P-Branes, Poisson-Sigma-Models and Embedding Approach to (p+

a r X i v :h e p -t h /9703099v 2 17 M a r 1997Preprint TUW 96/31

hep-th/9703099

P-Branes,Poisson-Sigma-Models and Embedding Approach to (p +1)-Dimensional Gravity

Igor A.Bandos 1and Wolfgang Kummer 2

1

Institute for Theoretical Physics,

NSC Kharkov Institute of Physics and Technology,

310108,Kharkov,Ukraine

e-mail:kfti@rocket.kharkov.ua,

2Institut f¨u r Theoretische Physik,

Technische Universit¨a t Wien,

Wiedner Hauptstrasse 8-10,A-1040Wien

e-mail:wkummer@tph.tuwien.ac.at Abstract A generalization of the embedding approach for d-dimensional gravity based upon p -brane theories is considered.We show that the D -dimensional p -brane coupled to an antisymmetric tensor ?eld of rank (p +1)provides the dynamical basis for the description of d =(p +1)dimensional gravity in the isometric embedding formalism.”Physical”matter appears in such an approach as a manifestation of a D -dimensional antisymmetric tensor (generalized Kalb-Ramond)background.For the simplest case,the Lorentz harmonic formulation of the bosonic string in a Kalb-Ramond background and its relation to a ?rst order Einstein-Cartan approach for d =2dimensional gravity is analysed in some detail.A general Poisson-sigma-model structure emerges.For the minimal choice of D =3an interesting “dual”formulation is found which has the structure of a Jackiw-Teitelboim theory,coupled minimally to a massive scalar ?eld.Our approach is intended to serve as a preparation for the study of d -dimensional supergravity theory,either starting from the generalized action of free supersymmetric (d ?1)-branes or

D (d ?1)-branes,or from the corresponding geometric equations (’rheotropic’conditions).PACs:11.15-q,11.17+y,04.60.Kz,04.50.+h,11.10.Kk.11.30-j.

1

1Introduction

As summarized,e.g.in ref.[1],the embedding approach to gravity has a long history,beginning at least from the famous book of Eisenhart[2].”From time to time,some interesting results would be derived in this way,but they would also be directly derivable from the Riemannian metric,interest in the embedding method thus subsiding again”[1].Most results obtained by this approach for General Relativity(GR)can be found in[3,4].

More recently the problems related to the quantization of GR and some early successes in string theory encouraged Regge and Teitelboim to study the possibility of a’string-like’description of gravity[5]as an alternative for the’intrinsic’description in terms of Riemannian geometry.This idea generated renewed interest in the search for a dynamical basis regarding the old embedding method[3,4].Since then this approach[5]has been developed in several papers[6,7,8,9],where e.g.in[6]the model for gravity provided by a free bosonic p-brane action in curved(but conformally?at)target space-time was considered.

It should be stressed that at the classical level p-brane(and D p-brane)theories in these approaches have only matter(and gauge?eld)degrees of freedom,with gravity involving only auxiliary non-propagating ones.But a kinetic term for gravity should appear in the e?ective action of the quantum theory after the integration over the matter?elds,yielding propagating gravity as a result of quantum e?ects(see,for example,the discussion of this point in[6]).Hence all the embedding models could be regarded as particular realizations of the concept of gravity induced by quantum corrections[10,11].

Recent progress to understand some nonperturbative features of superstring theories[12,13, 14,15,16,17,18,19](and refs.therein),and especially the concept of’p-brane democracy’[15] adds reasons to search for an adequate description of Nature in terms of the embedding approach and its supersymmetric generalization.Our present paper intends to prepare the ground for such investigations.

An important motivation for our study was the observation that a close relation of the Lorentz-harmonic formulation of bosonic string theory[20,21]exists with the Poisson-sigma-models(PSM)[22,23,24],in which the uni?cation of all types of matterless2-dimensional gravity(including dilaton gravity,models with dynamical torsion and spherically symmetric4d gravity)has been achieved.For our present purposes we mainly need from that approach the fact that vielbeine and spin connection are used systematically as independent variables in a ?rst order formulation.We shall demonstrate here that indeed a PSM-like structure appears naturally in the twistor-like Lorentz harmonic formulation of strings[20,21]after a change of variables corresponding to the transition to the so-called(Lorentz-)analytical coordinate basis [25,26,27,28,29,30,31,32].

The general model appears when the string interacting with a Kalb-Ramond(KR)back-ground is considered.A key point is that the interaction with the bosonic string does not put any restriction on the background at the classical level1.This is why the interaction with a KR-background produces an analog of an arbitrary potential involved in the PSM-action [22,23,24],whereas“matter”type interactions are induced by the transversal components of the target space coordinates.

In principle,D=3string theory is enough to describe arbitrary curved2-dimensional space-time.The deep reason for this is given by the general theorem about local isometric embedding of analytical d=(p+1)-dimensional manifold into D-dimensional?at Minkowski space-time [2,34,35,4].2It states that such an embedding is always possible(at least locally)when

d(d+1)

D≥

3This result have been brie?y reported in[38],where some possible application of the embedding approach for F-theory[39,40,41,42,43]were considered

4The extrinsic geometry of D=4string in a KR?eld was studied in[44](see also[45,46]),as well as in recent work[49].In[50]string theory for an anti-de-Sitter background was investigated and it was demonstrated that the minimality condition for the embedding is broken in such a case.

5From the point of view of supergravity the existence of both possibilities to describe world-volume gravity, namely by considering curved target space with nontrivial gravity as well as the?at one with KR?elds only, seems to be very natural.Indeed,at high dimensions(D=10,11)the only supermultiplet containing the KR ?elds is the supergravity multiplet which involves gravity too.

3

approach to p-branes(extended objects with p space like dimensions of the world volume). Moving frame variables(Lorentz harmonics[25,28,29,30,20,37,31,21,48])are introduced here.

The moving frame(Lorentz harmonic)formulation of the p-brane theory[20,37,21]in the D-dimensional generalized Kalb-Ramond(GKR)background is described in detail in Section 3.It is proved that it provides the dynamical ground for d=p+1dimensional gravity in the embedding formalism[3,4].

In Section4the bosonic string interacting with the KR?eld is studied as a model for d=2 gravity.We shortly describe a Lorentz harmonic formulation of free bosonic strings[20,21]and discuss the PSM-like structure of this formulation appearing in the world-sheet(’analytical’) basis.The integration with the KR background shows that a model for a general type of d=2 gravity with matter is obtained in this manner.As a simplest example we treat the gravity models inspired by free string theory.

Supersymmetric generalizations of our approach and general directions of further research are described in the Conclusion.

2The Embedding Method for d-Dimensional Gravity

2.1Metric Approach

For GR in d dimensional space-time M d is obtained by specifying the matter,calculating its energy-momentum tensor T mn(x)≡e m a e n b T ab(x),and use the latter as a source for the Einstein ?eld equation

R ac bc?1/2δa b R dc dc=1/2T a b(x)(2) where the curvature two-form is de?ned by6

R ab(d,d)≡d?ab??ac∧?b c(3)

≡1/2dx m∧dx n R ab nm(x)≡1/2e d∧e c R ab cd(x)

The spin connections?ab are supposed to be the ones,constructed from the vielbein?elds (e a=dx m e a m)

e m a?mbc=

=e m a e n b?[m e n]c?e m b e n c?[m e n]a+e m c e n a?[m e n]b(4) as a consequence of vanishing torsion

T a≡D e a≡de a?e b∧?a b=0,(5) with

T a≡1/2dx m∧dx n T a

nm ≡1/2e b∧e c T a

cb

the torsion two-form on M d.

So the curved space-time M d is described as the solution of Einstein equation(2),(3),(4), (5)for a given matter distribution speci?ed by the expression for energy-momentum tensor T ab(x).

≡0.

4

To?nd exact solutions of the Einstein equations(2)for d=4embedding methods were widely used in the past[3,4].They are based on general theorems[2]stating that any analytical d-dimensional curved space-time M d can be considered as a subspace in?at D-dimensional pseudo-Euclidean space-time R1,D?1with D≤d(d+1)/2at least locally7.The metric of M d is induced by the embedding

ds2=dx m dx n g mn(x)=dx m dx n?n X nηn,(6) i.e.it is expressed in terms of derivatives?m=?/?x m of the coordinate functions de?ning M d parametrically as the subspace in R1,D?1:

X m(x n)(7) Here X m=0,1,...,(D?1))denote Cartesian coordinates of R1,D?1,

ηn≡diag(+1,?1,...,?1)is the?at Minkowski metric,x m(m=0,1,...,(d?1))represent local holonomic coordinates of M d.

This description of d-dimensional space-time is similar to the one used for the world sheet of strings(d=2)and for the world volume of p-brane(d=p+1)theories.Further steps towards the description of the embedding method of GR imply the introduction of an extrinsic geometry formalism for M d.They are similar to the ones performed in the geometric approach for strings and p-branes[44,45,21]and will be considered in the same manner[21].

2.2Extrinsic Geometry in Cartan Variables

For our present purpose the vielbein formalism turns out to be more suitable than the metric one.To understand the embedding conditions in this case,let us use an appropriate local Lorentz(SO(1,D?1))transformation to adjust to each point of M d a local moving(co-)frame of R1,D?1

E a u a,a

m,u i m

m n

m

b

(x)are parallel to M d and(D?d)vectors u i m

u a m

u i m

,u i m m

7Here we will restrict ourselves by considering the case of local embedding of analytical spaces only.The number of additional dimensions(D?p)being necessary for global embeddings can be much higher[4,3].Moreover,even if one replaces the requirement for M d to be analytically embedded by the requirement of di?erentiability(C∞), he gets D≤d(d+3)/2instead of D≤d(d+1)/2[35,7].

8Writing(10)we identify the Lorentz group of M d with the SO(1,d?1)subgroup of the R1,D?1Lorentz group SO(1,D?1)and,in such a way,break SO(1,d?1)?SO(1,d?1)?SO(D?d)gauge symmetry of the considered construction up to SO(1,d?1)?SO(D?d).

5

u a m b=ηab(12)

u a m j=0(13)

u i m j=?δij.(14) Due to the natural SO(1,d?1)?SO(D?d)gauge invariance of the present construction,the vari-ables u a m are simply interpreted as homogeneous coordinates of the coset SO(1,D?1)

∧du a m

∧du i m

(x n)from our further considerations.

To obtain the di?erentials of the moving frame variables u a=(u a m)we shall take into account the orthonormality conditions(9)or(12)-(14).Using the unity matrix decomposition

δn=u a u n=u a m a?u i m,(17) being equivalent to(9)((12)-(14)),the di?erentials of the moving frame vectors u a=(u a m) become

du a=u b?a(d)?

du a mωa b(d)+u i m=a;

(18)

du i m A ji+u m=i

Here

?a(d)=??b(d)≡u a du b= ωa b(d)f ai(d)

?f bj(d)A ji(d) (19) is the so(1,D?1)valued Cartan1-form.In(19)it is decomposed(in a SO(1,d?1)?SO(D?d)gauge covariant way)into d×(D?d)covariant forms(forming the basis of the coset SO(1,D?1)

du m

SO(1,d?1)connection1-forms

2

ωab≡u a m b,(21) and(for D?d>1),(D?d)×(D?d?1)

m

du

?From(19)the Cartan forms?a satisfy the Maurer-Cartan equation

d?a??a∧?c=0(23) which now just re?ect the?atness of the D dimensional embedding space.

The SO(1,d?1)?SO(D?d)gauge covariant splitting of the connection form?a(19) induces the splitting of(23)into the following set of equations for the forms f ai,ωab,A ij:

D f ai≡d f ai?f bi∧ωa b+f aj∧A ji=0(24)

R ab≡dωab?ωac∧ωb c=f ai∧f bi(25)

R ij≡dA ij+A ik∧A kj=f i a∧f aj(26) Eqs.(24),(25),(26)give rise to Peterson-Codazzi,Gauss and Ricci equations,respectively, of the subspace embedding theory[2,4](see also[21]).

The covariant di?erential D used in Eq.(24)include the formωab and A ij as SO(1,d?1) and SO(D?p)connections.So its pull-back onto the surface M d(D=dx m D m=e a D a) considered as a covariant di?erential on M d implies that spin connections and gauge?elds are being induced by the embedding.

Using this di?erential we can investigate the integrability conditions for Eqs.(10)and(11) in manifestly SO(1,d?1)?SO(D?d)gauge invariant form

D E a≡dX m=D e a,(27)

D E i=dX m=0.(28) The covariant di?erentials D for the moving frame variables u a m from(18)are expressed in terms of SO(1,D?1)

≡du a mωa b(d)=u i m

≡du i m A ji=u m

9The proper de?nition of the second fundamental form is K i≡dx n dx m K i

≡e b e c K i ab,K i mn∝?m?n X m

mn

with?n X m=0[2].

7

e a m e b n K i ab=K i nm appearing in the decomposition o

f the pull-back of the covariant connection form f ai(see[21])

f ai=e b K abi.(34) Equations(24)–(26),(32),(33)are known from the Classical Theory of Surfaces[2]and describe an arbitrary surface embedded into?at D-dimensional space-time.

The main extrinsic curvatures of the surface h i are the traces of the second fundamental form matrix K abi

h i≡K abiηab=e m a f ai m≡f ai

a

.(35) They can be used to de?ne the embedded surface M d.In particular,the world volume of the free bosonic p-brane(p=d?1)is considered to be a minimal surface,i.e.is de?ned by

h i=0

.In the embedding method of GR the description of the curved space time M d is achieved by the use of the extrinsic geometry equations together with the algebraic equation

f ai

c

f ci c?f ai b f ci c?1/2δa b(f di c f ci f?f di d f ci c)≡

K ai c K ci b?K ai b K ci c?1/2δa b(K fi

c K ci f?K fi

f

K ci c)(36)

=T a b(x).

This can be derived from the Einstein equation(2)when the Gauss equation(25)is taken into account.

3Bosonic P-Branes with Kalb-Ramond Background and d= p+1Gravity

The main result of this section is that arbitrary nonvanishing extrinsic curvature h i=0can be produced by a D=(p+1)(p+2)/2-dimensional p-brane theory interacting with a GKR background,i.e.with an antisymmetric tensor?eld of rank(p+1).

This statement seems to be universal,i.e.independent of the special formulation of p-brane theory.But for the proof,the extrinsic geometry formalism(i.e.the so-called geometric approach [44,45,47,21,51,52])for p-branes in a GKR-background should be developed.The Lorentz harmonic formulation[20,37,21]is most suitable to do this,because it produces the master equations of the geometric approach(rheotropic conditions in the terminology of ref.[48])as e.o.m.-s(see below).

We will demonstrate that for a D-dimensional p-brane interacting with an antisymmetric tensor?eld B m

p+1

(X m

(p+1)2?a

0...a p

u a0m p(x)×(37)

H m

2

...m

1

m p+2(X)≡

≡(?m

2

m p+2(X)??m1m p+2(X)+...).(38) This means that

8

?any curved space-time M d can be identi?ed locally with some world volume of such a type.

?matter is the manifestation of the GKR background in such an approach.

3.1Action Functional

Let us consider a p-brane interacting with an antisymmetric tensor gauge?eld of the rank(p+1) (GKR?eld)

B p+1=dX m1B m

p+1

(X)(39) in the twistor-like Lorentz harmonic formulation[37,21].

Then the action functional

S p+1=S p+1

0+S p+1

int

(40)

is the sum of the free p-brane action[20,37,21]

S(p+1) 0=?

1

(p+1)e a∧e a1∧...∧e a p)?aa

1...a p

(which represents the bosonic limit of the generalized action for super-p-branes[48]),and of an interaction term

S p+1

int

= M p+1B p+1.(42) In(41),(42)the Lagrangian(p+1)forms are integrated over a world volume M p+1of the p-brane(which we will identify with curved d=p+1dimensional space-time)whose local coordinates are denoted byξm(a=0,1,...,p)(and will identi?ed with the x m coordinates from the previous section).

For simplicity in(41)we?x the”cosmological constant”(being proportional to the inverse p-brane tension,i.e.to the Regge slope parameterα′for the string case)to be equal to one.

e a=dξm e a m are world volume vielbeine(a=0,1,...,p),E a(a=0,1,...,p)are pull-backs o

f (p+1)1-forms(10)from the target space vielbein(8).The general target space vielbein E a

by Lorentz rotations represented by the matrix(9).

The variation of the moving frame vectors u a and u i which does not break the orthonormality conditions(9)(or(12)–(14))reduces to generalized Lorentz(SO(1,D?1))transformations

δu a=u b iδ?a≡u b?a(δ)

? δu a mωa b(δ)+u i m=a;

δu i m a f ai(δ)?u j m=i

(43)

where the parameters of unconstrained variations iδ?a=?a(δ)=(f ai(δ),ωab(δ),A ij(δ))can be considered as contractions of the Cartan forms(19)(or(20)-(22))with the variation symbol δ,or equivalently as Cartan forms depending on the variation symbolδinstead of the external di?erential d.

Similar expressions for the external di?erential of the vectors u a are obtained from(18)and are equivalent to the de?nition of the connection forms(19)(or(20),(21),(22)).

9

3.2Equations of Motion

Taking into account the constrained nature of the harmonic variables(9)(or(12)–(14))the variation of the action(40)can be written as

δS= M p+1[?1

p!f ai(δ)∧E i(d)e a1∧...∧e a p?aa

1...a p

(44)

+ 1

p!D E a(δ)e a1∧...∧e a p?aa

1...a p

+E a(δ)i u a H p+2,

where

H p+2≡dB p+1=

1/(p+2)!dX m1H m

p+2

(X)(45) is a GKR?eld strength and

i c a H p+2≡

1

p+2∧...∧dX m1H m

p+2

(X).(46) The contraction of basic1-forms with the variation symbolδ

E a(δ)≡iδE a≡δX m

E i(δ)≡iδE i≡δX m

ωab(δ)≡iδωab≡u a m

f ai(δ)≡iδf ai≡u a m=δu am

A ij(δ)≡iδA ij≡u i m,(47) together with the variation of the world volume vielbein

δe a=dξmδe a m

are taken as a basis in the space of variations.

The absence of the parameters A ij(δ)in(44)re?ects the SO(D?p?1)gauge invariance of the action(40).The SO(1,p)gauge invariance manifests itself as the presence of the parameter ?ab(δ)in the combination

δe a p?e bωa p

b

(δ)

only.Hence we can compensate SO(1,p)(pseudo-)rotations of harmonic variables by(pseudo-)rotations of the world volume vielbein

δSO(1,p)e a p=e bωa p

b (δ)=e b iδωa p

b

.

The latter can be interpreted as world volume Lorentz transformations.Therefore,the SO(1,p) subgroup of the target space Lorentz group SO(1,D?1)is identi?ed as a group of gauge symmetries of the action(40)with the world volume Lorentz group SO(1,p).

10

As a consequence,the world volume spin connections w ab are singled out which are induced by the embedding,i.e.

w ab=ωab,(48) whereωab is the pull-back of the SO(1,p)connection form(21).

The variation E a(δ)≡iδE a=δX m is related to the parameter of general coordinate (reparameterization)invariance.For this purpose the equivalent language of Noether identities is much more convenient,i.e.to analyse the interdependence of the e.o.m.-s as a result of such gauge symmetries(see below).

Now let us consider the e.o.m.-s which follow from(44)in detail starting with the variation of the world volume vielbeinsδe a and the rest of the admissible variations of moving frame variables characterized by parameter iδf ai.From(19)the Cartan forms10lead to the nondynamical equations(10),(11)which are the master equations of the geometric approach[44,45,21]as well as for the embedding method of GR[3,4](cf.section2):Fromδe a follows

(E a?e a)∧e a1∧...∧e a p?aa

1...a p

=0

or

E a≡dX m=e a,(49) whereas from iδf ai=f ai(δ)we obtain

E i∧e a1∧...∧e a p?1?aa

1...a p

=0

or

E i≡dX m=0.(50) Eqs.(49)and(50)precisely lead to the relations used in section2:The pull-backs of the(p+1) basic one forms E a of the target space become tangent to the world volume and coincide with world volume vielbeine e a(which in turn are introduced by embedding on the shell of the rheotropic conditions),and the pull-backs of the remaining(D?p?1)basic1-forms E i vanish. Thus the(D?p?1)vectors being dual to these1-forms become orthogonal to the world volume.

As demonstrated in section2,the selfconsistency conditions for the rheotropic relations(49), (50)are identical to eqs.(32),(33),respectively.

The covariant di?erential D appearing in eqs.(32)and(33)is the pull-back of the one de?ned in Eq.(18)(i.e.with SO(1,p)and SO(D?p?1)connections induced by embedding w ab=ωab, B ij=A ij).

The variation E a(δ)≡iδE a=δX m does not lead to independent e.o.m.-s.In accordance with the second Noether theorem this means that E a(δ)is related to(p+1)parameters of the gauge symmetry of the p-brane action,namely reparametrization symmetry(or general coordinate invariance for the world volume).

This can be seen from the e.o.m.for E a(δ)11

(?1)p+1

1

a H p+2=

1

p+1...dX m a H m

1

...m

10And corresponding to the SO(1,D?1)

is proportional to

∝u m2a

2u m m

p+2

,

on the surface of the rheotropic conditions(49),(50)(where dX m a).The latter expression vanishes also identically as an antisymmetric tensor of rank(p+2)with(p+1)valued vector indices.

We emphasize the interesting fact that all the independent equations considered above((49), (50))remain the same for the case of a free p-brane theory.

In the following we drop the rheotropic conditions(49),(50)and consider only their integra-bility conditions(32),(33)together with(23)(equivalent to((24)–(26))which can be solved by expressions(19)(or(20)–(22))for connection forms in terms of moving frame variables.

The embedding of an arbitrary(p+1)-dimensional subspace into?at space-time of dimension D≥(p+1)(p+2)/2is encoded in Eqs.(32),(33),(24)–(26).

To further specify the subspace under consideration it is necessary to de?ne its main extrinsic curvatures

h i≡ηab K bai,(52)

i.e.the traces of the second fundamental form matrix.This is just implied by the last e.o.m. following from iδE i=E i(δ)in(44),

f ai∧e a1∧...∧e a p?aa

1...a p

=

=?

1

(p+2)2?a

0...a p

u a0m p(x)×(54)

×H m

2

...m

(ξ).This means that our model—at least locally—provides the basis for a description of arbitrary curved space-time of dimension d satisfying D≥d(d+1)/2as a subspace in D-dimensional?at space-time.

Of course,to solve these equations in the general case some kind of’selfconsistency?eld technique’need be developed.

From the point of view of recent developments in string theory,the most interesting ones are possibly related to systems with a solitonic solution similar to the generalized magnetic monopoles[53]as a source for generating the main curvature.Further investigations along this line seem to be promising.

4Bosonic String in KR Background and Two-Dimensional Grav-ity

We now treat the simplest nontrivial example of the models proposed above.Of course most results follow from the general considerations of Section3.But we take the opportunity to introduce a light-like basis,most useful in d=2,and to also?x some notations for later

12

purposes.In an analytical basis of the target space(39)we?nd a close relation to PSM-type models[22,23,24]after incorporation of the Maurer Cartan equations into the action by Lagrange multipliers.

For the free string a suitable change of variables leads to the so-called Jackiw-Teitelboim model with an intriguing’picture-duality’property emerging for such strings in D=3.

4.1Twistor-like Action

In light-like notation the action describing the bosonic string interacting with a KR background reads

S=?1/2 M2(E+(d)e?(d)?E?(d)e+(d)

?e+(d)e?(d))+ M2B2(d,d),(55) where the second term involves the KR?eld B n(X(ξ))

B2≡B2(d,d)≡1/2dX m B n(X)

=dξm dξn?m X m B n(X(ξ)).(56) M2is a bosonic world sheet with local coordinates{ξm}={τ,σ},e±=dξm e±m≡e0±e1are the covariant light cone components of the zweibein1-form of the string world sheet.The signs in the superscript(+and?)denote the weight with respect to the action of the world sheet Lorentz group SO(1,1).

E+=dX m,(57)

E?=dX m,(58) are the pull-backs(onto the world sheet)of two basic1-forms of target space-time written in the light-like notation

E a=(1/2(E++E?),1/2(E++E?)),a=0,1.

Together with the other(D?2)forms E i(11)

E i=dX m,

they form the basis(8)of the space cotangent to target space-time.This basis di?ers from the holonomic basis dX m

+u?m,1/2(u+m)).(59)

m

The vectors u±,u i can be identi?ed with the SO(1,D?1)

u m u m u m

u m u m

du m

f?i≡u?m i,(63) the SO(1,1)connectionωab(21)having the simple representationωab∝ω?ab

ω≡1

du m

du m

2

f?i∧f+i(68)

f ij≡dA ij+A ik∧A kj=?f?[i∧f+j](69) Thus the variations of movin

g frame vectorsδu a=u b iδ?a=u b?a(δ),being necessary for the derivation of the e.o.m.-s of the action(55),become

δu±mω(δ)+u i m

=1/2e?u±m

.Substituting this expression into Eqs.

(62),(63)we?nd that the coset forms f±i are related to the second fundamental form of the embedded surface

K i mn∝?m?n X m(?n X m=0)

(whose traces de?ne the main curvatures[2])by

f±i=dξm?±i m?±i m∝K i mn e n±.(75) The evident symmetry of the second fundamental form K i mn=K i nm with respect to permu-tations of the world sheet vector indices implies that the f±i satisfy

f?i∧e++f+i∧e?=0?f?i?=f+i+≡1/2h i(76) which indeed also follows from selfconsistency conditions of the rheotropic relations(71).On the other hand,(72)and(73)yield vanishing world sheet torsion

T±≡D e±=de±?e±ω=0.(77)

14

The only proper dynamical equation appears as a result of the variation with respect to the X variable.Its independent part

f+i∧e??f?i∧e+≡e+∧e?h i=2i u i H3

≡dX m(ξ)u il n(X(ξ))(78) can be extracted by contraction with u i m

u m?u k n(X(ξ))(79)

2

through the pull-back on the KR?eld strength

H3≡dB2=1/3!dX m∧dX k n.(80) Thus,the main curvature of the world volume can be assumed to be an arbitrary function of the world volume coordinates by appropriately choosing the KR?eld.In accordance with the general theorem[3,2]mentioned in the previous Section,this means that this model even describes arbitrary two-dimensional surfaces already for the case of D=3target space dimensions.

4.2Bosonic String and Poisson-Sigma-Models

During recent years it became evident that all2d covariant models of gravity,including gener-alized dilaton models and models with gauge?elds etc.are special cases of so-called Poisson-Sigma-Models(PSM)[22,23]

S P SM= M2(Y C d A C+P BC(Y)A B∧A C)(81)

where P BC(Y)is a Poisson-structure on M2,the(zero-forms)Y represent PSM target space coordinates,A B are one-forms.Generally valid properties like the existence of absolute conserva-tion laws[22,54],special behaviors of quantized systems of this type[22,55],etc.are well-known by now.Here we shall argue that that the considered bosonic string action acquires the structure of an action functional for d=2gravity in the PSM approach.This structure becomes evident after a change of variables is carried out and an integration by parts is performed.We recall that the PSM-action for d=2gravity has the form of?rst order Einstein-Cartan theory[23]

S P SM=

=?1/2 ?X+?D e???X??D e++?X⊥d?ω(82)

?e+e?V(?X⊥,?X+?X?)

where X±,?X⊥are Lagrange multipliers,?ωis a world sheet spin connection(regarded as inde-pendent1-form variable)and,de?ning a covariant di?erential?D,

?D e±=de±?e±∧?ω.

V(?X⊥,X+X?)is an arbitrary function.By the e.o.m.forδX±the derivative of V de?nes world-sheet torsion

?T±≡?D e±=de±?e±?ω=e+e??V

and fromδ?X⊥Riemannian curvature

?V(?X⊥,?X+?X?)

R≡?d?ω=

;u±m)→(X±,X i;u±m)(85)

X±≡X m,X i≡X m

should be made.It corresponds to a transition to an analytical basis[32,25,28]of the target Minkowski space-time.The pull-backs of the target space vielbein forms E±are expressed in terms of these variables are expressed as

E+=D X+?X i f+i,E?=?D X?+X i f?i.(86) In(86)D is the world sheet covariant derivatives involving the pull back of the Cartan formω(64)as the spin connection(this is possible due to the identi?cation of the SO(1,1)subgroup of the target space structure group SO(1,D?1)with the world sheet Lorentz group,which corresponds to the property of the Lorentz harmonic formulation[20,37,48]):

D X+≡dX+?X+ω,D X?≡dX?+X?ω,(87) For?elds with SO(D?2)indices the pull-backs of(65)are used as”internal”SO(D?2) connections in D

D X i≡dX i+X j A ij.(88) Substituting expressions(86),(87)into Eq.(55)and integrating by parts(neglecting the surface terms for simplicity12)we obtain the action functional for the bosonic string in the form

S=?1/2 X+D e??X?D e++

+X i(f?i∧e+?f+i∧e?)?e+∧e? + B2.(89) Extracting the world sheet volume two-form e+∧e?in the last two terms we arrive at

S=?1/2 X+D e??X?D e++

+e?e+[1+X i((f?i?+f+i+)] + B2(90) which is similar to but not identical to the PSM-action(82).

The?rst(free)part of this action involves the target space coordinate?elds X±,X i(the latter with SO(1,D?1)Lorentz indices)and the pull-backs of one-forms

f±i=e+f±i

++e?f±i

?

=dξm f±i

m

(91)

ω=e+ω++e?ω?=dξmωm(92) only.In this approach the latter variables are still constrained to satisfy(23)(or equivalently (66)-(69)),i.e.we do not need any reference to the explicit expressions(19)(or(62)-(65))for them in terms of harmonic variables.Varying the action with respect to f of(91)andωthis must be taken into account.δ?a(d)can be determined from(23)contracted with the variation symbol,

iδ(d?a??a∧?c)=0,(93) i.e.

δ?a(d)=d?a(δ)+?a(δ)?b(d)??a(d)?c′(δ),(94) and thus

δf±i=D f±i(δ)?f±iω(δ)+f±j A ji(δ),

δω=dω(δ)?1/2f?i f+i(δ)+1/2f?i(δ)f+i,

dA ij=D A ij(δ)?f?[i f+j](δ)+f?[i(δ)f+j].(95) The independent variations are produced by the contractions of the corresponding components of the spin connection one-forms with the variation symbolδ

iδf±i=f±i(δ),iδω=ω(δ),iδA ij=A ij(δ),(96) which parameterise the Lorentz group algebra so(1,D?1).

To vary the second term of the action(involving the KR?eld which is assumed to be dependent on the coordinates X m

∧dX n H k m

∧E b(δ)H c a

≡E a u a

are now expressed in terms of the analytical basis coordinates by(86)and

E i=D X i?1/2X+f+i?1/2X i f?i.(98) The action(90)expressed in that basis contains X±which play the role of Lagrange multipliers for the condition of vanishing torsion for the induced connectionω(77)

T±≡D e±=de±?e±∧ω=0,(99)

17

just as in Einstein-Cartan action(82)[22,23,24]for torsionless gravity.

The?eld X i is a Lagrange multiplier for the condition de?ning the main curvatures of the world sheet.In the case of the free string(B2=0)they simply vanish

h i≡f?i e+?f+i e?=0?

h i≡f?i?+f+i+=0(100) and this means that the world sheet is embedded into the?at D-dimensional Minkowski space-time as a minimal surface.

In the corresponding case of a nonvanishing KR background one obtains in the analytical basis by(97),(78)or(79)

h i≡f?i++f+i?=2f+i+=

1

i u n+H

k m(ξ))

depends not only on the coordinates X±,X i,but also on the moving frame degrees of freedom (X m+1/2X?u+m)which can be regarded as related to1-form variables f±i,ω,A ij constrained to satisfy(23).

The e.o.m.-s appearing as a result of variations with respect to the Cartan forms and the vielbeine e±are nondynamical

E i≡DX i?1/2X±f?i=0(102)

E+≡DX+?X i f+i=e+(103)

E?≡DX??X i f?i=e?.(104) These’rheotropic’conditions are similar to the algebraic equations appearing in the PSM ap-proach[22,23].Finally also(66)–(68)must be taken into account.In addition the forms f±i satisfy(76)

f?i∧e++f+i∧e?=0

or

f?i?=f+i+≡1/2h i(105) which follows from selfconsistency conditions of the rheotropic relations(71).

The Riemann curvature is determined in terms of SO(1,D?1)/(SO(1,1)×SO(D?2)) connection forms f±i(i=⊥for D=3)by the Gauss equation(68)

R=dω=1/2e+∧e?(f?i

+f+i

?

?(h i)2)(106)

(Here Eq.(76)is taken into account,the Ricci tensor which is proportional to?R should not be confused with R in(106)).

In several respects the KR?eld assumes the role of the potential V in the Einstein-Cartan approach for d=2[22,23].Its dependence on moving frame variables(related to the connection forms)gives the possibility to describe not only matterless gravity,but any two-dimensional gravity in accordance with the general theorems about local isometric embedding[2,4]mentioned above.Still,this approach so far did not achieve an action with fully unconstrained variables and with the expected full PSM structure.

18

4.3Unconstrained Variables and’Picture Duality’

In4.2we identi?ed the analytical basis coordinates X±,X i with the Lagrange multipliers?X of the Einstein-Cartan approach[23,22].Now we show that the picture could be changed to a ’dual’one containing more straightforward counterparts of the?X?eld.In fact this’dual’picture appears when the Einstein-Cartan approach is incorporated at a more basic level.

We now introduce also the Maurer-Cartan Equations(66)–(68)with Lagrange multipliers into the free string action((89)with B2=0)

S= (?1/2X+D e?+1/2X?D e++1/2e+∧e?+

+X i(f?∧e+?f+∧e?)+

+Y+i D f?i+1/2Y?i D f+i+Y(dω?1/2f?∧f+)

+Y ij(dA ij+A ik∧A kj+f?[i∧f+j]).(107) The last two lines of(107)have the form of Einstein-Cartan-like model(82)without potential, but for the vielbein replaced by the pull-back of the SO(1,D?1)/[SO(1,1)×SO(D?2)]coset covariant forms f±i.The?rst and the third line are just the actions for two Einstein-Cartan gravity models,and the second line and the second line produces an interaction of these two models,gluing them together.

In fact,the entire action(107)represents a general PSM model(81)with one forms

A B= e±,f±i,ω,A ij .

For the simplest case D=3the number of the forms f±is the same as of vielbein forms e±, the gauge?elds A disappear,and the structure of the action functional

S= (?1/2X+D e?+1/2X?D e++1/2e+∧e?

X⊥(f?∧e+?f+∧e?)

+Y+D f?+1/2Y?D f++Y(dω?1/2f?∧f+)(108) becomes almost symmetric with respect to replacement e by f and coordinate?elds X by Lagrange multipliers Y.

Hence,supposing the forms f±to be linear independent,we could consider them in the cotangent space as an alternative basis to the e±.Passing from the e-basis to the f-basis we thus obtain another picture of the model,which in some sense is dual to the initial one.As a consequence of this we will be able to exclude vielbeins e and variables X±from the action.

4.4Dual picture for free string theory and Jackiw-Teitelboim model

The e.o.m.-s from variation with respect to the vielbein e±are purely algebraic

e?=D X??X i f?i,e+=D X+?X i f+i,(109) and,hence,can be substituted back into the action so that e±are eliminated.The same is true for the variations with respect to the coordinate?elds X±

f?i∧(D X i?1/2X?f+i)=0(110)

f+i∧(D X i?1/2X+f?i)=0(111)

19

Eqs.(110),(111)are solved by

X?=?(2D X i∧f?i)

R

(D+X i f?i

?

?D?X i f?i

+

)(112)

X+=?(2D X i∧f+i)

R

(D?X i f+i

+

?D+X i f+i

?

)(113)

where

R≡1/2(f+i

+f?i

?

?f+i

?

f?i

+

)

can be regarded as the Ricci curvature scalar pided by the square root of the metric,or as some counterpart of the determinant of the metric in the dual picture. E.g.in the simplest D=3case indeed

R≡1/2(f++f???f+?f?i+)=det f++f?+

f+?f??

follows.Substituting(109),(110),(111)into(107)we arrive at

S′= (Y ij(dA ij+A ik∧A kj+f?[i∧f+j])(114) +1/2?Y+i D f?i+1/2?Y?i D f+i+?Y dω+1/2?Y f+i∧f?i

+

1

2R (D?X j f+j

+

?D+X j f+j

?

)(D X i∧f?i) +

1

coupled action for a massive’pre-matter’scalar?eld interacting with’f-gravity’.Thus this action describes a Jackiw-Teitelboim model[58]coupled to matter which has provided a very useful laboratory for the study of the quantization for gravity.

It is important that the dependence of the action functional on the?eld X⊥is bilinear. So,in a quantum theory,the integration over this?eld still would provide a relatively simple e?ective action.

It should be stressed that in the derivation of the action(114)from(107)we have used only nondynamical equations.Hence the same can be done for the more general case of arbitrary KR background.The result will be the sum of the action(114)with the same interaction term as in(55).However,to simply integrate out the X⊥?eld for an arbitrary KR background will not be possible in general.

As a particular example,the complete solution of the e..o.m.-s for the action(116)in a conformal gauge is presented in Appendix A.

5Conclusion and Outlook

In this paper we have shown that D-dimensional p-brane theory interacting with a(p+1)-rank antisymmetric tensor(GKR)?eld represents a dynamical system providing a model for description of a general type of d=(p+1)dimensional gravity in the frame of the isometric embedding formalism[2,3,4],if the number of dimensions D of the target?at space time satis?es D≥(p+1)(p+2)/2.

This has been done using the moving frame(Lorentz harmonic)formulation[20,37,21]of the bosonic p-brane theories which produces the master equations of the so-called geometric approach[44,45,46,47],[21,59,52,38]as e.o.m.-s.d-dimensional”physical”matter appears in such models as a manifestation of the GKR background.

As a simple example we have studied a model for d=2gravity provided by D-dimensional string theory in more detail.We found that the model possesses a PSM structure[22,23,24,54] in the general case of arbitrary D.Also a deeper relation between a D=3string model with the PSM action for2-dimensional matterless gravity[22,23,24]appeared.The simplest model of a free bosonic string was shown to be equivalent to a Jackiw-Teitelboim model[58].

For d=4our model realizes the idea of Regge and Teitelboim for a’string-like’description of gravity[5]and provides a dynamical ground for description of General Relativity within the embedding approach[3,4].In our framework the Universe can be considered as a3-brane in D=10dimensional space-time with a rank-4antisymmetric tensor background.Matter in this Universe appears as a manifestation of D=10GKR?eld.

It seems to be more than a coincidence that the number of target space-time dimensions D=10is distinguished as a critical dimension of superstring theory too,inspiring speculations about a relation of the model considered here with string theory.

As it is well known[60],in the type IIB superstring spectrum a self-dual four-form gauge ?eld appears.Moreover,among the so-called string solitons in IIB superstring theory there is a3-brane[60].In accordance with the Mantonen-Olive conjecture[61],the dual theory,where solitons become fundamental objects,should exist.Such a dual theory is just the one of a (Dirichlet N=2super-)3-brane,being under active investigation now[62,63,64,65,66,67]. A4-form GKR gauge?eld can be coupled naturally to this3-brane.For nontrivial GKR background the embedding of the3-brane into?at10-dimensional Minkowski space–time should be nonminimal and should describe arbitrary curved4-dimensional Einstein space-time which may be suitable as a model for the Universe.Thus a(simpli?ed)model of the e?ective action for such a’solitonic’Universe seems to be covered by our approach.

It is interesting that the idea of the embedded Universe,explored here seems to attract renewed interest also from the?eld theoretical point of view.Recently a new study[68]devoted

21

本文来源：https://www.bwwdw.com/article/7gmq.html