with defect angle

Df s 2p Ž 1y'g . .

Ž 4.

The parameter g varies from 0 to 1, and is related to the point mass m at the origin ms

'

1y g 4GNŽ3.

.

Ž 5.

We will denote these spaces by Xg . Note that the mass m is the DeserJackiw mass which is different from the BTZ definition of mass, denoted by M, which is given by M s ygr Ž 8GNŽ3. . .

Ž 6.

BTZ and AdS3 as end-points Recall that the metric of pure AdS 3 is given by 2

2

ž

ds s ydt 1 q

r2 l2

/ ž q dr

2

1q

r2 l2

y1

/

q r 2 df 2

Ž 7.

while the metric for BTZ black holes is given by 2

ds s y

r2 l2

yMq yJ

qr2

ž

2r

J

ž / /

2 2

dt q

2r

dt q d f 2

r2 l2

yMq

J

ž / 2r

2 y1

dr 2

2

,

Ž 8.

where M and J refer to the mass and the angular momentum of the BTZ black hole Ž J ( M .. It is clear that the space Xg becomes AdS 3 for g s 1, whereas for g s 0 it becomes BTZ with M s 0 Ž M s 0 ´ J s 0.. We have already remarked that AdS 3 and BTZ are supersymmetric solutions of Ž1.. We will now show that the entire family of spaces Xg can be obtained as supersymmetric solutions of Ž1..

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It is known w20x4 how to embed the conical spaces Ž3. as supersymmetric solutions in Ž2,0.-type AdS 3 supergravity. The solution looks like l Am dx m s y g d f , Ž 9. 2 where Am is the UŽ1. gauge field appearing in the graviton supermultiplet. The embedding into N s Ž4,4. is a straightforward extension of the above, l Am3 dx m s y g d f , Am" dx m s 0, Ž 10 . 2 where the superscripts 3," refer to the R-parity group SUŽ2.. It can be explicitly checked that the equations of motion following from Ž1., as well as the Killing spinor equations ensuring supersymmetry are satisfied by the above solution. The equations of motion reduce to those of the UŽ1. problem w20x with the ansatz Ž10.. As far as the Killing spinors are concerned, the solution in our case is a doublet constructed out of the solution of the UŽ1. problem. The Killing spinor equation in the present case is Dn e s En q

v a bn g a b 4

1

y

i e an g a q An3 e s 0. 2l l

Ž 11 .

The solution is given by

es

e1 , e 1)

ž /

Ž 12 .

where e 1) is the complex conjugate of e 1 , the latter being the Killing spinor in the UŽ1. problem w20x. Connection with (1,1)-type AdS3 supergraÕity We emphasize that extended supersymmetry is quite essential to construct the conical spaces as supersymmetric solutions. The Killing spinors have holonomies under both the spin connection and the gauge connection. Under either one of them the spinors are quasiperiodic, corresponding to the fact that the Ž1,1.-type Killing spinors constructed using the formulae of w3x, which see only the spin connection, are quasiperiodic. With extended supersymmetry Ž i.e., Ž p,q .-type AdS 3 supergravity, with either p or q or both greater than 1. there is a UŽ1. gauge field. The holonomy under this gauge connection cancels that under the spin connection, making the Killing spinors periodic and hence globally defined. The NS or R boundary conditions of the spinors of w3x for g s 1,0 now refer to the condition that the holonomy under the gauge connection, represented by the Wilson line i W s Tr exp Am dx m , Ž 13 . l

H

is y1 or 1 respectively. 4

We thank P.Townsend for pointing out Ref. w20x to us.

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Conics as one-parameter family of gauge transforms It is clear from the above discussion that the family of solutions can be parameterized uniquely by the value of the Wilson line W Žthe metric is fixed once we specify this.. We can, therefore, change from the classical solution Xgs0 to Xg by making a ‘‘gauge transformation’’ U Ž t ,r , f . s exp i z Ž f . T 3 ,

Af ™ Uy1Af U q iUy1Ef U,

z Ž f . s f'g .

Ž 14 . The reason why the Wilson line W changes is that U is not single-valued. In other words, Eq. Ž14. describes an improper gauge transformation. The equations of motion demand that the metric changes appropriately from its value at g s 0 to the value at g . It is interesting to note that this change in the metric can also be understood as an Žimproper. SLŽ2, R . = SLŽ2, R . gauge transformation. Following w21,22x we combine the dribein emM and the spin connection v NMm into an SLŽ2, R . = SLŽ2, R . vector potential A". For the metric Ž3., it turns out to be A"' Ama" T adx m s

1

ž

y g e . r dt "

'

"d r

2 y g e " r dt "

'

.d r

/

,

Ž 15 .

where r s g sinh r and t "s t " f . One can show that this can be transformed to the value at g s 0 by the following SLŽ2, R . = SLŽ2, R . gauge transformation y1 A"™ Vy1 " A" V "q V " dV " ,

V2 "'exp

1 2

V "s V1 " V2 " ,

V1 "'

z Ž f . s1 .

ž

e . r r2 0

0 , e " r r2

/

Ž 16 .

z Ž f . being the same function as in Ž14.. Here s 1 is the Pauli matrix. As emphasized before, the gauge transformation of the metric can be regarded as a consequence of the gauge transformation Ž14. and the equations of motion. For comparison with the boundary CFT, we note that the gauge transformation Ž14. is implemented in the quantum theory by the operator $

Uˆ sexp i g f J3 ,

'

Ž 17 .

$

where Ja,a s 1,2,3 are generators of the SUŽ2. part of SUŽ1,1 <2..

3. Correspondence with spectral flow We have already remarked that the Brown-Henneaux Virasoro algebra can be supersymmetrized w3x by embedding the AdS 3 or BTZ solutions in N s 1 supergravity, and that the realizations of the superconformal algebra corresponding to the AdS 3 and BTZ solutions respectively map to the NS and R sector of the boundary conformal field theory. In the context of extended supersymmetry, as we have remarked above, the

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AdS 3 and BTZ solutions correspond to a Wilson line W equal to y1 or 1. In order to facilitate comparison, let us define the gauge-invariantized fermion

c˜ ' U w A x c ,

U w A x s exp

i

P

HA l P

.

Ž 18 .

0

The AdS 3 and BTZ solution correspond to antiperiodic or periodic c˜ respectively. It is this c˜ of supergravity that maps to the fermion c of CFT. Now, we know that there is a one-parameter flow, called spectral flow, between periodic and antiperiodic boundary conditions in the case of N s 4 superconformal theory w11x. This corresponds to quasiperiodic boundary condition on c ,

c Ž ze 2 p i . sexp Ž iph . c Ž z . .

Ž 19 .

Let us compare this flow with the one-parameter flow caused by the Wilson line ŽEqs. Ž14. and Ž17... We note that the gauge-invariantized fermion c˜ satisfies, under f ™ f q 2p

c˜ Ž ze 2 p i ,r ™ ` . sexp Ž ip'g . c˜Ž z . ,

Ž 20 .

where z s e iŽ tq f .. This suggests that the Hilbert space corresponding to spectral flow is the realization of boundary SCFT for the conical spaces, with the identification

h s 'g .

Ž 21 .

The fact that this is the right correspondence follows by noting that spectral flow in the SCFT is defined in terms of the generator w11,12x $

Uˆ sexp i g f J3 ,

'

Ž 22 .

$

where J3 is the SUŽ2. R-parity current. Thus the SUŽ2. generator for spectral flow Ž22. is the same as the SUŽ2. generator Ž17. in the bulk that changes the conical defect angle. This is exactly as it should be for the proposed AdSrCFT correspondence to work. The equality of Ž17. and Ž22. under the AdSrCFT correspondence proves our assertion. A rather important consequence of the above correspondence is that we know exactly what state in the CFT corresponds to a point mass in the bulk that creates the conical singularity, namely it is the state < g : ' Uˆ <0: .

Ž 23 .

This suggests a boundary representation in terms of CFT vertex operators of a point mass in the bulk and consequently a representation of their scattering in terms of CFT correlation in the sense of w8,9x. We will make some more remarks on this in Section 6. Related remarks also appear in w25x in a somewhat different context. We note that we are parameterizing the flow in supergravity by the Wilson line only, by adopting the attitude that the metric gets fixed as a consequence of the equation of motion. For example, the ADM mass for the metric corresponding to the Wilson line Ž10. is M s ygr8GNŽ3. .

Ž 24 .

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The counterpart of this statement in CFT is that by performing a spectral flow along J 3 we automatically change the value of L 0 on the ground state from 0 to c L0 s y h 2 . Ž 25 . 24 Similar remarks apply to L0 . Using Ž2. and the fact that c s 6Q1 Q5 Žsee Section 5. for the AdS 3 and BTZ points, it is easy to see that Ms

L0 q L0

. Ž 26 . l We see that Ž24. and Ž25. agree with Ž26. if we use g s h 2 , which is the same condition as in Eq. Ž21.. This provides additional support to our proposed correspondence. This matching has recently also been mentioned in w14x. We will continue to explore this correspondence in the rest of the paper in the context of the Euclidean free energy. 4. The Euclidean free energy of asymptotically AdS 3 solutions In this section we will compute the Euclidean free energy of conical spaces Žalso of AdS 3 and BTZ. following the method of Gibbons and Hawking w23x. Let us recall that the free energy is given by Z ' exp yb F Ž a . s Da w fields x exp w yS x ,

Ž 27 .

H

where S is now the Euclidean version of the action written in Ž1.. Boundary conditions In the above equation, a denotes boundary conditions on the fields. As noted in w17x, the boundary Žcorresponding to r ™ `. is T 2 , coordinatized by f and the Euclidean time t ' yit, with appropriate identifications. For the AdS 3 solution, the identifications are Ž t , f . ' Ž t q b , f q 2p . , Ž 28 . Ž where b denotes inverse temperature and is arbitrary. For the BTZ solution with mass M and Euclidean angular momentum JE s iJ . the identifications are Ždictated by smoothness of metric. Ž t ,f . ' Ž t q b0 ,f q F . , Ž 29 . where b 0 and f are given by

b0 s

2p rq l 2 2 2 rq y ry

rys yi

l2M 2

,

Fs

ž(

1q

2p < ry < l 2 2 2 rq y ry

JE2 M 2l2

,

rqs

l2M 2

ž ( 1q

1q

JE2 M 2l2

1r2

/

,

1r2

y1

/

.

Ž 30 .

In anticipation we note that implication of the identifications is that the boundary geometry is that of a torus with the modular parameter proportional to b 0 q iF .

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Therefore the partition function of the dual CFT need to be evaluated on the torus with the modular parameter proportional to b 0 q iF . The boundary condition on the fields is that the bosonic fields must have the same values at identified points whereas the fermion fields must have the same value up to a sign Žperiodic ŽP. or antiperiodic ŽA... Since we are talking about identifications on a two-torus, the fermion boundary conditions can be a s Ž P , P . , Ž P ,A . , Ž A,P . , Ž A,A . , Ž 31 . where the first entry denotes boundary condition in the f direction and the second entry denotes boundary condition along the t direction. For the conical spaces, the identification is the same as for AdS 3 , except that the specification of the functional integral includes the Wilson line ŽEq. Ž10.. as a part of the boundary condition. Saddle points We will evaluate Ž27. by finding saddle points of the action subject to specific boundary conditions. These are Euclidean versions of the classical solutions that we have described in the previous section. By virtue of the equation of motion R s y6rl 2 , the Euclidean action S of a classical space-time X is simply its volume times a constant. To be precise, 1 SŽ X . s Vol Ž X . , 4p l 2 GNŽ3. Vol Ž X . s

b

2p

R

H0 dtHr drH0

'

df g .

Ž 32 .

0

The ranges of f ,t follow from the identifications mentioned above. The lower limit r 0 of the r-integral is identically zero for AdS 3 and the conical spaces, whereas for BTZ it denotes the location of the horizon Žthe Euclidean section is defined only up to the horizon.. The upper limit R is kept as an infrared regulator to make the volume finite. We will in practice only be interested in free energies relative to AdS 3 and the R-dependent divergent term will disappear from that calculation. Free energy of BTZ We will now compute the free energy of BTZ relative to AdS 3 . The AdS 3 solution can be at any temperature b while the temperature of the black hole is fixed to be b 0 . To compare with the AdS 3 background one must adjust b so that the geometries of the two manifolds match at the hypersurface of radius R Žin other words, we must use the same infrared regulator on all saddle points of the functional integral.. This gives the following relation:

(

b0 s b

1 q l 2rR 2 1 y M 2 l 2rR 2

S Ž BTZ . y S Ž AdS 3 . s s

,

Ž 33 . 1

4p l 2 GNŽ3. 1 4p l 2 GNŽ3.

3

3

HBTZd x'g yHAdS d x'g 3

2 pb 0 Ž R 2yrq . ypb R 2 .

Ž 34 .

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Substituting the value of b 0 in terms of b and taking the limit R ™ ` we obtain S Ž BTZ . y S Ž AdS 3 . s

1 4p l 2 GNŽ3.

pb 0 l 2yp 2 rq l 2 .

Ž 35 .

For convenience let us define the left and right temperatures w17x as

bqs

2p l 2 rqq ry

2p l 2

bys bqs

,

Ž 36 .

rqy ry

Using these variables the difference in the action becomes S Ž BTZ . y S Ž AdS 3 . s

p

1 4p l

2

GNŽ3.

2

Ž bq qby . l 2yp 3 l 4

ž

1

bq

1 q

by

/

.

Ž 37 .

It is easily seen that the BTZ black hole dominates when Ž1rbqq 1rby . is much larger than Ž bqq by . and vice versa. Thus at high temperatures we can ignore the first term in the above equation,

p 2l2

S Ž BTZ . y S Ž AdS 3 . s y

4GNŽ3.

ž

1

bq

1 q

by

/

.

Ž 38 .

Using Eq. Ž2., the high temperature partition function of BTZ is, therefore, given by yln Z s p 2 Q1 Q5 l

1

ž

bq

1 q

by

/

.

Ž 39 .

Free energy of conical spaces We denote the conical spaces by Xg . The volume of Xg is given by Vol Ž Xg . s

R

bg

2p

H0 rdrH0 dtH0

d f s pbg R 2 .

Ž 40 .

Once again, bg is determined in terms of b by the requirement that the hypersurface r s R that acts as an infrared regulator has the same 2-geometry as the corresponding surface in AdS 3 . This gives

(

(

bg R 2rl 2 q g s b R 2rl 2 q 1 .

Ž 41 .

Using this, it is easy to find Vol Ž Xg . y Vol Ž AdS 3 . s pb

l2 2

Ž 1yg . .

Ž 42 .

This leads to, by Ž32., the following expression for the Euclidean action

b S Ž Xg . y S Ž AdS 3 . s

8GNŽ3.

Ž 1yg . .

Ž 43 .

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In the next section we will compare with the free energy computed from the boundary CFT. For that comparison it will turn out to be more appropriate to consider as reference space-time the BTZ black hole with J s M s 0 Žwhich is simply the space X 0 , also denoted BTZ 0 .,

b S Ž Xg . y S Ž BTZ 0 . s yln Z Ž Xg . y Ž yln Z Ž X 0 . . s y

8GNŽ3.

g.

Ž 44 .

5. The partition function from the CFT The aim of this section is to calculate the partition function of the Ž4,4. CFT on the orbifold T 4 Q 1 Q 5rSŽ Q1 Q5 .. The partition function will depend of the boundary conditions of the fermions of the CFT. Different bulk geometries will induce different boundary conditions for the fermions of the CFT. We will first calculate the partition function when the bulk geometry is that of the BTZ black hole. CFT partition function corresponding to BTZ The fermions of the CFT are periodic along the angular coordinate of the cylinder if the bulk geometry is that of the BTZ black hole. This can be seen by observing that the zero mass BTZ black hole admits killing vectors which are periodic along the angular coordinate w3x. Therefore the zero mass BTZ black hole correspond to the Ramond sector of the CFT. The general case of the BTZ black hole with mass and angular momentum correspond to excited states of the CFT over the Ramond vacuum with L0 q L 0 s Ml,

L 0 y L 0 s JE ,

Ž 45 .

where M and JE are the mass and the ŽEuclidean. angular momentum of the BTZ black hole. Therefore the partition function of the BTZ black hole should correspond to Z s TrR Ž e 2 p it L 0 e 2 p it L 0 . .

Ž 46 .

The Hilbert space of the CFT on the orbifold T 4 Q 1 Q 5rSŽ Q1 Q5 . can be decomposed into twisted sectors labeled by the conjugacy classes of the permutation group SŽ Q1 Q5 .. The conjugacy classes of the permutation group consists of cyclic groups of various lengths. The various conjugacy classes and the multiplicity in which they occur in SŽ Q1 Q5 . can be found from the solutions of the equation Q1 Q 5

Ý

nNn s Q1 Q5 ,

Ž 47 .

ns0

where n is the length of the cycle and Nn is the multiplicity of the cycle. The Hilbert space is given by Hs

[

m

Ý nNn s Q1 Q5 n ) 0

S N n HŽPn.n .

Ž 48 .

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S NH denotes the symmetrized product of the Hilbert space H , N times. By the symbol HŽPn.n we mean the Hilbert space of the twisted sector with a cycle of length n in which only states which are invariant under the projection operator Pn s

1

n

Ý e 2p i kŽ L yL . 0

n

Ž 49 .

0

ks1

are retained. The values of L 0 or L0 in the twisted sector of length n is of the form prn where p is positive integer. This projection forces the value of L 0 y L0 to be an integer on the twisted sector. It arises because the black hole can exchange only integer valued Kaluza–Klein momentum with the bulk w19x. The dominant contribution to the partition function arises from the maximally twisted sector. That is, from the longest single cycle of length Q1 Q5 . It is given by Zs

Ý d Ž Q1 Q5 n q m . d Ž m . e 2p i nt e 2p i mt r Q Q ey2p i mt r Q Q , 1

5

1

Ž 50 .

5

m, n

where d’s are the coefficients defined by the expansion ZT 4 s

2

Q 2 Ž 0
s

h3Žt .

Ý d Ž n . e 2 p it n .

Ž 51 .

n00

In the above equation, ZT 4 is the partition function of the holomorphic sector of the CFT on T 4 . We will first evaluate the sum `

P Ž m,t . s

Ý d Ž Q1 Q5 nqm . e 2p i nt .

Ž 52 .

ns0

For large values of Q1 Q5 we can use the asymptotic form of dŽ Q1 Q5 n q m.

ž (

d Ž Q1 Q5 n q m . ;exp 2p Q1 Q5 n q m .

Ž 53 .

/

Substituting the above value of dŽ Q1 Q5 n q m. in P Ž m,t . we obtain a sum which has an integral representation as follows: `

P Ž m,t . s

Ý e 2p'Q Q nqm q2p i nt q d Ž m . 1

5

ns1

i sP

2

`

Hy`

dwcothpv e 2 p 'iQ 1 Q 5 v qm y2 p vt q d Ž m . y

e 2 p 'm 2

,

Ž 54 .

where P denotes ‘‘principal value’’ of the integral. We are interested in the high temperature limit of the partition function. The leading contribution to the integral in the limit t ™ 0 is P Ž m,t . ; ip Q1 Q5rt e ip Q 1 Q 5 r2 tyi2 p mt r Q 1 Q 5 .

(

Ž 55 .

Substituting the above value of P Ž m,t . the partition function becomes `

(

Z s ip Q1 Q5rt

Ý d Ž m . ey2 p i mt r Q Q ; exp Ž ip Q1 Q5 Ž 1r2t y 1r2t . . . Ž 56. 1

ms0

5

J.R. DaÕid et al.r Nuclear Physics B 564 (2000) 128–141

140

Thus the free energy at high temperatures is given by yln Z s

yip Q1 Q5 2

ž

1

1 y

t

t

/

.

Ž 57 .

This exactly agrees with Ž39. with the identification t s i bqrŽ2p l .. CFT partition function corresponding to conical spaces We will focus on the low temperature Žlarge b . behaviour. At low temperature, using t s i brŽ2p l . Ž b real in this case., the partition function is given by Z ' Tr exp 2p it L 0 y 2p it L0 . s exp yb Ž E0 q E0 .

Ž 1 q O Ž exp wybD x . . , Ž 58 .

where E0 , E0 represent the ground state values of L 0 , L 0 , and D represents the excitation energy of the first excited level. According to our proposal, the boundary CFT for conical spaces corresponds to the Hilbert space of spectral flow satisfying the relation Ž21.. By using Ž25. and Ž26., we find that yln Z s

b Ž E0 q E0 . l

b sbMs

8GNŽ3.

Ž yg . ,

Ž 59 .

which agrees exactly with Ž44. Žnote that for the zero-mass BTZ solution the above expression vanishes..

6. Concluding remarks We have argued that point mass geometries in three-dimensional quantum gravity with negative cosmological constant are part of the phenomenon of AdSrCFT correspondence. The boundary CFT corresponds to spectral flow with the spectral flow parameter identified appropriately with the point mass, or equivalently ŽEq. Ž21.. with the defect angle of the conical singularity. The above identification also leads to a correspondence between vertex operators of the boundary CFT with states in the bulk that corresponds to the point mass, thus leading to a possible description of scattering of such point masses in terms of CFT correlations. In this context it would be very useful to connect with the work of w5,6x where multi-point-mass classical solutions were discussed. In particular, the spatial geometry of such solutions were obtained by quotienting two-dimensional hyperbolic space by appropriate sub-groups of SLŽ2, R .. For example, quotienting by sub-groups that generate elliptic isometries gives rise to point particles, while those with hyperbolic isometries lead to black holes. The interesting thing about these multi-body solutions is that they are generally not static, and therefore can be used to study black hole formation by collision of point particles, or more complicated processes like collisions of black holes w5x. In the light of the AdSrCFT correspondence, it would be interesting to know how such processes manifest themselves in the CFT picture. More precisely, what effect does quotienting the

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spatial slice have on the CFT? We have found a hint to this answer in this paper, for the simple case of a single point particle: it corresponds to a certain vertex operator in the SCFT Žrepresentation of a point mass in the bulk as vertex operator in a boundary Liouville theory has been discussed recently in w24x.. The understanding of more general quotients holds the promise of providing insights into some very interesting processes in Ž2 q 1.-dimensional gravity. Another interesting point to emerge in this paper is that the partition function of the dual CFT agrees exactly with that of the supergravity at high temperatures. This is remarkable as the partition function of the CFT was computed in weak coupling and the supergravity result is expected to be a prediction for the CFT result at strong coupling. Note that this agreement works for arbitrary angular momentum, in particular for J s 0 which is far from extremality and hence is far from the BPS limit.

Acknowledgements We would like to thank P. Townsend for useful discussions. S.W. would like to thank the ASICTP for their warm hospitality.

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x

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