Yang-Mills string theories

Nuclear Physics B125 (1977) 52-60 © North-Holland Publishing Company

YANG-MILLS STRING THEORIES *

Cosmas K. ZACHOS California Institute of Technology, Pasadena, California 91125 Received 12 April 1977

It is shown that the addition of a "colored" Yang-Mills supermultiplet to the locally supersymmetric spinning string Lagrangian does not yield a viable new string model, even though supersymmetry is preserved. However, a Yang-Mills extension of the Veneziano string that yields the color singlet sector of the Bardak$i-Hatpern model is constructed.

1. Introduction Within the past year, the three known string theories have been formulated in terms of two-dimensional field theories with gravity [ 1 ]. The "gravity" fields lack dynamical degrees of freedom and merely act as auxiliary fields, whose equations of motion reduce to Virasoro conditions for the string in a certain gauge. It has been conjectured [2] that this procedure might be applicable to the spinning string Lagrangian, further augmented by a supersymmetric Yang-Mills multiplet [3], to yield a modified string model with a "color" degree of freedom. It is shown here that the non-dynamical equations of motion of this extended theory cannot be interpreted as extra gauge conditions, since they do not close into the Virasoro algebra of the spinning string. Consequently there is no string theory corresponding to the field theory in question, even though the theory itself is locally supersymmetric and perfectly well defined. As a side remark, it is pointed out that if supersymmetry is forfeited, then the fermion sector of the Yang-Mills theory together with the boson sector of the string (i.e., the plain Veneziano string) can be joined to yield a string theory which reproduces the color-singlet sector of the Bardakci-Halpern dual model [4]. It also becomes clear why inclusion of the colored sector of the model prevents its formulation as a string model. In sect. 2 we exhibit how the extra constraints arise in the theory, as well as how commutator anomalies in the boson sector preclude their closure into the gauge al* Work supported in part.by the US Energy Research and Development Administration under Contract E-(11-1)-68. 52

C K. Zachos / Yang-Mills string theories

53

gebra of the simple spinning string theory. In sect. 3 we demonstrate the closure of these color gauges in the Yang-MiUs fermion sector. Finally, in sect. 4 we discuss their significance in restricting the construction of a dual model.

2. The extra gauge conditions The underlying idea of the theory to be considered is to combine two massless field theories which describe the spinning string and a Yang-Mills supermultiplet. Out of the new theory the gauge conditions are obtained, which are essential for decoupling ghosts in the spectrum of the conjectured string model. Closure of these constraints to give an infinite Lie algebra is a necessary condition for the construction of a string model, because a proliferation of gauge operators upon commutation would destroy the Fock space. These field theories are two-dimensional, as they are intrinsic to the world sheet swept out by the evolving string. In the case of the pure spinning string, the string variables ¢ and X are ten-dimensional vectors that describe the position and spin state of the string. Their contracted Lorentz indices are omitted for notational convenience. The Yang-Mills multiplet fields have the usual spins in the two-dimensional sense, but are nevertheless Lorentz scalars. Thus all Greek indices appearing in the following treatment refer to the space (o) and time (r) coordinates of the two-dimensional manifold. Latin indices are reserved for the internal symmetry, sometimes referred to as "color". The spinning string theory expressed as a two-dimensional gravity theory [1 ] is l--

£st = V(--~gUVOu~Ov~ + l ix~/ " OX + ~u'yVTUX(av~ + ~X~v)) .

(2.1)

The Lagrangian is locally supersymmetric. It contains the supergravity doublet: Vau and ~bu, that play the role of gauge fields. The "zweibein" field Vau is needed because Fermi fields are present. They anchor the gamma matrices present on a fiat Lorentz frame: 3"" O = "[aVala~#. The metric isguv = V ~ V a v , and V = det Vfl = X/~-g. The Fermi fields X and ~bu are Majorana. The following two-dimensional Yang-Mills Lagrangian [3] is globally supersymmetric: •YM

=

1

.~-a7 . oab~kb + lg#V(Dup)a(Dvp)a V ( - 41F ~av F #va + ~I l~, a--b

c

+ ~fa~cP X 3'5X }.

(2.2)

The fermion fields ~ka a r e Majorana, and all fields are in the adjoint representation, so that: Duab = Ourab --fabeA~, to accomodate supersymmetry. The field strength tensor is defined as Fgv = 3vA~ - 3vAa~ + fabcAuAb v'c By additionally coupling these fields to the supergravity doublet in a suitable fashion it is possible to make the supersymmetry local. The Lagrangian that achieves

54

C K.

Zachos / Yang-Mills string theories

this is IZ~f l p aUv-lg#va + {i~a~/. DabXb + ½gUV(Dup)a(Dvp)a '~YM = -t.--~,* 1

a

b

e

(2.3)

--

+ ]fabc P ~t 3'5~ + iFuuat~uTv Xa - ~t~T" (Dp)a~/u~/5)ta + ½~a2ta~ud/u} •

The invariance of this theory has not been proved in full detail, but it has been explored to a sufficient degree to assure its correctness. The theory to be considered as a possible string theory is the sum of eq. (2.1) and eq. (2.3), and it is locally supersymmetric. Regarded as a field theory it represents "colored" matter interacting with two-dimensional gravity. Note that the supergravity fields lack kinetic terms, and therefore dynamical degrees of freedom, because in two dimensions the Rarita-Schwinger Lagrangian for flu is zero, and the Einstein Lagrangian is a total divergence. The invariance of the action of eq. (2.1) plus eq. (2.3) can be exploited to justify choosing the gauge of the supergravity fields in the equations of motion [ 1 ] such that flu = 0 a n d g uv = A(x)r/uC A(x) cannot be set equal to 1, as conformal invariance [5] is broken by the kinetic term of the gauge vector fields, r/is the two-dimensional Minkowski metric: ~rr = -rloo = 1, rlro = riot = O. It ~11 now be shown that there is no string model obtainable from this theory, since the usual gauge conditions cannot be defined. The difficulty occurs in the boson sector and it will become evident that inclusion of fermions does not alleviate the trouble. Thus one may focus on (2.4) The equations of motion and boundary conditions obtained by varying ~, P, Au and guy are, respectively, t~¢~ = o ,

~o0(O = o, 7r) = o ,

(2.5)

D u D U P = O,

3aP(a = 0, rr) = 0 ,

(2.6)

-(DUFuv) a = fabcpb(DvP) e ,

FUV(a = O, rr) = O,

(2.7)

0 = 3uC3vc~ - irtuv~l

3,~¢3x¢ + DuPD~,P -,~rluvrl 1

-- A~°P FuoFvo + ~ A71~o~TKx~7ooFKoFKo •

D~PD~P

(2.8)

The last equation, the equation of motion of the auxiliary field guy, is interpreted after quantization as a constraint, i.e., it only holds acting on a physical state analogously to the Gupta-Bleuler condition: 3UAulphys) = 0). Its various components

C K. Zachos / Yang-Mills string theories

55

are easiest to deal with in lightcone coordinates (rT+_ = ~7-+ = I, r~++= rT__ = 0): a ± $ a , $ + D ± P D ± P = O, a

(2.9)

a

AFg_Fg_ = 0,

(2.10)

whence F~-_ = 0 since the group metric is positive definite, i.e., the first expression is a sum of squares. Note that with the addition of gauge fields Ag, an extra constraint, eq. (2.10), is obtained from a previously redundant component of the gravity equation of motion. This constraint, eq. (2.10), combined with eq. (2.7) yields the color current constraint *

fa~d'e(D+_e) C = 0 .

(2.11)

The main role o f A u in the theory is generating this constraint. It only modifies the dynamics o f P i n the following trivial sense: if one chooses the gauge A o = 0, it follows that Da = ao and F = 0 =OoAr, so that A r ( r ) is space independent. For Po a solution of DPo = 0, T

P - exp [ f

(2.12)

dt fCACr(t)]Po ,

o

is a solution o f D r D r P = O2aP, so that all reference to A r may be eliminated. The scalar fields, subject to the above boundary conditions, may be expanded in normal modes, 4) = C -

iaor + ~ n~0

-an - COS 170 /7

pa = C a _ ioflor + ~

e inr

_aa_ cos no e inr ,

(2.13)

n#=O /7

where the n take integer values. The canonical commutation relations for the fields and their conjugates give [an, am] = nSm+n,O ,

[aan, abm] = nSab~m+n, 0 ,

lag, am] = o .

(2.14)

The constraints, eq. (2.9) then reduce to L m e im(r+-a) = 0 , rrl =--

(2.15)

~

* It c a n b e c h e c k e d t h a t t h e r e are n o o r d e r i n g a m b i g u i t i e s t h a t m i g h t c a u s e t r o u b l e u p o n q u a n t i z a t i o n : F # ~ ' l p h y s ) = 0 i m p l i e s i n d e e d DuFlaVl p h y s ) = 0.

56

C.K. Zaehos / Yang.Mills string theories

where 00

L m = ]1

~

(OtnOLm_n + OntOm t _n):aa

(2.16)

n=--oo

are the familiar Virasoro operators which obey the algebra [Ln, Lm] = (n - m ) L m + n + a c-number times ~ m , - n •

(2.17)

The constraints, eq. (2.11) can be re-expressed in the form

n,k¢O

(fabc n1 ¢xb~c ~ e+_2ino) = O. n t X k - n ] eik(r+°)(1 +

(2.18)

Unfortunately, in contrast to the previous constraints, this expression cannot be resolved into "color" gauges, because of the effects of the last factor. One might have arrived, in a variety of ways (e.g., by dropping the troublesome factor in eq. (2.18)) to the guess that T f f - I72fabc E

be n , -1 OLnOtk_ n4~O 17

(2.19)

are gauge operators. Nevertheless, even if they did arise in a natural way, which they do not, their algebra would still not close. A more systematic way to show that a string interpretation is not possible, is to work directly with the fields and their canonical conjugates: [I =- Or~ ,

11a -- ~rpa .

(2.20)

The a interval may be doubled to [ - n , 7r], subject to the boundary conditions of the fields, so that the gauges can be defined alternatively as the Fourier coefficients of the constraints (2.9) and (2.11), do e i n ° ( ( ~ j ) + I1)2 + (~oP a + IIa)2) , --Tr

/r

r a rabc f --

do eina(ebI] c + Pboaec).

(2.21)

lr

The L n gauges close into the algebra, eq. (2.17), but commutation of Ta's generates further non-trivial operators. If these are included into the algebra, their commutation generates further operators and so on. An infinity of gauge operators is unac-

CK. Zachos / Yang-Mills string theories

57

ceptable for a string model, as mentioned before. This proliferation will be shown next to be absent in the fermion sector, where the "color" current constraints Ta do not contain derivatives 3o like those of eq. (2.21). It is easy to check that these derivatives are responsible for generating further operators with more derivatives when commuted with one another.

3. The fermion sector The supersymmetric theory, eq. (2.1) and eq. (2.3), does not reproduce a string model, since the boson part of the Ta gauges leads to the explosion of extra operators described above. Clearly, then, the construction of a string model with YangMills gauge fields necessitates dropping the P fields, and, as a result, supersymmetry as well. The Lagrangian then reduces to the usual Veneziano string, together with a set of "colored" fermion fields, which no longer need to be Majorana or in the adjoint representation of the Yang-Mills group, since supersymmetry has been sacrificed: £F = X / ~ ( ~ a u O Ouo - 4" l-r:a#u-r.uua + ~ i ~ a T . (O)k)a}

(3.1)

The equations of motion and boundary conditions resulting from the variation of ~, X, At, and V~ respectively, are

De = 0,

3a¢(o = 0, ~r) = 0 ,

"rUD~,X O,

~(o= 0)= 70~(0),

=

faac~a?u~ c = 2i(DvFUV) a , 0 = au~,~

(3.2)

~(o = n) = - 7 o ~ ( n ) ,

(3.3)

Fu~(a= O, n)= 0 ,

(3.4)

1 ~ch 1.--- ~r~u,~? ~K~ax~ + ~tXTuDuk - ½iguuX? "Dk

1

~

- g°OFuoF~p + ~ g u ~ o r

~Ko

•

(3.5)

Eq. (3.5) may be symmetrized with respect to/2, u since the antisymmetric part can be shown to vanish using eq. (3.3). As in the previous section, eq. (3.5) is split into a±~3+_¢ + l ikT+_D+)~ = 0 ,

(3.6)

Fa_F~_ = 0,

(3.7)

whence F+a_ = 0 ,

58

CK. Zachos / Yang-Mills string theories

transforming eq. (3.4) into the extra constraint, (3.8)

fabc~,b('~O + "/1))k c = 0 .

After elimination o f A a by use of the same arguments as before, the Dirac field X is expanded in normal modes, subject to the boundary conditions in eq. (3.3), -ina

(3.9)

einr( e ] ~a _ X//~ ~_ln ba \ e `ha ] ' __1

where n is half odd integer. This specific form of the spinor depends on our representation of the two-dimensional gamma matrices. The canonical anticommutation relations yield (barn, b ?nb}

= ~ab~rn,n

(3.10)

.

Then eq. (3.6~can be written as Lme

(3.11)

irn(r+°) = O ,

m

where Ln = ~

~

OlKO~n--k+

k integer

2

,,,u m-t''faha~,m+n,

(3.12)

rn h.o. integer

which satisfy eq. (2.17). Furthermore, eq. (3.8) reduces to

T a e im(r+-°) = 0 ,

(3.13)

m

with Tan = ~

b~fabk+n .

(3.14)

k

Here,Jgc is the representation of the group generators in the basis spanned by the fermions. Since the number of fermions is no longer fixed by supersymmetry, the representation is arbitrary. In this case we obtain a consistent string theory because the operators exist. Also, the operators, eq. (3.14) are acceptable gauges because the algebra closes [ Tan, Lm ] = n Ta+m , [Tan, T ~ ] = fal~cT~+m c + c number .

(3.15)

CK. Zachos / Yang-Millsstring thegries

59

This algebra may be represented in the canonical formalism, upon the formal extension of the cr interval )t(-a) ~- -7o~(O) (consistent with the boundary condition), and then forming the Fourier coefficients of eqs. (3.6) and (3.8) in the extended interval. The interesting conclusion about the operators Tna in eq. (3.15) is that they can only be represented with anticommuting oscillators *.

4. The import of the algebra The extended algebra, eqs. (2.17) and (3.15) suggests the existence of a dual model whose physical states are annihilated by these operators. The emission vertex for such a model should have conformal spin one as usual, so that the operators Ln - Lo + 1 pass through a sequence of vertices and propagators unaltered and hence annihilate all physical states. Likewise, the vertex sought should commute with the additional operators Tna, so that they too annihilate all physical states formed by a succession of vertices and propagators acting on some physical state. This second requirement however, restricts the vertex to emission of "color" singlet states only. As an example, consider the vertex

Va(k, O) = wa(o) e ik'Q(°) ,

(4.1)

where Q is the usual function of the a's, and

wa(o) = H*(O)/'aH(O) = ~

e -iOm Tffn

,

(4.2)

m

has conformal spin one, so that

[Ln

-

Lo, wa(o)] =

[Ln

-

Lo, ~

Tm a ] = nWa(O).

(4.3)

?n

However, this vertex is clearly unacceptable, since it gives zero when evaluated between physical states. The only allowed vertices commuting with the Tna's must be singlets, such as

W(O) =- ~

~'t~'ta~uz+,n=- ~

/,m

Tm •

(4.4)

rn

(Note that T,n is not a gauge operator.) Thus, by [Tfl, Tin] = 0, ITa, W(0)] = 0 ,

(4.5)

* Note that the corresponding Tnt operators of the SU(2) "string" cannot be represented by commuting oscillators [6].

60

C.K. Zachos / Yang-Mills string theories

and eq. (4.4) is a sensible vertex, as eq. (4.3) still applies. The more general point to be made is that a two-dimensional field theory with color gauge fields produces " c o l o r " gauges through its Maxwell equations. Thus on. ly color singlet states can be physical in the ensuing string model. It is a pleasure to thank Dr. J. Schwarz for his assistance, and Drs. L. Brink and J. Schonfeld for discussions.

References [1] L. Brink, P. Di Vecchia and P. Howe, Phys. Letters 65B (1976) 471; S. Deser and B. Zumino, Phys. Letters 65B (1976) 369; L. Brink and J. Schwarz, Nucl. Phys. B121 (1977) 285. [2] L. Brink, J. Schwarz and J. Scherk, Nucl. Phys. B121 (1977) 77. [3] S. Ferrara, Nuovo Cimento Letters 13 (1975) 629. [4] K. Bardakci and M. Halpern, Phys. Rev. D3 (1971) 2493; J. Schwarz, Phys. Reports 8 (1973) 269. [5] D. Freedman, Stony Brook preprint SUSB-ITP-76-39. [6] M. Ademollo et al., Nucl. Phys. B i l l (1976) 77.