The N = 8 supergravity in superspace

Volume 88B, number 3,4 PHYSICS LETTERS 17 December 1979

THE N = 8 S U P E R G R A V I T Y IN S U P E R S P A C E

Lars BRINK and Paul HOWE Institute o f Theoretical Physics, S-412 96 GSteborg, Sweden

Received 1 October 1979

The N=8 supergravity is presented in superspace. The theory is on-shell and has the expected local SU(8) and global E 7 invariances.

The largest possible extended supergravity theory with a linearly realized spectrum is the N = 8 theory which has been presented in great detail by Cremmer and Julia [1 ]. They obtained the theory by dimen- sionally reducing the 11 -dimensional simple supergravity and found as a result that the corresponding four-dimensional theory had a local SU(8) invariance and a global (non-compact) E 7 invariance. Extended supergravity for arbitrary N has been studied in superspace by MacDowell [2], but his approach is based on the "natural" extended supergroup Osp(N; 4) and therefore leads to the N--- 8 theory with a local SO(8) gauge invariance (with a corresponding propagating field as opposed to the SU(8) case where the gauge field is composite). In this letter we shall take a more pragmatic point of view and build the SU(8) gauge invariance into the tangent space. In this way we are able to describe t h e N = 8 theory o f ref. [1] as a geometri- cal theory in superspace (indeed, we can proceed in a similar fashion for N < 8, but we shall focus exclu- sively on the maximal value of N here). We feel that it is important to know the superspace theory as it may lead us to as yet unknown results (e.g. auxiliary fields), facilitate quantum calculations via a lagrangian and supergraphs [3], and may suggest possible altera- tions to existing theories.

The superspace under consideration has coordinates m = (xU, 0 ~ , 0 ~ ' ) . We summarize index and other conventions in table I for convenience and the basic fields are the vielbein which relates the coordinate one-forms to the tangent space basis* 1

,1 For a detailed exposition of differential forms in superspace see, e.g., Wess [7].

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Table 1 Conventions a)

A. Indices Tangent Curved space space

All MNPQ ¢ff[ Q'~ Spinor and internal ABC DEF ~ c ~ Spinor abcdef ~ Internal (SU(8)) i] ... t ~ n Vector uvw ... z I~ u SL(8, R), E7 IJKL -

B. Summation convention and spinors

X M Y M = XUYu + X A Y A - x A ' Y A , '

X A = X a, X A ' = X a'i,

YA: r;, = to,i,

Xab, = (oU)ab , Xu ,

(oU)ab , = (1,~),

1 U O C' (oUO)ab=~ {( lo )ac,(O )b - u ~ o ) ,

X a = ~abXbi , tab _

metric ~uo = (1, -1, -1 , -1),

;) • e 0 1 2 3 = 1

euoxy.

a) Complex conjugation replaces spinor indices by primed. ones and raises or lowers internal ones, e.g. (xa) += ~a t.

E M = d z C ~ Ec ,~ M ,

and the connection one-form

aM N = ~ a . . . r t tJ v .

(1)

(2)


Under the tangent space group, prototype vectors transform as

8 x M = X N L N M, 8XM=--LMNXN , (3)

with

Luv =-Lvu , LA B = 8{La b + 6abLl,

= x J r b' LA ,B' vi,_, a, + 6a,b'Lli ,

- L _ l , uv, L = - , Lab - ba- 2(0 )ab uv' La'b' -La'b' (4)

' = - i " = 0 Ltj -L] , Lti ,

and all others vanishing. Thus Lab corresponds to SL(2, C) and L z] to SU(8). The connection one-form f2MN has the same symmetry properties as LM N, although it transform inhomogeneously,

8aM N= -LMQaQ N + ~2MQLQN-dLM N . (5)

With the aid of ~2 one can therefore form an exterior covariant derivative, for example

DX M = dX M + xN~2NM. (6)

From E M and ~M N we construct the torsion two- form

T M = DE M = ½ E Q ̂ E P TpQ M , (7)

and the curvature

RM N = d~M N + ~M Q ̂ ~2Q N = ½ E Q ̂ E P RpQM N, (8)

the latter having the symmetries OfLM N on the indices M, N. From the properties of exterior derivation one may derive the Bianchi identities

DT M = EN ̂ RN M , (9)

DRMN =0. (10)

In components, (9) reads

(MNP, Q}

= RMNe o -- D M TNpQ - TMN s Tspt2 + cyclic = 0 ,

(11)

where the cyclic sum is generalized (one obtains a plus sign when permuting two spinorial indices and a minus sign otherwise). A useful theorem [4] assures us that if (9) is satisfied then so is (10) if one uses the com- mutation properties of two D's. Furthermore, it has been shown [4] that the torsion completely deter- mines the curvature via (9), because of the latter's symmetry properties (4). The problem is then to find torsions satisfying (9) and our solution is summarized in table 2 where we have already eliminated any auxiliary fields. We see that all components of the torsion and therefore of the curvature (table 3) are expressible in terms of a single superfield Wi/ka and its derivatives. Wis totally antisymmetric on ijk and its leading components (0 = 0) therefore corresponds to a set of Weyl spinors transforming under the 56 of SU(8). In addi-

Table 2 T o r s i o n s

P , . t

= _ V _ T W=o, T A B U _ • u i TAB C =eab~qkc • TAB u TAB C= TAB ' C ' - TAu - uo - - l ( a )ab,8],

(aU)aa,TuB C' = ieab~la ,c']k + i6a,C'Niak,

Mabi/=lD~aWb)ijk, N f k _ 1 ]klrnnpqr . . . . . . ab - - ~-~ e )Vlmnan, pqrb ,

( oU) aa, T uBC = _ 1 , . rc ] + -~4 i S ik [ eab JCa, + 8aC J ba, ] 21 ~ab ~ a' k

,iklw/klo, 4a,= a,kk, (aU)aa '(°v)bb' ruo c = Cab ~ 'b 'Ck + ea'b' {UabCk + SaC Ubk + 8 b c Uak }

Viabc , 1 ~ i lm. , + I.-~ilmn.., - i I . fiilmn,., c 1 Ai r~lmn = 5 w c, lV*lmab ] l / ~ c , ( a Wb)lm n, --f~l c'c Wlmn s 4 ~ lmn"c '

~ , b , c , = - l fftgla,mNb,c,)l m - 1D( ,kM~k,), Pba,i]kl = -½iDa,iW/klb,

i _ 1 - i . . . a - t l t a ) ~ ~imnfi~pqra' A ]kl - - ~ Da W]kl = ~" ~" j ~]klmnpqr "" a . . . .

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Volume 88B, number 3,4

Table 3 Curvatures.

PHYSICS LETTERS 17 December 1979

. . . . m . ,

1 i I ^ i j t + 1 i I i RAB' ,cd=~eac(Jdb ' ] g~] db'S ~ e a b ( J c b ' / - g 6 / J c b ' } ,

( v .bb '~i _ 3. - " 1. i " " Cr } l(ao, C d - a l ( e a c ¢ b d + , a d , b c ) U l b , + ~ , ( , c d V ~ b b , + 2ebdVlcb , - 4eadV~cb,),

(ov)bb'Ri _ . - i i i av, c'd' - leabUb'c'd' + eb'd'Rabc' + eb'c'Rabcd' '

R i = 1. i 1. ufi abc' 41Vabc' - 41Cab c "

ou)aa'.oo.bb'R + ea,b,Aabcd, ( ) uv,cd = eabBa'b'cd

= 1 . . . .~ km + 4 N _ k m ~ . . . . m 31Da'(cWb'qkWd)ijk+31Wb ' Da'(cWd)i]k Ba,b,cd -i~14McdkmMa,b , ca a o t¢ 2. - "" 2. - ilk

5 j k j m 7 +2 o ~ ijkl .,. - Z (ca' m d)b' k + iT4J(ca 'Jd)b ' - 3" da'i]kl'cb ' - ( a ' o b ' ) ] ,

1 k Aabcd = i~ [2D(cUabd) k + 4N(acMbd)k m + eacAbd + ebcAad + eadAbc + ebdAac] ,

1 . - . . . . - ' " - s . i . a a ' / A b d = r z l O b a , W a l l k W d i ] k - f i i f f /a i fkDba, Wdifk - 3 6 i w a q k D d a , Wbi]k + ~ e b d a a a , j a i

1 ebdJaa , j aa '+ l e ~. i]~a'b . . . . l~_ M 9 ~ N. ,ai l 17~8 6 bd a'b' i ] - ~ " e b d a e i ] Nacq + ' ~ M . . . . l-~ c'ifkl v

ataq o) - 9"d "c'bi/kl '

R A f t k l =-aab'tfl- k " +~6]dab, i - k -2~jdab'll 1¢_ i _-201aab,l̂ i- k +] 4¢~l ' tab '] - - i2°] i I ~i.kJab,+_6~jl_k.i-Olaab,,

Jab ikl.l = ~ ikm W b' f lma'

= _ f i ikmn 1 ~i-~ kmnr 1 _k~ imnr~.. RAukl u Whnna - 3 ° l ~ t Wmnra + 661 ru Wmnr'

R i . = 2 ~ i l m n o uv l ] ' [ u "v]hnn "

tion to determining the torsions and curvatures, there are numerous relations between the derivatives of the field W. In fact, there are very few more independent objects occurring as derivatives, amongst them

1 . -

ea'bi jk l = - '~ 1 G ' i l¥1klb ' (12)

and

Mabij = 1O~a Wb)ij k . ( 1 3 )

P is totally antisymmetric on ijkl and is in addition taken to be self-dual,

p-- "" = l , = i j k l m n p q p (14) - ab 'qkl 4! ~ * mnpqab ' "

It therefore corresponds to a set of scalar field strengths (the scalars being considered as Ramond fields [5]) transforming under the 70 of SU(8). M is symmetric on ab and antisymmetric on i] and we can therefore

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define

l¢o ~abM , (15) ~ijuv = 2" uo j abij

as a set of spin-one field strengths which transform as a 28 under SU(8) and which are "space-time" complex self-dual

~ril"uv = ½ eu vxy ~rxyq = i ~ i juv " ( 1 6 )

There are two further combinations which do not directly enter the equations of motion, namely U i abc and Aabcd (tables 2 and 3, respectively). The former corresponds to that part o f the Rarita-Schwinger field strength which does not appear in the equation of motion whilst the latter corresponds to the superspace Weyl spinor. (By the Rarita-Schwinger field strength we mean the curl of the Rarita-Schwinger field.)

The fields W, P, M, Uaibc and Aabcd are in fact the


only independent combinations. That is to say, one may build up the entire super field W from the above objects and their vector derivatives evaluated at 0 = 0 [6]. The spin-2 and spin-3/2 fields themselves occur to zeroth order in the vielbein, which may be chosen to be

EuU = e u, E A = t ~ u A ' EuA' = ~ A' ,

E _ ~ B = 6 q ~ B, E ~ B ' = E c { , B = E q { U = E ~ , u = o ,

E ~ , B' = - 6 ~ , B' , (17)

where euU is the x-space vielbein and ~tz A the Rar i t a - Schwinger fields which transform as an 8 under SU(8) (it is a set of Weyl spinors, with ~tt A' transforming according to the 8* representation). We have thus suc- ceeded in reproducing the N = 8 spectrum, namely one spin-2, 8 spin-3/2, 28 spin-l, 56 spin 1/2 and 70 spin-0, although the spin-1 and spin-0 fields appear only as field strengths.

Since the auxiliary fields have been explicitly eliminated and because our formalism is manifestly super- symmetric, the remaining fields must be on-shell (otherwise the number of bosonic and fermionic degrees of freedom would not match). The actual equations of motion are, however, also derivable from the Bianchi identities. Since they are of necessity highly compli- cated, we shall briefly sketch how this may be done, postponing the details to a future report.

The spin-2 equation may be gotten by computing the x-space curvature at 0 = O,

Ro~,,x y = Ez, N E~M R MNx y l0 =0 ' (I g)

and hence the Ricci tensor. Likewise, the spin-3/2 equation may be obtained from Tuo C at 0 = O. We have

Tuo c = EuC'ttt E ~ a q z E qt t c - u ~ v

TuoC lo=O = euU eoVav ~bu c - u ~" v + other terms, (19)

from which we directly find the spin-3/2 field strength and hence the Rarita-Schwinger equation.

The spin-1 equation, together with the appropriate Bianchi identity (see below) are both contained in the identity (uv C, D' ) which contains no curvatures, whilst the spin-l/2 equation may be extracted from the identity (AB u, D') , also containing no curvatures. Finally, the spin-0 equation is most easily obtained

by directly differentiating P via its definition (12) and using commutators.

Thus, we have the correct spectrum for N = 8, of necessity on-shell, but we have seen no trace of the scalar fields themselves or of the spin-one potentials. Neither have we seen any E 7 invariance. Indeed, in claiming that the spin-1 field strength is Mabi /we have really cheated a little as it is not difficult to see that a local U(1) gauge invariance is incompatible with the SU(8) local invariance - one cannot directly define potentials in an SU(8) covariant fashion, and the Bianchi identity alluded to above is somewhat spe- cious, as it has additional terms over and above the normal ones. Thus we see that there must be a global symmetry and we shall now indicate that the results we have so far actually imply its existence as well as that of the missing fields. Introducing the one-form P (table 4) and defining

(20)

we find that, as a consequence of the Bianchi identities (9) and our solution, these fields automatically satisfy

DPi]kl = 0, RiJkl =p i lmn A P m n k l . (21)

(21) can be rewritten in block matrix notat ion as

Table 4 The one-form P and the field strength F.

Pjktm = E UPu/klm + EA XA/ktm - EA'YA'jklm'

XAjklrn = 26i. Wklm" , IJ la

YA 'jklrn = ( 2 / 4 ! ) e i/klmrst Wa 'rst ' Pu given in table 2,

~jklm = (1/4 !) jklmpqrSPpqrs,

1 N M Fkl= ~E A E FMNkl,

FAB, kl .= - 2ieab6i(kl],

FAB, kl = FA,B,,k l = FuA,k I = O,

(oU)aa,FuB,,kl = Ca, b, W]kla ,

(oU) aa,(a°)bb,F uo, k l = - ie ab.~a,b,kl - ie a,b,Mabkl .

271


or, with

^ Q

~ = ( p 0 ) , (23)

as

d ~ - ~ A ~2 = 0 . (24)

Hence, ~2 can be considered as a connection one- form having zero curvature. From the structure of (23) we see that the dimension of the Lie-algebra is 70 + 63 = 133, and so the relevant group is, as anticipated, E 7 , 2 . (Although the original Bianchis (9) are of course not covariant under a local E 7.) From (24), we see that

is pure gauge and hence has the form

= v - l d V , (25)

where V is an element of the group E 7 in the funda- mental (56) representation. Hence V is of the form

• UIJ.. V= ( q ~IJi]~

VIJij ~iji j ], (26)

where each entry is a 28 X 28 matrix, being skew in both the 8 × 8 pairs I J, i]. It may be thought of as a kind of vierbein-like object where the local SU(8) acts on the lower case indices to the right and the global E 7 acts on the capital indices to the left by

= (2611K AJ] L] ~IJKL 6V l V = - A V ' (27)

--\~IJKL 26[/K ~" LIjI

where AIj is traceless skew-hermitian, and ~ self-dual,

~IIgl KI LI = ~V. ellglK1L lI2J2K2L2 ]~I2J2K2L2 . (28)

Hence ~ is E 7 invariant, but transforms inhomogeneously on the principal diagonal under SU(8), and the SU(8) connection is thus determined in terms of V. With the aid of V, we can also solve the spin-1 problem. Introducing the two-form Fi/(table 4) we find

4:2 Details of E 7 can be found in ref. [1]. Our only change is to essentially transpose everything.

that it satisfies tile identity (as a consequence of (9) and table 2)

DFii = Pi/kl A ff'kl. (29)

(The uvw component of this equation is the "Bianchi" identity for i/alluded to above.) If we now set

(Fi/' yi /) = (FIj ' pIJ) v ' (30)

then, in terms ofFi j , eq. (29) becomes simply

dFij = 0 . (31 )

From which we conclude that there exists potential one-forms and corresponding U(1)invariances. Under E 7 we have

6(F, F) = (F, Y3A, (32)

as required so that Fi/is E 7 invariant. Finally, we note that F is complex,

FI s = ai J + iHIJ, fflS = ai J _ iHIJ, (33)

where GIj (H 1J) transforms under the 28 (respectively, the contragredient 28) of the SL(8, R) obtained from A by restricting A/j , ~-'IJKL to be real. However, we do not have 56 spin-1 fields, as the relevant dynamical part ofFi/is ~ruv, i / (16) and is self-dual. Hence G and H are essentially dual versions of one another.

This concludes the presentation of our results con- cerning the N = 8 theory in superspace, which are in exact correspondence with the x-space theory [1 ]. A more detailed report in which we shall also discuss the auxiliary fields is in preparation.

References

[1] E. Cremmer and B. Julia, LPTENS 79/6, to be published. [2] S. MacDowell, Phys. Lett. 80B (1979) 212; and to be

published. [3] W. Siegel, Harvard preprint HUTP-79/A005. [4] N. Dragon, Z. Phys. C2 (1979) 29. [5] M. Kalb and P. Ramond, Phys. Rev. D9 (1974) 2773. [6] L. Brink, M. Gell-Mann, P. Ramond and J.H. Schwarz,

Phys. Lett. 74B (1978) 336; 76B (1978) 417; J. Wess and B. Zumino, Phys. Lett. 79B (1978) 394.

[7] J. Wess, Topics in quantum field theory and gauge theories (Salamanca, 1977) ed. J.A. de Azc~rraga.

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The N = 8 supergravity in superspace