Volume 176, number 3,4 PHYSICS LETTERS B 28 August 1986

T H E L I G H T - C O N E GAUGE M - i GENERATOR A N D INVARIANT STRING FIELD THEORY

T. JACOBSON, R.P. W O O D A R D

Department of Physics, University of California, Santa Barbara, CA 93106, USA

and

N.C. TSAMIS

Department of Physics, Stanford University, Stanford, CA 94305, USA

Received 5 May 1986

It is shown how the light-cone gauge M i generator can be used to infer the gauge symmetry of invariant string field theory.

The fact that light-cone gauge string field the- ory (1.c.g.s.f.t.) [1] is Poincar6 invariant, but not manifestly so, strongly suggests that it is the gauge fixed form of a gauge invariant theory. Moreover, the 1.e.g. generators of the Poincar~ group contain significant information about the nature of the gauge symmetry. These generators must agree with the action of the Poincar6 group induced from the invariant theory via gauge fixing (we assume that there is only one realizaton of the Poincar6 group on the gauge fixed theory) ,1. Since 1.e.g. is dis- rupted by a naive M -i transformation, the 1.e.g. M i generator will act on the field to induce a naive Lorentz transformation plus whatever field dependent gauge transformation is necessary to restore light-cone gauge. From this restoring gauge transformation one can infer much about the true symmetry of the invariant theory.

To illustrate this point imagine, in analogy with the situation actually prevailing in string theory,

,l In an earlier version of this paper we overlooked this argument and suggested that the I.c.g. M i generator pro- vides a check on proposed invariantizations of l.c.g.s.f.t. The above argument shows that a theory with manifest Poincar6 invariance will necessarily satisfy this check if it also gauge fixes to 1.c.g.s.f.t.

that we have been able to guess the Yang-Mil ls action in 1.e.g. [2]. From this we could compute to arbitrary loop order, but suppose we sought the invariant theory. We could first determine the M - ' generator by demanding the closure of the Poincar6 algebra, the other generators of which are trivial to pick off from the gauge fixed action. Then, acting this generator on the fields we would find

i[M -i, A,j] = ( x ' 3 - x - 3 i ) A , j - 6 / A , + [ ~ ]

+aj(1/0 )A.,

+gLb, A,,, (1) where

A°+ [.4] -= ( l /a_) ~ ' -&

-&f.e,.(1/O 2_)(el h . 3 A ), (2)

and A-. is the tranverse part of A.u = (A.+, A._, .4o).

The first term in (1) effects an M i transfor- mation on the coordinate dependence of A,,j. The presence of 8j in the second term allows us to identify it as the rotation of the vector component into the i direction. Of course there is no A . ( =

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Volume 176, number 3,4 PHYSICS LETTERS B 28 August 1986

- A , + ) componen t a m o n g the 1.c.g. variables A , , but the expression A,+[A] in (2) is in fact what one would obtain in the invariant theory by solv- ing for Ao+ using the gauge condit ion A , = 0 together with the A , _ equation of motion.

The remaining two terms are not associated with a naive Lorentz transformation, and there- fore can only be interpreted as the transverse piece of the gauge t ransformation that restores 1.c.g. The structure of these terms

8A<,j = 0 , [ (1 /0 ) A , i ] + gf, ho [ (a /o ) A m ] A , ,

(3)

contains essentially all information about the true gauge symmetry of the theory. We need only recognize the Aai-dependent gauge parameter 0,, = - ( 1 / O )A,~, and extend the index j to include the + and - components , to recover f rom (3) the general Yang-Mi l l s t ransformation

8 A,t* = - at*O, - gfu,,flh A (t*. (4)

The idea now is to carry out the above analysis in l.c.g.s.f.t, in order to discover the gauge symme- try of the invariant string theory. Unfor tunate ly the interacting 1.c.g. M -i generator is not pre- sently known, however there appears to be no serious obstacle to obtaining it even as we could have done in our Yang-Mi l l s example. Here we shall carry out the analysis in the free case, where both the M -* generator and the gauge symmetry are already known, the latter from the work of Banks and Peskin [3] and others [4]. The calcu- lation provides useful guidance for the interacting case because

(1) it reveals how the full restoring transforma- tion is related to its transverse part, and

(2) the gauge parameter of the restoring trans- formation is probably identical in the interacting case. This indeed occurs in the Yang-Mi l l s ex- ample above because the g-dependent term of 8A,, is proport ional to A , , and we start off with A,_ = 0 in 1.c.g.

We adopt the notat ional conventions of ref. [3]. The invariant, open bosonic string field • is writ- ten as

~ = f i inA m, ..... .t-~ ~.~t*, at*. ~(o) (5) t * l " ' ' t * n \ a 0 / U - - m l " " " - - r t l n "

n = 0

The symmetry of Banks and Peskin is

O 9

( / ) ' = ( b + E L ,O,, (6) n = 1

where the functionals O,,[X] are arbitrary trans- format ion parameters. Light-cone gauge is most easily attained by first imposing Virasoro gauge:

L,(b = 0 V n > 0 , (7)

and then using the residual symmetry with the equations of mot ion (in the critical dimension and with unit intercept) to impose ,2

+ a . (b = 0 V n > 0 . (8)

In terms of component fields (8) implies

A . . . . . . . . . . . " = 0 , (9) t * 2 • " t*,~

which, coupled with (7), determines the A +t*/ . ,,, in terms of the transverse fields.

The free light-cone gauge theory obtained by second quantizing the G G R T formalism [8] in- volves a field - let us call it 'it' - which depends only upon the transverse string coordinates A'(o) plus the zero-mode light-cone ones x0 -+. Its usual expansion is very similar to that for ~b:

L *n ~1 "-'ran vtt = I A i . . . . i,, ( X o ) O~-m " I I ~ 1 " -- m ,,

n=O

x e x p - ~'~ ~ , . ~ , . n = l

(10)

Note that the transverse component fields of and '/" are the same. This suggests the operat ion we call " tak ing the transverse part" :

~/"= t r ( ~ ) . (11)

Because '/" has fewer degrees of freedom than q),

,2 The fact that (7) could be imposed via a Banks-Peskin transformation was realized years before the connection with gauge fixing. The relevant projection operator was constructed by Brower and Thorn [5]. That there is suffi- cient residual symmetry to impose (8) follows from a rein- terpretation of the famous no-ghost theorem [6]. See also ref. [7] for discussions of the gauge fixing.

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Volume 176, number 3,4 PHYSICS LETTERS B 28 August 1986

the inverse mapp ing

+ £ i n - - - l t m ~ . . . m , r T l / '~ n A + t ~ 7.. ~,, l A l t X 0 )

n = l

X o/+ m la~2m 2 OL ~ ,,m q~<o) (12) ' ' " -- n

is obviously only defined when q~ obeys the 1.c.g. condi t ions (7), (8) ,3. For such q) we shall need the relation

( 1 / p + ) L ~ g " = t r ( ~ - q ) ) Vn > 0, (13)

which follows f rom (7) and (8). The free 1.c.g. genera tor ,//g-~ is obta ined by

second quantizing the first quant ized genera tor M -~ (for the form of M -~ see e.g. ref. [9]). Thus i J / - ~ acts on the field by commuta t ion in the same way M - i does on the first quantized wave functional, yielding

i [ J { ', ,/,1 = ( x ' 3 - - x - 3 ' ) ' / "

+ 7 , , = , 1 ~ l c C . L ~ . - 7 , ,= ,1 ~ 1Lt,,,c(~..

(14)

(We have d ropped the subscript " 0 " on the zero mode coordinates.)

Expression (14) is the sum of three distinct terms. We shall now demons t ra te how it t ranspires that the first two subject the zero- and h igher-mode coordinates to a naive M -~ boost , and that the third term represents the restoring Banks -Pesk in t rans format ion t runcated to l ight-cone gauge. Imagine the field ' / ' as the transverse par t of a field q) obeying the 1.c.g. condit ions (7), (8). As- suming q) to t ransform as a Lorentz scalar (q)'[A~,,X ~] = q)[X~]) a naive M i boost induces the following infinitesimal change:

anaive (/~ = ( x i O - - - X a ' ) ~ ~)

+ ~ 1 ,,,~2q~ - ~ 1 , , = 1 - n ~ ~,,%q). (15)

F/ - n = l

Taking the transverse par t of (15) and using (13) conf i rms that the first two terms of (14) are the transverse par t of a naive M -i boost.

The field dependent Banks -Pesk in t ransforma- tion 6q~ needed to restore light-cone gauge (8) must satisfy ,4

a+(¢ + 8~.i~@ + 8@) = 0 Vn>O. (16)

+q~ =.0 we F r o m this together with (15) and a ,

infer the relation

<,+8¢ = - a , , ~ Vn > 0. (17)

The solution to (17) is

; l=1

a.= ~ (_p+)-i , / = 0

x E . . . E r n l = l m / ~ l n + m 1 + . . . + m l

X a',, +.,, + ... +.,, (18b)

To see this note that

a,,+3q~= £ [a + , L _ , ] A q ~ (19a) n = l

e~ + n A,,,+,,

= - a~q~. (19c)

The last eciuality follows f rom the definit ion (18b) of Am+,, by interchanging the order of summat ion over n and l.

Now f rom (18) we see that the transverse par t of the restoring gauge t ransformat ion is given by

tr 6¢ = ~ - Lt_r,,o~i, tr cb. (20) n = l

This is just exactly the third term of (17), with g" = tr q), so we have conf i rmed that the Banks -Pesk in symmet ry (6) is consistent with the free l.c.g. M ' generator.

,3 In that case q'[XJ I ~,, =0v,, > o

one has '/'[ ~'](Xo ~ , xo ) = tr q)[X 1 = •4 R e s t o r i n g (8) t u r n s o u t a l so to p r e s e r v e (7), s ince L , , 8 ~ = 0

fo r the 6 4 o f (18) in t he cr i t ical d i m e n s i o n w i t h un i t intercept.

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Volume 176, number 3,4 PHYSICS LETTERS B 28 August 1986

The interacting theory possesses a symmetry (as yet probably unknown but see ref. [10]) which admits the gauge A " '"2 . . . . . . . . 0. The equations of

--/'2 ... ~t,,

motion (plus residual gauge conditions) then de- m l m ~ . . . m n - termine A+~27. ~, ' [A], although the solution will

differ from the free one. When this is substituted into the invariant action, the 1.c.g. theory of Kaku and Kikkawa should emerge. Our procedure for working backwards involves seven steps:

(1) Determine the M - ' generator. (2) Act it on the field and decompose into

components ,5 (3) Recognize the ( x ' O - - x - 3 ' ) part from the

interacting hamiltonian and x 0 dependence. (4) Recognize the constraint part from the

tensor structure

i[ M -i, Aj~'['.),, "'° ] = (x 0 coordinate rotation)

{a i m l m 2 . . . . 'n -~- . . . +,WArn , , m, . . . . ',,-~] - ~ v j A + / . . . . . i , vj , , + j . . . . . / , , , j

+ (restoring transformation). (21)

(5) Regard whatever is left as the restoring transformation.

(6) Identify the infinitesimal parameter in this and replace this functional of the fields by an arbitrary parameter to recover the general trans-

t:erse transformation. (7) Extend the ranges of all free and contracted

indices in this to obtain the covariant transforma- tion.

We strongly suspect that the gauge parameters of the interacting restoring transformation will equal those of the free theory. The reason is that an infinitesimal M -~ transformation must be a symmetry of the gauge fixed action which con- tains no terms of more than quartic order in the fields. Assume the full symmetry to be of the form 8A = 3 0 - g 1 9 A . The free 19 goes like A; if we allow even the lowest-order corrections ( - gA z) then it is clear that the M i variation of A will go like 8A - g2A3 + lower-order terms. (22)

,5 This last should not be thought of as an additional chore since the M ' generator will almost certainly have to be expressed in component fields to be well defined - even as is the case with the interacting hamihonian.

It is very difficult (though not impossible) to see how this can be a symmetry of an action which goes like

S - A 2 + gA 3 + g2A4. (23)

Consider two examples from field theory: Yang- Mills and gravity. As discussed, the infinitesimal parameter of the former does equal its free limit. For gravity the interacting parameter differs from the free one, but the gravitational action is non- polynomial in the usual graviton field.

Finally, it is clear from localizing the free the- ory [11] that the invariant interacting theory will ultimately be expressed in terms of a field which depends upon ghost coordinates in addition to the string configuration X ~ ( o ) . The form depending upon X"(o) alone will be obtained by partial gauge fixing and integrating out the auxiliary (component) fields. Our method of inferring the unknown symmetry will yield this (still invariant) form of the theory, to which the (presumably easier) process of localization can then be applied.

References

[1] M. Kaku and K. Kikkawa, Phys. Rev. D10 (1974) 1110, 1823.

[2] E. Tomboulis, Phys. Rev. D8 (1973) 2736. [3] T. Banks and M.E. Peskin, Nucl. Phys. B264 (1986) 513. [4] W. Siegel, Phys. Lett. B142 (1984) 276; B151 (1985) 391,

396; M. Kaku, Gauge field theory of covariant strings, CCNY preprint (March 1985); A. Neveu and P.C. West, Nuel. Phys. B268 (1986) 125.

[5] R.C. Brower and C.B. Thorn, Nucl. Phys. B31 (1971) 163. [6] R.C. Brower, Phys. Rev. D6 (1972) 1655;

P. Goddard and C.B. Thorn, Phys. Lett. B40 (1972) 235. [7] C.B. Thorn, Phys. Lett. B159 (1985) 107;

M.E. Peskin and C.B. Thorn, Equivalence of the light-cone formulation and th~ gauge-invariant formulation of string dynamics, SLAC preprint SLAC-PUB-3801 (October 1985).

[8] P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. B56 (1973) 109.

[9] J.H. Schwarz, Phys. Rep. 89 (1982) 223. [10] E. Witten, Noncommutative geometry and string field

theory, Princeton University preprint (October 1985); H. Hata, K. Itoh, T, Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B172 (1986) 186; A. Neveu and P. West, Phys. Lett. B168 (1986) 192.

[11] W. Siege] and B. Zwiebach, Gauge string fields, Berkeley preprint UCB-PTH-85130 (July 1985).

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