Volume 246, number 1,2 PHYSICS LETTERS B 23 August 1990

Morphisms between supersymmetric and topological quantum field theories *

Roger Brooks and David Kastor Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA

Received 27 May 1990

We find that topological invariants, isomorphic to Donaldson polynomials, exist in chiral superfield theories. Twists between

these invariants and those of the corresponding TQFT are given. In the topological sigma model, anticommuting charges of

integer spin are found which together with the BRST charge, fill out a D = 2, N = 2 supersymmetry algebra.

Supersymmetric quantum mechanics has been used

to study the topology of manifolds. For example, the calculation of the index Tr(-)F leads to proofs of the Atiyah-Singer index theorem [l] and the Morse inequalities [2]. This connection between topology and quantum field theory has been furthered through the introduction of a class of theories known as topo-

logical quantum field theories (TQFTs) [3]. Corre- lators of observables in these theories depend on the

topological class of the field configuration under con- sideration (i.e. on the choice of in and out vacua). They are independent of the differentiable structure of the manifold. In chiral superfield theories (xSTS) a similar situation arises. In ref. [4] it has been shown that the correlators of the lowest components of chiral

superfields are independent of the positions of the operators.

In this letter we will discuss how to make the

connection between XSTS and TQFTs precise. We will look at the examples of D = 2, N = 2 supersym- metric non-linear sigma model and D =4, N=2 supersymmetric Yang-Mills theory.

TQFTs can be constructed as the BRST gauge fixing

of a local shift (topological) symmetry under which only the homotopy class of a field configuration is preserved [5,6]. Schematically, one has S%(X) = 4”(x). In the BRST quantized theory, 9 carries

* This work is supported by the Department of Energy, con-

tract DE-AC03-76SF00515.

opposite Grassmann statistics to 9. If 9 is a commut- ing boson then 9 is Grassmann odd. In one and two space-time dimensions where spin is of no physical consequence, the anticommuting ghost f, fermion identification becomes meaningful. Then the topo- logical transformation is suggestive of a supersym- metry transformation.

The actions of TQFTs are of the form STOP= I, [Q, K}. M is the space-time manifold. The quan- tity K depends on the metric on the manifold through the definition of inner products of the fields. These fields constitute positive and negative ghost number multiplets under a BRST charge, Q. A BRST invariant

vacuum is constructed by inserting ghost zero-modes into the naive vacuum as is done in string theory. By differentiating the partition function Z with respect to the metric one finds

for some AOb. Correlation functions of the metric

independentobservables in the theory may be shown [3] to be topological invariants. This proof employs the BRST invariance of the vacuum and the transfor- mation laws of the various fields. It also assumes that the measure of Z is independent of the metric.

Let us now look at supersymmetric field theories. We quickly find an analogous result to eq. (1). Con- sider, for example, D = 4, N = 1 supersymmetry with

0370-2693/90/$03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland) 99

Volume 246, number 1, 2 PHYSICS LETTERS B 23 August 1990

supercharge Q~. It is well known that the energy- momentum tensor may be written as T~b=

i!6 tr(y(~r[(~, ~b)}) SO that for a supertranslationa--lly invariant vacuum, (T~b)= 0 [7]. As in (1), one takes Z to be metric independent . Next consider a chiral superfield theory with b a ~ = 0 . Let ~ [ = ¢ b and D ~ I = @~. Then for correlat ion functions of ~b:

a ~ ( 4 , ( x , ) . . . 4,(x,,))

= < , / , ( x , ) - - . [ 0 ~ , ¢,~} • • • ,/,(x,,)>

=0 . (2)

This result holds for general chiral superfields [4]. As the metric on the manifo ld was not used, and the part i t ion function is metric independent , such a corre- lation function is a topological invariant.

The presence of topologica l invariants stems from

the fact that the ene rgy-momentum tensors of these theories are given by the action of the supercharge on something. A general argument proceeds as fol- lows. Cons ider an action for bosonic and fermionic fields with canonical kinetic terms. Let the action, S, be constructed to be invariant under the action of a charge Q: [Q, S ] = 0 . Also require that the energy- momentum tensor, T,,b, may be written as T,b = [Q, A,,b}. This is immediate if S = [Q, g*} for some gauge fixing fermion ~. Applying the rules of canoni- cal quant izat ion we arrive at

[ ~ , 4)} = - i v o 4,, (3)

where M=- Y x R ~, • is a generic field and Vo is covariant with respect to some local symmetry which will not concern us. Construct an observable, O, which is a singlet of the local symmetry and is independent of the metric on M. Furthermore, let it be a C k function in any of the fields 4) for which [Q, ~} = 0 and for k large enough. It then follows that

a_~(O) = ([iPo, O})= ([i½Q, [()~, O}}) = 0. (4)

Thus the VEV of O is independent of the differenti- structure on the manifold M. In general, it will depend on the topological class of M. Note that if Ao, is a conserved current, eq. (3) yields a supersymmetry algebra, albeit unconventional . In supersymmetry we will see that the mechanism which allows for

topological invariants [more general than (2)] is the existence of a propagat ing field which transforms chirally under the supersymmetry algebra.

Given that topological invariants exist in xSTs, we are led to inquire if the latter are related to the observables of TQFTs. By relat ionship we will mean a morphism, termed a " twis t" S-, which makes the diagram

~TOP ~r ) °~SUP

l l gr

TQFT ) N = 2×ST

commutative. Here ¢WTOp¢SUP) is the space of topologi- cal invariants constructed as correlat ion functions in the TQFT (×ST). Henceforth, we will refer to this diagram as the twist diagram. We remark that 9- is purely a mathematical operat ion. Its physical inter- pretat ion, if any, has yet to be unveiled.

The physical states o f a TQFT [8] comprise a BRST complex and are the ground states of the hamil tonian. They will be in one to one correspondence with the ground states of the corresponding supersymmetr ic theory.

We will now illustrate these general remarks with examples in D = 2 and D = 4.

D=2. At tree level, topological sigma models [3] (TSMs) can be formulated on an arbitrary almost complex manifold. In the case of K~ihler manifolds, there is a relat ion to N = 2 supersymmetr ic sigma models. Prior to our work, this was realized by twist- ing the two-dimensional Lorentz algebra [U(1)] with the internal U(1) of the N --2 algebra and discarding two of the four supercharges. This gave a construct ion of the world-sheet scalar BRST charge in terms of two of the remaining supersymmetry generators [3,9]. We find that the TSM is invariant under vector charges which form a full supersymmetry algebra with the BRST charge. Furthermore, certain corre- lators in the supersymmetr ic theory are shown to be topological and may be twisted into those of the TSM.

The superspace action for an N = 2 model for maps from the world-sheet to a K~ihler manifold M is Ssup-- ~ I d 2 o - d40K(@, ~ ) , where the @~ are chiral superfields and K(@, ~ ) is the K~ihler potential on

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M. This action reduces to the component expression

Ssu P = f d2o r (g,j O~dp' O~ ~bz-i½g~j ¢~@t ¢'

~ ' -= o=6' + a~6~rJ~ ~ ,

~ ¢ ' --- at ¢ ' +at CKF;,~ ¢'. (5)

The N = 2 supersymmetry algebra in two dimensions contains two left-handed supercharges Q±, two right-handed supercharges t~±. We use the standard superconformal notation with the + indices denoting R-symmetry weights and the barred/unbarred denot- ing handedness. In this way the charges satisfy the algebra

[Q+, Q_} =i 2 a=, [(~+, (~_} =i 2 at, (6)

where all other commutators are zero. Ssup is invariant under the non-trivial N = 2 supersymmetry transformations for the left-handed fields

[ Q + , ~ ' } = i ¢ ~, [Q_ ,¢r}=icr ,

[Q_,¢'}=2G~', [Q+,¢r}=2GCr,

[Q+, @'}= F' , [ Q - , O r } = F r,

F' = i F~K~J~ K. (7)

The transformations for the right-handed fields are obtained by interchanging barred and unbarred indices. The actions of the Lorentz U(1) generator J and the U(1) R-symmetry generators on the fields can be summarized in the form ~(R,/~, J),

4,(0, o, o); ~,'(1, o, ½); ¢r(-1 , O, ~);

q~' (0, 1, -½); ~ r (0 , -1 , -½),

Q+(1, 0,½); t~+(O, 1, -½); Q_(-1, 0,½);

(~_(0, -1 , -½). (8)

The topological sigma model action for maps from the world-sheet, .Y, to a K~ihler manifold M [3] is given by

SToP = f d2o" (g~i G qb I Ot c/r r-'ll~gl.r p zZ ~ X i

-l~gtyP~=X -7~R~jKCX X P= Pc), ~ t X s = - atx~+a~ C e I ~ c x c ,

O - r F ~ L (9) ~ X *=-3zX ~ + ~ ~:LX •

The fields X and p are anticommuting world-sheet scalars and vectors, respectively. The TSM action is invariant with respect to the following non-trivial BRST transformations

[Qt, 6 r }= ix r, [Qr,~b'}=ix' ,

[Q,,p'=}=2a:6', [Qr, ptz} = -IFjKX" ' JPzK,

[Q,,p[}=-irjex¢p~, [Qr, p[}=2ae~b r. (10)

We have found that the action is also invariant under the new vector charges Qz and Q~ with

[Q~,6r}=ip[, [Q=, q~'} = ip'=,

[Qt,x '}=2a~ch ', [Q~,x'}=iFJKx~p~,

[Qt,xr}=irfxxi#~, [Qz,xT}=2O=ck r. (11)

The BRST and vector charges satisfy the supersym- metry algebra

[O,, O=} =i 2 0=, [O,, Or} =i 2 at, (12)

with all other anticommutators being zero. In par- ticular each Q is nilpotent. This is an on-shell closure of the algebra. Presumably, with the introduction of the BRST auxiliary field, this algebra will close off- shell much as a supersymmetry algebra closes off- shell with the appropriate introduction of auxiliary fields. The left and right ghost number (~3t, ~3,) and Lorentz (J) weights of the various fields and operators are as follows with the notation ¢'(~,, ~r, J):

~b(O, O, 0); pzX(-1, O, 1); pC(O, -1 , -1) ;

X'(0, 1, 0); Xr(1, O, 0),

Q,(1, 0, 0); Qr(0, 1, 0); Qt(O, -1 , -1) ;

Q=(-1, o, 1). (13)

Now re-define the Lorentz generator to be

j,__- j + ½(~ _ cg,), (14)

and identify R --- -~1 and/~ --- ~,. Then eqs. (12) and (13) become identical to eqs. (6) and (8), respectively, with the renamings

p'~-= ¢', pl--- ~ , x ' --- ~', x ~--- ¢~,

Q,~-Q-, Qr-=0+, Q t ~ O - , Q,=-Q+. (15)

One also verifies that STOO~ Ssuv.

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We have defined the map in the lower branch of the twist diagram. Let us now find the topological invariants of the N = 2 supersymmetric sigma model. One can show that, for example, that given an element A~y of H(~'I)(M), the operator

O1o) 1'1) ---- aly ~bl ~ ], (16)

is an element of a Q+@ Q_ cohomology. (This means it is a twisted chiral operator in the language of ref. [10].) Our notation is ta(R,a) where k denotes the real ,./(k) degree of O on Z with exterior derivative d. Further- more, the ascent equations

d O l o ) I'1) = i [ Q _ , ~.J (l)[)(°'l)/J - i [ 0 + , O~1)~'°)}, Ol°il)=-Atyd~ t d~b -r, O1~-)1,1) =- Ary ~b Y d~b ',

dO(O,,)_-i[(~+, r~(o,o)l ,,./(1 ) - - v ( 2 ) J,

dOlh 1'°)= - i [ Q + , r~(o,o)~. ~J(2) J,

O(0,o) = Aty dtb I ^ d~b y, (17) (2) - -

imply that the correlation functions of k-cycle integrals of the O operators are independent of the points on the manifolds. Eqs. (16), (17) may be gen- eralized to elements of H(P'q)(M). In fact given such a form, there are two complexes of operators O which may be formed. For example, in addition to O~o) l'l),

/"~(1,--1)= Atygdq~y which is an element there is also '-'co) - of a Q+~)Q_ cohomology. Under the twisting, the set of operators O (O) maps into the BRST (anti- BRST) observables of ref. [3] with positive (negative) ghost number. Thus at least at tree level, the twist diagram commutes. Also note that eq. (16) is the zero- momentum limit of the vertex operator for the axion- like field in the M a x K compactification of the superstring [11].

Having discussed the two-dimensional theories, we now make the following remarks about the D = 1 topological sigma model [12] and its relationship to N = 2 supersymmetric quantum mechanics. The lower branch of the twist diagram between topologi- cal quantum mechanics and N = 2 supersymmetric quantum mechanics was discussed at length in ref. [8]. The supercharges were realized as linear combi- nations of the BRST and anti-BRST charges. Based on this or by dimensional reduction of our D = 2 results, it is trivial to see that there are topological invariants of the form O~=A ....... (x) II~=l q~' in both theories. The a~ are vector indices on M and

Ak ~ Hk(M). The fields ~0 "~ are ghosts for the TQFT and fermions for the supersymmetric theory and are interpreted as sections of the pull-back ch*(TM) of the tangent bundle on M ( x c d~: R ~ M). The twist of ref. [8] relates them and allows us to conclude that the twist diagram is commutative.

D - - 4. Gaugino condensates of D = 4, N = 1 SYM are known to form topological invariants [4]. However, these invariants are unrelated to the Donaldson polynomials of TYM. The relationship is between D = 4, N = 2 SYM and TYM. The twisting, at the lagrangian level, between these theories has been discussed in refs. [3,13]. We will show that topological invariants exist in the untwisted super- symmetric theory. If one chooses, these may be twisted to the Donaldson polynomials.

The D =4 , N = 2 SYM multiplet [14,15] contains one spin-1 field, A~, two spin-0 fields, C and C*, two spin-½ fields, A_~ and an auxiliary field, Bah. Apart from the gauge group, the bosons are singlets under a rigid, internal SU(2)I group for which the gaugino is a doublet (labelled by the " a " index, g --- aa) and B is a triplet. Additionally, the theory is invariant under a rigid U(1) group for which the charges of the set of fields (Ao, (7, A~) are (0, 2, 1). The super- symmetry transformation laws which will be relevant to our discussion are

[Q~, Fob} = io'[_,l~d ~Io]A ~,

[ Q~, A~_} = io'~,~@~C6~,

[0_~, C} =0. (18)

Let us, for simplicity, take the Yang-Mills gauge group to be SU(2). Take the object 12(0)---Tr(C2). From eq. (18) one immediately sees that [O-~, O(o)} = 0. Given this, the following ascent equations may be derived:

dO(o) = [Q-~, ~'~(1)~},

d~(2) = [ (~-~, ~(3)_~},

d~(3)_~ = [Q_rt , ~(4)},

dO(,)~ = [Q-~, O(2)t~_~},

d/~(4) = 0, d$'~(4)~/j = 0,

where

O(o) = Tr(C2),

O(I)_~ = - i ½ Tr(cr~A-" C) dx g,

(19)

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A O(2)~_ ~ ~ ~'~(2)C_&0 --F O ( 2 ) ~ ,

g~(2) = ~ Tr(F~b C + ½A~cr~bA ~) dx -~ A dx °,

~(2)_a~ = ~ T r ( c r ~ A ~Av b) dx ° A dx b,

.0~3)_~ = i ~6 Tr(F,b %~aA -~) dx a A dx ~ A dx -c,

=i ~2 Tr(F.~C~(~l~la)a ~(b Co)~) dx ~ ̂ dx b ̂ dxL

~(4) = ~ Tr(F.b F~a) d xo ^ dx -b ̂ dx -~ ̂ dx -~,

~(~(4)_&~ = 2~(~(4)C_&~ • (20)

Here CoO is the complex conjugate of the anti- symmetric symbol of SL(4, C) ~ SU(2)®SL(2, C) defined by C_~_~ ~ Ca~ cab. o.b is defined as tr~b~ ----= 40"[~[t~&O'_b] with {tr~,~rb}=-2~7~6~. These ex- pressions were derivable, in part, because C is chiral. It is the lowest component of a chiral superfield (in N = 2 superspace), W [15]: ~7_~W=0. The entire geometry o f D = 4, N = 2 SYM superspace may be specified in terms of this superfield and its hermitian conjugate. This leads us to conjecture that eqs. (19), (20) (which carry topological information only) may be obtained from Tr( W :) by the standard superspace projection techniques in the spirit of the construction of the Donaldson polynomials for TYM given in ref. [8]. Furthermore, although we have not checked it, we do expect that the g2(o) may be generalized to Tr (C 2") for a rank n group. There will be a similar generalization for the remaining g2(k)'S. The U(1) charge of each O(k) is (4--k) . The space, ~Wsup, of topological invariants for D = 4, N = 2 SYM is com- posed of various Lorentz scalar products of homology k-cycles of the O(k)'S in an analogous manner to the construction of the Donaldson maps given in ref. [3]. It would be interesting to understand the connection between these invariants, moduli space, supersym- merry breaking, etc.

The twisting to D = 4 TYM is achieved by first identifying the SU(2)~ index a with the dotted SU(2)R index d. That is, with SO(4)~SU(2)L®SU(2)R as the Lorentz group on the four-manifold (with euclidean signature), the symmetry group of D = 4 , N = 2 supersymmetry SO(4)®SU(2)I®U(1) becomes SU(2)L®SU(2)D®U(1) [3]. Here SU(2)D is the diagonal part of SU(2)a®SU(2)~. This means that the supercharge becomes Q_~ = Qa a ~ O~a. The latter is then identified as the anticommuting, scalar

BRST charge. Similarly, the spin- t gaugino field is identified as an anticommuting vector field, qJ_~, through the twisting ~ A -~ = ~ A "a-~ ~ A ~ Oa. Accordingly, the supersymmetry transformations (18) become

[Q, Fab} = i@E_~ ~1 , [Q, ~a} = - ~ _ ~ ,

[Q, ~b} = 0, (21)

where we have identified the spin-0 field C as the commuting, scalar field ~b of the TYM ghost multiplet. The ghost numbers of the TYM fields are the same as the U(1) charges of the corresponding fields in the SYM muitiplet. As the construction of the Donaldson polynomials is based solely on eq. (21), it is clear that the twisting leads to them. One may also explicitly check that under this twisting, eq. (20) leads to the Donaldson polynomials of the TYM theory. For example, the most complicated expression in (20), namely/2(3)_~_~_~ evaluates to (up to a normalization) Tr(i~b ^ F) which is W~3) in ref. [3]. Thus we have found the twisting 3 : °Wsup~ ~WTow from the space of topological invariants of D = 4, N = 2 SYM to the corresponding space in TYM. Given the construction, in ref. [13], of the D = 4 TYM action from that of D = 4, N = 2 SYM, we have found strong evidence that the twist diagram is commutative.

In taking only the diagonal sum of the SU(2)'s, we have disposed of the extra charges. However, these may be recycled in the following way. Generally, with the identification of the SU(2)I index as a SU(2)R index, we will have O_~ Q@Q~b and Q_~ Q~ where Qah is antisymmetric and self-dual. Based on our experience with the vector charges of the TSM (sec- tion 3), we expect that these will also generate a symmetry of the TSM theory and form a supersym- metry algebra.

Our discussion of the twist diagram has been at the tree level. The TSM is conformally invariant at the one-loop level only if the target manifold is Ricci scalar flat [16]. Ricci flat Kfihler manifolds meet this criterion. This is the condition for the vanishing of the one-loop fl-function of the N = 2 supersymmetric sigma model [17]. This constraint on the geometry of the target manifold of the TSM, a theory which is supposed to be topologically invariant, is surprising. It arises from the metric dependence of the measure of the partition function. Nevertheless, it suggests

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Volume 246, number 1, 2 PHYSICS LETTERS B 23 August 1990

that the twist d i ag ram is c o m m u t a t i v e also at o n e - l o o p

o rde r for Ricci fiat K~hle r mani fo lds . Fo r the D = 4

T Y M theory, the o n e - l o o p / 3 - f u n c t i o n for the gauge

coup l ing i s / 3 ( g ) = -[2g3/(4~)z]ca2'~J(G) [6]. This is

exact ly the o n e - l o o p / 3 - f u n c t i o n o f D = 4, N = 2 pure

S Y M (no hypermul t ip le t s ) [18] and the result also

suggests that the Y a n g - M i l l s twist d iagram is c o m m u -

tat ive at o n e - l o o p order. In conc lus ion , we w o u l d like to stress the fo l lowing

points . There are genera l classes o f topo log ica l

invar iants in the D = 2, N = 2 non - l i nea r s igma m o d e l

and pure D = 4, N = 2 SYM. These invar iants are

i s o m o r p h i c to the genera l i zed D o n a l d s o n poly-

nomia l s o f TQFTs . Twists be tween the respec t ive

×SFTs and T Q F T s have been defined. G r o u n d states

o f ×SFTs are in one to one c o r r e s p o n d e n c e with the

phys ica l states o f the co r r e spond ing TQFTs . It is an

in teres t ing ques t ion as to h o w the twist ing p r o c e d u r e

i m p l e m e n t s the res t r ic t ions on the phys ica l Hi lber t

spaces.

We thank J. Louis for useful discussions.

References

[1] L. Alvarez-Gaum6, Commun. Math. Phys. 90 (1983) 161; D. Friedan and P. Windey, Nucl. Phys. B 235 (1984) 395.

[2] E. Witten, J. Diff. Geom. 17 (1982) 661. [3] E. Witten, Commun. Math. Phys. 117 (1988) 353; 118

(1988) 411. [4] V. Novikov, M. Shifman, A.I. Vainshtein, M.B. Voloshin

and V.I. Zakharov, Nucl. Phys. B 229 (1983) 394; V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 229 (1983) 384, 407; G.C. Rossi and G. Veneziano, Phys. Lett. B 138 (1984) 195;

D. Amati, G.C. Rossi and G. Veneziano, Nucl. Phys. B 249 (1985) 1; S. Cecotti, L. Girardello and A. Pasquinucci, Nucl. Phys. B 328 (1989) 701.

[5] L. Baulieu and I.M. Singer, Nucl. Phys. B (Proc. Suppl.) 5B (1988) 12; J.M.F. Labastida and M. Pernici, Phys. Lett. B 212 (1988) 56; D. Birmingham, M. Rakowski and G. Thompson, Nucl. Phys. B 315 (1988) 577.

[6] R. Brooks, D. Montano and J. Sonnenschein, Phys. Lett. B 214 (1988) 91.

[7] S. Ferrara and B. Zumino, Nucl. Phys. B 87 (1975) 207; B. Zumino, Nucl. Phys. B 89 (1975) 207.

[8] R. Brooks, Nucl. Phys. B 325 (1989) 481. [9] E. Witten, On the structure of the topological phase of

two dimensional gravity, IAS preprint # IASSNS-HEP- 89/66 (December 1989).

[10] S.J. Gates Jr., C.M. Hull and M. RoLek, Nucl. Phys. B 248 (1984) 157.

[11] X.-G. Wen and E. Witten, Phys. Lett. B 166 (1986) 397. [12] L. Baulieu and B. Grossman, Phys. Lett. B 212 (1988) 351;

D. Birmingham, M. Rakowski and G. Thompson, Nucl. Phys. B 315 (1989) 577; A. lwazaki, Prog. Theor. Phys. 81 (1989) 1104; J.M.F. Labastida, Commun. Math. Phys. 123 (1989) 641; L. Baulieu and I.M. Singer, Commun. Math. Phys. 125 (1989) 227.

[13] A. Karlhede and M. Ro~ek, Phys. Lett. B 212 (1988) 51; J. Yamron, Phys. Lett. B 213 (1988) 325; A. Galperin and O. Ogievetsky, Holonomy groups, com- plex structures and D=4 topological Yang-Mills theory, Karlsruhe preprint KA-THEP-l-1990 (January 1990).

[14] S. Ferrara and B. Zumino, Nucl. Phys. B 79 (1974) 413; P. Fayet, Nucl. Phys. B 113 (1976) 135.

[15] R. Grimm, M. Sohnius and J. Wess, Nucl. Phys. B 133 (1978) 275.

[16] D. Montano and J. Sonnenschein, Nucl. Phys. B 313 (1989) 258.

[17] L. Alvarez-Gaum6, D.Z. Freedman and S. Mukhi, Ann. Phys. 134 (1981) 85.

[18] P.S. Howe, K.S. Stelle and P.C. West, Phys. Lett. B 124 (1983) 55.

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