Born-Infeld strings between D-branes

PHYSICAL REVIEW D, VOLUME 61, 066004

Born-Infeld strings between D-branes

K. Ghoroku*Department of Physics, Fukuoka Institute of Technology, Wajiro Higashi-ku 811-02, Fukuoka, Japan

K. Kaneko†

Department of Physics, Kyushu Sangyo University, Matsukadai, Higashi-ku, 813-8503, Fukuoka, Japan~Received 31 August 1999; published 25 February 2000!

We examine the solutions of the world-volume action for a D3-brane being put near other D3-branes whichis replaced by the background configuration of bulk space. It is shown that the BPS solutions, which have beengiven by Gauntlettet al., are not affected by the D3-brane background, and they are interpreted as dyonicstrings connecting two branes. On the contrary, the non-BPS configurations are largely influenced by thebackground D-brane, and we find that the solutions with pure electric charge cannot connect two branes. Thesesolutions correspond to the bound state of a brane and antibrane which has been found by Callan andMaldacena.

PACS number~s!: 11.25.2w, 11.15.2q, 11.27.1d

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I. INTRODUCTION

Recently, classical solutions of the Born-Infeld~BI! ac-tion, which is considered as the world-volume action of tD-brane, have been studied@1–5# in flat background spaceThey are classified as Bogomol’nyi-Prasad-Sommerfi~BPS! and non-BPS solutions. The stable BPS solutionsinterpreted as strings which connect two D-branes separby an infinite distance, while the non-BPS solutions areprotected from dynamical fluctuations since they are notpersymmetric. Among them, non-BPS solutions with puelectric charge have attracted attention@1#. It can be regardedas the half of the string which connects a brane and antibwith a finite distance to form a bound state. The stabilitysuch a non-BPS configurations has been studied fromviewpoint of the quantum mechanics@1,6#.

It will be interesting to examine these solutions from tviewpoint of SU(N) Yang-Mills theory which can be constructed by a stack of D3-branes. In this direction, soprogress has been made by studying the world-volume acof test brane embedded in the background of D-bra@7–12#. The situation of the symmetry breaking SU(N)→SU(N21)3U(1) is realized by setting one of the branfar away from the others. Then we can say that the classsolutions of the world-volume action represent the solisolutions such as monopoles or dyons appearing in theAbelian Yang-Mills theory. It would be meaningful to studthe non-BPS solutions which are nonsupersymmetric stions obtained in the background of the type-IIB superstrtheory, because they are expected to play some role insupersymmetric Yang-Mills theory.

The purpose of this paper is to study the classical sotions of the world-volume action in a situation where the tD-brane is set parallel to a background D-brane~s!. In thenext section, the model is given. In Sec. III A, we show tBPS and non-BPS dyonic solutions in the flat background

*Email address: [email protected]†Email address: [email protected]

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Secs. III B and IV, we solve the case of the D-brane baground. We find that the BPS solutions are not affectedthe background, but the configuration of non-BPS solutiois influenced by the background. Especially, the pure elecsolutions cannot be reached at the position of the backgrobrane~s!. Conclusions are given in the final section.

II. D-BRANE ACTION TO BE SOLVED

Even if we consider the supersymmetric case, the feronic coordinates are not necessary to obtain a classical stion of the D-brane action. So we neglect them, andbosonic part of the effective action of a D-p-brane beingcoupled to the background can be written as follows:

Sp1152TpE dp11jS e2fA2det~Gmn1Fmn!

11

~p11!!e i 1••• i p11Ai 1••• i p11D , ~1!

whereTp denotes the tension of the D-p-brane and

Fmn5]mAn2]nAm ~2!

is the field strength of a U(1) gauge field residing in tworld volume. We also neglect here the antisymmetric tenBMN , which should be added toFmn , since the brane we arconsidering here has no Neveu-Schwarz–Neveu-Schw~NS-NS! charge. The metricGmn and the (p11)-form arethe pullback of the space-time metricGMN and the Ramond-Ramond~R-R! (p11)-form, respectively,

Gmn5GMN]mXM]nXN, ~3!

Ai 1••• i p115] i 1

XM1•••] i p11

XM p11AM1•••M p11. ~4!

The embedding of the world volume into the target spais described by XM(jm) as a function of the(p11)-dimensional world-volume coordinatesjm.

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K. GHOROKU AND K. KANEKO PHYSICAL REVIEW D 61 066004

Our purpose is to solve the equations of motion forabove actionSp11 under the target space configuration fF, GMN , and AM1•••M p11

, which are obtained as the D

p-brane configuration by solving the ten-dimensional supgravitational theory. Before solving the equations of tD-brane action in this way, we briefly review the D-braconfigurations.

They are obtained by solving the supergravity effectaction corresponding to the superstring theory@13,14#. Thed-dimensional coordinatesxM are denoted by separatinthem into tangential (xm) and the transverse (ym) parts to theD-p-brane asxM5(xm,ym), wherem50;p and m5p11;d21. Then the D-brane configurations are given as flows:

ds25e2Ahmndxmdxn1e2Bdmndymdyn, ~5!

Ap1152eCdx0`dx1`•••`dxp, ~6!

wherey5Admnymyn and

e2A(y)5H~y!21/2, e2B(y)5H~y!1/2, ~7!

eC(y)5H~y!2121, e2F5H~y!(32p)/2, ~8!

H~y!5112QkTpG~y!, ~9!

G~y!5H @ duyu dV d11#21 d.0,

21

2plog uyu d50,

~10!

and d5d2p23. HereVq52p (q11)/2/G„(q11)/2… denotesthe area of a unitq-dimensional sphereSq.

III. SOLUTIONS OF THE D-BRANE ACTION

Here we discuss the classical solutions ofSp11 in thebackground D-brane~s! which is set parallel to the test branconsidered now. The classical equations ofSp11 are solvedfor p53 by adopting the static gauge for the diffeomorphisinvariance, for which the world-volume coordinatesjm areequated with thep11 spacetime coordinates as

XM5jm, M50,1, . . . ,p. ~11!

The remaining coordinatesXm are treated as scalar fieldsthe brane world volume, but we retain only one coordinamong them as a field which is responsible for the confirations of the brane and it is denoted here byX(j). Thiscoordinate is taken in the direction perpendicular to the otbrane.

Then we can writeSp11 with p53 as follows:

S452T3E d4jS 1

H~AD21! D , ~12!

where the field-independent constant is subtracted fromabove Lagrangian density for brevity, and

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D5~12HEW 2!„11H~¹X!2…1H2~EW •¹X!22HX2

12H2EW •~BW 3¹X!X1HBW 2~12HX2!

1H2~BW •¹X!22H2~BW •EW !2, ~13!

H5112QkT3G~Xm2X!, ~14!

where Xm denotes the distance between two branes. TU~1! gauge fields are denoted by the conventional elecmagnetic fieldsEW andBW .

A. Solutions in the flat background

First, we solve the equations forH51: the case of aninfinite distance between the two branes. The solutionsrestricted to the static one. In this case,D is written as

D5~12EW 2!@11~¹X!2#1~EW •¹X!21BW 2

1~BW •¹X!22~BW •EW !2, ~15!

and the following equations are obtained:

¹•$@~12EW 2!¹X1~EW •¹X!EW 1~BW •¹X!BW #/AD%50, ~16!

¹•$@„11~¹X!2…EW 2~EW •¹X!¹X1~BW •EW !BW #/AD%50, ~17!

¹3$@BW 1~BW •¹X!¹X2~BW •EW !EW #/AD%50. ~18!

Since we are interested in isotropic solutions, we takefollowing Ansatz:

EW 5 f ~r ! r , BW 5g~r ! r , ¹X5X8~r ! r , ~19!

wherer represents the unit vector in the radial direction, aa prime denotes differentiation with respect tor. Then, Eqs.~16! and ~17! are expressed as

¹•H X8A 11g2

12 f 21X82r J 50, ~20!

¹•H fA 11g2

12 f 21X82r J 50. ~21!

From Eqs.~20! and~21!, the functionf is proportional toX8:

f 5aX8, ~22!

wherea is a constant. Since theAnsatz~19! satisfies auto-matically Eq.~18! for any g, we cannot determine the function g since the number of independent equations is reduto 2. So we need a furtherAnsatzor assumption to obtain thesolution. We now assume that the magnetic fieldBW is givenby some appropriate form as an arbitrary external field; thwe can solve Eq.~20! or ~21! as follows:

X~r !5Er

`

drA

A~11g2!r 42r 04

, ~23!

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BORN-INFELD STRINGS BETWEEN D-BRANES PHYSICAL REVIEW D61 066004

where A denotes an integral constant andr 045(12a2)A2.

Here we consider the purely electric (g50) case, namelywithout the external magnetic field; then the solution~23!reduces to the BPS state fora51 and to the non-BPS solution given in Ref.@1# for aÞ1.

Next, consider the case adding one moreAnsatzto Eq.~19! as follows:

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Bt

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g5bX8, ~24!

whereb(Þ0) is a constant. In this case, the string is consered as the source of both the electric and the magncharges, so we call this type the dyonic string. The equais solved as

X~r !5Er

`

drF2$12~12a2!A2/r 4%1A@12~12a2!A2/r 4#214b2A2/r 4

2b2 G 1/2

. ~25!

nt

then,ur-

For a21b251, this solution reduces to the BPS dyonic slution

X~r !5A

r. ~26!

In this case, the Bogomol’nyi equations are satisfied by sration of bound

E>a~EW •¹W X!1b~BW •¹W X!. ~27!

The solutions obtained here are classified in the paramspacea andb as shown in Fig. 1.

There are many non-BPS dyonic string solutions, asome typical ones are shown in Fig. 2 along the line inparameter space shown in Fig. 1.

B. Solutions in the background D-brane„s…

Here we consider the solutions for the case ofHÞ1. Inthis case, the Wess-Zumino term in the actionS4 given inEq. ~12! cannot be neglected as in the previous section.see its important meaning, we notice thatS4 vanishes forEW

5BW 50. This is a result of the cancellation between theand the Wess-Zumino~WZ! terms, and it reflects the facthat there is no force between two parallel branes@15#. We

FIG. 1. The static solutions are classified in thea-b plane. Theyare represented by three types of solutions:~A! The pure electricsolution, ~B! the dyonic solution, and~C! the BPS solution. Theplus symbols denote the points on the lineb51.5a for ~a! a50.3, ~b! 0.5, ~c! 0.7, and~d! 0.9.

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therefore expect that the WZ term would play an importarole in solving the equations forHÞ1.

Here we solve the equations of motion ofS4 in the back-ground configuration of the parallel D3-brane~s!, so the forceis absent for the trivial solution ofEW 5BW 50. But it is ex-pected that the nontrivial solutions, which are obtained inbackground of flat space-time by solving the BI actiowould be affected by the background configuration. The sviving solutions should be deformed and restricted.

The equations are written as follows:

]H

]XF11¹•FW 250, ~28!

¹•

1

AD$@11H~¹X!2#EW 1H@~BW •EW !BW 2~EW •¹X!¹X#%50,

~29!

¹31

AD$~BW 1H@~BW •¹X!¹X2~BW •EW !EW #%50, ~30!

where]H/]X 54Q/(Xm2X)5 and

FIG. 2. The typical dyonic non-BPS solutions, forb51.5a, A51. The curves represent for~a! a50.3,~b! 0.5,~c! 0.7,and ~d! 0.9 in Fig. 1. The boundary conditions are taken asX(4)5A/4 for all curves.

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F151

H2 ~AD21!11

2HAD$EW 2@11H~¹X!2#

2~¹X!2~12HEW 2!22H@~BW •¹X!2

1~EW •¹X!22~BW •EW !2#2BW 2%, ~31!

FW 251

AD$~12HEW 2!¹X1H@~EW •¹X!EW 1~BW •¹X!BW #%.

~32!

Here the time dependence of the fields is neglected sincesolutions are restricted to the static one. Obviously the efof the background configuration appears throughH, and Eqs.~28!–~30! are respectively reduced to Eqs.~16!–~18! for H51. The most characteristic feature is seen in the first teof the left-hand side~LHS! of Eq. ~28!. And the WZ term isalso included inF1 of this term. ForQ50, this term disap-pears, and the second term reduces to Eq.~16! given in theflat background. The first term therefore represents a neappearing constraint due to the nontrivial background cfiguration. Because of this term, the above equations casolved withAnsatze ~19! and ~22! only since the new constraint determines the magnetic fieldg(r ) as seen below.

We solve the above equations by taking theAnsatz~19!used in the previous subsection. Then the above equaare rewritten as follows:

]H

]XF11¹•S X8

AD~11Hg2! r D 50, ~33!

¹•S f

AD~11Hg2! r D 50, ~34!

¹3S f

AD@11H~X822 f 2!# r D 50, ~35!

where

F151

2H2AD@2~D2AD !1H~ f 22X822g2!

12H2g2~ f 22X82!#, ~36!

D5~11Hg2!@11H~X822 f 2!#. ~37!

The same way as in the previous subsection, the numbeequations is reduced to 2 since Eq.~35! is satisfied by anyfunctional form of f , g, andX.

Then we need one moreAnsatzwhich gives one relationamong these three functions. To find similar solutions toone obtained in the previous subsection, we take the folling Ansatz:

f 5aX8. ~38!

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It should be noticed that this was not an ansatz but a soluof the equations in the previous subsection. Under thisAn-satz, Eq. ~34! is solved as

f

AD~11Hg2!5

A2

r 2 , ~39!

whereA2 is an integral constant. The Eq.~33! is rewritten as

F150, ~40!

since the second term of this equation vanishes, i.e.,¹F250. We notice that the above equation~40! does not appeain the case of the flat background. As a result, the magnfield g(r ) cannot be determined. In the present case, weobtain the following solution due to Eq.~40!:

X5A2

r, f 52a

A2

r 2 , g52A12a2A2

r 2 . ~41!

This solution, which has been obtained by a different methin @10#, represents the dyonic string with both electric amagnetic charges, and the BPS bound is saturated bysolution since it is represented as

EW 5cos~u!¹X, BW 5sin~u!¹X, ~42!

by the parametrization cosu5a (<1). Here, we notice thefollowing points: ~i! The Ansatz~38!, which yields a non-BPS solution in the case of the flat background, leads toBPS solution due to Eq.~40!. ~ii ! The form of the BPS so-lution obtained is independent of the parameter of the baground since it is not deformed by the background confirations. ~iii ! The effect of the background configuratioappears indirectly through the induced world-volume mein the brane action. Namely,r has a minimum valuer 05A2 /Xm @10#, where the solution touches the opposbrane~s! sinceX(r 0)5Xm . At this point, the proper distancin the world volume of the brane becomes infinite sinceinduced metric diverges. Then we arrive at the configuratof the dyonic string which connects two branes with a findistance.

From the fact~iii !, the energy of the string part is obtaineas a finite value by integrating the energy density ofsystem in the ranger 0,r ,` as follows:

E54pA2T3Xm , ~43!

which represents the energy of the string of the lengthXmwith the tension of 4pA2T3. It is expected that this objecwith a finite energy would appear as a dyon in Yang-Mitheory.

IV. NON-BPS SOLUTIONS

We now turn to the non-BPS solutions. As seen in Sec.A, the non-BPS solutions obtained in flat space are separinto two groups according to properties. One is the pure etric string which connects to the antibrane to form a boustate. The second is the dyonic string which could arrive

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the other brane~s! at infinite distance. In the case of finitdistance, also, these two types of solutions can be foundtaking otherAnsatze than the one given in the previous setion, Eq. ~38!.

A. Pure electronic case

First we consider the case of the pure electric solutiThis is obtained by taking the followingAnsatz:

g50. ~44!

In this case, we can obtain the following equation forX fromEqs.~33! and ~39!:

X912

rX82

X8

2~ ln D !852

2Q~AD21!2

~Xm2X!5H2, ~45!

where

D511HX82

11H~A2 /r 2!2. ~46!

It is easily seen that Eq.~45! is reduced to the equation of thflat background in the limit ofQ50 (H51). In fact, weobtain the following solution forQ50,

X85c1 /r 2

A12c2 /r 4, ~47!

wherec1 and c2 are constants depending on the boundconditions. This is equivalent to the solution~23!.

Equation~45! for HÞ1 is highly nonlinear, so it is diffi-cult to solve it analytically. Then we give the solutions nmerically.

The resultant solutions are shown in Fig. 3, whereBPS solutionX5A2 /r is also shown by the dotted line fothe comparison. In Fig. 3, two types of non-BPS solutio

FIG. 3. The typical two types of non-BPS solutions, forQ51, Xm55, A251. The boundary conditions are taken asX(10)50.1 and~a! X8(10)520.00974,~b! 20.00948,~c! 20.00923,~d!20.00898 for the first type, and~e! X8(10)520.0101, ~f!20.011,~g! 20.0115 for the second type.

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are seen. They are obtained here by giving the boundconditions at an appropriate point (r 5r B) as

X~r B!5A2

r B, X8~r B!52g

A2

r B2 , ~48!

whereg is a parameter. The solutions of the first group~a!–~d! are obtained forg,1, and the second group~e!–~g! arefor g.1. We notice that the BPS solution is obtained fg51. In this sense, the BPS solution is the critical one whseparates the two types of solutions.

The first type of solution covers all regions ofr, and itends atr 50 with a finite value ofX(0) and negative valueof X8(0). So theshape of this configuration has a cusp ar50 and the configuration is singular at this point.

The second type of solution is bounded asr>r 1, wherer 1varies depending on the boundary conditions andX8(r 1)5`. Because of this property, the solution can be connecto the other half with opposite orientation@1# to make thebound state of brane and antibrane.

These solutions have the same qualitative properties wthe one obtained in the flat background. However, theyaffected by the nontrivial background in the present caThe most prominent influence is seen in the fact that thsolutions could not reach the position of the other brane~s!,X5Xm . This is understood as follows. From Eqs.~37! and~39!, we obtain

f 25H211X82

H21~r 2/A2!211. ~49!

Then we find the relationf 5X8 at X5Xm , whereH2150,and we obtain the BPS solution from this condition. Thimplies that only the BPS solution can arrive atX5Xm . Sothere is no non-BPS electrics string state, which connetwo parallel branes with a finite energy. Such an objecrestricted to the BPS saturated strings.

As for theQ dependence of the solutions, we can seetypical effect from the solutions as shown in Fig. 4, whethe second type of solutions for two differentQ values is

FIG. 4. The typical non-BPS solutions forXm55, A251, ~a!

Q51 and~b! Q570 with the boundary conditionsX(10)50.1 andX8(10)520.0101.

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given. TheQ dependence is small, but it can be seen nearend point of the solution. The largerQ becomes, the widethe radius of the tube grows, and the end point of the strconfiguration goes back. This implies that the bound statthe brane and antibrane would be pushed to vanish throthe annihilation of them near the branes.

B. Dyonic case

Next, we consider the dyonic non-BPS string configutions under the D-brane background. The equation forsolution is obtained by taking theAnsatz ~24!, g5bX8,which is given in Sec. III A to obtain the dyonic solution ithe flat background. In the present case, Eq.~22! is not useddifferently from the the case of Sec. III A In terms of thAnsatz~24! and Eq.~39!, which is the solution of Eq.~34!,we obtain

f 2511HX82

11H@b2X821~A2 /r 2!2#S A2

r 2 D 2

. ~50!

FIG. 5. The typical dyonic non-BPS solutions forXm55, A2

51, ~a! b50.2, ~b! b50.6, ~c! b51.0, and~d! b51.4 with theboundary conditionX(10)50.1 andX8(10)520.0115.

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Substituting Eqs.~50! and ~24! into Eq. ~33!, we can solveEq. ~33! with respect toX(r ) by rewriting it as the differen-tial equation ofX(r ). We solve the differential equation numerically.

We first discuss the dyonic solutions which correspondthe solutions~g! in Fig. 3 at the limit ofb50. Namely, wetake the same boundary conditions as the one of~g! in Fig. 3in solving the equation considered here forbÞ0. In Fig. 5,the solutions are shown for various values ofb. The b de-pendence of the solutions is seen from Fig. 5, and the beiors are similar to the case of the flat background.

These solutions exceedXm which was the bound for thenon-BPS solution ofb50. This property is also seen in thcase of the flat background, where this configuration exteto X5` at r 50. But the shapes of this configurations aaffected by the D-brane background. This point is differefrom the case of the flat background. This is seen fromresults shown in Fig. 6, in which theQ dependence for thesolution is presented. Although theQ dependence is smallwe can see it near the neck (0.2,r ,6) of the throat byextending the scale ofX(r ). This Q dependence is the maidifference from the BPS solutions which are independenthe background.

We should notice the following fact. For any solution, thmetric of the world-volume action is determined from thsame D-brane background, and it has a singularity aX5Xm . So the configuration should be cut at this point; thwe can say that the configuration obtained here represthe dyonic string which connects two D-branes with a findistance. But these solutions are not the BPS states, so sdynamical corrections would modify the configurations otained here. This is a dynamical problem, which wouldrelated to nonsupersymmetric Yang Mills theory. This pois open here.

Finally, we comment on the solutions ofbÞ0 which arereduced to the solutions belonging to the group of~a!–~d! ofFig. 3 for b50. The results are shown in the Fig. 7 fovarious values ofb. In this case, the qualitative behavior othe solutions is the same as the one ofb50.

FIG. 6. TheQ dependence of the dyonic non-BPS solutions forXm55, A251, b50.8 with the boundary conditionsX(10)50.1 and

X8(10)520.0115. From the above curve, the curves forQ51, 101, 201, 301 are shown.

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V. CONCLUSIONS

We have given the solutions of the world-volume actiof a D3-brane which is placed parallel to the backgrouD3-brane~s!. The equations are first solved in the flat bacground, and two kinds of non-BPS solutions are shown,the pure electric and the dyonic solutions. They smootapproache the BPS solutions in a special limit of the paraeters.

Both the BPS@10# and non-BPS solutions are also o

FIG. 7. Theb dependece of the dyonic non-BPS solutions

Xm55, A251, Q51 with the boundary conditionsX(10)50.1and X8(10)520.009974. The curves for~a! b50.2, ~b! 0.6, ~c!1.0, and~d! 1.4 are shown.

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tained in the D-brane~s! background. The BPS solution hathe same functional form as the one obtained in the flat baground and it can extend to infinity. But it should bbounded at the position of the opposite brane~s! since theproper distance in the world volume of the brane becominfinite there. As a result, a finite energy of this BPS stateobtained.

On the other hand, both types of non-BPS solutionsaffected by the D-brane background. Especially the electtype solutions cannot arrive at the opposite brane~s! sincethey are pushed back by the background configurationsthis sense, this type of non-BPS solution cannot be conered as strings which connect two branes. One of theseBPS solutions can be interpreted as half of the bound statthe brane and antibrane as in the case of the flat backgroIt is observed that the distance between the brane andantibrane of this bound state becomes shorter when it nthe background branes.

Although the configurations of the dyonic non-BPS sotions are also affected by the D-brane background, tcould extend overX5Xm andX becomes infinite atr 50. Onthe other hand, the induced metric of the world-volumetion of the D-brane is independent of the solutions, soconfigurations of the solutions are bounded atX5Xm as inthe case of the BPS solutions. Then we arrive at the consion that the configuration of non-BPS dyonic string conects two D-branes with a finite distance and with a finenergy. This configuration, however, receives a dynamcorrection because of no supersymmetry, and this wouldrelated to the dynamics of nonsupersymmetric Yang-Mdynamics.

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Born-Infeld strings between D-branes