N=4 SUPERGRAVITY COUPLED TO N=4 MATTER ... - Google Sites

Nuclear Physics B185 (1981) 403-415 ~) North-Holland Publishing Company

N=4

S U P E R G R A V I T Y C O U P L E D TO N = 4 AND HIDDEN SYMMETRIES

MATTER

Ali H. C H A M S E D D I N E 1

CERN, Geneva, Switzerland Received 25 September 1980 (Revised 22 January 1981) We present the N = 1 supergravity in 10 dimensions obtained by truncating the reduced N = 1 supergravity from 11 dimensions. This is further reduced to 4 dimensions to give SU(4) supergravity coupled to six SO(4) vector multiplets. As the reduction is from 10 dimensions, the theory is expected to have the symmetry SL(6R)gtobat x SO(6hocab but we give a theoretical argument that this can be extended to SO(6, 6) x SU(1, 1)~lobal and SO(6) × SO(6) × U(1)local.

1. Introduction The study of the dual spinor model and interacting closed strings revealed the existence of N = 1 supergravity in 10 dimensions and predicted the necessary fields [1]. This knowledge was then used in constructing the pure SU(4) supergravity [2]. F r o m a different angle, after the establishment of N = 1 supergravity in 11 dimensions [3], it was argued by Scherk that this can be truncated to give the 10-dimensional theory [4]. It is also known that in four dimensions, after reduction from ten, the model describes SU(4) supergravity coupled to six N = 4 vector matter multiplets. However, neither the theory in ten dimensions, nor in four dimensions, are fully worked out. O u r interest in this model stems from the fact that N = 4 supersymmetry is the largest one that admits matter representations, with matter defined as consisting of particles of m a x i m u m spin 1 [5]. Thus the ten-dimensional theory provides an example of an unrestricted scheme coupling supergravity to an arbitrary n u m b e r of matter multiplets. This may be considered as an advantage over the N = 8 supergravity. Other uses of this model for the spinning string theory are advocated by Schwarz [5]. Taking the N = 8 supergravity as an example [6], it is natural to try to find the hidden symmetries that may be present. One can argue, guided by the work of C r e m m e r and Julia [6], that the reduction from ten dimensions will result in an obvious SO(6)~ocal× SL(6R)globa~ symmetry. The presence of the large n u m b e r of scalars (38 with duality transformations and 37 without), is, by the " e x p e r i m e n t a l " 1 Address after 1 January 1981: Northeastern University, Boston, MA. 403

404

A.H. Chamseddine / N = 4 supergravity

rules of extended supergravity models [7-9], an indication of a larger symmetry. We argue that the symmetry can be enlarged to the non-compact groups SO(6, 6 ) x SU(1, 1) globally and SO(6) x SO(6) x U(1) locally with a duality transformation to the antisymmetric field (but only SO(6, 6 ) x U(1) global and S O ( 6 ) x SO(6) local without). On the negative side it was shown in the calculation of the scalar-scalar and p h o t o n - p h o t o n interaction of 0 ( 4 ) s u p e r g r a v i t y coupled to self-dual 0(4) matter multiplets that the results are not finite, implying the non-vanishing of the/~ function, although it does vanish for pure 0(4) supergravity and pure 0(4) matter [10]. The plan of this p a p e r is as follows. In sect. 2, we start with the N = 1 supergravity in 11 dimensions and show the consistent truncation that leads to the N = 1 theory in 10 dimensions. In sect. 3 we reduce the theory to four dimensions and identify the physical fields. Since we did not obtain all the interaction terms here, we shall only give in sect. 4 the group theoretical arguments that led us to identify the groups.

2. N = 1 supergravity in 10 d i m e n s i o n s

We start with the now familiar N = 1 lagrangian in 11 dimensions [6], using the same notation and conventions. The fields are the elfbein e ~ A, the Rarita-Schwinger Majorana field OM, and the antisymmetric potential AA~Np. The supersymmetry transformation rules are: 8e~ A = --i~FA~bM ,

3tSAMN P = ~EF[M N ~//P] , 8 0 M = D M (tO) e + 1 ~ i (FNPORM -- 8FPQR8 N ) EFNPQR .

(2.1)

The definitions of the supercovariant connection ~)MABand field strength FNPQRa r e given in ref. [6]. Using part of the local SO(1, 10) symmetry we can bring the elfbein to the form a

Bit

where ~, ~, . . . . = 6, i . . . . . 9 are curved indices, and a,/~ . . . . = 0, 1 . . . . . 9 are flat indices. Note that we reserve the dotted numbers for curved indices and the undotted for flat ones. By splitting the eleventh index explicitly the other fields decompose as AMNP = (A,~o, A , o l f ) ,

(2.3) We are not interested in keeping all these fields, as this will correspond to the known N = 8 supergravity in four dimensions (and N = 2 supergravity in 10 dimensions), and we have to set some fields to zero. Due to the special properties of ten

405

A . H . Chamseddine / N = 4 supergravity

dimensions, we can impose the Majorana and Weyl ccmsitions simultaneously on the spinors. Thus each of the Majorana spinors 0., 01f and e, split into two MajoranaWeyl ones. We can show that keeping only the left-handed components of 0 . and e and the right-handed component of ~lf, and setting B . n and A . , . o to zero, form a consistent truncation. In other words, we now have YnO~, = 0 , ,

YH01~ = - 0 , f ,

The = e,

(2.4)

where F 11 = - / 1 1

(Fll) 2 = -1,

= i'Yll ,

('~11) 2 =

+1.

The truncation is consistent because 6B~ al = - i e F l l O~ = f:6~ = O,

(2.5)

3-

~eFt,,vOol= O,

6A.~o =

and these vanish due to the condition (2.4). Moreover, the handedness of 011 and 011 is preserved as the quantities o3,~ H, o3~f~0, and F~,xo all vanish. We shall write the new transformation rules after the reduction of the theory, as some field redefinitions and scalings are needed. As the dimensional reduction technique is explained in detail in ref. [6], we shall only indicate the necessary steps so that the reader can keep track of the field redefinitions, and without falling into repetition. We note that, apart from the equivalent theory containing the opposite handed spinors, the truncation (2.4) is unique. The bosonic lagrangian in 11 dimensions is -1VllR

(to ) - ~ VllFMNpOF

M~PO

2

+ (-~

e

M~ M

"'" ~FMr..~FM~...M~AM~M~oM~ .

(2.6) To reduce the gravitational sector we define the Weyl scaling for the 10-bein: c~

1

c~

el0~, = q~e~, ,

(2.7)

and we obtain 1

I

132

- z V x l R (to) = -ZVloRlo(to) + ~(Z) (01o~ log

~)2.

(2.8)

The remaining parts of (2.6) give 1 117 --3/2 /a,;< vA 0o'1-', IU' i~ VlOq~ gioglogxol'~,,olf.~',