Ž .Physics Letters B 481 2000 369–378

The mass formula for a fundamental stringas a BPS solution of a D-brane’s worldvolume

Takeshi Sato 1

Institute for Cosmic Ray Research, UniÕersity of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba, 277-8582 Japan

Received 21 January 2000; received in revised form 3 April 2000; accepted 3 April 2000Editor: H. Georgi

Abstract

Ž .We propose a generalized ‘‘mass formula’’ for a fundamental string described as a BPS solution of a D-brane’sworldvolume. The mass formula is obtained by using the Hamiltonian density on the worldvolume, based on transformationproperties required for it. Its validity is confirmed by investigating the cases of point charge solutions of D-branes in a

Ž .D-8-brane i.e. curved background, where the mass of each of the corresponding strings is proportional to the geodesicŽ .distance from the D-brane to the point parametrized by the regularized value of a transverse scalar field. It is also shown

that the mass of the string agrees with the energy defined on the D-brane’s worldvolume only in the flat background limit,but the agreement does not always hold when the background is curved. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

To gain better understanding of string theoriesand M-theory, intersecting branes have played animportant role, and worldvolume analyses have beenpowerful approaches to investigate the intersecting

w x w xbranes 1–5 . In Refs. 1–3 solutions of worldvol-Ž .ume field theories of branes in flat backgrounds

with nontrivial worldvolume gauge fields were ob-tained. In the case of D-branes, it was shown that anappropriate excitation of one of the transverse scalarfields is needed in order to obtain a supersymmetricŽ .i.e. BPS point charge solution of the worldvolume

w xgauge field 1 . Each of the solutions was interpretedas a fundamental string ending on the D-brane on the

1 E-mail: tsato@icrr.u-tokyo.ac.jp

Ž .basis of the fact that the regularized energy of eachsolution defined on the worldvolume is proportional

Ž .to the regularized value of the scalar field, which isconsidered to be the length of the string 2. Thisinterpretation is also consistent with the charge con-

w x Ž w x.servation suggested in Ref. 6 see also Ref. 7 .These analyses are very important in that they madeclear ‘‘how one of intersecting two branes is de-scribed’’ from the viewpoint of another brane’sworldvolume.

In fact, however, the energy defined on the brane’sworldÕolume does not always agree with the target-space mass of the string, though the agreement of

w xthe two holds for the case of Ref. 1 . We can easilyunderstand this by considering the fact that the

2 Since the energies of the point charge solutions are infinite,Ž .UV regularization is needed to calculate them.

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00432-9

worldvolume energy depends on the definition of thetime of the worldvolume, while the mass of thestring should not depend on it. That is, the derivationof the mass of the string must be considered morecarefully. The main purpose of this paper is topresent a generalized ‘‘mass formula’’ of a funda-mental string described as a BPS solution of aD-brane’s worldvolume, which holds independent ofdefinition of its worldvolume time.

Our idea to obtain the mass formula is as follows:the mass of the string does not always agree with theworldvolume energy, to be sure, but it is also truethat the two are very close to each other, since thetwo give the same results at least in the cases

w xdiscussed in Ref. 1 . So, we construct the massformula in a heuristic way, by using the Hamiltoniandensity defined on the worldvolume, based on theinvariance of the mass under the coordinate transfor-mations of the worldvolume. Moreover, we also

Žconstruct explicitly point charge solutions with ap-.propriate excitations of single transverse scalar fields

of branes’ worldvolumes in a curÕed background 3.There are two advantages to consider these solutions:First, in each of the cases, the time component of the

Žinduced worldvolume metric g as well as spatial˜00.ones becomes nontrivial, So, the differences origi-

nating from the contribution of g become apparent˜00

in discussing the mass formula. Second, in each caseof the solutions, the mass of the correspondingfundamental string should be proportional to thegeodesic distance from the brane to the point param-

( )eterized by the regularized Õalue of the scalar field,and that the proportional coefficient should be anappropriate tension of the string, as discussed in Ref.w x10 . This requirement is very tight. So, once we findout the quantity which gives the mass stated above, itis expected to be the correct mass formula, even if itis constructed by hand. We will construct it based on

Žthis idea. The discussion on the worldvolume inter-pretation of the string mass will be given finally in

.Section 3.The worldvolume theories we discuss here are the

two cases: those of a test D-4-brane and a testŽ .D-8-brane both embedded parallel to a subspace of

3 Ž .Some worldvolume solutions of branes in a curved branew xbackgrounds are discussed for other purposes in Refs. 8,9 .

the worldvolume of the D-8-brane backgroundw x Ž .11,12 i.e. a massive IIA background . First, wepresent the two reasons to choose this background:One is that this background has only one transversecoordinate, leading to the fact that the harmonicfunction depends linearly on the coordinate. Only inthis case, we can obtain explicitly the exact solutionsof the worldvolumes without any extra assumptionŽ .as we will see later . We note that obtaining theexplicit form is is crucial not only in order to findout the mass formula by hand and but for otherdiscussions. Another reason is that choosing thisbackground, we can see the supersymmetry pre-served in the solutions by using superalgebras in amassive IIA background via brane probes’’ in Ref.w x 413 . From the supersymmetry, we can confirm thatthe solutions are BPS states and that their target-spaceinterpretation is consistent. Next, we explain why wechoose the two worldvolume theories embedded inthose ways: This is because at least one overall

Ž .transverse space is needed after embedding testD-branes’ worldvolumes in order to obtain super-symmetric point charge solutions. In the case of a

Ž .D-2-brane a D-6-brane , only the intersection withŽ .the background D-8-brane on a string a 5-brane

leads to the preservation of supersymmetry 5, but,there is no overall transverse space. A D-0-brane isnot adequate to this worldvolume description sincethere is no world space. So, we consider the abovetwo cases, which preserve at this moment 1r4 and1r2 supersymmetry, respectively.

Concrete procedures are in the following: In eachof the two cases we construct explicitly an pointcharge solution with an appropriate excitation of theonly overall transverse scalar field. The consistencyof the interpretation of each solution as a fundamen-tal string is confirmed in two ways: by discussing itsbehavior in the flat background limit and by check-ing its preserved supersymmetry. Then, we proposethe generalized mass formula, and examine whether

4 w xIn the previous paper 9 we have confirmed that the super-symmetry preserved in the solution of brane’s worldvolume can

Žbe derived via ‘‘superalgebras in brane backgrounds’’ see alsow x.14 .

5 Ž < .We do not discuss bound states like 6 D6,D8 since theD-8-brane solution is ‘‘singular’’ just on the D-8-brane hyper-surface.

it gives the geodesic distance multiplied by an appro-priate tension of the string. Moreover, we also dis-cuss the condition that the mass of the string agreeswith the energy defined on the D-brane’s worldvol-ume.

The organization of this paper is as follows: InSection 2 we construct the point charge solutions anddiscuss their mass formula. In Section 3 we give theconclusion and some discussions on the consistencyand the interpretation of the mass formula

The notations in this paper are as follows: We use‘‘mostly plus’’ metrics for both spacetime andworldvolumes. We denote coordinates of each p-

i j Žbrane’s worldvolume as j ,j , . . . i, js0, 1, . . . ,. m n Žp , those of 10D spacetime as x , x , . . . m,ns0,

. a b. . . , 9 , fermionic coordinates as u ,u , . . . , andthose of superspace as Z M. We use hatted letters

ˆŽ .M,m,a , . . . for all the local Lorentz frame indicesˆ ˆŽ .and under-barred letters m,i for spatial indices

Ž .but not time one , respectively. We denote gamma� 4matrices as G , which are all real and satisfy G ,Gm m nˆ ˆ ˆ

s2h . G is antisymmetric and others symmetric.ˆm n 0ˆ ˆ0Charge conjugation is CCsG .

2. Point charge solutions of 10D IIA D-branes’worldvolumes and their mass formula

In this section we construct the point chargesolutions corresponding to fundamental strings anddiscuss their mass formula.

The D-p-brane action in a general 10D massivew x w xIIA background 15,12 takes the form 16,17

S sSBI qSWZD p D p D p

pq1 yfsyT d j e ydet g qFF˜( Ž .Hp i j i j

FF˜ <qT CeH Ž pq1. - formp

m pr2q V dV , 2.1Ž . Ž .pr2q1 !Ž .

m nwhere g sE E h is the induced worldvolume˜i j i j m nˆ ˆ ˆm M m Nˆ ˆmetric where E sE Z E where E is the super-i i M M

vielbein. f is the dilaton field and FF are thei j

components of a modified worldvolume 2-form fieldstrength

˜FFsdVyB 2.2Ž .2

˜of the worldvolume 1-form gauge field V where B2

is the worldvolume 2-form induced by the super-˜space NS-NS 2-form gauge potential B . C is a2

˜Ž r . Žformal sum of worldvolume r-forms C rs1, 3,.5, 7, 9 induced by the superspace R-R r-form gauge

potentials C Ž r .. m is a mass parameter which is thedual of the 10-form field strength F Ž10. of a R-R

Ž9. w x Ž9-form C 18,12 . T is the ‘‘formal’’ but notp.physical tension of the D-p-brane which is given by

w x 616

1T s . 2.3Ž .pp 2pŽ .

Ž .We take the background of the action 2.1 to bew xthe D-8-brane solution given by 11,12

ds2 sH er2dx mdxnh qHy5 e r2y2dy2 ,mn

ef sH 5e r4 , C Ž9. sH e , 2.4Ž .01 PPP 8

m n Ž .where x and x m,ns0, . . . ,8 are the space-time coordinates parallel to the D-8-brane and y is asingle transverse coordinate. e is a nonzero parame-ter which cannot be determined by the equations ofmotions of 10D massive IIA supergravity. We note

Ž .that the solution 2.4 with esy1 is the standardform of the D-8-brane solution since it is obtainedvia T-duality from the other D-p-brane solutionsw x Ž .12 . HsH y is a harmonic function on y. In thispaper we set

m< <H y sc q y , 2.5Ž . Ž .1 < <e

which means that the D-8-brane lies at ys0. Wechoose c )0 and m)0 to avoid a singularity at1

ys0 and to get a real dilaton. We note that theŽ .solution 2.4 becomes the flat spacetime metric in

the massless limit

m ™ 02.6Ž .½ c ™ 1 via diffeomorphism .Ž .1

Ž . er8The Killing spinor of 2.4 has the form ´sH ´0

where ´ has a definite chirality, i.e. G ´ sq´0 y 0 0ˆfor y)0 and G ´ sy´ for y-0.y 0 0ˆ

We first consider a point charge solution of aD-4-brane worldvolume parallel to the background

6 We choose aX s1 in this paper.

Table 1

Ž .background D8 at ys0 : 0 1 2 3 4 5 6 7 8 –Ž .worldvolume D4 at ys y : 0 1 2 3 4 – – – – –0

fundamental string: 0 – – – – – – – – 9

D-8-brane. Since the solutions we construct here is abosonic one, we set fermionic coordinates u to bezero. Moreover, we consider the ansatz

° i ix sj is0,1, . . . ,4Ž .5 6 7 8x , x , x , x : constants~ 2.7Ž .ysy r )0Ž .

¢V sV r , V s0,Ž .0 0 i

24 iwhere r is defined as r' S j . We note that( Ž .is1

Ž .the upper two columns of 2.7 mean that the D-4-brane is embedded in the 1234-hyper-plane. That is,from target-space point of view, we consider theintersection of three branes shown in Table 1, where

Ž 9 .y is a positive constant and x sy .0

Then, since V sus0, SWZ does not contributei D4

to the equations of motion, and the equations ofmotion to solve are given by

BI BI BI BId LL d LL d LL d LLsE , sE .i jm md x dE x d V dE Vi i j i

2.8Ž .So, let us examine more about LL BI The inducedworldvolume metric g is given by˜i j

yH er2 0g s ,˜ er2 y3ey2i j 0 H P d qH E yE yž /i j i j

2.9Ž .5e r2w y3 ey2whose determinant is det g sH 1qH˜i j

Ž .2 x Ž .= E y . At this moment, det g qFF arising in˜i j i j

LL BI is very complicated. However, setting the con-dition

E ysE V , 2.10Ž .i i 0

Žresults in the simple form of the determinant: det gi j. 5e r2 7qFF sH . Then, the equations of motioni j

7 It is shown that no supersymmetry is preserved in this caseŽ .without 2.10 , by using a superalgebra via brane probe appearing

Ž Ž .. Ž .later Eq. 2.20 . So, 2.10 is considered to correspond to theBPS condition.

Ž .2.8 become the following two simple equations

E Hy3 ey2E y s0 2.11Ž .Ý Ž .i ii

E Hy2 ey1E y s0. 2.12Ž .Ý Ž .i ii

ŽŽ .2.11 arises from the 9th component of the formerŽ . Ž .of 2.8 , 2.12 from the time component of the latterŽ . .of 2.8 , and the others are solved. So, requiring the

two equations to be compatible, it needs to holdesy1, and the two are combined into one equation

E HE y s0. 2.13Ž .Ž .Ý i ii

We note that the harmonic function gives nontrivialŽ .contribution to 2.13 , which means that the equa-

tions of motion of the D-4-brane worldvolume areaffected by the background D-8-brane. By usingŽ . Ž .2.5 and y)0, the Eq. 2.13 is written as

2c2 1E yq s0. 2.14Ž .Ž .Ý i ž /mi

We choose the boundary condition

y ™ y )0Ž .0 2.15Ž .½ V ™ 00

for r™`, which means that the D-4-brane lies atysy . Then, the solution is obtained, with the fol-0

lowing unusual form, as1r22c c c1 2 1

y r s qy q y ,Ž . 0 2ž /m mr1r22c c c1 2 1

V s qy q y qy , 2.16Ž .0 0 02ž / ž /m mr

where c is a constant proportional to the electric2

charge of the gauge field. The electric charge Q is1w xdefined as 3

Q s wD , 2.17Ž .H1S3

where S is the n-sphere, w is the worldvolumenHodge dual and D is the 2-form defined by Di j s

d LL1 D 4y . Then, we haveT d F4 i j

Q s w HdV smc V , 2.18Ž . Ž .H1 2 3S3

where V is the volume of the unit n-sphere. Wen

note that in this definition the string coupling g sefs

is included in Q since LL BI is proportional to the1 D4

inverse of g .s

Since the solution has been obtained explicitly,we next give some pieces of evidence that the solu-tion corresponds to the fundamental string ending onthe D-4-brane. First, we discuss the massless limit of

Ž .the solution 2.16 . We assume here that the chargeŽQ is independent of m. The validity of the assump-1

.tion is discussed later. Then, in the massless limitŽ . Ž . Ž .2.6 , the solution 2.16 with 2.18 behaves as

Q Q1 1y™y q , V ™ . 2.19Ž .0 02 22V r 2V r3 3

Ž .The right hand side of 2.19 is exactly the solutionof a D-4-brane’s worldvolume in the 10D flat space-time, which corresponds to the fundamental stringw x1 . So, it is expected to correspond to a fundamentalstring. Next, we check the preserved supersymmetryof the solution by using ‘‘superalgebras in brane

w xbackgrounds via brane probes’’ 14,13,9 . The super-algebra in a D-8 brane background via a D-4-brane

w xprobe is given in Ref. 13 as

Qq,Qq s2 d4j P CCG mŽ .� 4 abHa b mMM4

4 Ž0.iq2 d j PP E y CCG GŽ .H i y 11 a bMM4

2T4 5r4 m m1 4q H dx PPP dxH4! MM4

= CCG GŽ .m PPP m 11 a b1 4

q2T H 5r4dx mdydV CCGŽ .H4 m y a bMM4

25r4q2T H dV CCGŽ . Ž .H ab4 11MM4

qOO u 2 , 2.20Ž . Ž .1q Gq ywhere Q s Q is the supercharge preserved ina a

2Ž .2.4 and MM is the worldspace of the D-p-brane.p

Ž0. iPP is almost equivalent to the conjugate momen-Žtum of V . The contributions of the Chern-Simonsi

.term are subtracted. Substituting the solution for the

Ž .right hand side of 2.20 , the superalgebra can bewritten as

1qCCG Gˆ ˆ ˆ ˆ1234 11q q 4 1r4Q ,Q s4T d j H� 4 Ha b 4 ž /2MM ab4

1qCCG Gy 11ˆ25r4qH E y .Ž .i ž /2 ab

2.21Ž .

Since the three gamma matrix products G ,CCG Gˆ ˆ ˆ ˆy 1234 11ˆŽ Ž ..and CCG G arising in 2.21 commute with eachy 11ˆ

other, all of them can be simultaneously diagonal-ized. Since the square of each matrix product isequal to the identity and each is traceless, both of the

1q CCG G1q CCG G y 11ˆ ˆˆ ˆ ˆ1234 11matrices and are projection op-2 2

erators. So, we conclude that the solution has 1r8supersymmetry, hence is consistent with the target-space interpretation. This also shows that the solu-tion is a BPS state.

Now, we discuss the mass of the string. Let usfirst consider the energy of the solution. For thispurpose, we pass to the Hamiltonian formalism as

w xdone in Ref. 1 . If we assume that V is purelyelectric and that only the scalar y is excited, SD4

reduces to

2 2 25 2 2S s yT d j 1 y H F 1 q H E y q H F E y y HyŽ . Ž . Ž . ˙(� 4 � 4HD 4 4 0i i 0i i

2.22Ž .

Ž .where H is the harmonic function . The canonicalmomenta of y and V , are defined respectively asi

T Hy4Ps ,

2 2 22 21y H F 1 q H E y q H F E y y HyŽ . Ž . Ž . ˙(� 4 � 40i i 0i i

2T H F 1 q H E y y HE y F E yŽ . Ž .w x� 44 0i i i 0 j j

P s .i2 2 22 21y H F 1 q H E y q H F E y y HyŽ . Ž . Ž . ˙(� 4 � 40i i 0i i

2.23Ž .

4The Hamiltonian H is constructed as H'H d j HHMM4

where HH is the Hamiltonian density given by

2 2 2y2 y1 2 y2 y1 y2HHsT 1 q H E y 1 q T H P q T H P q T P E y .Ž . Ž . Ž .Ž .(� 44 i 4 4 i 4 i i

2.24Ž .

We note that P is subject to the constraint E P s0.i i iŽ .Substituting the solution 2.16 for H, we can obtain

the energy of the solution E defined on the world-volume. We note that for a BPS solution like thiscase, it generically happens that the square root of Hbecomes a perfect square and that the energy be-comes a sum of the two parts: the part originatingfrom the D-p-brane and that from the string. so, wedenote the first part of HH as HH and the second part1

as HH . Concretely, the energy in this case takes the2

form

24EsT d j 1qH E y 'E qE . 2.25Ž .Ž .H4 i 1 2MM4

The first term E is the ‘‘energy’’ of the D-4-brane1

itself, and the second term E is the energy of the2Ž .excitation i.e. the string , both defined on the world-

volume. Since we are interested in the second part,we compute only E here. E is infinite in this case,2 2

but if we regularize it by introducing a small parame-ter d , we can get the energy for rGd as

` 23E sT dV r dr c qmy r E y rŽ . Ž .Ž . Ž .H H2 4 3 1 id

sT Q y d yy . 2.26Ž . Ž .Ž .4 1 0

Ž .That is, the energy is again proportional to thedifference of the coordinate. Thus, we conclude thatthe energy defined on the brane’s worldÕolume doesnot agree with the mass of the string in the case ofD-branes in curÕed backgrounds. We note that thisresult is rather reasonable, in a sense, in the case ofg /1, because the energy has the same transforma-˜00

tion property as E under the reparametrization of0

j 0.Now, we construct the mass formula. Since it

should be invariant under the reparametrization ofj 0, we propose the generalized mass formula M fora string described as a solution of a D-p-brane’sworldvolume, as

p 00(Ms d j yg HH , 2.27Ž .˜H 2MMp

where HH is the second part of the Hamiltonian2Ždensity defined on the D-p-brane originating from

.the excitation corresponding to a string .

Let us calculate the mass of the string in this case,based on the formula. Substituting the solution forŽ .2.27 , we find

24 5r4M d sT d j H E yŽ . Ž .H4 iMM4

` 25r43sT dV r dr c qmy r E yŽ .Ž . Ž .H H4 3 1 id

5r84T Q c4 1 22s c qmy qŽ .1 0½ 525m d

5r4y c qmy . 2.28Ž . Ž .1 0

On the other hand, the geodesic distance l from theŽ .D-4-brane lying at y to the point parametrized by0

Ž .y d is given by

Ž .y dl y d ; y ' g dyŽ .Ž . (H0 y y

y0

5r4c14 1r4s m y d qŽ .5 ž /m

5r4c1y y q . 2.29Ž .0ž /m

So, we obtain the proportional relation:

MsT Q P l y d ; y . 2.30Ž . Ž .Ž .4 1 0

Furthermore, we can show that the coefficientT Q reproduces the tension of the fundamental string4 1

correctly. To derive this, we discuss the unit electricŽ . Ž .charge for a 1,0 i.e. a fundamental string. First,

we review the discussion about the case of pointcharge solutions of the D-p-brane in the flat space-

w x Ž w x.time 1 and Ref. 19 . Let us consider a tripleŽ . Ž .junction of strings: a 0,1 string, a n,0 string and a

Ž .n,1 string. If the string coupling g is small, its

holds

2 2DT g nŽ .ss , 2.31Ž .

T 2Ž0,1.

Ž .where T is the tension of a p,q string and DTŽ p,q .is the additional tension DT'T yT . On theŽn,1. Ž0,1.other hand, the solution of a D-1-brane worldvolumecorresponding to the above string junction is given in

w xRef. 19 . In the flat background with the ansatz that

0 0 1 1 9 Ž 1. mx sj , x sj , x sy j , x s constant for ms2, . . . ,8, and us0, the D-1-brane action is writtenas

T1 2 2 22 (S sy d j 1q E y y F y E y .Ž . Ž . Ž .HD1 1 01 0gs

2.32Ž .

The solution of the D-1-brane’s worldvolume as thetriple string junction with an electric charge q is1w x19

yq j for j )01 1 1y j sV j s 2.33Ž . Ž . Ž .1 0 1 ½ 0 for j -0.1

The energy of the solution can be computed by usingthe Hamiltonian, and the additional tension DT isalso derived from this correctly. By taking into

Ž . w xaccount the bending of the n,1 string 19 , it isŽ .Ž .2obtained as DTrT s 1r2 F . Comparing thisŽ0,1. 01

Ž .with 2.31 , the ‘‘charge quantization condition’’ŽF sq sg n is deduced for a point charge q for01 1 s 1

.an integer n . By T-dualizing with respect to thedirections of x m for ms2, . . . ,8, the charge quanti-zation condition for a electric point charge q of the1

D-p-brane’s worldvolume is shown to be

1q ' F sg n. 2.34Ž .H1 0 r spy1

S2pŽ . py1

Next, we discuss the case which is related byŽ .T-duality to the case of 2.16 . Let us suppose a

D-1-brane parallel to a subspace of the worldvolumeof a D-5-brane background, and that some number nof fundamental strings are absorbed in the D-1-brane.ŽThis is also a BPS configuration since 1r8 space-

w x .time supersymmetry is preserved 20 . If we con-sider a D-5-brane background solution, the stringcoupling g becomes a local function on the trans-s

a Ž f y1r2verse coordinates y g se sH where H issa.the harmonic function on y , and so is the tension

T . If the ‘‘test’’ D-1-brane is put near the D-5-Žn,1.brane, the string coupling g is considered to bes

sufficiently small around the D-1-brane. So, the Eq.Ž . y1r22.31 with g sH holds on the basis of thes

same discussion. On the other hand, suppose that aelectric point charge qX is added to the D-1-brane’s1

worldvolume with an excitation of a scalar field y9.Then, the D-1-brane action is

2 222 9 9( Ž .S syT d j 1q H E y y H F y H E y .Ž . Ž .HD1 1 1 01 0

2.35Ž .

If we assume the existence of the correspondingpoint charge solution, the additional tension due tothe field strength is derived in the same way, as

DT 1 2 21y1s PH P s PH F .Ž . Ž .1 0122T 2 TŽ .Ž0,1. 1

2.36Ž .Ž .So, comparing this with 2.31 we have the charge

quantization condition H 1r2F sng for an integer01 s1n. By using the the ‘‘electric induction’’ D sy P01T1

d LLD1 Ž .sHF in this case it can be rewritten in a01dF01

more generic form as

D sn. 2.37Ž .01

The higher dimensional D-brane cases are related toŽ . 82.37 by T-dualities . So, we have

1D sn. 2.38Ž .H 0 rpy1

S2pŽ . py1

This is the ‘‘generalized’’ charge quantization condi-tion for the point charge of the D-p-brane parallel to

Ž .the worldvolume of a D- pq4 -brane background.Ž .Let us return to to the case of 2.16 . the left hand

Ž .side of 2.38 with ps4 is equivalent to the chargeŽ . Ž .3Q of 2.16 multiplied by 1r 2p . So, the unit1

Ž .charge Q for a 1,0 string in this case is Q s1 1Ž .32p . That is, the unit charge is the same as the one

Žin the case of the flat background, This result isconsistent from physical point of view, since the unitcharge Q is considered to be independent of the1

.background. Thus, the proportional coefficient ofŽ . Ž .2.30 for a 1,0 string is obtained as T Q s1r2p ,4 1

which is exactly the tension of the string. So, theŽ .quantity M defined in 2.27 certainly gives the mass

of the string correctly!As another case, we consider the case of a test

D-8-brane parallel to the background D-8-brane. The

8 We note that the background fields should also be trans-formed by T-dualities simultaneously.

Ž .worldvolume action is given in 2.1 for ps8. Inthis case, we consider the ansatz

° i ix sj is0,1, . . . ,8Ž .~ysy r )0Ž . Ž .¢V sV r , V s0,Ž .0 0 i

28 iwhere r is defined as r' Ý j . Then, com-( Ž .is1

˜Ž9.bined with us0, only the term including C inSWZ does contribute to the equations of motionD8

˜Ž9. eŽ .C s H . The expression of the induced01 PPP 8Ž .worldvolume metric g is the same as 2.9 except˜i j

Žfor the range of the indices in this case i,js. Ž .1,2, . . . ,8 . Setting the same condition as 2.10 makes

Ž .the determinant of g qFF simple, such as˜i j i jŽ . 9e r2det g qFF sH . Then, we find the equations˜i j i j

of motion

E Hy2 ey2E y s0, 2.39Ž .Ý Ž .i ii

E Hyey1E y s0. 2.40Ž .Ý Ž .i ii

So, these two equations are again compatible only ifesy1, and the equations to solve become a singleequation

2E ys0. 2.41Ž .Ž .Ý i

i

We note that unlike the D-4-brane case, the har-Ž .monic function H does not appear in 2.41 . So,Ž .choosing the same boundary condition as 2.15 , the

solution is obtained easily as

cX cX2 2

ysy q , V s , 2.42Ž .0 06 6r r

where cX is a constant proportional to the electric2

charge of the solution. By using the definition of theŽ .charge similar to 2.17 , we have the electric charge

QX :1

1 d LLD4X XQ 'y w s w dV s6c V .Ž .H H1 2 7ž /T dFS S8 i j7 7

2.43Ž .

We note that the form of this point charge solutionŽ .2.42 is completely the same as that of the D-8-braneworldvolume in the flat background.

Here, We derive the preserved supersymmetry ofŽ .the solution 2.42 in the same way. 1r4 supersym-

metry is expected to be preserved. The superalgebrain a D-8 brane background via a D-8-brane probe isw x13

Qq,Qq s2 d8j P CCG mŽ .� 4 abHa b mMM8

8 Ž0.iq2 d j PP E y CCG GŽ .H i y 11 a bMM8

2T8 5r4 m m1 5q H dx PPP dx dyH5! MM8

=dV CCGŽ .m PPP m y a b1 5

2T8 5r4 m m1 4q H dx PPP dxH4! MM8

2= dV CCG GŽ . Ž .m PPP m 11 a b1 4

q2T H 5r4dx mdyH8MM8

3= dV CCGŽ . Ž .m y a b

45r4q2T H dV CCGŽ . Ž .H ab8 11MM8

qOO u 2 . 2.44Ž . Ž .The momentum P includes the following twom

terms:

T8Ž0.P sP qm m 8!

=Hy2 mye 0 i1 PPP i8E xn1 PPP E xn 8e ,i i mn PPP n y1 8 1 8

2.45Ž .

where P Ž0. is the contribution of SBI and them D8WZ Ž 01 PPP 8 .second term is that of S where e s1 .D8

Substituting the solution for the right hand side ofŽ .2.44 , the superalgebra can be written as

Qq,Qq� 4a b

1qCCG G2 y 11ˆ8 1r4s4T d j H E y .Ž .H8 i ž /2MM ab8

2.46Ž .

By the same discussion as done in the D-4-branecase, it is shown that 1r4 supersymmetry is pre-served in this configuration, which is consistent withthe spacetime interpretation. So, we interpret the

Ž .solution 2.42 as a fundamental string again.Ž .Applying the mass formula 2.27 to this case, we

obtain the result:

200 8 1r4(M' yg HH sT d j H E y˜ Ž .H H2 8 iMM MM8 8

` 21r47sT dV r dr c qmy r E yŽ .Ž . Ž .H H8 7 1 id

sT QXP l y d ; y . 2.47Ž . Ž .Ž .8 1 0

Based on the same discussion as that done in theD-4-brane case, the unit electric charge QX for a1Ž .1,0 string ending on a D-8-brane can also be

X Ž .7derived as Q s 2p . So, the tension of the string1Ž .is reproduced correctly again, and 2.27 also gives

Ž .the mass of the string for the solution 2.42 cor-Ž .rectly. Therefore, we conclude that 2.42 is the

correctly generalized mass formula, which holdswhen the background of the D-brane is curved. Wenote that the energy defined on the worldvolumeagain gives the difference of the coordinate y multi-plied by T QX . That is, this case is another example8 1

that the worldvolume energy does not agree with themass of the string.

3. Summary and discussions

Ž .In summary, we have proposed 2.42 as thegeneralized mass formula which holds when thebackground of the D-brane is curved, and haveshown explicitly using the two examples that theformula certainly gives the mass of a fundamentalstring described as a BPS solution of a D-brane’sworldvolume. In addition, based on the obtainedformula, we can see that the mass of the stringagrees with the worldvolume energy only in the

Žcases g sy1 where g is the induced worldvol-˜ ˜00 i j.ume metric . which include the case discussed in

w xRef. 1 .Here, we discuss the consistency of the mass

formula from another point of view, especially focus-00(ing on the factor yg . Suppose we consider the˜

D-4-brane embedded parallel to the D-8-brane back-Ž .ground 2.4 with no excitation of worldÕolume fields

Ž i i 5 8i.e. x sj for is0, . . . ,4 and x , . . . , x , y: con-.stants . Then, on the analogy of the mass formula of

Ž . w xa p,q string given by Sen in Ref. 10 , the target-space mass m of the D-4-brane should be propor-D4

tional to its spatial volume element measured by thegeodesic distances in the spacetime. So, it should begiven by

X4 yfm s d j T e det g , 3.1Ž .˜H (D4 4 i jMM4

where gX

is the induced world-space metric of the˜ i j

D-4-brane. In fact, the mass m obtained in thisD4

way can be shown to agree with the quantity MD4

defined as

4 00 4 1r4(M s d j yg HH s d j H P1 , 3.2Ž .˜H HD4 1 ž /MM MM4 4

Žwhere HH is the first part i.e. originating from the1.D-4-brane of the Hamiltonian density of the solu-

Ž .tion 2.16 defined on the D-4-brane. This meansthat the information of the D-4-brane mass can also

Ž .be extracted from the solution 2.16 by integratingthe Hamiltonian density multiplied by the factor

00(yg with respect to world-space coordinates.˜00(Thus, the factor yg arising in the formula is˜

consistent in this sense, too.Finally, let us discuss the worldvolume interpreta-

Ž .tion of the mass formula 2.27 . If we define a0worldvolume proper time t as dt' yg dj , the( ˜00

00(Hamiltonian density multiplied by yg might be˜regarded as the energy density defined with respectto t . So, we might say that from worldvolume pointof view, the mass of the string is interpreted as ‘‘theenergy defined with respect to the worldvolumeproper time’’.

Acknowledgements

I would like to thank Taro Tani for useful discus-sions and encouragement. I would also like to thankY. Imamura, Tsunehide Kuroki and Shinya Tamurafor useful discussions in computing the energy of thesolution.

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