Volume 132B, number 4,5,6 PHYSICS LETTERS 1 December 1983

A FULL SUPERSPACE ACTION FOR OFF-SHELL N = 2 SUPERGRAVITY

Alice ROGERS 1 Theoretical Physics Group, Blackett Laboratory, Imperial College o f Science and Technology, London SW7 2BZ, UK

Received 10 August 1983

Off-shell N = 2 supergravity is formulated in terms of an action which is the integral of a simple scalar function over the full superspace of the theory. This is an extended superspace, with an extra complex bosonic dimension corresponding to the central charge, which gives the superspace measure the correct dimension for constructing the action.

Although the first supergravity theory [1] was written down as long ago as 1976, progress in quantis- ing such theories and in particular in establishing whether they have the desired ultraviolet properties, has been held up in the case of extended supergravity theories (particularly for N/> 3) by the lack of off- shell versions of the theories, and by the lack of an expression for the action in terms of a full superspace integral.

In this paper off-shell N = 2 supergravity is formu- lated in terms of a simple action which is an integral over the full superspace of the theory - which is an extended superspace with an additional complex bo- sonic coordinate to the off-shell "central charge". This version of the theory is extremely simple, it has the additional merit of making further superfield tech- niques possible, and giving some indication of how the off-shell problem for N ~> 3 might be resolved. It has been recognised for some time [2,3] that central charges can play an important part in off-shell ex- tended supergravity theories, and the analysis of Tay- lor [4] and Rivelles and Taylor [5] shows that central charges are essential when N is greater than 2. In this paper one sees from the geometric point of view the role of the extra dimensions corresponding to the cen- tral charge, in enabling one to construct a simple ac- tion - while for higher N one may expect the extra dimensions not to be an optional extra but to be essential, both for the building of off-shell torsions

1 SERC research fellow.

and curvatures consistent with the Bianchi identities and for building actions.

A simple and elegant formulation of minimal off- shell N = 1 supergravity, based on a very simple super- space action, was given by Wess and Zumino [6,7] ; (the same theory was derived in component form by Stelle and West and by Ferrara and Van Nieuwen- huizen [8]). More recently, the new minimal version of off-shell N = 1 supergravity (originally presented by Sohnius and West [9]) has been formulated in terms of a superspace action, both at the linear level by Bed- ding and Lang [10] and at the full non-linear level by Howe et al. [ 11 ].

There are various component formulations of off- shell N = 2 supergravity [ 12] ; among versions formu- lated in superspace, including actions generating the dynamics, are that by Breitenlohner and Sohnius [13] and Sokatchev [14]. I f one tries to extend the ap- proach used for N = 1 supergravity, where superspace has 4 real even dimensions x m (m = 0, 1, 2, 3) and one complex odd two-component spinor, dimensions 0 u

(together with complex conjugate 0u), to the super- space usually used for N = 2 supergravity - which has four real even x m and 2 odd complex two-component spinors 0t u (t = 1,2) - one immediately finds that the measure dz = dx dO dO has dimension zero (in mass units). This is because

dimensionx m = - 1 , dimension 0~ = - ½ , (1)

and, as a consequence of Berezin integration,

0 .031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland 333

Volume 132B, number 4,5,6 PHYSICS LETTERS 1 December 1983

dimension d0~ = -dimension 0~. (2)

Thus, to construct a dimensionless action of the form

(where g is the gravitational constant and E is the berezinian superdeterminant of the vielbein) a super- field B of dimension - 2 is required [15]. While such a prepotential may exist, a formulation involving pre- potentials has not yet been found - also, for further extended supergravities, where the dimension of the measure becomes progressively more positive, super- fields of more and more negative dimension are required; these are further and further away from the geometrical objects of the theory, the torsions and curvatures in superspace, all of which have dimension at least zero. The authors of refs. [13,14] avoid the problem by integrating over submanifolds. An alter- native possibility is to introduce extra dimensions into superspace, leading to a measure with more tractable dimensions. This possibility is explored in this paper, and leads to an extremely simple result.

The starting point is on-shell N = 2 supergravity in the extended superspace formulation due to Howe and LindstrSm [16] and Kallosh [17];(these authors treat N = 8 supergravity, the explicit truncation to N = 2 is given by Howe in ref. [18]). Extended super- space has the usual 4 real even coordinates x m and 2 odd complex two-component spinor coordinates 0t g (with complex conjugate oftt); and also an extra even complex coordinate ytU = _yUt. (This apparently redundant labelling o fy coordinates by tu is conve- nient for keeping track of indices.) The entire set of coordinates is denoted z M.

The coframe bundle is reduced to a SL(2, C) × U(2) bundle, with the preferred set of one-forms E A = (E a, E~, Ei]) related to the coordinate one-forms dz M by the "sehrvielbein" matrix

E A = dz M EM A . (4)

(The tangent space index conventions are: general A, B, ..., vector a, b ..... spinor a, fl ..... internal i , ] . . . . . )

The action of SL(2, C) × U(2) on the E A is defined by

8E A = EBLB A , Lab = - L b a , (5)

1 b - - ' "

La13 = : (o a ) ~ L a b , Li] = - L l i ,

a] - - - ' ~ & i ,

LiJkl = - Z k l ij = 2~ [~L j] i] , (5 con 'd )

with all other components of LA B vanishing. The geometry is completed by the introduction of a con- nection one-form CA B and corresponding torsion and curvature,

T A = D E A , RA B = DeA B . (6)

The on-shell theory is now specified by giving the val- ues of the torsion and curvature; it is evident from the Bianchi identities

D T A = E B A RB A , DRA B = 0 , (7)

that the various components of torsion and curvature are highly interdependent, and in fact all the compo- nents of torsion and curvature can be expressed in terms of a single complex superfield lkIa#ij (~&~ij)

which is symmetric in a/3 and antisymmetric in ij; it is sufficient to specify the following torsions:

Dimension zero. The only non-zero torsions are:

• i = --lad c~ 5 ] ,

T/~ M = -2 iec~6~ j] (8a)

(together with their complex conjugates). Dimension half:

All torsions are zero. (8b)

Dimension one. The only non-zero torsions are:

T#~ 7~ i]= iepr~I~.~ i / , TZ d i'~k = i e ~ M j ~ ik (8c)

(together with their complex conjugates). (Here two- component spinor notation is used for SL(2, C) vec- tors with Va ~ Vc& = Vaoac,&. The conventions used are those of Howe [ 18]). On shell the extra dimension does not lead to extra physics, and the component version of the theory (including the field equations, which are built in to the superspace version by virtue of the Bianchi identities) is obtained as in Brink and Howe [19], except that we now consider values of the superfields and the sehrvielbein when 0 = 0 = y = y = 0, and we choose a gauge in which the sehrvielbein EM A takes the form

334

Volume 132B, number 4,5,6 PHYSICS LETTERS 1 December 1983

l em a t~ m ~ ~m ki Ami f Am ' 7 \ t

i ! 5u6 i 0 . . 0 0 o o o ,

o o q , . l \ \ 0 0 0 0 0

(9)

where em a is the vielbein, ~m c~ the Rari ta-Schwinger field and Am i / the spin one "potential" .

This approach is slightly different from that adopted in refs. [16,18], where it is required that "O/OY TM, O/a2tu is zero on all fields. The two give equivalent theories on-shell, but off-shell the approach taken here seems to be the only practicable one, and to resemble more closely the N = 1 superspace meth- od.

These on-shell constraints on the torsion must now be relaxed to give an off-shell representation of the supersymmetry transformations; (this process has been carried out by Howe [18] in a slightly less ex- tended superspace). For the superspace of this paper, the off-shell constraints on the torsion are:

Dimension zero. The only non-zero torsions are:

i . • . i i j _2iec~6~j] T~(3j d=--lOda3~j , T~ 3kl = (10a)

(together with their complex conjugates) Dimension half:

All torsions are ze ro . (10b)

Dimension one:

Tab c = O. (10c)

Combining these constraints with the Bianchi identities, one Finds that all superspace torsions and curvatures can now be expressed in terms of the super- field M (present in the on-shell theory) together with a real zero-dimensional scalar superfield C. The non- zero dimension-one torsions are

T3~ "r~ if = i ( e 3~ M ~ ij + e ~J~ 8N~/~),

T~3 i .~k = i(e¢TM3~ik + e3~ 8N~rik) ,

T~d 7 8 = - 6 i G,rd e~ ,

where

(11)

(12)

(13)

(14)

G c ~ = l i . - D iD ) C. (15)

For future reference we note that

Dk ^d] .r~va~ = 0 , (16)

and that

i ]_ i ]] - ( 1 7 )

so that the on-shell theory is obtained if Go3 = 0. Tile final stage is the dynamics. With the extra

complex bosonic dimension, the superspace measure dz = dx dO d0 dy dfl has the correct dimension for S = •-2 f BE dz to be an action i fB has dimension zero. Here E is the berezinian superdeterminant of the "sehrvielbein" matrix. It will now be shown that the choice

S = f c E ~ . (18)

To obtain the field equations, one uses the method of Wess and Zumino [7] to extremise the action (18) subject to the constraints on the torsion (10a)-(10c) .

First, let

HA B = EAM 6EM B (19)

and

~2AB C = 6 qSAB C , (20)

then

6 E = E H A A ( - 1 ) IAI (21)

(where IAI is the Grassmann degree of A). Also

6TBC A = DBHC A - (_ I ) IBI ICI DcHB A + TBCDHD A

- HBDTDc A -- (_I ) IBIICIHcDTDB A

- [~BC A -- (--1) IBI ICI £2CB A ], (22)

It is clear from (22) that the constraints on the torsion restrict the possible values of HA B and 6C (the incre- ment in the field C). The general solution fo rH, 6C (up to a gauge transformation [7]) is

HA A (--1) IAI = -(DkDtJ(i4)LC~(/

+ DkkD(JI~])~k. k.) + (DiDt~i _ D~iI)~) VM ' (23) tJ tl 1)

k . " a(j" I~ n ( i r~ i )T & (24) 6 C = Dc~D~(iD])L ~ + "&'~3 ~ W '

335

Volume 132B, number 4,5,6 PHYSICS LETTERS 1 December 1983

where V a~, L a ( / 9 are arbitrary superfields with V a6 hermitian and L c~U ~ symmetric in ft. This leads to

similar formulation of off-shell N = 3 supergravity, will shortly be presented.

5S = r E ( H A A ( - - 1 ) I A I c + ~iC) de

= f E [ _ ( D k D . ,.D~, L . ( / ~ + D&kD(iD~/)~¢, k) j a pLl 1) ~, u J

The author is very grateful to Paul Howe and Kelly Stelle for useful discussions.

References

+ (DaD~ V a~ - D : D a V ~ ) ] C

n k n r~a.c~(/ ' i) r~ r~fiDI3J)-f& k (25) + ~ ' ~ ( i ~ j J b k 4- tJ&kt~ ~ ~(1" i) "

Integrating by parts (which can be done simply, since TAB B = 0 [7]) we find 6S = zero only if

D~'D~) DakC = 0 , (26)

and

(DgDIJ i - DiJi ~ ) C = 0 . (27)

Eq. (27) tell us that

G ~ = 0 .

and thus the on-shell theory is obtained as required. To summarise, it has been shown that the formu-

lation for on-shell N = 2 supergravity in extended superspace [16 -18 ] can be enlarged to an off-shell theory in which the action is an integral over the full superspace. This is a very simple formulation, and should lead to improved superfield techniques for investigating the quantum properties of the theory; it gives an entirely new version of off-shell N = 2 super- gravity with two central charges; [ they are not how- ever true central charges since they do not commute

with the U(2) symmetry group]. As well as the intrinsic interest of the result, it

gives some indications of possibilities for higher N. Already the algebraic analysis of Taylor [4] and Rivelles and Taylor [5] shows that one must expect to need 'extra off-shell symmetries. The corresponding extra dimensions bring both an increased possibili ty of building off-shell torsions and curvatures, and of

building actions. A more detailed paper, including details of the

torsions and curvatures, and possible approaches to a

[1] S. Ferrara, D.Z. Freedman and P. Van Nieuwenhuizen, Phys. Rev. D13 (1976) 3214; S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335.

[2] B. de Wit and S. Ferrara, Phys. Lett 81B (1979) 317. [3] M. Sohnius, K.S. Stelle and P.C. West, Phys. Lett. 92B

(1980) 123. [4] J.G. Taylor, Phys. Lett. 105B (1981) 429,434; 107B

(1981) 217. [5] V.O. Rivelles and J.G. Taylor, Phys. Lett. 104B (1981)

131. [6] J. Wess and B. Zumino, Phys. Lett. 66B (1977) 361. [7] J. Wess and B. Zumino, Phys. Lett. 74B (1978) 51. [8] K.S. Stelle and P.C. West, Phys. Lett. 74B (1978) 330;

S. Ferrara and P. Van Nieuwenhuizen, Phys. Lett. 74B (1978) 333.

[9] M.F. Sohnius and P.C. West, Phys. Lett. 105B (1981) 353.

[10] S.P. Bedding and W. Lang, Nucl. Phys. B196 (1982) 532.

[11] P. Howe, K.S. Stelle and P. Townsend, Phys. Lett. 107B (1981) 420.

[12] E.S. Fradkin and M.A. Vasiliev, Phys. Lett. 85B (1979) 47; B. de Wit, J.W. Van Holten and A. van Proeyen, Nucl. Phys. B167 (1980) 186; B. de Wit, R. Phillipe and A. van Proeyen, Nucl. Phys. B219 (1983) 143; M. Ro~ek and U. LindstriSm, ITP Stockholm preprint 4.

[13] P. Breitenlohner and M.F. Sohnius, Nucl. Phys. B178 (1981) 151.

[14] E. Sokatcher, Phys. Lett. 100B (1981) 466. [15] R. Kallosh, Phys. Lett. 99B (1981) 122;

B. Zumino in: Unification of the fundamental particle interactions, eds. S. Ferrara, J. Ellis and P. Van Nieuwen- huizen (Plenum, New York, 1980).

[16] P. Howe and U. LindstriSm, Nucl. Phys. B181 (1981) 487.

[17] R. Kallosh, lecture 1981 Trieste Workshop on super- gravity.

[18] P. Howe, Nucl. Phys. B199 (1982) 495. [19] L. Brink and P. Howe, Phys. Lett. 88B (1979) 268.

336