BRST-symmetries in free string field theory

Volume 177, number 1 PHYSICS LETTERS B 4 September 1986

BRST-SYMMETRIES IN FREE STRING FIELD THEORY

Hideaki AOYAMA

Department of Physics, Northeastern University, Boston, MA 02115 :, USA and L vman Laboratory of Physics, Harvard UniversiO', Cambridge, MA 02138, USA

Received 29 May 1986

A unitary field transformation useful for examination of the BRST-symmetries in free bosonic open-string field theory is introduced. It is similar to Siegel and Zwiebach's transformation, but leads to a hermitian BRST-charge in contrast to theirs. The transformed BRST-transformation factorizes into two parts, making the gauge-unfixing procedure rather trivial. Both the minimal-type gauge-invariant action and Witten's action are dealt with.

1. Introduction. Covariant string field theory has been rapidly advanced in recent years with hopes that it will provide a sound and useful formulation of string theories. It was started by Siegel's idea [1] that the world-sheet BRST-symmetry [2,3] would produce the spacetime BRST-symmetry. His free action for the real bosonic open- string field q~[X(o), C(o), C(o)] is given as

where ~ is the second-quantized BRST-charge derived from the world-sheet BRST-charge QB [~B =- f(QB ~) 8/8q~1, and c is the zero-mode of the ghost coordinate. This action is BRST-invariant, i.e. guage-fixed, due to the nilpotency of the BRST-symmetry, ~ = 0. The free gauge-invariant actions were found by a couple of groups using different approaches [4-6]. Although these groups used quite different approaches, the resulting action containing only the dynamical fields (gauge fields and Stuckelberg fields) all agree. Eventually, an elegant formulation using some auxiliary field was found [7,8]. Siegel and Zwie- bach [4] used a field transformation (hereafter called SZ-transformation) • = K ~ to truncate the BRST-multiplet consistently to obtain a minimal gauge-invariant action. Their transformation was

useful because the BRST-transformation for the transformed fields is similar to the standard BRST-transformation in Yang-Mills theories.

Although SZ-transformation served its purpose fully, it has one undesirable feature; it ( K ) is not unitary. As a result, the BRST-transformation on the transformed fields is not unitary, the transformed BRST-charge Q B ( = K 1QBK ) is not hermitian. For this reason, the SZ-transformation is sometimes not suitable for further application. The purpose of this paper, then, is to introduce a unitarized transformation, which enables us to overcome the above problem and leads to a re- fined treatment for BRST-symmetries. We will find that the resulting fields are separated into two sets that are not mixed by the BRST-transformation, as in the standard BRST-transformation in Yang Mills theories. We will also see that in the new framework the "physical" BRST-symmetry defined in ref. [9] leads to a trivial gauge-unfixing procedure for both the minimal action [4-6] and Witten's action [7,8].

2. The unitary transformation. The first quantized BRST-charge for the open bosonic string is expanded with respect to the ghost zero-mode c as [2],

Q~ = (~ /~c)H + cT+ + Q+. (1)

1 Work supported by the National Science Foundation, grant PHY-83-05734.

The operator H is a singlet of SU(1, 1) whose generators are T+ and T_~. The operator Q+ is an

30 0370-2693/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


upper component of a doublet. They satisfy, Q2+ = , HT+. The "inverse operator" T+ 1 satisfies,

T + T + I = I - 8 _ , T 7 1 T + = l - 8 + ,

T+ 8 + = 8 T + = 0 , T+ 1 8 = 8 + T + 1 = 0 , (2)

where T denotes the total "isospin" of SU(1, 1). The operators H, Q+, T and O/Oc are hermitian, while T+, T> c and [c, O/Oc] are anti-hermitian. Therefore, QB is hermitian. (For details, see refs. [4,9].)

The SZ-transformation is defined by the following,

= K~), K= I + TT_~Q+ a/Oc. (3)

Since K 1 = 1 - T+IQ+ 3/Oc and K t = I - Q + T+ 10/~c, K is not unitary. The transformed BRST-charge Oa - K 1QuK is given by

QB = (0 /0c ) 8+~I+cT++½(8++8_)Q+

+½[c, ~)/~)c1(8_- 8+)Q+, (4)

where It=-H+ Q+T+IQ+. The first term in fact violates hermiticity *~

We introduce the transformation given by,

U=- I + { T+I, Q+ } 3/3c,

U- ' = a t = 1 - {T+', Q+} O/Oc. (5)

The transformed BRST-charge Q,u = U 1QuU is given by

Qu = (~ /0c ) [ 8+ /4 8_ +(1 - 8+)H(1 - 8 )]

+cT++½(Q+ 8 + 8 + Q + + 8 Q+

+ Q + 8 + - 2Q+) + ½[c, O/Oc](Q+8_

-8+Q+ +8_Q+-Q+8+). (6)

Note that the second-quantized charge is transformed as,

= f (7)

The operator Qu in (6) can be written in a follow-

*~ Since the first term is proportional to the (~/~c) 8+, this affects only the 8+ ~p-component, and not to the physical sector. Therefore, this is not a fundamental problem, although this is undesirable for practical purposes as ex- plained earlier.

ing way:

Os=PoO,Po+P, OBpl(=O0+O,), (8)

where Poa are the hermitian projection operators,

Po-8+(0/0c)c+8 c(~/Oc), P~=-a-Po. (9)

The usual relation P , ~ F 8, /~ guarantees the in- dividual nilpotency, Q~Q/= 0. The string field = U(b is decomposed into two sectors:

sector0: n 0=P0 ~ = ~ + + c ~ _ ,

sector 1: ,i~a = P1 ~ = ~ + c~+. (10)

It follows that $2

~ _ - 8 q~, ¢ + - = ( 1 - 8 _ ) ~ ,

¢+---8+¢, ~ _ ~ ( 1 - 8 + ) ~ . (11)

The BRST-transformation 8.~)= ( 0 . ~ for sector 0 is,

8u~_ = c 8_ Q+q~_,

8 ,~+ = , 8+(Hq,_ + O+~+), (12)

and for sector 1,

8uq~ + = , ( 1 - 8 ) ( - O + q ~ + - T + ~ ),

8 ,~_ = , ( 1 - 8 + ) ( H O + - O+~_) . (13)

This "factorization" property of the new represen- tation of the BRST-algebra simplifies the analysis in the following sections.

3. The minimal gauge-invariant action. After the U-transformation, Siege's action is given as & = ( ~ 8 , f~O&), where O - U t c ( a / O c ) U . In terms of the new fields, it is decomposed as

S~ = S~°°) + &(ol)+ S~,,), (14)

where S~ ~j) is a bilinear of the fields ~, and ~j,

s 0o, =

S~°"=f[q, (H+2Q+T+IQ+)O++g,_Q+¢ 1,

(15)

,2 The massless modes in q, are the gauge field A t and the- ghost field c. Similarly, q,+ contains the antighost field ~, 4'- contains the Nakanishi-Lautrup field [10] B.

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Volume 177, number 1 PIIYSICS LETTERS 13 4 September 1986

Let us discuss the gauge-unfixing procedure using the new representat ion. In ref. [9], we showed that the BRST-symmet ry given by ~B is not relevant for the second quant izat ion. It is because of the two problems in the second quant iza t ion; (i) if an act ion is wri t ten as {~ , * } with a ni lpotent exact ~, the v a c u u m - v a c u u m ampl i tude (and other "phys i ca l " ampl i tudes) vanishes, (ii) the func- t ional integrals vanish because of f [ d ~ + ] l = 0 . The trivial solut ion to the second prob lem is sim- ply to drop the ~+-integrat ion. This is equivalent to a gauge-fixing of a extra (rather trivial) local symmet ry [1] 8s~b + = X [Ss(other fields) = 0]. Since this gauge-fixing breaks the BRST-invariance, the first p rob lem is at the same t ime avoided. Let us take the following gauge-condi t ion,

~ + = 0 . (16)

(This gauge-condi t ion cor responds to the a = 1 case in ref. [9].) Al though the original BRST-sym- merry is no longer exact, there remains an exact BRST-symmet ry (named "phys i ca l " BRST-symmetry), which is a subgroup of [the original BRST-symmetry] × [the extra local symmetry]. It is given by the following:

sector O: 8phy4, = t 8 Q+4, , 8phy~+ = 0,

s e c t o r l : 8ph74 ,+=~(1- -8 ) ( - Q + 4 , + - T + ~ ),

8phy ~ = e ( 1 - a + ) ( H 4 , + - Q + ~ ).

(17)

(We denote the cor responding charge by O@phy. ) Note that the n i lpotency of this t ransformat ion tr ivial ly follows from that of (12) and (13).

The gauge-unfixing procedure is ra ther trivial in this formalism: The fields in sector 1 are either of ghos t -number nonzero (4'+) or auxil iary ( ~ ) . Since this sector is decoupled from the other sector, we can s imply set 4,+ = ~b = 0. After this one step, we are left with S, ~°°). This term is exactly the minimal gauge- invar iant action. (That is, it is not necessary to impose T~ = 0 as an addi t ional con- straint.) This is because the opera to r 8 + / ) 8 gives T~ ~< 0 and T 3 >/0 ([H, T3] = 0), which then leads to T = T 3 = 0 [9]. The physical BRST- t rans forma- tion (17) for 4, is closed and is " u p g r a d e d " to the gauge t ransformat ion , ~(gauge) ~T ,04 , = gv.oQ + A.

Because of the factor izat ion proper ty (12) and (13), S~ 11), S~ °t~ and S~ 11t are separa te ly invar iant under ~ ,3. Therefore, the act ion is still invar iant even if the three terms are given arb i t ra ry coefficients. However, among three coefficients, the over-al l factor renormal izes h and therefore is irrelevant, another coefficient can be absorbed by a renormal iza t ion of the fields in sector 1 (relat ive to sector 0). Therefore, we are left with a BRST- invar iant act ion with one free parameter , which we write as S , ,= S,~°°' + S~°l)+ o¢S (11). This is a

general izat ion of the c~-gauge Y a n g - M i l l s action. In a Y a n g - M i l l s theory, the sum of the gauge-

fixing act ion and the ghost act ion is wri t ten as { ~ , * }, an an t i commuta to r of a relevant ( that is, quan tum-mechan ica l ly exact) BRST-charge and a bi l inear of fields. The gauge- invar iant term cannot be wri t ten as such. This p roper ty guarantees the gauge- invar iance of the physical S-matr ix ele- ments, once interact ions are in t roduced [11]. The same proper ty holds for our actions. The term S, (°11 should be expressed as an an t i commuta to r of ~phy and a bi l inear of fields ~0 and ~1. In compo- nents, the possible terms are 4, F4,+ and 4, G~+ , where F has to have 8T 3 = - 1 / 2 and G, 8T~ = 0. Most simple opera tors possible for F are Q+ T+ 1 and T~lQ+ and for G, 1. The lat ter two have no contr ibut ion . Calcula t ing the an t i commuta to r for the first operator , we find that

Fo r S~ ~1~, similar considera t ion leads to

S~,11~={~phy, f(4,+T+lQ+4,++½4,+~_)}. (19)

The fact that Sf °°~ cannot be writ ten as a s imilar an t i commuta to r is obvious from the cons idera t ion in the massless sector [9] ,4

:~3 For example, for the massless sector, the two terms in S~0~ contains ic 02c and B ~A, respectively. The first term in Sf m contains the B 2 term, while the other terms in S~ ~ have no contribution to the massless sector.

,4 Since ~÷ does not contribute to (18) or (19), these hold even when ~phy is replaced by ~B- Since Siegel's action itself is written as an anticommutator using ~ , this means that the gauge-invariant piece can also be written similarly. In fact, we find that

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4. Witten's action. Witten's action is defined to be the following,

= f f (20)

where P projects out the contributions of fields of nonzero ghost number. Similarly to (14), Sw is decomposed as

S~ = S,(~ °°, + S~ u>, (21)

where

x.oo. = (22)

and

v"= f(½.+.<o.+-.+o+<

- ½S- T+Sr,. , /2S-)

=

x ( S _ + o+r+tq~+). (23)

The term Sw m°) is actually exactly equal to the minimal action S~ °°). In order to prove this, let us expand the ~-field by its total isospin:

ep= ~ ~, (@,=Sr,,ep). (24) 2t=0

Although [H, T] ~ 0, an identity /1 8v. o = 8r,o/~1 = aT,oH 8r. o holds [4,9]. Therefore, the contributions to S~ °°) are from q,o/1 8r,.oq~ o and ep,~I 8r,,oq~,,(tt' 4: 0). For the latter, we find

~a,H 8r~.0q, ,, c( ~, 8r,.,T+~IT + 8r _~q~,,. (25)

Since the following identity holds,

T+~IT+ = - T+Q2+ + T+Q+T+'T+Q+

= - T + Q + 8+Q+=0, (26)

we find that (25) vanishes. Thus S~ °°) = Ss m°) ,5 Terao and Uehara [12] introduced an auxiliary

field X in order to discuss the gauge-unfixing. We

,5 Incidentally, S~ H) is purely auxiliary because of (23). This proves the equivalence of S~ and the minimal gauge-invariant action S~ °°).

take the following action,

S,~=-s~-f½(@_-2_)T+ 8,-3,_1/2(S_-2 ). (27)

(The operator 8v,_1/2 is not in their original action, but is necessary as we shall see later.) This action is invariant under the "extended" BRST- transformations, where 85~ are taken to be the same as 8~/. (We use the same symbols ~s and "~phy to denote the corresponding charges.) They showed that by gauge-unfixing Stu, one obtains the Witten action

S, = f ½~x~xP~x, (28)

where ~x =- R ~ = X + cep, ~x - R'~BR' using the S-2 exchanging operator

R -= • 8/8 S + S 8/82- (29)

(Note that R e= 1 guarantees ~ 2 = 0 . ) The relation between Stu and S,o of (28) is then

st . = Sw + s ? " + s ~ ' " - a s ''1', (30) where

as'l"=f(½ep+ H 8T,,0q,+ +0+Q+ 8r,, ,/25<

+S_T+ 8T3 _1/2X_

+½~_T+ 8r," , /2S-) . (31)

Using the identities,

{~'phy' f~+s-}= f(*+ "~+-}-S-T+S' ) '

= f ( < - < - . + 0 + S + + + 0 + 2

+S r+2_),

{~p,,,, f½ep+rT~'Q+~,+} = f ( - < - < +.+e+S ). (32)

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we find that ,6

As(ll)={,, ph ) , a r , /25-+++

+ ½~+ T+ 1Q+ 8r~,0~ +). (33)

(Note that unless the cons t ra in t 8.+,. ~/2 is im- posed in the added term in (27), AS would con- tain a XX-term that cannot be writ ten as an anti- commuta tor . )

It is interest ing to note that ~x used for the const ruct ion of the Wit ten act ion (28) is not the BRST-charge ob ta ined by gauge-fixing it. The gauge-fixed act ion Stu is invar iant under ~B and ~phy, but not invariant under ~x" The BRST-invar iance of S w under ~B is guaranteed by

~ B ~ x = ~ x ~ B = 0, (34)

which follow from ~ B R = ~ . Similar relat ions hold for "~phy. Although ~x and ~B are related by the opera to r R, they are essential ly different second-quan t ized operators .

In conclusion, we have in t roduced and ex- amined a uni tary t ransformat ion, which has many desi rable features. The new t ransformat ion leads to a factorizable, hermit ian BRST-charge, which allows a trivial gauge-unfixing procedure. Siegel's act ion (in the Siegel gauge) is general ized to an

,6 Similar analysis leads to the following for S~ in (20)-(23),

S w = { ~ , , f ' 2 ( + ~+-q '+ ST,. 1/2~

-- 9 + T ; I Q + 8r , .0~ + )}.

c~-gauge case. The gauge-fixing and ghost act ion for both the minimal gauge- invar iant act ion and the Wit ten act ion are writ ten as an an t i commuta - tor of the relevant BRST-charge ~phy and a bi l inear of fields, as required for any non-abel ian theory. It would be interest ing to app ly the new t ransformat ion to the p roposed interact ing theories [7,13,14]. Since the factorizat ion proper ty holds for interact ing Y a n g - M i l l s theories, we hope that the formal ism developed in this let ter will be useful for the full interact ing theories.

References

[1] w. Siegel, Phys. Lett. B 151 (1985) 391, 396. [2] M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. [3] K. Fujikawa, Phys. Rev. D 25 (1982) 2584:

S. Hwang, Phys. Rev. D 28 (1983) 2614. [4] W. Siegel and B. Zwiebach, Nucl. Phys. B 263 (1986) 105. [5] T. Banks and M. Peskin, Nucl. Phys. B 264 (1986) 513. [6] K. Itoh, T. Kugo, H. Kunitomo and H. Ooguri, Progr.

Theor. Phys. 75 (1986) 162. [7] E. Witten, Princeton University preprint (October 1985). [8] A. Neveu, H. Nicolai and P.C. West, Phys. Lett. B 167

(1986) 307. [9] H. Aoyama, preprint NUB-2694-HUTP-86/A031 (April

1986). [10] N. N akanishi, Progr. Theor. Phys. 35 (1966) 111;

B. Lautrup, K. Dan. Vidensk. Selsk. Math. Fys. Medd. 34 (1967) 1.

[11] T. Kugo and I. Ojima, Phys. Lett. B 73 (1978) 549; Progr. Theor. Phys. Suppl. 66 (1979).

[12] H. Terao and S. Uehara, Hiroshima preprint RRK 86-3 (1986).

[13] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B 172 (1986) 186, 195; Kyoto University preprint KUNS-829 (1986).

[14] A. Neveu and P. West, Phys. Lett. B 168 (1986) 192.

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BRST-symmetries in free string field theory