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Four-dimensional High-Branes as Intersecting D-Branes

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UG-7/96MRI-PHY/96-27September 1996hep-th/9609056Four-dimensional High-Branes as Intersecting D -Branes E.BERGSHOEFF,M.DE ROO Institute for Theoretical Physics University of Groningen,Nijenborgh 49747AG Groningen The Netherlands and S.PANDA Mehta Research Institute of Mathematics &Mathematical Physics 10Kasturba Gandhi Marg Allahabad 211002India ABSTRACT We show that a class of extremal four-dimensional supersymmetric

“highbranes”,i.e.string and domain wall solutions,can be interpreted as intersections of four ten-dimensional Dirichlet branes.These d =4solu-tions are related,via T -duality in ten dimensions,to the four-dimensional extremal Maxwell/scalar black holes that are characterized by a scalar cou-pling parameter a with a =0,1/√ 3.

1.Introduction

The construction of solutions to the Einstein equations,including mat-

ter in the form of scalar?elds and gauge?elds,has advanced rapidly in

recent years due to the developments in string theory.Solution generating

transformations,which follow from the T and S dualities of string theory,

produce quite intricate p-brane solutions from relatively simple ones1.

Some extremal solutions can be understood in terms of bound states[2,

3,4,5]or as intersections of D-branes(M-branes)in ten(eleven)dimensions

[6,7,8,9,10,11].The construction of lower dimensional solutions in terms

of ten-dimensional ones suggests a possible p-brane classi?cation scheme.In

this letter we provide further support for this point of view by showing that

not only black holes(p=0)but also a class of extremal string(p=1)

and domain wall(p=2)solutions in four dimensions can be understood as

intersections of ten-dimensional D-branes.Furthermore,we show that all

p=0,1,2solutions are related to each other via T-duality in ten dimensions,

thereby providing a unifying picture for all these d=4solutions.

The black hole solutions to string e?ective actions in diverse dimensions

have drawn the most attention in this context2.The four supersymmetric,

single charge,extreme dilaton black hole solutions in four dimensions can

all be understood in terms of the intersection of Dirichlet3-branes in ten

dimensions.This intersection is described by a ten-dimensional metric con-

taining four independent harmonic functions[7,10].The single charge black

hole solutions that are characterized by a scalar coupling parameter a with √3can be obtained by the identi?cation or absence of some a=0,1/

of these harmonic functions.Black hole solutions in terms of four harmonic

functions were obtained earlier in[2,3],and generalized to p>0-branes

in[4,5].These solutions generically contain several scalar?elds,and are

generalizations of a class of solutions discussed in[13].

General p-branes in d dimensions can be divided in two categories:(i)

“low-branes”(p=0,...,d?4),which couple to the fundamental(or dual)

gauge?elds of the underlying supergravity theory and(ii)“high-branes”

(p=d?3,d?2)which,in contrast to the low-branes,are not asymptotically

?at.The p-branes whose charge is carried by a NS-NS gauge?eld have been

understood for some time.The p-branes coupling to RR?elds or their

duals are represented by the so-called Dirichlet(D)-branes[14,15].In ten

dimensions D-branes exist for all p=0,...,9.They correspond to solutions

of the IIA(IIB)supergravity theories for p even(odd),with the following

string frame metric and dilaton:

ds2S,10=H?1/2dx2(p+1)?H1/2dx2(9?p),

e2φ=H(3?p)/2,(1) where H is a harmonic function depending on the9?p transverse coordi-

nates.These D-branes are related by T-duality,which turns a p-brane into a p+1-brane solution,by[16,17]:

g′μν=gμν,g′xx=1/g xx,e2φ′=e2φ|g xx|?1,(2) where x is one of the transverse coordinates,and it is understood that H is independent of x.This duality transformation acts between the IIA and IIB theories.

The four-dimensional black holes can be understood as an intersection of four D3-branes:3⊥3⊥3⊥3[7,10].We will construct the four-dimensional high-branes(strings and domain walls)in terms of similar intersections of d=10D-branes.That will be done in the next section.In section3we will discuss our results.

2.Strings and domain walls as D-brane intersections

The metric and the dilaton of intersecting D-branes in ten dimensions have a structure involving products of the individual harmonic functions. The possible intersections of two D-branes were investigated in[8,9].If a p+r and a p+s brane intersect over a p-brane,the metric has an overall world volume part(p+1coordinates),and at most9?p?r?s overall transverse coordinates.Generically,an intersection of N D-branes leads to a con?guration with32/2N unbroken supersymmetry generators3.

The3⊥3⊥3⊥3intersection has the property that each pair of3-branes intersect over a1-brane,and that these1-branes intersect over a0-brane. There are then one overall world-volume(time–)coordinate with metric component(H1H2H3H4)?1/2and three overall transverse coordinates with metric components(H1H2H3H4)1/2.The harmonic functions depend on the overall transverse coordinates only.Reduction over the remaining six dimensions results in a four dimensional solution involving four intersecting 0-branes,and produces the solution of[2,3],involving three scalar?elds4.

Using T-duality over the overall transverse directions,we can turn the 3⊥3⊥3⊥3intersection into a4⊥4⊥4⊥4and5⊥5⊥5⊥5intersection which intersect over a1-brane and2-brane,respectively.The reduction of these T-dual solutions to four dimensions naturally results into string and domain wall solutions involving four independent harmonic functions.Given the fact that3⊥3⊥3⊥3is a solution in d=10(which indirectly follows from [2,3]),and that T-duality and reduction does not change this property,the four-dimensional strings and domain walls,involving four scalars and four vector?elds,also satisfy the equations of motion.

The ten-dimensional solution with four intersecting(p+3)-branes(p=

0,1,2)has the following string frame metric and dilaton:

ds 2S,10=1H 1H 2H 3H 4dx 2(p +1)

? H 3H 4

H 2H 4H 2H 3H 1H 2H 1H 3H 1H 4

|g | R +1

2(p +5)!

e ?pφ/2F 2(p +5) .(7)

In the reduction to four dimensions we parametrize the metric as follows:

g E,10= e 2Σg E,4

00h ,(8)

where h is the six-dimensional internal metric and

e ?2Σ=(det h )1/2.(9)

This leads to the following four-dimensional lagrangian for metric and scalar ?elds:L E,4= 2

(?φ)2+2(?Σ)2?15

For even p this is

IIB supergravity,for which there is no action.Here we employ the pseudo-action de?ned in [18],which one can freely use in dimensional reduction.

The reduction of the ten-dimensional gauge?elds is straightforward. Here we only discuss the coupling of the resulting d=4gauge?elds to the scalars.The four harmonic functions H i contain four charges Q i,which correspond to the solution for four d=4(p+2)-form tensor?eld-strengths. Therefore the reduction of the ten-dimensional(p+5)-form?eld-strength comprises three direct,and three double dimensional reductions.The so-lution(3,5)tells us for which coordinates there is a double dimensional reduction:4,5,6for H1,4,8,9for H2,5,7,9for H3and6,7,8for H4.

The resulting d=4action takes on the form:

L E,4= 8 (?χ)2+(?σ)2+(?τ)2 +

(?1)p+1

H1H2,eσ=

H2H4

H1H4

.(15)

For simplicitly,we refrain from giving the expressions for the(p+2)-form ?eld-strengths.

The multiscalar action(12)and the solution(13-15)reproduce the solu-tions given in[4,5](where also the solution for the gauge?elds are given). Special cases are obtained by setting one or more of the harmonic functions equal to unity.In this way one?nds solutions with one,two or three indepen-dent harmonic functions.If we identify the remaining harmonic functions, the resulting solutions can be expressed in terms of a single scalar?,and the action takes on the form

L E,4= 2(??)2+(?1)p+1

and for which the solution reads:

ds2E,4=H(p?1)N/2dx2(p+1)?H(p+1)N/2dx2(3?p),

√N2(p2?1)+4N.(18) F0,1,...,p,i=

These results agree with the single harmonic solutions obtained in[19].The domain wall solutions(p=2)were also obtained in[20](for a recent review, see[21]).

3.Discussion

In this letter we have reduced a ten-dimensional solution by standard Kaluza-Klein techniques to four dimensions.The resulting solution can be interpreted as a con?guration of four p-branes(p=0,1,2)in four dimen-sions.The four-dimensional metric components are the overall world-volume and the overall transverse components of the ten-dimensional metric.

Note however,that the four dimensional theory which we obtain for p= 2contains four three-index gauge?elds,or four independent cosmological constants after dualization of F i,(4).Clearly the d=4lagrangian is then not a standard d=4supergravity lagrangian6.The three-index gauge?elds arise from the Kaluza-Klein reduction of the d=10six-index gauge?eld (which is the dual of the IIB RR two-index gauge?eld).This is the source of the cosmological constants in the domain wall case.

Alternatively,one might have performed a duality transformation in d= 10,turning the six-form gauge?eld into a two-index?eld.Then,in the reduction to d=4,no3-form gauge?elds appear,and,at?rst sight,no cosmological constants.However,now the reduction gives rise to additional scalars,which exhibit a shift symmetry that can be used in a Scherk-Schwarz reduction[22]to generate independent cosmological constants.This version of the Scherk-Schwarz technique was recently applied to the d=10IIB theory[23],and is extensively discussed,in the context of domain walls in diverse dimensions,in[24].Cosmological constants can also be obtained directly in supergravity theories by suitably chosen gaugings of its global symmetries[25].In fact,it turns out that the results of the Scherk-Schwarz reduction can be recaptured by such gaugings in lower dimensions[26].

It is natural to generalize the results of the present work to six dimen-sions.To obtain d=6solutions with a similar structure we can only use intersections of two D-branes in d=10[8].The intersection of two(p+2)-branes in d=10gives rise to p-brane solutions in d=6with p=0, (4)

In this case the lagrangian is

L E,6= 4(?χ)2+

e?(p?1)φ e?χF21,(p+2)+eχF22,(p+2) .(19)

The solution for the metric and scalars is

ds2E,6=(H1H2)(p?3)/4dx2(p+1)?(H1H2)(p+1)/4dx25?p,

eφ=(H1H2)(1?p)/4,eχ=

[3]M.Cvetic and A.A.Tseytlin,Solitonic strings and BPS saturated dy-

onic black holes,hep-th/9512031.

[4]N.Khviengia,Z.Khviengia,H.L¨u and C.N.Pope,Intersecting M-

branes and Bound States,hep-th/9605077.

[5]M.J.Du?and J.Rahmfeld,Bound states of black holes and other

p-branes,hep-th/9605085.

[6]G.Papadopoulos and P.K.Townsend,Phys.Lett.B380(1996)273.

[7]A.A.Tseytlin,Harmonic superpositions of M-branes,hep-th/9604035;

I.R.Klebanov and A.A.Tseytlin,Intersecting M–branes as four-

dimensional black holes,hep-th/9604166.

[8]K.Behrndt,E.Bergshoe?and B.Janssen,Intersecting D–Branes in

Ten and Six Dimension,hep-th/9604168.

[9]J.P.Gauntlett,D.A.Kastor and J.Traschen,Overlapping Branes in

M–Theory,hep-th/9604179.

[10]V.Balasubramanian and https://www.sodocs.net/doc/394987017.html,rsen,On D-Branes and Black Holes in

Four Dimensions,hep-th/9604189.

[11]K.Behrndt and E.Bergshoe?,A Note on Intersecting D-branes and

Black Hole Entropy,hep-th/9605216.

[12]A.A.Tseytlin,Composite black holes in string theory,talk at the

Second International Sakharov Conference in Physics,Moscow1996, gr-qc/9608044.

[13]H.L¨u and C.N.Pope,Multi-scalar p-brane solutions,hep-th/9512153.

[14]J.Polchinski,Phys.Rev.Lett.75(1995)184.

[15]J.Polchinski,S.Chaudhuri and C.V.Johnson,Notes on D-branes,

hep-th/9602052.

[16]E.Bergshoe?,C.M.Hull and T.Ort′?n,Nucl.Phys.B451(1995)547.

[17]E.Bergshoe?and M.de Roo,Phys.Lett.B380(1996)265.

[18]E.Bergshoe?,H.J.Boonstra and T.Ort′?n,Phys.Rev.D53(1996)

7206.

[19]H.L¨u,C.N.Pope,E.Sezgin and K.S.Stelle,Nucl.Phys.B456(1995)

669.

[20]M.Cvetic,Phys.Lett.B341(1994)160.

[21]M.Cvetic and H.H.Soleng,Supergravity domain walls,

hep-th/9604090.

[22]J.Scherk and J.H.Schwarz,Phys.Lett.B82(1979)60.

[23]E.Bergshoe?,M.de Roo,M. B.Green,G.Papadopoulos and

P.K.Townsend,Nucl.Phys.B470(1996)113.

[24]P.M.Cowdall,H.L¨u,C.N.Pope,K.S.Stelle and P.K.Townsend,

Domain walls in Massive Supergravities,hep-th/9608173.

[25]A.Salam and E.Sezgin,Supergravities in diverse dimensions,North-

Holland/World Scienti?c(1989).

[26]E.Bergshoe?,M.de Roo and E.Eyras,in preparation.

[27]R.R.Khuri and T.Ort′?n,Phys.Lett.B373(1996)56;R.R.Khuri and

T.Ort′?n,Nucl.Phys.B467(1996)355.

若变量均已正确定义并赋值

若变量均已正确定义并赋值,以下合法的C语言赋值语句是(c)。 A、x+n=I; B、x==5; C、x=n/2.5; D、5=x=4+1; 下述错误的C语言常量是:( c). A、0xf B、5. C、090 D、.25 根据下面的程序,使a=123,c1='o',c2='k'哪一种是正确的键盘输入方法?(b )。(规定用字符串[CR]表示回车,U表示空格) main() {int a; char c1,c2; scanf("%d%c%c", &a,&c1,&c2); } A、123UoUk[CR] B、123ok[CR] C、123Uok[CR] D、123oUk[CR] 已知键入a=4,b=3,执行下面程序输出的结果是( b). main() { int a,b,s; scanf("%d%d",&a,&b); s=a; if(a>b) s=b; s=s*s; printf("%d\n", s); } A、13 B、9 C、16 D、15 执行下面程序段后,i的值是( a). int i=10; switch(i) {case 9: i+=1; case 10: i--; case 11: i*=3; case 12: ++i; } A、28 B、10 C、9 D、27 以下不是无限循环的语句为（d）。

A、while(1){x++;} B、for(;;x++=i); C、for(i=10;;i--) sum+=i; D、for(y=0,x=1;x>++y;x=i++) i=x 下面程序段运行的结果是(a ). x=-1; do{ x=x*x; }while (!x); A、循环执行一次 B、有语法错误 C、死循环 D、循环执行两次若有说明:int a[][3]={1,2,3,4,5,6,7};则a数组第一维的大小是（a）。 A、3 B、无确定值 C、4 D、2 若定义int a[12]={1,2,3,4,5,6,7,8,9,10,11,12};char c='a',d,g; 则数值为4的表达式是 ( d). A、a[4] B、a[g-c] C、a['d'-'c'] D、a['d'-c] 以下叙述中正确的是（c）。 A、可以在一个函数中定义另一个函数 B、main()函数必须放在其它函数之前 C、构成C程序的基本单位是函数 D、所有被调用的函数一定要在调用之前进行定义下述函数的功能是(c ). int fun(char *x) { char *y=x; while(*y++); return y-x-1; } A、比较两个字符串的大小 B、求字符串存放位置 C、求字符串的长度 D、将字符串x连接到字符串y后面有关"文件包含"命令说法,下列叙述正确的是( a). A、将预处理的结果作为一个源程序编译,得到一个目标文件. B、一个 include 命令可指定多个被包含文件. C、将预处理的结果作为多个文件编译. D、被包含的文件可以是源文件或目标文件. 在声明语句:int *f();中，标识符f代表的是（a）。 A、一个返回值为指针型的函数名 B、一个用于指向函数的指针变量 C、一个用于指向一维数组的行指针 D、一个用于指向整型数据的指针变量

Four-dimensional High-Branes as Intersecting D-Branes

若变量均已正确定义并赋值

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