Volume 151B, number 3,4 PHYSICS LETTERS 14 February 1985

NON-MINIMAL N = 1 SUPERGRAVITY AND BROKEN GLOBAL SUPERSYMMETRY ~

B.B. DEO 1 and S James GATES Jr. 2 International Centre for Theoretical Physics, Trieste, Italy

Received 29 October 1984

We discuss the spontaneous breakdown of global de Sitter supersymmetry that occurs by taking the global limit of pure non-minimal N = 1 supergravity with a cosmological term.

An unusual situation [1 ] occurs when a cosmological term is added to the action for pure N = 1 non-minimal supergravity [2]. The mere presence of the cosmological term implies that N = 1 supersymmetry must undergo a spontaneous breakdown as the supergravity theory approaches the limit of global de Sitter supersymmetry. We will use superspace methods to elucidate how this phenomenon occurs.

First, it is useful to discuss how this breakdown is inferred from known results. For s superspace with a global de Sitter geometry, the graded commutator algebra of supercovariant derivatives 7.4 (see ref. [1 ] for notation and conventions) takes the form

[V,~, VI3} =-2XMa#, [V,,, V&} = iVa, [Vt~, Vb} =-iXC/~V~, [Va,Vb}=2XX(C~Mao+CflM~,h), (1)

where k is a constant. Due to the last result, we see that the curvature scalar C~Rab a# is the only non-vanishing part of the full curvature tensor Rab.r8 and takes the value -1 2 Ikl 2. Thus, the geometry of x-space has a constant negative curvature. On the other hand, in non-minimal supergravity (Breitenlohner's auxiliary fields) the full local commutator algebra is given by

[V a, V~) = ½ T(aV~) - 2(/~ + ½ T'~T)Mu#, [Va, V&) = iV a - -~(T V,~ + ~Va) ,

[Va, Vb) = - i C ~(/~ + { TTT)V~ + iC ~Ow~V,y - i½(V~T~ - ~73T ~ + T~.~)V - i[(V~- T~)/~] M ~

-- iCa#(v'rgs~)M78 + i C # { W ~ g - ~ [VTGTg - (Vg - Tg) (R + { T 8 T~)].~hg), (2)

where for simplicity we have omitted the explicit form of [Va, V b) . But this may be found from

[%, V b} = [-i([V~, IVy, % ) ) - ~ [~7~, % ) - (V b ?~) V~) + h.c.], (3)

which follows from the Bianchi identities. Although (2) and (3) correspond to the Breitenlohner auxiliary fields, we have chosen different conventional constraints [3] and furthermore chosen the n = -1 form [4] of the theory. The results (2) and (3) reduce to (1) when

W # 7 = G a = T =0, /~=k . (4,5)

But for non-minimal n = -1 supergravity/~ is related to T~ by

Research supported in part by the US National Science Foundation under grant PHY81-07394. I Permanent address: Physics Department, Utkal University, Bhubaneswar, 751004, India. 2 Address after 1 September 1984: Physics Department, University of Maryland, College Park, MD 20742, USA.

0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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~ - - 1 Ot R - - ~ V T (6)

and this equation cannot be satisfied in the limit where (4) and (5) are valid. Thus a global de Sitter supergeometry (X 4= 0) must break supersymmetry. We want to investigate how this comes about.

The superspace action for pure n = -1 supergravity is given by [5]

S S G = - ~ f d 8 z E -1 , (7)

and the cosmological term takes the form

S c = ~ f d 8 z E - I R - 1 + h.c.. (8)

Initially, it appears that an understanding of the symmetry breaking requires a very complicated analysis of the details of a full supergravity theory. This can be carried out using, for instance, the superconformal tensor calculus [6 [. However, there is an alternate way to proceed. The solution of the constraints for supergravity was solved some time ago; first for the minimal theory [7] and later for other versions [8,9]. Because these solutions are available, they can be used to analyze (7) and (8) in a more expedient manner.

As shown previously [8,9], N = 1 supergravity can always be written in terms of an axial vector supergravity superfield H a [10] and some density-type compensator. These density-type compensators should not be confused with the covariant type (referred to as tensor type in ref. [ 1 ] ) that have been introduced in superconformal ten- sor calculus methods and in superspace [ 1 ]. It is known that the x-space Weyl tensor occurs in Wa0 v while the x- space traceless Ricci tensor occurs in G a [11]. The facts that the background values (4) for these superfields vanish and that only Ta enters (6) imply that Watt. t and G a play no role in the symmetry breaking process. In terms of unconstrained superfields, setting H a to zero eliminates these unnecessary multiplets. (In ordinary gravity, this is analogous to setting the vierbein equal to a scalar compensator field e m = ~0-1(x)6 m .) Thus, the analysis of (7) and (8) requires only the compensating superfield contributions. Using the results from ref. [1 ], these are found to be

E -1 = (TY) -1, /~ = --52 [T ln(T2~)] , (9)

where T is a complex linear superfield, i.e. ~2 T = 0. The total compensating superfield action [the sum of (7) and (8)] takes the form

S=-~ fd8z[ .C 1 + X(.C 2 + .C3)], (10)

£1 = ( T T ) - I ' £2 = (T~')-I [D2(T ln(T2T))] ' £3 = (T~')-I [D2(Tln(~'2T))]" (11,12,13)

This may be regarded as a problem in global supersymmetry. When X = 0, (10) takes the form of a rigid supersym- metric non-linear sigma model. We have recently studied such theories [12] and will make use of our previous re- suits.

To begin we define the component fields by the projection techniques [13]

A ( x ) - T I , ~ = D a T I , F = D 2 T I , Pa=F)&DTI, p~=D~TI, 3o,=½b&D,~F)&TI. (14)

The "physical" scalar Re(A) corresponds to ~o 2, the dilatation compensator. The "physical" scalar, Im(A), is the chiral U(1) compensator. Finally the "physical" spinor, ~a, is the S-supersymmetric compensator. The remaining components pa, F, Pa and/~ are the "non-conformal" auxiliary fields of the Breitenlohner formulation. (In the full supergravity theory it is always possible to fix a gauge where A = 1 and ~a = 0.) The definitions in (13) imply that under a supersymmetry variation we f'md

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6QA=-eaou-e&~&, 6Q~a = - ~ ( i aa.4 + Pa), 6Qpa=eaF-e&Pa . (15)

The complete action in (10) is quite complicated. To make progress we consider the purely bosonic theory (see also the appendix). This yields

s = - ~ f d 4 x { [1 + X(A]Aff" + A/AF)] [g,AX(~A) (Oa A) + ½ g,AA (OaA) (~a A) + lg,x .~(oa2 ) (a a A)

+gA~(FF -PaPa) - ~gAAPa(Pa --i2~aA ) - ~ g ~ P a ( / 5 a + i2aa.4)] }

f d4x {A-3if-1 [h, Aff(aaA)(a a A-) + ½ h AA(aaA)(aaA ) + ½ h j j ( a a Z ) ( ~ X ) + h A~(PF - PaPa) K 2 ' ,

l_t, pa(p i 23aA) _ 1 t, - _pa(p h.c.}, --2",AA V.a-- 2",AA ~ a + i 2 a a "~)] + (16)

where the functions g and h are given by

g(A ,A) =- (A/T) -1 , h(A ,_~) =A ln(A2A). (17)

The potential in this model is obtained by neglecting all dependence on a a A and Pa. Thus,

V(A, A, F, F) = --g,Aff FP -- 3,(.~A-lg,A ff + A -3h,Ax)F - X(Aff- l gAx + i f -3 ~ Aff)ff ' (18)

and variation with respect to ff implies that

F = -2X A -3/T-5. (19)

At this point, it is clear why the non-minimal supergravity theory with a cosmological term is very different from the minimal theory. In the (superconformal) gauge where A = 1, eqs. (15), (19) imply that the auxiliary spi- nor Pa transforms as

6Qpa = - 2 X e + ... (20)

under a local supersymmetry transformation after eliminating F which is equivalent to/~1. But the transformation law in (20) identifies Pa as a goldstino which may be gauged away. In approaching the global limit, the degrees of freedom represented by p~ are absent (Oa is equivalent to Tal) and the inequality of fermionic versus bosonic degrees of freedom results in a broken global de Sitter supersymmetry.

It is also clear why the minimal theory [7,14] does not possess this symmetry-breaking mechanism. In this theory, when the auxiliary field S acquires a vacuum value, it is the "physical" spinor of the compensating chiral multiplet that transforms as a goldstino. But the "physical" spinor is already the S-supersymmetry compensator. So in fixing the superconformal gauges, it must necessarily vanish already and there is no problem with Fermi- Bose counting. The local theory may be taken to a globally supersymmetric limit.

In closing we note that the situation with the new flexible supergravity theory [9] is unclear. We know from our previous work [12] that this theory possesses two chiral scalar superfields analogous to R and two indepen- dent auxiliary spinors p~ and p~ in addition to the "physical" spinor of the compensating multiplet. This suggests that even more complicating symmetry-breaking can occur in this theory.

Note added. After the completion of this work we were informed that similar conclusions had been reached by Karlhede, Grisaru and Ro6ek [15].

The authors would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. Comments by M. Ro6ek are gratefully acknowledged.

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Appendix. Building blocks for component actions. Here we give some details for calculating the component ac- tions in the text. We begin with (10), replace the 0 integration by D's and evaluate the D's on the integrand

s = - - fd4x (1 + ~k[(D2~22)-I + (D 2 ~ 2 ) - 1 ] (D 2 D 2 ~ l )

_ X ( ~ 2 ~ 2 ) - 2 [(~2ff~ 1) ( D 2 ~ 2 ~ 2 ) + (D ~ ~ 2 ~ 2 ) (D ~2ff~2) ]

- X(D 2 ~2 ) - 2 [ (D2~I ) (D2D2 ~2 ) + (D&D2~I)(D&D2~2)] + X(D2~2) -3 (D2~l ) (D~ 'D2f~2) (D D2f~2 )

+ X(D2~2)-3(D2£'21 ) (D&D2~2) (D&D2~2)}, (A.1)

~21 _= ( T ~ ) - I , ~2 2 - T l n (T2 T). (A.2)

Using the definitions in (13), we evaluate (A.1) in terms of component fields. We find

~2~2i1 [2i,~ff + -&- 1 -&- - ' - = ni, A~ ~ p& + i~'2i,AA ~ ~& + ~ ~i,.~Xp°~p&, (A.3)

DaD2[2il = -~2i,,~03 a + i{ Oa~& ) + ~2i, A~LOaF + ~&Pa + P&(Pa - i DaA)] + ~2i, A,4(~ p+ P&Pa )

i +i a + ~ ~2i, AA A ~ i,71,~(~,.,p p&), (A.4)

where the subscripts following the commas denote partial differentiation. The fourth derivative terms D2D2~2i are extremely complicated and have been completely described previously (see eq. (2.14) in ref. [11]). Here we merely give the purely bosonic terms

D202~ i I = ~2i, A~(OaA )(Oa.4 ) + ~ ~2i, AA (OaA)(OaA) + ½ ~2i,~(OaA) (O aA) + ~2i, A.3(FP -PaP a)

__ 21 "'i, A A O patov a - i 20 a A) _1~ [2i ,~ ffa(ff a + i 20 a ,4). (A.5)

[1] SJ. Gates Jr., M.T. Grisaru, M. RoEek and W. Siegel, Superspace (Benjamin-Cummings, Menlo Park, 1983). [2] P. Breitenlohner, Phys. Lett. 67B (1977) 49; Nucl. Phys. B124 (1977) 500. [3] SJ. Gates Jr. and W. Siegel, Nucl. Phys. B163 (1980) 519. [4] W. Siegel, Harvard preprint HUTP-77/A068 (November 1977), unpublished. [5 ] W. Siegel, Harvard preprint HUTP-77/A080 (December 1977), unpublished;

J. Wess and B. Zumino, Phys. Lett. 74B (1978) 49. [6] E. Bergshoeff, M. de Roo, J.W. van Holten, B. de Wit and A. Van Proeyen, Superspace and Supergravity, eds. S.W. Hawking

and M. Ro6ek (Cambridge U-P., London, 1981). [7] W. Siegel, Nucl. Phys. B142 (1978) 301. [8] W. Siegel and S.J. Gates Jr., Nuel. Phys. B147 (1977) 77;

S.J. Gates Jr., M. RoEek and W. Siegel, Nucl. Phys. B198 (1982) 113; V.I. Ogievetsky and E. Sokatehev, Supergravity as a theory of an axial superfield, Seminar on Quantum gravity (Moscow, December 1978).

[9] A. Galperin, V. Ogievetsky and E. Sokatchev, J. Phys. A15 (1982) 3785 ; JINR preprint E2-83-589 (1983). [10] V. Ogievetsky and E. Sokatchev, Nucl. Phys. B124 (1977) 309. [11] S. Ferrara and B. Zumino, Nucl. Phys. B134 (1978) 301. [12] B.B. Deo and SJ. Gates Jr., Comments on non-minimal scalar multiplets, MIT preprint, Mathematical Department (May

1984). [13] J. Wess and B. Zumino, Phys. Lett. 79B (1978) 394. [ 14] S. Ferrara and P. van Nieuwenhuizen, Phys. Lett. 74B (1978) 333;

K. Stelle and P. West, Phys. Lett. 74B (1978) 330. [15] A. Karlhede, M.T. Grisaru and M. RoEek, unpublished.

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