Volume 91B, number 3,4 PHYSICS LETTERS 21 April 1980

ELEVEN-DIMENSIONAL SUPERGRAVITY ON THE MASS SHELL IN SUPERSPACE

Lars BRINK and Paul HOWE Institute o f Theoretical Physics, S-412 96 G6teborg, Sweden

Received 15 February 1980

The eleven-dimensional supergravity theory is presented in superspace on the mass shell

One of the major problems of the N = 8 super- gravity theory that has yet to be resolved is the auxiliary field structure. The theory was first dis- covered in x-space by Cremmer and Julia [ 1], who used the method of dimensional reduction, and then recast in superspace language by the authors [2] with the goal of obtaining the auxiliary fields as a primary motivation. This problem has, as yet , proved to be somewhat more intractable than anticipated and it is for this reason that we propose to discuss the theory in yet another setting, viz. a superspace with eleven commuting and thir ty-two anticommuting coordi- nates. The 11-dimensional x-space theory is known [3] and was employed by Cremmer and Julia in their construction of the four-dimensional version. The ad- vantage of this seemingly roundabout route is that there are (formally) less fields to deal with in eleven dimensions, and it would be convenient if this state of affairs continued off of the mass shell. Whether this is so or not is a moot point, but, as we shall show, the on-shell superspace theory is much simpler in form than the corresponding four-dimensional version.

As usual, we suppose that the superspace has lorentzian tangent space structure, so that the basic fields are the vielbein, EM A , and the connection, ~'2MA B and the corresponding 1-forms ,1

~2 Latin (greek) indices refer to vectorial (spinorial) quanti- ties, whilst capitals refer to both. Letters from the begin- ning of the alphabet are used to denote tangent space in- dices, and letters from the middle refer to curved space ones. For a detailed treatment of lorentzian superspace formalism, see ref. [4].

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E A dzEM_M A , (1)

~'~A B = dz M ~-2MAB

The lorentzian assumption implies

~ab = --~2ba ' ~ u b = 0 , (2)

1 b ~'2~13 = ~(I "a )a3~"2ab , ~2a~ = 0 .

From these basic fields one forms the torsion,

T A - L ~ - C r - B T A -2 ,~ ,-, IBC = D E A = d E A + E B ~ B A , (3)

and the curvature

RA B 1EDL, C R B_ d a A B + a A C a d (4) = ~ r, CD,A -

the latter having the same symmetry properties as ~2A B. One then has the Bianchi identities

D T A = EB R B A , (5)

D R A B = O, (6)

or, in component form

(RAB D - D A T B C D - T A B E T E c D) = 0 , (7) (ABC)

~_J (DARBCD E + TABFRFcD E) = 0 , (8) (ABC)

where ~(ABC) denotes the graded cyclic sum. For the tangent space group structure (2), (8) is satisfied if (7) is.

In eleven dimensions, the dynamical fields of super- gravity in x-space are the graviton, the gravitino and a 3-form potential Xmn r [3], the latter being subject to

Volume 91B, number 3,4 PHYSICS LETTERS 21 April 1980

the gauge transformation

~Xmnr = D[m Ynr], Ynr = - Y r n • (9)

However, the geometrical fields EM A and g2MA B can- not contain such a field as may be seen from the fol- lowing simple argument: let ~A(x, O) be the parameter of a general coordinate transformation and LAB(x, O) the parameter of a Lorentz transformation, then we have

5Eu A = ~u~ A + ... , (10)

6~uA B = --OuLA B + ....

Thus, in a 0-expansion of the parameters, only the leading (0 = 0) terms do not appear in (10). All the remaining parameter components merely translate

irrelevant fields in E,, A,,. g2uA B and hence physically the only x-space gauges are ~A 10 =0 and LAB[o=o , i.e. x-space coordinate transformations, local supersym- metry and local Lorentz transformations. Therefore, to accommodate Xmn r, we introduce the 3-form potential

X = ( t / 3 ! ) E C E B E A XAB C , (1 1)

with the corresponding gauge transformation

5X = d Y , Y = ~EBE A Y A B , (12)

and field-strength 4-form H,

H = dX = (1 /4! )EDECEBE A HABCD . (13)

The Bianchi identity for H is

d H = O , (14)

or, in components

(DAHBCDE + TAB F HFCDE) (ABCDE)

-- ~ . (--1)BC+BD+CETADFHFBEC = 0. (15) (ADBEC)

Our problem is now to solve the coupled Bianchi iden- tities (7) and (15) simultaneously, and we shall do this by employing dimensional arguments. One can assign dimensions to the fields either as their four-dimen- sional ones (as we are ultimately interested in dimen- sionally reducing the theory) or by observing that in eleven dimensions • has dimension 9/2 (length). Either way, when we factor out K, canonical bosons

will be left with dimension 0, and canonical fermions with dimension - 1 / 2 (length). As the physical fields themselves are not gauge invariant and as we are on the mass shell, we may conclude that TAB c, HABCD only contain constants, the field strengths for Xmn r (dimension -1 ) , the gravitino ( - 3 / 2 ) and the graviton ( - 2 ) and also the equations of motion which have dimension - 2 , - 3 / 2 and - 2 , respectively. We are therefore forced to choose (up to a normalization ,2)

T~C = - i ( P c £ e ' (16)

T~S/= Tab c = O,

and

n~e,.r8 = Huo.rc = Habcd = 0 , (17 )

Ha~cd = i(Fcd)a¢~ .

Further, although TabC has dimension - 1 we may always choose the connection such that it vanishes ,a On the other hand, Ta~7 may contain nabcd linearly and utilising the R ~ , c d and Ra~,.r8 identities from (7) and the DaH~cde identity from (15) one finds

1 H tpbcd'~ + t--- t r ", 14bcde Ta/3y = 3-6 abcdt Jfl~, 2sSt'abedel[3.r*" •

(18)

We therefore have only one torsion remaining, namely Tab'Y of dimension - 3 / 2 , and from the Rc~b,c d and Rab,,y8 components of (7), together with the Dc~Hbcde component of (15) one finds

Tab s = ~ i ( p c d ) ~D~Habcd , (19)

together with

' (20) DaHbcde = - ~ ( F [ b c P fg) J D {3 Hde ] fg ,

and

(pabc) e Tbc~ = O . (21)

Eq. (20) shows that Do, Habcd is entirely taken up by

,2 We use real p matrices (Majorana representation) and

i.al...an = F[alFa2 ... Fan I .

An independent set is given by ra, i.ab, r, abc (symmetric) and C, r abc, r abcd (skew-symmetric) where C is the charge conjugation matrix (= p0).

,3 These torsion components are therefore the same as in the four-dimensional case [5].

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Volume 91B, number 3,4 PHYSICS LETTERS 21 April 1980

the gravitino field strength ,4 (19), whilst (21) is the spin 3/2 equation of motion. The final identi ty to be solved involves Rab,c d itself and we have

Rab,y 8 = D a Tb,r~ - O b Ta~/8 + O~/Tab 8

+ Ta3,e Tbe ~ - Tb.re Tae a . (22)

Contracting this equation with (Pcd) y8 , we find Rab,c d and, in particular, the spin-2 equation of motion is

Rab - l rlabR = - ~ ( 4 H a c d e H b cde

-- ½rlab H c d e f H c d e f ) , (23)

where Rab = f lbdRab,cd, R = rlabRab . The remaining components o f R a b , c d are taken up by the Weyl ten- sor. If we contract (22) with (1-'c)~ we obtain the equation of motion for the Xmn r field,

_ ] f f Hel . . .e4Hfl . . . f4 Da Habcd 36 • 48 ebcdq. . .e4 1... 4 (24)

Since the only new component o f n a b c d at the level of two D's is the Weyl tensor, the only candidate for a new field at the level of three D ' s is contained in the left-hand side of (25). However, the right-hand side of (25) consists of vector derivatives and bilinear combi- nations of known fields, thus justifying our assertion.

To summarize, the mass-shell 11-dimensional super- gravity in superspace is described by a coupled system of the geometric variables (vielbein and connection) and a 3-form potential X. The torsions, curvatures and components of the 4-form H are all expressible in terms of a single superfield Habcd whose independent components are n a b c d (0 = 0), that part of the gravi- tino field strength which does not vanish as a conse- quence of the equation of motion (21) and the Weyl tensor. The equations of motion (21), (23) and (24) are precisely those that one derives from the x-space action, although expressed in covariant form (the spin 3 / 2 - m a t t e r couplings appear when one transforms from the tangent space basis to the x-space curved basis by means of the vielbein).

Further contractions merely solve for the remaining components of D r Tab 8 in terms of vector derivatives and bilinear combinations o f Habcd. Thus D a D ~ H c d e f is entirely taken up by terms of this type and the Weyl tensor. Hence, we have shown that the torsions, cur- vatures and components Of HABCD that are not zero or constants are components of the superfield Habcd. To show that there are no new independent compo- nents when one attacks nabcd with a third spinorial derivative one may use (8) * s. One has

De~Rbc, d e = DbRcw, d e - DcRc~b,de + Tb,~R3,cd e

- Tca' tRTb,d e - Tbc#R#a, de . (25)

,4 By the gravitino field strength we mean the covariant curl of the spin-3/2 field. Not all of the components of this ob- ject are zero when the equation of motion (21) is satisfied.

,s This method of identifying the components of a superfield by taking spinorial derivatives is due to Wess and Zumino [6].

N o t e added : After the complet ion of this work, we learned from Sergio Ferrara that similar results had been obtained by a group in Paris.

References

[1] E. Cremmer and B. Julia, Phys. Lett. 80B (1978) 48; Nucl. Phys. B159 (1979) 141.

[2] L. Brink and P. Howe, Phys. Lett. 88B (1979) 268; P. Howe, in: Supergravity, eds. P. van Nieuwenhuizen and D.Z. Freedman (North-Holland, Amsterdam, 1979).

[3] E. Cremmer, B. Julia and J. Scherk, Phys. Lett. 76B (1978) 409.

[4] R. Grimm, J. Wess and B. Zumino, Nucl. Phys. B152 (1979) 225.

[5] J. Wess and B. Zumino, Phys. Lett. 66B (1977) 361; R. Grimm, J. Wess and B. Zumino, Phys. Lett. 73B (1978) 415.

[6] J. Wess and B. Zumino, Phys. Lett. 79B (1978) 394.

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