April, 1988
IASSNS-HEP-88/13
A GAUGE INVARIANT ACTION IN
TOPOLOGICAL QUANTUM FIELD THEORY
J. M. F. LABASTIDA † ∗ and M. PERNICI †
The Institute for Advanced Study
Princeton, NJ 08540, USA
ABSTRACT
A gauge invariant lagrangian whose BRST gauge fixing corresponds to
the theory recently proposed by Witten in the framework of topological quantum field theory is presented. The solutions to the field equations corresponding to this lagrangian consist of self-dual Yang-Mills fields.
†
Research supported by U. S. DOE contract DE-AC02-76ER02220
∗
On leave from Instituto de Estructura de la Materia, CSIC, Serrano 119, 28006 Madrid, Spain
In recent years we have witnessed an important development in the study of the topology
of low dimensional manifolds due to the works of Donaldson[1,2] and Floer[3] . These works
make use of the self-dual Yang-Mills equations and, in particular, Floer’s work has been
interpreted by Atiyah[4] in terms of a non-linear field theory involving Yang-Mills fields. The
relation observed between Donaldson’s approach applied to manifolds with boundary and
Floer’s theory has led Atiyah to conjecture that the field theory involved in his interpretation
of Floer’s theory must be an approximation of a relativistic quantum field theory which
may provide a field theory interpretation of Donaldson’s theory. Witten has presented this
relativistic field theory in a recent paper[5] , introducing the framework of topological quantum
field theory[5,6] . He has shown that the Donaldson invariants can be obtained via a path
integral formulation of his theory. Witten’s action possesses a BRST invariance and so it
must correspond to the BRST quantized form of a gauge invariant theory. To find such a
gauge invariant theory is an important matter because, on one hand, we may gain insight
on the structure of actions involved in topological quantum field theory and, on the other
hand, we may find a very useful tool to compute Donaldson invariants.
In this letter we present a lagrangian which leads to Witten’s lagrangian after BRST
quantization. Our construction is inspired by the work done in ref.[7] , where quadratic
actions for self-dual fields in D = 4n + 2 dimensions in Minkowski space are considered. In
particular, it is shown there that to a pair of self-dual tensor fields a Kähler-Dirac ghost is
associated, following the approach of ref.[8] . The motivation of that work was to obtain the
gravitational anomaly corresponding to a pair of self-dual tensors from an action principle.
In the construction presented here we take the formulation in ref.[7] and we write it in D = 4
Euclidean dimensions. It corresponds to one-forms or vector fields. We keep only one of
1
the vector fields in the pair, and we interpret the self-dual part of the field-strength of the
other one as an auxiliary field. Once this auxiliary field is integrated out, the resulting
theory contains a gauge invariance which upon BRST gauge fixing leads to the formulation
presented in ref.[5] .
The BRST quantization that will be carried out in this paper uses the Batalin-Vilkovisky
algorithm[9] . In the process we find similar patterns to the ones found in the BRST quantization of the Freedman-Townsand model[10] and Witten’s open string field theory[11] . These
two theories have been also BRST quantized using a heuristic modified Faddeev-Popov procedure (in ref.
[12]
and
[13]
, respectively).
Our BRST quantization will consider all the gauge invariances present in the theory.
Witten’s formulation[5] , however, though it corresponds to a BRST quantized theory, possesses still Yang-Mills gauge invariance, i.e., the theory is only partially BRST fixed. One
should do a further BRST quantization corresponding to this invariance to obtain the full
BRST quantized theory. We will show, using the Batalin-Vilkovisky algorithm[9] , that the
BRST quantization of the action presented in this paper and the full BRST quantized theory
proposed by Witten are equivalent. Actually, we are able to formulate a BRST quantized
theory whose corresponding BRST transformations are nilpotent off-shell, contrary to the
case of Witten’s formulation which possesses nilpotency only on-shell. The presence of an
additional auxiliary field permits the realization of nilpotency off-shell. Of course, after
integrating out this field, this formulation and Witten’s become identical.
Let us consider a gauge group G and let Aα be the corresponding Yang-Mills connection
with field strength, Fαβ = [Dα , Dβ ], where Dα is the ordinary covariant derivative (Dα φ =
2
∂α φ+[Aα , φ]). Our starting point is the following lagrangian in D = 4 Euclidean dimensions,
1
1 (+)
(+)
Lclas =Tr (Fαβ F αβ + Fαβ F̃ αβ ) + ( Gαβ − Fαβ )G(+)αβ
4
2
1
(+)
(+)
= Tr (Fαβ − Gαβ )(F (+)αβ − G(+)αβ ) ,
2
(+)
(+)
(1)
(+)
where Gαβ is an auxiliary field antisymmetric in its Lorentz indices (Gαβ = −Gβα ). In (1)
we use the notation
(+)
(−)
Aαβ =Aαβ + Aαβ ,
1
(±)
Aαβ = (Aαβ ± Ãαβ ),
2
1
Ãαβ = αβγδ Aγδ ,
2
(2)
so the auxiliary field Gαβ is algebraically self-dual:
Gαβ =
1
(+)
αβγδ Gγδ = Gαβ .
2
(3)
The lagrangian (1) is invariant under the following gauge transformations† ,
δAα = − Dα ω + α ,
δGαβ
1
=D[α β] + αβγδ Dγ δ + [ω, Gαβ ],
2
(4)
where ω is the the usual G gauge parameter and α is another gauge parameter. The
lagrangian (1) also possesses a global scale invariance where the scaling dimensions for the
fields (Aα , Gαβ ) are (1, 2).
In (1) the topological invariant term
1
8
Tr F ∧ F may be dropped without disturbing
any of the considerations that follow, since such a term, being topological invariant, is
certainly invariant under both types of gauge transformations. However, we will include
it in our formulation to obtain the theory that is used in ref.[5] to deal with Donaldson
invariants.
†
In our conventions A[αβ] = 12 (Aαβ −Aβα )
3
So far we have been assigning the role of auxiliary field to Gαβ . The reason for this is
that from (1) Gαβ has an algebraic field equation. However, from (4) one may conclude that
it is some kind of gauge field. This type of field appears typically in lagrangians for chiral
bosons[14] , although in that case one has cubic terms in the action. A quadratic version in
that context was studied in ref.[15] that in fact motivated the work[7] that has inspired the
construction presented here.
Let us analyze the degrees of freedom in the lagrangian (1). The Gαβ field equation is
(+)
Fαβ − Gαβ = 0. The gauge transformations (4) are just enough to choose a gauge where
Gαβ vanishes leaving untouched the parameters corresponding to the G gauge symmetry.
Though the field Gαβ has 3 components, while the gauge parameter α has four, not all of
these components are effective regarding gauge invariance. The lagrangian (1) with gauge
transformations (4) possesses an on-shell gauge invariance of the gauge invariance. Namely,
if α = Dα Λ and ω = Λ, both fields, Aα and Gαβ , remain invariant on-shell under gauge
transformations:
δAαβ =0,
δGαβ on-shell
(+)
=[Fαβ − Gαβ , Λ]
(5)
on-shell
= 0.
In this gauge the solutions to the Euler-Lagrange equations from (1) are self-dual Yang-Mills
fields and they still possess full Yang-Mills gauge invariance.
Our next task is to carry out the BRST quantization of (1). The naive Faddeev-Popov
procedure can not be applied to this theory because the second generation gauge invariance
observed in (5) is realized only on-shell. We will use the Batalin-Vilkovisky algorithm[9] . In
their classification, the theory at hand corresponds to a first-stage reducible theory due to
the presence of the second generation gauge invariance observed in (5). The first step in
the BRST quantization procedure consist of introducing ghost fields corresponding to the
4
different gauge invariances: α → ψα , ω → c and Λ → φ. The first two are Grassmann odd
and possess ghost number +1; the third one, since it corresponds to the second generation, is
Grassmann even and has ghost number +2. These ghost fields and the classical fields Aα and
Gαβ , whose ghost number assignment is zero, are collectively called Φamin . For each field in
Φamin a corresponding antifield, which has opposite statistics and ghost number equal to the
ghost number of the field minus 1, is introduced. These antifields are (A∗α , G∗αβ , ψα∗ , c∗ , φ∗ ),
have ghost numbers (-1,-1,-2,-2,-3), and are collectively denoted by Φa∗
min .
The main idea of the Batalin-Vilkovisky algorithm consists of the construction of an
action-like object, S(Φa , Φa∗ ), where Φa and Φa∗ denote collectively sets of fields which
a
a∗
include Φamin and Φa∗
min respectively. S(Φ , Φ ) has ghost number zero and satisfies the
master equation,
∂r S ∂l S
∂r S ∂l S
−
= 0,
∂Φa ∂Φa∗
∂Φa∗ ∂Φa
(6)
where the left and right derivatives are relevant for differentiation with respect to Grassmann
odd fields. In addition, S(Φa , Φa∗ ) must satisfy the following boundary conditions,
a
S(Φ , 0) = Sclas =
∂l ∂r S =(Dα δx )ij ,
j
∂Ai∗
∂c
α
∗
Φ =0
∂l ∂r S = − iδαβ δ ij δx ,
j ∂Ai∗
∂ψ
α
β Φ∗ =0
∂l ∂r S = − (Dα δx )ij ,
∂ψαi∗ ∂φj ∗
Φ =0
d4 x Lclas ,
(7)
∂l ∂r S = − f ijk Gkαβ δx ,
j
∂Gi∗
∂c
αβ
∗
Φ =0
ij
∂l ∂r S 1
σ ω
=
−
i
D
δ
+
D
δ
,
αβσω
[α
β]γ
γ δx
j
2
∂Gi∗
αβ ∂ψγ Φ∗ =0
∂l ∂r S = − δ ij δx ,
∂ci∗ ∂φj ∗
Φ =0
(8)
where i, j, k are group-theoretical indices which label the adjoint representation of the gauge
group G and f ijk are the structure constants. The boundary conditions (8) are obtained
5
from the form of the gauge transformations. In this way, the first four conditions in (8)
correspond to the gauge transformations (4) while the last two correspond to the second
generation on-shell gauge invariance observed in (5)‡ . In addition to the construction of
S(Φa , Φa∗ ) a gauge fixing function, Ψ, must be chosen, which consists of fields multiplying
the selected gauge conditions. These fields (antighosts and extraghosts fields) belong to the
set Φa . No antifields enter in Ψ. This gauge fixing function must be non-degenerate i.e.,
there must be no redundant gauge conditions. In first-stage reducible theories this function
is typically degenerate and one is forced to introduce a new field called extraghost to make
Ψ non-degenerate. As we will see below our gauge conditions are such that there is not need
for the extraghost.
Once the object S(Φa , Φa∗ ) is constructed and the gauge-fixing function Ψ chosen, the
quantum action is given by
Sq = S(Φa ,
δΨ
).
δΦa
The BRST transformations for the fields in Φa are,
S
∂
r
δB Φa = a∗ ∂Φ a∗
Φ
(9)
,
(10)
δΨ
= δΦ
a
where is the constant Grassmann odd BRST parameter. From this we see that the BRST
transformations of the fields whose corresponding antifield does not enter in S(Φa , Φa∗ )
are zero. The algorithm guaranties invariance of the quantum action (9) under (10) and
nilpotency on-shell of the BRST transformations (10).
The construction of S(Φa , Φa∗ ) involves first a minimal part which deals only with Φamin
a
a∗
a
a∗
and Φa∗
min : S(Φmin , Φmin ). The way to solve for S(Φmin , Φmin ) verifying (6)-(8) consists of
‡
In the BRST fixing described here we have introduced constant factors as the i’s and minus signs in (8) to
converge to the notation in ref.[5] . All amounts to field redefinitions by constant factors respect to the ordinary
BRST procedure.
6
writing down an expansion of S(Φamin , Φa∗
min ) in powers of antifields with arbitrary coefficients
involving the fields, and with ghost number zero. The solution turns out to be
a
a∗
S(Φmin , Φmin ) = Sclas + d4 x Tr A∗α (Dα c − iψ α ) − G∗αβ (2iDα ψ β + [c, Gαβ ])
i
− ψα∗ (Dα φ − {c, ψ α }) − c∗ (iφ − cc) − φ∗ [c, φ] − {G∗αβ , Gαβ∗ }φ .
2
(11)
Notice that a term quadratic in antifields appears in (11). The origin of this term is related
to the fact that the BRST transformations that one would obtain using (10) and considering
only the linear terms in the antifields in (11) are not nilpotent off-shell (as one could have
guessed from (5)). As a consequence, the master equation is not satisfied when having only
linear terms in the antifields in (11). Notice that, at linear level in antifields, from (10)
follows that the master equation (6) vanishes if the BRST transformations from (10) were
nilpotent off-shell. The quadratic term in antifields in (11) is enough to cancel the terms
which are left in the master equation from the linear part.
Next, we must choose the gauge-fixing function Ψ. To do this one first introduces
antighost fields which enter in Ψ in a linear form and whose coefficients correspond to the
gauge-fixing conditions. Our choice of gauge fixing-conditions consists of the one discussed
above for Gαβ and two Lorentz-type conditions. Namely, we choose Ψ to be,
Ψ=
1 αβ
1
χ Gαβ + b∂ α Aα − λDα ψα ,
2
2
(12)
where the antighost fields are: χαβ , which is self-dual Grassmann odd with ghost number
-1; b, which is Grassmann odd with ghost number -1; and λ, which is Grassmann even with
ghost number -2. The gauge-fixing function Ψ has ghost number -1. Since there are not
redundant gauge conditions in (12), the gauge fixing function Ψ is non-degenerate and so
we do not need to introduce the extraghost which is usually present in first-stage reducible
7
theories. This ends the minimal construction in the Batalin-Vilkovisky algorithm. Now,
using (9) and (10) one can obtain the minimal form of the quantum action and the BRST
transformations.
The Batalin-Vilkovisky algorithm guarantees BRST invariance of the quantum action
and nilpotency on-shell of the BRST transformations. One can verify that this is indeed the
case for the minimal construction consisting of S(Φamin , Φa∗
min ) in (11) and the gauge fixing
function (12). However, the algorithm possesses some freedom in constructing S(Φa , Φa∗ ).
One can always introduce new fields in Φa , new antifields in Φa∗ , and add a new part to
S, say, ∆S, satisfying the master equation. The standard choice[9] for ∆S is a product of
antifields associated to the antighost fields and Lagrange multipliers (the π’s in ref.[9] ). These
Lagrange multipliers impose the gauge-fixing conditions after varying them in the quantum
action (9). However, one is free to add to this standard choice other terms as long as (6) is
satisfied. In fact, to identify our BRST formulation with Witten’s theory we must add new
terms in ∆S.
Let us introduce antifields χ∗αβ , b∗ and λ∗ associated to the antighosts fields χαβ , b and λ.
The antifields χ∗αβ and b∗ are Grassmann even with ghost number 0, while λ∗ is Grassmann
odd with ghost number 1. Corresponding to these antifields we introduce the Lagrange multipliers fields dαβ , d and η. The first two are Grassmann even with ghost number 0, and η is
Grassmann odd with ghost number -1. The field dαβ is self-dual. Furthermore, a Lagrange
multiplier antifield, η ∗ , Grassmann even and with ghost number 0, is also introduced. Collecting all the fields, the full Φa is (Aα , Gαβ , ψα , c, φ, χαβ , b, λ, dαβ , d, η) with ghost numbers
(0, 0, 1, 1, 2, −1, −1, −2, 0, 0, −1), while the full Φa∗ is (A∗α , G∗αβ , ψα∗ , c∗ , φ∗ , χ∗αβ , b∗ , λ∗ , η ∗ ) with
ghost numbers (−1, −1, −2, −2, −3, 0, 0, 1, 0). Notice that the fields dαβ and d do not have
8
antifields and so their BRST transformations will be zero. Given this new field content we
find that
∆S =
(+)
d4 x Tr χ∗αβ dαβ + b∗ d − 2iλ∗ η + χ∗αβ 2(Fαβ − Gαβ ) + {c, χαβ }
1
− λ [c, λ] + η ( [φ, λ] + {c, η}) ,
2
∗
(13)
∗
satisfies the master equation (6).
The quantum action is easily obtained using (9). We find from (1), (11) and (13),
Sq = d4 x Lq = S(Φamin , Φa∗
min ) + ∆S
1
1
(+) (+)αβ
4
αβ
Fαβ F
− Gαβ G
+ iDα ψβ χαβ − iηDα ψα + λDα Dα φ
= d x Tr
2
2
i
i
1
αβ
α
αβ
α
α
α
+ Gαβ d − λ{ψ , ψα } − φ{χ , χαβ } + d∂ Aα − ib∂ ψα + b∂ Dα c .
2
2
8
(14)
The BRST transformations are derived using (10) from (11) and (13),
δB Aα =(iψα − Dα c),
i
1
δB Gαβ =i D[α ψβ] + αβγδ Dγ ψ δ + [χαβ , φ] + [c, Gαβ ],
2
2
(+)
δB χαβ =dαβ + 2(Fαβ − Gαβ ) + {c, χαβ },
δB dαβ =0,
δB ψα =(−Dα φ + {c, ψα }),
δB φ =[c, φ],
δB λ =(2iη + [c, λ]),
1
δB η =( [φ, λ] + {c, η}),
2
δB c =(cc − iφ),
δB b =d,
δB d =0.
9
(15)
One can verify by explicit computation that the transformations (15) are actually nilpotent
on-shell as guaranteed by the Batalin-Vilkovisky algorithm. The field equations must be
used to achieve nilpotency on the fields Gαβ and χαβ .
The BRST quantized theory obtained in (14) and (15) is equivalent to the one proposed
by Witten[5] once the BRST quantization concerning the G gauge symmetry is carried out in
this last theory. It is simple to observe that BRST gauge fixing Witten’s theory in the Lorentz
gauge, ∂ α Aα = 0, one obtains the last three terms in (14) and the BRST transformations
(15), except for the presence of the fields Gαβ and dαβ . Notice that the last three are the
typical of Yang-Mills except for the term −iφ in the transformation of the ghost. All the
other fields get new terms in their BRST transformations corresponding to the identification
ω → c. However, our BRST quantized theory (14) and (15) contains two fields, Gαβ and
dαβ , which are not present in Witten’s formulation. Using the field equations of Gαβ and
dαβ in (14) and (15) we obtain a formulation which is identical to Witten’s. In this case
the BRST transformations are nilpotent off-shell for all the fields except on χαβ . The field
equation of ψα must be used to obtain nilpotency on that field.
The BRST transformations (15) are nilpotent off-shell except for terms involving the
field dαβ in the double transformations of Gαβ and χαβ . If one sets dαβ = 0 in (14) and
(15) one obtains a BRST formulation whose BRST transformations are nilpotent off-shell.
However, one can not integrate out dαβ without integrating out Gαβ . While the Gαβ -field
equation is dαβ = 2Gαβ , the dαβ -field equation is Gαβ = 0, and so both fields are set to zero.
Nevertheless, it is important to notice that taking dαβ = 0 and keeping Gαβ in (14) and (15)
one obtains a BRST formulation whose BRST transformations are nilpotent off-shell.
The BRST quantized theory in (14) and (15) with dαβ = 0 is a very unusual one. It is
10
simple to verify using (15) that the quantum lagrangian in (14), can be written as Lq =
(+)
δB Θf = {Q, Θf }, where Θf = Tr 14 χαβ (Fαβ + Gαβ ) − 12 λDα ψα + b∂ α Aα . Since the last term
in Θf corresponds to the ones originating the last three terms in (14), i.e., from the BRST
gauge fixing of the G gauge symmetry, it follows that Witten’s lagrangian, after adding the
(+)
extra field Gαβ can be written as δB Θ = {Q, Θ}, where Θ = Tr 14 χαβ (Fαβ +Gαβ )− 12 λDα ψα .
This property may be related to the fact that the classical action (1) vanishes if one integrates
out the auxiliary field.
As we discussed above, the classical lagrangian (1) is invariant under global scale transformations. This invariance should be maintained through the BRST quantization. In fact,
this is the case. The scaling dimensions of the fields (Aα , Gαβ , χαβ , ψα , φ, λ, η, c, b, d) are
(1, 2, 2, 1, 0, 2, 2, 0, 2, 2).
The Batalin-Vilkovisky algorithm possesses some freedom in choosing ∆S above, as well
as the gauge-fixing function Ψ. The only requirements are that ∆S must satisfy the master
equation (6) and that Ψ must not contain redundant gauge conditions. Here, our choices (12)
and (13) lead to the equivalence between the BRST quantized theory from (1) and Witten’s
theory. Other choices are certainly possible. The simplest one consists of (12) and a product
of antifields, corresponding to the antighost fields, and Lagrange multipliers (the first three
terms in (13)). However, there are other choices which lead to extensions of Witten’s full
BRST quantized theory (they contain some extra auxiliary fields) and to a different set of
BRST transformations. It would be interesting to know which choices are better suited to
carry out the analysis on Donaldson invariants discussed in ref.[5] .
We would like to conclude with some final remarks. Since the classical action (1) van
We would like to thank E. Witten for pointing this out to us.
11
ishes when the auxiliary field is integrated out there are not physical degrees of freedom in
the theory. The only excitations can be topological. This naive argument must be made
more precise by studying the path-integral measure and by taking into account the gauge
invariance. The BRST gauge fixing constitutes a way of dealing with this and, as shown in
ref.[5] , only the ground state is a Q-physical state. There are no physical excitations in the
BRST fixed theory and only zero modes may contribute to the path integral.
Since the classical lagrangian is essentially zero, it is reasonable that the quantum lagrangian can be written as a BRST transformation. It will be interesting to characterize the
type of theories where this phenomena occur. Certainly, it is not common to all the systems
containing no physical excitations. One may conclude this from the form of the quantum
action corresponding to the N = 2 spinning string or some systems involving chiral bosons.
In those cases one has a cancellation of degrees of freedom similar to the one in the theory
at hand when constructing the Q-cohomology. However, the quantum lagrangian can not be
written as a BRST transformation.
Additionally, also based on the fact that the classical lagrangian is essentially zero one
may argue that the theory is invariant under metric deformations. In the quantum lagrangian
this follows from the simple fact that since it can be written as a BRST transformation, the
corresponding energy momentum tensor must have the same property.
It would be very interesting to study other types of theories which contain no physical
degrees of freedom but just zero modes. It is straightforward to generalize this theory to
dimension D = 4n in the abelian case. For instance, for D = 8, one would have a classical
lagrangian L = (F (+) − G(+) )2 , with F = dA, and A a three-form. The auxiliary field can
be eliminated using the gauge invariance following a similar procedure to the one in ref.[7] .
12
In the BRST gauge fixing one finds sets of ghost and auxiliary fields which can be arranged
in bispinors (Kähler-Dirac ghosts).
In D = 2 a model similar to the one considered in this paper was studied in ref.[15] in
Minkowski space. It was noted there that the model was physically unacceptable because
of the presence of a field with the wrong sign in the kinetic term. If one continues such a
field to obtain the right sign of the kinetic term one would expect to violate CPT. In the
second paper of ref.[6] , Witten has studied two-dimensional models which correspond to the
non-linear realization of the model described above. In these models the kinetic terms have
the right sign but there is a CPT violating term. This identification leads us to conjecture
that Witten’s two-dimensional instantons are non-linear generalizations of chiral bosons, and
as such they might have applications in string theory. We intend to pursue these ideas in
due course.
We would like to thank E. Witten and D. Zanon for very useful discussions.
13
References
[1] S. Donaldson, An Application of Gauge Theory to the Topology of Four Manifolds,
J. Diff. Geom. 18 (1983) 279; The Orientation of Yang-Mills Moduli Spaces and 4Manifold Topology, J. Diff. Geom. 26 (1987) 397; Polynomial Invariants for Smooth
Four Manifolds, Oxford preprint
[2] D. Freed and K. Uhlenbeck, Instantons and Four Manifolds, Springer-Verlag, 1984
[3] A. Floer, An Instanton Invariant for Three Manifolds, Courant Institute preprint (1987)
[4] M.F. Atiyah, New Invariants of Three and Four Dimensional Manifolds, to appear in
the Symposium on the Mathematical Heritage of Hermann Weyl (University of North
Carolina, May 1987) ed. R. Wells et al.
[5] E. Witten, Topological Quantum Field Theory, IAS preprint, IASSNS-HEP-87/72,
February 1988
[6] E. Witten, Topological Gravity, IAS preprint, IASSNS-HEP-87/2, February 1988; Topological Sigma Models, IAS preprint, IASSNS-HEP-87/7, February 1988
[7] J.M.F. Labastida and M. Pernici, Phys. Lett. 200B (1988) 501
[8] I.A. Batalin and E.S. Fradkin, Nucl. Phys. B279 (1987) 514.
[9] I.A. Batalin and G.A. Vilkovisky, Phys. Rev. D28 (1983) 2567
[10] S.P. de Alwis, M.T. Grisaru and L. Mezincescu, Brandais preprint, October 1987, BRX
TH-235.
[11] M. Bochicchio, Phys. Lett. 192B (1987) 253, 188B (1987) 332]
[12] S.P. de Alwis, M.T. Grisaru and L. Mezincescu, Phys. Lett 190B (1987) 122
[13] C.B. Thorn, Nucl. Phys. B287 (1987) 61
[14] W. Siegel, Nucl. Phys. B238 (1984) 307
14
[15] J.M.F. Labastida and M. Pernici, Nucl. Phys. B297 (1988) 557
15