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Gauge fixing: chiral model versus string theory
Volume 247, number
PHYSICS
1
LETTERS
6 September
B
1990
Gauge fixing: chiral model versus string theory A.P. Demichev and M.Z. Iofa Institute for Nuclear Physics, Moscow State University, SU-I 19 899 Moscow, USSR Received
27 April 1990
The chiral model is discussed as an analog to the string theory to study anomalies a counterpart of the conformal gauge rendering the anomalous WI and also another the anomaly in the chiral model is shown to be the same as in the string theory.
1. The BRST formalism provides a convenient and effective method to quantize gauge theories. This method was applied successfully to string theory also, for example, to define the physical sector of the theory, to study decoupling of spurious, etc. [ l-41. However, in conformal gauge in high orders of string perturbation theory the BRST symmetry proves to be anomalous. In the path integral approach this anomaly was studied earlier for the Lagrange BRST transformations [ 51 in refs. [6,7]. In this paper, in the framework of the functional approach, we discuss the BRST quantization of a simpler model which provides an analog to the bosonic string. In this model the fields defined on twodimensional euclidean space R2 take their values in the finite-dimensional Lie group G. Taking the gauge group of the theory to be a subgroup H c G, the physical degrees of freedom are elements of the homogeneous space M=G/H [8]. The central point of this paper is the investigation of the BRST Ward identity. Due to the fact that the ghost action is quadratic in ghosts and antighosts it is possible to verify the WI explicitly. We show that in different gauges the WI may be both anomalous and non-anomalous. In particular, in the gauge which can be viewed as a counterpart of the conformal gauge gag=&, the WI proves to be anomalous. As in the case of string theory the anomaly is shown to be a consequence of the non-invariance of the integration domain over antighosts with respect to BRST transformations. We discuss also an example of the gauge in which the WI has no anomaly. To explain the difElsevier Science Publishers
B.V. (North-Holland)
in BRST Ward identities (WI). We construct gauge with non-anomalous WI. The source of
ference between these gauges it should be noted that the operator P in the ghost lagrangian L, = CPc in the first case acts from the Lie algebra al H into the algebra al G and has zero modes, while in the second case it acts from al H into al H and has no zero modes. In string theory the same property has the harmonic gauge. 2. Let G be a compact semi-simple Lie group with the algebra al G, H c G be a subgroup of G with the algebra al H and M = G/H be a homogeneous space of the group G. For sake of definiteness we consider G as a matrix group. Let {LA} be a left-invariant basis of al G normalized by the condition Sp LALB= -S,,. Left-invariant forms are defined as U-‘dU=QAL,; U= exp ( uALA). The metric on the group is Sp( U-‘dU) We introduce
(U-‘dU)+
=QAQB6,,
the connection
D, U= a, U-k UA, ,
.
(1)
on G
A,, =A, ‘h,
(2)
where the {hi} form a basis for al H, and we construct the chiral lagrangian [ 5 ] L=TSp(U-‘D,U)(U-‘D,U)+ =T(S2,*S,*+29,‘A,‘+A,‘A,‘).
(3)
3. Next we consider the BRST quantization of the theory with the lagrangian (3). The infinitesimal gauge transformation is 6U= U6a = Uk’hi
.
(4) 41
Volume 247, number 1
PHYSICS LETTERSB
We introduce the ghost fields ci(z) and define BRST transformations of the chiral field U(z) and of the ghost as
sU= Uc= Uc% , sc'= - l-2f'jkcJck
(5)
where f gjk are the structure constants of the algebra al H. Now we introduce a gauge condition. By analogy with the conformal gauge in string theory ga~~ob= 0 (~,abis a reference metric, gab is a dynamical metric; the metrics g and ~ have the same Teichmtiller parameters and ~ defines a slice in the space of metrics) one can take the gauge condition U - O= 0, where 0 are elements of G parametrized as 0 = exp (l,u ~) with la tangent to M. Here the chiral field U corresponds to the dynamical metric g, the coordinates u ~ in the space M correspond to the Teichmtiller parameters, and the group H plays the role of the gauge group Diff~
s#(z) = i b ( z )
(6)
we have L ~ = s Sp [e( U - O) ] = Sp[ib( U - 0) ] - S p ( e U c ) .
(7)
Any element U~G can be uniquely written in the form U= Oexp(y~hg). On the Lie group the gauge condition U - 0 = 0 admits the unique solution y~=0. For small values of yg the gauge-fixing term may be written as Sp lib( U - 0 ) ] = --ibASp(lA Uhj) yi
=- ibAPAyj ,
(8)
where PAj is a rectangular matrix. The ghost lagrangian takes the form
Lgh=Sp(eUc)=e~Pajc j , P~j=Sp(lAUh~).
(9)
4. Now we are in a position to write down the expression for the generating functional for the Green 42
6 September 1990
functions. The invariant metric (1) generates the Haar measure d # ( U ) . The generating functional is
Z [ J , . . . ] = f DI~(U) [c,c,b] ×exp(-~
d2z(Lc, +Lgf + L j ) ) .
(10)
Here
Ls=UAJA +eAfln+ ~],Ci •
( 11 )
D u (U) is the functional measure corresponding to the Haar measure d/z(U) and D [g, c, b] is the functional measure to be specified later. Let P+ be adjoint to the operator P in (9) acting as
f d2z FaPAjfJ= f d2z p+jAFaf J ,
(12)
where F A, fJ are any integrable functions on ~2. Since Pai is a rectangular matrix, the equation P +jAFA= 0 has r ~ (dim G - d i m H) solutions. In the same way as in the string theory it is convenient to use the field O(z) as a source for the auxiliary field b(z) [6]. However, since the coordinates u a (a = 1.... , dim M) of the fields U and 0 in M coincide, only gauge but not complete variations of 0 make sense. Making a gauge variation of the field O: 8 0 = Of y= 06yihi, one obtains ( 8 / 6 y 0 f d2z Sp [b( U - 0) ] = - P ~ b ". This expression means that the field 0 may be used as a source only for the component bs of the auxiliary field orthogonal to k e r P +. To obtain all the components of the field b(z) we modify the gauge-fixing term of the action: Sgr = s f d 2z{Sp [g( U - 0) ] + T"¢ ~ ( F ~ - P ~ ) }, (13) Here the T ~ are constant sources, {F~} and {P~} form the basis for k e r P + and k e r P + correspondingly. Varying eq. ( 13 ) with respect to T ~ one obtains the projections 6~ = f d 2z bAFaA of the auxiliary field on kerP +
8Sgf - - i6, + ~ d 2z (ibaF~ -- cASFaA ) 8T ~
( 14 )
Turning offthe sources, i.e. at S j = 0 , T " = 0 , the generating functional Z[ J, ...] coincides with the partition function. Since the components ¢, = fdZz (AF, A
Volume 247, number 1
PHYSICS LETTERS B
of the antighost do not enter the action in (10) (see (9) ), in order to obtain a non-zero expression for the partition function one must integrate over the fields (A orthogonal to k e r P +. In other words, decomposing the antighost as (A =gA +g,~F,A, one must require the measure D[(, c] to be of the form D [ & , c] (dg~) R((~), where fR(e,~) de~¢0. Here the indices are raised and lowered by the flat Killing metric OAB. At small values o f y i up to terms of order y2 o n e can take in (8) PAj instead of PAj. The components b~ do not enter the action and to obtain a finite partition function one must take the functional measure D[b] in the form D [ b ] = D [ b + ] (db=) N(b,~), where f N(b~) db~ < ~ . One can express b A and g$ as b A =pAin,; 6A =pAi#~ and integrate over all the values of n ~ and f~ without any further restrictions. In these variables the generating functional (10) reads
Z = f D # ( U ) D [ f ' , #, n'] X e x p ( - S ¢ l - f d2z[in'P~KAjyJ-f~P,AKajc j
+O(y2)l+Sj),
(15)
where
KAi =Pro + To,( S F ~ / SY i) ly=o.
(16)
Integrating over n ~ one obtains 8(y ~) (recall that U= 0exp(yih~)). Determinants appearing from integration over n ~and F, # cancel each other. Thus, we get
Z= f Dp( U) 6(y) exp(-ScI + ~ d2z uAJA) X exp(W),
(17)
where
W= - f d 2z [7~M~AflA-- (fiM~nF~A) ( • - l ) ,,fl,]. (18) Here M ~A is the Green function of the operator Km ( 16 ) satisfying the equations KAiMJA:(~i j , KAiMiBm(~A B--SaASaB ,
(19)
where the {S~} form a basis of k e r K +. As T ' ~ 0
6 September 1990
KAi , M jA and S ~ become PAt, N iA and F~A and satisfy equations similar to (19). In (18) we denoted
~ o = f d 2z F~ASn o, flu= f d 2z fla F A a. 5. Now we shall derive the BRST WI. Since we are interested in the ghost part of the action, we shall put JA = 0. This simplification is quite possible because the WI must be valid at all the values of the sources. Making the change of functional variables corresponding to the BRST transformations of the fields and recalling that Scl + Sgh is BRST invariant we have ( sgj ) = f d 2z ( - #ifijkCJC k + ibaflA )
= t2(oo)Z ,
(20)
Here (...) denotes functional average, the subscript (op) indicates that the corresponding operator is expressed in terms of functional derivatives with respect to the sources. In the case anomaly is absent one has ~(op)Z= O. A non-zero expression for t2(op)Z means that the functional measure is not BRST-invariant i.e. that the WI is anomalous. We argue that in our case ~2(op)Z¢ 0. This fact is established by a direct calculation analogous to that in string theory [ 7 ]. For the sake of simplicity we shall put T ~ = 0 in our final expressions. Now we shall outline the scheme of this calculation. One has
d Zz b~oo)flA - b (op)fl~ 5
+ d2zflA(z) NiA(z) 5yi(z~--~,
(21)
where b(aop) is defined using (14). Substituting this expression in (20) and finding the action of functional derivatives on W with the use of eqs. ( 19 ) for the Green functions, one obtains ~2(op)Z = fl,(b(%p)- f d2wdZz#k(w) NkA(w)
x aP~(w)
6yi(z)
Nm(z) fiB(z)) Z.
(22)
Here the caret over an operator means that the corresponding expression refers to the field U. The RHS of eq. (22) can be written as 43
Volume 247, number 1
fla f d 2 z
S(t~AFaA ) (op)Z .
PHYSICS LETTERS B
(23)
Thus, we see that the WI is anomalous; at JA = T " = 0 the anomaly is given by (23). The source o f the anomaly (23) is the same as in string theory [ 7 ]. One integrates effectively over the antighosts restricted by the condition fd2zcAFaA=O i.e. over ~A ± k e r P + . But this condition is not BRST-invariant or, in other words, the effective integration domain changes under the BRST transformations. 6. From the preceding discussion it can be seen that the anomaly is independent of a particular choice o f the measure but results from the form o f the gauge condition. Usually, in field theory in which a Lie group H acts freely in the space of fields ~ the gauge conditions f~(x)=0 ( a = l, ..., d i m H ) are introduced by functions f a (x) mapping ~ into ~dirn H [ 9 ]. In our case, even in the vicinity of the slice y i = 0 one has U - U = O6yihiCHand thus the gauge condition does not define a map from G into ~dim H. This observation suggests that one can obtain a non-anomalous WI in another suitably chosen gauge. For this sake consider the gauge h - h ~ = 0 where hEH is defined by the equation U = Oh, and h is a fixed element in H. In this gauge the operator Po in the ghost action is a square matrix and thus has no zero modes (except for certain submanifolds of zero measure in group space). One can derive the WI in this gauge and ver-
44
6 September 1990
ify that in this case the anomaly is absent. The same argument can be carried over to string theory. An example of such a gauge is given by harmonic gauge ~a~a (gbc--½gb~Pqgpq)= 0; R (g) = R (~) where ~ is a fixed metric with a constant negative curvature [ 10 ]. Repeating the calculations similar to the case of the chiral model, one can verify that in this gauge the WI is non-anomalous. Details of this investigation will be reported elsewhere
References [ 1 ] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, Vols, 1, 2 (Cambridge U.P., Cambridge, 1987). [2] E.D'Hoker and D.H. Phong, Rev. Mod. Phys. 60 (1988) 917. [3] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [4] H. Sonoda, Phys. Lett. B 184 (1987) 336. [5] L. Baulieu, Phys. Rep. 129 (1985) 1. [ 6] P. Mansfield, Nucl. Phys. B 283 (1987 ) 55 I. [ 7 ] A.P. Demichev and M.Z. Iofa, Phys. Lett. B 236 ( 1990 ) 17. [8] H. Eichenherr and M. Forger, Nucl. Phys. B 155 (1979) 381. [9]B.S. De Witt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965 ). [ 10] D.Z. Freedman, J.I. Latorre and K. Pilch, Nucl. Phys. B 306 (1988) 77; L. Baulieu, W. Siegel and B. Zwiebach, Nucl. Phys.B 287 (1987) 93.
PHYSICS
1
LETTERS
6 September
B
1990
Gauge fixing: chiral model versus string theory A.P. Demichev and M.Z. Iofa Institute for Nuclear Physics, Moscow State University, SU-I 19 899 Moscow, USSR Received
27 April 1990
The chiral model is discussed as an analog to the string theory to study anomalies a counterpart of the conformal gauge rendering the anomalous WI and also another the anomaly in the chiral model is shown to be the same as in the string theory.
1. The BRST formalism provides a convenient and effective method to quantize gauge theories. This method was applied successfully to string theory also, for example, to define the physical sector of the theory, to study decoupling of spurious, etc. [ l-41. However, in conformal gauge in high orders of string perturbation theory the BRST symmetry proves to be anomalous. In the path integral approach this anomaly was studied earlier for the Lagrange BRST transformations [ 51 in refs. [6,7]. In this paper, in the framework of the functional approach, we discuss the BRST quantization of a simpler model which provides an analog to the bosonic string. In this model the fields defined on twodimensional euclidean space R2 take their values in the finite-dimensional Lie group G. Taking the gauge group of the theory to be a subgroup H c G, the physical degrees of freedom are elements of the homogeneous space M=G/H [8]. The central point of this paper is the investigation of the BRST Ward identity. Due to the fact that the ghost action is quadratic in ghosts and antighosts it is possible to verify the WI explicitly. We show that in different gauges the WI may be both anomalous and non-anomalous. In particular, in the gauge which can be viewed as a counterpart of the conformal gauge gag=&, the WI proves to be anomalous. As in the case of string theory the anomaly is shown to be a consequence of the non-invariance of the integration domain over antighosts with respect to BRST transformations. We discuss also an example of the gauge in which the WI has no anomaly. To explain the difElsevier Science Publishers
B.V. (North-Holland)
in BRST Ward identities (WI). We construct gauge with non-anomalous WI. The source of
ference between these gauges it should be noted that the operator P in the ghost lagrangian L, = CPc in the first case acts from the Lie algebra al H into the algebra al G and has zero modes, while in the second case it acts from al H into al H and has no zero modes. In string theory the same property has the harmonic gauge. 2. Let G be a compact semi-simple Lie group with the algebra al G, H c G be a subgroup of G with the algebra al H and M = G/H be a homogeneous space of the group G. For sake of definiteness we consider G as a matrix group. Let {LA} be a left-invariant basis of al G normalized by the condition Sp LALB= -S,,. Left-invariant forms are defined as U-‘dU=QAL,; U= exp ( uALA). The metric on the group is Sp( U-‘dU) We introduce
(U-‘dU)+
=QAQB6,,
the connection
D, U= a, U-k UA, ,
.
(1)
on G
A,, =A, ‘h,
(2)
where the {hi} form a basis for al H, and we construct the chiral lagrangian [ 5 ] L=TSp(U-‘D,U)(U-‘D,U)+ =T(S2,*S,*+29,‘A,‘+A,‘A,‘).
(3)
3. Next we consider the BRST quantization of the theory with the lagrangian (3). The infinitesimal gauge transformation is 6U= U6a = Uk’hi
.
(4) 41
Volume 247, number 1
PHYSICS LETTERSB
We introduce the ghost fields ci(z) and define BRST transformations of the chiral field U(z) and of the ghost as
sU= Uc= Uc% , sc'= - l-2f'jkcJck
(5)
where f gjk are the structure constants of the algebra al H. Now we introduce a gauge condition. By analogy with the conformal gauge in string theory ga~~ob= 0 (~,abis a reference metric, gab is a dynamical metric; the metrics g and ~ have the same Teichmtiller parameters and ~ defines a slice in the space of metrics) one can take the gauge condition U - O= 0, where 0 are elements of G parametrized as 0 = exp (l,u ~) with la tangent to M. Here the chiral field U corresponds to the dynamical metric g, the coordinates u ~ in the space M correspond to the Teichmtiller parameters, and the group H plays the role of the gauge group Diff~
s#(z) = i b ( z )
(6)
we have L ~ = s Sp [e( U - O) ] = Sp[ib( U - 0) ] - S p ( e U c ) .
(7)
Any element U~G can be uniquely written in the form U= Oexp(y~hg). On the Lie group the gauge condition U - 0 = 0 admits the unique solution y~=0. For small values of yg the gauge-fixing term may be written as Sp lib( U - 0 ) ] = --ibASp(lA Uhj) yi
=- ibAPAyj ,
(8)
where PAj is a rectangular matrix. The ghost lagrangian takes the form
Lgh=Sp(eUc)=e~Pajc j , P~j=Sp(lAUh~).
(9)
4. Now we are in a position to write down the expression for the generating functional for the Green 42
6 September 1990
functions. The invariant metric (1) generates the Haar measure d # ( U ) . The generating functional is
Z [ J , . . . ] = f DI~(U) [c,c,b] ×exp(-~
d2z(Lc, +Lgf + L j ) ) .
(10)
Here
Ls=UAJA +eAfln+ ~],Ci •
( 11 )
D u (U) is the functional measure corresponding to the Haar measure d/z(U) and D [g, c, b] is the functional measure to be specified later. Let P+ be adjoint to the operator P in (9) acting as
f d2z FaPAjfJ= f d2z p+jAFaf J ,
(12)
where F A, fJ are any integrable functions on ~2. Since Pai is a rectangular matrix, the equation P +jAFA= 0 has r ~ (dim G - d i m H) solutions. In the same way as in the string theory it is convenient to use the field O(z) as a source for the auxiliary field b(z) [6]. However, since the coordinates u a (a = 1.... , dim M) of the fields U and 0 in M coincide, only gauge but not complete variations of 0 make sense. Making a gauge variation of the field O: 8 0 = Of y= 06yihi, one obtains ( 8 / 6 y 0 f d2z Sp [b( U - 0) ] = - P ~ b ". This expression means that the field 0 may be used as a source only for the component bs of the auxiliary field orthogonal to k e r P +. To obtain all the components of the field b(z) we modify the gauge-fixing term of the action: Sgr = s f d 2z{Sp [g( U - 0) ] + T"¢ ~ ( F ~ - P ~ ) }, (13) Here the T ~ are constant sources, {F~} and {P~} form the basis for k e r P + and k e r P + correspondingly. Varying eq. ( 13 ) with respect to T ~ one obtains the projections 6~ = f d 2z bAFaA of the auxiliary field on kerP +
8Sgf - - i6, + ~ d 2z (ibaF~ -- cASFaA ) 8T ~
( 14 )
Turning offthe sources, i.e. at S j = 0 , T " = 0 , the generating functional Z[ J, ...] coincides with the partition function. Since the components ¢, = fdZz (AF, A
Volume 247, number 1
PHYSICS LETTERS B
of the antighost do not enter the action in (10) (see (9) ), in order to obtain a non-zero expression for the partition function one must integrate over the fields (A orthogonal to k e r P +. In other words, decomposing the antighost as (A =gA +g,~F,A, one must require the measure D[(, c] to be of the form D [ & , c] (dg~) R((~), where fR(e,~) de~¢0. Here the indices are raised and lowered by the flat Killing metric OAB. At small values o f y i up to terms of order y2 o n e can take in (8) PAj instead of PAj. The components b~ do not enter the action and to obtain a finite partition function one must take the functional measure D[b] in the form D [ b ] = D [ b + ] (db=) N(b,~), where f N(b~) db~ < ~ . One can express b A and g$ as b A =pAin,; 6A =pAi#~ and integrate over all the values of n ~ and f~ without any further restrictions. In these variables the generating functional (10) reads
Z = f D # ( U ) D [ f ' , #, n'] X e x p ( - S ¢ l - f d2z[in'P~KAjyJ-f~P,AKajc j
+O(y2)l+Sj),
(15)
where
KAi =Pro + To,( S F ~ / SY i) ly=o.
(16)
Integrating over n ~ one obtains 8(y ~) (recall that U= 0exp(yih~)). Determinants appearing from integration over n ~and F, # cancel each other. Thus, we get
Z= f Dp( U) 6(y) exp(-ScI + ~ d2z uAJA) X exp(W),
(17)
where
W= - f d 2z [7~M~AflA-- (fiM~nF~A) ( • - l ) ,,fl,]. (18) Here M ~A is the Green function of the operator Km ( 16 ) satisfying the equations KAiMJA:(~i j , KAiMiBm(~A B--SaASaB ,
(19)
where the {S~} form a basis of k e r K +. As T ' ~ 0
6 September 1990
KAi , M jA and S ~ become PAt, N iA and F~A and satisfy equations similar to (19). In (18) we denoted
~ o = f d 2z F~ASn o, flu= f d 2z fla F A a. 5. Now we shall derive the BRST WI. Since we are interested in the ghost part of the action, we shall put JA = 0. This simplification is quite possible because the WI must be valid at all the values of the sources. Making the change of functional variables corresponding to the BRST transformations of the fields and recalling that Scl + Sgh is BRST invariant we have ( sgj ) = f d 2z ( - #ifijkCJC k + ibaflA )
= t2(oo)Z ,
(20)
Here (...) denotes functional average, the subscript (op) indicates that the corresponding operator is expressed in terms of functional derivatives with respect to the sources. In the case anomaly is absent one has ~(op)Z= O. A non-zero expression for t2(op)Z means that the functional measure is not BRST-invariant i.e. that the WI is anomalous. We argue that in our case ~2(op)Z¢ 0. This fact is established by a direct calculation analogous to that in string theory [ 7 ]. For the sake of simplicity we shall put T ~ = 0 in our final expressions. Now we shall outline the scheme of this calculation. One has
d Zz b~oo)flA - b (op)fl~ 5
+ d2zflA(z) NiA(z) 5yi(z~--~,
(21)
where b(aop) is defined using (14). Substituting this expression in (20) and finding the action of functional derivatives on W with the use of eqs. ( 19 ) for the Green functions, one obtains ~2(op)Z = fl,(b(%p)- f d2wdZz#k(w) NkA(w)
x aP~(w)
6yi(z)
Nm(z) fiB(z)) Z.
(22)
Here the caret over an operator means that the corresponding expression refers to the field U. The RHS of eq. (22) can be written as 43
Volume 247, number 1
fla f d 2 z
S(t~AFaA ) (op)Z .
PHYSICS LETTERS B
(23)
Thus, we see that the WI is anomalous; at JA = T " = 0 the anomaly is given by (23). The source o f the anomaly (23) is the same as in string theory [ 7 ]. One integrates effectively over the antighosts restricted by the condition fd2zcAFaA=O i.e. over ~A ± k e r P + . But this condition is not BRST-invariant or, in other words, the effective integration domain changes under the BRST transformations. 6. From the preceding discussion it can be seen that the anomaly is independent of a particular choice o f the measure but results from the form o f the gauge condition. Usually, in field theory in which a Lie group H acts freely in the space of fields ~ the gauge conditions f~(x)=0 ( a = l, ..., d i m H ) are introduced by functions f a (x) mapping ~ into ~dirn H [ 9 ]. In our case, even in the vicinity of the slice y i = 0 one has U - U = O6yihiCHand thus the gauge condition does not define a map from G into ~dim H. This observation suggests that one can obtain a non-anomalous WI in another suitably chosen gauge. For this sake consider the gauge h - h ~ = 0 where hEH is defined by the equation U = Oh, and h is a fixed element in H. In this gauge the operator Po in the ghost action is a square matrix and thus has no zero modes (except for certain submanifolds of zero measure in group space). One can derive the WI in this gauge and ver-
44
6 September 1990
ify that in this case the anomaly is absent. The same argument can be carried over to string theory. An example of such a gauge is given by harmonic gauge ~a~a (gbc--½gb~Pqgpq)= 0; R (g) = R (~) where ~ is a fixed metric with a constant negative curvature [ 10 ]. Repeating the calculations similar to the case of the chiral model, one can verify that in this gauge the WI is non-anomalous. Details of this investigation will be reported elsewhere
References [ 1 ] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, Vols, 1, 2 (Cambridge U.P., Cambridge, 1987). [2] E.D'Hoker and D.H. Phong, Rev. Mod. Phys. 60 (1988) 917. [3] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [4] H. Sonoda, Phys. Lett. B 184 (1987) 336. [5] L. Baulieu, Phys. Rep. 129 (1985) 1. [ 6] P. Mansfield, Nucl. Phys. B 283 (1987 ) 55 I. [ 7 ] A.P. Demichev and M.Z. Iofa, Phys. Lett. B 236 ( 1990 ) 17. [8] H. Eichenherr and M. Forger, Nucl. Phys. B 155 (1979) 381. [9]B.S. De Witt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965 ). [ 10] D.Z. Freedman, J.I. Latorre and K. Pilch, Nucl. Phys. B 306 (1988) 77; L. Baulieu, W. Siegel and B. Zwiebach, Nucl. Phys.B 287 (1987) 93.