Volume 182, number 3,4 "PHYSICS LETTERS B 25 December 1986

R A M O N D - R A M O N D C L O S E D S T R I N G F I E L D T H E O R Y

Alessandro B A L L E S T R E R O a and Ezio M A I N A b,a

a Istituto Nazionale di Fisica Nucleare, Sezione di Torino, 1-10125 Turin, Italy b Dipartimento di Fisica Teorica, Universith di Torino, 1-10125 Turin, Italy

Received 9 September 1986

It is shown that a new consistent truncation for the Ramond field allows the construction of an action for the Ramond-Ramond sector of the closed string. The action is gauge invariant and can be gauge fixed to the light-cone de- grees of freedom. It can be extended to the Ramond-Neveu Schwarz sector.

The field theory of the Ramond-Neveu Schwarz and R a m o n d - R a m o n d sectors o f closed strings is still un- satisfactory, even in the non-interacting case. As for the open string, the difficulties derive from the existence of commuting zero modes. Therefore, at each mass level, there can be, in principle, an infinite number of F a d d e e v - Popov ghost fields. Two approaches have been proposed to tackle the problem. One of them [ 1,2] is based on consistent truncations of the Ramond string field. The other [3] introduces "picture changing" operations to deal with the whole set of zero mode excitations.

In this paper we propose an action for the R - R and R - N S sectors which makes use of truncated fields. Previous at tempts [4,5] at constructing the R a m o n d - R a m o n d field theory suffer from various and severe

limitations. In both these papers the field ~b is assumed to satisfy ( F R - FL)I@ = 0. In a general gauge this con- dition involves time derivatives contrary to the analogous one (K R - KL)I@ = 0 for bosonic strings. The model of ref. [4] does not possess any gauge invariance and completely ignores the commuting zero modes and half the anticommuting ones. This leads to the definition of a reduced BRST charge, to be used in the action, only vaguely related to the true charge. In ref. [5] the basic fields and the gauge parameters are truncated rather dras- tically, allowing for at most one antighost creation operator.

For the action we present the two equations, F R kb) = 0, F L kb) = 0, which are obtained as equations of mot ion (after gauge fixing) on the physical fields. The model is gauge invariant, although the gauge parameters must satis- fy some constraints as will be explained later in detail. The Ramond field ~b and the gauge parameter A have the most general expansion in ghos t -ant ighos t operators compatible with their ghost number.

As is well known, the most fruitful approac h to the gauge covariant string field theory is based on the BRST formalism. The R - R BRST charge, with the notat ions of ref. [ 1 ] is

Q = CLK L + CRK R - CLTL -- C-RTR +QL +OR + e L F L + e R F R - e-LSL - ~RSR -- ~-Le2L -- C-R e2 . (1)

In the critical dimension D = 10, Q2 = 0. This implies, for example,

~2 S 2 F 2 , L = K R , L , Q R , L = K R , L T R , L + F R , L S R , L , R , L = 0 , S R , L = ½ [ T R , L , F R , L ] . (2)

The commutat ion relations for the zero modes are

{CR,L' C'R,L) = [eR,L, eR,L] = 1, (3)

all others being equal to zero. We make use of two vacua 10), l~) which are annihilated by all Cn, -dn, e n, en for n > 0, with

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (Nor th -Hol land Physics Publishing Div is ion)

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Volume 182, number 3,4 PHYSICS LETTERS B 25 December 1986

10>t = <~1, i'~>t = <01.

The zero modes act on the vacua as follows:

~-R 10) = ?R 10) = ?R I~) = (01 e R = 0 , %10) = c L 10) = c L I~) = (01gL = 0 .

To be consistent with the commutat ion relations (3) this requires

(0[CRC-L[0) = i .

The ghost number operator G is

(4)

(5)

(6)

G ~ R _-=R - R R - + 1- R~R = (C_ne n - c nC n ) ~(c c - ~ - R e R ) - - ~ (e_nenR--R +-dR_neRn)_½(eRgR+gReR)+(R_+L) . (7) n>0 n>0

In the right sector we truncate the field as suggested in ref. [1]. Expanding in zero modes we have

q~RI0)- [q~ + (e R + CRFR)q~I 10). (8)

In the left sector we define a new truncation as foUows:

~bL[0)= (~b L + eL *L + C-L~b L + e-LUL~b L + e-2LSL~bL + e-2LCLSLq~L)[0). (9)

Therefore we construct the following R - R string field with G = 0:

~b[0) = [~b I + (e R + CRFR)~b ] + gLib2 + (e R + eRFR)e~LdP'2 + ~L~b3 + (e R + eRFR)e-Ldp' 3 + gL~-L~b4

- - - - t --=-2 t + (e R + CRFR)eLCLdP4 + e2LSL~b 5 + (e R + CRFR)eLSLdP5 + e-2c-LSL~ 6 + (e R + CRFR)g2gLSL¢~] 10). (10)

These truncations are consistent in the sense that acting on fields of this form with Q one gest states with the SaBle form.

We can then consider the following action:

S = i<0lgR~CR~-LQ~bl0)+ c.c. (1 1)

Performing the zero modes algebra it reads: t

S = ~-lFR~bl + ~I(QR + aL)~bl + ~I (QR + 0L)~bl 1--, -- g~b 1 { T R , F R ) ~ 1 + ~lgL~b; + ~]gL~b 3

+ IFL I + iFL 2 + c.c. (12)

A nice feature ofeq . (12) is the fact that ~b 2, ~b~, ¢3, ~b~ appear only as Lagrange multipliers, giving as equations of motion

FL~b I = FLq~ ] = 0 . (13)

Eliminating the multipliers, eq. (12) reduces to ~ t 1 t

S ' = ~lFR~bl + ~1(0 R + QL) 1 + ~ i ( 0 R + aL)~bl -- 2 ~1 {TR' FR}~bl ' (14)

which, after gauge fixing, will give the remaining equation of motion for the physical field ~1. It is interesting to notice the similarity of eq. (14) with the action proposed for the open Ramond string in refs.

[4,1 ]. C ovariant string field theories are invariant under transformations of the following kind:

8¢10) = QAI0). (15)

In some formulations it is necessary to impose conditions on the gauge parameters A in order to get the desired invariance.

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Volume 182, number 3,4 PHYSICS LETTERS B 25 December 1986

A possible way of removing these restrictions has been shown in ref. [6]. In the present case, choosing the field A to be

A]0) = [A 1 + (e R + CRFR)A ] + e-LA2 + (e R + CRFR)eLA' 2 + C-LA3 + (e R + CRFR)~LA' 3 + ~-L~-LA 4

+ (eR + CRFR)eLCLA4 + e2SLA5 + (e R + CRFR)-(2SLA'5 + e2LC-LSLA 6 + (e R + CRFR)e-2-(LSLA'6[ 10), (16)

the component fields transform as ! t t t

8~b I = K L A 3 - TRFRA 1 +FLA 2 + S R A 1 +(~A1, 6~b] =KLA 3 + F R A 1 +FLA 2 + d A ' 1 ,

6~b 2 =KLA 4 - TRFRA ~ + 2FLSLA 5 - SLA 1 +SRA ~ + 0A2, 6 ~ =KLA ~ +FRA 2 + 2FLSLA ~ - SLA ] + 0A~

8~ 3 = T R F R A ~ - T L A 1 - F L A 4 - 2 S L A 5 - S R A ~ - Q A 3, 6¢)~=-FRA 3 - T E A ] - F L A ~ - 2 S L A ~ - d A ~ ,

8¢,4 = TRFRA'4 - TLA2 + 2SLFLA6 + SLA3 - SRA -- 0A4, = - F R A 4 - TLA'2 + 2SLFLA'6 +SLA'3 - OA'4

8q~ 5 =KLA 6 +TRFRA ~ - A 2 - S R A ~ - Q A S , 6~b~=KLA ~ - F R A 5 - A ' 2-QA'5,

8q~ 6 =--TRFRA ~ - TLA 5 +A 4 + S R A ~ +dA6, 6~b~ =FRA 6 - TLA ~ + A ~ + d A ~ , (17)

with d = QR + QL" Plugging these transformations into the action (1 2), one gets the following variation:

8S=~IFRKLA3 -- ^2 ' + ~)1QLA1 + ~1FRFLA2 + ~3KL [KLA3 - TRFRA] + FLA2 + SRA] + (QR + 0L)A1]

+ ~i(QR + QL)KLA3 + ~ i (dR + 0L)FLA2 + ~ i Q 2 a l + q52FL [KLA3 - TRFRAI +FLA2 +SRAi + (dR +QL)

+ ~3KL [KLA~ + FRA1 + FLA~ + (QR + dL)A]] - ½~-] {TR, FR) (KLA; + FLA~)

t ^ A t + el(dR + dL)(KLA~ + FLA~) + ~2FL [KLA~ + FRA1 + FLA2 + (QR + QL) 1] + c.c. (18)

Therefore the action is invariant provided

FLAI0) =- 0 . (19)

In order to get a sensible string field theory, it must be possible to eliminate all longitudinal and ghost excitations of the fields. For the action (14) this can be done, fully exploiting the residual gauge invariance, using the methods of refs. [7,4]. The end result is

S" = ~llFR cY, (20)

where cT is purely transverse. it is interesting to notice that the expression (11) can also provide an action for the closed string with Ramond

right-movers and Neveu-Schwarz left-movers. It is sufficient to redefine the action of the zero modes on the vacuum according to

b-RI0) = C-R [~')= CLI0)= CL["(~)= e-R ]0)= (01e R = 0 , (21)

and to assume the following consistent truncation for ~:

~]0) = [~b 1 + (e R + CRFR)~ ] + CL~2 + (eR + CRFR)CLdP'2] ]0). (22)

One must, of course, use the appropriate BRST charge:

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Volume 182, number 3,4 PHYSICS LETTERS B 25 December 1986

Q=CLKL +CRKR -- CLTL -- CRTR +QL + QR +eRFR -- eRSR -- C-R e 2 . (23)

In components, then, the action becomes

SR-NS = 61FR{bl + 61(QR +QL) 1 + ~I(0R + {~L)¢I --2¢1 {TR' FR~bl + ~IKLq]2 + ~IKL¢2 + c.c. (24)

The action (11) is invariant only for a restricted class of gauge parameters and treats the right- and left-movers in a nonsymmetrical way. Forther work, aimed at removing these defects, is in progress.

In conclusion we want to stress that new consistent truncations and an appropriate choice for the vacuum, can provide powerful tools for constructing actions involving Ramond fields.

References

[ 1] Y. Kazama, A. Neveu, H. Nicolai and P. West, preprint CERN-TH. 4301/85 (1985). [2] Y. Kazama, A. Neveu, H. Nicolai and P. West, preprint CERN=TH. 4418/86 (1986). [3] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [4] T. Banks, M.E. Peskin, C.R. Preitschopf, D. Friedan and E. Martinec, Nucl. Phys. B 274 (1986) 71. [5] H. Aratyn and A.H. Zimerman, Intern. J. Mod. Phys. A 1 (1986) 421. [6] A. Ballestrero and E. Maina, Phys. Lett. B 180 (1986) 53. [7] M.E. Peskin and C.B. Thorn, Nucl. Phys. B 269 (1986) 509.

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