Nonlinear realization of partially broken N=2 superconformal symmetry in four dimensions

19 February 1998

Physics Letters B 420 Ž1998. 69–76

Nonlinear realization of partially broken N s 2 superconformal symmetry in four dimensions Yoshinori Gotoh a , Tsuneo Uematsu a

b,1

Graduate School of Human and EnÕironmental Studies, Kyoto UniÕersity, Kyoto 606-01, Japan b Department of Fundamental Sciences, FIHS, Kyoto UniÕersity, Kyoto 606-01, Japan Received 15 July 1997 Editor: M. Dine

Abstract We investigate the nonlinear realization of spontaneously broken D s 4 N s 2 superconformal symmetry. We particularly study Nambu-Goldstone degrees of freedom for the partial breaking of N s 2 superconformal symmetry down to N s 1 super-Poincare´ symmetry, where we get the chiral NG multiplet of dilaton and the vector NG multiplet of NG fermion of broken Q-supersymmetry. Evaluating the covariant derivatives and supervielbeins for the chiral as well as the full superspace, we obtain the nonlinear effective lagrangians. q 1998 Elsevier Science B.V.

1. Introduction In the last few years there has been much interest in the partial breaking of extended supersymmetry. This has some relevance for the N s 2 supersymmetric gauge theories which have recently attracted a great deal of attention in the context of studying the non-perturbative properties of field theories as well as string theories. Hughes, Liu and Polchinski w1x first pointed out the possibility of the partial breaking by giving the argument based on supersymmetry current algebras evading the existing no-go theorem for the partial

1

Supported in part by the Monbusho Grant-in-Aid for Scientific Research No. C-09640345 and for Scientific Research on Priority Areas No. 09246215. 0370-2693r98$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 8 5 - 8

breaking. From the view point of linearly realized models, they also constructed the four-dimensional supermembrane solution of the six-dimensional supersymmetric gauge theory, in which the second supersymmetry, in the equivalent four-dimensional N s 2 theory, is spontaneously broken. While, Antoniadis, Partouche and Taylor w2x introduced the electric and magnetic Fayet-Iliopoulos terms in the N s 2 gauge theory of abelian vector multiplet and have shown that there occurs spontaneous breaking of N s 2 to N s 1 supersymmetry. This partial breaking induced by the Fayet-Iliopoulos terms has also been obtained by taking the flat limit of the N s 2 supergravity theories w3x. On the other hand, Bagger and Galperin w4,5x have studied the nonlinear realization of N s 2 supersymmetry partially broken down to N s 1 supersymmetry w6,7x. They obtained the Nambu-Goldstone

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Y. Gotoh, T. Uematsur Physics Letters B 420 (1998) 69–76

ŽNG. multiplet both for the cases; chiral multiplet w4x and vector-multiplet w5x, and discussed the nonlinear transformation laws as well as the low-energy effective lagrangians w8x. Here in this paper, we shall investigate the nonlinear realization of D s 4 N s 2 superconformal symmetry which is realized for the case of vanishing b-function in the N s 2 supersymmetric QCD, for example. We study the spontaneously breaking of this symmetry down to N s 1 super-Poincare´ symmetry. We find that there appear two independent N s 1 NG multiplets: a chiral multiplet and a vector multiplet. The vector multiplet consists of a NG fermion of the broken second Q-supersymmetry and an extra vector field. The chiral multiplet consists of dilaton, axion and dilatino which are associated with the broken dilatation, chiral UŽ1. rotation and the first S-supersymmetry generators. We note that the NG fermion of the broken second S-supersymmetry can be expressed as the derivative of the true NGfermion of the second Q-supersymmetry. We next study the effective interaction for the system of the dilaton multiplet coupled to the NG-vector multiplet. It is intriguing to examine whether Born-Infeld like action would emerge in the presence of the dilaton multiplet, as in the case studied in Ref. w5x.

2. Partial breaking of N s 2 superconformal symmetry In this section, we consider the N s 2 superconformal algebra and its spontaneous breaking w9x. We then determine the the relations among NG fields corresponding to the broken generators. 2.1. N s 2 superconformal algebra The D s 4 N s 2 superconformal group is usually denoted by SUŽ2,2r2. ŽSee for example w10x., the generators of which are those of conformal group: translation Pm , Lorentz rotation Mmn , conformal boost Km , and dilatation D operators; together with Q-supersymmetry generators: Qa A , Qa˙A Ž A s 1,2.; S-supersymmetry generators: S a A , SAa˙ Ž A s 1,2. and SUŽ2. generators, TAB, the chiral UŽ1.R charge A,

and the UŽ2. charge BAB ' TAB q 16 dAB A. Some relevant commutation relations are the following: Pm , Kn s 2 i Ž gmn D y Mmn . ,

 Qa A ,QaB˙ 4 s 2 dABsm a a˙ P m ,  SaA ,Sa˙ B 4 s 2 dBAsm a a˙ K m i

3

w Qa A , D x s Qa A , w Qa A , A x s Qa A , 2

2

i

SAa˙ , D s y SAa˙ , 2 bB

 Qa A ,S 4

s dAB

Žs

3

w Sa A, Ax sy Sa A 2

mn

.

b b a Mmn y 2 i da D

y 4dab BAB

Qa A , Km s sm a a˙ SAa˙ , SAa˙ , Pm s sma˙ a Qa A Qa A ,TBC s dAC Qa B y SAa˙ ,TBC s dAC SBa˙ y TAB ,TCD

1 2

d BC Qa A ,

1

d C S a˙ 2 B A s dAD TCB y dCB TAD

Ž A, B,C, D s 1,2 . . Ž 1.

2.2. Nambu-Goldstone multiplet Now we identify the independent N s 1 NG multiplets for the partially broken N s 2 superconformal algebra. We find that there appear two independent NG multiplets; a chiral multiplet and a vector multiplet. In our case where D s 4 N s 2 superconformal group is spontaneously broken down to D s 4 N s 1 super-Poincare´ group, we have the broken generators Qa 2 ,Qa2˙ , S a 1,S1a˙ , S a 2 ,S2a˙ , Km , D, A, T and T. We assign the corresponding NG particles, xa , xa˙ , ca 1 , c a˙ 1, ca 2 , c a˙ 2 , fm , s , r , Õ and Õ, respectively. To discuss the relationships among the above NG fields, we consider the charge commutation relation between an unbroken charge and a broken charge which generates another broken charges. We note that these relations impose the constraints on NG fields w11,12x. Let us take, for example, one of the commutation relations of N s 2 superconformal algebra Pm , Kn s 2 i Ž gmn D y Mmn . .

Ž 2.

Y. Gotoh, T. Uematsur Physics Letters B 420 (1998) 69–76

where Pm and Mmn are unbroken, while Kn and D are broken. First we consider the Jacobi identity for Pm , Kn and s Pm , Kn , s q w Kn , s x , Pm q

s , Pm , Kn

s 0.

Ž 3.

Next we take the vacuum expectation value of Ž3. ²0 < Kn , Em s <0: s y2 gmn ²0 < w D, s x <0: / 0.

This equation shows that fm ; Em s , thus fm is not an independent NG degree of freedom. Further relations on NG particles are obtained from the following commutation relations in the same manner. S2a˙ , Pm s sma˙ a Qa 2 ,

w Qa 1 ,T x s Qa 2 ,

Ž 5.  Qa 1 ,S b 2 4 s y4dab T  Qa 1 ,S b 1 4 s Ž s mn . ab Mmn y 2 i dab D y 4dab B11 Qa 1 , Km s

1 2

˙ Q ,c fm ; smaa  a 1 a˙ 1 4 ,

fm ; Em s .

Ž 8.

These transformations Ž8. show that there exists another independent N s 1 NG multiplet Ž s q i r .Ž x, u , u . ' f Ž x, u , u . which contains component fields w s s q i r where s is the dilaton and r is the axion , and ca 1 isthedilatino. We will impose the chirality condition on f Ž x, u , u .. It will turn out that f Ž x, u , u . is a NG-background chiral superfield with spin 0 and 12 fields. We shall call this multiplet as dilaton multiplet.

sm a a˙ S1a˙ , 3. Nonlinear realization

Pm , Kn s 2 i Ž gmn D y Mmn . .

Ž 6.

We notice that Ž5. gives rise to the constraints on NG particles x a Ž x ., Õ Ž x . and ca 2 Ž x .. The Jacobi identities lead to

c 2 a˙ ; sma˙ aE mxa , Õ ;  Q1a , xa 4 ,

as a derivative of the NG fermion corresponding to the broken Q-supercharge. Actually this statement is proved by using a spectral representation of the correlation function of supercurrents, which will be discussed elsewhere w13x. On the other hand we notice that Ž6. gives rise to the constraints on NG particles s Ž x ., r Ž x . and ca 1Ž x .. The Jacobi identities lead to

ca 1; w Qa 1 , s q i r x ,

Ž 4.

71

c 2 a˙ ; Q1a˙ ,Õ .

Ž 7.

These transformations Ž7. show that there is an independent N s 1 NG multiplet x a Ž x, u , u . which contains component fields x a , Õ and ca 2 , moreover we can identify these fields with Abelian gaugino, D-term and the derivative of gaugino, respectively. Now we impose the chirality condition on x a Ž x, u , u .. Thus, x a Ž x, u , u . is a vector multiplet with spin 12 and 1 fields. The spin 1 field does not really correspond to a NG degree of freedom, but a superpartner of the NG fermion. For this reason, we may call this multiplet as NG-Maxwell multiplet. From the argument based on the first equations of Ž5. and Ž7., we expect that if Q-supersymmetry is spontaneously broken, then the corresponding S-supersymmetry is broken as well and the NG fermion corresponding to the broken S-supercharge is written

As we have seen in the previous section, there are two independent N s 1 NG multiplets f Ž x, u , u . and x a Ž x, u , u . in the system. Next we shall write down the effective action for these multiplets by using the method of nonlinear realization w14–17x. 3.1. Coset construction First we consider the coset construction for the nonlinear realization for the symmetry characterized by the group G broken down to a subgroup H. In our case the relevant left-invariant coset representative is given by LŽ x , u , u . s T F U

Ž 9.

where T s exp Ž ix P P q i u A Q A q i uA Q A .

u 2'x ,

u2 'x

F s exp Ž i f P K q i cA S A q i c A SA . =exp Ž i s D q iÕ1 B11 q iÕ 2 B22 . U s exp Ž iÕT q iÕT . .

Ž 10 .

Y. Gotoh, T. Uematsur Physics Letters B 420 (1998) 69–76

72

Here we note that the coset space is parametrized by the N s 1 superspace coordinates X A s Ž x, u , u . as well as by the NG superfields f Ž x, u , u ., PPP ,Õ Ž x, u , u ., the first components of which correspond to the NG fields discussed in the previous section. Under the left multiplication of a group element g g G, L transforms as L ™ gL s LX h

Ž hgH . ,

Ž 11 .

from which we obtain the transformation laws of the NG fields. We calculate the Cartan differential 1-form as Ly1 dL s iD x P P q iDu A Q A q iDuA Q A q iDf P K q iDcA S A q iDc A SA q iDs D q iD Õ 1 B11 q iD Õ 2 B22 q iD ÕT q iD ÕT q

1 2

vmn M mn

Ž 12 . where D x m s exp Ž ys . dx m q idu As muA q iduA s mu A ' exp Ž ys . dl m Du a C s WAB UBC du a A y i c Asm

ž

A

a

/

dl m

B

Dua˙ C s Ž W † . B Ž Uy1 . C dua˙ A y i Ž cA sm . a˙ dl m Ds s d s y 2 dl P f y du AcA y duA c

A

Dr s d r y 2 dl m c Asm cA q idu AcA y iduA c

ž

/

A

B

UAB ' expi Ž Õt q Õt . A cos'ÕÕ

s

( i

Õ Õ

WAB ' exp y

r'

1 6

sin'ÕÕ 1 2

(

i

Õ Õ

sin'ÕÕ

cos'ÕÕ

0

that we only consider the case where SUŽ2. = UŽ1. space-time symmetry is broken to UŽ1.. Namely, the three generators; T s B21 , T s B12 and A s 3Ž B11 q B22 . are broken and T3 s 12 Ž B11 y B22 . remains unbroken. If T3 were broken, we would need another real scalar as well as one fermionic superpartner because of N s 1 supersymmetry. For this reason we take T3 to be unbroken in our present study. This amounts to take Õ 3 s 0 and WAB is reduced to expwy 12 Ž s y 3i r .x dAB. 3.2. CoÕariant deriÕatiÕes and constraints In this subsection we determine the constraints which realize the relation of the NG fields discussed in Section 2 in the framework of nonlinear realization. In general the constraints should be invariant under the nonlinear transformations. First we introduce the superspace coordinates X A s Ž x m , u a , ua˙ . and the supervielbein EMA defined as D X A s dX M EMA ,

D X A' Ž D x m , Du a , Dua˙ . ,

dX M ' Ž dx m , du a , dua˙ . .

Ž 14 .

We next introduce the covariant derivatives of NG superfields j Ž x, u , u . which transform linearly under the full group and are obtained from the Cartan differential 1-form. The constraints are represented as DA j Ž x, u , u . s constant, where Dj M Dj DA j s s Ž Ey1 . A . Ž 15 . A DX dX M In our present case we set the following constraints: DA x a s 0, DA xa˙ s 0 Ž A s m , a , a˙ . DA s s 0, Da r s 0, Da˙ r s 0. Ž 16 . We will see that these constraints realize the relations Ž7., Ž8. and further impose the chirality conditions on x Ž x, u , u . as well as on f Ž x, u , u .. Before we solve Ž16., we define covariant derivatives using the supervielbein matrix elements Ž e mm,e am ,e a˙ m . as m

Dm ' Ž ey1 . m Em ,

B

 Ž s y 3i r . 1 y iÕ 3t 3 4

Da ' Ea y e am Dm

, A

m

s Da y i Ž Da xs mx q Da xs mx . Ž ey1 . m Em ,

Ž Õ1 q Õ 2 . , Õ 3 ' Õ1 y Õ 2 .

Ž 13 .

We take Q2 a to be the broken supercharge while keeping Q1 a unbroken. Here one should also note

D a˙ ' E a˙ y e a˙ m Dm m

s D a˙ y i D a˙xs mx q D a˙xs mx Ž ey1 . m Em . Ž 17 .

ž

/

Y. Gotoh, T. Uematsur Physics Letters B 420 (1998) 69–76

where Em s ErE x m , Ea s ErEu a, E a˙ s ErEua˙ ,

supervielbeins we obtain the nonlinear effective lagrangians for the dilaton multiplet as well as the NG-Maxwell multiplet. From Ž12.-Ž14., the explicit expression for the supervielbein for the full superspace is given by

e mm ' dmm q i Ž Em xs mx q Em xs mx . , m

Ž ey1 . m e mn s dmn , e am ' i Ž s mu . a q i Ž Ea xs mx q Ea xs mx . , e a˙ m ' i Ž s mu . ˙ q i Ž E a˙xs mx q E a˙xs mx . , a

Ž 18 .

and Da , D a˙ are the flat N s 1 differential operators. These covariant derivatives coincide with those introduced in w5x. Now we solve Ž16. and find ˙ D bx a s 0 ,

1

ca˙1s y

4

1 4

Ž 19 .

U12 U22

Db˙ x a˙ ,

(

Õ

b˙

Ž Yy1 . a˙ i Dm Ž xs m . b˙ y

ž

tan2'ÕÕ s Dx q

Õ

ž

= 1y

fm s

1 2

Dma

Gm a˙

Bam a˙ m

Eaa a˙ a

Ha a˙

C

J aa˙˙

F

0

Ž 24 .

where A mm s ey s e mm ,

Bam s ey s e am ,

C a˙ m s ey s e a˙ m

1

Dma s ey 2 Ž sy3 i r . Em x a U21 y ie mm c Asm

ž

U12 U22

1 4

g d

1 2

Dg xd D x Da˙ xb˙

/

UA1

1

yie am c Asm

ž

a

/

UA1

1

F a˙ a s ey 2 Ž sy3 i r . U21E a˙x a y ie a˙ m c Asm UA1

ž

Db˙ f ,

Dx Da xb D ax b ˙ D a˙x b

a

/

Eaa s ey 2 Ž sy3 i r . U11d aa q U21Ea x a

Ž 20 .

1

Ž 21 . i



EMA s

A mm

b˙

Ž Yy1 . a˙ Db˙ f q i Dm Ž xs m . g˙ Db˙ x g˙ ,

Yb˙a˙ ' db˙a˙ q

ca˙2 s y

˙ D bf s 0,

73

/

2

Gm a˙ s ey 2 Ž sq3 i r . Em xa˙ Ž Uy1 . 1 A

yie mm Ž cA sm . a˙ Ž Uy1 . 1

/

,

Dm s q Ž Dm x . c 2 q Ž Dm x . c 2 .

1

Ž 22 .

2

Ha a˙ s ey 2 Ž sq3 i r . Ž Uy1 . 1 Ea xa˙ A

yie am Ž cA sm . a˙ Ž Uy1 . 1

Ž 23 .

Note that the conjugated relations also hold for Ž19.-Ž22.. The two equations of Ž19. are the chirality conditions for x a Ž x, u , u . and f Ž x, u , u .. The Eq. Ž20. shows that c 1 a Ž x . is the superpartner of w Ž x .. From Ž21. c 2 a Ž x . is seen to be the derivative of xa˙ Ž x .. The Eq. Ž22. shows that Õ Ž x . is given as the auxiliary scalar which appears in the second component x a Ž x, u , u .. The last Eq. Ž23. indicates that fmŽ x . is the derivative of s Ž x .. These relations are in accordance with Ž7. and Ž8.. 3.3. SuperÕielbeins and effectiÕe lagrangians In this subsection we present supervielbeins for the full as well as the chiral superspace. Using these

1

1

2

Ja˙a˙ s ey 2 Ž sq3 i r . Ž Uy1 . 1 da˙a˙ q Ž Uy1 . 1 E a˙xa˙ A

yie a˙ m Ž cA sm . a˙ Ž Uy1 . 1

Ž 25 .

From these results, the full superdeterminant is found to be y2 y2 s

sdet E s e

det Ž

e mm

i

(

. 1 y Dx 2

Õ Õ

tan'ÕÕ

.

Ž 26 . The simplest invariant action for the dilaton multiplet is obtained from the above superdeterminant, by forming the invariant phase volume.

Hd

4

xd 2u d 2u sdet E s d 4 x LD q PPP

H

Ž 27 .

Y. Gotoh, T. Uematsur Physics Letters B 420 (1998) 69–76

74

where the nonlinear lagrangian LD turns out to be i ) LD s eyw y w yE mw )Em w q Em c 1s mc 1 2 i i y c 1s mEm c 1 y Ž Em w y Em w ) . c 1s mc 1 2 2 1 1 2 2 q F) q Žc 1. F q Ž c1 . Ž 28 . 2 2

ž

/ž

/

where we denote the first, second and auxiliary components of the superfield f by w , c 1 and F, respectively. w and c 1 are NG fields, and F does not correspond to any broken generator. Eq. Ž28. is the same effective lagrangian for the spontaneous breaking of N s 1 superconformal to N s 1 superPoincare´ symmetry discussed in w11x. Next we introduce the chiral superspace in the NG background w5x, Ž x Lm , u a ., as x Lm s x m y i us mu y i xs mx .

The supervielbein for the chiral superspace in NG background is defined as Du

Ž Dx ,

a

. sŽ

dx Lm ,

du

' Ž dx Lm ,

a

.

ž

A mm Bam

Cma Daa

/

d u a . EL

Ž 30 .

A mm s ey s e Lmm ,

e Lmm s dmm q 2 i EmL xs mx

Bam s ey s e Lma ,

e Lma s 2 i

1

1 2 Daa s e

m

Ž s u . a q 2 i Ea xs x

ž

EmL x a U21 y ie Lmm

3

1

1q

2

˙

DŽ a˙ xb˙ . D Ž a˙x b . q PPP

Ž 35 .

where DŽ a˙ xb˙ . '

1 2

2

žD˙ x ˙ q D ˙ x ˙ / s a

b

b

Ž Uy1 . 2

a

D

DŽ a˙ xb˙ .

s DŽ a˙ xb˙ . q PPP

Ž 36 .

with i 2

(

Õ Õ

tan'ÕÕ Dx .

Ž 37 .

We will see that Ž35. satisfies Ž34. up to second order of the superfields. In this case the action becomes 4

x L d 2u

1 3

ey3 f det Ž e Lmm .

a A

UA1

žc s / m

/

= 1y

Ž s y3 i r .

žd ž

a 1 a U1 q Ea

x a U21

= 1q

a

/

UA1 .

/

= 1y

2

(

Õ

tan'ÕÕ Dx

ž ž

= 1y .

Ž 32 .

= 1y

The invariant action for the chiral superspace has the following form : x L d 2u sdet EL f Ž DA j ,F . q h.c.

1 2

(

Õ

tan'ÕÕ Dx ˙

DŽ a˙ xb˙ . D Ž a˙x b . q h.c.

H

y1

Õ

2

Õ

s d 4 x L d 2u ey3 f Ž 1 q 2 i EmL xs mx q PPP .

sdet EL s ey3 sy3 i r det Ž e Lmm . i

i

Ž 31 .

The chiral superdeterminant is given by

4

1

y1

Ž s y3 i r .

yie Lma c Asm

Hd

fs

Hd

m

Ž 34 .

Now we construct the invariant action for the NG-Maxwell multiplet which can be obtained by the standard prescription Ž33. in the nonlinear realization. We take the function f to be

Ds1q

where

2 Cma s e

Da˙ sdet EL f Ž DA j ,F . s 0.

Ž 29 .

ELAM

m

where j is any NG superfield, F is a spectator superfield which transforms linearly in the nonlinear realization and f Ž DA j ,F . satisfies the chirality condition

Ž 33 .

1 4

Dx Dx q PPP

1

Ž Dx . 4

1

2

q 2

/

˙ Da˙ xb˙ D a˙x b

/

qh.c. ' d 4 x L LN G M

H

Ž 38 .

Y. Gotoh, T. Uematsur Physics Letters B 420 (1998) 69–76 ˙ where we have used the relation: DŽ a˙ xb˙ . D Ž a˙x b . s 1 a˙ b˙ 2 Da˙ xb˙ D x y 2 Ž Dx . . Following w5x, we represent xa in terms of Wa as follows:

xasWa q

1 4

D 2 Ž W 2 . Wa y iWs m WEmWa q O Ž W 5 . .

Ž 39 . where Da s Ea y iŽ s mu .a Em , D a˙ s E a˙ y iŽ s mu . a˙Em , and Wa s iD 2 Da V. V is the vector superfield. To the order of x 2 s W 2 q PPP , we get LN G M s d 2u ey3 f

H

1 3

ž

1y

1 4

D 2 Ž W 2 . q PPP

q h.c.

/ Ž 40 .

where the integrand turns out to be chiral to this order. Because of the nonlinear transformation invariance, the choice of f ŽDA j . is unique to the order of x 2 . Note that we have used the Bianchi identity, DW s yDW. The total lagrangian L s LD q LN G M provides the kinetic term for the NG-Maxwell multiplet by using the equation of motion for F and F ) . Hence we get

½

y d 4 xey4s 1 q

H

1 4

Fm n F m n y

i 2

Em ls ml

i 1 q ls mEm l y D 2 q O Ž Fm4 n . 2 2

5

Ž 41 .

where la is the fermionic component of Wa , by the interplay of Ž28. and Ž40.. Note that the free kinetic term of the vector multiplet cannot be obtained by the usual construction as Hd 4 xd 2ux x q h.c., because of the nonlinear transformation invariance. Here we have some comments on scale Žlength. dimensions of NG fields. As can be seen from the coset construction Ž9. Ž10., the superfield x has a scale dimension 1r2. So the gauge field strength Fm n as well as the auxiliary field D have vanishing scale dimensions and l has a scale dimension 1r2. Thus this action is invariant under the nonlinear scale transformation: x m ™ e k x m, where the dilaton transforms as s ™ s q k .

75

4. Conclusion In this paper we have investigated the nonlinear realization of 4-dimensional N s 2 superconformal symmetry which is partially broken down to N s 1 super-Poincare´ symmetry. We have particularly studied the nonlinear effective interaction of the NG degrees of freedom which consist of a chiral multiplet as well as a vector multiplet. One should note the connection with the breaking of D s 4 N s 2 superconformal to D s 4 N s 2 super-Poincare´ symmetry, discussed in w12x. In that case there appear a N s 2 multiplet consisting of dilaton s , axion r , dilatinos c i , c i Ž i s 1,2. and vector gauge field Am . The effective lagrangian is similar to the sum of Ž28. and Ž41.. In that case if N s 2 supersymmetry breaks down to N s 1, this multiplet may split into two N s 1 multiplets; the chiral multiplet and the vector multiplet which are obtained in this paper. Finally, a comment on the Born-Infeld action w18,19x is in order. It is not straightforward to extend the Born-Infeld action for the partially broken N s 2 super-Poincare´ symmetry discussed in w5x where the dilaton multiplet is coupled to the NG-Maxwell multiplet. It might be possible to derive the Born-Infeld action by summing up the higher order powers of Fm n in Ž41., which is now under investigation.

References w1x J. Hughes, J. Polchinski, Nucl. Phys. B 278 Ž1986. 147; J. Hughes, J. Liu, J. Polchinski, Phys. Lett. B 180 Ž1986. 370. w2x I. Antoniadis, H. Partuche, T.R. Taylor, Phys. Lett. B 372 Ž1996. 83; I. Antoniadis, T.R. Taylor, CPTH-PC445.0496, hep-thr9604062, 1996; H. Partouche, B. Pioline, CPTHPC496.0297, hep-thr9702115, 1997. w3x S. Ferrara, L. Girardello, M. Poratti Phys. Lett. B 376 Ž1996. 275; M. Poratti, hep-thr9609073, 1996; P. Fre, ´ L. Girardello, I. Pensando, M. Trigiante, hep-thr9611188, 1996. w4x J. Bagger, A. Galperin, Phys. Lett. B 336 Ž1994. 25. w5x J. Bagger, A. Galperin, Phys. Rev. D 55 Ž1997. 1091. w6x J. Bagger, J. Wess, Phys. Lett. B 138 Ž1984. 105. w7x S. Samuel, J. Wess, Nucl. Phys. B 221 Ž1983. 153. w8x J. Bagger, Nucl. Phys. B ŽProc. Suppl.. 52A Ž1997. 362. w9x P. Fayet, Nucl. Phys. B 149 Ž1979. 137. w10x M. Sohnius, Phys. Rep. 128 Ž1985. 39. w11x K. Kobayashi, T. Uematsu, Nucl. Phys. B 263 Ž1986. 309.

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Y. Gotoh, T. Uematsur Physics Letters B 420 (1998) 69–76

w12x K. Kobayashi, K.H. Lee, T. Uematsu, Nucl. Phys. B 309 Ž1988. 669. w13x Y. Gotoh, T. Uematsu, in preparation. w14x S. Coleman, J. Wess, B. Zumino, Phys. Rev. 177 Ž1969. 2239; C. Callan, S. Coleman, J. Wess, B. Zumino, Phys. Rev. 177 Ž1969. 2247. w15x D.V. Volkov, Sov. J. Particles and Nuclei 4 Ž1973. 3.

w16x V.I. Ogievetsky, in Proc. of 10th Winter School of Theoretical Physics in Karpacz, Wroclaw, 1974, vol. 1, p. 117. w17x T. Uematsu, C.K. Zachos, Nucl. Phys. B 201 Ž1982. 250. w18x M. Born, L. Infeld, Proc. Roy. Soc. ŽLondon. A 144 Ž1934. 425. w19x S. Cecotti, S. Ferrara, Phys. Lett. B 187 Ž1986. 335.