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Gauged N=4 matter couplings
Volume 156B, number 5,6
PHYSICS LETTERS
27 June 1985
GAUGED N = 4 MATrER COUPLINGS M. D E R O O Institute for Theoretical Physics, P. 0. Box 800, 9700 A V Groningen, The Netherlands Received 26 February 1985
The N = 4 Yang-Mills multiplet is coupled to N = 4 conformal supergravity. The action has a local U(4)×G symmetry, where G is the Yang-Mills gauge group. The action and supersymmetry transformation rules are presented in the Poincar6 gauge, and properties of the scalar potential are discussed.
In a recent paper [ 1 ], the complete lagrangian for an arbitrary number o f N = 4 abelian vector multiplets coupled to N = 4 conformal and Poincar6 supergravity was established. If the number of vector multiplets is n +6, then six o f these multiplets are used in the transition from N = 4 conformal to Poincar6 supergravity, and the remaining n correspond to physical vecto r multiplets. The scalar fields in such theories parametrize the manifold M = [SO(n, 6)/SO(n) × SO(6)] X [SU(1,1)/U(1)]. In the present work, I extend these results to the non-abelian N = 4 matter multiplet. The N = 4 Yang-Mills multiplet [2] has the following transformation rules under global supersymmetry ~ A lz = ~i'y iz ~Oi + -~i'yiz ~ i , 6 ~ i = - o u v e i F u v*-
27%d
_ 2ei[t~ik,
ki] ,
~dpi/ = -6[i~] l -- ei]kltk ~l i .
(1)
The fields take values in the Lie-algebra o f a local symmetry group G, with generators T I (1 = 1, ...,P) [TI, Tj] = f l / T K ,
Tr T I T J = rlZj,
(2)
with real structure constants f. The metric 7/can be taken diagonal. The fields ~i/satisfy ~i] =--~i]TI = --q~/i,
q~i]I~ ( 6 ) *
=1
j]kl~lkl "
(3)
I follow the conventions o f ref. [1 ]. In this paper I couple the multiplet (1) to N = 4 conformal supergravity [3]. The superconformal multiplet has an SU(1,1) X U(1) symmetry, which in re f. [ 1 ] played a crucial role in the construction o f the transformation rules and action. The scalar fields o f the Weyl multiplet, ~a (a = 1, 2), which satisfy
transform as a doublet under SU(1,1). For abelian vector multiplets in a superconformal background the SU(1,1) symmetry also acts as duality transformations on the field strengths. In the non-abelian case this symmetry will be broken due to the appearance of the vector potential Av. in covariantizations. Nevertheless, the result of ref. [1 ] is a useful starting point for the present work. The Q- and S-supersymmetry transformations (with parameters e i and ri i) of the N = 4 Yang-Mills multiplet in a superconformal background turn out to be
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
331
Volume 156B, number 5,6
PHYSICS LETTERS
27 June 1985
8A u = ~(~iyu~k i - 2~i~/u~ii + -diTuAi~ ii) + , ~ * ( g i T u q / - 2gi~ui¢i/ + ~iTuN¢i]) , 8 ~0i = --OabeiF;b -- 2 ~ i j e i + EijCkek + ~ ei~. ~] -- e]Ai~ j + 1 Taek~lTaAi¢k I _ 2eii¢ik, Ck/] ¢,, _ 2 ~ / / ~ , 6~i1 = g[i~/] -- eiikt~kt~ l .
(5) A
All derivatives are covariant with respect to Yang-Mills and all superconformal symmetries, and Fuu is the completely covariant Yang-Mills field strength. In (5)/~ appears in the form
Fab -'+ = (1/alp) F.+a b + (¢b*/cb)K-~b '
Kab + = -- ~1AiOab ~ki + Tab ij¢ tl"" .
(6)
The fields Ai, Ekl and Tab # are matter fields of the N = 4 Weyl multiplet [3]. The scalars Ca of the Weyl multiplet appear in the combination dp= ¢1 + ¢2,
~* = ¢1 - ¢2 •
(7)
The commutator of two transformations (5) closes on the superconformal algebra [3], a Yang-MiUs gauge transformation and the fermion field equation. As in ref. [1], the field equations determine a superconformally invariant action, which now contains a number of new terms due to the non-abelian structure. I will present the result only in the Poincar6 gauge, in which the superconformal D and S symmetries are broken by the conditions Tr ¢i/¢ # = - 6 ,
Tr ¢i/~ j = 0 .
(8)
The action can be simplified considerably by using some information from the equations of motion. It contains terms which are linear in the fields DiJkl and Xikl of the Weyl multiplet, and the equations of motion of these fields imply a more stringent condition on ¢i/and ~/ [ 1 ] : Tr ¢i]¢ kl
=
xl , - ~ ,1,xk[iv/]
Tr ¢i/~ k = 0 .
(9)
On using the conditions (9) the Poincar6 action takes on the following form: e-1 ~ = Tr [ - ¼FuvFU v+( 1/&) (¢1 _ ¢2 ) _ ~ fji.y # cOp 1~i _ ¼ cO t~¢ijcO t~¢i/_ ( 1/~b)/~;~gt~ v _ 1 Ekl ~k ~0l
+ ¼ ei/kl~io. T/k~O l - ½(¢b*[¢)K+uuK u"+] - ~R(¢o) - ! ,,-l~uvxa ,Zi ~ cn ,t, + ~ ( g k t e kt + 4 D a C D a ¢ a - 2 AkTU cO#A k) + Tr [ _ ~ ¢ i / [ ~ i , ~0/] + ~bA/c~iwik + ~ IdPl2 W/W/i] + ~ ~*EktXkl + Tr [-~iTuTz'dg/cO,,¢ii - dP~kTu~k/Wk / - (1/2cb)F+b(~iuOabTU~b i - ~iuTUOabA/Cbi/+ 2~ai~bbpi/)]
+ lcbl~ukTUAlXkl--s'~_2,~.Z ~uk _uu.,. u ~,l Av.kl + 1_ 4 e-leu,hat.kuk Tv~ l c t k cO a¢i] + -4 eao~okTu¢~q~aAk .
.
.
.
.
%'~
+ h.c. + four-fermion terms.
1
--
(10)
I have introduced the following tensors which depend on the scalar fields ¢i/:
W / = [¢ik,¢k]],
Xi/=TrCkl[¢ik,¢l]]
.
(11)
They transform under SU(4) as the 15- and 10-dimensional representations, respectively. In (9) the derivatives cOu contains local Lorentz, SU(4), U(I) and G covariantizations. The four-fermion terms are the same as in the abelian case [1]. In the Poincar6 gauge the supersymmetry transformations are a combination of (5), and a compensating transformation required to maintain the conditions (8). In the present form the invariance of (10) under supersymmetry requires the use of the equations of motion ofEi], Tabi] and the SU(4) gauge field Vu~I.. This is due to the fact that I have used (9), which are equations fo motion and not invariant under supersymmetry, to simplify the action. Only after elimination of E, T and Vis the action fully invariant. These fields are then replaced by 332
Volume 156B, number 5,6
PHYSICS LETTERS
E / / = T r ~ i ~ k j --~¢b 4 . Xi] ,
"+ + ~(~P/dP*)Tr~iOab~l , Tabii : (1/cb*)Tr~iiFab
Vu/j = Tr [AuW] i + ~w a ~.ik ~u gWj .~ k -- ~1 ~ iT# ~j ] -- ~ A i T# A/ -- traces•
27 June 1985 (12,13) (14)
Compared to ref. [ 1 ], the final form of the action now contains a scalar potential Z?V and a collection of masslike terms Z?M . These are easily derived from (10) and (12)--(14): e - l Z ? v : Idpl2(--4 s k l x k l + ½Tr WJw]i) ,
(15)
e-IZ?M = Tr [1 dpxk l ~k ~Ol _ dp~i/ [~ i, ~/] + dPAk ~ IWlk - dP~uI"/u ~k W? ] "
+ ½¢}~ttkTUAixkl -- ] ~*~ukOttV~ul X k l + h.c.
(16)
The final form of the supersymmetry transformations is
8qbo~ = -eiAieafJq~3,
8~i] = -e[i~]] -- ei]kl ~kt~l,
5A i = 2ea~dpa~j(~aei_~ ep• Xi/et " + ~1 ei/ktOabel(I/CP)Tr~klp~ + ~2.terms, ~i = - - 2 ¢ ~ i ] J -- 2~P*E/'[Wi] + -~(~ik x k ] + Xik~k])] -- °ab ei(1/~) (/7+b + ~kl r r ~klI?+b) -- "ylZe]Tr(~kl~lJP[il)~bk] ] + At//, ~2-telTfls , 8A# = dp(-~i3,~ i - 2ei~Jq~i] + eiTtzAj~ ij) + @ * ( c i T ~ i - 2 e i ~ j ~ i] + ei'y#l~ij) , fig i = 2 cZ)uei - ~ "rJb*XiJe/ - %bTuei(1/ep) Tr (~ilpab • ~ " - Tr (¢~kcl)uOjk)el - ~1 ~ a ~-~ ~uOae i -- eijkt~I. t k , k A l + qj2, A2.terms,
e~ a = ~i'ya~ui + h.c.
(17)
In this form the theory has local G, SU(4) and U(1) symmetry. However, these will in general not be preserved by solutions of the constraints (4) and (9). Let us therefore consider the restrictions on G, which follow from the constraint (9). The condition on the scalar field can only be solved if the metric r0j(2 ) has at least six negative eigenvalues. If furthermore one requires that the scalar kinetic terms all have the canonical sign, then there can be no more than six negative eigenvalues. In particular, for P = 6 all eigenvalues of rllj must be negative, and if one normalizes to rlII = - 1 the constraint (9) is solved by (T I are represented by 4 X 4 matrices, a, b = 1, ..., 4)
(~bij)ab = ¼(1 + i)ea[l~j] b -- ~(1 --i)eijab ,
(~i)ab = O .
(18)
This solution preserves a local SO(4) group, and leads to the gauged N = 4 supergravity theory of Freedman and Schwarz [4]. The tensors W and X are now given by
(W/)ab = ½6a[iel] b , Xi/= --~(1 + i)ei] ,
(19)
and the scalar potential term (15) is e - l Z ? v = + lcb[2 .
(20)
Eq. (20) takes on a more familiar form if the constraint on ~a, eq. (4), is solved by setting ~b1 = (1 - 1/12) -~/2 ,
¢2 = Z(1 - IZI2)-1/2 ,
(21)
thus breaking the local U(1) symmetry. In this parametrization (20) reads e - l . ~ v = I1 - / 1 2 / ( 1 -
[Zl2).
(22) 333
Volume 156B, number 5,6
PHYSICS LETTERS
27 June 1985
The corresponding potential is unbounded below, and has no critical points. Nevertheless, the equations o f motion have a stable electrovac solution [5]. In the general case one is interested in critical points o f the potential (15) in the (~i/directions, in view o f the possibility of supersymmetry breaking. This aspect is presently being investigated. Variation o f (15) yields
e-18Z?V = 1412 Tr 6q~iiZi/,
(23)
with
zi/= ]x[ikWk/] _ ~ ei/klXl m Wkm + [Wk [i, q~/]k] ,
(24)
Since 8~ii can be restricted to those variations which preserve the constraint (9), a sufficient condition for a critical point is that Z ii is proportional to ~ij. One easily verifies that Z i/vanishes for the solution (18). By contracting with ~# one finds that Z q = 0 implies
]X#Xi/ = Tr W/iw j ,
(25)
so that in this class o f critical points the value of the potential V = -.6? 7 is negative. This is a necessary condition for unbroken supersymmetry [6]. After completion o f this work I learned that similar results have been obtained by Bergshoeff, Koh and Sezgin [7], using a different method.
References [1] M. de Roo, Groningen preprint, Nucl. Phys. B, to be published. [2] L. Brink, J. Seherk and J. Sehwarz, Nuel. Phys. B121 (1977) 77; F. Gliozzi, D. Olive and J. Seherk, Nuel. Phys. B122 (1977) 253. [3] E. Borgshoeff, M. de Roo and B. de Wit, Nuel. Phys. B182 (1981) 173. [4] D.Z. Freedman and J. Sehwarz, Nucl. Phys. B137 (1978) 333. [5] D.Z. Freedman and G.W. Gibbons, Nucl. Phys. B233 (1984) 24. [6] S. Ferrara, Phys. Lett. 69B (1977) 481. [7] E. Bergshoeff, I.G. Koh and E. Sezgin, ICTP Trieste preprint, in preparation.
334
PHYSICS LETTERS
27 June 1985
GAUGED N = 4 MATrER COUPLINGS M. D E R O O Institute for Theoretical Physics, P. 0. Box 800, 9700 A V Groningen, The Netherlands Received 26 February 1985
The N = 4 Yang-Mills multiplet is coupled to N = 4 conformal supergravity. The action has a local U(4)×G symmetry, where G is the Yang-Mills gauge group. The action and supersymmetry transformation rules are presented in the Poincar6 gauge, and properties of the scalar potential are discussed.
In a recent paper [ 1 ], the complete lagrangian for an arbitrary number o f N = 4 abelian vector multiplets coupled to N = 4 conformal and Poincar6 supergravity was established. If the number of vector multiplets is n +6, then six o f these multiplets are used in the transition from N = 4 conformal to Poincar6 supergravity, and the remaining n correspond to physical vecto r multiplets. The scalar fields in such theories parametrize the manifold M = [SO(n, 6)/SO(n) × SO(6)] X [SU(1,1)/U(1)]. In the present work, I extend these results to the non-abelian N = 4 matter multiplet. The N = 4 Yang-Mills multiplet [2] has the following transformation rules under global supersymmetry ~ A lz = ~i'y iz ~Oi + -~i'yiz ~ i , 6 ~ i = - o u v e i F u v*-
27%d
_ 2ei[t~ik,
ki] ,
~dpi/ = -6[i~] l -- ei]kltk ~l i .
(1)
The fields take values in the Lie-algebra o f a local symmetry group G, with generators T I (1 = 1, ...,P) [TI, Tj] = f l / T K ,
Tr T I T J = rlZj,
(2)
with real structure constants f. The metric 7/can be taken diagonal. The fields ~i/satisfy ~i] =--~i]TI = --q~/i,
q~i]I~ ( 6 ) *
=1
j]kl~lkl "
(3)
I follow the conventions o f ref. [1 ]. In this paper I couple the multiplet (1) to N = 4 conformal supergravity [3]. The superconformal multiplet has an SU(1,1) X U(1) symmetry, which in re f. [ 1 ] played a crucial role in the construction o f the transformation rules and action. The scalar fields o f the Weyl multiplet, ~a (a = 1, 2), which satisfy
transform as a doublet under SU(1,1). For abelian vector multiplets in a superconformal background the SU(1,1) symmetry also acts as duality transformations on the field strengths. In the non-abelian case this symmetry will be broken due to the appearance of the vector potential Av. in covariantizations. Nevertheless, the result of ref. [1 ] is a useful starting point for the present work. The Q- and S-supersymmetry transformations (with parameters e i and ri i) of the N = 4 Yang-Mills multiplet in a superconformal background turn out to be
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
331
Volume 156B, number 5,6
PHYSICS LETTERS
27 June 1985
8A u = ~(~iyu~k i - 2~i~/u~ii + -diTuAi~ ii) + , ~ * ( g i T u q / - 2gi~ui¢i/ + ~iTuN¢i]) , 8 ~0i = --OabeiF;b -- 2 ~ i j e i + EijCkek + ~ ei~. ~] -- e]Ai~ j + 1 Taek~lTaAi¢k I _ 2eii¢ik, Ck/] ¢,, _ 2 ~ / / ~ , 6~i1 = g[i~/] -- eiikt~kt~ l .
(5) A
All derivatives are covariant with respect to Yang-Mills and all superconformal symmetries, and Fuu is the completely covariant Yang-Mills field strength. In (5)/~ appears in the form
Fab -'+ = (1/alp) F.+a b + (¢b*/cb)K-~b '
Kab + = -- ~1AiOab ~ki + Tab ij¢ tl"" .
(6)
The fields Ai, Ekl and Tab # are matter fields of the N = 4 Weyl multiplet [3]. The scalars Ca of the Weyl multiplet appear in the combination dp= ¢1 + ¢2,
~* = ¢1 - ¢2 •
(7)
The commutator of two transformations (5) closes on the superconformal algebra [3], a Yang-MiUs gauge transformation and the fermion field equation. As in ref. [1], the field equations determine a superconformally invariant action, which now contains a number of new terms due to the non-abelian structure. I will present the result only in the Poincar6 gauge, in which the superconformal D and S symmetries are broken by the conditions Tr ¢i/¢ # = - 6 ,
Tr ¢i/~ j = 0 .
(8)
The action can be simplified considerably by using some information from the equations of motion. It contains terms which are linear in the fields DiJkl and Xikl of the Weyl multiplet, and the equations of motion of these fields imply a more stringent condition on ¢i/and ~/ [ 1 ] : Tr ¢i]¢ kl
=
xl , - ~ ,1,xk[iv/]
Tr ¢i/~ k = 0 .
(9)
On using the conditions (9) the Poincar6 action takes on the following form: e-1 ~ = Tr [ - ¼FuvFU v+( 1/&) (¢1 _ ¢2 ) _ ~ fji.y # cOp 1~i _ ¼ cO t~¢ijcO t~¢i/_ ( 1/~b)/~;~gt~ v _ 1 Ekl ~k ~0l
+ ¼ ei/kl~io. T/k~O l - ½(¢b*[¢)K+uuK u"+] - ~R(¢o) - ! ,,-l~uvxa ,Zi ~ cn ,t, + ~ ( g k t e kt + 4 D a C D a ¢ a - 2 AkTU cO#A k) + Tr [ _ ~ ¢ i / [ ~ i , ~0/] + ~bA/c~iwik + ~ IdPl2 W/W/i] + ~ ~*EktXkl + Tr [-~iTuTz'dg/cO,,¢ii - dP~kTu~k/Wk / - (1/2cb)F+b(~iuOabTU~b i - ~iuTUOabA/Cbi/+ 2~ai~bbpi/)]
+ lcbl~ukTUAlXkl--s'~_2,~.Z ~uk _uu.,. u ~,l Av.kl + 1_ 4 e-leu,hat.kuk Tv~ l c t k cO a¢i] + -4 eao~okTu¢~q~aAk .
.
.
.
.
%'~
+ h.c. + four-fermion terms.
1
--
(10)
I have introduced the following tensors which depend on the scalar fields ¢i/:
W / = [¢ik,¢k]],
Xi/=TrCkl[¢ik,¢l]]
.
(11)
They transform under SU(4) as the 15- and 10-dimensional representations, respectively. In (9) the derivatives cOu contains local Lorentz, SU(4), U(I) and G covariantizations. The four-fermion terms are the same as in the abelian case [1]. In the Poincar6 gauge the supersymmetry transformations are a combination of (5), and a compensating transformation required to maintain the conditions (8). In the present form the invariance of (10) under supersymmetry requires the use of the equations of motion ofEi], Tabi] and the SU(4) gauge field Vu~I.. This is due to the fact that I have used (9), which are equations fo motion and not invariant under supersymmetry, to simplify the action. Only after elimination of E, T and Vis the action fully invariant. These fields are then replaced by 332
Volume 156B, number 5,6
PHYSICS LETTERS
E / / = T r ~ i ~ k j --~¢b 4 . Xi] ,
"+ + ~(~P/dP*)Tr~iOab~l , Tabii : (1/cb*)Tr~iiFab
Vu/j = Tr [AuW] i + ~w a ~.ik ~u gWj .~ k -- ~1 ~ iT# ~j ] -- ~ A i T# A/ -- traces•
27 June 1985 (12,13) (14)
Compared to ref. [ 1 ], the final form of the action now contains a scalar potential Z?V and a collection of masslike terms Z?M . These are easily derived from (10) and (12)--(14): e - l Z ? v : Idpl2(--4 s k l x k l + ½Tr WJw]i) ,
(15)
e-IZ?M = Tr [1 dpxk l ~k ~Ol _ dp~i/ [~ i, ~/] + dPAk ~ IWlk - dP~uI"/u ~k W? ] "
+ ½¢}~ttkTUAixkl -- ] ~*~ukOttV~ul X k l + h.c.
(16)
The final form of the supersymmetry transformations is
8qbo~ = -eiAieafJq~3,
8~i] = -e[i~]] -- ei]kl ~kt~l,
5A i = 2ea~dpa~j(~aei_~ ep• Xi/et " + ~1 ei/ktOabel(I/CP)Tr~klp~ + ~2.terms, ~i = - - 2 ¢ ~ i ] J -- 2~P*E/'[Wi] + -~(~ik x k ] + Xik~k])] -- °ab ei(1/~) (/7+b + ~kl r r ~klI?+b) -- "ylZe]Tr(~kl~lJP[il)~bk] ] + At//, ~2-telTfls , 8A# = dp(-~i3,~ i - 2ei~Jq~i] + eiTtzAj~ ij) + @ * ( c i T ~ i - 2 e i ~ j ~ i] + ei'y#l~ij) , fig i = 2 cZ)uei - ~ "rJb*XiJe/ - %bTuei(1/ep) Tr (~ilpab • ~ " - Tr (¢~kcl)uOjk)el - ~1 ~ a ~-~ ~uOae i -- eijkt~I. t k , k A l + qj2, A2.terms,
e~ a = ~i'ya~ui + h.c.
(17)
In this form the theory has local G, SU(4) and U(1) symmetry. However, these will in general not be preserved by solutions of the constraints (4) and (9). Let us therefore consider the restrictions on G, which follow from the constraint (9). The condition on the scalar field can only be solved if the metric r0j(2 ) has at least six negative eigenvalues. If furthermore one requires that the scalar kinetic terms all have the canonical sign, then there can be no more than six negative eigenvalues. In particular, for P = 6 all eigenvalues of rllj must be negative, and if one normalizes to rlII = - 1 the constraint (9) is solved by (T I are represented by 4 X 4 matrices, a, b = 1, ..., 4)
(~bij)ab = ¼(1 + i)ea[l~j] b -- ~(1 --i)eijab ,
(~i)ab = O .
(18)
This solution preserves a local SO(4) group, and leads to the gauged N = 4 supergravity theory of Freedman and Schwarz [4]. The tensors W and X are now given by
(W/)ab = ½6a[iel] b , Xi/= --~(1 + i)ei] ,
(19)
and the scalar potential term (15) is e - l Z ? v = + lcb[2 .
(20)
Eq. (20) takes on a more familiar form if the constraint on ~a, eq. (4), is solved by setting ~b1 = (1 - 1/12) -~/2 ,
¢2 = Z(1 - IZI2)-1/2 ,
(21)
thus breaking the local U(1) symmetry. In this parametrization (20) reads e - l . ~ v = I1 - / 1 2 / ( 1 -
[Zl2).
(22) 333
Volume 156B, number 5,6
PHYSICS LETTERS
27 June 1985
The corresponding potential is unbounded below, and has no critical points. Nevertheless, the equations o f motion have a stable electrovac solution [5]. In the general case one is interested in critical points o f the potential (15) in the (~i/directions, in view o f the possibility of supersymmetry breaking. This aspect is presently being investigated. Variation o f (15) yields
e-18Z?V = 1412 Tr 6q~iiZi/,
(23)
with
zi/= ]x[ikWk/] _ ~ ei/klXl m Wkm + [Wk [i, q~/]k] ,
(24)
Since 8~ii can be restricted to those variations which preserve the constraint (9), a sufficient condition for a critical point is that Z ii is proportional to ~ij. One easily verifies that Z i/vanishes for the solution (18). By contracting with ~# one finds that Z q = 0 implies
]X#Xi/ = Tr W/iw j ,
(25)
so that in this class o f critical points the value of the potential V = -.6? 7 is negative. This is a necessary condition for unbroken supersymmetry [6]. After completion o f this work I learned that similar results have been obtained by Bergshoeff, Koh and Sezgin [7], using a different method.
References [1] M. de Roo, Groningen preprint, Nucl. Phys. B, to be published. [2] L. Brink, J. Seherk and J. Sehwarz, Nuel. Phys. B121 (1977) 77; F. Gliozzi, D. Olive and J. Seherk, Nuel. Phys. B122 (1977) 253. [3] E. Borgshoeff, M. de Roo and B. de Wit, Nuel. Phys. B182 (1981) 173. [4] D.Z. Freedman and J. Sehwarz, Nucl. Phys. B137 (1978) 333. [5] D.Z. Freedman and G.W. Gibbons, Nucl. Phys. B233 (1984) 24. [6] S. Ferrara, Phys. Lett. 69B (1977) 481. [7] E. Bergshoeff, I.G. Koh and E. Sezgin, ICTP Trieste preprint, in preparation.
334