Volume 180, number 1,2 PHYSICS LETTERS B 6 November 1986

CLOSED BOSONIC AND NEVEU-SCHWARZ FREE STRINGS AND THEIR GAUGE INVARIANCE

Alessandro BALLESTRERO a and Ezio MAINA b,a Istituto Nazionale di Fisica Nucleare, Sezione di Torino, 1-10125 Turin, Italy

t, Dipartimento di Fisica Teorica, Universitgt di Torino, 1-10125 Turin, Italy

Received 8 July 1986

A gauge-invariant action is presented for the free closed bosonic and Neveu-Schwarz string field theory which does not require any a priori constraint on the fields or the gauge parameters.

All open-string free field theories have been shown to be described by the action [ 1 ]

A-- ( z I Q I z ) , (1)

where Q is the appropriate BRST charge. The BRST charge for the closed bosonic and

Neveu-Schwarz string is

R R L L R L Q = c o L +coL +g2 +g2 +g~MR+(t6ML

= d + L + + d L - + g 2 + d + M + + d - M , (2)

with

=(cR a+-=(eR+ eb)/,,/5,

L+-=(LR-~-LL)/N/2 , M++-=(MR-~-ML)/N~ ,

~e'~= ~ R ~ ~¢'~L , (3)

and

[d-+,d-+]+ =1 , (4)

all other anticommutators vanishing. We define, in close analogy with ref. [ 2 ], the zero

modes vacuum [ - + ) by

d I - + ) = d - I - + ) - - 0 ,

( - + Id + 3 - I - + ) = i , (5)

all other zero mode vacuum expectation values being equal to zero.

The action for the bosonic string cannot be of the form (1), since the integration measure is even in

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ghost variables. In ref. [ 1 ] the string fields are assumed to satisfy

L [ z ) = O and the d , d - operators are simply dropped introducing a correspondingly reduced (~ charge, leading to the action

A = ( z I Q . I x ) (6)

with gauge invariance

0 I x ) = ( ~ I A ) • (7)

Neveu and West [ 2 ] propose the following action and gauge invariance:

A = i ( z I ~ I - Q I z ) , (8)

~ l Z ) = Q I A ) • (9)

This invariance, however, holds only if the gauge parameters satisfy L - I A) = 0.

It appears then, that the action (8) and the trans- formation (9) correspond to a partially gauge-fixed theory analogous to electromagnetism in the Lorentz gauge in which we can still perform a guage transfor- mation c~A~ = 0~e provided [] e = 0.

In ref. [3] de Alwis and Ohta generalize the Neveu-West action introducing string fields and gauge parameters with a larger number of compo- nents. Their gauge fixing, though, appears rather involved and the condition L - I A ) = 0 has to be imposed in order to avoid nonlocal terms in the residual gauge tranformations.

We propose the following action, introducing an

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Volume 180, number 1,2 PHYSICS LETTERS B 6 November 1986

additional field [ r/):

A= ½i(zl[tT-, Q] I z ) + ½ i ( z I Q L - Irl)

_ ½i(rllZ-Qlz ) (10)

with gauge transformations

~ I x ) = Q I A ) , ~lr / ) - - IA) , (11)

where IX) and It/) have ghost number 0 and - 1 respectively. It is evident that the action (10) is invariant under (l l ) for a general gauge-parameter field I A ) of ghost number - 1.

I Z) and [A) can be expanded in zero modes as

I z ) = ( ~ + g / d + + ~ d - + ~ d + d - ) I - + ) , (12)

IA)=(Aj+A2d + +A3d-+A4 d+ 3-) I - + ) • (13)

The action (10) can be easily gauge fixed to the Neveu-West one, using all gauge parameters I A) with L - ]~) # 0 to impose L - I t / ) =0, eliminating all the components of [ r/) which appear in the action.

The corresponding Faddeev-Popov ghost action is ( E [ L - [ E ) which is nondynamical and can be discarded.

The resulting action

A = ½ i ( z l [ d - , Q] Iz )

=OL+O-(bL-~+2Og2gt-gtL-~-~M+q/ (14)

is still invariant under

6 I x ) = Q I A ) , L - I A ) = O .

We

(15)

notice that [ d - , Q ] = 2 d - Q - L - and that

( z I L - I Z) vanishes for any I Z) with ghost number zero.

Written as in (14), it is clear that this action is completely equivalent to the one proposed in ref. [ 1 ] since ~ and g7 merely act as Lagrange multipliers to eliminate all components of q~ and ~t which do not satisfy L - ~ = L - q/= 0.

The resulting string field, I Z' ) , can then be writ- ten as

Iz ' )=(~+~ud +) I - + ) . (16)

Under (15) with gauge parameters (13) it trans- forms according to

~q)=I2AI+M+A2, ~tg=-L+AI+I2A2, (17)

which are exactly the gauge transformations given in ref. [1].

In conclusion, we have built an action which is invariant under general gauge transformations and can be straightforwardly gauge fixed to the actions proposed in refs. [ 1,2] which are essentially the same. We think that our result could be relevant for a better understanding of interacting string field theory.

We gratefully acknowledge several useful conver- sations with M.L. Frau.

References

[ 1 ] See eg. T. Banks, M.E. Peskin, C.R. Preitschopf, D. Friedan and E. Martinec, SLAC-PUB.-3853 (1985), and references therein.

[2] A. Neveu and P. West, CERN-TH.4358/86 (1986). [3] S.P. de Alwis and N. Ohta, Phys. Lett. B 174 (1986) 388.

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