Strings on AdS wormholes

Strings on AdS wormholes

Mir Ali,1,* Frenny Ruiz,1,† Carlos Saint-Victor,1,‡ and Justin F. Vazquez-Poritz1,2,x1Physics Department, New York City College of Technology, The City University of New York 300 Jay Street,

Brooklyn, New York 11201, USA2The Graduate School and University Center, The City University of New York 365 Fifth Avenue, New York, New York 10016, USA

(Received 28 May 2009; published 10 August 2009)

We consider the behavior of open strings on anti-de Sitter wormholes in Gauss-Bonnet theory, which

are the Gauss-Bonnet gravity duals of a pair of field theories. A string with both endpoints on the same

side of the wormhole describes two charges within the same field theory, which exhibit Coulomb

interaction for small separation. On the other hand, a string extending through the wormhole describes

two charges which live in different field theories, and they exhibit a springlike confining potential. A

transition occurs when there is a pair of charges present within each field theory: for small separation each

pair of charges exhibits Coulomb interaction, while for large separation the charges in the different field

theories pair up and exhibit confinement. Two steadily-moving charges in different field theories can

occupy the same location provided that their speed is less than a critical speed, which also plays the role of

a subluminal speed limit. However, for some wormhole backgrounds, charges moving at the critical speed

cannot occupy the same location and energy is transferred from the leading charge to the lagging one. We

also show that strings on anti-de Sitter wormholes in supergravity theories without higher-derivative

curvature terms can exhibit these properties as well.

DOI: 10.1103/PhysRevD.80.046002 PACS numbers: 11.25.Tq, 04.50.Gh, 04.65.+e

I. INTRODUCTION AND SUMMARY

The AdS/CFT correspondence [1–3] can be used tostudy certain strongly-coupled gauge theories. In particu-lar, the behavior of open strings can be related to that ofparticles in the field theory. The original correspondencewas between type IIB string theory onAdS5 � S5 and four-dimensional maximally supersymmetric N ¼ 4 SUðNÞYang-Mills theory [1]. In the large N and large ’t Hooftcoupling � limit, the string theory can be approximated byclassical supergravity. A large number of generalizations tothe original AdS/CFT correspondence have been made [4].For instance, string theory on an asymptotically anti-deSitter (AdS) background corresponds to a field theoryundergoing a renormalization group flow from a conformalfixed point in the ultraviolet regime.

A quantum theory of gravity such as string theory gen-erally contains higher-derivative corrections from stringyor quantum effects, which correspond to 1=� or 1=Ncorrections in the field theory. However, not much isknown about the precise forms of the higher-derivativecorrections, other than for a few maximally supersymmet-ric cases. The most general theory of gravity in higherdimensions that leads to second-order field equations forthe metric is described by the Lovelock action [5]. Thesimplest such higher-derivative theory of gravity is knownas the Einstein-Gauss-Bonnet theory, which only containsterms up to quadratic order in the curvature.

Our primary motivation for considering the Einstein-Gauss-Bonnet theory is that it contains Lorentzian-signature five-dimensional asymptotically locally AdSwormhole solutions which do not violate the weak energycondition, so long as the Gauss-Bonnet coupling constantis negative and bounded according to the shape of thesolution [6,7]. In particular, we consider the static worm-hole solutions that were found in [8,9] which connect twoasymptotically AdS spacetimes each with a geometry atthe boundary that is locally R�H3 or R� S1 �H2,where H2 and H3 are two- and three-dimensional (quo-tiented) hyperbolic spaces, respectively. These back-grounds do not contain horizons anywhere, and the twoasymptotic regions are causally connected. The proof thatthe disconnected boundaries must be separated by blackhole horizons assumes that the Einstein equations hold[10], and therefore does not automatically carry over tohigher-derivative gravity theories such as the Einstein-Gauss-Bonnet theory.If one is able to apply the standard AdS/CFT correspon-

dence to these AdS wormholes, then string theory on sucha background corresponds to two quantum field theories(assuming that each boundary admits a well-defined fieldtheory). Each theory undergoes a renormalization groupflow from a conformal fixed point in its UV limit. However,we would like to emphasize that the AdS/CFT correspon-dence has not actually been tested for gravitational back-grounds of the Einstein-Gauss-Bonnet theory and may notbe valid. We are taking some liberty in applying thestandard AdS/CFT correspondence in this context, sincethe duality is between a gauge theory and a ten-dimensional string theory, whereas these five-dimensional

*[email protected]†[email protected]‡[email protected]@citytech.cuny.edu

PHYSICAL REVIEW D 80, 046002 (2009)

1550-7998=2009=80(4)=046002(13) 046002-1 � 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.80.046002

backgrounds have not been embedded within string theory.We are making the working assumption that the low-energy effective five-dimensional description of gaugetheory/string theory duality has a sensible derivative ex-pansion in which the higher curvature terms are systemati-cally suppressed by powers of the Planck length. Aftermaking the appropriate field redefinitions, the curvature-squared terms in the action appear in the Gauss-Bonnetcombination. Thus, within the limitations of the derivativeexpansion, one can use the AdS/CFT correspondence todetermine the properties of gauge theories dual to Einstein-Gauss-Bonnet gravitational backgrounds [11].

In fact, one might expect that within the vast stringlandscape there are higher-derivative corrections whichlead to similar backgrounds with asymptotically AdS re-gions with multiple disconnected boundaries. Concreteexamples of such backgrounds already exist within stringtheory, including a multiboundary orbifold of AdS3 [12]and the Maoz-Maldacena wormhole [13]. Although theorbifold of AdS3 clearly has lower dimensionality whilethe Maoz-Maldacena wormhole has a Euclidean signature,these backgrounds serve as toy models with which todemonstrate that open strings apparently exhibit somefairly universal behavior on AdS wormhole-type back-grounds. This indicates that our findings regarding thebehavior of open strings on AdS wormholes in theEinstein-Gauss-Bonnet theory could also apply to similarbackgrounds which may arise in string theory.

In this paper, we consider the behavior of open strings onthese AdS wormholes. The string endpoints lie on probeD-branes and correspond to charges within one or the otherof the field theories. We refer to these charges as type 1 andtype 2 charges. If both of the string endpoints lie on thesame D-brane and are both located on the same side of thewormhole, then this corresponds to a type 1 or a type 2charge-anticharge pair. On the other hand, if a string isextended through the neck of the wormhole such that itsendpoints lie on opposite sides, then this corresponds toone of the charges being type 1 while the other is type 2.

Summary. The expectation value of a rectangular Wilsonloop can be computed from the proper area of the stringworldsheet [14,15], from which the energy of a pair ofcharges can be read off. Applying this prescription to AdSwormholes, one can recover the result that a pair of chargesof the same type exhibit Coulombic attraction in the UVlimit of the field theory being probed by a string which isfar from the neck of the wormhole. We generalize thisprescription for the case of two charges which do not livewithin the same field theory. Even though the string end-points lie on opposite sides of the wormhole, there is arectangular contour within the physical spacetime whosehorizontal sides run along the distance between the twocharges and whose vertical sides run along time. We findthat two charges of different types exhibit confinementwith a springlike potential. Thus, the strength of the inter-

action can be parametrized in terms of an effective forceconstant. A pair of heavy charges exhibits a weaker inter-action than do lighter charges. We also find that thesecharacteristics are shared for open strings on a multiboun-dary orbifold of AdS3 [12] as well as on the Maoz-Maldacena wormhole [13]. For a family of AdS wormholesin the Einstein-Gauss-Bonnet theory [8,9], the effectiveforce constant can be tuned by adjusting a parameterassociated with the apparent mass on each side of thewormhole.We find that there is a rather curious transition which

involves a quadruplet of charges that consists of a charge-anticharge pair within each field theory. For small charge-anticharge separations, each pair exhibits Coulomb inter-action and does not interact with the pair of charges in theother field theory. However, for large separation, it is thecharges of different types that interact and exhibit confine-ment whereas the pairs of charges of the same type nolonger interact with each other. In this sense, each pair ofcharges of the same type exhibits the feature of effectivelyhaving a screening length, even though both field theoriesare at zero temperature.We also consider the behavior of steadily-moving

strings. In particular, demanding that the string configura-tion is timelike imposes constraints on the mass parametersof the corresponding charges, as well as on their separa-tion. We find that the charges have an upper bound on theirspeed which depends on the mass parameter of one of thecharges (which charge depends on a parameter associatedwith the wormhole geometry). This speed limit is generallyless than the speed of light, which is a result of the fact thatthe proper velocity of the string endpoints is greater thanthe physical velocity in the field theory [16].In addition, we consider the behavior of strings on an

Einstein-Gauss-Bonnet wormhole which connects anasymptotically locally AdS spacetime with another non-trivial smooth spacetime at the other asymptotic region.Even though one of the asymptotic regions of the Einstein-Gauss-Bonnet wormhole is not AdS, we find that openstrings on this background share many of the same char-acteristics as open strings on a multiboundary orbifold ofAdS3 [12]. This may be an indication that some aspects ofthe behavior of open strings on wormholes are fairlyuniversal, regardless of the asymptotic geometries.Namely, we find that a pair of steadily-moving charges

of different types can occupy the same location with notransfer of energy or momentum between them, providedthat their speed is less than a certain critical speed. Thiscritical speed coincides with the speed limit of the chargesand monotonically decreases with the mass parameters. Ifthe charges move at this critical speed, then they exhibit aseparation gap which increases with the mass parameter ofthe lagging charge but is not so sensitive to the massparameter of the leading charge. Also at this critical speed,energy and momentum are transferred from the leading

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charge to the lagging one. Note that the procedure forcalculating the rate of energy and momentum transferparallels the way in which the rate of energy loss of aquark moving through a strongly-coupled plasma wascalculated via a dragging string moving on a five-dimensional AdS black hole background [17–19].Curvature-squared corrections of the background wereconsidered in [20–22]. However, the case of an AdS blackhole corresponds to energy being transferred from thequark to the plasma, whereas in the present case of anAdS wormhole the energy is transferred between twodifferent types of charges. We find that the rate at whichenergy is transferred actually decreases with the massparameter of the lagging charge. While this is not neces-sarily an unexpected feature of a wormhole background, itis a rather surprising characteristic on the field theory side.

Wormholes of various dimensionality which connectAdSn � Sm in one asymptotic region to flat spacetime inthe other have been constructed in Einstein gravity [23,24].It would be interesting to see if open strings on thesewormholes have similar features to what we have foundin this paper, even though the notion of time tends to bedifferent for the two asymptotic regions.

This paper is organized as follows. In Sec. II, we providegeneral formulas for the AdS wormhole metric, stringembeddings, and equations of motion. In Sec. III, weanalyze the behavior of static string configurations on afamily of AdS wormholes that arise in the Einstein-Gauss-Bonnet theory. In Sec. IV, we consider steadily-movingstring configurations on these AdS wormhole backgrounds.In Sec. V, we consider strings on an Einstein-Gauss-Bonnetwormhole for which only one asymptotic region is AdS.Lastly, in Sec. VI, we consider strings on AdS wormholesthat can be embedded within string theory.

II. GENERALITIES

A. Wormholes in five-dimensional Einstein-Gauss-Bonnet theory

We are making the assumption that the low-energyeffective five-dimensional description of gauge theory/string theory duality has a sensible derivative expansion[11]. After field redefinitions, the leading terms in theaction take on the Einstein-Gauss-Bonnet form:

I ¼ �Z

d5xffiffiffiffiffiffiffi�g

p �12

‘2þ R

þ �‘2ðR2 � 4R��R�� þ R��R

��Þ þ � � ��; (2.1)

where � is related to the Newton constant, ‘ corresponds tothe AdS curvature scale at leading order, and � is thedimensionless Gauss-Bonnet coupling. In order to makethe above derivative expansion, we require that� � 1. Fora fixed value of �, there is a family of wormhole solutionsto (2.1) which is described by the metric [8,9]

ds2 ¼ ‘2ð�cosh2ð� 0Þdt2 þ d2 þ cosh2d�23Þ:(2.2)

The base manifold �3 can be either H3=� or S1 �H2=�,where H3 and H2 are hyperbolic spaces and � is a freelyacting discrete subgroup of Oð3; 1Þ and Oð2; 1Þ, respec-tively. The hyperbolic part of the base manifold must bequotiented so that this geometry describes a wormholerather than a gravitational soliton with a single conformalboundary. The neck of the wormhole is located at ¼ 0,which connects two asymptotically locally AdS regions at ! �1. The geometry is devoid of horizons and radialnull geodesics connect the two asymptotic regions in afinite time �t ¼ . These wormholes can evade the proofthat the disconnected boundaries must be separated byblack hole horizons [10], since we are dealing with ahigher-derivative theory of gravity. No energy conditionsare violated by these wormholes, since the stress-energytensor vanishes everywhere [8,9]. The stability of thesewormholes against scalar field perturbations has been dis-cussed in [25].It has been shown that gravity pulls towards a fixed

hypersurface at ¼ 0, which lies in parallel with theneck of the wormhole [8,9]. In particular, timelike geo-desics are confined, and oscillate about this hypersurface.Although the wormhole is massless, for nonzero 0, themass of the wormhole appears to be positive for observerslocated on one side and negative for the other. The parame-ter 0 is related to the apparent mass on each side of thewormhole; for 0 ¼ 0, the wormhole exhibits reflectionsymmetry.If we can apply the AdS/CFT correspondence to this

background, then this is the gravity dual of two interactingfield theories on R� �3. Except in the UV limit, theconformal symmetry of both theories is broken by thelength scale associated with �3 (H3 and H3 must have

the radius 1 and 1=ffiffiffi3

p, respectively) as well as by the

parameter 0. By computing the boundary stress tensors[26], one can determine how 0 affects the expectationvalue of the stress tensor of each field theory.

B. String embeddings and equations of motion

The dynamics of a classical string on a spacetime withthe metric G�� are governed by the Nambu-Goto action

S ¼ �T0

Zd�d�

ffiffiffiffiffiffiffi�gp

; (2.3)

where ð�; �Þ are the world sheet coordinates, gab is theinduced metric, and g ¼ detðgabÞ. For a map X�ð�;�Þfrom the world sheet into spacetime

� g ¼ ð _X � X0Þ2 � ðX0Þ2ð _XÞ2: (2.4)

where _X � @�X, X0 � @�X, and _X � X0 ¼ _X�ðX�Þ0G��.

We will consider a string which is localized at a point inH2 and moves along the S1 direction, which we label x.

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Choosing a static gauge where � ¼ and � ¼ t, we findthat the equation of motion for a string on the backgroundmetric (2.2) is given by

@

@

�cosh2ð� 0Þcosh2x0ffiffiffiffiffiffiffi�g

p�� cosh2

@

@t

�_xffiffiffiffiffiffiffi�g

p�¼ 0;

(2.5)

where

� g

‘4¼ cosh2ð� 0Þ þ cosh2ð� 0Þcosh2x02

� cosh2 _x2: (2.6)

It will also be useful to express the equation of motion inthe alternative gauge � ¼ x and � ¼ t, for which

@

@x

�cosh2ð� 0Þ0ffiffiffiffiffiffiffi�g

p�� cosh2

@

@t

�_ffiffiffiffiffiffiffi�g

p�

¼ cosh coshð� 0Þ sinhð2� 0Þ � cosh sinh _2

þ coshð� 0Þ sinhð� 0Þ02; (2.7)

where

� g

‘4¼ cosh2cosh2ð� 0Þ � cosh2 _2

þ cosh2ð� 0Þ02: (2.8)

The general expressions for the canonical momentumdensities of the string are given by

0� ¼ �T0G��

ð _X � X0ÞðX�Þ0 � ðX0Þ2ð _X�Þffiffiffiffiffiffiffi�gp ;

1� ¼ �T0G��

ð _X � X0Þð _X�Þ � ð _XÞ2ðX�Þ0ffiffiffiffiffiffiffi�gp :

(2.9)

The energy density and density of the x component ofmomentum on the string world sheet are given by 0

t and0

x, respectively. Thus, the total energy and momentum ofthe string are

E ¼ �Z

d�0t ; p ¼

Zd�0

x: (2.10)

We will also make use of the fact that the string tension is

given by T0 ¼ffiffiffiffi�

p=ð2‘2Þ, where � is the ’t Hooft cou-

pling of the field theory.

III. STATIC STRINGS

A. Straight strings

The simplest solution is a constant x ¼ x0, for which thestring is straight, as shown in Fig. 1. Suppose that theendpoints are located at ¼ �1 and ¼ 2, where wewill always take 1; 2 > 0. We will refer to the chargesassociated with the endpoints at ¼ �1 and ¼ 2 as atype 1 charge and a type 2 charge, respectively. Also, wewill refer to the radial locations of the string endpoints 1

and 2 as the mass parameters of the charges. TheD-branes (or at least the part that lies closest to the neckof the wormhole) are denoted by the surfaces at the stringendpoints and the surface at ¼ 0 represents the fixedhypersurface towards which gravity pulls. From (2.10), wefind that the static string has vanishing momentum, and itsenergy is given by

E ¼ T0‘2ðsinhð1 þ 0Þ þ sinhð2 � 0ÞÞ: (3.1)

From the equation of motion (2.7) expressed in thealternative gauge, we find that there is also a solution atconstant ¼ 0=2. However, since this would presumablycorrespond to an extended object in the field theory, wewill not consider this string configuration further.

B. Curved strings

Curved strings can either have both endpoints on thesame side of the wormhole or else on opposite sides, asshown in Fig. 2. We will first consider the first case, whichcorresponds to a pair of charges of the same type. Thismeans that the string dips down toward the wormhole andthen comes back up. Wewill denote the turning point of thestring by t, and L is the distance between the endpoints ofthe string. For simplicity, we will take the endpoints to be

ρ 0

ρ ρ0

ρ ρ1

ρ ρ2

Type 1 Charge

Type 2 Charge

FIG. 1 (color online). A static string stretching straight througha wormhole.

L

L

FIG. 2 (color online). A string with both endpoints on the sameside of the wormhole (left) and on opposite sides (right).


046002-4

at ¼ 2, so that we have a pair of type 2 charges. For 2,t � 1, the string is not sensitive to the presence of thewormhole and so it can be found that its energy goes as1=L. A Coulomb potential energy is consistent with thefact that the theory is conformal in the extreme UV regime.If the masses associated with the charges are decreased, or

if the distance between the charges is increased, then thestring dips closer to the wormhole. Then, terms in theenergy that are of higher order in L will become important.From (2.5) and (2.6), we find the distance L between the

string endpoints is given by

L ¼ 2 coshðt � 0Þ cosht

Z 2

t

d

coshffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosh2ð� 0Þcosh2� cosh2ðt � 0Þcosh2t

p : (3.2)

The above integration region assumes that 2 >t. As wewill see shortly, there are string solutions for which 2 <t, in which case the upper and lower integration boundsmust be interchanged.

For 2 >120, the reality of L implies that the turning

point is located in the interval 2 >t � 120. Similarly, a

string with endpoints at 2 <120 has a turning point in the

interval 2 <t 120. This means that if the turning

points are within the interval 0<2 <0=2, then thestring will bend away from the center of the wormhole.Moreover, if the endpoints are located on the opposite sideof the wormhole as the hypersurface ¼ 0, then there arestrings which go through the neck of the wormhole andhave a midsection on the opposite side of the wormhole asits endpoints. All of these possibilities are shown in Fig. 3.

As shown in Fig. 4 for various values of 0, the turningpoint t monotonically decreases with L. In particular, as Lgets large the turning point asymptotically approaches 1

20

from above. Note that, unlike the case of strings on an AdSblack hole background, there is a single string configura-tion for each value of L. From the field theory perspective,this means that the charges do not exhibit a screeninglength but rather remain interacting no matter how farthey are from each other (which is not the case in thepresence of additional pairs of charges, as we will discussshortly). In particular, as the distance between the chargesincreases, the intercharge potential becomes ever moresensitive to the IR characteristics of that sector of thetheory. For vanishing 0 (black curve in Fig. 4), the poten-tial is partly determined by the extreme IR region of the

theory for large L. However, for 0 > 0, the large-distancepotential only reflects the features of the energy rangecorresponding to the region � 1

20.

We will now consider a curved string with endpoints onopposite sides of the wormhole, which corresponds to atype 1 charge and a type 2 charge. From (2.5) and (2.6), thedistance between the string endpoints in the x direction is

L ¼ CZ 2

�1

d

coshffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosh2ð� 0Þcosh2� C2

p : (3.3)

ρ 0

ρ12

ρ0

ρ 0

ρ12

ρ0

FIG. 3 (color online). Strings bend towards ¼ 120 regardless of where the string endpoints are located, even if this means that the

string bends away from the neck of the wormhole (left) or goes through the neck of the wormhole (right).

2 1 1 2ρt

2

4

6

8L

FIG. 4 (color online). The distance between string endpoints Lversus the turning point t for the case in which both endpointsare on the same side of the wormhole. We have set 2 ¼ 2 and0 ¼ 2 , 0, �2, and �4 (from right to left).


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For small separation L, we can expand in C � 1 so that

L CZ 2

�1

d

coshð� 0Þcosh2: (3.4)

We will now Euclideanize the metric (2.2) in order tocompute the energy of the string configuration in terms ofits action. The action for the string Euclidean world sheet isgiven by

SE ¼ T0

Zd�d�

ffiffiffig

p: (3.5)

The energy of a pair of charges is E ¼ SE=�t, where wetake the time interval�t ! 1. Then for a static string (2.5)and (2.6) become

x0 ¼ Cffiffiffig

pcosh2ð� 0Þcosh2

;

g

‘4¼ cosh2ð� 0Þð1þ cosh2x02Þ:

(3.6)

We find the energy of the string to be

E ¼ T0‘2Z 2

�1

dcosh2ð� 0Þ coshffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cosh2ð� 0Þcosh2� C2p : (3.7)

In the small separation limit C � 1,

E ¼ Estraight þ 1

2T0‘

2C2Z 2

�1

d

coshð� 0Þcosh2;

(3.8)

where Estraight is the energy of a straight string given by

(3.1). Using (3.4) to express C in terms of L, we find

E ¼ Estraight þ 12kL

2; (3.9)

where

k ¼ffiffiffiffi�

p2

�Z 2

�1

d

coshð� 0Þcosh2��1

; (3.10)

and we have used T0 ¼ffiffiffiffi�

p=ð2‘2Þ. Thus, for small L, a

pair of charges of opposite types are confined and exhibit a

springlike potential, as opposed to a pair of charges of thesame type which exhibit a Coulomb potential. Since we areat zero temperature, there is no screening length. The leftplot in Fig. 5 shows that the effective force constant kmonotonically decreases with the mass parameters 1

and 2, where we have taken 0 ¼ 0:1. This means thatheavy charges are not as sensitive to the confining potentialas lighter charges. We also find that k monotonicallyincreases with the 0 parameter, as shown in the rightplot of Fig. 5 for 1 ¼ 2 ¼ 1.For two sets of charge-anticharge pairs, an interesting

transition can take place. For instance, suppose we have apair of interacting type 1 charges and a pair of interactingtype 2 charges, described by the two curved strings shownin the left plot of Fig. 6. For simplicity, we will assume thatboth pairs of charges are a distance L apart. For small L,there is no interaction between the type 1 pair and the type2 pair. However, as L increases, the strings bend closer toeach other. Note that these strings cannot pass each other,since neither one can pass the radius ¼ 0=2. However,even before either string hits this bound, the string con-figuration in the right plot of Fig. 6 becomes the energeti-cally favorable one with the same set of endpoints. Thus,for large L, each type 1 charge becomes coupled to a type 2charge. This is similar to a screening length, in that the

k

ρ1

ρ2

0.2 0.4 0.6 0.8 1ρ0

0.75

0.8

0.85

0.9

0.95

1

k

FIG. 5 (color online). The left plot shows the effective force constant k versus 1 and 2 for 0 ¼ 0:1. The right plot shows k versus0 for 1 ¼ 2 ¼ 1.

L

L

L

FIG. 6 (color online). There is a critical distance Lcrit associ-ated with two sets of charge-anticharge pairs. For L < Lcrit, theleft configuration is energetically favorable while, for L > Lcrit,the right configuration is the favored one.


046002-6

charges of the same type no longer interact with each other.However, since we are at zero temperature, this is really afeature of having multiple pairs of charges close to eachother.

In principle, there is no need for us to assume that eachpair of charges are separated by the same distance L.Suppose that L1 and L2 denote the distance between thetype 1 charges and the type 2 charges, respectively. Then,in general, there is a critical admixture of L1 and L2 forwhich the aforementioned transition takes place. The onlydifference is that, past the critical lengths, each string‘‘connecting’’ a type 1 charge to a type 2 charge is nowcurved, rather than straight, as shown in the right plot ofFig. 2. If the endpoints are not held in place by an externalforce, then each string will straighten out.

Note that this type of transition can occur even if thereare two pairs of charges that are all of the same type, suchas is shown in Fig. 3. This indicates that such processesmay not be restricted only to field theories described byAdS wormholes.

IV. STEADILY-MOVING STRINGS

A. Straight strings

A steadily-moving straight string is given by xðt; Þ ¼x0 þ vt. For this solution, we find that

� g

‘4¼ cosh2ð� 0Þ � v2cosh2: (4.1)

For simplicity, we will take 0 � 0 in the remainder of thepaper. Then for v < vcrit � e�0 , �g > 0 everywhere forv < 1, which implies that no parts of the string move fasterthan the local speed of light. This is to be contrasted with astraight string steadily moving in an AdS black hole back-ground [17,19] which, for any nonzero velocity, has aregion with �g < 0 and is therefore not a physical solu-tion. On the other hand, for v ¼ vcrit, we require that 2 <0=2. For v > vcrit, g ¼ 0 at ¼ crit where

crit ¼ ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie0 � v

v� e�0

s; (4.2)

at which point the induced metric on the string world sheetis degenerate. We can still have steadily-moving straightstrings which lie entirely within the region crit < .However, for strings which have a region that lies within < crit, �g < 0 and the action, energy, and momentumare complex. This is a signal that this part of the stringtravels faster than the local speed of light and must bediscarded. In the next section, we will consider curvedstrings with v > vcrit which satisfy �g > 0.

For the special case of 0 ¼ 0, we have

� g

‘4¼ ð1� v2Þcosh2 > 0; (4.3)

for all v < 1. In other words, for this case the critical speed

vcrit ¼ 1 for which g ¼ 0 and the induced metric on thestring world sheet is degenerate.The energy and momentum of the steadily-moving

string are given by the integrals

E ¼ T0‘2Z 2

�1

dcosh2ð� 0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cosh2ð� 0Þ � v2cosh2p ;

p ¼ T0‘2v

Z 2

�1

dcosh2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cosh2ð� 0Þ � v2cosh2p :

(4.4)

For the special case of 0 ¼ 0, we find that

E ¼ Erestffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v2

p ; Erest ¼ T0‘2ðsinh1 þ sinh2Þ:

(4.5)

For nonvanishing 0, the expressions for E and p cannot beintegrated in closed form. Note that, since 1

t and 1x

vanish, there is no energy or momentum current flowingalong the string.We will now comment on the speed limit of these

comoving charges, which stems from the fact that theproper velocity V of the string endpoints differs from thephysical velocity v in the four-dimensional field theory[16]. From the metric (2.2), we see that

V ¼ coshi

coshði � 0Þv; (4.6)

where i ¼ �1 or 2. In order to avoid a spacelike stringworld sheet, we must have V 1. This corresponds to

v vmax ¼ coshð2 � 0Þcosh2

; (4.7)

where we have taken i ¼ 2, since 2 is closer to 0 � 0than is �1. As is the case for steadily-moving strings inAdS black hole backgrounds, the speed limit depends onthe mass parameter associated to the charge. This samespeed limit applies to the curved strings which we nowdiscuss. We see that vmax ¼ vcrit ¼ 1 for 0 ¼ 0. On theother hand, for 0 > 0, we find that vmax < vcrit < 1.

B. Curved strings

We will now consider steadily-moving curved strings,which are described by solutions of the form xðt; Þ ¼xðÞ þ vt. The term with the time derivative in (2.5)vanishes and we are left with

@

@

�cosh2ð� 0Þcosh2x0ffiffiffiffiffiffiffi�g

p�¼ 0: (4.8)

where

� g

‘4¼ cosh2ð� 0Þ þ cosh2ð� 0Þcosh2x02

� v2cosh2: (4.9)

The first integral of (4.8) is


046002-7

cosh2ð� 0Þcosh2x0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�g=‘4p ¼ C: (4.10)

where C is an integration constant. Thus,

x02 ¼ C2

cosh2ð� 0Þcosh2

��cosh2ð� 0Þ � v2cosh2

cosh2ð� 0Þcosh2� C2

�: (4.11)

Solving for �g gives

� g

‘4¼ cosh2ð� 0Þ

� cosh2

�cosh2ð� 0Þ � v2cosh2


�: (4.12)

For v vcrit (with the exception of 0 ¼ 0 and v ¼ 1)then �g > 0 everywhere along the string provided thatjCj<Ccrit where

Ccrit ¼ 12ðcoshð2min½2; 0=2� � 0Þ þ cosh0Þ: (4.13)

In the limit that 0 ¼ 0, solutions exist with jCj< 1 for allv < 1. However, for 0 ¼ 0 and v ¼ 1, the induced metricis degenerate everywhere, regardless of the value of C.Strings with v > vcrit can still have �g > 0 everywhereprovided that 2 < crit and jCj<Ccrit, where crit andCcrit are given by (4.2) and (4.13), respectively. From thefield theory perspective, this corresponds to an upperbound on the mass of the type 2 charge. For v > vcrit,one can also have a string that lives entirely in > crit,provided that jCj>Ccrit. This corresponds to a light-heavypair of type 2 charges with a lower bound on their masses.

The distance between the charges is given by

L ¼ CZ 2

�1

d

coshð� 0Þ cosh

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosh2ð� 0Þ � v2cosh2


s; (4.14)

For cases in which we require that jCj<Ccrit, there is anupper bound on L. However, if the endpoints of the stringare allowed to move freely, then the string will tend tostraighten out to the C ¼ 0 configuration so that it mini-mizes its energy. On the other hand, for the string corre-sponding to a light-heavy pair of type 2 charges movingwith v > vcrit, we must have jCj>Ccrit and there is acorresponding lower bound on L. This string cannotstraighten out without decreasing its speed.

From (2.9), we find that the rate at which energy andmomentum flow through the string can be written in termsof the constant C as

1t ¼

ffiffiffiffi�

p2

Cv; �1x ¼

ffiffiffiffi�

p2

C; (4.15)

where we have used T0 ¼ffiffiffiffi�

p=ð2‘2Þ. Thus, for nonzero

C, energy and momentum are transferred from the leadingcharge to the lagging one. If there is no external forcepresent, then the leading charge presumably slows downand the lagging charge speeds up until an equilibrium stateis reached corresponding to a steadily-moving straightstring.

V. STRINGS ON ANOTHER EINSTEIN-GAUSS-BONNET WORMHOLE

Thus far in this paper, we have discussed strings movingon the Einstein-Gauss-Bonnet wormholes which have twoasymptotically locally AdS regions. We now turn to an-other wormhole solution which has the metric [9]

ds2 ¼ ‘2ð�e2dt2 þ d2 þ cosh2d�23Þ: (5.1)

The geometry is locally AdS for ! þ1. The asymptoticgeometry for ! �1 is also smooth and can be roughlythought of as a five-dimensional Lorentzian analog of theSol manifold [27].As before, we will work in the static gauge ð�; �Þ ¼

ð; tÞ. The Nambu-Goto action on this background givesrise to the equation of motion

@

@

�e2cosh2x0ffiffiffiffiffiffiffi�g

p�� cosh2

@

@t

�_xffiffiffiffiffiffiffi�g

p�¼ 0; (5.2)

where

� g

‘4¼ e2 þ e2cosh2x02 � cosh2 _x2: (5.3)

A straight static string extending from ¼ 2 to ¼�1 has an energy

E ¼ T0‘2ðe2 � e�1Þ: (5.4)

Expressing the equation of motion in the alternative gaugeð�; �Þ ¼ ðx; tÞ, it can be shown that there are also straightstring solutions at constant ¼ 0.We will now discuss some properties of static curved

strings on this background. As was the case with theprevious class of wormholes, curved strings can haveboth endpoints on the same side of this wormhole or onopposite sides. In the former case, a string that is far fromthe neck of the wormhole has an energy E� 1=L, sincethis is the UV conformal regime of the theory. In this case,the pair of charges of the same type exhibit a Coulombpotential. As the string dips closer to the wormhole, IReffects show up as terms in the energy that are of higherorder in L. There is a one-to-one correspondence betweenL and the location of the turning point t. Thus, the chargesdo not exhibit a screening length.For a curved string with endpoints on opposite sides of

the wormhole,

L ¼ CZ 2

�1

d

coshffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2cosh2� C2

p : (5.5)


046002-8

For small separation L, this can be expanded in C � 1 sothat

L C

�sech2 � sech1 þ 2 arctan

�e1þ2 � 1

e1 þ e2

��:

(5.6)

Euclideanizing the metric in order to compute the energyof the string configuration in terms of its action, we find theenergy of a pair of charges to be

E ¼ T0‘2Z 2

�1

de2 coshffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e2cosh2� C2p : (5.7)

Expanding E in the small separation limitC � 1 and using(5.6) to express C in terms of L, we find

E ¼ Estraight þ 12kL

2; (5.8)


(5.4) and

k ¼ffiffiffiffi�

p2

�sech2 � sech1 þ 2 arctan

�e1þ2 � 1

e1 þ e2

��1:

(5.9)

Thus, for small L, a pair of charges of opposite types areconfined and exhibit a springlike potential. It can be shownthat k decreases monotonically with the mass parameters,and therefore heavy charges are less sensitive to the con-fining potential.

We will now consider steadily-moving straight stringswhich, from (5.3) with _x ¼ v and x0 ¼ 0, we find can existin the region > crit, where

crit ¼ ln

ffiffiffiffiffiffiffiffiffiffiffiffiv

2� v

r: (5.10)

For v ! 0, crit ! �1 while for v ! 1, crit ! 0. Forfixed 1, we can have steadily-moving straight strings onlyfor crit <�1, which corresponds to v < vcrit, where

vcrit ¼ 2

1þ e21: (5.11)

Interestingly enough, the critical velocity depends on themass parameter of the lagging charge, which in this case isthe type 1 charge. If 1 ¼ 0 then vcrit ¼ 1, and vcrit mono-tonically decreases with the mass parameter for 1 � 0.The energy and momentum of such a string are given by

E ¼ T0‘2Z 2

�1

de2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e2 � v2cosh2p ;

p ¼ T0‘2v

Z 2

�1

dcosh2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e2 � v2cosh2p :

(5.12)

We will now consider steadily-moving curved strings.For xðt; Þ ¼ xðÞ þ vt, we get

x02 ¼ C2

e2cosh2

�e2 � v2cosh2

e2cosh2� C2

�: (5.13)

Solving for �g yields

� g

‘4¼ e2cosh2

�e2 � v2cosh2

e2cosh2� C2

�: (5.14)

Strings can exist with �1 > crit, where crit is given by(5.10), provided that C< Ccrit, where

Ccrit ¼ 12ðe�21 þ 1Þ: (5.15)

Thus, along with the upper bound on the mass parameterassociated with the type 1 charge, there is an upper boundon the parameterC. This latter restriction corresponds to anupper bound on the distance between the charges

L < Ccrit

Z 2

�1

d

e cosh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2 � v2cosh2

e2cosh2� C2crit

s; (5.16)

along with the following upper bounds on the rate ofenergy and momentum flow between the charges:

1t <

ffiffiffiffi�

pv

4ðe�21 þ 1Þ; �1

x <

ffiffiffiffi�

p4

ðe�21 þ 1Þ:(5.17)

There can also be strings in the region < crit, whichcorrespond to light-heavy pairs of type 1 charges. In thiscase, crit corresponds to a lower bound on the associatedmass parameters, and jCj>Ccrit corresponds to a lowerbound on distance between the charges as well as the rateof energy and momentum transfer.Since the numerator and denominator in the expression

(5.14) are both positive for large and positive , it ispossible for �g > 0 even for strings passing through thecritical point crit, provided that the numerator and de-nominator both change signs there. This corresponds tothe case in which v � vcrit. Then no restrictions are placedon the mass parameters but we must have

C ¼ 1

2� v: (5.18)

This specifies the distance between the charges to be

L ¼Z 2

�1

d

e cosh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffive�2 þ vþ 2

ð2� vÞe2 þ 4� v

s: (5.19)

We have confirmed numerically that L increases mono-tonically with v for v � vcrit. The rate of the energy andmomentum flow along the string is given by

1t ¼

ffiffiffiffi�

pv

2ð2� vÞ � 1x ¼

ffiffiffiffi�

p2ð2� vÞ : (5.20)

From the field theory perspective, we have the followingpicture. There is no interaction between two coincident


046002-9

charges of opposite types comoving with a speed of v <vcrit. However, for v ¼ vcrit, the charges exhibit a separa-tion gap, which increases with larger v. This is shown inFig. 7 for 1 ¼ 2 ¼ 1 (bottom line) for which vcrit 0:0238, and 1 ¼ 1:5 and 2 ¼ 1 (top line) for whichvcrit 0:095. For a given v above the critical value, theseparation gap L increases with 1 and is less sensitive tothe value of 2. Also, for v < vcrit, no energy or momen-tum is transferred between the charges. However, for v ¼vcrit,

1t ¼

ffiffiffiffi�

p2

e�21 ; �1x ¼

ffiffiffiffi�

p4

ðe�21 þ 1Þ; (5.21)

which both decrease with the mass parameter of the lag-ging charge. This is certainly unexpected from the fieldtheory side. However, on the gravity theory side, this arisesas a peculiar property of the wormhole background.

As in the previous section, in order to avoid a spacelikestring, we must have the proper velocity V 1. From the

metric (5.1), this translates into v vcrit. Thus, eventhough �g > 0 for strings which travel at v > vcrit for anappropriate value of C, it turns out that such strings wouldstill be spacelike. Perhaps if a string attempts to movefaster than this speed limit, then the increased amount ofenergy and momentum transferred along the string, asshown in Fig. 7, would immediately cause it to slowdown. However, it would appear that it is possible forstrings to travel at the critical speed itself, so long as thereis a constant input of energy and momentum. The situ-ations for v < vcrit and v ¼ vcrit are depicted in Fig. 8.

VI. STRINGS ON ADS WORMHOLES INSUPERGRAVITY

Thus far, we have considered AdS wormholes which aresolutions to the Einstein-Gauss-Bonnet theory. One couldask which properties of strings on such backgrounds mightsimply be a reflection of the specific higher-derivativecurvature terms in the theory. We will now consider acouple of examples of AdS wormholes which can beembedded within string theory. The similar behavior ofstrings for these cases suggests that strings on AdS worm-holes share some universal features.

A. A multiboundary orbifold of AdS3

We will first consider the behavior of strings on a multi-boundary orbifold of AdS3 [12]. The metric is given by

ds23 ¼ ‘2ð�dt2 þ dx2 � 2 sinhð2Þdtdxþ d2Þ: (6.1)

In the static gauge ð�; �Þ ¼ ð; tÞ, the Nambu-Goto actionon this background gives the equation of motion

@

@

�cosh2ð2Þx0ffiffiffiffiffiffiffi�g

p�� @

@t

�_x� sinhð2Þffiffiffiffiffiffiffi�g

p�¼ 0; (6.2)

where

� g

‘4¼ 1� _x2 þ 2 sinhð2Þ _xþ x02cosh2ð2Þ: (6.3)

From (2.9) and (2.10), we find that the energy of astraight string extended from ¼ �1 to 2 is

E ¼ T0‘2ð1 þ 2Þ: (6.4)

If one expresses the equation of motion in the gaugeð�; �Þ ¼ ðx; tÞ, then it can be shown that there are alsostraight strings lying along constant for any value of .We will now consider a string with both endpoints on the

same side of the wormhole at ¼ 2. Since the geometryis asymptotically locally AdS, the endpoints of a string thatis far from the wormhole exhibit Coulomb behavior. Thedistance L between the endpoints is given by

L ¼ 2 coshð2tÞZ 2

t

d

coshð2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosh2ð2Þ � cosh2ð2tÞ

p ;

(6.5)

0.2 0.4 0.6 0.8 1v

1

2

3

4

L

FIG. 7 (color online). The separation gap L as a function ofvelocity v for 1 ¼ 2 ¼ 1 (bottom line) and 1 ¼ 1:5, 2 ¼ 1(top line).

v vcrit v vcrit

FIG. 8 (color online). A steadily-moving string can stretchstraight through the wormhole for v < vcrit, as shown in theleft plot. However, a steadily-moving string with v ¼ vcrit mustcurve in order to be physically acceptable, as is shown in theright plot. Strings with v > vcrit are spacelike.


046002-10

where the turning point t can lie within the region 2 >t � 0. Thus, all strings bend towards the wormhole andnone of them pass through the neck of the wormhole.

For a string with its endpoints on opposite sides of thewormhole, expanding in C for small L as in previoussections, we find

Estraight þ 12kL

2; (6.6)


(6.4) and

k ¼ffiffiffiffi�

p½tanhð1

2 Þ þ tanhð2

2 Þ�: (6.7)

While k decreases monotonically with the mass parame-ters, it asymptotically approaches the minimum value

kmin ¼ffiffiffiffi�

p2

; (6.8)

as the i ! 1.As in the previous examples, when there is a pair of type

1 charges and a pair of type 2 charges there can be atransition. Namely, the charges of the same type exhibitCoulomb interaction for small L and the opposite typecharges do not interact, whereas for large L the oppositetype charges are confined and the charges of the same typeare effectively screened.

We now turn to steadily moving straight strings, forwhich

� g

‘4¼ 1� v2 þ 2v sinhð2Þ: (6.9)

In order for�g > 0 everywhere along the string, it must liein the region bounded by > crit, where

crit ¼ 12 lnv: (6.10)

For v ! 0, crit ! �1 while for v ! 1, crit ! 0. Forfixed 1, we can have steadily moving straight strings onlyfor crit <�1, which corresponds to

v < vcrit ¼ e�21 : (6.11)

The critical velocity depends on the mass parameter asso-ciated with the lagging charge, which in this case is thetype 1 charge. If 1 ¼ 0, then vcrit ¼ 1. For fixed ’t Hooftcoupling, vcrit decreases with the mass parameter andvanishes for infinite 1.

We will now consider steadily moving curved strings.For xðt; Þ ¼ xðÞ þ vt, we get

x02 ¼ C2ð1� v2 þ 2v sinhð2ÞÞcosh2ð2Þðcosh2ð2Þ � C2Þ ; (6.12)

with

� g

‘4¼ cosh2ð2Þð1� v2 þ 2v sinhð2ÞÞ

cosh2ð2Þ � C2: (6.13)

Strings can exist with �1 > crit, where crit is given by(6.10), provided that C2 < 1. This bound on C results in

L <

�v

2arctan

��

v� sinhð2Þ�� 1 arctanð��Þ

þ 1

4ln

�v2 þ ð�þ 1Þ2v2 þ ð�� 1Þ2

��2

�1

; (6.14)

where

� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi��2 þ 2v sinhð2Þ

q; � ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� v2p ; (6.15)

along with the following bounds on the rate of energy andmomentum flow between the charges:

1t <

ffiffiffiffi�

pv

2; �1

x <

ffiffiffiffi�

p2

: (6.16)

There can also be strings in the region < crit, whichcorrespond to light-heavy pairs of type 1 charges. Thenthere is a lower bound on the associated mass parameters.Also, C2 > 1, which corresponds to a lower bound on L aswell as the rate of energy and momentum transfer.It is possible for �g > 0 even for strings with v � vcrit

which pass through the critical radius crit so long as thenumerator and denominator of�g both change signs there.While there are now no restrictions on the mass parame-ters, we must have

C ¼ 1þ v2

2v: (6.17)

This specifies the distance between the charges to be

L ¼ Im

�1

iþ varctanh

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ 2v sinhð2Þ � 1

piþ v

��2

�1

:

(6.18)

The rate of energy and momentum flow from the leadingcharge to the lagging one is given by

1t ¼

ffiffiffiffi�

p4

ð1þ v2Þ; �1x ¼

ffiffiffiffi�

p4v

ð1þ v2Þ: (6.19)

For v < vcrit, L ¼ 0. However, for v ¼ vcrit, L becomes

L ¼ Im

�1

iþ e�21

� arctanh

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie�41 þ 2e�21 sinhð22Þ � 1

piþ e�21

��: (6.20)

Likewise, no energy or momentum is transferred from onecharge to the other for v < vcrit. However, for v ¼ vcrit,

1t ¼

ffiffiffiffi�

p4

ð1þ e�41Þ; �1x ¼

ffiffiffiffi�

p2

coshð21Þ:(6.21)

Curiously enough, the rate at which energy is transferred


046002-11

decreases with the mass parameter 1 of the laggingcharge, whereas the rate at which momentum is transferredincreases with 1.

From the metric (6.1), the proper velocity of the stringendpoint at ¼ �1 is given by

V ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ 2v sinhð21Þ

q; (6.22)

where v is the physical velocity in the dual two-dimensional field theory. In order to avoid a spacelikestring, V 1, which corresponds to having v vcrit.Thus, it turns out that vcrit is the speed limit of the pairof charges.

B. The Maoz-Maldacena wormhole

We will now consider some static properties of stringson a Euclidean AdS wormhole solution arising in five-dimensional gauged supergravity [13]. The metric is givenby

ds25 ¼ d2 þ e2!�d 2 þ 1

4sin2 wawa

�; (6.23)

where

e2! ¼ffiffiffi5

p2

cosh2� 1

2; (6.24)

and wa are the left-invariant one-forms on S3. There arealso SOð6Þ gauge fields given by

AIJ� ¼ iAa

�LaIJ þ i ~Aa

�~LaIJ; (6.25)

where La and ~La are generators of SOð3Þ � SOð3Þ and

Aa ¼ cos2

2wa; ~Aa ¼ sin2

2wa; (6.26)

are an instanton and an anti-instanton which are SOð5Þsymmetric under rotations of S4. The corresponding gaugetheory isN ¼ 4 super Yang-Mills theory with an externalfixed gauge field coupled to the SOð6Þ currents. It wasshown explicitly that the supergravity background is ratherstable, and the field theory on each boundary is welldefined [13].

The action of the Euclidean string world sheet is givenby (3.5). We will consider a string whose world sheet liesalong two of the S3 directions, which we will refer to as tand x. Since the string lies at a point in the direction, wewill rescale the x and t coordinates to include constantfactors in the metric. In the gauge � ¼ and � ¼ t, wefind that the equation of motion is given by

@

@

�e4!x0ffiffiffi

gp

�þ e2!

@

@t

�_xffiffiffig

p�¼ 0; (6.27)

where

g ¼ e2!ð1þ _x2 þ e2!x02Þ: (6.28)

From (2.9) and (2.10), we find that the energy of a straight

string extended from ¼ �1 to 2 is

E ¼ �i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5

p � 1

2

sT0

�EllipticE

�i1;

2ffiffiffi5

pffiffiffi5

p � 1

�

þ EllipticE

�i2;

2ffiffiffi5

pffiffiffi5

p � 1

��; (6.29)

where EllipticE is the elliptic integral of the second kind. Eis a real function and increases monotonically with i.In the gauge � ¼ x and � ¼ t, the equation of motion is

@

@x

�e2!0ffiffiffi

gp

�þ e2!

@

@t

�_ffiffiffig

p�

¼ffiffiffi5

psinh2

2ffiffiffig

p ð ffiffiffi5

pcosh2� 1þ 02 þ _2Þ; (6.30)

where

g ¼ e2!ðe2! þ 02 þ _2Þ: (6.31)

We see that there is also a straight string solution atconstant ¼ 0.We will now consider string solutions with both end-

points on the same side of the wormhole at ¼ 2. Sincethe geometry is asymptotically hyperbolic, the endpointsof a string that is far from the wormhole exhibit Coulombbehavior. As the distance L between the endpoints isincreased, the turning point t gets closer to the wormhole.From (6.30), we find that

L ¼ 2e2!t

Z 2

t

d

e!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie4! � e4!t

p ; (6.32)

where !t ¼ !j¼t. By analyzing the above formula, we

see that there can be string solutions with 2 >t � 0. Itseems that all strings bend toward the wormhole and noneof them can pass through the neck of the wormhole, asopposed to the wormhole background discussed in Sec. III.For a string with its endpoints on opposite sides of the

wormhole

L ¼ CZ 2

�1

d

e!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie4! � C2

p : (6.33)

For small separation L, we can expand in C � 1 so that

L CZ 2

�1

de�3!: (6.34)

From the world sheet action given by (3.5), we find theenergy of the string to be

E ¼ T0

Z 2

�1

de3!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e4! � C2p : (6.35)

Expanding E in small C and then expressing C in terms ofL, we find

E Estraight þ 12kL

2; (6.36)


046002-12

where Estraight is the energy of a straight string which is

given by (6.29) and

k ¼ T0

�Z 2

�1

de�3!

��1: (6.37)

Thus, for small L, a pair of charges of opposite types areconfined by a springlike potential. It can be shown that kdecreases monotonically with the mass parameters, whichmeans that heavy charges are less sensitive to the confiningpotential.

When there is a pair of type 1 charges a distance L apartand a pair of type 2 charges also a distance L apart, thesame types of transitions can occur as with the previouslydiscussed wormholes. Namely, for small L only charges of

the same type interact with each other, whereas for large Lonly opposite type charges interact with each other. Thiswas discussed in a bit more detail in Sec. III B and illus-trated in Fig. 6.

ACKNOWLEDGMENTS

We would like to thank Philip Argyres, MohammadEdalati, and Hong Lu for useful correspondence. M.A.,F. R., and C. S.-V. are supported in part by the EmergingScholars Program. F. R. and C. S.-V. are supported in partby New York City Louis Stokes Alliance for MinorityParticipation in Science, Mathematics, Engineering andTechnology.

[1] J.M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998);Int. J. Theor. Phys. 38, 1113 (1999).

[2] S. S. Gubser, I. R. Klebanov, and A.M. Polyakov, Phys.Lett. B 428, 105 (1998).

[3] E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998).[4] O. Aharony, S. S. Gubser, J.M. Maldacena, H. Ooguri, and

Y. Oz, Phys. Rep. 323, 183 (2000).[5] D. Lovelock, J. Math. Phys. (N.Y.) 12, 498 (1971).[6] B. Bhawal and S. Kar, Phys. Rev. D 46, 2464 (1992).[7] H. Maeda and M. Nozawa, Phys. Rev. D 78, 024005

(2008).[8] G. Dotti, J. Oliva, and R. Troncoso, Phys. Rev. D 75,

024002 (2007).[9] G. Dotti, J. Oliva, and R. Troncoso, Phys. Rev. D 76,

064038 (2007).[10] G. J. Galloway, K. Schleich, D. Witt, and E. Woolgar,

Phys. Lett. B 505, 255 (2001).[11] A. Buchel, R. C. Myers, and A. Sinha, J. High Energy

Phys. 03 (2009) 084.[12] V. Balasubramanian, A. Naqvi, and J. Simon, J. High

Energy Phys. 08 (2004) 023.[13] J.M. Maldacena and L. Maoz, J. High Energy Phys. 02

(2004) 053.

[14] S. J. Rey and J. T. Yee, Eur. Phys. J. C 22, 379 (2001).[15] J.M. Maldacena, Phys. Rev. Lett. 80, 4859 (1998).[16] P. C. Argyres, M. Edalati, and J. F. Vazquez-Poritz, J. High

Energy Phys. 01 (2007) 105.[17] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz, and L.G.

Yaffe, J. High Energy Phys. 07 (2006) 013.[18] J. Casalderrey-Solana and D. Teaney, Phys. Rev. D 74,

085012 (2006).[19] S. S. Gubser, Phys. Rev. D 74, 126005 (2006).[20] K. B. Fadafan, J. High Energy Phys. 12 (2008) 051.[21] J. F. Vazquez-Poritz, arXiv:0803.2890.[22] J. Noronha, M. Gyulassy, and G. Torrieri,

arXiv:0906.4099.[23] H. Lu and J. Mei, Phys. Lett. B 666, 511 (2008).[24] A. Bergman, H. Lu, J. Mei, and C.N. Pope, Nucl. Phys.

B810, 300 (2009).[25] D. H. Correa, J. Oliva, and R. Troncoso, J. High Energy

Phys. 08 (2008) 081.[26] V. Balasubramanian and P. Kraus, Commun. Math. Phys.

208, 413 (1999).[27] W. P. Thurston, Three-Dimensional Geometry and

Topology (Princeton University Press, Princeton, 1997),Vol. 1.


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Strings on AdS wormholes