gauge theory of the string geodesic field

PHYSICAL REVIEW D VOLUME 47, NUMBER 6

Gauge theory of the string geodesic field

15 MARCH 1993

Antonio Aurilia*Department ofPhysics, California State Polytechnic University, Pomona, California 91768

Anais SmailagicInternational Center for Theoretical Physics, 11 strada Costiera, Miramare, 34100 Trieste, Italy

Euro Spallucci~Dipartimento di Fisica Teorica, Universita di Trieste, Istituto Nazionale di Fisica Nucleare (INFN1,

Sezione di Trieste, 34014 Trieste, Italy(Received 15 October 1992)

A relativistic string is usually represented by the Nambu-Goto action in terms of the extremal area ofa two-dimensional timelike submanifold of Minkowski space. Alternatively, a family of classical solu-tions of the string equation of motion can be globally described in terms of the associated geodesic field.In this paper we propose a new gauge theory for the geodesic field of closed and open strings. Our ap-proach solves the technical and conceptual problems affecting previous attempts to describe strings in

terms of local field variables. The connection between the geodesic field, the string current, and theKalb-Ramond gauge potential is discussed and clarified. A non-Abelian generalization and the generallycovariant form of the model are also discussed.

PACS number(s): 11.17.+y, 03.50.Kk, 11.10.Ef, 11.15.—q

I. INTRODUCTION, MOTIVATIONS,AND TECHNICAL BACKGROUND

The dynamics of a relativistic point-particle of mass mis encoded into an action which is essentially the properlength of the particle world line x"=X"(~),i.e.,

S=—I dw —X"X„= dzL X,X

particular, the tangent bivector in parameter space,

a aBT Bo'

is mapped by 0 into the tangent bivector

(1.2)

where X"(r) is the tangent four-vector. Other than itsgeometrical meaning, this action also exhibits invarianceunder time reparametrization r~~'(~). With such asymmetry, this elementary system represents the "proto-type" of any theory invariant under general coordinateredefinitions. The dynamical variables commonly used todescribe pointlike particles are either the Hamiltonianpair (X",P ), where P is the linear momentum conju-gate to the tangent vector X"according to P„=t3L /t)X",or the Lagrangian coordinates (Xt', X"). The two(dynamically equivalent) descriptions are related by theLegendre transform.

A string is the simplest generalization of a pointlikeparticle: it extends in one spatial dimension and spans,evolving in time, a two-dimensional world surface. If &is a domain in the space of the parameters P=(r, o ),which represent local coordinates on the Lorentzianstring manifold, and 0 is an embedding of & in the Min-kowski space M, then 0:gC&~Q(g)=X"(g)HM. In

Electronic address: AAURILLA@CSUPOMONA. EDU~Electronic address: ANAIS ~ ITSICTP.BITNET&Electronic address: SPALLUCCI N TRIESTE.INFN. IT

at each point of the embedded submanifold x"=X"(g)representing the string history 'N in Minkowski space-time. In the analogy with the point-particle case, areparametrization invariant action, which is proportionalto the world-sheet area, can be assigned to the string ac-cording to

SNo=, fdrdtrQ —(1/2)X" X„2&0!

= f drdtr LNo(X, X) . (1.3)

In the canonical approach based on the analogy withthe point-particle case, one defines two linear momentaP„and P„' canonically conjugated to X"=BX"/B~ andX'"=OX"/Bo., respectively, and then one develops thecorresponding Hamiltonian formalism. This is the usualstarting point to string normal modes decomposition andsubsequent quantization. However, this approach breaksreparametrization invariance on the world sheet from thevery beginning, so one of the main features of the modelis lost. In order to preserve this symmetry at all times,one has to treat ~ and o. on an equal footing, and this re-quires a noncanonical formulation of string dynamics,i.e., the introduction of new noncanonical variables. Inthis connection, the key remark is that in going from apointlike object to an extended system, quantities such asB,X" lose their physical meaning of "velocities"; rather,

47 2536 1993 The American Physical Society

47 GAUGE THEORY OF THE STRING GEODESIC FIELD 2537

~L Na X„,H„=Q( —1/2)X" X„

(1.4)

which involves both' P„and P„', and, according to Eq.(1.4), is proportional to the unit norm tangent element tothe string world sheet. Then, H„satisfies the generalizedmass-shell condition

(1.5)

which corresponds to the relativistic particle momentumconstraint p~"= —p . Similarly, in terms of the dynam-ical variables (X,II„„), the string equations of motiontake on the compact form

g~'b~g. II ()b~ =0,(II„,c)bX') ~a~=0 .

(1.6)

The embedding 0 maps the boundary of & into theworld lines of the string end points x"=X"(r,cT =cr, (r))=X", (v), x"=X"(r, o = cr ~(r ) )

=X2 (r).Thus, the first equation in (1.6) represents the conser-

vation of the area momentum along the string worldsheet, while the second provides a boundary condition atthe string end points.

they become projectors on the string world sheet. Tomaintain the analogy with the pointlike particle case andwith the geometric meaning of the velocity as tangent ele-ment to the world trajectory of the physical object, it ispreferable to define the string velocity as the tangentbivector (1.2). In this case, the conjugate dynamical vari-able is the area momentum [1,2]

the world sheet at any stage in the formulation of stringdynamics. This property is particularly desirable since itsuggests a novel approach to the quantum theory of ex-tended systems [2]; second, it lends itself to a straightfor-ward generalization to the case of submanifolds of higherdimensionality and, in the process, it enlightens the closecorrespondence between the theory of extended systemsand the Hamilton-Jacobi formulation of the mechanics ofpoint particles [3,4]. Against this background, our im-mediate objective is to show that the new choice of vari-ables (X",II ) enables one to cast the first equation ofmotion in (1.6) in the form of a Bianchi identity. Thisproperty, in turn, opens the way to the formulation ofstring dynamics as the gauge theory of an antisymmetrictensor field. Such a gauge formulation for strings is theprimary purpose of the paper. In physical terms, thepayo6' of this new gauge formulation is a mechanism ofmass generation for antisymmetric tensor fields (in thisspecific instance, the Kalb-Ramond field). The resultingequations describe massive spin-1 particles and representthe relativistic counterpart of the London equations ofsuperconductivity. Elsewhere we have speculated thatthis new mechanism of mass generation, when applied toan antisymmetric tensor field of rank 3, may play an im-portant role in cosmology in connection with the problemof production of dark matter in the early Universe [5].An equally interesting application of the gauge formula-tion of string dynamics is briefly discussed in Sec. III inconnection with the induced gravity program pioneeredby Zel'dovich and Sakharov [6] as a way to get aroundthe long-standing problem of quantizing general relativi-ty. There, we will argue that the Einstein and Kalb-Ramond terms are generated in the effective action forthe background fields as induced quantum terms describ-ing the low-energy behavior of the underlying quantumstring theory.

A. Motivation and objectives B. Technical background

To reiterate the main point of our introductory re-marks: the choice of dynamical variables (X",II„)offersseveral distinct advantages over the conventional choiceof canonical variables. First, it preserves covariance on

The area momentUm is simply related to the canonical mo-menta; in fact,

rr„.a,X =,6["] " 'B,X

a.x a,x.V —(1/2)X~ X

1 ~[ah] B,Xa p

Q —( 1/2)X Xp

B,Xg[ab] a )M

Q —(1/2)X X

where c),X„/Q —( 1/2)X X~:2n a'P, „, and P—o„=P„,

I')„—=I'„'.

Before we embark on a detailed discussion of the gaugeformulation of string dynamics, it may be helpful to ad-dress the main technical dif6culty that we need to over-come. Suppose that one is able to invert the relation be-tween x" and P=(r, o) so that c), in (1.6) is expressedthrough the chain rule as c), =(c)X /c)P)(c)/c)X ). Then,the first equation in (1.6) can be written as

(1.7)

However, from condition (1.5) we deduce thatII~ c} II„„=0. Then Eq. (1.7) can be fully antisym-metrized, yielding

X B(gH„ i=0 . (1.8)

Thus, whenever the matrix X is nondegenerate, Eq.(1.8) implies that II satisfies a Bianchi-type identity andtherefore can be written, at least locally, in terms of agauge potential B„(x). The above remarks are quite gen-eral and apply both to closed and open strings.

Unfortunately, this result hinges on two assumptionswhich are at least questionable.

2538 ANTONIO AURILIA, ANAIS SMAILAGIC, AND EURO SPALLUCCI 47

:—0, (1.9)

since any four-index totally antisymmetric tensor in twodimensions is identically zero. Therefore, Eq. (1.8) doesnot imply the existence of a gauge potential for the string,even locally.

With the above observations in mind, in the next sec-tions we shall discuss a Lagrangian field theory whichprovides a consistent description of string dynamics interms of local gauge fields. Presently, we shall briefly in-troduce the formal apparatus required to realize this pro-gram.

The mathematical object needed to give II„,(g) thestatus of a local variable, i.e., defined at any spacetimepoint rather than on the string world sheet alone, is theslope field or sheet field [3,2] 4"'(x ). The slope field is atotally antisymmetric tensor which assigns a tangentplane at any spacetime point. More precisely, because ofits geometrical meaning, the slope field is characterizedby the following properties: (a) when evaluated on thestring world sheet, the slope field coincides with thetangent element, i.e., @"(x =X)=X"; (b) it is orthogo-nal to its dual, i.e., 4" 4&„=0; (c) if it satisfies the Bian-chi identities, i.e., B~&+„ l=0, then it is called a geodesicfield [3].

Properties (b) and (c) imply that the two-form4= —,'N„(x)dx" hdx has rank two in four dimensions,rather than four, i.e., there exists a coordinate systemwhere N, can be written in terms of a pair of Clebsch [1]potentials S'(x ) and S (x ) as 4&(x ) =dS' h dS . Thisproperty is essential to solve problem (i) [8].

The role of the slope field in connection with string dy-namics has already been discussed by some authors, butmainly from a kinematical point of view as a useful de-vice to describe a family of minimal surfaces solving theclassical string equation of motion [1—3].

Property (c) led Nielsen and Olesen [9] to identify thestring field strength as the dual of a closed world sheet:

(i) In fact, in order to invert the relation x"=X"(r,o. )

one has to consider a two-parameter family of classicalsolutions x"=X"(~,o,$„$2) and assign to P„P2 the roleof additional coordinates in parameter space [7]. Then,the embedding functions establish a mapping betweenspacetime and parameter space which, however, may notbe one to one and is thus beset with integrability prob-lems. In any case, this kind of approach which describesstrings in terms of a pair of scalar fields, i.e., P„Pz is hardto interpret as a genuine gauge theory.

(ii) If the matrix X~ represents the tangent element tothe string world sheet, then it is degenerate. In fact,

detX~ =e e. . . .X»X"X«X)MVPO' P V P 0

dig('flgt hie g XPg X~pvpa a C e

XXPB,X ~„,.B,X~8„X B~XPB,X

F„=,'e—„ f d cr X~ 5 [x —X(cr)]D:aD=8

gapaPVP~ (1.10)

Here, the absence of a boundary guarantees that F„defined by (1.10) satisfies the Bianchi identity so that, atleast locally, a gauge potential can be defined:

~ ~~ a,F„.=2a„j~.=o=F„.=a„g.,

Therefore, in the Nielsen-Olesen formulation, F„ac-quires the meaning of a string geodesic field. However,the ansatz (1.10) has several drawbacks: first, it appearsthat only closed strings can be given a gauge-typedescription; second, it is not possible to prove that (1.10)is a solution of the field equations of A„; third, the degen-eracy of the matrix X" forbids a frame-independentderivation of the string equation of motion from the A„field equation.

Our main purpose is to derive a general gauge descrip-tion, valid for closed and open strings, which solves theabove technical and conceptual problems. The way out isto give up the ansatz (1.10) and to consider the slope andthe geodesic fields as basically distinct objects, which arerelated to one another by a set of classical field equationsderived from a suitable Lagrangian density. The slopefield accounts for the geometric properties (a) and (b); in-stead, the geodesic field represents the gauge partner ofthe slope field, and satisfies the Bianchi identity bydefinition. Then, the solutions of the field equations pro-vide the link between the two types of fields, and relategeometric features to gauge properties of the string.However, in order to implement this program, one has torevise, from the very beginning, the relation between theslope field and the string current; then, one has to discussthe connection between the slope field, the string current,and the Kalb-Ramond potential which mediates thegauge interaction between string elements.

Basically, the slope field is nothing but the generaliza-tion of the notion of Uelocity field in a continuous medi-um. Suppose that a spacetime region is completely filledwith a Quid of pointlike particles, each moving alongnonintersecting world lines. Rather than describing theAuid dynamics in terms of the motion of each microscop-ic constituent, one defines a regular vector field whichmatches at any point the four-velocity of the correspond-ing particle. This idea can be applied to the string theoryas well, in which case N" can be related to the stringcurrent

Ji' (x ) = f d $ 5 (x —Y(g') ) Y~ (1.12)

2Sometimes in the literature this property is referred to as thePliicker condition [3]. Indeed


J~ (x =X)=fd'g'fi'(X(g) —Y(g') ) Y"

=constX f d~p5 (g —g')1

Q —(1/2)X X~

=const'XII" (g), (1.13)

where we have explicitly regularized the distribution J" by assigning a physical width a to the string. Notice that thestring current satisfies the condition

&" (x )J„(x')= fd'g fd'g'&'(x —X(g))6'(x —Y(g') N(")fi(')~„,.a.xi'a, X a, Y~ad Y

(1.14)

again because there is no totally antisymmetric four-index tensor in two dimensions. The only relevantdifference between the string current and the slope field isthat J" (x) is a distribution difFerent from zero onlyalong the string world sheet, while 4&""(x) is a fielddefined over the whole spacetime manifold. The aboveresults are concisely expressed by the relationship,

J"'(x ) =4" (x)f d g 6 (x —X(g)), (1 15)

A. Closed strings

Let us consider the action

g f—d xQ —(1/2)W„W"

+ —fdxW" B(B),F„(x)=B(„B)(x),

(2.1)

which is our own definition of the slope field in terms ofthe string current. By comparing Eq. (1.12) with Eq.(1.15) one would be tempted to say that @i' (x ) is nothingby Y" with Y(g) replaced by x" [9]. This identification,however, must be made with caution. As a matter offact, the geodesic field resulting from a given family ofminimal surfaces which are solutions of the classicalstring equation of motion, should be constructed accord-ing to the following procedure: given a classical solutionX"(g), compute the corresponding area momentum, thenuse both the embedding equations x"=X"(f) and thestring equation of motion to write the area momentum asa function of x. Then, the resulting field is a smoothfunction of the coordinates and represents the "canoni-cal" extension of II„ in the sense that the two-formII(x ) —= —,'ll„+x~ 6 dx has rank two [3].

The rest of the paper is organized as follows. In Sec.II, we study a nonlinear Lagrangian for the geodesic fieldof both open and closed strings interacting with Kalb-Ramond and electromagnetic potentials. The on-shellequivalence of this model with ordinary string theory isshown. In Sec. III, we discuss the coupling of the stringgeodesic field to gravity, and its non-Abelian generaliza-tion.

II. THE STRING GAUGE FIELD STRENGTH

In this section we shall introduce a nonlinear Lagrang-ian for the string geodesic field which is partly suggestedby the physical interpretation of strings as extended soli-tons of an underlying local field theory, and partly by ear-lier investigation of string nonlinear electrodynamics byNambu [1,3] and Nielsen and Olesen [9]. The two casesof open and closed strings have to be discussed separate-ly.

where W„(x ) is a totally antisymmetric tensor and g is adimensional constant. Physical dimensions are assignedas follows: [ W„]= [E„]= [g ] =M . An action of thistype was proposed by Nambu as an effective apheliantheory interpolating between @CD and classical stringdynamics [3].

At this stage, the B field appearing in Eq. (2.1) is sim-ply a Lagrange multiplier enforcing a "transversality"condition for the 8'„ field; we shall see that on-shell the8„ field becomes the string gauge potential. In fact, byvarying the action (2.1) with respect to B„and W„, weget the following set of field equations:

+F„=O . (2.3)

The closed string appears as a special solution of (2.2):namely,

W" (x)=cf d $5 (x —X(g))X" =cJ" (x),JV

c =dimensionless const.(2.4)

The general solution of (2.2), i.e., Wi' =cJ~ +e" ~ 8 V,includes a "radiation part" described by a vector fieldVi (x ), which we set equal to zero everywhere in the fol-lowing discussion since our present purpose is to identifystringlike solutions of action (2.1).

The right-hand side of (2.4) is, except for a multiplica-tive constant, the current distribution associated with thetwo-dimensional manifold '}V representing the string his-tory. If the string is spatially closed, then NV=E andJi' has vanishing divergence Equation (2.3). is an alge-braic relation linking the slope field Wi' (x ) to the stringfield strength F„:

2540 ANTONIO AURILIA, ANAIS SMAILACiIC, AND EURO SPALLUCCI 47

„(x)F„(x)= —gQ —(1/2) k pk P

J„(x)

Q —(1/2)J pJ P

@„(x)

Q —(1/2)@ pN P(2.5)

from which it follows that the Hamilton-Jacobi (HJ)equation

'F —(x—)F" (x ) =g (2.6)

holds eueryiohere. Thus, F= ,'F„dx—"hdx' (which isclosed by definition) is a two form which satisfies (on-shell) the generalized HJ equation (2.6).

The net result of these manipulations is that while8'„(x ) is a singular field having support only along thestring history, F„ is defined over the whole spacetime

N„(x )II„(x) = cF„—(x ) = 2wa' Q ( 1/2)@ @~p(2.8)

Note that Eq. (2.7) implies that we identify the term g cwith the string tension. That this is indeed the case canbe verified directly by inserting solution (2.4) into action(2.1). This operation yields the classical effective actionfor X(g):

manifold. Furthermore, when evaluated on the stringworld tube, F„ is proportional to the area conjugatemomentum. In fact,

X„F„(x=X(g))=—g" = ——II

Q —(1/2)X pX"P

(2.7)

Conversely, Eq. (2.5) defines the area Jfeld II (x ) whichis the canonical (=rank-two) extension of the volumemomentum:

S' = cg —f d x ——4 p4 P f d $5 {x—X(g))1 /2

cg —fd"x f d $5 (x —X(g))Q —(1/2)X pX P

, f d gQ —(1/2)X pX p,27TCX

(2 9)

which represents the action for a free string with aneQectiue string tension I /Zmct'=cg .

Finally, as a consistency check, we wish to show thatthe gauge field representation of the string in terms ofF leads to the classical equations of motion (1.6). Tothis end, we recall that F„satisfies the Bianchi identitieseverywhere, so that, in view of Eq. (2.8),

it seems natural to ask what is the relationship, if any, be-tween the geodesic field associated to the string and itsgauge partner A„(x ). In other words, once it is accept-ed that the action (2.1) describes a theory of closed stings,the next step is to study how to introduce the interactionwith the Kalb-Ramond field.

Let us consider the model

B(iII„)(x) =0 (2.10)

at each spacetime point. Then we can project Eq. (2.10)along the string history, that is, we evaluate II„,(x) atx =X(g) and take the interior product with X ":

4"'= f d x —gQ —(1/2) W„W" + —W" 'd(„B )

1

grP g II IIV Pv 2X3t p.vp (2.12)

x'&a„ll,.,(g) =5('")a.x'a, x~a„ll„„,(g)

=5(")a,X~a.11„,(g) =0 . (2.11)

B. The gauge interaction

The last line in (2.11) is just the classical equation «motion (1.6) of the string.

where the coupling constant K has the dimension of mass,and H„p B[pA p] is the Kalb-Ramond field strength.The action (2.12) is invariant under the set of transforma-tions

6A = ——8 A1

IV v1 ~

Closed interstring interaction is known to be mediatedby the Kalb-Ramond gauge potential A„,(x) [10]. Thus,

5B„=A„+B„(P,

58'" =0,(2.13)

Note that second term in {2.1) does not contribute to (2.9)since B„J"=0.

which shows that under the generalized gauge transfor-mation of the Kalb-Ramond potential, the 8-field trans-forms as the corresponding Goldstone boson. From thisviewpoint, B„can be seen as the gauge part of A„or as


a Stueckelberg compensating field.Now the field equations become

turns out to be proportional to the Kalb-Ramond fieldstrength:

(2.14) a,„lI.„(x)= —«H„.,(x ) . (2.21)

g" +B(„8„)+K A „=0, (215)

Q —(1/2)W pW P

(2.16)

Finally, by projecting Eq. (2.21) on the string-worldsheet, we obtain

Again, the closed string appears as a special solution oftype (2.4) of Eq. (2.14), while the field strength of thestring, i.e., the geodesic field of the string, can be ab-sorbed into a redefinition of the Kalb-Ramond field A„which is gauge invariant under (2.13), and therefore canbe interpreted as the physical string field strength

H(—x )X~P~P (2.22)

which is the "Lorentz force" equation for the string.Furthermore, by substituting the solution (2.4) into Eq.(2.16), we obtain the Kalb-Ramond field equation coupledto the string current:

scA„:—vA„+B(„8 )= —g

Q —(1/2)W pW ~

—4p g

pv

(2.17)

(2.18)

H" ~=I7J ~ .P

C. Qpen string

(2.23)

X„A„,(x =X(g))=-

+—(1/2)X pX ~

(2.19)

The open case requires some further discussion. Infact, the current of an open string has a nonvanishingdivergence, and thus it does not satisfy Eq. (2.2). Moreprecisely

B„J"'=f dr g ( —I)'5 [x —X(r, o =o;(r))]If we insert the solutions of Eqs. (2.14)—(2.16) in action(2.12), and take into account that the field strengths H„of A„and H„of A„are the same, then we find theeffective action

dXd7 o cr

l

=—J (x), (2.24)

+—f d x J" A„—,f d x H H" ~,

(2.20)

which is just the usual Kalb-Ramond classical actioncoupled to a closed string, where 1/2m+' —=cg is theeffective string tension, and ~=crt is the effective cou-pling constant.

By inverting Eq. (2.19) one obtains for the string geo-desic field II„(x) = —KA„(x ). Then, the curl of II„(x)

where "x=X"( err =o.;(r)) represent the world lines ofthe two string end points o. =o.„o=o.2. Read from rightto left, Eq. (2.24) represents the old "trick" used by Diracto describe the electrodynamics of a pair of oppositepoint charges in terms of string variables [11]. Theboundary current J has vanishing divergence as a conse-quence of the identity B„BQ":—0. The above remarkssuggest the rationale to modify our gauge field formalismin order to be able to describe the open string as well.The action (2.12) has to be supplemented with termsdescribing the motion of the string end points and withan Abelian vector gauge potential to compensate for the"leakage" of symmetry through the boundary:

1/2

+ —W" (x)B(„8 )(x)+fB„(x)J"(x)1

+ —W" (x)A — H H" ~+eA J"——F — AK 1 1 f«2 " 2X3~ "~ " 4 " e e

po fdr g—Q —X)X,„,i =1,2

(2.25)


5 A p~ B[pA~]

6B„=B„O—~A„,

~A, =a,y+e

68'" =0,5X/'(r) =0 .

(2.26)

From the above gauge transformation it follows that5H" i'=0, but

e'0 [pA~] (2.27)

so that F does not represent a physical quantity. Gauge-invariant field strengths can be assembled as

where ~ is the Kalb-Ramond coupling constant, and po isthe mass of the particles located at the string boundaries.Note that the action depends now explicitly on B becauseof the coupling to the boundary current, whereas action(2.12) depends on B only through its field strength. Withhindsight one realizes that the action (2.25) is designed toensure that a special solution of type (2.4) still exists. Themodel is now invariant under the extended gauge trans-formation:

X;„P&p

— po Q —x, x,,(2.34)

W"'(x)=fJ" (x),J"'(x ) = J dr d o 5 (x —X(r, o ) )X"',

a„J"(x)=J (x) .

(2.35)

(b) Equation (2.32) describes the motion of the boundaryunder the combined action of the string geodesic fieldB~„B ] and the Lorentz force acting on the charged endpoints. (c) Equation (2.31) relates B„and A„ to theslope field:

N„(x )B(„B ]+vA„=—g

Q —(1/3!)4& ti@ ~

At this point several comments seem appropriate: (a)The coupled field equations above involve only gauge-invariant combinations of the various field variables.Equation (2.31) is the same as (2.15), and again relatesA„, and B„ to W„; as promised, Eq. (2.30) admits aspecial solution 8'" which is proportional to the currentof an open string having the two world lines x =Xi(r)and X~(r) as its only boundary:

1= ——11 (x)p~ (2.36)

5 F„— A„e

5(eF„+fa(„B,) ) =0,5(a(„B )+~A„,)=0 .

(2.28)

a H" ~ F —A~ =—fxJ—e e

(2.37)

where the effective string tension now is 1/2~a'= g f. —(d) Using the string solution (2.35), the above systein offield equations can be written in the form

The set of equations (2.28) displays the mixing of variousfields to form physical (=gauge-invariant) quantities;specifically, the first equation in (2.28) justifies, a pos-teriori, the nonstandard choice for the A„kinetic term in(2.25). Varying the action (2.25) with respect to A„„AB„,W„, and X"(r) we get the following set of field equa-tions:6 4'i'=0:

A

(2.29)

6ti 4'~ =0: a„W" (x ) =fJ (x ), (2.30)

6 S'~=0:8'

dP;„"=(—1)'(fa(„B,)+eF„)X,',r

n„W=0: aA ' p

(2.32)

(2.33)

W„(x )+a(„B )

+~A„=O, (2.31)+—(I/2)W t3W ~

5 4'~=0:x,~(~)

dP;„ =( —1)'(II„fvA„+eF„)x—;, i =1,2, (2.38)d7.

a F"~ A'~ =eJ~, —e

(2.39)

a(„II )= f~H„— (2.40)

a.a,H~'= a. F' f A ~ faaQ'— —e e

The interpretation of the above equations is as follows.First, Eq. (2.40), which is the covariant curl of Eq. (2.36),provides the actual link with string dynamics. Indeed, byprojecting Eq. (2.40) on the world sheet, one recovers theequation of motion (2.22) for the "body" of the string.Instead, Eq. (2.38) describes the dynamics of the stringend points acted upon by the "internal force" II„~,andby the "extended Lorentz force" (f~ A „eF„)X'. —Thus, Eqs. (2.38) and (2.40) are the equations that actual-ly govern the dynamics of the string.

The other two field equations tell us something new:first we note that Eq. (2.39) guarantees that H""~ is a reg-ular function. In fact, from Eq. (2.37),

where =f~J~ f~J~=0. — —(2.41)

GAUGE THEORY OF THE STRING GEODESIC FIELD 2543

Moreover, if one looks at (fz/e ) A '~

F ~+ (fK/e ) 2 '~ as a "gauge transformation" ofleaving the Kalb-Ramond field strength H" i'

unmodified, then Eq. (2.37) can be written as a Londontype equation

(2.42)

describing the propagation of a massive, spin-1 field cou-pled to its source J i'(x). So, the initially massless, spinless field 3„,because of its mixing with the vector gaugepotential 3„,acquires mass and spin: this is a peculiarmechanism through which tensor gauge potentials be-come massive [12]. Notice that the count of degrees offreedom is the same as in the conventional Higgs mecha-nism: a spin-1 gauge field with two degrees of freedomcombines with a massless spin-0 field thereby acquiringthe three degrees of freedom necessary to describe a mas-sive spin-1 particle. The difference here lies in the factthat the massless spin-0 field is itself a gauge field, i.e., therank-two antisymmetric Kalb-Ramond potentialOn physical grounds, Eqs. (2.37)—(2.40) evoke the famil-iar picture of Abrikosov vortices existing in type-2 super-conductors. What is at work here is a kind of relativistic

Meissner effect: we have a massive (superconducting)medium embedded into which are strings (vortices)compressed by the pressure of the surrounding mediuminto lines of electromagnetic Aux. From here twoscenarios come to mind. The first is the familiar one ofquark confinement (electric or magnetic) originally advo-cated by Nambu: as Aux tubes squeezed by the pressureof a mass-inducing medium, such strings constitute verysuitable traps for the hadron constituents (string endpoints). Alternatively, one may be tempted to associatesuch lines with "cosmic strings" as seeds of materialstructures in an otherwise featureless sea of dark matter.In either scenario, Eqs. (2.37) and (2.39) or Eq. (2.42) ap-ply to the surrounding background medium, whereasEqs. (2.38) and (2.40) govern the evolution of the string insuch a background. Clearly, if there is any element oftruth in the above scenarios, then two questions arise im-mediately: (i) What is the effect of gravity on the gaugeformulation of string dynamics? (ii) With an eye on thestandard model of particle physics, is it possible to attachinternal indices to the string geodesic field 8'"?

We will brieAy examine both questions in the next sec-tion. However, for completeness, we shall close this sec-tion with the observation that the equations of motion(2.37)—(2.40) can be derived from the effective action

S' = — d'gQ —(1/2)X X + f d x Ji' (x)A — H H"'~2&ex aP 2 ~ 2X3!

+fd x eA J"——F i'+1 .fr4 e

—po g fdr+ —X/X;„,

i =1,2(2.43)

which is obtained by inserting the solution (2.35) in theaction (2.12), and represents an open string with massivecharges at the end points. "Neutral" strings with mas-sive end points have been studied by several authors[13,14], mainly in connection with hadron dynamics [15].The novel feature of S' is the "residual gauge symme-try" [12]

A. Generally covariant formulation

For the sake of simplicity, we shall consider the in-teraction of closed strings with the gravitational field.Accordingly, we substitute in (2.1) the Minkowski metricwith g„(x ), replace ordinary derivatives with generallycovariant ones V„, and add the Einstein action. Thus, weare led to consider

n~„=a„y+e

(2.44)

surviving after the elimination of 8„ in favor of thestring variables.

4"=—g f d x&—g Q —(1/2)g„g W'" Wi'

+—fd x&—gW" V(8 )

III. GENERALLY COVARIANTAND NON-ABELIAN FORMULATION

F„.(x)=—a„a., (x)=—V,„a.,(3.1)

As we anticipated in the previous section, here we con-sider the possibility of extending the formalism describedso far in two directions: (i) coupling the system to gravity,and (ii) non-Abelian string geodesic fields.

4A similar picture can be constructed in terms of higher-dimensional extended objects, i.e., membranes or bags andhigher-rank antisymmetric tensor gauge fields [5].


Here, g= detg„. The corresponding set of field equa-tions represents the generally covariant generalization ofEqs. (2.2) and (2.3),

W" (x)=

c =dimensionless const,

f d g5 (x —X(g))X" =cJ""(x),g Jt

(3.6)

W„

Q —WPWp=F„

V W~=O . 8&—gW"=0,1

P Q gP (3.2)

(3.3)

which is nothing but the general covariant form of thestring current. To demonstrate the equivalence withgravity coupled to a closed string we will have to showthat (3.4) is the Einstein field equation with a stringsource on the rhs. To do that, we notice that the La-grangian in (3.5) vanishes on shell, i.e., 1.(F; W) =0.Then

supplemented by the Einstein equations

R„——,'g„R =8mGT„ (3.4)

where the energy-momentum tensor on the right-handside (RHS) is given by

T =g cJ„J

Q —(1/2)J PJp

X„Xd3( " 5 (x —X(g)),Q —(1/2)X»Xp,

g2 8„8'T ——2

2 Q —WpW+ W„F +g„ l. .

(3.5)

Then, a closed string appears as a special solution of (3.2),namely,

(3.7)

and the equivalence with general relativity becomes man-ifest once we show that the energy-momentum tensor hasvanishing divergence. This is indeed the case if X"(g)represents a classical solution of the string equation ofmotion. In fact,

x~~"

BXX'—2 2g ~ f d2(5[ma]

v' —gV 5 (x —X(g))

Q —(1/2)XPrXp

—2 2 BXXd 2g 5[ma]

v' —g Q —(1/2)XP&X„

a.x.i-5 (x —X(g)) —V'

Q —(1/2)XPrXp~

—2 2

V„T" = f d g 5 V'„(x —X(g))g Q —(1 /2)X PrX~p

5 (x —X(g))

(3.8)

a.X.X--2C2V „r~ = g ' d'g5™]a

P i/ g Q —( 1 /2 )XP~Xp

X 54(x —X(g) )

fd'g[5[-]a n-a.x.]5'(x —X(g)) =0 .g

(3.9)

Notice how the consistency condition (3.9) immediatelygives the equation of motion of the string as an alterna-tive to projecting the Bianchi identity for F„„on the

The first term in the last line of Eq. (3.8) is a pure surfaceintegral, which is zero for a closed string. Furthermore,antisymmetrized covariant derivatives can be replacedwith ordinary partial derivatives, and this yields thedesired result

string world sheet. This property represents a distinctadvantage of the generally covariant formulation of ourmodel. Ultimately, of course, the consistency conditionfor the Einstein equations coupled to matter, i.e., thatboth the Ricci tensor and the energy-momentum tensormust have vanishing covariant divergence, can be tracedback to the Bianchi identity for the Riemann tensor.

At this point one might elect to investigate the natureof the solutions of the classical system (3.1). However, inview of the equivalence that we have just established, adiscussion of some such solutions already exists in theliterature, especially in connection with cosmic strings[16]. Presently, what interests us is a deeper conceptualquestion: With an eye on quantum gravity and on its at-tending difficulties, is it really necessary to include fromthe very beginning the Einstein term in the action (3.1)?

We are partly led to this question by the result com-municated in a previous article [17]where we have shownthat general relativity may arise as the low-energy limit

GAUGE THEORY OF THE STRING GEODESIC FIELD 2545

+—f d x [v' —g W~ V(„8 )+ W~ K„], (3.10)

where g„(x) and IC„(x) represent, respectively, a sym-metric and an antisymmetric arbitrary external source;i.e., they are background fields implementing invarianceunder general coordinate transformations and extendedgauge invariance:

of a quantum theory of relativistic membranes. Indeed,the current attitude towards ultra-short-distance physicsis to replace local fields with extended objects, mainlystrings, as fundamental constituents of matter and totreat particle physics below some (string) energy scale asa local limit of the fundamental theory. In the followingwe shall argue briefly that our gauge formulation ofstring dynamics is perfectly consistent with this point ofview.

Suppose we start from the action

g—f d x& g—Q —(1/2)g g W"'W~

terms of the vibrating modes of a relativistic string [20].Further elaboration of this idea led to models of themeson as a pair of colored quarks, or monopoles, joinedby a thin fiux tube [21,22].

Therefore, if we believe that the geodesic field ap-proach discussed above is general enough to provide aconsistent description of elementary strings as well asgauge strings, then it should be possible to embody non-Abelian symmetries into the proposed approach. It turnsout, however, that there are severe restrictions on thefeasibility of this program. The rest of this section is de-voted to discuss this problem.

Again, we shall follow a line of reasoning similar to theone proposed in the previous section; i.e., we start fromthe non-Abelian current associated with a pair of"colored" pointlike objects,

dX,J'"=f dr g (—1)'5 (x —X;(r))p'[X;(r)]

(3.14)Bx)'

g„' (x')= g (x),ax~ 'a ~a"W'" (x')= W~ (x),ax~ ~x.

ax~B' (x') = 8 (x ),Bx'" (3.11)

which acts as the source of an SU(3) gauge fieldgoverned by the Yang-Mills action. p'[X] is the athcomponent of the Yang-Mills charge carried by the"quark" located at x"=Xi'(r).

By implementing again the "Dirac trick, "we write J'"as the gauge-covariant divergence of a singular Yang-Mills field

M„(x ) =A„(x )+B„P(x),nI~„.(x ) =a,„A.,

At this stage there is no relationship between g„,(x ) andthe physical spacetime metric, nor between X„(x) andthe Kalb-Ramond tensor potential. However, onceW""(x ) is eliminated from (3.10) by means of the formalsolution (2.4), we obtain the action for a string nonlinearcr model [18]:

G'" (x)= fdrdo p'[X]6 (x —X(g))X" (3.15)

x"=X"(r,cr =o;(r))=XI"(r), i =1,2 . (3.16)

In order to implement the Dirac relationship betweenG'" and J'" we require that p'[X] be covariantly con-stant [23], i.e.,

where x"=X~(r,o. ) describe a world sheet having thetwo world lines x"=X/'(r) as its only boundary, i.e.,

g '= —g c fd2$+ —(1/2)g„(X)g (X)X"X Dabpb[X] 0 DabGbPv Javp p (3.17)

+ —f d g K„,(X)X" (3.12)

Then, an effective action for the background fields is in-duced at the two-loop quantum level [19], and can becomputed in a perturbative expansion in powers of the in-verse effective string tension:

I = gc f d "x&—g[ag"—R„dH"'~K„—+O[(1/g c) ]] . (3.13)

Thus, in such an approach, the Einstein and Kalb-Ramond terms are recovered as induced quantum terms,describing the low-energy behavior of the underlyingquantum string theory.

&,(p')'=& (p')'=0 . (3.18)

It is also worthwhile to remark that the regularized formof G'" evaluated on the string world sheet factorizes inthe product of the color distribution times the (aphelian)volume momentum

This constraint is reasonable since p"[X] is not a dynami-cal variable, but rather an external source which can besuitably chosen. It also follows from (3.17) that(p'):—p'p' is independent of the world-sheet coordinates

B. Non-Abelian geodesic field

Spatially extended objects were introduced into ha-dronic physics after the recognition that the energy spec-trum of the dual resonance model could be interpreted in

5What we have in mind is the QCD string, so we considerSU(3), as the underlying symmetry group of strong interactions,and let the color index a=1, . . . , 8. In order to distinguishinternal indices we shall write them always as upper indices.Repeated indices are traced over with a Euclidean metric.


6'~ (x =X)= f d'g'p'[I ]5'(X(g)—I'(g')) I'~ Accordingly, we establish the following relation betweenthe string field G'" (x ) and the slope field:

=const X, f d'g'p'[ Y]5'(g —g')1

6'" (x)=N" (x )f d g p'[X]5 (x —X(g)) . (3.20)x~

Q —(1/2)X X~

=const'Xp'[X]II" (g) . (3.19)From the above equation one derives a formal expressionfor G'" squared:

6'" (x)G„' (x)=4" (x)C„(x)f d gd g'p'[X]p'[X]5 (x —X(g))5 (x —X(g'))l 2=4" (x )@„(x)(p') f d g 5 (x —X(g) )

which we shall use later on.The above remarks suggest we can write an SU(3)-invariant action for the string geodesic field as

(3.21)

4'= f d x —g ——W' W'~PV

1 /2

+—W'~ (x )D („B,) (x )+fB„'(x)J'"(x )1

+e A 'J'~ ——F'Q'"1P 4 )M

po f—dr g g —X[X,„,i =1,2

(3.22)

where J'"(x) is given by (3.14) and D(„B )(x) is thegauge-covariant derivative of the B field. The colored ob-jects in (3.22) transform under color internal rotations as

5W'~ (x)=f'A W'" (x)

6„' (x)

~—( 1/2)6" 6 b~&

=——,' B'„(x )B'i' (x ) =g (3.29)

5Ba (x ) DabAbv

5p'[X] =fb, &'p'[X]

(3.23)

II„' (x ) =— fB '" (x ),— (3.30)

Equation (3.29) allows us to introduce a colored geodesicfield

5 S'=0 D' 6 "'(x)=fJ' (x) (3.24)

G„',(x )5 4'=0: g +D („'B') =0, (3.25)

Q —(1/2)6 6

where fb, are the SU(3) structure constants. The corre-sponding field equations are

which on the string world sheet reads

p'[X]X11;.(x =X)= P,V

}/(p') Q —(1/2)X„.X„„p'[X]X„,

2~~ V' —( I/2)X,.X&(3.31)

dP;„ = ( —1)'p [X;](fD („"B")+eF„' )X;, (3.26)

6 4'=0: D'"F " =eJ'A p (3.27)

From the first two equations we derive the following for-mal solutions in terms of string variables:

and represents the area momentum of a colored stringwith an effective tension:

g'f217o! Q(p~)2

(3.32)

From (3.29), (3.30), and the constraint (3.17), we deducethat the color singlet p'[X]B'" (x ) satisfies the Bianchiidentity

W'"'(x ) =f6'" (x )BI~ B„' )=0. (3.33)

and

—' W'"'(x =X)=const Xp'[X]II" (g'), (3.28)Once evaluated along the world sheet, Eq. (3.33) yieldsthe equations of motion for X(w, cr ):


x"~a,~'[x]a„',, (g) =o, =-g'v'(p')'n(i')a, x~a„" =o .Q —( 1/2)X„X""

Therefore, the system of Eqs. (3.24) —(3.27) can now be written in the familiar "string" form:

s»")a,x~a„lI„.(g) =o,dP, „"=(—1)'(II„+ep'[X,. ]F„' )X,", i =1,2,r

D abFbpv

b'av

P

(3.34)

(3.35)

(3.36)

(3.37)

The first equation of this system is nothing but (3.34) written by taking into account Eq. (3.32), and describes a minimalsurface traced in spacetime by the evolution of the body of a string. Equations (3.36) represent a generalization of theWong equation [24], and describe the motion of the string end points under the infiuence of both the Yang-Mills fieldand the "internal Lorentz force" represented by II„. Finally, the colored string boundary enters Eq. (3.37) as thesource of the Yang-Mills field itself. The whole system of Eqs. (3.35)—(3.37) can be derived through variation of theefFective actioh functional obtained by inserting (3.28) and (3.29) into (3.22):

4'~= f d x g f't/—(p') Q —(I/2)@„,(x)@"(x)f d g5 (x —X(g))

+ed„'J'"— F„'g'"—" po—f dr g Q —X)X,„

i =1,2

= —fdr, f do Q —( I/2)x„x"'+p, o g +—X)X,.„i =1,2

d xF„' '" +e dr —I 'p'X; X,"A& X;i =1,2

(3.38)

The functional (3.38) is a generalization of the "massiveends" string action suggested by Chodos and Thorne[13], including color and interacting with a Yang-Millsfield.

Notice how the color degree of freedom disappearsfrom the Nambu-Goto action, having been completelyreabsorbed into the definition of the string tension. Infact, non-Abelian gauge symmetry is incompatible withreparametrization invariance for any kind of spatially ex-tended object [25], so that a true realization of a coloredstring seems to be impossible. In this connection, observethat in Eqs. (3.17) and (3.29), the color charge p'[X] sim-ply multiplies the ordinary (=Abelian) string variables,and thus remains localized at the pointlike string boun-daries. Accordingly, the coupling of the string with theYang-Mills field occurs only along the world lines of thepointlike boundaries.

Now, it would be tempting to inquire if a Kalb-Ramond interaction can be added in a sensible way, or inother words, if the gauge transformations (2.27) admit anon-Abelian generalization. The main problem is to in-troduce a suitable kinetic term for the internal algebravalued Kalb-Ramond tensor potential 3„' (x). In fact,the naive extension H'" ~H„' breaks the vector gaugeinvariance. Possible solutions to this problem arecurrently under investigation along two main lines ofthought: one is to implement the equivalence of a certainclass of non-Abelian Kalb-Ramond theories and chiral,nonlinear t7 models [26]; the second consists of compen-sating the lack of vector gauge symmetry by means ofsuitable matter fields [27]. The final goal is to introduce anew mass generating mechanism arising from the mixingbetween tensor and vector gauge bosons.

[1]Y. Nambu, Phys. Lett. 928, 327 (1980).[2] Y. Hosotani, Phys. Rev. Lett. 47, 399 (1981).[3] H. A. Kastrup, Phys. Lett. 828, 237 (1979); Phys. Rep.

101, 1 (1983); H. A. Kastrup and M. A. Rinke, Phys. Lett.1058, 191 (1981); M. Rinke, Commun. Math. Phys. 73,265 (1980); Y. Nambu, Phys. Lett. 1028, 149 (1981).

[4] A. Aurilia and E. Spallucci (unpublished).

[5] A. Aurilia and E. Spallucci, "The dual Higgs mechanismand the origin of mass in the universe" (unpublished).

[6] Ya. B. Zel'dovich, Pis'ma Zh. Eksp. Teor. Fiz. 6, 883(1967) [JETP Lett. 6, 316 (1967)]; A. D. Sakharov, Dokl.Akad. Nauk SSSR 177, 70 (1968) [Sov. Phys. Dokl. 12,1040 (1968)].

[7] Y. Hosotani, Phys. Rev. Lett. 55, 1719 (1985).[8] See H. A. Kastrup and M. A. Rinke, Phys. Lett. 1058, 191

(1981).[9] H. B. Nielsen and P. Olesen, Nucl. Phys. 857, 367 (1973).

[10]M. Kalb and P. Ramond, Phys. Rev. D 9, 2273 (1974).


[11]P. A. M. Dirac, Phys. Rev. 74, 817 (1948).[12]A. Aurilia and Y. Takahashi, Prog. Theor. Phys. 66, 693

(1981).[13]A. Chodos and C. B.Thorn, Nucl. Phys. B72, 509 (1974).[14] W. A. Bardeen, I. Bars, A. J. Hanson, and R. D. Peccei,

Phys. Rev. D 13, 2364 (1976).[15]I. Bars and A. J. Hanson, Phys. Rev. D 13, 1744 (1976).[16]See, for instance, Ruth Gregory, Phys. Rev. D 39, 2108

(1989), and various works quoted therein.[17]A. Aurilia, A. Smailagic, and E. Spallucci, Class. Quan-

tum Grav. 9, 1883 (1992).[18]G. C. Callan, D. Friedan, E. J. Martinec, and M. J. Perry,

Nucl. Phys. 8262, 593 (1985).[19]E. S. Fradkin and A. A. Tseytlin, Nucl. Phys. B261, 1

(1985); K. Kikkawa and M. Yamasaki, Prog. Theor. Phys.Suppl. 85, 228 (1985).

[20] See, for example, S. Mandelstam, Phys. Rep. 13C, 260(1974);J. Scherk, Rev. Mod. Phys. 47, 123 (1975).

[21] Y. Nambu, Phys. Rev. D 10, 4262 (1974); Phys. Rep. 23C,250 (1956).

[22] T. Eguchi, Phys. Lett. 59B, 73 (1975).[23] S. Ryang and J. Ishida, Prog. Theor. Phys. 66, 685 (1981).[24] S. K. Wong, Nuovo Cimento 65A, 689 (1970).[25] C. Teitelboim, Phys. Lett. 167B, 63 (1986).[26] D. Z. Freedman and P. K. Townsend, Nucl. Phys. B177,

282 (1981).[27] L. Smolin, Phys. Lett. 137B, 379 (1984).

gauge theory of the string geodesic field

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