Dual versions of higher-dimensional supergravities and anomaly cancellations in lower dimensions

Nuclear Physics B268 (1986) 532-542 © North-HoUand Publishing Company

D U A L V E R S I O N S OF H I G H E R - D I M E N S I O N A L SUPERGRAVITIES AND ANOMALY CANCELLATIONS IN L O W E R DIMENSIONS* Hitoshi N I S H I N O and S. James GATES, Jr.

Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742, USA 27 August 1985 Supergravity theories with higher-rank tensors dual to the ordinary second-rank tensors (dual versions) are given in D = 8 and 6. The special role of higher-rank tensors in the dual versions for anomaly cancellations in lower even dimensions is studied by inspecting examples of D = 10, N = 1 dual version compactified to D = 6, and D = 8, N = 1 dual version compactified to D = 4. The difference of dual versions from the ordinary versions in higher-dimensional supergravities is elucidated.

1. Introduction

Ten-dimensional N - - 1 (open and closed) superstring theories [1] have become promising for realistic model building, after the remarkable discovery of anomaly cancellations for the gauge group SO(32) or Es x Es by the Green-Schwarz mechanism [2]. These anomaly cancellations are achieved by the special dimensionality (496) of these groups, and a modified gauge and Lorentz transformation rule of the antisymmetric tensor B~,~ in the supergravity multiplet (e~', ~,, B,,~, X, ~b)

(1.1)

in the field theory limit [2, 3]. It has been recently pointed out in ref. [4] that the supergravity multiplet (1.1) is not the unique one in the low-energy limit, due to ambiguities in the field representations; there can be an alternative version (e~, t/,~,, M~,,...~,6,X, 4,)

(1.2)

in addition to the ordinary one (1.1). The field strength N,,,...,,7-----70tmM,,2...~,1 is essentially a dual form of G,,~p ~ 30t,,B,pl, so that we refer to this alternative version as the "dual version". It is to be emphasized that the dual version is also free of all anomalies, if we suppose the existence of a corresponding superstring theory, and therefore the validity of a modified gauge and Lorentz transformation rule of the M,,,...~ field [4]. Even though the anomaly 12-form of the dual version has the same structure, since the counter-term induced by the superstring effect for the anomaly cancellation * Research supported by NSF grant # PHY 84-16030. 532

H. Nishino, S.J. Gates Jr. / Anomaly cancellations

533

differs from the ordinary theory, we suspect that these two versions are inequivalent to each other at the quantum level. Another noteworthy difference of the dual version from the ordinary version is the absence of a topological restriction for compactifications. In the ordinary version, we have

O=fQdGO=fQ(trR2-~TrF 2)

(1.3)

for the background curvature and Yang-Mills field strength 2-forms on a compact boundary-less internal manifold Q. [5] This was simply the result of the modified field strength G = dB + tO3L-- ~0tO3v, d G = tr g 2 -~0 Tr F 2

(1.4) (1.5)

and Stokes' theorem. (We use here the same notation as in [4].) In the dual version, however, we have the modified field strength

l(I=dM + XT, d b / = ~ Tr F 4 - 721oo(Tr~2~2 -- I _

24oTrF2trR2+~trR4+~E(trR2)2.

(1.6) (1.7)

As far as compactifications into Minkowski space-time are concerned (i.e. vanishing R ' s and F ' s with four-dimensional indices), since d/(/0 is an eight-form, we do not obtain any topological restriction such as eq. (1.3) [4]. This is true even for the local form of eq. (1.3). According to the paper by Candelas et al. [ 6 ] , if we impose the surviving D = 4, N = 1 supersymmetry, the vanishing background value for G is possible [6]. This results in the local condition dGo = tr g 2-~o Tr F 2 = 0.

(1.8)

Thus in the ordinary version there arises a strong restriction on topological compactifications, which has been exhaustively studied by Pilch and Schellekens [7]. In the dual version [4], though the surviving D = 4, N = 1 supersymmetry in some cases reqires the similar constraint No = 0, the corresponding local form dN0 = 0 to eq. (1.8) is an eight-form; it is literally zero for the Minkowski background, as we have seen in eq. (1.7). Therefore, in the dual version we have no such local topological restriction as eq. (1.8) for D = 4, N = 1 surviving supersymmetry. In the ordinary version, the topological restriction (1.3) is important to guarantee the absence of 4-dimensioanl anomalies after compactifications [5]. Therefore, the absence of the topological restriction in the dual version seems contradictory with the anomaly freedom in 4 dimensions or any other lower even dimensions. In this paper, we clarify the low-dimensional anomaly freedom of the dual version by considering the role of the antisymmetric tensor M~,,...~ in the compactification into six dimensions, as an explicit example. As opposed to the assumption by

H. Nishino, S.J. Gates Jr. / Anomaly cancellations

534

Frampton and Yamamoto [8],* we show that the leading anomalies need not be necessarily cancelled. Since this special role of higher rank tensors in other dual versions of even-dimensional supergravities is to be important for anomaly-free realistic model building, we give explicit dual versions in dimensions D = 8 and 6. This paper is organized as follows. In sect. 2, we study the role of antisymmetric tensors in the anomaly-cancellations of the lower-dimensional theory, which is obtained after the compactifications of higher-dimensional dual supergravities. As examples, we see the cases of D = 10, N = 1 compactified into D = 6, and D = 8, N = 1 compactified to D = 4. Concluding remarks are given in sect. 3. In appendices A and B, We give explicit dual versions of D = 8, N = 1 and D = 6, N = 2. The possibility of duality transformation for the scalar (dilation) field 0~, which is universal in higher-dimensional supergravities, is mentioned in appendix C.

2. Higher-rank tensors and lower-dimensional anomaly cancellations In this section, we investigate the problem of lower-dimensional anomaly cancellations after the compactification of anomaly-free dual versions in higher dimensional supergravities. As a first example, we take the anomaly-free dual version of D = 10, N= 1 supergravity coupled to an E8 x Es or SO(32) Yang-Mills vector multiplet, and consider its compactification to six dimensions. According to the argument in ref. [9], the anomaly 8-form for spin 23-and spin ½ in the 6 dimensions, after a compactification on internal 4 dimensions M4 is in general obtained as Is -

1

2(4"rr) 6

+

(trR2_~oTrF2) f

JM4

1 L (tr Rg-

[~TrF2Fg_AtrRgTrF2_~trRgtrR2 ]

Tr Pg)[ Tr F4

(2.1)

(Tr F2)2+ tr R4+

(tr R2)21.

4

where Fo and Ro are background values and are functions only of points in M4, while F and R are functions of 6-dimensional points. In the ordinary D = 10, N = 1 version, the topological restriction (1.3) removes away the second integral o f leading anomalies. This is a desirable feature, because there can be no counter-term to cancel them in 6-dimensions. However, in the dual version, this integral remains, and gives the leading anomalies such as tr R 4 or Tr F 4. This seems contradictory with the anomaly freedom in the original ten dimensions. This paradox can be solved by considering the existence of the 6th rank tensor intrinsic in the original dual version. The consistent anomaly corresponding to /8 of (2.1) can b e written as [9, 10]

f [2(tO~L-~OtO~,y)X4+2(trR2-~TrF2)X~+X~],

(2.2)

* This does not n e c e s s a r i l y i m p l y d i s a g r e e m e n t w i t h t h e i r result, since in t h e i r case n o h i g h e s t - r a n k t e n s o r is p r e s e n t from the outset.

535

H. Nishino, S.Z Gates Jr. / Anomaly cancellations

where c is a constant, and X4-= I

[]Tr(F°2F2)-~tr R~Tr F2-~2tr R2tr R2],

(2.3)

M4

Xs- f

(tr R g - ~ Tr F 4 - r ~ - ( T r Fg)[~(Tr F2)2+~ tr R 4 + 5 ( t r

R2)2],

Ma

(2.4)

with x , = dX3,

8X~ = - d X ~ ,

Xs = dKT,

8x7 = - d X ~ ,

(2.5) tr R 2 =

t~tO3L=

dtO3L ,

--dtO21L ,

(2.6) Tr

F 2 = dto3v ,

•(.03y = - d t o / y

,

under the gauge-Lorentz transformation & If we consider the counter-term Sc=c f M

with the modified gauge-Lorentz transformation [2] 6 M = --2(tO2L--Y6W2v)X4-! 1 1 2(tr R 2 -Y6l Tr F 2)X2i - X~,

(2.7)

(2.8)

the anomaly G is completely cancelled out: 8Sc + G = 0. Accordingly, the modified field strength N of M is N = d M - Z ( t O , L - ~ t O 3 v ) X 4 - 2 ( t r R 2 - ~ Tr F 2 ) X 3 - X ~ .

(2.9)

This N satisfies naturally 8N = 0, d N = -4(tr R 2 - ~ tr FZ)x4 - Xs

(2.10)

= (terms only of field strengths ). • Recall the important fact that even the leading anomaly in Xs is completely removed by the counter-term of M, without any aid of the topological restriction (1.3). This situation is different from the ordinary version, such as discussed in ref. [8]. As another example, we show the anomly cancellation in 4 dimensions after the compactification of D = 8, N = 1 anomaly-free dual version [11].* * According to the argument given in ref. [12], for anomaly-free N = 1 supergravity in D = 8 itself, the chiral SO(2) subgroup of SO(n, 2) in the coset SO(n, 2 ) / S O ( n ) X SO(2) should not be gauged, because of mixed anomalies. (For the details, see ref. [12].) As long as SO(2) chiral subgroup is not gauged, the D = 8, N = 1 supergravity (ordinary as well as dual versions) is anomaly-free. Chiral compactification to D = 4 Minkowski space-time m a y be possible, even without SO(2) chiral gauging, owing to a positive definite potential.

536

H. Nishino, S.J. Gates Jr. / Anomaly cancellations

The 4-dimensional anomaly after the compactification is supposed to have the following general 6-form: 16 = A Tr F a + B tr R 2 Tr F ,

(2.11)

where A and B are constants, whose explicit forms are not essential for our argument. (For a semi-simple gauge group, Tr F = 0.) The corresponding consistent anomaly [9, i0] is G= c

f

1 iB(to2L 1 1 Tr F + to0~vtr R2)], [Ato4v-F

(2.12)

the first term is the leading anomaly. In the original D = 8, N = 1 dual version (see appendix A), we have a 4th rank tensor, which is available in order to cancel the anomaly G. In fact, the counter-term Sc = c I M

(2.13)

with the modified gauge-Lorentz transformation 8M=-Ato~y

l 1 Tr F + to~v tr R 2) --~B(to2L

(2.14)

can cancel G completely: 8so+ G = 0. The modified field strength is now N = d M - Ato3v -- ½ B ( t o 3 L Tr F + to iv tr R2).

(2.15)

We have thus seen that the higher-rank antisymmetric tensor M guarantees the complete cancellation of all anomalies in lower even dimensions. Since topological restrictions no longer play important roles, we also see that much wider •classes of anomaly-free' lower-dimensional theories emerge from the dual versions of higherdimensional supergravities. For example, we have more anomaly-free 6-dimensional models than given in refs. [6], [7] or [9]. 3. Discussions and concluding remarks In this p a p e r we have realized the importance of the higher-rank tensor in the dual versions of higher dimensional supergravities. In both cases of D = 10, N = 1 compactified to D = 6 and D = 8, N = 1 compactified to D = 4, the antisymmetric tensors M~,,...~ and M~,...~,, play a remarkable role in removing the leading (as well as non-leading) anomalies in the resultant lower-dimensional theories. Since there is no topological restrictions for compactifications, dual versions can give rise to wider classes of anomaly-free lower-dimensional theories than ordinary versions. The special roles of higher-rank tensors are universal, and become very important in anomaly-free realistic model building based on dual versions in higherdimensionsal supergravities. In this argument, we have imp!icitly assumed the absence of anomalies in lower dimensions after the compactification o f original anomaly-free theories. This is valid when we keep all the massive modes in the compactified internal space. If we

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H. Nishino, $..I. Gates Jr. / Anomaly cancellations

truncate away all those massive modes, we may see a similar anomaly cancellation only when the antisymmetric tensors survive as the zero modes in the lowerdimensional theory. Since this prescription is opposed to the spirit of Kaluza-Klein supergravity, we did not take this attitude in this paper. In appendix C, we showthe possibility of the duality transformation for the "zero rank", i.e. scalar field ~b. Although presently this duality transformation is not related to the anomaly problem, we show the appearance of interesting new non-polynomial interactions in the dual version. We acknowledge Dr. E. Sezgin for valuable discussions.

Appendix A DUAL VERSIONS OF D = 8, N = 1 SUPERGRAVITY In this appendix, we give explicit lagrangians and transformation rules for dual versions of simple supergravity coupled to vector multiplets in D = 8 which might be of great importance for realistic model building with chiral fermions. In eight dimensions, the coupling of n vector multiplets to the simple supergravity ( N = 1) is performed in ref. [ 11 ]. Following the technique of duality transformations preserving local supersymmetry developed by Nicolai and Townsend [13], we get the dual version with the field content* (e~', ~b,, X, M,I...~,,, A~, ¢~, Aa, or),

(A.1)

where M,,...,, has the field strength N,,...~5Ot,,M~,,...,,~1 dual to the G~,~,--30t~,B~pj+ (Chern-Simons) in the ordinary theory [ 11 ]. Our result for the invariant lagrangian for the dual version is e - l ~ dua! ~R ..L Io~, la'z'PD ,l, I ( r a I~? I12?l'~z'J I 1 --2cr.r 2 D=8, N=I

=

--~otl/~'y

3 -

I.r

a..,,z,~,Fp--~ e 1 --a

I,L

~/ja a

~v ,t

3

-~.le

2

1

/~(t~l.../~ s

ai

+~XY D~(+~A y D,)t -~(0~,tr) - z P , ( A ) P

I.~ai

(A)

-ztXy y ~O~.o~+2JA y y P,,a(A)~b. + ~

6

e-'~ No.~...,~s(+ iA Y9 Yt;~Y°~"~sY.~]~b'~

+ 2~y9 yAyO-,...o-sd/~ - ~ I 1X 9. . . .~. .

sx + i~a'y9"y°'r"°'sA a )

+ ,~3'~'~/~X+ 2/A"yx ¢'"qJxL7 - 2A~V"~xL7 - ,~"y~'"f--tAa) _

~2ti e-,,/2( ~ - # , % j . r 9 A

a C a __ i ~ 9

A a C a __ £ a ~ b

A b)

1 4" (4!) e~,,..4,,M . , . . . . , ) l u,-.r . ~i r . ,,~. , J

_~ e - , ~ C o C ~ _ _ _

+ (quartic terms). * For notations, we follow the ref. [11].

(A.2)

H. Nishino, S.Z Gates Jr. / Anomaly cancellations

538

The 2n scalars in the n vector multiplet s parametrize the coset SO(n, 2 ) / S O ( n ) SO(2), whose M a u r e r - C a r t a n form is (O.,,t'(A) L~( a, r O~. + f u r a ~ / ) LK a = \ pt,, o ( a ) a, b = 1 , . . . ,

x

P~o'(A)]

Qj(A)/' i , j = 1,2.

n,

(A.3)

The vector fields a~,~ are for the gauged (n +2) dimensional subgroup SO(l, 2) x H of SO(n, 2) (dim H = n - 1).* Main geometrical quantities are defined as follows: n

~ u - ( - - + " " " + ) = ~7A8,

Ft,, / = Or.A,! -o3~At~ t + fKL IA~, KA~ L , a u =- L / L j + L f l L ~ , £,

= L 1+ i~,gL~ ,

.~,°(A) - P~,'~(A)+ i ~ , e ~ 2 ( A ) ,

C a = fljKL1 IL2JLKa '

Cab i ~

fIjKLalLbJLKi,

(A.4)

Nt,,...~, ' =- OD,,Mt,~..4,~] .

For other conventions, see ref. [11]. The supersymmetry transformation rule is obtained as (up to fermion cubic terms) 8 e ~ = g.ym,p~,,

80" = - - i g x ,

gLI = - i~AaL~ ,

8A'~ = - ~ i

$L~ = ~3,qAoL~ ,

8L~ = - ig£tA ° ,

e - " / ~ ( e- L^ ' d / , , - ~"t-e L ' % , ) ( + t e"% , A ° L,,) '

8Mm...~, = 2i e ~ y 9 y [ t , , . . . t , 3 ~ , ] --½ e~t~y9yt,,...t~4X, 8~bt, = D~,e + ~2~i(%, ~ - 108~ ¢ ' ) f - , l e F ~ ' + ~

i

e

-o"

I •

+ ~1

/

1

e

-~

Tr..~ s

"y9~3/tL 1

o'/2 ~

.....

79~ '

5~

~--

--~Ott ' 7 I~V

~2...fsx

"~r

)EDI.q...~-s"

I

'eN~,...~s,

SAo = -½i¢'P~,o(A)e + ~_~1 .'cr/2"ygVyj-,Ia©'.l~gl~.+1~ • See the footnote on page 8.

i e:O'/2Cayg8 .

(A.5)

H. Nishino, S.J. Gates Jr. / Anomaly cancellations

539

Notice that the exponent in the kinetic term of M~,,...~,, is opposite to that of G~,,~ in the ordinary theory [11]. This is actually related to the existence of the scale transformation or-, or + c,

Mm ...,4

''> e ~M ~,~...~4,

.,4' . ~ -.* c_-,/2~t ~, ,

g --> e C / 2 g ,

(A.6)

where c is an arbitrary constant, while g is for any coupling constants for gauged non-abelian groups. The scaling weight of M~,,...~, is opposite to that of B~,~ in the ordinary theory. This situation is anlogous to the case of D = 10, N = 1 supergravities [3, 4].

Appendix B DUAL VERSION OF D=6,

N=2

In the case of D = 6 , N = 2 notations) rrl

SUPERGRAVITY

[14], we have the field content (see ref. [14] for

"

f

(e~, ft., B~,~,X, A~,, A', ~b~, ~a, i f ) ,

(B.1)

so that the dual version has the content

(e2, ~,., M . . X , A~, At, ~,~, ¢", 6 ) .

(B.2)

Notice that the dual version has the potential M~,~ of the same rank. The invariant lagrangian for the dual version is obtained as e- 1,~rual

1 1 2 1 - 2~r2 ~ 2 1 ,/~t~ I i~,p. v l D=6, N=2=+~R(t°(e))--~(O.C~) --i~¢ N~,.a-~e F~.--

-A

v

p,

a

a

- J~ e -*/'~( ~ ; . ? r'~ ~c '~+ ~r'~ ~c '~+ 2~;"~ ~ V.aA~°t) -~e

C C -~e

M~,.Fp,.F..+(quart~cs),

(83) where the signature is ~/m. = ( - - + + + + + ) , and N~,~p~ 3ah, M~]. For other conventions, see ref. [14].

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H. Nishino, S.Z Gates Jr. / A n o m a l y cancellations

The supersymmetry transformation rule for the dual D = 6, N = 2 system is (up to fermion cubic terms)* 8e~ = ~3/"$,,, aM,,. =

85 = x/~X,

e"n*(-~'.vt,.,q,,.,~-½~%,.x),

ax = +@v"~ 0,,4, +~ e-'~" y"~%N,,~,,, 8V&, = D,.((o(e))e - ~ e-~a%,,~,%,eN p', ,

aA t = + ~ 8dS' =

....

e */ a F.~'y~'"e -x/~ e-'t'/'/E T ' e C d ,

-- V a A e

a--Ill a ,

aq" = - (D~,qb~ ) V="A~/~'eA.

(B.4)

Notice the global scale invariance e-') 6 + c,

M~,~ --) e ./2c Mz~,

~ _ ^-c / J -2 ,t i A~,-.-.c eta,

g-->eC/~g,

(B.5)

where c is an arbitary constant, and g is any non-abelian coupling constant for any gauged subgroup of the isotropy groups of quaternionic Kiihler manifolds [15, 16]. The scaling weight for M . . is again opposite in sign to the for B~,. in the ordinary version, as in D = 10, N = 1 [3, 4] or D = 8, N = 1 supergravities. Even though M~,, has the same rank as B~.~, the lagrangian (B.3) for the dual version cannot be obtained merely by field redefinitions in the ordinary lagrangian. The difference in scaling weight between B~,. and M~,~ is also related to this fact.

Appendix C D U A L I T Y T R A N S F O R M A T I O N FOR D I L A T O N F I E L D

In this section, we try the duality transformation for the scalar field (dilation) ~b in D -- 10, N = 1 supergravity. Consider the ordinary pure D = 10, N = 1 supergravity [3], whose field content is

( e'~ ~b~,B~,~X, d) ).

(C.1)

Since the "field Strength" of ~b is a one-form, the dual field "potential" is expected to be 8-form; Then the dual version with respect to 4) has the field content

(e~, ~,, B~,X, M~,,...m).

(C.2)

* The quartie fermion terms in the lagrangian and cubic fermion terms in the transformation rule in the ordinary D = 6, N = 2 version [14] are under inspection [15].

H. Nishino, S.J. Gates Jr. / Anomaly cancellations

541

The lagrangian of ordinary D = 10, N = 1 pure supergravity [3] is rewritten in the following form:

e-,~o=+IR(eo(e))

,-i~b~y

..p

, " 2 D.(aJ(e))~bo-i~G.,, o

- ½2y~'D~. (to ( e ) ) x - ½(0~.6 )2 + "v~6~,y"y~'XO~¢

+ (quartic terms),

(C.3)

where/~..-= e*B,~ and (C.4)

t~,, o --- 3(atfl},p J - ~t~,,0pj¢)

contains a "Chern-Simons" form with ap4~. Notice that the dependence on ~b in the ordianry lagrangian through exponential functions is all absorbed into /~,,; only the field strength of ¢ appears in the iagrangian (C.3). This fact enables us to perform the duality transformation on ¢ [13]. We can replace a~,4> everywhere in the lagrangian (C.3) by a new independent field H~,,and add the constraint lagrangian 1

,~c = ~ e"'"'"'°" M ., .. aoH,. The field equation of H~., obtained from the i n v a r i a n t lagrangian ~o(a~.¢-~ H,.) + &meis then H ~

=

-~, ~"","oN .......+~-~.o(at"~"~- ~t""Ho~) +4~6.~"~"x.

(c.5)

This expression is very cumbersome, since H,, appears again on the r.h.s, of eq. (C.5). However, we can estimate H,, using eq. (C.5) in a symbolic manner: eq. (C.5) is rewritten

(8,,v + Pj,")S, =J~,,

P~ -= ~ . o ~ 1

Y~, - - ~

~

(C.6)

+-~8.v~ o~,

(c.7)

e,, vl...~9 N ....... q'- 4 ~ , ' Y- ~ . ' Y ' I vx + ~ B3 " ~ p a t , B", , 1 .

(C.8)

Therefore symbolically, we get H,, = (I - P + p2 - . . . ) ~ , j , ,

(C.9)

where P is regarded as a 10 x 10 matrix whose components are P~f. It is interesting to have such a non-polynomial structure for the fields B~,,, N~,,...~ and spinors ~b~, and X. Since the final lagrangian is too complicated, we do not dare to give it here. We can try the same duality transformation for the total system of D = 10, N = 1 supergravity coupled to abelian vector multiplets. In the non-abelian case, however, we encounter difficulty, since there remains explicit ~b-dependence in the Yang-Mills field strength, even after exponential scaling of A~: ,4~, e-¢'/2A~. =

542

H. Nishino, S.J. Gates Jr. / Anomaly cancellations

The case of other higher-dimensional theories of supergravity is almost the same, so that we do not give the explicit result here. Note added in proof In this paper we argued anomaly-freedom independently of field equations, because so far no completely supersymmetric field equations with superstfing corrections were given. If we also consider the M-field equation, there might arise the equation tr R 2 - 1 T r F 2 = (total divergence), for example in the dual D = 10, N = 1 theory [4]. However, at least in the ordinary theory [3] the six-dimensional anomalyfreedom requires eq. (1.3), therefore the manifold Q should always be boundaryless, while in the dual version [4], because the 6th rank tensor always cancels the leading anomalies, we allow any manifold M4 even with boundaries. In other words, the dual version has one less restriction on the global topology of boundary effects. References [1] M.B. Green and J.H. Schwarz, Nucl. Phys. BI81 (1981) 502; B198 (1982) 252; J.H. Schwarz, Phys. Reports 89 (1982) 223; M.B. Green, Surveys in high energy physics 3 (1983) 127 [2] M.B. Green and J.H. Schwarz, Phys. Left. 149B (1984) 117 [3] A.H. Chamseddine, Nucl Phys. B185 (1981) 403; E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Nucl. Phys. B195 (1982) 97; G.F. Chapline and N.S. Manton, Phys. Lett. 120B (.1983) 105 [4] S.J. Gates, Jr.'and H. Nishino, Phys. Lett. 157B (1985) 157 [5] E. Witten, Phys. Lett. 149B (1984) 351 [6] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985) 46 [7] K. Pilch and A.N. Schellekens, Nucl. Phys. B259 (1985) 637 [8] P.H. Frampton and K. Yamamoto, Phys. Lett. 156B (1985) 345 [9] M. Green, J.H. Schwarz, and P.C. West, Caltech prepdnt, CALT-68-1210; Nucl. Phys. B254 (1985) 327 [10] L. Alvarez-Gaum6 and E. Witten, Nucl. Phys. B234 (1983) 269; B. Zumino, Les Houches Lectures (1983); R. Stora, Carges¢ Lectures (1983); W.A. Bardeen and B. Zumino, Nucl. Phys. B244 (1984) 421; L. Alvarez-Gaum6 and P. Ginsparg, Harvard Univ. preprint HUTP-84/A016 [11] A. Salam and E. Sezgin, Phys. Lett. 154B (1985) 37 [12] A. Salam and E. Sezgin, ICTP, Trieste preprint, IC/85/19 [13] H. Nicolai and P.K. Townsend, Phys. Lett. 98B (1981) 257 [14] H. Nishino and E. Sezgin, Phys. Lett. 147B (1984) 187 [15] H. Nishino and E. Sezgin, in preparation (1985) [16] I.G. Koh and H. Nishino, Phys. Lett. 153B (1985) 45