IL NUOVO CIMENTO VOL. 106 A, N. 3 Marzo 1993

Manifestly Covariant, Gauge-Invariant Interacting Closed-String Field Theory (*).

B. F. L. WARD Department of Physics and Astronomy, The University of Tennessee Knoxville, TN 37996-1200

(ricevuto il 27 Maggio 1992; approvato il 17 Settembre 1992)

Summary. - - We argue that Witten's covariant open-string field theory already implies its covariant closed-string field theory analog. The respective analog closed- string field theory is explicitly exhibited.

PACS 11.17 - Theories of strings and other extended objects. PACS ll.10.Kh - Field theories in higher dimensions (e.g., Kaluza-Klein theories). PACS 11.90 - Other topics in general field and particle theory.

The advent of the promising superstring theories of Green and Schwarz [1] and of Gross et al. [2] has opened the possibility that all known elementary-particle interactions may indeed be manifestations of a single unified dynamical mechanism. Progress in realizing this unification will necessarily involve the step of solving the superstring theories to the extent required to derive their low-energy predictions from their 10-dimensional (or 4-dimensional) Planck scale string-theoretic for- mulations [3]. It is for this latter step that covariant string field theory, according to some, holds an amount of promise in particular, since one may hope to investigate nonperturbative phenomena which may be crucial to arriving at the correct vacuum about which to develop the low-energy limit of the respective superstring theory.

Accordingly, in a pioneering work, Witten[4] has introduced a manifestly covariant open-string field-theory formalism. The purpose of this paper is to show that this formalism already implies its closed-string analog. We begin with a brief review of some relevant background theoretical information.

More precisely, in the formulation of covariant open-string field theory given in

(*) Work supported by the Department of Energy, contract DE-AS05-76R03956.

2 1 - I1 Nuovo Cimento A 309

310 B.F.L. WARD

ref. [4], the key-ingredients are an algebra based on a �9 operation for multiplication of

string fields, an integration operation [ for the integration of string fields, and a

derivation Q of the �9 algebra with the nice property that Q is the BRST charge of the

respective open-string theory. The �9 and ~ operations are defined clearly for open

strings; naively, the restriction to open-string fields may traced to the use of a mid-point insertion in these operators-an open string has such a point, a closed one does not. We emphasize, however, that this argument is naive.

What we wish to point out is made manifest by the following. Consider the standard representation of an open string:

(1) Xopen - ~ X 0 + ~ n>0 ~ X~ cos(n~).

For such a string, Gross and Jevicki [5], following Witten, have presented a rigorous

realization of Witten's �9 and f operations in terms of their affect on the Fourier

components Xn in (1). Thus, one can take the work of Gross and Jevicki as a definition ~ J

of Witten's open-string field theory in terms of {Xn }. We do this. Now, consider the analog of (1) for a closed string:

(2) Xclosed = ~ X o + ~ n~>0 [Xn COS (2vto') + "~n sin (2nq)].

Is it possible to express (2) in terms of the modes in (1)? The mathematical result that {1 /V~,{V~/V~cosnz: n >0}} are the normalized solutions of a Sturm-Liouville problem guarantees that we can. We have

(3) Xel~ = -~= X~ + ~-~ ~" X~

if

(4) Xo = X'o

and, for n > O,

6'n, 2,~Xm + (1 - ~'n, 2.~)l(1 - ( - 1) 2"~+'~) 1 1 (5) Xn ~- 1 n + 2m n - 2m "~ "

The closed-string requirement Xclosed (0) = Xclosed (7~) follows from the result

1 | 1 (6) 2 k ~=-| k + m + l/2 =-0.

Thus, (3)-(5) are rigorously equivalent to the closed string in (2).

MANIFESTLY COVARIANT, GAUGE-INVARIANT INTERACTING CLOSED STRING ETC. 311

But, now, we invoke the powerful results of Gross and Jevicki to apply Witten's

string field theory methods to the system (3)-(5): the * and j operations are all

determined on the {X~}; the ,,mid-point, of (2) is identified as ~ = =/2 in the representation (3). Of course, the functional measure ~Xc~osed needs to be expressed in terms of 1-I ~ X n ; but this is easily done by the Jacobians

n

)) (7) [de t 3 X 2 n + l / a : ~ m = det 2n + 1 + 2m 2n + 1 - 2m "

These determinants, one of which is trivial, map 1-I ~ m 9~'m to 1-[ ~Xn. Their m n

inverses may be represented by standard methods. (We suppress the powers of the dimension of space-time in (7).) Hence, (3)-(7) make it clear that the formalism of Witten, as realized by Gross and Jevicki, applies to any type of string, closed or open. The detailed applications of covariant, gauge-invariant interacting closed-string field theory will be taken up, then, elsewhere(*).

Here, we will illustrate our results (3)-(7) by using them to compute the closed tachyon-tachyon-tensor vertex. Specifically, from (3)-(7) we have the oscillator relations, in the conventions of Gross and Jevicki,

(8)

op _ 1 ~ ( 2n + 1 _ 2n_+__l / . a2n+x- rdm=l. 2 n + l + 2 m 2 n + l - 2 m ]

[(1 l ) d c ( �9 _ _ ( ~ - ~ ) + - -

2-mm + 2 n + 1

op ~ -el + O(~ ~2n an

1 1

2m 2n + 1

n>~0,

where the superscripts are the notation op-= open and cl -- closed. From the formulae of Gross and Jevicki, we have the representation for the vertex in question

(9) F 1 o p , , x T r s op, s . . . . r s op~ _ i p~2 l+Uoo I ql>[~2)[q3)

k

(*) This result that (3)-(5) are equivalent to the closed string in (2) has also been independently pointed out by Siopsis (Phys. Rev. D, 39, 534 (1989)). Indeed, the latter author has also shown explicitly that Witteffs action gives a manifestly covariant closed-string field theory by constructing the respective QBRST and its representations for that theory as they are implied by (3)-(5). However, Siopsis was unable to show that the closed-string vertex operator which follows from Witten's theory via (3)-(5) is not the naive extension of Witten's theory to closed strings. Further, he was not able to calculate the complete expression for the tachyon-tachyon-tensor vertex and he was not able to compute the entire four-point amplitude of Virasoro and Shapiro (see the discussion below)�9 It is in these ways that our paper goes beyond the work of Siopsis.

312 B . F . L . WARD

where 70 is chosen so that the respective three-tachyon vertex is unity. The Nnr~ and U0o are all given in ref.[5]. If we take 1@3)= ~3~'~d'3-1, ac1:%'3 I P3; 0) in a standard notation, where P~ = 0 and ~ ' is the tensor polarization in the convention of ref. [5], then we find, after some Gross-Jevicki algebra,

(10) 1 n 1 (pL_p~)r 27 V-- ~" ~ (P1 _pR), 2 64rc 2

( )' 1 (ln(2_V3)+101n2) �9 4 + V ~ + ~

in the stated units in the notation of ref. [5] and the L-R conventions of ref. [6]. To our knowledge the result (10) is the first ever rigorous explicit prediction of a

closed-string vertex from manifestly covariant, gauge-invariant open-string field theory [7,8]. Further such predictions will appear elsewhere.

The correctness of the 3-point vertices implied by the Gross-Jevicki vertex operator, which acts on the first quantized Fock space, implies that amplitudes such as the Virasoro-Shapiro 4-point function are correctly reproduced by standard first quantized arguments. Indeed, although we have suppressed it in (9), the theory does give the correct vertex for 3 closed-string tachyons: as Witten has already emphasized, the effective tachyon emission vertex operator in the theory is the usual exp [iPX], if we ignore ghosts, which we may for this 3-point function. This then is the power of the Gross-Jevicki formalism: it gives the vertices from covariant gauge- invariant string field theory in terms of the familiar operators acting on first quantized string Fock space.

Recently, it has been emphasized by several authors [9] that, when one looks at the higher-point functions in a closed-string covariant gauge-invariant field-theoretic framework, one finds the necessity for higher-point vertices than the basic 3-point interaction term r ~. ~ in Witten's formulation of open strings would seem, naively, to imply. Thus, one may ask, are (9) and its consequences inadequate? The_answer lies in the basic relations (3)-(5) and the matrices exhibited in (7): 8X2n/SXm and

8X2,~+l/SX~. For, in Witten's theory, in either the open- or the closed-string representation the three-string vertex is

(11) O~a~<Tr/2

�9 *(x3 ( .) - x1 (= - *)) r (x1 (.)) ~2 (x2 (*)) ~3 (x3 (*))

(we suppress the ghost degrees of freedom for reason of pedagogy), where ~j (Xj) are the respective string fields. Here, we focus on the difference, for tachyon emission, between (11) in the closed-string representation (2) and (11) in the open-string

MANIFESTLY COVARIANT, GAUGE-INVARIANT INTERACTING CLOSED STRING ETC. 313

representation (3). In one case, (11) becomes

(12) S+illt~:t:t~:S~'~'~xj[YIm~(Yi'l'm-Yt'2'm§247 " j = 1 'odd ' '

m-Xl § Z m n'+modd 'n'

whereas, in the other, we have

(13) Ir m ) § Z ( X 1 , 2 n + I § 2n+l �9 n=O

( ) "~ X 2 , 2 m - Z 3 , 2 m § Z ( Z 2 , 2 n + l + Z3,2n+l)a2m, 2n+l �9 n=0

( ) "~ X 3 , 2 m - X1,2m § Z ( X 3 , 2 n + I § X1,2n+1)a2m, 2n+l ~I(XI((T))~2(X2((y))~3(X3(cT)), n=O

where we now define am, ~= (4/=)(V~)-~(1/( 2n + 2m) + 1/(2n - 2m)) and a2,~,2n+l = (2/=)(-1)m+~(V~)-~(1/(2n + 1 + 2m) + 1/(2n + 1 - 2m)). In (12) and (13) we see the fundamental difference between the naive extension of Witten's vertex to closed strings and the rigorous fully covariant extension via (3)-(8). For, in

(12), the coordinate difference ~'i, ~ - ~'i+1, m is not coupled to the coordinates Xi, n' +

+ X~+I, n' if m + n ' is even, whereas in (13) this difference, in view of (5), is coupled to --~ = (--) (--)__ X~.~,+Xi+~,,~, for all values of n'. Here ~ ' 4 , k - X l , k. This means that

configurations in which Xi, n' § X~§ ~, are nonzero generate nonzero values of X~, m -

-Xi+ l ,~ for all m in (13) but only for n ' + m = 2j, j integral, in (12). The missing support in the vertex (12) will mean that each n-point amplitude constructed from it will have a part of its configuration space absent [9]. This missing configuration space can be restored by adding appropriately chosen n-point vertices which would then generate the respective missing contributions in the naive extension of Witten's theory to closed strings. We emphasize that (13) already contains a full configuration space for the vertex so that our theory (3)-(9), (13) differs from the naive theory based on (12) by what amounts to an infinite number of higher-point vertices. Thus, we agree with the results in ref. [9] which show that a fully covariant, gauge-invariant interacting closed-string field theory which is based on the naive extension of Witten's vertex to closed strings must be nonpolynomial.

With such a discussion as we have presented, one may naturally exhibit the closed string S-matrix in terms of open-string theoretic constructs. For example, consider the famous result of Virasoro and Shapiro (see ref. [6] for a complete set of references) for the scattering of two closed tachyons, of four momenta kl and k2, to two closed

314 B . F . L . WARD

tachyons of four momenta -k3 and - k a, on shell. The tachyon emission vertex in Witten's theory is exp [ikX] as we have indicated and as one can see from (9). We should note that there is one implication which follows from Witten's work and which is also a consequence of the work of Gross and Jevicld. Namely, the three-string vertex, (01V] ~1 )]r ) ]~3 ), involves the implicit projection P onto the respective string Hflbert space, since P ] ~ i ) = ]~b~) implies that the vertex is also given by (0 ]PV(]~i))(P]C2))(PI~8)). Hence, from the operator methods, we see that PVP is actually the respective vertex operator. Here, for closed-string tachyon emission of 4-momentum k2 from 1~3) to form (~bl ] we have the vertex (~1 I:exp[ik2X]: 1~3) = = (~1 ]P :exp[ik2X]: PIr thus, P :exp[ik2X]: P is the actual vertex operator for closed-string tachyon emission. Hence, if we want to use Witten's theory to describe the Virasoro-Shapiro amplitude, Ba, we would write (here, (k2 ~ ga) denotes the usual term obtained from the first by permuting k2 and k3)

(14) B4=(O; -k l IP :exp[ik2X~ : P :L~ p - 1 :

P :exp [ik3X~ P I0; k4 ) + (kz "~ ka ),

using the standard operator vertex methods for the open-string theory, where the superscripts op again indicate that the operators are the open-string representation of the corresponding closed-string operators according to (2)-(5) and (8). To see that this result (14) is that of Virasoro and Shapiro, we note that X ~ = X cl, that :L~ p -

- 1 : = : L ~ 1 +y_~l--2: according to the usual closed string normal-ordering

prescription if we define L~ 1 and/,~1 to be the normal-ordered closed-string operator analogs of the classical Virasoro generators for the a cl and ~cl sectors respectively, and, finally, that the closed-string operator product (here, P = ~, ]i)dcl(il because {[i)d} is a complete set of closed-string states) i

(15) p :exp [ik2XCl]: p 1 ~ p : e x p [ i k 3 X d ] : p L~ 1 + L~ l - 2

is the same as

(16) p :exp[iMxr ~ [i)o~ c~(i I L~ ~ + 1 _ 2 P :exp[ik3Xd]: P"

This last form is important because all closed-string physical states respect the identity

(17) f(L~l _ ~1) - l i ) = I/)ol

for any f(z) such that f(O)= 0 and f ' (0)= 1, presuming that f is differentiable at

z = 0; for (L~ 1 - ~ 1 ) i i ) d = 0 for all closed-string physical states. Hence, (15) is the same as

A

(18) P:exp[ik~X~]: ~ ]i)c~d(i] f(L~l -["~) 1 P:exp[ikaX:,]:p.

MANIFESTLY COVARIANT, GAUGE-INVARIANT INTERACTING CLOSED STRING ETC. 315

The choice f(z) = (1/7:)sin (rzz) then gives finally for (15)

sin (T:(L~ 1 - ~1 )) 1 (19) P :exp [ik~XC~]: ~ li) d c~(i I

i ~(L~l _ ~_~1) L~ 1 -4- t ~ 1 -- 2 :exp [ikaX ~1 ] : P =

sin (~(L~ 1 - L~ 1 )) 1 = P :exp [ik2X d ] : :exp [ikaX cl ]: P .

7~(t~l _ t ~ l ) t~ l ~_ fj~l _ 2

On introducing this last result into (14), we recover the famous result of Virasoro and Shapiro

sin (=(L~ 1 - L~ l )) 1 (20) B4 = (0; - k l ]:exp[ik2Xr

7:(L~l _ ~ l ) L~ l "~ t ~ 1 -- 2

�9 :exp[iksXel]: 10; k4) + +(k2"-"ks).

This we have done completely from the operator methods for the open-string theory (*).

Note added. The question naturally arises as to the s trength of our closed-string vertex if the e lementary Witten 3-open string vertex has coupling go. From the representation (3), it follows that our 3-closed string vertex (01VI 1) c112} d 13} r is given by

8

~ j k=l

w h e r e l i) d - E ~,)r I i ' ) ~ I j} ~ . Thus, since ( 0 I V I il }op l i2)~ l ia)~ _ go it follows t h a t i ' , j

~op ~(2) ~; \op ~(3) {0IV[ 1} cl [2) cl [3) cl = 2 {0IV~ !I! [ii)~ I/i/ ~J2 [i2)~ J2/ i~/3 I ia)~ [Ja)~ - - go 2 �9 ~ e l J 1

{i~,j~}

since each term in the sum is - g~ (each term is the product of two elementary Witten 3-open string vertices, one for (0IV I i l )op li2 )op ]i~ )op and one for (O[Yljl)op ]J2 )op ]J3 )op ; for, I ik) ~ and Ilk, )op are created by commuting operators). Thus, our three closed- string vertex coupling constant dependence is consistent with the expectations of the lore. Here, ~q-) -'v are the coefficients in the representation of our typical closed-string state I/) cl via (3) in terms of the respective open-string states {1i) ~ } (**).

(*) Recently, Tye (Phys. Rev. Lett., 63, 1046 (1989)) has proposed a free closed string field theory based on the propagator A' = bob0 sin (=(L~ 1 + L~l))/(2r&~lL0d), where bo is the standard antighost operator. No successful vertex has been found. The calculation of B4 in our work suggests that Tye was on a good track, but without a vertex operator it is impossible to say how reliable this suggestion really is. He hopes to provide the vertex soon. (**) We thank Profs. J. Shapiro and J. Distler for useful discussions on this point.

316 s . F . L . WARD

The author is indebted to Profs. J. Ballam, S. D. Drell, S. B. Treiman and A. Salam for their continued encouragement in his efforts to apply the methods of the operator field to the problems of particle physics.

R E F E R E N C E S

[1] M. B. GREEN and J. H. SCHWARZ: Phys. Lett. B, 149, 117 (1984). [2] D. J. GROSS et al.: Phys. Rev Lett., 54, 502 (1985). [3] See, for example, H. KAWAI et al.: Nucl. Phys. B, 288, 1 (1987). [4] E. WITTEN: Nucl. Phys. B, 268, 253 (1986). [5] D. J. GROSS and A. JEViCm: Nucl. Phys. B, 283, 1 (1987); 287, 225 (1987). [6] M. B. GREEN et al.: Superstring Theory, Vols. 1 and 2 (Cambridge University Press,

Cambridge, 1987). [7] Other efforts to construct closed string physics via Witten's formalism are those by J. A.

SHAPIRO: Rutgers preprint RU-87-46, 1987; A. STROMINGER: Phys. Rev. Lett., 58, 629 (1987); and, P. MANSFIELD: Oxford preprint 84/88, 1988, and references therein.

[8] Recently, several authors (C. WENDT: SLAC-PUB4447, 1987; A. R. BOGOJEVIC et al.: Brown preprint, 1988) have questioned the completeness of Witten's superstring action and his attendant gauge trasformation law. The results in our analysis are unaffected by such questions.

[9] M. S ~ I and B. ZWEmACI4: Ann. Phys., 192, 213 (1989); T. KuGo et al.: Phys. Lett. B, 226, 48 (1989); M. KAKU: City Coll., N.Y., preprint, May, 1989; see also M. E. PESI(IN: unpublished.