Fermionic string field theory

Volume 168B, number 1,2 PHYSICS LETTERS 27 l:ebruary 1986

FERMIONIC STRING FIELD THEORY

Andr6 LECLAIR

l.vman Lahorato O" of Physws, Ilart,ard Unit'er.vity. Camhrtdge. MA 02138. USA

Received 7 November 1985; revised manuscript received 6 December 1985

A spacetime Lorentz-invariant, gauge-fixed field theor) for the NSR fermionic string is formulated using BRST gauge fixing. It is shown that, in order for the fields not to violate the spin statistics theorem, the states must be G-parity even.

1. Introduction. Up to date, string theories have developed primarily at the level of first quantization [ l ]. Much of relativistic particle mechanics may be derived with the first quantized approach, i.e., where the dynamical variables are the position of the particle, but some issues such as spontaneous symmetry break- ing require a field theoretic formulation to be trans- parent. It is possible that future progress in string theory may require a field theory of strings [2,3], for example to strengthen, or modify, the study of vacuum configurations in ref. [4]. We already know of one example where the field theory for the string de- viates significantly from a naive point particle field theory - this is the case of the Yang-Mills transformation properties of the antisymmetric field Buv(x ) of the superstring that are crucial for the anomaly cancellations [5].

The approach that will be followed in second quantizing the NSR fermionic string will be to use BRST gauge f'txing, as was done for the bosonic string by Siegel [2]. The resulting action will be gauge Fixed, and the first quantized ghost "coordinates" will become the ghost fields. It will be seen that the G- parity truncation [6] of the theory is required by the spin-statistics theorem for the fields. In choosing to work with the NSR formulation of the superstring, we give up manifest global spacetime supersymmetry; however, the model has its advantages for second quantization. Firstly, the gauge algebra is relatively simple, as compared with the algebra for Siegel's pro- posed covariant action [7]. Secondly, the second quantization of an action with superspace variables

0370-2693/86/S 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

leads to string superfields, which are likely to suffer from the problems with superspace formalisms in ten dimensions.

The outline of the paper is as follows. Section 2 is a review of the BRST formalism. In sections 3 and 4 the BRST analysis of the fermionic string is performed. The free field theory is constructed in section 5, an the lowest mass sectors examined.

2. The BRSTformalism. For a general gauge theory, the BRST procedure [8] determines a BRST- invariant lagrangian that contains gauge fixing terms and ghost terms. If ¢(x) is a physical variable, then under BRST transformations it undergoes a gauge transformation with gauge parameter ea(x) replaced by Xca(x), where X is an anticommuting constant, and ca(x) is a ghost. Thus ghosts have the opposite statistics of the generators. If

6¢ = X6"¢, (1)

then the BRST transformations of the ghosts ca(x) are defined such that

66¢= 0 (nilpotency). (2)

The full lagrangian is given by

22= 220 + 22GF + "/2FP = "C0 + 6(ca Fa) , (3)

where the F a are gauge fixing functions and the ~a are antighosts.

In the quantum theory, BRST transformations are generated by BRST charge Q, that can be calculated via the Noether procedure. Since we will be con-

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Volume 168B, number 1,2 PHYSICS LETTERS 27 February 1986

structing a BRST invariant field theory, we will mainly be interested in Q. It has been shown [9] that i fG i are the gauge generators and

{Gi, G/] = f~i1.Gk , (4)

then

a = ciGi + ~ i ( - lfaigkffi/cici, (5)

where 9k is the momentum conjugate to ghost Ck, and n i = - 1 if G i is a fermionic generator, zero otherwise.

As an illustrative example that will be useful later, consider the particle version of the fermionic string

[101

S= f drlIkUku/e-i¢u,i,u-(i/e)XYcu¢ul , (6)

where ¢ ~ are anticommuting, e is a vielbein, and k the superpartner of e. The action is invariant under r-reparameterizations and supergauge transformations:

6x u = ia¢u , 8 ¢ u = (a/e)(.~ u - ~ i X ¢ u ) ,

8 e = i a k , 6~,= 2&. (7)

In the gauge e = 1, ;~ = 0, the constraints are p . ¢ = 0 and p2 = 0, and quantization gives ( ¢ u , CV) = rtuv. Introducing the anticommuting ghost c for r-reparameterizations, and the commuting ghost e for supergauge transformations

Q = cp 2 + ep" ¢ + ½eel , (8)

where {c, c-) = 1. The hamiltonian is independent of ¢ u , c, and ~,

thus the vacuum is degenerate. ¢ u can be represented by Dirac matrices 7u/x/~, and the vacuum is a Majorana spinor since CU is real. The degeneracy with respect to c, ~ is contained within the degeneracy with respect to ~u , since {c, CU} = ~ , q,u) = 0. c and ~" are represented by 7/V~, where (3, u, 7-} = 0.

To construct a field theory, we require the action

S =fdx d ¢ dc de ~ + D ~ , (9)

to the BRST invariant. The integrals over ¢ and c are interpreted as inner products over the spinor vacuum UaU3 = 5 °e3, U// = ~LUR + URUL . This is sensible, since integration over x is an inner product over the spacetime part o f the Hilbert space. ~ is to span the first

quantized Hilbert space, and is thus ¢(x) ® lu) with lu) the spinor, and ~(x) independent o f c , ~ , ¢ u . If (D, Q] = 0, which is automatic if D ={ O, Q] for some operator O, then S is BRS'I" invariant. Let O = -x /~b/3e , then

D = i¢ - x / 2 e ~ , (10)

S = f d x de( fi~ I ~ ( i ~ ' - x/2-e~)¢~ I u~)

= f d x de~i,a'¢, (11)

since ¢7~ = 0. The e integration is trivial since e is a constant by its equation o f motion and (e le) = 1. It is no surprise that the ghosts amounted to nothing since the Dirac lagrangian has no gauge invariances.

3. BRST analysis o f the fermionic stn'ng. We will consider only open strings. The action in superconfor- mal gauge is [12] (2w~' = 1)

S = ~fdodr(~ba~xuabX u + i,~uoaaa,I,u), (12)

where Cu is a world sheet Majorana spinor, and the O a are the two-dimensional Dirac matrices. Defining

P± =(?+-X' ) /x /~ (X '= 3 o X ) , (13)

where 9Du is the momentum conjugate to XU, the constraints are

l "2 T++ =G+ = ~ [P~ + ½ i ~ ( 3 0 + 31)¢21 = 0 ,

T_ _ = G_ = ½ [p2_ + ~_ i¢~(30 _ 31)kO~ l = O. (14)

jO = 3bXU ob oO¢u = 0 , (15)

where ¢ = ( .2)" Using the equations o f motion

(3 0 + ~ 3 1 ) ¢ = 0 (16)

03 = pop1), the following constraints can be written as

1 "2 G+ = i ( e z + i ¢ 2 . ,1,2 ) = 0 ,

G_ = ½(P2_ _ i ¢ 1. ¢~) = 0 , (17)

4 = J 1 =ib_" ¢ 1 = 0 ,

4 =-J2 =.b+- ¢ 2 = 0 .

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If

{c,(o), c/(o')] = ~ ( o , o') [c~(o) + Gk(o') ] ,

t h e n

(18)

Q = fdociGi + i(- 1) ni f dO~k(O) o o

x ( f ~(o,o')c/(o')do') d(o)do. (19)

Using the canonical commutations

l x v ( o ) , 9 ~ ( o ' ) l = i 6 ( o - o ' ) n v ~ , (20)

(,Iq(o), q,7(o')} = 6 i / ~ a ( o - o ' ) ,

we calculate

f+++ = - f - _ = i a 6 ( o - o ' ) l O o ,

f i-I =f~22 = 6(O -- O') ,

-[l_(oo') = o ' ) = -fl+2(o', o) = ,,)

= ½ i ( a / a o - ~ a/ao')6(q- 03. (21) The supergauge transformations

6XU = g ~ u , 6 ~ u = - iOaXUpae , (22)

are generated by ig J 0 = e , J 2 - e2J 1 where e = (~1) is a world sheet Majorana spinor. The ghosts multx- plying the generators J 1 and J2 are thereby identified, and making up a commuting two-dimensional spinor. Define the momentum to e l , e 2 and e ' l , e'2- Note that G + ( - o ) = G _ ( o ) , and due to boundary conditions at O = 0, rr, xIt 1 (O) = '~II2(--O), implying dl (o) = d 2 ( - o ) . Introducing ghosts c ± for G±, it is known that c + ( - o )

= c - ( o ) . It will be shown in the next section that e l (o ) = - e 2 ( - o ) as a consequence of the equations of motion for these ghosts and their boundary conditions. As a result we can extend the o integration to - n , and using eq. (19), write

I t

e = f do[c (G+ ic '6/6c - "~e') - - f f

+ e (J+ e6 /5c + ½ ~'c')] , (23)

where primes indicate a /ae , (5/6 c(o) ,c(o ' )} = 6 ( o - o ' ) , and Q has been normalized for later convenience.

As in the bosonic string, it is presumed that the nilpotency condition of BRST transformations, Q2 = 0, will restrict the spacetime dimension to 10 and the normal ordering constants to be their usual values, in each sector separately * 1

4. Mode expansions• The operators X, P, c, and xI, have the usual mode expansions in e-in o. The mode expansions consistent with boundary conditions for the commuting world sheet spinor ghost e must be derived from its action. The Faddeev-Popov ghost lagrangian comes from fixing the gauge for Xa, the gravitino superpartner to the vierbein eaa. Supercon- formal gauge is p~p~Xct = F0 = 0 .3 = 0, 1 gives the same gauge fixing function

F = k 0 -P ;~ I = 0 , (24)

"~GF + ~ F P = - 6 ( e T F ) • (25)

Thus

Z?FP = - - e T g v = e'T(00 - - P 0 1 ) e , (26)

since

g~,~ = - a ~ e , (27)

if the conformal factor is constant. (This is permissible since Q2 = 0 impfies no conformal anomalies.) The equations of motion are

(a 0 - a l ) e 1 = 0 , (a 0+ a l ) e 2 = 0 , (28)

(a 0 - a l ) ~ 1 = 0 , (a 0+ al)~" 2 = 0 . (29)

The boundary conditions

e'l 6el - e'26e216 = 0 (30)

require e I = +e 2 at o = 0, n. To summarize the mode expansions for the variables in Q:

I ~-x en e - in° - - en e - In° , e = , ~= 1 i ~ ~ "

xpu = 1 n~ dnVe_ino,

n ~ z / 2 , ( 3 l )

,1 Upon complet ion o f this work , two preprints were received that show Q2 = 0 implies d = 10 and v0 NS = 1/2, v0 R = 0. See ref. [11] .

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Volume 168B, number 1,2 PIIYS1CS LETTERS 27 February 1986

,6u= 1 ~ % U e _ i n o X/21r n

I e_ O l c - - ~ c . ' ~c v T / . '

n E Z , (31 cont 'd)

with (anti) commutations

. u /a v [an, a m ] = n6 m,_nrl "v , { d . , dm} = rl"V~m,__n ,

(Cm, "Cn} = ~rn,-n , [en, era] = ~m,-n , (32)

~ :-.,/'~ia/a~., d~--v"lx/2, ~o = O/OCo, gO = - o / 8 e 0 - (32)

The zero mode dependence of Q is what is needed to construct a BRST invariant field theory. The result of the calculation is

Q = c 0 [Iq - ( N - / 1 0 ) l - TalOc o + e0(6¢ - ~ )

+ ~ VOlOeo - % % a/OCo + O , (33)

with

N=: ~ ~a n 'a_ n + n c n c _ n - n e _ n e n +½nd_ndnCn: n ¢ O

T = ~ - 2 n C n C _ n + 2 e n e _ n , V=: ~ n c n e _ n : , n>0 n (34)

1 n '~ C n ". = : ~ % " d_n + 2enc_n + ~ -n n ~ O

The normal orderings are defined by

0 = c n 10) = b~n 10) = en 10) = en 10) for n < 0 , (35) 0 = %Ul0)=dn~t0), f o r n > 0 .

/a 0 is a constant that arises from normal ordering am- biguities, d~ has been replaced by q~u/x/~ in anticipa- tion of constructing the field theory.

5. Field theories. The BRST invariant free action is of the form

(36)

where (D, Q] = 0, and primes indicate no integration over zero modes. The functiona 1 integrals for the non-

zero modes will be interpreted as inner products over the Hilbert space.

Consider first the Neveu-Schwarz sector defined bydnUl0) = aural0) = 0 for m ~ ( Z > 0) ,n E( (Z + ½ ) > 0). If D = {0, Q], then 0 is the unique operator that gives the correct action for the tachyon. As in the bosonic string,

=I DNS [Co ~/~c O, Q] (37).

= ~ c0 [D - ( N - u0)l + i r o/at0 + ~ e0e0 o /a t0 .

In this sector/~0 = ~, and the only vacuum degeneracy is with respect to c o and c0. Thus~ = ~ + c0~b. Integrating over Co, and dropping the e 0 integration

SNS =fdx Tr{~ ¢[F1 - (N - ½)]¢ + ½ ~ ( T + 1 ) 4 } . (38)

The two lowest mass sectors are contained in

= ~b(x) + AU(x)dU_l/2 + e l /2r l (x ) + ~'l/2~(x), (39)

= i -~ l /2B(x) .

Performing the inner products, the action is

s =f d x ½ dp(x)(V1 + ½)dP(x) + X A u ( x ) F I A u ( x )

- ~ ( x ) D r l ( X ) + ~B2(x) . (40)

There are several things to notice. If cb is to have a timed Grassman-parity ,2 , that is, if q5 is to be either commuting, or anticommuting, then the fields ~(x) and AU(x) have opposite Grassman-parity, since AU(x) multiples an anticommuting operator dU_l/2 . If we require AU(x) to be a commuting vector field, then the tachyon be forced to be an anticommuting field. In fact, once AU(x) is fixed to be commuting, there are many integer spin fields in the theory that are forced to obey Fermi statistics. These are unacceptable and are truncated from the theory by requiring states to be G-parity even:

GNS = ( - - I ) E ~ ' = ' / 2 d - k ' d k - I , (41)

since ~ff=l/2d_k • d k counts the number of fermions.

• 2 Historically, string theories were first used to model strong interactions and G-parity was related to familiar phenomeno- logical G-parity. In the context of this work, it appropriately refers to Grassman-parity.

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It may be argued that there are two string fields o f opposite parity. Consider then the field with parity complementary to the above choice. Upon throwing out the wrong statistics fields, we are left with a tachyon, and a commuting massless scalar. This is not sensible since we know that the theory at the first quantized level does not contain a massless scalar. Also note that as in the case of the bosonic string, the action for AU(x) is gauge fLxed (otherwise .~ =

F~vFUV) and the fields r/(x) and ~(x) are the conse- quential Faddeev-Popov ghosts. The fields 77 and F? are required to be anticommuting since they multiply commuting ghosts el/2 and e'l/2 in eq. (39), and are the only massless ghost fields in the theory. The non- propagating BRST auxiliary field B(x) is determined to be commuting since g' has opposite statistics to ~, and B(x) multiples commuting ghost e+l/2"

In the Ramond sector the vacuum is degenerate with respect to d~, c 0, c" 0, and thus has the same structure as for the fermionic particle treated in section 2. There is again a unique BRST invariant operator D that gives the correct action for the spinor vacuum.

D R = [a/ae 0, O] = (i~ - N) - 2e0~, (42)

and the action is

--f x ® ® SR

X (filcb÷(i~ - ?~)¢tu) . (43)

Consider the lowest mass sectors:

v ~a L luR)) (44) qb=(¢c*+t~U_l~+d lqS~)( lua)+

where L - R denote left-right. If we require that and ~a has Ftxed Grassman-parity, then the fields ~u ~u

canot both be anticommuting fields. However, we can not remove~udU__ 1 from the theory otherwise qb+Nqb = 0, since N contains a - I " d l + d- la l . Note also that ifcb --- aU_l ~z~ulu~) + dUl ~ l u ~ ) t h e n cI,N~ = 0 since O/LlU L) = O. Thus the only way to impose Grassman-parity consistently with the inner product on the vacuum is to assign lu L) and lu R) opposite parity. Then

a, = + lug>. + lug> (45)

has fLxed parity and all the fields are anticommuting as they should be. Note that the massless spinor is forced to be Weyl. The action is

s = f d x i~q~L + ~U (ij~ -- 1) q s ] , (46)

where qz~ and q/~ have been combined into a single spinor. The action for ,,pu is that o f a massive Rar i ta - Schwinger field in the gauge 7u~ u = 0 [13]. There is one more physical field with m 2 = 1 ; it is a spinor that appears in the ghost sector along with the bona fide spinor ghosts. The m 2 = 1 ghost states are

Cl(lU L) + lUR) ), C'I(iUL)), eI(lUL)+ lUR)),

e'l(lUL ) + lUR) ) •

Bearing in mind the G-parity of the ghosts Cl, Cl, e l , e'l, and the G-parity assignments o f lu L) and lUR) , we see from the form (45) that these states correspond to two commuting spinor ghost fields of opposite chirality, and two physical anticommuting spinor fields also o f opposite chirality. Presumably the anticommuting spinor is the 7" xl, component of ~u .

Some higher mass states will also violate spin- statistics. Right-handed states with an odd number o f d_ k operators, and left-handed states with an even number ofd_k operators are consistent with the form (45). This truncation is performed by requiring

G R = "y(- 1 ) x~*--1 d_k "dk ,

to be even. It is interesting that to construct a sensible field

theory, that is one in which the fields have an inter- pretation as physical or Faddeev-Popov ghosts, the G-parity truncation was necessary, whereas it was originally required to remove inconsistencies in scat- tering amplitudes. In any case the string field will need to be o f fixed parity once interactions are intro- duced for it.

It has recently been shown [2] that all the infor- mation necessary to construct the gauge invariant theory is contained in the BRST analysis. The key to the analysis is to shift the BRST auxiliary fields in the string action so that they become BRST invariant. The action for the Ramond sector, eq. (43), only displays the propagating fields and will need to be modified to construct the gauge invariant theory* 3. Work is in progress.

,a The action for the Ramond sector given in reL [14] may have the correct BRST auxiliary fields to construct the gauge invariant theory. This can only be decided by ex- amining the higher mass sectors. In any case this action has the same propagating fields as ours.

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This research is supported in part by the National Science Foundation under Contract No. PHY.82- 15249. I would like to thank L. Alvarez-Gaum~, G. Moore and P. Nelson for useful discussions.

References

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[21 W. Siegel, Phys. Lett. 149B (1984) 157,162; 151B (1985) 391,396; W. Siegel and B. Zwiebach, Berkeley preprint UCB4XrH - 85/30.

[3] T. Banks and M. Peskin, in: Anomalies, geometry and topology, ed. A. White (World Scientific, Singapore, 1985), to be published; M. Kaku, in: Anomalies, geometry and topology, ed. A. White (World Scientific, Singapore, 1985), to be published; D. Friedan, String field theory, Chicago preprint EFI 85- 27 (April 1985); A. Neveu and P.C. West, CERN preprint CERN-TH 4200• 85 (June 1985).

[4] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Santa Barbara preprint (December 1984).

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[10] L. Brink, P. Di Vecehia and P. Howe, Nucl. Phys. BI18 (1977) 76.

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[13] P. Van Nieuwenhuizen, Phys. Rep. 68 (1981) 191. [14] H. Terao and S. Uehara, Phys. Lett. 168B (1986) 70.

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Fermionic string field theory