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PHYSICAL REVIEW D, VOLUME 65, 026004
Exact string solutions in nontrivial backgrounds
P. Bozhilov*Department of Theoretical and Applied Physics, Shoumen University ‘‘Konstantin Preslavsky,’’ 9712 Shoumen, Bulgaria
~Received 27 March 2001; published 21 December 2001!
We show how the classical string dynamics in aD-dimensional gravity background can be reduced to thedynamics of a massless particle constrained on a certain surface whenever there exists at least one Killingvector for the background metric. We obtain a number of sufficient conditions, which ensure the existence ofexact solutions to the equations of motion and constraints. These results are extended to include the Kalb-Ramond background. TheD1-brane dynamics is also analyzed and exact solutions are found. Finally, weillustrate our considerations with several examples in different dimensions. All of this also applies to thetensionless strings.
DOI: 10.1103/PhysRevD.65.026004 PACS number~s!: 11.25.2w, 04.25.2g, 11.10.Lm, 11.27.1d
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I. INTRODUCTION
The string equations of motion and constraints in curvspace-time are highly nonlinear and,in general, not exactlysolvable. Different methods have been applied to solve thapproximately@1–5# or, if possible, exactly@6–11#. On theother hand, quite general exact solutions can be foundusing an appropriate ansatz, which exploits the symmetof the underlying curved space-time@9,12–34#. In mostcases, such an ansatz effectively decouples the dependon the spatial world-sheet coordinates @12–26# or thedependence on the temporal world-sheet coordinatet@19,22,27–32#. Then the string equations of motion and costraints reduce to nonlinear coupledordinary differentialequations, which are considerably simpler to handle thaninitial ones.
In this article, we obtain some exact solutions of the clsical equations of motion and constraints for both tensilenull strings in aD-dimensional curved background. Thisdone by using an ansatz, which reduces the initial dynamsystem to the one depending on only one affine paramThis is possible whenever there exists at least one Killvector for the background metric. Then we search for sucient conditions, which ensure the existence of exact stions to the equations of motion and constraints withouting particular metric. These results are extended to inclthe Kalb-Ramond two-form gauge field background. TD-string dynamics are also analyzed and exact solutionsfound. After that, we give several explicit examples in fofive, and ten dimensions. Finally, we conclude with a discsion of the derived results.
II. REDUCTION OF THE DYNAMICS
To begin with, we write down the bosonic string actionD-dimensional curved space-timeMD with the metric tensorgMN :
S5E d2jL, L52T
2A2ggmn]mXM]nXNgMN~X!,
*Email address: [email protected]
0556-2821/2001/65~2!/026004~12!/$20.00 65 0260
d
m
yes
nce
-
e
-d
aler.g-u--e
ere,-
]m5]/]jm, jm5~j0,j1!5~t,s!, m,n50,1,
M ,N50,1, . . . ,D21, ~1!
where, as usual,T is the string tension andg is the determi-nant of the auxiliary metricgmn .
Here we would like to consider tensile and null~tension-less! strings on equal footing, so we have to rewrite the ation ~1! in a form in which the limitT→0 can be taken. Tothis end, we set
gmn5S 21 l1
l1 2~l1!21~2l0T!2D ~2!
and obtain
L51
4l0gMN~X!~]02l1]1!XM~]02l1]1!XN
2l0T2gMN~X!]1XM]1XN.
The equations of motion and constraints following from thLagrangian density are
]0F 1
2l0~]02l1]1!XKG2]1F l1
2l0~]02l1]1!XKG
11
2l0GMN
K ~]02l1]1!XM~]02l1]1!XN
52l0T2@]12XK1GMN
K ]1XM]1XN1~]1ln l0!]1XK#,
gMN~X!~]02l1]1!XM~]02l1]1!XN
1~2l0T!2gMN~X!]1XM]1XN50, ~3!
gMN~X!~]02l1]1!XM]1XN50, ~4!
where
GMNK 5
1
2gKL~]MgNL1]NgML2]LgMN!
©2001 The American Physical Society04-1
ris
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a
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th
n
par-
atz
the
d.iled
tter.
er,
t
P. BOZHILOV PHYSICAL REVIEW D 65 026004
is the connection compatible with the metricgMN(X). Wewill work in the gauge lm5const in which the Euler-Lagrange equations take the form
~]02l1]1!~]02l1]1!XK1GMNK ~]02l1]1!XM
3~]02l1]1!XN5~2l0T!2~]12XK1GMN
K ]1XM]1XN!.
~5!
Now we are going to show that by introducing an appropate ansatz, one can reduce the classical string dynamicthe dynamics of a massless particle constrained on a cesurface whenever there exists at least one Killing vectorthe background metric. Indeed, let us split the indexM5(m,a), $m%Þ$B% and let us suppose that there existnumber of independent Killing vectorshm . Then in appro-priate coordinateshm5]/]xm and the metric does not depend onXm. In other words, from now on we will work withthe metric
gMN5gMN~Xa!.
On the other hand, we observe that
XM~t,s!5F6M@w6~t,s!#, w6~t,s!5~l162l0T!t1s
are solutions of the equations of motion~5! in arbitraryD-dimensional backgroundgMN(XK), depending onD arbi-trary functionsF1
M or F2M ~see also@6#!. Taking all this into
account, we propose the ansatz
Xm~t,s!5C6m w61ym~t!, C6
m 5const,
Xa~t,s!5ya~t!. ~6!
Inserting Eq.~6! into constraints~3! and~4! one obtains~theoverdot is used ford/dt)
gMN~ya!yMyN62l0T
3C6m @gmN~ya!yN62l0TC6
n gmn~ya!#50,
C6m @gmN~ya!yN62l0TC6
n gmn~ya!#50.
Obviously, this system of two constraints is equivalent tofollowing one:
gMN~ya!yMyN50, ~7!
C6m @gmN~ya!yN62l0TC6
n gmn~ya!#50. ~8!
Using the ansatz~6! and constraint~8! one can reduce theinitial Lagrangian to get
Lred~t!}1
4l0@gMN~ya!yMyN22~2l0T!2C6
m C6n gmn~ya!#.
It is easy to check that the constraint~7! can be rewritten asgMNpMpN50, where pM5]Lred/] yM is the momentumconjugated toyM. All this means that we have obtained a
02600
-toinr
e
effective dynamical system describing a massless pointticle moving in a gravity backgroundgMN(ya) and in a po-tential
U}T2C6m C6
n gmn~ya!
on the constraint surface~8!.Analogous results can be received if one uses the ans
Xm~t,s!5C6m w61zm~s!, Xa~t,s!5za~s!. ~9!
Now putting Eq.~9! in Eqs. ~3! and ~4! one gets (8 is usedfor d/ds)
@~2l0T!21~l1!2#gMNz8Mz8N14l0T@~2l0T7l1!
3C6m gmNz8N12l0TC6
m C6n gmn#50,
l1gMNz8Mz8N1@~l172l0T!
3C6m gmNz8N72l0TC6
m C6n gmn#50.
These constraints are equivalent to the following ones:
gMN~za!z8Mz8N50,
C6m FgmN~za!z8N1
2l0T
2l0T7l1C6
n gmn~za!G50.
~10!
The corresponding reduced Lagrangian obtained withhelp of Eqs.~9! and ~10! is
Lred~s!}~l1!22~2l0T!2
4l0 FgMN~za!z8Mz8N
22S 2l0T
2l0T7l1D 2
C6m C6
n gmn~za!Gand a similar interpretation can be given as before.
In both cases—the ansatz~6! and the ansatz~9!, the re-duced Lagrangians do not depend onym andzm, respectively,and their conjugated generalized momenta are conserve
Let us point out that the main difference between tensand null strings, from the point of view of the reduceLagrangians, is the absence of a potential term for the la
Because the consequences of Eqs.~6! and~9! are similar,further considerations will be based on the ansatz~6!.
III. EXACT SOLUTIONS IN CURVED BACKGROUND
To obtain the equations which we are going to considwe use the ansatz~6! and rewrite Eq.~5! in the form
gKLyL1GK,MNyMyN64l0TC6m GK,mNyN50. ~11!
At first, we setK5m in the above equality. It turns out thain this case the Eqs.~11! reduce to
d
dt@gmnyn1gmaya62l0TC6
n gmn#50,
4-2
te
e
-
g a
on-
x-
enet-
EXACT STRING SOLUTIONS IN NONTRIVIAL BACKGROUNDS PHYSICAL REVIEW D65 026004
i.e., we have obtained the following first integrals~constantsof the motion!
gmnyn1gmaya62l0TC6n gmn5Am
65const. ~12!
They correspond to the conserved momentapm . From theconstraint~8! it follows that the right-hand side of Eq.~12!must satisfy the condition
Am6C6
m 50.
Using Eq.~12!, Eqs.~11! for K5a and the constraint~7!can be rewritten as
2d
dt~haby
b!2~]ahbc!ybyc1]aV654] [a~gb]mkmnAn
6!yb
~13!
and
habyayb1V650, ~14!
where
hab[gab2gamkmngnb ,
V6[Am6An
6kmn1~2l0T!2C6m C6
n gmn ,
and kmn is by definition the inverse ofgmn : kmlgln5dnm .
For example, whengMN does not depend on the coordinayq
hab5gab2gaqgqb
gqq,
whengMN does not depend on two of the coordinates~sayyq
andys)
hab5gab2gaqgssgqb22gaqgqsgsb1gasgqqgsb
gqqgss2gqs2
,
and so on.At this stage, we restrict the metrichab to be a diagonal
one, i.e.,
gab5gamkmngnb , for aÞb. ~15!
This allows us to transform further equations~13! and obtain~there is no summation overa)
d
dt~haay
a!21 ya]a~haaV6!1 ya(
bÞaF]aS haa
hbbD ~hbby
b!2
24] [aAb]6haay
bG50, ~16!
where we have introduced the notation
Aa6[gamkmnAn
6 . ~17!
02600
In receiving Eq.~16!, the constraint~14! is also used aftertaking into account the restriction~15!.
To reduce the order of the differential equations~16! byone, we first split the indexa in such a way thatyr is one ofthe coordinatesya, and ya are the others. Then we imposthe conditions
]aS haa
haaD50, ]a~hrr y
r !250, ] r~haaya!250, Aa65]a f 6.
~18!
The result of integrations, compatible with Eqs.~14! and~15!, is the following
~haaya!25Da~yaÞya!1haa@2~Ar62] r f
6!yr2V6#
5Ea~yb!,
~hrr zr !25hrr H S (
a21DV62(
a
Da
haaJ
1F(a
~Ar62] r f
6!G2
5Er~yr !, ~19!
whereDa , Ea , andEr are arbitrary functions of their arguments, and
zr[ yr1
(a
hrr~Ar
62] r f6!.
To find solutions of the above equations without choosinparticular metric, we have to fix all coordinatesya exceptone. If we denote it byyA, then theexact solutions of theequations of motion and constraints for a string in the csidered curved background are given by
Xm~XA,s!5X0m1C6
m ~l1t1s!
2EX0
A
XA
k0mnFgnA
0 7An6S 2
hAA0
V60D 1/2Gdu,
Xa5X0a5const for aÞA,
t~XA!5t06EX0
A
XAS 2hAA
0
V60D 1/2
du, ~20!
whereX0m , X0
A , andt0 are arbitrary constants. In these epressions
hAA0 5hAA
0 ~XA!5hAA~XA,X0aÞA!
and analogously forV60, k0mn , andgnA
0 .
IV. TURNING ON THE B FIELD
Here we are going to obtain exact string solutions whthe background also includes the Kalb-Ramond antisymm
4-3
b
rsthe
(ulte
icen
ric
P. BOZHILOV PHYSICAL REVIEW D 65 026004
ric gauge fieldBMN(X). To this end, we start with thebosonic part of the Green-Schwarz superstring action
S152T
2E d2j@A2ggmn]mXM]nXNgMN~X!
2«mn]mXM]nXNBMN~X!#. ~21!
Varying Eq.~21! with respect toXM andgmn , we obtain theequations of motion
2gLK@]m~A2ggmn]nXK!1A2ggmnGMNK ]mXM]nXN#
51
2HLMN«mn]mXM]nXN, H5dB,
and the constraints
~gklgmn22gkmg ln!]mXM]nXNgMN~X!50.
In the gaugegmn5const and using Eq.~2!, the Euler-Lagrange equations can be rewritten as
gLM@~]02l1]1!22~2l0T!2]12#XM1GL,MN@~]02l1]1!
3XM~]02l1]1!XN2~2l0T!2]1XM]1XN#
52l0THLMN]0XM]1XN, ~22!
and theindependentconstraints take the same forms~3! and~4! as before. Putting the ansatz~6! into Eq.~22! one obtains
gKLyL1GK,MNyMyN12l0TC6m ~HKmN62GK,mN!yN50.
~23!
Now we suppose that the tensor fieldBMN has the samesymmetry as the background metricgMN , i.e., ]mgMN5]mBMN50. Then it follows that
gmNyN12l0TC6n ~Bmn6gmn!5Bm
65const. ~24!
are first integrals of the equations of motion~23!. These con-served quantities are compatible with the constraint~8! whenBm
6C6m 50. Using Eq.~24!, the equations of motion forya
and the other constraint~7! can be transformed into
2d
dt~haby
b!2~]ahbc!ybyc1]aVB
654] [aBc]6 yc, ~25!
habyayb1VB
650, ~26!
where now
VB6[~2l0T!2C6
m C6n gmn1~Bm
622l0TBmlC6l !kmn
3~Bn622l0TBnrC6
r !,
Ba6[gamkmnBn
612l0T~Bal2gamkmnBnl!C6l .
The comparison of Eqs.~25! and ~26! with Eqs. ~13! and~14! shows that the former can be obtained from the latterthe replacements
02600
y
V6°VB6 , Aa
6°Ba6 ,
i.e., these equalities are form invariant. Therefore, the fiintegrals~19! will have the same form as before under tsame conditions on the metrichab and withBa
65]a f B6 . The
corresponding generalization of the exact solutions~20! forBMN(X)Þ0 will be
Xm~XA,s!5X0m1C6
m ~l1t1s!2EX0
A
XA
k0mnFgnA
0 7~Bn6
22l0TBnl0 C6
l !S 2hAA
0
VB60D 1/2Gdu,
Xa5X0a5const for aÞA,
t~XA!5t06EX0
A
XAS 2hAA
0
VB60D 1/2
du. ~27!
We note that until now we have used the string framesmodel! metric. It would be useful to have the obtained reswritten in Einstein frame metric, i.e., the frame in which thD-dimensional Einstein-Hilbert action is free from dilatonscalar factor. In particular, we will need it in one of thfollowing sections. InD dimensions, the connection betweethese two type of metrics is@35#
gMN[gMNstring5expS a~D !
2f DgMN
E , ~28!
where for the string
a~D !5A 8
D22,
andf(X) is the dilaton field. With the help of Eq.~28!, onecan check that Eq.~27! will give the right formulas for theexact solution in the Einstein frame, if one takes all metcoefficients in this frame and replacesVB
60 with
expS a~D !
2f0DVB
60
in the expression forXm, and with
expS 2a~D !
2f0DVB
60
in the expression fort, wheref05f0(XA), and
VB6[~Bm
622l0TBmlC6l !kE
mn~Bn622l0TBnrC6
r !
3expS 2a
2f D1~2l0T!2C6
m C6n gmn
E expS a
2f D .
4-4
xil
e
ti
xtr
-
eld
of
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EXACT STRING SOLUTIONS IN NONTRIVIAL BACKGROUNDS PHYSICAL REVIEW D65 026004
Let us finally give the induced metricGmn which ariseson the string worldsheet in our parametrization of the auiary metric gmn given by Eq.~2! and after taking into ac-count the ansatz~6!. It is
G005@~l1!22~2l0T!2#G11, G015l1G11,
G115C6m C6
n gmn~Xa!.
V. EXACT D-STRING SOLUTIONS
In this section our aim is to consider theD-string dynam-ics in nontrivial backgrounds. We will use the action
SD52TD
2 E d2j exp~2af!A2KK mn
3~Gmn1Bmn1~gsTD!21Fmn!, ~29!
introduced in @36#, which is classically equivalent to thDirac-Born-Infeld action
SDBI52TDE d2j exp~2af!
3A2det~Gmn1Bmn1~gsTD!21Fmn!.
The notations used in Eq.~29! are as follows.TD5T/gs istheD-string tension, whereT5(2pa8)21 is the~fundamen-tal! string tension andgs5exp f& is the string coupling ex-pressed by the dilaton vacuum expectation value^f&.Gmn(X)5]mXM]nXNgMN(X),Bmn(X)5]mXM]nXNBMN(X),andf(X) are the pullbacks of the background metric, ansymmetric tensor, and dilaton to theD-string worldsheet,while Fmn(j) is the field strength of the worldsheetU(1)gauge field Am(j). K is the determinant of the matriKmn ,K mn is its inverse, and these matrices have a symmeas well as antisymmetric part
K mn5K (mn)1K [mn][gDmn1vmn,
where the symmetric partgDmn is the analogue of the auxil
iary metricgmn in the string actions~1! and ~21!.To proceed further, we set in Eq.~29!
gDmn5S 21 l1
l1 2~l11l2!~l12l2!1~2l0TD!2D ,
vmn52l2«mn, ~30!
and obtain the Lagrangian density
LD5exp~2af!
4l0$gMN~]02l1]1!XM~]02l1]1!XN
2@~l2!21~2l0TD!2#gMN]1XM]1XN
12l2@BMN]0XM]1XN1~gsTD!21F01#%.
The constraints that follow from here are
02600
-
-
ic
gMN~]02l1]1!XM~]02l1]1!XN
1@~2l0TD!21~l2!2#gMN]1XM]1XN50,
gMN~]02l1]1!XM]1XN50,
BMN]0XM]1XN1~gsTD!21F012l2gMN]1XM]1XN50.
We will use the last constraint to express the worldsheet fistrengthF01 through the background fieldsgMN ,BMN andpartial derivatives of the received solutions forXM. Theother constraints we will solve together with the equationsmotion.
As until now, we will work in the gauge in which theLagrange multipliers are fixed:l0,1,25const. In this case, theequations of motion forAm
]mFl2exp~2af!
2l0gsTDG50
are identically satisfied when the dilaton is also fixed:f5f05const. The remaining equations are those forXM andthey are
gLM$~]02l1]1!22@~2l0TD!21~l2!2#]12%XM
1GL,MN$~]02l1]1!XM~]02l1]1!XN
2@~2l0TD!21~l2!2#]1XM]1XN%
5l2HLMN]0XM]1XN. ~31!
The following step is to apply our ansatz~6!. However, in theD-string case it does not work properly—a little modificatiois needed. Actually, now the background independent stions of the equations of motion~31! are
XM~t,s!5F6M@~l11A6!t1s#,
A656A~2l0TD!21~l2!2,
where againF6M are arbitrary functions of their argument
Therefore, the appropriate ansatz is
Xm~t,s!5C6m @~l11A6!t1s#1ym~t!, Xa~t,s!5ya~t!.
~32!
The insertion of Eq.~32! into the Eqs.~31! reduce them tothe following ones:
gKLyL1GK,MNyMyN1C6m ~l2HKmN12A6GK,mN!yN50,
which possess the first integrals
gmNyN1C6n ~l2Bmn1A6gmn!5Bm
65const, Bm6C6
m 50.
In full analogy with the previously considered case, tequations of motion forya can be transformed into the form~25!, where now instead ofVB
6 andBa6 we have
4-5
one
ua-
ne
e
e
ns.tics
P. BOZHILOV PHYSICAL REVIEW D 65 026004
VD6[A6
2 C6m C6
n gmn1~Bm62l2BmlC6
l !
3kmn~Bn62l2BnrC6
r !,
BaD6[gamkmnBn
61l2~Bal2gamkmnBnl!C6l .
The corresponding exact solution is
Xm~XA,s!5X0m1C6
m ~l1t1s!2EX0
A
XA
k0mnFgnA
0 7~Bn6
2l2Bnl0 C6
l !S 2hAA
0
VD60D 1/2Gdu,
Xa5X0a5const for aÞA,
t~XA!5t06EX0
A
XAS 2hAA
0
VD60D 1/2
du,
F01
gsTD5l2C6
m gmn0 C6
n 2~Br62l2Brl
0 C6l !k0
rmBmn0 C6
n
7~BAn0 2gAr
0 k0rmBmn
0 !C6n S 2
hAA0
VD60D 21/2
,
f5f05const. ~33!
VI. EXAMPLES
Let us first give an explicit example of an exact solutifor a string moving in four-dimensional cosmological Kasntype background. Namely, the line element is (x0[t)
ds25gMNdxMdxN52~dt!21 (m51
3
t2qm~dxm!2, ~34!
(m51
3
qm51, (m51
3
qm2 51. ~35!
For definiteness, we chooseqm5(2/3,2/3,21/3). The metric~34! depends on only one coordinatet, which we identifywith Xa5ya(t) according to our ansatz~6!. Correspond-ingly, the last two terms in Eq.~16! vanish and there is noneed to impose the conditions~18!. Moreover, the metric~34! is a diagonal one, so we havehaa5gaa521. Takingthis into account, we obtain the exact solution of the eqtions of motion and constraints~20! in the considered particular metric expressed as follows:
Xm~ t,s!5X0m1C6
m ~l1t1s!6Am6I m~ t !, t~ t !5t06I 0~ t !,
I M~ t 
tio
l-
-
isis
he
EXACT STRING SOLUTIONS IN NONTRIVIAL BACKGROUNDS PHYSICAL REVIEW D65 026004
ut~ t !2t0u →t→0 3
5@~A16!21~A2
6!2#1/2t5/3,
X1,2~ t,s! →t→0
X01,21C6
1,2s73A1,2
6
@~A16!21~A2
6!2#1/2t1/3,
X3~ t,s! →t→0
X031C6
3 H s73l1
5@~A16!21~A2
6!2#1/2t5/3J ;
ut~ t !2t0u →t→` 3
2uA36u
t2/3,
X1,2~ t,s! →t→`
X01,21C6
1,2S s73l1
2A36
t2/3D ,
X3~ t,s! →t→`
X031C6
3 s73
4t4/3.
The asymptotic behavior of the same solution as a funcof the worldsheet time parametert is given by
t~t
erotsre
t i
a
el-
ol-ed
gein
s
on-
P. BOZHILOV PHYSICAL REVIEW D 65 026004
exact null string solution (T50) for a gravity backgroundwith arbitrary, but differentqm . It is given by Eq. ~36!,where
I M~ t !5const2Ap
A26 (
k50
`~A3
6/A26!2k
k!G~1/22k!
tP
P
3FS 1/21k,P
2~q22q1!;
32~q22q3!k13q222q11122qM
2~q22q1!;
2S A16
A26D 2
t2(q22q1)D ,
P[2~q22q3!k1q21122qM ,
qM5~0,q1 ,q2 ,q3!, for q1.q2 ,
and
I M~ t !5const1Ap
A16 (
k50
`~A3
6/A16!2k
k!G~1/22k!
tQ
Q
3FS 1/21k,Q
2~q12q2!;
32~q12q3!k13q122q21122qM
2~q12q2!;
2S A26
A16D 2
t2(q12q2)D ,
Q[2~q12q3!k1q11122qM , for q1,q2 .
Because there are no restrictions onqm , except q1Þq2Þq3, the above probe string solution is also valid in genalized Kasner type backgrounds arising in superstring cmology@37#. In string frame, the effective Kasner constrainfor the four-dimensional dilaton-moduli-vacuum solution a
(m51
3
qm511K, (m51
3
qm2 512B 2,
212A3~12B 2!
<K<211A3~12B 2!, B 2P@0,1#.
In the Einstein frame, the metric has the same form, bunew, rescaled coordinates and with new powersqm of thescale factors. The generalized Kasner constraints aremodified as follows:
(m51
3
qm51, (m51
3
qm2 512B22
1
2K2, B21
1
2K2P@0,1#.
02600
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n
lso
Actually, the obtained tensionless string solution is also revant to considerations within apre-big-bangcontext, be-cause there exist a class of models for pre-big-bang cosmogy, which is a particular case of the given generalizKasner backgrounds@37#.
Our next example is for a string moving in the followinten-dimensional supergravity background given in Einstframe @35#:
ds25gMNE dxMdxN5exp~2A!hmndxmdxn
1exp~2B!~dr21r 2dV72!,
exp@22~f2f0!#511k
r 6, f0 , k5const,
A53
4~f2f0!, B52
1
4~f2f0!,
B0152expF22S f23
4f0D G .
All other components ofBMN as well as all componentof the gravitinocM and dilatinol are zero. If we param-etrize the sphereS7 so that
g102 j ,102 jE 5exp~2B!r 2)
l 51
j 21
sin2 x102 l , j 52,3, . . . ,7,
g99E 5exp~2B!r 2,
the metricgMNE does not depend onx0, x1, and x3, i.e., m
50, 1, and 3. Then we setya5y0a5const for a54, . . . ,9
and obtain a solution of the equations of motion and cstraints as a function of the radial coordinater:
Xm~r ,s!5X0m1C6
m ~l1t1s!6I m~r !, t~r !5t06I ~r !,
I m~r !5hmnEr 0
r
duFBn612l0T«nkC6
k exp~2f0/2!
3S 11k
u6D G S 11k
u6D W621/2,
I 3~r !5B3
6
s2 Er 0
r du
u2W6
21/2, s[)l 51
6
siny0102 l ,
I ~r !5exp~f0/2!Er 0
r
duW621/2,
where
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EXACT STRING SOLUTIONS IN NONTRIVIAL BACKGROUNDS PHYSICAL REVIEW D65 026004
W65H FB0612l0TC6
1 exp~2f0/2!S 11k
u6D G 2
2FB1622l0TC6
0 exp~2f0/2!S 11k
u6D G 2J S 11k
u6D2S B3
6
s D 2 1
u21~2l0T!2exp~f0!H @~C6
0 !22~C61 !2#
3S 11k
u6D 21
2~C63 s!2u2J .
This is also the solution in the string frame, because we h
o
g
02600
ve
one and the same metric in the action expressed in twoferent ways.
The above solution is extremely simplified in the tensioless limitT→0. Let us give the manifest expressions for thcase. Forr 0,r , they are
limT→0
I m~r !5hmnBm6~J01kJ6!, lim
T→0I 3~r !5
B36
s2J2,
limT→0
I ~r !5J0,
where
Jb~r !52A p
~B06!22~B1
6!2
3H 1
r b21 (n50
` GS 26n1b25
4 D ~k/r 6!n
~6n1b21!GS 122n
2 DGS 22n1b25
4 D Pn(26n1b21)/4,2n21)S 122
d
kr 4D
21
r 0b21 (
n50
` GS 2n1b13
4 D ~2d/r 02!n
~2n1b21!GS 122n
2 DGS 6n1b13
4 D Pn(2n1b21)/4,2n21)S 122
k
dr 0
24D J ,
d[~B3
6/s!2
~B06!22~B1
6!2,
and Pn(a,b)(z) are the Jacobi polynomials. To obtain the s
lution for r 0.r , one has to exchanger andr 0 in the expres-sion for Jb.
Now let us turn to the case of aD string living in five-dimensionalanti–de Sitter space-time. The correspondinmetric may be written as
g0052S 11r 2
R2D , g115S 11r 2
R2D 21
,
g225r 2sin2 x3sin2 x4, g335r 2sin2 x4, g445r 2,
whereK521/R2 is the constant curvature. NowgMN doesnot depend onx0, x2 andBMN50. If we fix the coordinatesx3,x4 and use the generic formula~33!, the exact solution asa function ofr[x1 will be
X0~r ,s!5X001C6
0 ~l1t1s!7B06
3Er 0
r
duS 11u2
R2D 21
~g00VD60!21/2,
-X2~r ,s!5X0
21C62 ~l1t1s!6
B26
c2 Er 0
r du
u2~g00VD
60!21/2,
t~r !5t06Er 0
r
du~g00VD60!21/2,
c[sinx03sin x0
4 ,
F01~r !52gsTDl2H F S C60
R D 2
2~cC62 !2G r 21~C6
0 !2J ,
where
g00VD605F ~B0
6!22S B26
cRD 2
1~A6C60 !2G2S B2
6
c D 2 1
u2
1A62 F2S C6
0
R D 2
2~cC62 !2Gu21S A6
R D 2F S C60
R D 2
2~cC62 !2G u4.
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at
e-
f t
aus
,the
th.ualero.
eld
so
P. BOZHILOV PHYSICAL REVIEW D 65 026004
This solution describes aD string evolving in the subspac(x0,x1,x2).
Alternatively, we could fix the coordinatesr 5r 0 ,x45x04
[c0 and obtain a solution as a function of the coordinx3[u. In this case, the result is the following:
X0~u,s!5X001C6
0 ~l1t1s!7B0
6%
g000 E
u0
u
du~2VD60!21/2,
X2~u,s!5X021C6
2 ~l1t1s!6B2
6
%E
u0
u du
sin2 u~2VD
60!21/2,
t~u!5t06%Eu0
u
du~2VD60!21/2, %[r 0sinc0 ,
F01~u!5gsTDl2@~C60 !2g00
0 1~%C62 !2sin2 u#,
where
2VD605@~B0
6!2g110 2~A6C6
0 !2g000 #
2~A6C62 % !2sin2u2
~B26/% !2
sin2 u.
This is a solution for aD string placed in the subspace dscribed by the coordinates (x0,x2,x3).
Another possibility is when the coordinatesr and u arekept fixed, while the coordinatec is allowed to vary. How-ever, it is easy to show that in this case the dependence osolution onc will be the same as onu, with some constantschanged.
Finally, we will give an example of exact solution forD-string moving in a nondiagonal metric. To this end, letconsider the ten-dimensional black-hole solution of@38#. Instring frame metric, it can be written as@39#
ds25S 11r 0
2 sinh2a
r 2 D 21/2S 11r 0
2 sinh2 g
r 2 D 21/2
3H 2dt21~dx9!21r 0
2
r 2~coshxdt1sinhx dx9!2
1S 11r 0
2 sinh2a
r 2 D @~dx5!21~dx6!21~dx7!21~dx8!2#J1S 11
r 02 sinh2 a
r 2 D 1/2S 11r 0
2 sinh2 g
r 2 D 1/2
3F S 12r 0
2
r 2D 21
dr21r 2 dV32G ,
02600
e
he
exp@22~f2f`!#
5S 11r 0
2 sinh2 a
r 2 D 21S 11r 0
2 sinh2 g
r 2 D . ~37!
The equalities~37! define a solution of type IIB string theorywhose low-energy action in Einstein frame containsterms
E d10xA2gFR21
2~“f!22
1
12exp~f!H82G , ~38!
whereH8 is the Ramond-Ramond three-form field strengThe Neveu-Schwarz three-form field strength, the self-dfive-form field strength, and the second scalar are set to zTherefore, in the solution~33!, we also have to setBMN50.Besides, our solution corresponds to a constant dilaton fif5f0. If we identify f0[f` , this leads toa5g in Eq.~37!. On the other hand, we have not included in theD-stringaction ~29! the Ramond-Ramond two-form gauge field,we have to setH850. It follows from here thata5g50@39#. All this results in a simplification of the metric~37!,and in this simplified metric the exactD-string solution as afunction of the radial coordinater reads
Xm~r ,s!5X0m1C6
m ~l1t1s!6I m~r !, m50,2,5,6,7,8,9,
X3,45X03,45const, t~r !5t06I ~r !,
where
I 052Er 0
r
du@B06g992B9
6g09#g11W 21/2,
I 95Er 0
r
du@B06g092B9
6g00#g11W 21/2,
I l5Bl6E
r 0
r du
gll0W 21/2, l 52,5,6,7,8,
I 5Er 0
r
duW 21/2,
F015gsTDl2C6m C6
n gmn0 ~r !,
W5F ~B96!22
~A6C60 !2
g11Gg001F ~B0
6!2
2~A6C6
9 !2
g11Gg9922S B0
6B961
A62 C6
0 C69
g11Dg09
21
g11H (
l 55
8
@~A6C6l !21~B6
l !2#
1S B62
c D 2 1
u21~A6C6
2 c!2u2J .
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E
no
tial-
sio
he
gitsein
hicw
leic
th
i-
dk-
gn
ndt
try
in
o
au
za-
ofx-ts.ugh
ck-
de
rgy
tric.
al
ureionen
ve-rld-
c-
-ld
d toion. Inllfor
ot
ed
actencal
EXACT STRING SOLUTIONS IN NONTRIVIAL BACKGROUNDS PHYSICAL REVIEW D65 026004
Actually, after fixing the dilaton to a constant andH8 to zero,we have obtained a solution for aD string moving in ten-dimensional Einstein gravity background, as is seen from~38!. That is why we are now going to consider the casef5f0 ,H8Þ0. To this aim, we have to add a Wess-Zumiterm to the action~29!, describing the coupling of theDstring to the Ramond-Ramond two-form gauge potenBMN8 (H85dB8). It can be shown that this leads to the folowing change in the equations of motion~31!:
l2HLMN→l2HLMN12l0m exp~af0!HLMN8 ,
wherem is theD-string charge (m56TD from the require-ments of space-time supersymmetry and worldsheetk invari-ance of the superD-string action!. Then one proceeds abefore to obtain an exact solution of the equations of motand constraints. This solution is given by Eq.~33!, where thereplacement
l2BMN→l2BMN12l0mexp~af0!BMN8 ,
must be done. Now we can put there the background~37!with a5gÞ0 to obtain an explicit probeD-string solution.
VII. DISCUSSION
In this article we performed some investigation on tclassical string dynamics inD-dimensional~super! gravitybackground. In Sec. II we begin with rewriting the strinaction for curved background in a form in which the limT→0 could be taken to also include the null string caknown to be a good approximation for string dynamicsstrong gravitational fields. Then we propose an ansatz, wreduces the initial dynamical system depending on tworldsheet parameters (t,s) to the one depending only ont,whenever the background metric does not depend on atone coordinate. An alternative ansatz is also given, whleads to a system depending only ons.
Let us note that the usually used conformal gauge forauxiliary worldsheet metric gmn corresponds to l1
50, 2l0T51. However, the latter does not allow for a unfied description of both tensile and tensionless strings.
The ansatz~6! and the ansatz~9! generalize the ones usein @12–32# for finding exact string solutions in curved bacgrounds. It is also worth noting that Eqs.~6! and ~9! arebased on the obtained string solutionsF6
M(w6), which donot depend on the background metric. In conformal gauthey reduce to the solutions for left or right movers, knowto be the onlybackground independentnonperturbative so-lutions for an arbitrary static metric which are stable ahave a conserved topological charge which are thereforepological solitons@6#.
In Sec. III, using the existence of an Abelian isomegroupG generated by the Killing vectors]/]xm, the problemof solving the equations of motion and two constraintsD-dimensional curved space-timeMD with metric gMN isreduced to considering equations of motion and one cstraint in the cosetMD /G with metric hab . As might beexpected, an interaction with an effective gauge field appein the Euler-Lagrange equations. In this connection, let
02600
q.
l
n
,
ho
asth
e
e,
o-
n-
rss
note that if we write downAa6 , introduced in Eq.~17!, as
Aa65Aa
nAn6 ,
this establishes a correspondence with the usual KaluKlein type notation and
gMNdyMdyN5habdyadyb1gmn~dym1Aamdya!
3~dyn1Abndyb!.
In the remaining part of Sec. III, we impose a numberconditions on the background metric, sufficient to obtain eact solutions of the equations of motion and constrainThese conditions are such that the metric is general enoto include in itself many interesting cases of curved bagrounds in different dimensions.
In Sec. IV, the previous results are generalized to inclua nontrivial Kalb-Ramond background gauge fieldBMN ,which arises in the supergravity theories—the low-enelimits of superstring theories. TheBMN is restricted to de-pend on the same coordinates as the background meThere are also indirect restrictions ongMN and BMN . Theyfollow from the condition on the effective one-form potentiBa
6 in the equations of motion~25! to be oriented along oneof the coordinate axes, with all other components being pgauges. We explain how the derived probe string solutalso looks in the Einstein frame metric—the one used whsearching for brane solutions of the string-theory effectifield equations. At the end of the section, the induced wosheet metric is given and it depends only on thegmn part ofthe background metric, which corresponds to its Killing vetors ]/]xm.
In Sec. V, we consider theD-string dynamics. In an appropriate parametrization of the auxiliary worldsheet fieK mn and with the help of a modified ansatz, we succeedereduce the task of finding exact solutions to the applicatof methods used for this purpose in the previous sectionthat case, the limitTD→0 could also be taken and this wigive a solution of the equations of motion and constraintsa tensionless Dstring. However, the tensionlessD string isnot a null string, i.e., the induced worldsheet metric is ndegenerate. Really, the induced metric is
G005@~l1!22~l2!22~2l0TD!2#G11,
G015l1G11, G115C6m C6
n gmn~Xa!,
detGmn52@~l2!21~2l0TD!2#G112 .
Consequently, whenTD50, detGmn is still different fromzero. Moreover, the tensile fundamental string can be viewas a particular case of the tensionlessD string@40#. Indeed, itis not difficult to show by using Eqs.~2! and ~30! that thefundamental string action~21! can be obtained from theD-string action~29! by settingAm50, f50, l252l0T, andtaking TD→0.
The next section is devoted to four examples of exstring solutions in different backgrounds in four, five, and tdimensions. As an example of a solution in a cosmologi
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ot
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P. BOZHILOV PHYSICAL REVIEW D 65 026004
type background, we consider four-dimensional Kasspace-time. The next example is for a string moving in tdimensional supergravity background. TheD-string solutionsare illustrated with two examples—in five-dimensionanti-de Sitter and in ten-dimensional black-hole bagrounds. It is evident from the solution~27! and from thefirst two examples that the~exact! null string dynamics incurved backgrounds,@41–43,20,44–48#, is simpler than thetensile string one. Moreover, the tensionless string doesinteract with theBMN background, according to the actio~21!. This is a consequence of the condition fork invarianceof the Green-Schwarz superstring action, the bosonicwhich is Eq.~21!. Actually, the null superp-brane actions arek invariant in flat space-time without Wess-Zumino term@49,50#. The D-string case is different. Thetensionless Dstring does interact with the Kalb-Ramond background.
Let us finally note some specific features of the receivsolutions. The parametrization of the worldsheet fieldsgmn
andK mn is such that they allow for a unified description
d.
. A
02600
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l-
ot
rt
d
the tensile and tensionless string solutions. The ansatz~6!and the ansatz~32! contain the terms62l0TC6
m t andA6C6
m t, respectively, which disappear in the expressio~27! and~33! for the exact (D)-string solutions as a functionof one of the coordinates. There, they are replaced bycorresponding integrals. The other part of the backgrouindependent solution,C6
m (l1t1s), is a particular case osuch solution for the nullp branes@51,25#. Another distin-guishing feature of our exact solutions is that we do nrestrict ourselves to the usually usedstatic gauge Xm(j)5jm, when probe brane dynamics is investigated. It is psible to impose this gauge on the solutions and this wresult in additional conditions on them.
ACKNOWLEDGMENT
This work was supported in part by a Shoumen Universgrant under Contract No. 20/2001.
B
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