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Remarks on the BRST current for strings
Volume 186, number 3,4
PHYSICS LETTERSB
12 March 1987
REMARKS O N T H E BRST CURRENT FOR S T R I N G S Robert MARNELIUS Institute of TheoreticalPhysics, S-412 96 G~teborg, Sweden Received 17 November 1986
It is shown that the appropriate conserved BRST current in the bosonic string theory in the critical dimension may be defined to be the BRST transform of the ghost number current. This BRST current is therefore alwaysa gaugegenerator. It is also shown that the antieommutator of this BRST current is anomalous.
In this note some properties of local currents within the first quantized bosonic string theory in arbitrary dimensions is worked out using the operator formalism. In particular the properties of the BRST current are discussed. It is, e.g., shown that the BRST current may be defined to be the BRST transform of the ghost number current in the critical dimension. This definition leads to a modified form of the naive BRST current which I later have seen argued for in other papers [ 1,2 ]. This work was inspired by the fascinating paper by Witten on the BRST formulation of the interacting theory [ 3 ], a paper which also emphasized the local properties of the string theory. Let me start with the gauge-fixed lagrangian for the free bosonic string [ 4 ] (see also refs. [ 3,5 ] ) La(z,tr)=½0~x.O~x+ 2b,~O~c# + O a b ~ c p ,
(1)
where x~'(z,a) are the space-time coordinates of the string, and c~(z,a) and b ~ ( z , a ) are the ghost and antighost fields, respectively. The latter fields are odd elements o f a Grassmann algebra, c ~ is chosen to be real and b ~ imaginary, b,~ is symmetric and traceless. The two-dimensional indices ot and fl are governed by the metric r/11= - ~/22= 1, ~/12= ~/2~= 0. 0 , is the derivative with respect to ~ (C°-=z, ~ l - a ) . a has the standard finite range. (Notice that L# differs only by a divergence from the simpler expression ..~'= ½Oo,x.O'~x+b~O'~c a. The choice (1) is only made in order to have the canonical energy-momentum tensor below.) The equations of motion are given by 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
n x " =0,
O~b,~p=O,
O'~ca +O# c ~ - ~ l ~ Ovc v = 0 ,
(2)
where [] -= 0~0 '~. The last two equations imply •b,~ =[]:=0.
(3)
In addition we have the boundary conditions dlX ~'=btrc r = 0
at a = 0 , n
(4)
for open strings and periodic boundary conditions for closed strings. I shall now consider some of the important invarlances of the lagrangian (1) and the corresponding conserved current densities. Translation invariance in the z,a-space leads to a conserved energy-momentum tensor. The canonical energy-momentum tensor is given by
T~# = T ~ + T ~ ,
(5)
where
T~# =O'~x.OP x -
½tl'~#Ovx.Or x ,
T ~ = 2b~vOP c r + O#bavc r - 2rla#b:,60rc '~ - r i g o r b~,rc6 .
(6)
There is also a conserved symmetric energym o m e n t u m tensor which is traceless, O '~# - T ~ #
+0~,
(7)
351
Volume 186, number 3,4
PHYSICS LETTERS B
12 March 1987
On shell, i.e. by means of eqs. (2), we have T~# = 0 °~. The lagrangian ( I ) is also Lorentz invariant and dilatation invariant in the internal z,~r-space. The corresponding conserved currents are
mental reparametrization invariance of the original string theory a n d j ~ belongs to the large class of generators which are particular to the BRST formulation. Now all formula are most conveniently written in terms of light-cone indices since the equations of motion (2) are directly solved in terms of the lightcone coordinates ~ ± = z +_a. Thus, we replace the metric r/,~p by the light-cone one rhb with t/++ =r/_ _ =0, r/+ _ = ½. Eq. (2) becomes then
ga#r=~rOa#-~#O ~,
O+O_xu=O,
O ~ - b~'vOPc~ + ba~O~'c~ + O~b~ac r -rl"a br~Orc a . (7 cont'd)
j~=~aO'~a.
(8)
O_b++=O+b__=O,
These kinematical invariances are only special cases of the more general dynamical conformal invariance. Most important is the fact that the lagrangian (1) is BRST invariant, i.e., it is invariant under
and the above considered conserved currents take the following form on shell:
8~x u =2c~ 0~,x ~, 8~c ~ =2c~ 0~c '~,
0 ± + = T ~ + +T~_+ ,
8ab~ = -20~
,
(9)
where 2 is a constant odd Grassmann parameter. Notice that this BRST transformation is only nilpotent on shell. Even though 82x " = 8 2 c ~ = 0, 8 ~ b,~a = 0 is only obtained by means of the equations of motion (2). The corresponding conserved current is given by
j~( z,tr) = ( T~'~ + ½T~h~)C~ = T~pc p +b'~rOpcrc p -byaOYcPc '~ .
(10)
Another important invariance is the invariance under the ghost transformations
8cxU=O,
8cCa=~c%
$cb~p=-~b~,
O_c + =O+c- = 0 ,
(14)
T~+_ =(d+_x):, T~+_ =2b++0+c +-+O+_b+_+c+- , j f f = ( 2 T ~ ± + T ~ + _ ) C +-, j ~ = 2 b ± ± c +-.
(15)
Eqs. (14) together with the boundary conditions for open and dosed strings yield the solutions c ± = c ± ( ~ - - ) , b±_+=b±_+(~ -+) and xU=x~(~+)+ x ~ ( ~ - ) , where c -+, b_+± and XLR are periodic with period 2n. Their Fourier decompositions are given by
x~(t)=~4~(qU+pUt+i~. 1 a~e-int) n,,o n
(11)
where e is an infinitesimal constant. The corresponding conserved current is
x~(t):~4~(q~+p~l+i~olb~ne-int
j~ =b'~pc p .
C+.t.
(12)
) ,
x-~ + e _ i n t n
There are of course more invariances and corresponding conserved currents. Actually an infinite number since we are dealing with a free theory. However, in the following I will concentrate on the above important ones. In particular I shall emphasize the on-shell Poisson bracket relations
0 '~a={QS, b'~P}, Jg={QB,j~.},
(13)
and their corresponding commutators. (Qa is the BRST charge.) Eq. (13) tells us that O aa a n d j ~ are gauge generators. O ~p is associated with the funda352
b±±(t)= with
l_~.~,'1~ ._ +-e - int
2zt~
"
'
(16)
the reality properties a~° -- a ~- n , bun* /~ "1-* + b-n, qff =r/-+-n and ~ ' = ~ - n . (Notice that b• a was chosen to be imaginary.) For open strings we have x~ = x f~, c + = c - and b+ + = b_ _. Also the currents (15) are periodic with period 2ft. (Notice that the plus components only depend on ~ + and the minus components only on ~ - . ) Hence, we
Volume 186, number 3,4
PHYSICS LETTERS B
may write
12 March 1987
[ a u, a~ ] _ = [ b u, b~ ] _ = -- nnUUt~n,_m ,
TX + ( t ) = l Z L ~ e -2to.
-i"'
[qU,p~]_ =_inU~,
'
1 - 6 1 - e -i"t ,
T~+(t)=~-~
1 -- + int , j + a ( t ) =~-~n~a~ne-
j + c ( t ) = ~ -1~_Q e ~.e - int ,
(17)
where
L+" =21 ~a~a_.m~.
,
[n~,, ~ 2 ] + = 8 . , - m •
(21)
Notice that a~ =-b~-pU, qU, n~ and ~ff are hermitean while a U tu- a- _ . , b ut-hu.- v _ . , n f f t = n + . and ~ + t = ~ z+ . . The hermitean operators build up the vacuum state space on which a u, bU., n + and ~ + act as annihilation operators for positive n and as creation operators for negative n. The quantum current (17) must be normal ordered. The resulting Fourier coefficients take the following form: Qc. =i½ (To ~ . - ~.r/o)
a ~ - p u,
+ i ~ ( r / _ k ~ . + k - - ~._k~/k) , k=l
L y =~1 ~bUmb._m" u,
b~ =pU , G~ = - in Qc. + R. ,
+
+
G~ = ~ ( m - 2 n ) ~ m n Y _ m , n
~ k( ~n_knk"l-n_k~n+k) ,
Rn = -
k=l
+ + + -~G n+- - m n m+-) , Qff.= L~ . [L n--mnm m
Qa. = ½(Knno + noKn)
~ + + Q~_n -.~- i~-..~ 2.,~'n_mnUn.
(18)
rn
+ ~ (K._krlk+q_~K.+k),
This is for closed strings. For open strings all the plus components are equal to the minus ones, and b u = a u. The fundamental nonzero Poisson brackets are easily obtained from the lagrangian (1). They are
{xU( a), OoX~( a') } =nU"~( a - a ') , (boa(a), cP(a')} = O ~ O ( o ' - a ') ,
(19)
which in turn lead to the following nonzero Poisson brackets for the Fourier coefficients in (16) ( a ~ b~ -pU):
{aU~, a~ } = {b u, bL } = - iml~ ~.,_ m , {qU,p~}=rl~,
{r/+, ~ + } = i O . , _ m ,
(20)
In the quantum theory these Poisson brackets are replaced by the nonzero fundamental commutation relations (h = 1 )
k=l
Kn-Ln+½Gn
,
L . = ½~ :aga._m~: +fla..o,
(22)
m
where the superscripts + are suppressed and where fl = Oto for open strings and fl = ao/2 for closed strings. With a little algebra one may show that QBn is unaffected if Kn above is replaced by
K.=L.+R.,
(23)
which leads to a simpler expression for QB.. The following nonzero transformation formulas are straightforward to derive [Qcn, rim]-
=inn+m,
[ Q e n , "-~rn] = - i ~ . + . . ,
(24)
353
Volume 186, number 3,4
PHYSICS LETTERS B
.u -/~ [L,,, a,,,]_ -ma,,+,,,,
[L., qU] _ =iaU,
where the anomalous terms are underlined. Notice the BRST charge QB is equal to Qffo + Qgo for closed strings and to QBo for open strings. The last formula above is therefore consistent with the fact that Q2 is zero for D = 2 6 and a o = 1 or 2 for open and closed strings, respectively [ 4,5 ]. Now the above currents are not physical operators or, equivalently, BRST-invariant operators in general. In fact, only when Q2 is zero do we have such currents, which are related to the ones considered above. We have then, e.g.,
[G., q,.] _ = (2n+m)q.+m,
[G., ~ m ] _ = ( m - n ) t ~ . + , , ,
,
[ QB,,,q u] _ =i~aun_kqk, [QB., a ~ ] _ =rnY~aU.,,+._krlk, k
[ Q B . , ?~m] + = ~ k ~ n + m - - k ~ k , k [Qbn,~m]
+ : -Ln+m
£.=-
- Gn+m -inQcn+m
12 March 1987
•
[QB, ~ . 1 + = L . + G . ,
0B. =i[Qn, Q c . ] - = Q B . - ] n 2 , . ,
(24 cont'd) With a little further algebra I found the following current algebra (the nonzero commutators): [ Q c n , Q c m ] - = - nrJ . . . . .
[Qc., G,.] _ = - n Q c . + m -i½n2O. - , . ,
B. = [QB, q.] + = ~krl.--krlk, k
(26)
where/~, is just the total energy-momentum tensor O + +. B. is a current of the form j+ oct + 0 +c +. Owing to the form of these currents they generate gauge transformations in the physical subspace and they satisfy a closed commutator algebra [6]. The nonzero elements are
[Qc., QBml- =iQB.+m--in(~n+m)rl.+,,, ,
[E., ff.m] _ = ( m - n )E.+m, [L.,L,.,]_ = ( m - n ) L . + , , ,
[ (~B., (~Bm]+ = --2nmB,,+m,
+ ~n(On 2+ 24//-D)6._,.,, [G.,G,,,]
[OB., /~m]- =
[L., Bin]_ = (2n+m)B,,+m ,
=(m-n)G,,+,,,
[QB., L..]_ = ~ ( m - n+ k)L.+m_kqk k
~m(Om 2+24//-D)q.+,.,
[QB., G,,,]_ =
-nQB,,+,,,
-- ~ ( rn--n+ k)Ln+,,,_k~Ik k
+ ~m(13m z - 1 + 9m 2n) t/.+,,,, [QB., QBm]- = ~ { ( D - 2 6 ) k
3+ [3nm(2-D)
j~ = i [ Q B , j ~ ] + 18(n 2 + m 2) + (24fl+2--D)]k}q.+m_krlk, (25) 354
(27)
from which one infers that the BI~ST current satisfies an anomalous anticommutator. (Classically we have {QB., QBm}=0.) The B. current was only included in (26) due to this anomaly. Thus, in D = 2 6 with Oto= 1 or 2 the physical currents out of the considered ones are the total energy-momentum tensor and the BRST current provided it is modified according to (26). This modification was also found in ref. [1] (p. 107) by the requirement [(2B., Em]=--n(~B.+m. (Notice that [ QB.,Lm] = --nQB.+,.+~m2nq,,+,,,.) In conclusion I have found that the BRST current jg may be defined by
-~n(13n2-1)O.,_m ,
-
- n(~B.+m,
(28)
when the quantization is anomaly free, i.e., when Q~ =0. Notice that j~ is well defined. One consequence of (28) is that j~ generates parts of the large
Volume 186, number 3,4
PHYSICS LETTERS B
References
gauge invariance under (Qnl p h y s ) = 0) I phys ) ~ 1 phys ) + Q I
) •
12 March 1987
(29)
It would be quite interesting to find out how general a relation o f the form (28) is. I believe it is valid for more general models than the simple string model considered here (e.g. the case o f ref. [ 2 ]). It should be noticed that the definition (28) requires the conservation o f both the ghost current and the BRST current. Thus, the ghost charge Qc must be conserved and the BRST charge Qa must have ghost number one. However, this has always been required.
[ 1] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [21 T. Banks, D. Nemenschansky and A. Sen, Nucl. Phys. B 277 (1986) 67. [3] E. Witten, Nucl. Phys. B 268 (1986) 253. [~] S. Hwang, Phys. Rev. D 28 (1983) 2614. [5] M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. [6] R. Marnelius, Phys. Lett. B 99 (1981) 467; Acta. Phys. Polon. B 13 (1982) 669.
355
PHYSICS LETTERSB
12 March 1987
REMARKS O N T H E BRST CURRENT FOR S T R I N G S Robert MARNELIUS Institute of TheoreticalPhysics, S-412 96 G~teborg, Sweden Received 17 November 1986
It is shown that the appropriate conserved BRST current in the bosonic string theory in the critical dimension may be defined to be the BRST transform of the ghost number current. This BRST current is therefore alwaysa gaugegenerator. It is also shown that the antieommutator of this BRST current is anomalous.
In this note some properties of local currents within the first quantized bosonic string theory in arbitrary dimensions is worked out using the operator formalism. In particular the properties of the BRST current are discussed. It is, e.g., shown that the BRST current may be defined to be the BRST transform of the ghost number current in the critical dimension. This definition leads to a modified form of the naive BRST current which I later have seen argued for in other papers [ 1,2 ]. This work was inspired by the fascinating paper by Witten on the BRST formulation of the interacting theory [ 3 ], a paper which also emphasized the local properties of the string theory. Let me start with the gauge-fixed lagrangian for the free bosonic string [ 4 ] (see also refs. [ 3,5 ] ) La(z,tr)=½0~x.O~x+ 2b,~O~c# + O a b ~ c p ,
(1)
where x~'(z,a) are the space-time coordinates of the string, and c~(z,a) and b ~ ( z , a ) are the ghost and antighost fields, respectively. The latter fields are odd elements o f a Grassmann algebra, c ~ is chosen to be real and b ~ imaginary, b,~ is symmetric and traceless. The two-dimensional indices ot and fl are governed by the metric r/11= - ~/22= 1, ~/12= ~/2~= 0. 0 , is the derivative with respect to ~ (C°-=z, ~ l - a ) . a has the standard finite range. (Notice that L# differs only by a divergence from the simpler expression ..~'= ½Oo,x.O'~x+b~O'~c a. The choice (1) is only made in order to have the canonical energy-momentum tensor below.) The equations of motion are given by 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
n x " =0,
O~b,~p=O,
O'~ca +O# c ~ - ~ l ~ Ovc v = 0 ,
(2)
where [] -= 0~0 '~. The last two equations imply •b,~ =[]:=0.
(3)
In addition we have the boundary conditions dlX ~'=btrc r = 0
at a = 0 , n
(4)
for open strings and periodic boundary conditions for closed strings. I shall now consider some of the important invarlances of the lagrangian (1) and the corresponding conserved current densities. Translation invariance in the z,a-space leads to a conserved energy-momentum tensor. The canonical energy-momentum tensor is given by
T~# = T ~ + T ~ ,
(5)
where
T~# =O'~x.OP x -
½tl'~#Ovx.Or x ,
T ~ = 2b~vOP c r + O#bavc r - 2rla#b:,60rc '~ - r i g o r b~,rc6 .
(6)
There is also a conserved symmetric energym o m e n t u m tensor which is traceless, O '~# - T ~ #
+0~,
(7)
351
Volume 186, number 3,4
PHYSICS LETTERS B
12 March 1987
On shell, i.e. by means of eqs. (2), we have T~# = 0 °~. The lagrangian ( I ) is also Lorentz invariant and dilatation invariant in the internal z,~r-space. The corresponding conserved currents are
mental reparametrization invariance of the original string theory a n d j ~ belongs to the large class of generators which are particular to the BRST formulation. Now all formula are most conveniently written in terms of light-cone indices since the equations of motion (2) are directly solved in terms of the lightcone coordinates ~ ± = z +_a. Thus, we replace the metric r/,~p by the light-cone one rhb with t/++ =r/_ _ =0, r/+ _ = ½. Eq. (2) becomes then
ga#r=~rOa#-~#O ~,
O+O_xu=O,
O ~ - b~'vOPc~ + ba~O~'c~ + O~b~ac r -rl"a br~Orc a . (7 cont'd)
j~=~aO'~a.
(8)
O_b++=O+b__=O,
These kinematical invariances are only special cases of the more general dynamical conformal invariance. Most important is the fact that the lagrangian (1) is BRST invariant, i.e., it is invariant under
and the above considered conserved currents take the following form on shell:
8~x u =2c~ 0~,x ~, 8~c ~ =2c~ 0~c '~,
0 ± + = T ~ + +T~_+ ,
8ab~ = -20~
,
(9)
where 2 is a constant odd Grassmann parameter. Notice that this BRST transformation is only nilpotent on shell. Even though 82x " = 8 2 c ~ = 0, 8 ~ b,~a = 0 is only obtained by means of the equations of motion (2). The corresponding conserved current is given by
j~( z,tr) = ( T~'~ + ½T~h~)C~ = T~pc p +b'~rOpcrc p -byaOYcPc '~ .
(10)
Another important invariance is the invariance under the ghost transformations
8cxU=O,
8cCa=~c%
$cb~p=-~b~,
O_c + =O+c- = 0 ,
(14)
T~+_ =(d+_x):, T~+_ =2b++0+c +-+O+_b+_+c+- , j f f = ( 2 T ~ ± + T ~ + _ ) C +-, j ~ = 2 b ± ± c +-.
(15)
Eqs. (14) together with the boundary conditions for open and dosed strings yield the solutions c ± = c ± ( ~ - - ) , b±_+=b±_+(~ -+) and xU=x~(~+)+ x ~ ( ~ - ) , where c -+, b_+± and XLR are periodic with period 2n. Their Fourier decompositions are given by
x~(t)=~4~(qU+pUt+i~. 1 a~e-int) n,,o n
(11)
where e is an infinitesimal constant. The corresponding conserved current is
x~(t):~4~(q~+p~l+i~olb~ne-int
j~ =b'~pc p .
C+.t.
(12)
) ,
x-~ + e _ i n t n
There are of course more invariances and corresponding conserved currents. Actually an infinite number since we are dealing with a free theory. However, in the following I will concentrate on the above important ones. In particular I shall emphasize the on-shell Poisson bracket relations
0 '~a={QS, b'~P}, Jg={QB,j~.},
(13)
and their corresponding commutators. (Qa is the BRST charge.) Eq. (13) tells us that O aa a n d j ~ are gauge generators. O ~p is associated with the funda352
b±±(t)= with
l_~.~,'1~ ._ +-e - int
2zt~
"
'
(16)
the reality properties a~° -- a ~- n , bun* /~ "1-* + b-n, qff =r/-+-n and ~ ' = ~ - n . (Notice that b• a was chosen to be imaginary.) For open strings we have x~ = x f~, c + = c - and b+ + = b_ _. Also the currents (15) are periodic with period 2ft. (Notice that the plus components only depend on ~ + and the minus components only on ~ - . ) Hence, we
Volume 186, number 3,4
PHYSICS LETTERS B
may write
12 March 1987
[ a u, a~ ] _ = [ b u, b~ ] _ = -- nnUUt~n,_m ,
TX + ( t ) = l Z L ~ e -2to.
-i"'
[qU,p~]_ =_inU~,
'
1 - 6 1 - e -i"t ,
T~+(t)=~-~
1 -- + int , j + a ( t ) =~-~n~a~ne-
j + c ( t ) = ~ -1~_Q e ~.e - int ,
(17)
where
L+" =21 ~a~a_.m~.
,
[n~,, ~ 2 ] + = 8 . , - m •
(21)
Notice that a~ =-b~-pU, qU, n~ and ~ff are hermitean while a U tu- a- _ . , b ut-hu.- v _ . , n f f t = n + . and ~ + t = ~ z+ . . The hermitean operators build up the vacuum state space on which a u, bU., n + and ~ + act as annihilation operators for positive n and as creation operators for negative n. The quantum current (17) must be normal ordered. The resulting Fourier coefficients take the following form: Qc. =i½ (To ~ . - ~.r/o)
a ~ - p u,
+ i ~ ( r / _ k ~ . + k - - ~._k~/k) , k=l
L y =~1 ~bUmb._m" u,
b~ =pU , G~ = - in Qc. + R. ,
+
+
G~ = ~ ( m - 2 n ) ~ m n Y _ m , n
~ k( ~n_knk"l-n_k~n+k) ,
Rn = -
k=l
+ + + -~G n+- - m n m+-) , Qff.= L~ . [L n--mnm m
Qa. = ½(Knno + noKn)
~ + + Q~_n -.~- i~-..~ 2.,~'n_mnUn.
(18)
rn
+ ~ (K._krlk+q_~K.+k),
This is for closed strings. For open strings all the plus components are equal to the minus ones, and b u = a u. The fundamental nonzero Poisson brackets are easily obtained from the lagrangian (1). They are
{xU( a), OoX~( a') } =nU"~( a - a ') , (boa(a), cP(a')} = O ~ O ( o ' - a ') ,
(19)
which in turn lead to the following nonzero Poisson brackets for the Fourier coefficients in (16) ( a ~ b~ -pU):
{aU~, a~ } = {b u, bL } = - iml~ ~.,_ m , {qU,p~}=rl~,
{r/+, ~ + } = i O . , _ m ,
(20)
In the quantum theory these Poisson brackets are replaced by the nonzero fundamental commutation relations (h = 1 )
k=l
Kn-Ln+½Gn
,
L . = ½~ :aga._m~: +fla..o,
(22)
m
where the superscripts + are suppressed and where fl = Oto for open strings and fl = ao/2 for closed strings. With a little algebra one may show that QBn is unaffected if Kn above is replaced by
K.=L.+R.,
(23)
which leads to a simpler expression for QB.. The following nonzero transformation formulas are straightforward to derive [Qcn, rim]-
=inn+m,
[ Q e n , "-~rn] = - i ~ . + . . ,
(24)
353
Volume 186, number 3,4
PHYSICS LETTERS B
.u -/~ [L,,, a,,,]_ -ma,,+,,,,
[L., qU] _ =iaU,
where the anomalous terms are underlined. Notice the BRST charge QB is equal to Qffo + Qgo for closed strings and to QBo for open strings. The last formula above is therefore consistent with the fact that Q2 is zero for D = 2 6 and a o = 1 or 2 for open and closed strings, respectively [ 4,5 ]. Now the above currents are not physical operators or, equivalently, BRST-invariant operators in general. In fact, only when Q2 is zero do we have such currents, which are related to the ones considered above. We have then, e.g.,
[G., q,.] _ = (2n+m)q.+m,
[G., ~ m ] _ = ( m - n ) t ~ . + , , ,
,
[ QB,,,q u] _ =i~aun_kqk, [QB., a ~ ] _ =rnY~aU.,,+._krlk, k
[ Q B . , ?~m] + = ~ k ~ n + m - - k ~ k , k [Qbn,~m]
+ : -Ln+m
£.=-
- Gn+m -inQcn+m
12 March 1987
•
[QB, ~ . 1 + = L . + G . ,
0B. =i[Qn, Q c . ] - = Q B . - ] n 2 , . ,
(24 cont'd) With a little further algebra I found the following current algebra (the nonzero commutators): [ Q c n , Q c m ] - = - nrJ . . . . .
[Qc., G,.] _ = - n Q c . + m -i½n2O. - , . ,
B. = [QB, q.] + = ~krl.--krlk, k
(26)
where/~, is just the total energy-momentum tensor O + +. B. is a current of the form j+ oct + 0 +c +. Owing to the form of these currents they generate gauge transformations in the physical subspace and they satisfy a closed commutator algebra [6]. The nonzero elements are
[Qc., QBml- =iQB.+m--in(~n+m)rl.+,,, ,
[E., ff.m] _ = ( m - n )E.+m, [L.,L,.,]_ = ( m - n ) L . + , , ,
[ (~B., (~Bm]+ = --2nmB,,+m,
+ ~n(On 2+ 24//-D)6._,.,, [G.,G,,,]
[OB., /~m]- =
[L., Bin]_ = (2n+m)B,,+m ,
=(m-n)G,,+,,,
[QB., L..]_ = ~ ( m - n+ k)L.+m_kqk k
~m(Om 2+24//-D)q.+,.,
[QB., G,,,]_ =
-nQB,,+,,,
-- ~ ( rn--n+ k)Ln+,,,_k~Ik k
+ ~m(13m z - 1 + 9m 2n) t/.+,,,, [QB., QBm]- = ~ { ( D - 2 6 ) k
3+ [3nm(2-D)
j~ = i [ Q B , j ~ ] + 18(n 2 + m 2) + (24fl+2--D)]k}q.+m_krlk, (25) 354
(27)
from which one infers that the BI~ST current satisfies an anomalous anticommutator. (Classically we have {QB., QBm}=0.) The B. current was only included in (26) due to this anomaly. Thus, in D = 2 6 with Oto= 1 or 2 the physical currents out of the considered ones are the total energy-momentum tensor and the BRST current provided it is modified according to (26). This modification was also found in ref. [1] (p. 107) by the requirement [(2B., Em]=--n(~B.+m. (Notice that [ QB.,Lm] = --nQB.+,.+~m2nq,,+,,,.) In conclusion I have found that the BRST current jg may be defined by
-~n(13n2-1)O.,_m ,
-
- n(~B.+m,
(28)
when the quantization is anomaly free, i.e., when Q~ =0. Notice that j~ is well defined. One consequence of (28) is that j~ generates parts of the large
Volume 186, number 3,4
PHYSICS LETTERS B
References
gauge invariance under (Qnl p h y s ) = 0) I phys ) ~ 1 phys ) + Q I
) •
12 March 1987
(29)
It would be quite interesting to find out how general a relation o f the form (28) is. I believe it is valid for more general models than the simple string model considered here (e.g. the case o f ref. [ 2 ]). It should be noticed that the definition (28) requires the conservation o f both the ghost current and the BRST current. Thus, the ghost charge Qc must be conserved and the BRST charge Qa must have ghost number one. However, this has always been required.
[ 1] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [21 T. Banks, D. Nemenschansky and A. Sen, Nucl. Phys. B 277 (1986) 67. [3] E. Witten, Nucl. Phys. B 268 (1986) 253. [~] S. Hwang, Phys. Rev. D 28 (1983) 2614. [5] M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. [6] R. Marnelius, Phys. Lett. B 99 (1981) 467; Acta. Phys. Polon. B 13 (1982) 669.
355