Volume 66B, number 1 PHYSICS LETTERS 3 January 1977

S U P E R G R A V I T Y A N D L O C A L E X T E N D E D S U P E R S Y M M E T R Y

S. FERRARA 1, J. SCHERK Laboratolre de Physique Thgorique de l'Ecole Normale Supgrieure 2,

24, rue Lhomond, 75231 Paris Cedex 05, France

and

B. ZUMINO CERN, Geneva, Switzerland

Received 19 November 1976

3 We consider the possibility of having supergravity theories with several (N > 2) Majorana spin g gauge fields. From

the spmor dual model one can infer the existence of these theories with (Yang-Mills) matter couphng up to N = 4 spin 3 ~- gauge fields. In the present note we construct the simplest of these theories, Le. pure supergravaty with a triplet of

3 spm ~- gauge fields. This theory contains, as subcases, complex supergravity and ordinary supergravlty coupled to the vector multiplet.

It has been shown recently, that it is possible to construct locally supersymmetric theories [1 ] with

3 more than one spin ~- gauge fields [2]. In these theo- • 3

des, the spin ~ fields gauge an extended global super- symmetry [3] in which the spinor charges Q i (i = 1, 2, .. N) belong to the vector representation of 0(N). For example, complex supergravity [2] gauges the global supersymmetry algebra with 0(2) as a real internal symmetry. On the other hand, it has been shown very recently that the spinor dual model [4] in 10 dimensions, (through compactification of 6 dimen- sions [5]) contains four dimensional supergravity theo- ries which gauge an 0(4) global supersymmetry [6]. While the sector of closed strings contains supergravity, the open string sector contains a Yang Mtlls matter supermultiplet with global 0(4) extended supersym- metry [6]. As a consequence, it can be inferred that supergravity theories coupled to matter exist with, at

3 least, 4 Majorana spin 7 gauge fields (even though they have not yet been constructed). Interestingly enough, N = 4 is also the largest number compatible with the presence of a matter multiplet. Pure super- gravity theories (without matter) could also exist for

1 Present address" Frascatl National Laboratories, Frascati, Italy.

2 Laboratoire propre du C.N.R.S. associ4 ~ l'Ecole Normale Sup~neure et ~ l'Unwerslt~ Paris-Sud.

N = 5, 6 and 8. For N > 8 no local theory with a local supersymmetry can be constructed.

These assertions are a simple consequence [7] of the fact that for N spinor charges t l , the hellclty con- tent ~ of a massless irreducible multiplet is given by I~1, I~1 - ½, , I~1 - (N/2) which implies that matter (I Xmaxl = 1) exists up to N = 4, and supergravity (I Xmax I = 2) up to N = 8. Also the chiral multiplet (i Xmax I 1 = 7), and therefore an explicit mass term can only exist for N = 1, 2.

In the present letter, we construct the simplest example of this hierarchy of supergravity theories,

• 3 namely pure supergravity, with a triplet ff~ of spin gauge fields (0(3) supergravity). This is the last theory in which the complications due to scalar particles [8] do not appear, and, moreover, it contains, as subcases, complex supergravity [2] and ordinary supergravlty coupled to the Abehan vector multiplet [9, 10].

The O(3) pure supergravity theory is obtained by coupling the gauge multlplet of ordinary supergravity

1 The multlphcity of the one particle states of a massless irre- duoble multlplet can be easily obtained by means of the Wlgner method of reduced representations applied to global extended supersymmetry [71. In general, the multiphcity of states having hellcitles h = hma x - (N/2) and k = (N/2) - kma x - (K/2) (K = 0,1, .. N) IS given by NW/(KV(N- K)!). If hma x = N/4, these states coincide and the representation Is self-conJugate.

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Volume 66B, number 1 PHYSICS LETTERS 3 January 1977

2 3 [1] ( , 7) to two splnor multiplets [11] (}, 1) and one vector multlplet (1 t , g), or equivalently, by cou- pling the gauge multiplet of complex (0(2)) super- gravity [2] (2, a a 7, 7, 1) to its spinor multiplet (-~, 1, 1, 7). Its particle content is therefore: two singlets of

• 3 spin 2 and }, and two triplets of spin ~ and 1, respec- tively *2.

The Lagrangian, as well as the transformatmn laws, can be entirely fixed: by 0(3) invariance, by the fact that it must reproduce, as special cases, the 0(2) pure supergravity [2] as well as ordinary supergravaty cou- pled to the vector multiplet [9, 10], and by the power- ful requirement that the algebra of the transformation laws must close.

The complete Lagrangian is given by the sum of two pieces

£G =/~ + £4 (I)

in which d~ is the rmnimally coupled Lagranglan

1 - - l * * v p o - - l = - ( e / 2 K 2)R (e, Co) - -~ C u e 75 7uDa 40

- ~ e X T * * D u X - 4 ~, ~, * **v" po

+ (K /4 )e ~iu'r'~yg~/**XFi ~ (2)

- (K/2V~)f#k ~ [eFiUV + ~-75 F':Uvl ,~ ,

and £4 contains "seagull" terms:

£4 = _ L K 2 T , i ,t,] r,,g~,M,t, vl _ ~**14vt ) +e**Voo~loT 5 4Jo]

- (KZ/2x/r2)f i l k XTuoo° q2~ ~ 1 4 ~ (3)

+ (K2/4) e ~oo°7**X~7oX.

All latin indices i,], k run from 1 to 3 . f q k is the completely antlsymmetric tensor with 3 indices with f123 = 1. da is the c o n n e c t i o n w i t h to r s ion

6a**ab = cOOab + K**ab,

K**a b 1 2 -1 i _ ~ .Yb4la + ~t 4~] = 4 K [4**"/a4b a')'**

- ~ K 2ee**ab T X-75 7 T X,

/)p is the gravitational covariant derivative with tor-

4=2 Interestingly enough, thxs is the ftrst theory in which a non-gauge particle (spin }) belongs to the gravitational multiplet.

sion

/Sp = ap + ~ ~pab °~b- (5)

P~v is given by euvpoFPal where Fioa = apAio - OoAi o and e = de t ~u" We use the same conventions as in refs. [2, 8, 91.

The previous action gwen by £G is invariant under the following transformation laws.

Beau = K ~iTa 4 iu, (6)

~Ai u = X/~ f q k u * k u - ~iTux , (7)

8X = ouve iP '"~, (8)

8 4~ = (2/K)D**e' + (K/4) (X75 7 ° X) 3, 0 7**75 e'

+ (I/V:}) f ilk OOo 7** ek ~ o (9)

+ ( K / 2 V c } ) f Ok [(t~/uTP X) 7p ek + (~/uTp 75 X)75 7pek]

where

l ~ v - D **A vi _ D v A ~" (10)

/?** is the supercovariant derivative defined by:

^ i - i _ ( K / 2 x / ' ~ ) f i / k ~ 1 4 ~ . ( l l ) D **A v - O **A v + (K/2)~/uT.X

It is trivial that, m the case 41 = 4**, 4 2 = ¢**, if3 = X = 0 =A 1 =A2"A 3 =A. one recovers the 0(2) theory [2], and m the case"4 2 = Cu 3 = 0, ¢,u 1 = 4** ; A 2 = A 3 = 0 one recovers the Maxwell-Einstein theory of ordinary supergravity [9].

There are terms of new type both in the Lagrangian and in the transformation laws which were not present in previous models. In the Lagrangian, it is the X43 term• This term can be simply understood by noticing that the equation of motion of the spin fields is the free Dirac equation with gravitational supercovariant derivative. The new terms in the field transformation laws are the 42 term in 8X (eq. (8)) and 4X in 8 4 (eq. (9))•

In this model, the commutator of two local super- symmetry transformations with parameters e~ (x), e~(x) gives • (0 a general coordinate transformation of parameter

~"(x)= -i . i 2e27 e l,

(ix) a local Lorentz transformation of parameter

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Volume 66B, number 1 PHYSICS LETTERS 3 January 1977

t l k - t k ^1 ¢Oab(X ) = ~UKua b + KvC2 f [e2elF/ab

+ 1 -i k "~ ~ee2~/5e 1 Flabl,

(hi) a supersymmetry transformation of parameter

J ( x ) = - ' - +

- v5x( 5qk)l,

(w) a gauge transformation (acting on A/u) o f param- eter

N = (2V'2 /K) fq l¢ ~12 e~ - ~aA~.

All these transformations are field dependent. How- ever, notice that the first term in the gauge transfor- matron survives when the fields are set equal to zero. This shows that the vector fields A/u are gauge fields associated with three central charges [12] (i.e. com- muting with all the elements of the algebra) which arise in the ant icommutators of the supersymmetry transformations. Because of this, these central charges are Abelian. The second term in the field dependent supersymmetry transformation (iii) was absent in pre- vious models, and was crucial to fix the correct trans- formation laws for the gauge field ¢~.

Using the procedures of refs. [1 ,2 , 3] , the action can be shown to be locally supersymmetrlc to all orders in the grawtatlonal coupling constant K. For instance, in checking the variation, we note that terms of new type are present, in particular of the form F2Xe, F'A 2 ~e, •3 ~e, ~4Xe. All these terms vanish

separately. We now make some general comments about this

theory. We observe that, although the theory is global- ly 0 (3) invariant, the vector fields A/u do not gauge the 0 (3) group, but as explained above, the central charges. It is an open problem to see whether a gauge coupling can be introduced to provide a non Abehan 0(3) gauge symmetry, consistent with local supersym- metry t3 .

Matter couphng of the 0 (3) supergravlty would re-

3 ~3 A consxstent minimal coupling for the spin ]- gauge field of ordinary supergravity has been recently introduced by Freedman [13] m the Maxwell-Einstein theory ofref. [9]. Thts coupling, however, reqmres an enormous cosmologi- cal term if the gauge coupling is a~umed to be of the order of the electric charge.

qmre the introduct ion of a Yang-Mllls multiplet. The latter contains a vector, four Majorana spinors, six spinless particles, all massless and in the adjoint repre- sentation of the Yang-Mills group. Incidentally such a multiplet turns out to have a bigger (0(4) ) global super- symmetry [6].

As far as renormahzation is concerned, it has been shown that pure supergravity theories are one-loop re- normalizable [14] and may be also two-loop renor- mahzable [15 ]. The same reasoning has good chances to apply to this theory. If this is so, the Maxwell- Einstein system of ordinary supergravlty [9, 10], which has been proved to be non-renormalizable [16], is promoted to a renormalizable theory after two-spin (3 , 1) multlplets have been added.

One of us (S.F.) would like to thank Professor Murray Gell-Mann for illuminating discussions on extended supersymmetry. He also acknowledges use- ful conversations with Professors D.Z. Freedman and P. Van Nieuwenhuizen.

References

[1] D.Z. Freedman, P. Van Nieuwenhmzen and S. Ferrara, Phys. Rev. D13 (1976). S. Deser and B. Zummo, Phys. Lett. 62B (1976) 335.

[2] S. Ferrara and P Van Nieuwenhuizen, ITP-SB-76-48 preprmt, to be published.

[3] A. Salam and J. Strathdee, Nucl. Phys. B80 (1974) 499; Nucl. Phys. B84 (1975) 127. B. Zummo, m: Proc. of the 17th Int. Conf. on High energy physics, London, 1974, ed. J.R. Smith (Rutherford Lab. Chllton, Didcot, U.K.), P. Fayet, Nucl. Phys. Bl13 (1976) 135; M. Gell-Mann and Y. Ne'eman, unpubhshcd.

[4] A. Neveu and J.H. Schwarz, Nucl. Phys. B31 (1971) 86; A. Neveu and J.H. Schwarz, Phys. Rev. D4 (1971) 1109; P. Ramond, Phys. Rev. D3 (1971) 2415.

[5] E Cremmer and J. Scherk, Nucl. Phys. B103 (1976) 399. [6] F. Ghozzl, D. Olive and J. Scherk, PTENS preprint 76/21,

to be pubhshed in Phys. Lett. F. Ghozzl, D. Olwe and J. Scherk, CERN preprint TH/2253, to be published.

[7] M. Gell-Mann and Y. Ne'eman, unpublished. [81 S Ferrara, F. Ghozzi, J. Scherk and P. Van Nteuwenhmzen,

PTENS 76/19 preprint, to be published in Nucl. Phys.; S. Ferrara, D.Z. Freedman, P. Van Nieuwenhuizen, P. Breitenlohner, F Ghozzi and J. Scherk, ITP-Sb-76-46 preprlnt, to be pubhshed m Phys. Rev.

[9] S. Ferrara, J. Scherk and P. Van Nleuwenhuizen, Phys. Rev. Lett. 37 (1976) 1035.

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Volume 66B, number 1 PHYSICS LETTERS 3 January 1977

[10] The coupling of supergravlty to the vector multiplet has been also derived in ftrst order formahsm: S. Ferrara, F. Ghozzi, J. Scherk and P. Van Nieuwenhuizen, PTENS 76/19 preprlnt, to be published m Nucl. Phys.; D.Z. Freedman and J. Schwarz, ITP-SB-76-41 preprint (1976), to be pubhshed; S. Deser and B. Zumlno, unpublished.

[11] V.I. Ogievetsky and E. Sokatchev, JETP 23 (1976) 66; Dubna preprint, to appear m Nucl. Phys. B.

[12] R. Haag, J.T. Lopuszanskl and M. Sohnius, Nucl. Phys. B88, (1975) t57.

[13 ] D.Z. Freedman, ITP-SB-76-50 preprmt (1976). [14] M.T. Grisaru, P. Van Nieuwenhuizen and J.A.M.

X?ermaseren, ITP-SB-76-52 preprint (1976). [ 15 ] M.T. Grisar u, Brandeis preprmt (1976). [16] P. Van Nieuwenhuizen and J.A.M. Vermaseren, ITP-SB-

76-44 preprmt (1976), to be published in Phys. Lett.

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