Strings in five-dimensional anti-de Sitter space with a symmetry

Strings in five-dimensional anti-de Sitter space with a symmetry

Tatsuhiko Koike*

Department of Physics, Keio University, Yokohama 223-8522, Japan

Hiroshi Kozaki+

Department of Applied Chemistry and Biotechnology, Niigata Institute of Technology, Kashiwazaki, Niigata 945-1195, Japan

Hideki Ishihara‡

Department of Mathematics and Physics, Graduate School of Science, Osaka City University, Osaka 558-8585, Japan(Received 8 October 2007; published 2 June 2008)

The equation of motion of an extended object in spacetime reduces to an ordinary differential equation

in the presence of symmetry. By properly defining the symmetry with the notion of cohomogeneity, we

discuss the method for classifying all these extended objects. We carry out the classification for the strings

in the five-dimensional anti-de Sitter space by the effective use of the local isomorphism between SOð4; 2Þand SUð2; 2Þ. In the case where the string is described by the Nambu-Goto action, we present a general

method for solving the trajectory. We then apply the method to one of the classification cases, where the

spacetime naturally obtains a Hopf-like bundle structure, and find a solution. The geometry of the solution

is analyzed and found to be a timelike, helicoidlike surface.

DOI: 10.1103/PhysRevD.77.125003 PACS numbers: 11.27.+d, 04.50.�h, 98.80.Cq

I. INTRODUCTION

Existence and dynamics of extended objects play im-portant roles in various stages of cosmology. Examples ofextended objects include topological defects, such asstrings and membranes, and the Universe as a whole em-bedded in a higher-dimensional spacetime in the context ofthe brane-world universe model [1].

The trajectory of an extended object forms a hypersur-face in the spacetime which is determined by a partialdifferential equation (PDE). For example, a test string isdescribed by the Nambu-Goto equation, which is a PDE intwo dimensions. Because the dynamics is more compli-cated than that of a particle, one usually cannot obtaingeneral solutions. One way to find exact solutions is toassume symmetry. The simplest solutions to such a PDEare homogeneous ones, in which case the problem reducesto a set of algebraic equations. However, the solutions donot have much variety and the dynamics is trivial.

One may expect that if we assume ‘‘less’’ homogeneity,the equation still remains tractable and the solutions haveenough variety to include nontrivial configurations anddynamics of physical interest. The cohomogeneity-oneobjects give such a class, which helps us to understandthe basic properties of the extended objects and serves as abase camp to explore their general dynamics. For a string,stationarity is a special case of the cohomogeneity-onecondition. Some stationary configurations of the Nambu-Goto strings are obtained in the Schwarzschild spacetime

[2]. Even in the Minkowski space, many nontrivialcohomogeneity-one solutions of the string were recentlyfound [3,4]. A cohomogeneity-one object is defined,roughly speaking, as the one whose world sheet is homo-geneous except in one direction. Then, any covariant PDEgoverning such an object reduces to an ordinary differen-tial equation (ODE), which can easily be solved analyti-cally, or at least numerically. A solution represents thedynamics of a spatially homogeneous object, or the non-trivial configuration of a stationary object, depending onwhether the homogeneous ‘‘direction’’ is spacelike ortimelike. The case of the null homogeneous directionshould also give new intriguing models.In this paper, we treat strings in the five-dimensional

anti-de Sitter space AdS5. The choice of this spacetime isto meet the recent interest in higher-dimensional cosmol-ogy, including the brane-world universe model, and instring theory, though the method developed here is appli-cable to any background spacetime. A particular examplewhich has recently been attracting much attention is thestring in a spacetime with large extra dimensions, whichare suggested e.g. by the brane-world model. A detailedinvestigation [5] suggests that the reconnection probabilityfor this type of string is significantly suppressed. Then,contrary to what had usually been believed, the strings inthe Universe can stay long enough to be considered sta-tionary. Therefore, classifying cohomogeneity-one stringsand solving dynamics thereof are important for examiningthe roles of the string in cosmology. We first give theclassification of all cohomogeneity-one strings, which isvalid for any covariant equation of motion. Then, in thecase of Nambu-Goto strings, we give a general method forsolving the trajectory. The method can be easily applied to

*[email protected][email protected]‡[email protected]

PHYSICAL REVIEW D 77, 125003 (2008)

1550-7998=2008=77(12)=125003(9) 125003-1 � 2008 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.77.125003

the cases of other equations of motion. We demonstrate theprocedure and give explicit solutions in some particularcases.

In the classification, we make use of the local isomor-phism between SOð4; 2Þ and SUð2; 2Þ in an essential way.The latter group is easier to treat because the dimension-ality of the matrix is lower and because the Jordan decom-position of complex matrices is simpler than that of realones. Therefore, though a similar classification of Killingfields is found in the literature in the context of construct-ing quotient spaces of the anti-de Sitter space [6], wepresent an alternative proof based on the classification ofH-anti-self-adjoint matrices in the Appendix.

In Sec. II, we give a method for the classification of allcohomogeneity-one strings in general, and a method forsolving the equations of motion for Nambu-Goto strings.The latter can be easily applied to other equations ofmotion. In Sec. III, the useful relation of the isometrygroup SOð4; 2Þ0 and SUð2; 2Þ is briefly explained. Wegive the classification of the cohomogeneity-one stringsin the anti-de Sitter space in Sec. IV. In Sec. V, we dem-onstrate the method presented in Sec. II with an example.There we solve the Nambu-Goto equation and examine thegeometry of its world sheet. Section VI is devoted to aconclusion.

In this paper, a spacetime ðM; gÞ is a manifold Mendowed with a Lorentzian metric g. We denote by G theidentity component of the isometry group of ðM; gÞ, andby g its Lie algebra. We use the units in which the speed oflight and Newton’s constant are one.

II. GENERAL TREATMENT OFCOHOMOGENEITY-ONE STRINGS

In this section, we develop a general method for classi-fying cohomogeneity-one objects and solving their dynam-ics in an arbitrary spacetime ðM; gÞ. Let us start with thedefinition of the cohomogeneity-one objects. We say that am-dimensional hypersurface S in M is of cohomogeneityone if it is foliated by ðm� 1Þ-dimensional submanifoldsS� labeled by a real number� and there is a subgroupK ofGwhich preserves the foliation and acts transitively on S�.In particular, the hypersurfaces S� are embedded homoge-neously in M. A cohomogeneity-one object has a worldsheet which is a cohomogeneity-one hypersurface. In thispaper, we focus on the case that the extended objects arestrings, so that m ¼ 2, and K is a one-parameter groupð��Þ�2R of isometries.

First, let us consider how to classify the cohomogeneity-one strings. Given a one-dimensional subgroupK � G anda point p 2 M, the equations of motion determine a uniqueworld sheet of a cohomogeneity-one object. The dynamicsof the two strings can be considered the same if there is anisometry sending one of their trajectories, S, to the other,S0. In this paper, we identify the two dynamics if we can doso gradually, namely, if there is a one-parameter group of

isometries ð�0�Þ�2½0;1� such that �0

0 is the identity and

�01ðSÞ ¼ S0. We therefore classify the cohomogeneity-

one strings up to isometry connected to the identity. Wedo this by classifying the Killing vector fields �, whichgenerate K, up to scalar multiplication and up to isometry.Namely, � and a�� are equivalent if there exists � 2 Gand a � 0. To put it more algebraically, the task is to findg=AdG up to scalar multiplication.Second, let us give a formalism to solve the dynamics

and the configuration of the cohomogeneity-one strings.We assume that the string is described by the Nambu-Gotoaction

S ¼ZS

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�gabdxadxb

q:

The orbit space of the string with the symmetry group K isdefined by O :¼ M=K, i.e., by identifying all the pointson each Killing orbit in M. The submanifolds S� men-tioned above are the preimages ��1ðxÞ of a point x 2 O.One can endow O with a metric h so that the projection�: ðM; gÞ ! ðO; hÞ is an orthogonal projection or, moreprecisely, a Riemannian submersion. The metric h is givenby

hab :¼ gab � �a�b=f; (1)

where f :¼ �a�a. This metric has the Euclidean signatureif the Killing vector � is timelike, i.e., if f < 0, and theLorentzian signature if � is spacelike, i.e., if f > 0.Carrying out the integration along � in the Nambu-Gotoaction, one obtains

S ¼Zc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�fhabdxadxb

q; (2)

where c is a curve on O. Thus the problem of the stringreduces to finding geodesics on the orbit space O with themetric �fh. For convenience, we adopt a modified action

S ¼Zcd�

�� 1

�fhab _x

a _xb þ �

�; (3)

where an overdot denotes the differentiation by �. Theaction (3) derives the same geodesic equations as (2) andretains the invariance under reparametrization of �. Thefunction � is the norm of the tangent vector.The two-dimensional world sheet of the string is the

preimage ��1ðcÞ of the geodesic c on ðO;�fhÞ: However,it is sometimes more convenient to find a lift curveC onMwhose projection �ðCÞ is a geodesic on ðO;�fhÞ than tofind a geodesic on ðO;�fhÞ (Fig. 1). The Hopf string inSec. V is such an example. In this case, the trajectory of thestring is given by

S ¼ ��1ð�ðCÞÞ ¼ f��ðCð�ÞÞ; ð�; �Þ 2 R2g: (4)

Note that the last expression in (4) depends on the objectsin M only. Thus the trajectory S can be viewed as a

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foliation by mutually isometric curves �� C labeledby �.

After one obtains the solutions of the equation of mo-tion, one may want to classify their trajectories up toisometry. This can be done by identifying C (or S) whichare related by homogeneity-preserving isometries. We saythat an isometry � is homogeneity-preserving if it pre-serves the action of K, i.e., if it satisfies

� � K ��1 ¼ K: (5)

The homogeneity-preserving isometries form a group. Inalgebraic terms, the group is the normalizer of K in thegroup G of isometries on M, which is denoted by NGðKÞ.Its Lie algebra is the idealizer of k in g which is denoted byNgðkÞ.

We note that in the special case that � commuteswith the action of K, i.e. when � is in the centralizerZGðKÞ of K in G, the squared norm of � must be invariantunder �. This can be seen from ��f ¼ ��ðgab�a�bÞ ¼ð��gabÞ�a�b þ gabð��aÞ�b þ gab�

að��bÞ ¼ f, wherewe have used ��gab ¼ gab and ��a ¼ �a.

The whole procedure of solving the dynamics is explic-itly carried out, as an example, in Sec. V.

III. AdS5 AND ITS ISOMETRY GROUP

Hereafter in this paper, we assume that the spacetimeðM; gÞ is the five-dimensional anti-de Sitter space AdS5,

or its universal cover gAdS5. The former space has closedtimelike curves which, in the latter space, are ‘‘opened up’’to infinite nonclosed curves. The latter is usually moresuitable when we discuss cosmology, but we will notdistinguish them strictly in the following.

The space AdS5 is most easily expressed as a pseudo-sphere,

� ¼ �1 (6)

in the pseudo-Euclidean space E4;2 whose metric is dS2 ¼l2d � d , where we have used complex coordinates :¼ð 0; 1; 2ÞT 2 C3, and have defined � :¼ y� and� :¼ diag½�1; 1; 1�.The isometry group of AdS5 is SOð4; 2Þ acting on

ðs; t; x; y; z; wÞT 2 R6, where 0 :¼ sþ it, 1 :¼ xþ iy,and 2 :¼ zþ iw. In the classification of the strings, how-ever, we take advantage of the isomorphism SOð4; 2Þ0 ’SUð2; 2Þ=f�1g and work with SUð2; 2Þ. Let V be the vectorspace whose elements are complex symmetric matrices ofthe form

p ¼0 ð 0Þ� � 2 ð 1Þ�

ð 0Þ� 0 2 ð 2Þ� 2 � 1 0 0

�ð 1Þ� �ð 2Þ� 0 0

26664

37775

¼ si�z � �y þ t1 � �y þ xi�y � �z þ y�x � 1

� zi�y � �x þ w�x � �y; (7)

where �x, �y, and �z are the Pauli matrices and 1 is the

2� 2 identity matrix. The action of an element ofSOð4; 2Þ0 on E4;2 corresponds to the action of U 2SUð2; 2Þ on V in the following way [[7], p. 106]:

p� UpUT: (8)

The Lie algebra suð2; 2Þ of SUð2; 2Þ consists of thematrices X satisfying Xþ Xy ¼ 0, where :¼diag½1; 1;�1;�1�. The explicit form is

X ¼ ��y �

� �; (9)

where � is a 2� 2 complex matrix, and and � are 2� 2anti-Hermitian matrices. The infinitesimal transformationfor (8) is given by the action of X 2 suð2; 2Þ as

p� Xpþ pXT ¼ fXS; pg þ ½XA; p�; (10)

where XS :¼ ðX þ XTÞ=2 and XA :¼ ðX � XTÞ=2 are thesymmetric and antisymmetric parts, respectively, of X. Thecorrespondence between the suð2; 2Þ and soð4; 2Þ infini-tesimal transformations is given in Table I, where X ¼ðe1 � e2Þ=2. In the table, Jxy denotes the rotation in the

xy plane, L denotes the rotation in the st plane, Kx denotesthe t boost in the x direction, ~Kw denotes the s boost in thew direction, etc.

IV. THE CLASSIFICATION

In this section, we obtain the classification of thecohomogeneity-one strings in AdS5. As discussed inSec. II, the classification is to find g=AdG up to scalarmultiplication, whereG ¼ SOð4; 2Þ0. Because SOð4; 2Þ0 is

FIG. 1. To solve a trajectory of the cohomogeneity-one stringis to find a curve C in M which projects to a geodesic c on O.

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isomorphic to SUð2; 2Þ=f�1g as is seen in Sec. III,soð4; 2Þ=SOð4;2Þ0 is isomorphic to suð2; 2Þ=AdSUð2;2Þ.Thus the classification is to find suð2; 2Þ=AdSUð2;2Þ up to

scalar multiplication. However, the equivalence classessuð2; 2Þ=AdSUð2;2Þ are known, as in the lemma below, so

that we can easily classify the cohomogeneity-one stringsby further identifying the equivalence classes by scalarmultiplications.

We begin by introducing some terms, which is necessaryto state the lemma. Let H be an invertible Hermitianmatrix. The H-adjoint of a square matrix A is defined byA? :¼ H�1AyH. A matrix A is called H-self-adjoint whenA? ¼ A, A is called H-anti-self-adjoint when A? ¼ �A,and A is called H-unitary when AA? ¼ A?A ¼ 1. We saythat matrices A and B are H-unitarily similar and writeAH B if there exists an H-unitary matrix W satisfyingB ¼ WAW�1. In these terms, SUð2; 2Þ is the group ofunimodular -unitary matrices and suð2; 2Þ is the Liealgebra of traceless -anti-self-adjoint matrices. Thus,from the discussion in Sec. II, our task of classifyingcohomogeneity-one strings is to classify the elements ofsuð2; 2Þ up to the equivalence relation and up to scalarmultiplication.

Let us introduce another equivalence relation that isclosely related to the one above. Let ðA;HÞ be a pairconsisting of a complex matrix and an invertibleHermitian matrix H. The pairs ðA;HÞ and ðA0; H0Þ aresaid to be unitarily similar if there is a complex matrixW such that A0 ¼ WAW�1, H0 ¼ WHWy [8]. This is an

equivalence relation and will be denoted by ðA;HÞ ðA0; H0Þ. Note that A A0 is equivalent to ðA;Þ ðA0; Þ. Let A be an H-self-adjoint matrix. Then if � is aneigenvalue of A, so is its complex conjugate ��. Let J0ð�Þbe the Jordan block with eigenvalue � and let

Jð�Þ :¼�J0ð�Þ � is real;diag½J0ð�Þ; J0ð��Þ� � is nonreal:

(11)

Now we can state the lemma [8].Lemma.—If A is H-self-adjoint, then ðA;HÞ ðJ; PÞ

with

J ¼ diag½Jð�1Þ; . . . ; Jð�Þ�; (12)

P ¼ diag½ 1P1; . . . ; �P�; P�þ1; . . . ; P�;

j ¼ �1; Pj ¼0 1

. ..

1 0

264

375; (13)

where �1; . . . ; �� are the real eigenvalues of A,��þ1; �

��þ1; . . . ; �; �

� are the nonreal eigenvalues of A,

and the size of Pj is the same as that of Jð�jÞ.For any X 2 suð2; 2Þ, there is a pair ðJ; PÞ in the lemma

such that ðX=i; Þ ðJ; PÞ, because X=i is -self-adjoint.We will denote the type of X by

Type ðXÞ :¼ ð 1d1; . . . �d�jd�þ1=2; . . . ; d=2Þ; (14)

where dj :¼ dimJð�jÞ. [If there is either no real (� ¼ 0) or

no nonreal (� ¼ 4) eigenvalues, we put a 0 in the corre-sponding slot.] We combine all the types with the same djand call it the (major) type ½d1; . . . d�jd�þ1=2; . . . ; d=2�,and we call ð 1; ; �Þ the minor type. In the theorembelow, Jxy denotes spatial rotations in the xy plane, Kzdenotes the boost with respect to the time t in the zdirection, ~Kw denotes the boost with respect to the time sin the w direction, L denotes the rotation in the st plane,etc.Theorem.—Any one-dimensional connected Lie group

of isometries of AdS5 is generated by one of the nine types

TABLE I. Correspondence between the suð2; 2Þ and soð4; 2Þtransformations. For example, ð1=iÞ � �x 2 suð2; 2Þ corre-sponds to Jyz 2 soð4; 2Þ.e1, e2 1 �x �y �z

1=i Jyz Jzx Jxy�x Kw ~Kx

~Ky~Kz

�y � ~Kw Kx Ky Kz�z=i L Jwx Jwy Jwz

TABLE II. The classification of cohomogeneity-one strings. The type of the generator ofSUð2; 2Þ and the corresponding Killing vector fields � on AdS5 are given.

Type Killing vector field �

ð4j0Þ Kx þ ~Ky þ Jxy þ Lþ 2ðJyz þ KzÞð�3;�1j0Þ Kx þ ~Ky þ Jyz � Jxw þ aðJxy � L� JzwÞð2; 2j0Þ Kx þ Lþ aJyzð2;�2j0Þ Kx þ Jxy þ aJzwð2; 1; 1j0Þ Kx þ ~Ky þ Jxy þ Lþ aJzw þ bðJxy � LÞð1; 1; 1; 1j0Þ aLþ bJxy þ cJzw (a2 þ b2 þ c2 ¼ 1)ð2j1Þ Kx þ ~Ky þ Lþ Jxy þ aJzw þ bðKy þ ~KxÞð1; 1j1Þ Kx þ ~Ky þ aJzw þ bðL� JxyÞð0j2Þ Kx þ Jxy þ a ~Kz (a � 0)ð0j1; 1Þ aKx þ b ~Ky þ cJzw (b � �a, a2 þ b2 þ c2 ¼ 1)


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of � in Table II up to isometry of AdS5 connected to theidentity, where a, b, c are real numbers, and the doublesigns must be taken in the same order in each expression.

The proof is given in the Appendix. Note in Table II thattype ð0j1; 1Þ would become type ð1; 1; j1Þ (with b ¼ 0) ifone set b ¼ �a, and that type ð0j2Þ would become typeð2;�2j0Þ (with a ¼ 0) if one set a ¼ 0.

V. THE HOPF STRING

In this section, we choose a type from the classifiedstrings in the theorem and find its trajectory. We assumethat the string obeys the Nambu-Goto equation and applythe general procedure presented in Sec. II. The examplealso shows that working with the lift curves as explained inSec. II can make the calculations and geometric interpre-tation of the trajectory simple and transparent.

We shall say that a Hopf string is a cohomogeneity-onestring which is homogeneous under the change of theoverall phase in the complex coordinates defined inSec. III:

� ei� ; � 2 R: (15)

This isometry is the simultaneous rotations in the st, xy,and zw planes. The Killing vector field � is proportional toLþ Jxy þ Jzw and falls into type ð1; 1; 1; 1j0Þ with the

condition a ¼ b ¼ c. The Killing orbits are closed time-

like curves in AdS5. In the universal cover gAdS5, they arenot closed and the string solution represents a stationarystring.

Let us find the configurations of the Hopf string bysolving the action principle (3) and finding the geodesicson ðO;�fhÞ. We first see that the orbit space ðO; hÞ is aRiemannian manifold, since � is timelike. Then, from thefact that f ¼ �a�a is constant (which we set �1), we findthat solving the geodesics on ðO;�fhÞ is nothing butsolving geodesics on ðO; hÞ. One could either introducesome coordinate system onO to solve (3) directly or makean ansatz with some coordinate system on AdS5 to solve(2). Both methods work well but would lead to somewhatcomplicated equations. In what follows, we would takeadvantage of the symmetry, especially the complex struc-ture, of E4;2 and find the lift curves on the spacetime AdS5

which project to the geodesics on ðO;�fhÞ, as was ex-plained in Sec. II.

The metric h in (1) for the Hopf string is the usual flatmetric d � d with the contribution from the phase changebeing subtracted. With the constraint (6), h can be writ-ten as

h ¼ l2d � ð1� PÞd ; (16)

where P :¼ � � is the normal projection along . This isthe same as the Fubini-Study metric on a projective spaceCP2 except that we started with an indefinite scalar product� ¼ diag½�1; 1; 1� in (6) and in dS2 ¼ l2d � d , while the

usual Fubini-Study metric is defined by means of apositive-definite scalar product. We shall also call h theFubini-Study metric here and shall denote the Riemannianmanifold ðO; hÞ by CP2�. The fibration CP2� ’ AdS5=Uð1Þis the generalization of the Hopf fibration to the case ofindefinite scalar product [9]. Thus the problem of findingNambu-Goto strings has been reduced to solving geodesicson CP2�.Our action (3) for the Hopf string becomes

S ¼ZCd�

�1

�_� ð1þ � Þ _ þ �þ�ð1þ � Þ

�; (17)

where � is a Lagrange multiplier. This is the action forgeodesics on O written in terms of the coordinates in

E4;2. The action (17) has a Uð1Þ gauge invariance ð�Þ �ei�ð�Þ ð�Þ [10] which corresponds to the freedom in thechoice of a lift. This gauge degree of freedom is used tosimplify the calculation. In particular, we shall show thateach geodesic on O for the Hopf string can always bewritten in a proper gauge as the projection of a geodesic onAdS5.The Euler-Lagrange equations are the constraint (6) and

_� ð1þ � Þ _ ¼ �2; (18)

��1

�ð1þ � Þ _

�� þ 1

�_� _ þ� ¼ 0: (19)

Multiplying both sides of (19) by � from the left and usingthe constraint (6), one obtains an equation which merelydetermines �. On the other hand, the time derivative of (6)implies that � _ is pure imaginary. This value can be

changed by the gauge transformation � ei�ð�Þ ð�Þ.We can always choose the gauge Re � _ ¼ 0 which, underthe constraint (6), implies

� _ ¼ 0: (20)

Geometrically, (20) means that the curve C on M ishorizontal (namely, it is orthogonal, with respect to g) tothe fiber ��1ð� � Cð�ÞÞ at each point on C. Multiplyingboth sides of (19) by 1þ � from the left, and using (6)and (20), one obtains the geodesic equation for the Fubini-Study metric,

ð1þ � Þ� _

�

�� ¼ 0: (21)

Choosing the parameter of the curve to be the properlength so that � 1, one can write (21) in a particularly

simple form. Since (18) and (20) imply � € ¼ � _� _ ¼�1, (21) yields

€ ¼ : (22)

One can immediately solve the equation to obtain

ð�Þ ¼ A cosh�þ B sinh�; (23)


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�AA ¼ �1; �AB ¼ 0; �BB ¼ 1; (24)

where A, B 2 C3. The projection � � C of the curvesC: �� ð�Þ expressed by (23) are geodesics on O.

Some remarks are in order. First, the geodesics on thefour-dimensional manifold O should contain seven inde-pendent real constants: the initial position and the directionof the initial velocity. One sees that � � C actually con-tains seven independent real constants since we have 12real constants, four constraints (24), and one redundancy,i.e., the phase of ð0Þ. Second, the lift curve (23) is ahorizontal geodesic on AdS5. A special feature of theHopf string is that one can always choose a lift curveC—the horizontal lift in this case—of a geodesic c onthe orbit space ðO;�fhÞ so that C is also a geodesic onðM; gÞ. Third, a horizontal geodesic C on AdS5 is theintersection of AdS5 and a two-dimensional plane throughthe origin in E4;2, which corresponds to the great circle inthe case of positive-definite metric. Thus the hyperboliccurve (23) is unique up to isometry, for any choice of A andB. Furthermore, C is a Killing orbit on AdS5.

Now the world sheet S of the Hopf string can be writtendown easily. From (4), (15), and (23), we have

ð�; �Þ ¼ ei�ðA cosh�þ B sinh�Þ; (25)

where A and B satisfy the condition (24).To describe geometry of the world sheet S in more

detail, let us introduce a new time coordinate T on AdS5

defined by

T ¼ arg 0 ¼ argðsþ itÞ: (26)

In gAdS5, T runs from �1 to 1. The T ¼ constant hyper-

surfaces embedded in gAdS5 are Cauchy surfaces. TheKilling field � ¼ d=d� drives the simultaneous rotationsin the xy and zw planes while going up along the T axis.Thus the world sheet of the Hopf string can be viewedpictorially as the surface swept by a boomerang (23) flyingup while rotating (Fig. 2).

Let us reduce the degrees of freedom of A and B in (23)by the homogeneity-preserving isometries and canonical-ize them, as was explained in Sec. II. The Lie algebra NgðkÞof the homogeneity-preserving isometries is the vectorspace spanned by

�; L� Jxy; Lþ Jwz; Jyz þ Jwx; Jzx þ Jwy;

~Kz þ Kw;Kz � ~Kw; ~Kx þ Ky; Kx � ~Ky:(27)

In fact, all generators (27) commute with �. The isometriesgenerated by (27) map the solution (25) to another isomet-ric one. First, using the isometries generated by L, Jxy, and

Jwz, one can make a general A 2 C3 in (24) to be real, i.e.,to have no t, y, w components. Then, by using ~Kz þ Kwand ~Kx þ Ky, one has A ¼ ð1; 0; 0ÞT . Next, we canonic-

alize B by the isometries, which leaves this A unchanged. Itfollows from �AB ¼ 0 that B ¼ ð0; B1; B2Þ. By using Jxy

and Jwz, one can make B1 and B2 real. Finally, by usingJzx þ Jwy, one has B ¼ ð0; 1; 0ÞT , where � 2 R. As a

result, the trajectory (25) can be written up to isometry as

Txyzw

0BBBBB@

1CCCCCA ¼

�sinh� cos�sinh� sin�

00

0BBBBB@

1CCCCCA; (28)

where we have used T ¼ argðsþ itÞ. In particular, theworld sheet has no parameter and is unique. We can there-fore say that the Hopf string has rigidity.Figure 2 shows the world sheet of the Hopf string. This is

a helicoid swept by a rotating rod passing through the Taxis. This surface is periodic in the T direction with period�. A similar helical motion of an infinite curve in theMinkowski space has a cylinder outside of which thetrajectory becomes tachyonic (spacelike). In the anti-deSitter case, however, the trajectory is always timelikebecause the physical time passing in the unit interval ofT becomes large when the curve is far from the T axis inFig. 2.Let us summarize some special features of the Hopf

string. (i) The Killing vector � has a constant squarednorm. (ii) The orbit space ðO;�fhÞ for the Nambu-GotoHopf string inherits the complex structure of E4;2, overwhich AdS5 admits a Hopf fibration. (iii) The orbit spaceðO;�fhÞ is homogeneous and is highly symmetric.(iv) The world sheet of the string is homogeneously em-bedded and is flat intrinsically. (v) The world sheet of thestring is rigid, i.e., it is unique up to isometry.Among anti-de Sitter spaces, a Killing field satisfying (i)

or (ii) exists only in the odd-dimensional ones. In the case

-5

0

5x

-5

0

5y

-5

0

5

T

-5

0

5x

-5

0

5y

FIG. 2 (color online). The world sheet of the Hopf string in thecoordinates ðT; x; yÞ. The other coordinates z and w vanish.


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of AdS5, the only Killing vector satisfying (i) is Lþ Jxy �Jzw up to scaling and rotation of the spatial axes [11].

Condition (i) is partially a reason for (ii) and (iii). In thecase of the Hopf string, the homogeneity-preservingisometry group NGðKÞ equals the centralizer ZGðKÞ. Onthe other hand, ZGðKÞmust preserve f (Sec. II). Thus (i), ingeneral, suggests high symmetry of ðO;�fhÞ. In the caseof the Hopf string, the isometry group of the orbit space isan eight-dimensional group. In fact, the vector fields (27),except the first one �, form a closed Lie algebra and act onðO;�fhÞ as Killing fields.

As for (iv), one finds that the resulting world sheet (28)for the Hopf string is invariant under the infinitesimalisometry ~Kx þ Ky of AdS5. Since � and ~Kx þ Ky com-

mute, the world sheet S is acted on by R2 and is homoge-neous. This implies that S is flat intrinsically; namely, S isthe two-dimensional Minkowski space embedded in AdS5.This can also be verified by a direct computation of theintrinsic metric.

The high symmetry (iii) implies (v) for the Hopf string.Incidentally, stationary strings in AdS4 [12] do not haverigidity. They would most naturally correspond in AdS5

to the cases � / Lþ bJxy, which are of the same type

ð1; 1; 1; 1j0Þ as the Hopf string but with differentparameters.

These facts suggest that the Hopf string is similar to thestring with simple time translation invariance in theMinkowski space. The Hopf string is the only solution inAdS5 which shares all of the properties (i), (iii), (iv), and(v) with the flat string in the Minkowski space.

VI. CONCLUSION

The cohomogeneity-one symmetry reduces the partialdifferential equation governing the dynamics of an ex-tended object in the spacetime M to an ordinary differen-tial equation. With applications in higher-dimensionalcosmology in mind, we have presented the procedure toclassify all cohomogeneity-one strings and solve theirtrajectories with a given equation of motion. The formeris to classify the Killing vector fields up to isometry, andthe latter is to solve geodesics on the orbit space ðO;�fhÞwhich is the quotient space of M by the symmetry groupK. We have carried out the classification in the case that thespacetime is the five-dimensional anti-de Sitter space, byan effective use of the local isomorphism of SOð4; 2Þ andSUð2; 2Þ and of the notion of H similarity. Assuming thatthe string obeys the Nambu-Goto equation, we have solvedthe world sheet of one of the strings, which we call theHopf string, in the classification. The problem has reducedto finding geodesics on the orbit space ðO; hÞ. By using atechnique similar to the one used in quantum informationtheory and working on the lift curves in M, we haveobtained a new solution which describes the trajectoriesof the Hopf string. The solution represents timelike heli-

coidlike surfaces around the time axis, which is unique upto isometry of AdS5.We can say that the Hopf string is the simplest example

of a string in anti-de Sitter space which corresponds to astraight static string in Minkowski space. The Killingvector field defining the symmetry of the string is homo-geneous in the spacetime and has a constant norm. Thisgreatly simplifies solving the geodesics on the orbit space,and the world sheet becomes homogeneous and rigid, aswe have seen in Sec. V. The simplicity of the Hopf stringssuggests that they were common in the Universe andplayed significant roles, if the Universe is higher-dimensional or is a brane world.We would like to remark that, although we now have all

types where the equations of motion reduce to ordinarydifferential equations, this does not, in general, implysolvability. The solvability problem is nontrivial andstrongly related to the structure of the orbit spaces. Asystematic analysis will be presented in a future work.Finally, we would like to remark that the classification

presented here will be the basis for that of higher-dimensional cohomogeneity-one objects. The procedureis the following: (i) For each of the Killing vector fields� classified in Table II, enumerate how one can add new

independent Killing vector fields �ð1Þ; . . . ; �ðnÞ such that

�; �ð1Þ . . . ; �ðnÞ form a closed Lie algebra k0; (ii) reducethe degrees of freedom of k0 by using the isometries whichpreserve �, thus classifying the Lie algebras k0;(iii) examine the orbits in the spacetime generated by k0.

ACKNOWLEDGMENTS

The work is partially supported by Keio GijukuAcademic Development Funds (T. K.).

APPENDIX: PROOF OF THE THEOREM

Let X be an -anti-self-adjoint matrix X. The lemmaimplies that ðX=i; Þ ðJ; PÞ with some ðJ; PÞ. On theother hand, if ¼ WPWy, the definition of unitary simi-larity implies ðJ; PÞ ðWJW�1; Þ. Thus ðX=i; Þ ðWJW�1; Þ so that X iWJW�1. We therefore cancarry out the classification by the following procedure:(i) Enumerate ðJ; PÞ in the lemma such that there existsW satisfying ¼ WPWy, (ii) construct X0 ¼ iWJW�1,(iii) translate X0 back to the Killing vector field � inSOð4; 2Þ by Table I.In some cases, however, the canonical pairs ðJ; PÞ and

ðJ0; P0Þ correspond to X0’s which generate an identical Liegroup. This happens when ðJ0; P0Þ ð�J; PÞ with a non-zero real number�. Thus it is important to know how a pairð�Jjð�jÞ; PjÞ can be canonicalized. For �> 0, we simply

have ð�Jjð�jÞ; PjÞ ðJjð��jÞ; PjÞ, so that they generate

an identical group. Thus we focus on ð�Jjð�jÞ; PjÞ in the

following. When dj is odd, we have


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ð�Jð�jÞ; PjÞ ðJð��jÞ; PjÞ: (A1)

This can be seen by applying a similarity transformationwith diag½1;�1; 1; �. When dj is even, we have

ð�Jð�jÞ; PjÞ ðJð��jÞ;�PjÞ; (A2)

which can be shown by applying a similarity transforma-tion with diag½1;�1; 1; �, etc. In the special case ofdj ¼ 2 and �j 2 C, not only (A2) but also (A1) holds

because �Jð�jÞ ¼ Jð��jÞ.The relation between ðJ; PÞ and ðJ;�PÞ is also impor-

tant. Let us show that their corresponding Killing vectorfields are related by a reflection r: ðt; xÞ � ð�t;�xÞ,which is a transformation in SOð4; 2Þ which is not con-nected to the identity (and hence is not used in the equiva-lence relation ). When ðJ; PÞ ðX0; Þ, we haveðJ;�PÞ ð�X0

0; Þ with X00:¼ �UXU�1 and U :¼ �y �

�x, becauseUUy ¼ �. On the other hand, one can read

off from (7) that the transformation p� �UpðU�1ÞT ¼�UpUT is a reflection along the t and x axes. Thus theKilling vector field � corresponding to X0 and the one �0corresponding to X0

0 are related by �0 ¼ r��.Let us find the relation of the minor types within each

major type by using the results above. We denote by anequal sign that two minor types are related by -unitary

similarity, which should be considered identical, and by scthat two minor types are related by a scalar multiplication.

For type ½4j0�, it follows from (A2) that ðþÞsc ð�Þ, which isinvariant under r (though the parameters change). For type½3; 1j0�, there are two minor types, ðþ�Þ and ð�þÞ, whichare not related by scalar multiplication but by the reflectionr (and hence not equivalent in the classification). For type

½2; 2j0�, it follows from (A2) that ðþþÞsc ð��Þ, which isinvariant under r. By a simple reordering, we haveðþ�Þ ¼ ð�þÞ, which is invariant under r. For type½2; 1; 1j0�, by reordering, there are at most two minor types,

ðþ þ�Þ and ð� �þÞ. Furthermore, we have ðþ þ�Þsc ð� �þÞ, by applying (A2) to all blocks. It is invari-ant under r. Type ½1; 1; 1; 1j0� has only one minor type (by

reordering). For type ½2j1�, we have ðþÞsc ð�Þ by applying(A2) to the first block and (A1) to the second block,yielding ðdiag½J1; J2�;diag½�P1;P2�Þ ð�diag½J01; J02�;

diag½P1;P2�Þ. Type ½1; 1j1� has a unique minor type (byreordering). Type ½0j2� and type ½0j1; 1� have a uniqueminor type.Let us demonstrate the concrete calculation for type

½2j1� (the other types can be found in a similar manner).We have, because J is traceless,

J ¼ diag

��a 10 a

�;�aþ bi;�a� bi

�;

where a and b are real numbers, and

P ¼ diag

��0 11 0

�;

�0 11 0

��:

As discussed above, however, it suffices to consider theplus sign. Let us choose W¼S23 diag½Rð�=2Þ;Rð��=2Þ�where

Rð�Þ ¼ cos� � sin�sin� cos�

� �

and

S23 ¼1 0 0 00 0 1 00 1 0 00 0 0 1

26664

37775:

Then

X0 ¼ iWJW�1

¼iðaþ 1=2Þ 0 i=2 0

0 �ia 0 b�i=2 0 iða� 1=2Þ 00 b 0 �ia

26664

37775

¼ �að1=iÞ � �z þ b�x � 1� �z2

� ð�y þ ð�z=iÞÞ � ð1þ �zÞ4

:

From Table I, we find that X0 corresponds to the soð4; 2Þtransformation �¼� ~KwþKzþLþJwzþaJxyþbðKw�~KzÞ, where we have rescaled � (by �4) and redefined aand b.

[1] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370(1999).

[2] V. P. Frolov, V. Skarzhinsky, A. Zelnikov, and O. Heinrich,Phys. Lett. B 224, 255 (1989); V. P. Frolov, S. Hendy, andJ. P. De Villiers, Classical Quantum Gravity 14, 1099(1997).

[3] K. Ogawa, H. Ishihara, H. Kozaki, H. Nakano, and I.Tanaka, in Proceeding of the 15th JGRG Workshop,

Tokyo, 2005, edited by T. Shiromizu et al., p. 159.[4] H. Ishihara and H. Kozaki, Phys. Rev. D 72, 061701(R)

(2005).[5] M.G. Jackson, N. T. Jones, and J. Polchinski, J. High

Energy Phys. 10 (2005) 013.[6] S. Holst and P. Peldan, Classical Quantum Gravity 14,

3433 (1997).[7] I. Yokota, Classical Simple Lie Groups (Gendai-


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Sugakusha, Kyoto, 1990) (in Japanese).[8] L. Gohberg, P. Lancaster, and L. Rodman, Matrices and

Indefinite Scalar Products (Birkhauser Verlag, Basel,1983).

[9] In quantum mechanics, the global phase of the state vectoris irrelevant so that one works on the projective space. Oneobtains the Fubini-Study metric on CPn by subtracting thecontribution of the phase change from the usual innerproduct on the Hilbert space of state vectors. See J.Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697(1990); for a physical application, see, e.g., A. Carlini,A. Hosoya, T. Koike, and Y. Okudaira, Phys. Rev. Lett. 96,060503 (2006). The geodesic equation is derived and

solved in a similar manner.[10] Also, the variable � changes by the gauge transformation:

�� ð1=�Þð2 _� Im � _ þ _�2 � Þ.[11] If one also admits the spatial reflection, an isometry which

is not connected to the identity, one sees that Lþ Jxy þJzw is the only Killing vector of constant norm.Alternatively, one can treat the case with Lþ Jxy � Jzwin the same manner as in the present section by consider-ing ð 0; 1; ð 2Þ�Þ instead of ð 0; 1; 2Þ.

[12] A. L. Larsen and N. Sanchez, Phys. Rev. D 51, 6929(1995);H. J. de Vega and I. L. Egusquiza, Phys. Rev. D54, 7513 (1996).


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Strings in five-dimensional anti-de Sitter space with a symmetry