Conformal and poincaré tensor calculi in N=1 supergravity

Nuclear Physics B226 (1983) 49-92 © North-Holland Pubhshlng C o m p a n y

C O N F O R M A L AND POINCARI~ T E N S O R CALCULI IN N = 1 SUPERGRAVITY Taichlro K U G O t and Shozo U E H A R A

Departmentof Phvstcs, Kyoto Umverslty, Kvoto 606, Japan Received 17 August 1982 (Revised 21 January 1983)

We present the superconformal tensor calculus for N = 1 supergravlty in a complete form, ~rreduclble multxplets, their multlphcatlon and embedding formulae and m v a n a n t acnon formulae It is further clanfied in detail how the vanous versions of N = 1 Polncar~ supergravlty 0 e with different sets of auxlhary fields) are reproduced from the u m q u e superconformal theory The tensor calcuh for all the known versions of Pomcare supergravlty are denved exphcltly

1. Introduction It is now widely known that the tensor calculi in various versions of Pomcar~ supergravlty are systematically derivable from the unique tensor calculus m conformal supergravHy. Conformal supergravlty has larger symmetries but closes with fewer fields than Poincar6 supergravity, and hence it is much easier to find its tensor calculus. Such a point of view has been established through the long efforts of many authors including Kaku, Townsend, van Nieuwenhulzen, Ferrara, Grisaru, de Wit, van Holten and van Proeyen [1-4]. The higher the N of supergravity becomes, the more the power of this conformal approach becomes apparent; indeed thts approach was pursued systematically to N = 2 and 4 supergravity by de WLt et al. [4], and led to fruitful results. N = 1 supergravity, conformal as well as Poincarr, is now regarded as being completely understood, at least as far as its kinematical aspects are concerned. Nevertheless, there have appeared some new versions of N = 1 Polncar6 supergravity (i.e. with new sets of auxahary fields) in recent literature [5] At present the following versions are known as N = 1 Poincar6 supergravlty: (A) old minimal [6]: 12 + 12; (B) Sohnlus-West new nunimal [7]: 12 + 12; i Present address CERN, Geneva, Switzerland

50

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N = 1 supergravtty

(C) de Wit-van Nieuwenhuizen [8]: 16 + 16; (D) Breitenlohner [9]: 20 + 20; (E) Rivelles-Taylor new non-minimal I, II [5]: both 20 + 20.

(1.1)

The numbers n + n here denote the numbers of Bose and Fermi degrees of freedom. H o w version A is deduced from conformal supergravity was first clarified by Kaku and Townsend [1], and by Ferrara, Grisaru and Nieuwenhuizen [2], and was fully spelled out in van Nieuwenhuizen's review article [10]. Recently, versions B and D were also shown to be derivable similarly by de Wit and Ro~ek and by Siegel and Gates [3]. But B and D have not been worked out in detail and there remain m a n y points to be noticed. Therefore, in this paper we thoroughly discuss all five versions A - E and present the details, deriving their tensor calculi and actions in a unified way based on superconformal tensor calculus. Precisely speaking, versions C and E above are not versions of Poincar6 supergravity in the proper sense: (i) First, "auxiliary" fields in version C are not all truly auxiliary, 1.e. some of them have kinetic terms. But we have included it since it can be derived from conformal theory in quite the same way as other versions and ~t corresponds to an old minimal supergravity in the presence of a "Fayet-Ihopoulos" term for which the tensor calculus has to be substantially modified. (ii) Second, the two " n e w non-minimal versions" E turn out to have reducible sets of auxaliary fields, contrary to the original authors' claim. We d~scuss these "versions" here since their reducibility is quite obscure until we clarify the superconformal structure. Superconformal tensor calculus in N = 1 supergravity is more or less known already due to the m a n y works of the above authors. But unfortunately the results are scattered over m a n y papers and no literature seems to have appeared which gives that tensor calculus in its complete form. In view of its importance in m a n y applications, we present the totality of superconformal tensor calculus in a form as complete as possible; some parts are mere collections of known formulae and some are the original contributions of this paper. This is done in sect. 2. In sect. 3, the tensor calculi of Poincar6 supergravity are derived exphcltly for versions A - D above. By clarifying the relation between conformal Poincar6 multiplets, we deduce all the formulae in Polncar6 theories, multiphcatlon rules, embeddrag formulae, acnon formulae, etc. Sect. 4 is devoted to the discussion of the two " n e w non-minimal (20 + 20) versions" E which were recently constructed by Rivelles and Taylor [5] at the linearlzed level. The full non-linear structure of these versions is clarified easily in our conformal framework, and then the reducible character of the auxiliary field sets is made manifest. We shall see that these two versions are, in fact, reducible to the old minimal and new minimal versions, respectively. Combined with the arguments of Rivelles and Taylor [5], this implies that there exast only three irreducible sets of

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51

N = 1 supergravt(v

auxaliary fields with spins up to 1 for N = 1 Poincar6 supergravity, that is, the three versions (A) old minimal, (B) new minimal and (D) Breitenlohner's already exhaust all the possibilities.

2. Superconformal tensor calculus As stated m the introduction, we first present here the superconformal tensor calculus in N = 1 supergrawty in its complete form. We, therefore, also include here the recapitulation of many known results mainly due to the works of Kaku, Townsend and van N~euwenhulzen [1]. For the notation and conventions we follow throughout this paper, see van Nieuwenhulzen's review article [10]. 2 1 SUPERCONFORMAL ALGEBRA

Conformal supergravity is the gauge SU(2,211), the generators XA of which are supersymmetry ones Q and S, and a chlral transformation parameters eA are defined,

theory of the superconformal algebra conformal ones Pm, M~,,, K,,, and D, two symmetry one A. The gauge fields hff and respecnvely, as

h . -=h~,XA=e~,P,,,+~og A m m. M,,,,,+~b~,Q+U; -~ Kin+ b~,D+CpgS+A~A, e=--eAxA = ~ m P ~ + ) t m " M . , , , + g Q + ~ < K , , , + ) t D D + ( S + O A ,

(2.1) (m>n). (2.2)

The gauge transformation 6(e) and curvature (field strength) R.v A are given as usual:

6ga~g~(e)hA=E60(eB)h~=O~,e A + n~e , e c~ Joe A =(D~,e) A ,

(2.3)

B B C A R . / = O~h'~ - O.h~ + h~h.fc8 .

(2.4)

Here the structure constants lAB c defined through the superconformal algebra [X A, XB] = f A S X c are given explicitly in table 1 together with the exphclt forms of R t~vA .

Up to here the Pm transformation is an internal transformation quite irrelevant to the general coordinate transformation. In order to convert the former into the latter, the following three constraints are imposed on the curvatures [1]:

R.~'(P) =0,

(2 5a)

R~,~(Q)y ~ = 0,

(2.5b)

R j ~ " ( M ) e m % , , v - ½Rx~(Q)I,.~p ~ - ltee~,.o,,RP"(A) = 0.

(2.5c)

These constraints give algebraic equations for the M. .... S and K., gauge fields ~0~"",

52

T Kugo, S Uehara / N = 1 supergrawty TABLE l

Superconformal algebra and curvatures [M.,.,M.~]=3,,rMm, +3,~,M,,r-3.,rM.,

[P.,,Mr,]=8.,rP,-8.,,P.

[P.,, P.,] = 0.

[K.,,M.A=8,,,~K,-Sm,Kr, [P.,,D]=P,.,, [K.,,,P,,]= [S.P.,]='f,.,Q,

IS, D]= -t2S.

[Q. A] = - ]t¥5 Q, {O,O}='2(y"'c

[S,M.,,,]=%,,,S,

{S,S}=-'~(v°'C

[Q. D] = '2Q,

')P,,,, ')to ....

')M.,,,-(t75C ' ) A ,

{Q.S}='z(C-')D-(a'"C = O~. ....

-K.,.

2(Sm.O+ Mm,,),

[O.M.,,,l=o.,,,,O,

R/'"(M)

[K.,,K.I=o,

[K.,.D]=

[S,A]=~,75S.

[Q. K,.,] = - v . , S ,

8,,,M.,r.

(re>n)

~o"'%/-2(C/;'-~2/;' ) & o " ' % - ( ~ ) ,

R~,/"(P) = O~e;'- w/'"e,,~ + [email protected]"'¢. + e~"b,,- (tx ~ v), R j " ( K ) = O.f~"- %""f,,t, - '40t,Y " % - f ; % -

(Ix~ ~),

Ru~(D)=O,,bt, + 2e.,t, fY'+ 'zf.% R~oo(O)=(D~,L+,~,.+~bS

~

(t t ~ v).

'.A~,~.,~)o (Ix~),

R~,. ,~( S) = ( DyCp~,- ~ . 7 . , f [ " - ~bFP~+ '4A,,UPtdYs),~ - (Ix ~ P), R.~( A) = O.A. - ~ , v 5 % - (ix "-" ~)

{Pt, and f.", respectively, whach are solved to yield mn nan n m m n tOt' = --tO t' (e, qJ)+(e~,b -e~,b ),

=- % + . 6/'At'=_e

.),

'd'"o°ysy.Qp¢o,

=- e

(2.6)

.,ooS "°,

®.¢.-(O~,-½tOt,'o+½b~,-¼t75A~,)qJ

(2 ~,

7)

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N = 1 ~upergravttv

e;'f,,,~= -~(R~,-~g~R)+}RT,~,(Q)T~+x-~tk,~(A),

53

(2.8)

w h e r e / ~ denotes R~:~mn(M)emXe,~ with f~" put equal to zero. Since the constraints (2.5) are not lnvanant only under the Q transformation, the actual Q transformations, 6e, of these dependent gauge fields cox'", cp, and f ~ (determined by the ones of independent fields e~', ~ , A,, be), are no longer identical with the original group rule 6~(e) in (2.3): the differences 6~(e) = 6Q(e) - 3~(e) are [1, 10]

6~( e)6°~,mn= ½Rmn( Q )Y~e, 6~(e) q~. = ¼17"{75R.u(A) +

(2.9)

½k.~,(A))e,

8b(e)f ~ = _ , ,~¢ov,o, ,,,,

. m,~¢ovtc~

(2.10) (2.11)

Here and hereafter the superscript "cov" means the "additional" Q-covariantization which ~s always necessary in the presence of denvatlve terms of those dependent gauge fields; in the above case, due to the 0cp terms in R,,(S),

Rc~°"(S) =R,,,(S)-¼1(fJ,,ya(ysRx,(A)-

½/~x,(A))- (g~/,))

(2.12)

By the help of Bianchi identities g'"P°D~Rpoa = 0 and others, one can show that the constraints (2.5) imply the following equalities which will be useful below:

Ru.(D)=-~lh~,~(A)=½(R~7(M)-R~7(M)), R~°V(M) +

R~°~(M) = 0,

(2.13a) (2.13b)

R,~( Q )o ~'~= d'"P°R~a( Q )7, = 0,

(2.13c)

where

RC°Vm,,(M) = R~,~,,,n(M ) + ½R,,,.(Q)(y~, - 7u~p~). The following equation will also be useful: (2.14) The identity t~

r

A

A

= ~ s ( ~ " hB)h~ + ~R.S B

(2.15)

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gives the meaning of a general coordinate transformation to the Pm transformation as

a.(4 m

A_

m v

em)hA -ESB,(

m B'

hm)hA - ~ R~. A,

(2.16)

B'

where the index B' means all transformations other than the Pm transformation. The (Q, Q) antlcommutator onginally yielded the Pm transformation, but now gives the following P,~ transformation: 6 ~ ( ~ m ) = 6GC (~me~n) --

Y'~6A,(~mhA,'),

(2.17)

A' [ ~ Q ( E l ) , ~ Q ( E 2 ) ] = ~j~(/E2~{mE1 )

(2.18)

on any independent gauge fields. The elimination of the ~ R p.vA term in (2.17) was attained by the constraints (2.5) a n d / o r the change of Q-transformation (2.9)-(2.11). Use has been made of equality (2.13c) there. Hence, the commutator algebra between two transformations other than the Pm transformation now obeys the group rule [SA'(eA'),

6B'( eW) ] = E 6C( eA'e"'/,'K c)

(2.19)

C

with the understanding that 6c = 6~ when C = Pro. The commutators including/5 are also easily calculated by using (2.19) and constraints (2.5) and are found to become [a~(~m), aA'(eA')] = E a . ( ~ ' % A L ' P . , " ) + a~ '

E

B

B'=M,S,K

m t/ A"

m'

E

a . , ( ~ ' a ~ ( e ) h~'),

(2.20)

8B,(6b(~, • +)~2. hB' - (1 ~ 21),

B'=M,S,K

(2.21) where the second terms of (2.20) are present only when A' is a Q-supersymmetry transformation. The algebra (2.19)-(2.21) holds for any independent gauge fields and hence also for any function of these fields. The transformation of matter fields is determined such that the algebra (2,19) holds for them, and hence the whole commutation relations (2.19)-(2.21) hold also for the matter multiplets. For the fields q~ carrmg only flat indices (i.e. general coordinate scalar fields), the/5 m transformation defines a conformal covariant derivative D~,:

In terms of this covariant derivative, the commutators (2.20) and (2.21) are rewritten

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55

as

[Dm~,8.'(~')] =ES.( E f~'e., )+8~' h'

E

O

B

8.,(8~(e)h.~,),

(2.23)

B'=M.S,K

[D,~,D/,~]=Y'.6A,(R,,,,,A') + A'

E

~B,(~Q(~m)hn

B'

-(m(-)n)).

B'=M,S,K

(2.24) In (2.23) the parameter e A' is understood not to be differentiated. These indeed are useful formula in treating covarmnt derivatives. For instance, the conformal covariant d'alembertian can be easily evaluated by using the formula (2.23):

rq'0 = D,~,Dm, ep = OmD`m0 _ E S A , ( h A ') D,~flp A"

=

__

r, ( _[_ ~ (0 m e~n )e,,,D~ep c~l.tDy~

__

~ ~A' E ( D=aA,( h,. ) -- [ DL, aA.( I~A.,')I)e~ A'

=e

'O~,(eg~'~D:,f)-w.,

(e)D~O--EDLSA,(hm)O A'

+ E 6B(hAm'f,'e,,f)O + A',B

Z

6e'(8~(+,,)hBm')O •

(2.25)

B'=M,S,K

H e r e / ~ ' means that ham'is not differentiated by D m, and use has also been made of the formula n # - - ( 0 m e~)emVn+O"V~+w mm n (e)V,,=e-lc),(eg"~V,),

(2.26)

which holds for any vector quantity Vm. 2 2 MATTER MULTIPLETS

2.2.1 Complex vector mulnplet c~'(w, n). A most general multiplet (without external mdmes) is a complex vector mulnplet c~-= [C, ~, .cJC,K , 633,,, A, @]*. In general, mulnplets in superconformal theory are characterized by two parameters, w and n, called conformal (Weyl) and chiral weights which specify the properties under the dilatation D and chlral A transformations. We always define these weights by the transformation property of the first component of the multlplet: 8D(x

)c = w ,oc,

A(o)c =

(2.27)

The complex vector mulnplet cVwlth weights w and n transforms as: * We denote complex (or Dlrac) fields by s~npt letters C, ~, etc, whale real (or Majorana) fields are denoted by ordinary atahc letters C, Z, etc But, excepnonally only for the A-component, we denote the complex (Dxrac) field by A and the real (Majorana) field by X

56

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Q-transformatmn (2.28a) (2.28b) (2.28c) ~e(~)~c= - ½ e ( a ~ + a ) ,

(2.28d)

~ e ( 0 % = - ~ e ( D ; ~ + Vma),

(2.28e) (2.28f) (2.28g)

where (2.29a)

D~,(~ = Om~- ½f ml~'5~ -- wbm~- ½,nAm@,

(2.29b)

+ (3,.y 5 - ½m)Am~Z, Dm~, , ~ = D,~j~,,+~+m(D,~,~+.y,,A)~c

- (w+ 1)b,,~ _ ~,,,( +~__+1 w 2 v5) ),~ (2.29c)

_ ½mAroOn - 2mCem,f,,, DmA = D,,~A - 1 ( o . ~ + 'VS@)+m -- (W + ~)bmA

+ ½(lY59C+ ~ + ~ - O~G, ys)(W + nVs)(F,, - (½m + 3 , y s ) A , A + em~f;(w +

n'}'5)Y,~,

(2.29d)

with the definition D,~ - em~'(0~,- %,k/6Mk0. S-transformatmn (2.30a)

as(t)c=0, a s ( ~ ) ~ = - , ( n + wvs)C~,

(2.30b)

8 s ( f ) ~ = (( w - 2 2) --5-- ~v5+ ~, (( w - 2 n 6s(~)%= ---5-- + 2vs) ~'

(2.3Oc)

(2.30d)

\

.@ 8,(~)a =

-[w+l -

n ) Tm~;,

½(,-~0c + ~ + ~

6s (~') @ = ½(("~5 + ' n ) D ~

-

a'c,~,~)(w + ,~)~,

+ (("¢5 w + m ) A .

(2.30e) (2.30f) (2.308)

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57

Other transformations D, K m and A are given in table 2. These transformation propernes are almost identical with those in the rigid superconformal case if the usual denvative 0~ is replaced by the superconformal covariant one D,~. Only one difference appears in the definition of oy~, in (2.28f) above owing to the noncommutatlvlty of two D,~'s:

3m,

D~,~.-D~,~,,,+~tem.

[Dk,D[]G

=Dm@ . - D~@r, + ½ R , , , ( Q ) ~ + ½ w R m , ( A ) G -

¼nemnklRkl(A)G,

(2.31)

where formula (2.24) and the constraints (2.5) are used. In order to check that the algebra (2.19)-(2.21), or (2.18) in particular, holds on the mulnplet ~ , we need m a n y equalines derivable from the constraints (2.5), for instance,

( ymokt~£ )( Rk,( Q )y,,e) = O,

(2.32a)

emnkIDnRkl(A)=O,

(2.32b)

em~k'(D~Rkz(Q) + R~,~v(s) y,) = 0,

(2.32c)

R'm°~(S)o "" = 0

(2.32d)

as well as (2.13).

2 2.2 Real vector multtplet V (n = 0).

In the case when the chlral weight vanishes (n = 0), the complex vector mulnplet decomposes into two irreducible real vector multtplets V~ and V2: ~ = I"1 + iV2. Thus, the real vector mulnplet V = [C, Z, H, K, B m, ~, D] exists for arbitrary conformal weight but has vanishing chlral weight n = 0. Their components are all real (or Majorana) fields. The transformanon property of V is just gwen by that of q[wlth n = 0. 2.2.3 Gauge multtplet G (w = n = 0). For the real vector multlplet V with vanishing conformal weight w = 0, we can define a submultiplet which we call the gauge multtplet. Define

B E =- e,m B , , - ½f, Z;

(2.33)

then its Q-transformation becomes (2.34) and the last derivative term can be regarded as a gauge transformanon. The subset G = [BE, X, D] indeed spans a closed multlplet under all superconformal transformations by virtue of w = n = O. Other members of this gauge multlplet G transform

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T Kugo, S Uehara / N = 1 supergravtty TABLE 2 D, Kin, and A transformations on a complex vector mulnplet

a,)(~,,,) c = wXDC,

~A(O)C= "m(~O ,

~D(x~)~=(w+½)x~,

ao(xo)gC=(w + I)xo%, ~ (0)~c = ( - ~%+ 8A(0) ~.,

",,~c)o,

ao(),D)~=(w + 1) x,,%

= ~' i n ~ m O ,

a , , ( x , , ) ~ = (w + 1) x , , ~ ,

<(O)A = (-~,. + ~,~,) aO,

aoCXo)a = ( w + ~ ) x o a ,

6 A ( 0 ) 6 ~ = '2tn6)O,

~o(xD)o~=(w+ 2)x.~,~,

m ( 8K(~Tv')@ = -2w~KD,~g2m~Km 6~m,

C~,g, % and ~ are 8,v inert

u n d e r the Q - t r a n s f o r m a n o n

as

6Q(e)h = ½(o. F ~ + lysD)e,

(2.35a)

8o(~)D = ½~,VsOcX,

(2.35b)

w h e r e F,', a n d D~)~ n o w t a k e s i m p l e f o r m s since w = n = 0.

F ~ =- ( OrB" + ½f ,'/, X ) - (it ~ v ) , D~ X = D ~ X - ½( a . F ~ + t~,sD ) q,~ - ~b~X - 34t'YsA~X.

(2.36) (2.37)

G is S and K m inert. 2 2.4. Chtral multtplet Y. ( w = n). W h e n w = _+ n, a s m a l l e r m u l t i p l e t exasts t h a n the g e n e r a l c o m p l e x v e c t o r m u l t l p l e t W. It is a r i g h t - h a n d e d chlral multlplet ~. = [(~, 6)RX, ~](62 R - ½(1 + ~'5)) w h e n w = n, a n d a l e f t - h a n d e d chtral m u l t i p l e t Y.* = Ida*, @ L X , ~ * ] ( ~ L ------½(1- VS)) w h e n w = - n . ( T h e 4 - c o m p o n e n t X is a M a j o r a n a

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59

field.) Their embedding into the complex vector multlplet is

¢~-(~ ) =

[ ( 2 , -- I @ R X , -- ~ , l ~ ,

,D~(2,0,O],

(2.38a)

¢~(~*) = [ (2" I~LX, --~*, --lO~*, --/Din(2*,0,0 ] .

(2.38b)

Their transformation rules can therefore be read from those of 'V. In particular, the Q and S transformations are given explicitly by 8Q+s(2- [SO(e) + 6 s ( f ) ] (2 = ½g@RX,

(2.39a)

~Q+S@RX = @R(D~(2 + ~ ) e + 2W(2@R~ ,

(2.39b)

6Q+S~3 = ½gD~@RX + (1

-

W)(@RX

,

(2.39c)

where (2 40a)

D,~(2 = 0 , . ( 2 - ½f,,,@RX -- wbm(2- ½twAm(2,

D~@RX = @R[ DY, x - ( D ~(2 + ff)+m - ( w + ½)bmx - t ( ½ w - ¼) A m X - 2W(2Cpm] . (2.40b) In rigid supersymmetry, Y. and Y.* are the multiplets satisfying Dqp = 0 and D~* = 0, respectively. The components (2 and ff are complex and are usually written m terms of real component fields as (2=½(A+tB),

~=½(F+

lG).

(2.41)

We also use these real notations below. 2.2 5 Linear mulnplets ~ (w - n = 2), L (w = 2, n = 0). In rigid supersymmetry, a hnear mulUplet ~s a multlplet subject to a constraint D D ~ ) = 0, which may be ~mposed by using a right-handed chlral multiplier q~ as

f d4x DD(cpJDID¢) = f d4x D D D D ( c p ~ )

(2.42)

m the action. In conformal supersymmetry (yet rigid), the action (2.42) is superconformal lnvarlant only when q0q) has conformal weight 2 and chlral weight 0. Since the right-handed chiral multiplet q0 has weights satisfying w = n, the conformal and chlral weights of the hnear multiplet • must satisfy w-n=2.

Tbas is also the case in local conformal supersymmetry. This hnear multlplet is

60

T Kugo, S Uehara / N = 1 supergravlty

complex in general and denoted by fi = [C, ~ , 3C, ~m, X]. [The component fields are all complex (or Dirac) except for the last one, X, which is Majorana.] We call it a complex linear multiplet. The transformation properties of the components will be manifest if we see how It is embedded into the general complex vector multiplet:

w(e) = [G, s, %, -,%, % , (1 - -~s)x + W(~,~ z, - ,z~), -G~c-

,DJ"],

(2.43) where Z~ and Z 2 a r e two Majorana components of %: ~ ; - Z~ + tZ 2. The complex linear multlplet of opposite "chirality" (hence satisfying w + n = 2) corresponding to DD~* -- 0 is also possible, but we denote it by ~*, simply regarding it as a complex conjugate of the above linear multiplet of "normal chlrality" (w - n = 2). 2,2.6. Real hnear multlplet. This can exist only when the chlral weight n is zero, and hence must have conformal weight 2 (w = 2, n = 0). Since it is real, it satisfies both conditions D Dq~ = D D ~ = 0 in the rigid case notation. It is denoted by L = [C, Z, B,,] and embedded into the real vector multiplet as

v ( t ) = [c, z,o,o, B~,-O~z, -Gcc].

(2.44)

Here the vector component B,, is subject to the constraint

D~B~=O.

(2.45)

Interestingly this constraint is solvable as m the global case (cf. ref. [4]), due to the equality DCB m =

--

n

I

O•(eem"(Bm + ZOmn ~ ) + ~le

# u p a

~b~yp~oC } .

(2.46)

--

In proving this the following identities will be helpful:

D,~V'=e

--

I

O~,(eg V~)+½f/.V+mVm+ 3bmV " /xe

le""Ooy,+,(fUy~+O)+Om"+t(fmy'~n)+2Omn~/

n

(2.47)

(~b'T~k)--0, --

m

m

(2.48)

where the former holds for any vector quantity Vm and the latter is derivable from the useful idennty [101 (2.49) In any case, due to (2.46), the constraint equation (2.45) is solved m terms of an

T Kugo, S Uehara / N = I supergrawtv

61

anttsymmetrtc tensor a ~.: B m = e -lemt~ 8~,po[O~aoo_~~flyo+oC)~

_ ZomnlP-

n.

(2.50)

a,~ transforms as

8a( e)a,. = -- ½go,.ysZ + ¼/~" (V.¢r - Vu+tt)C,

(2.51)

and remains invarlant under S, D, K m and A transformations. The commutator [6o, 60] on a . . is modified by a vector gauge transformation term 8g(Ao)a.~ = O~,A~ -

O~A.:

[~Q(el),~Q(e2)]a~v=6~(~m)a~ +(~g(--½t~,C--~Xahp)a~v, with ~m

(2.52)

I - Ym81 . ~-82"

2 3 CHIRAL MULTIPLET FROM OTHER MULTIPLETS

The embedding of various multlplets into the vector one is already stated in the above. We discuss the embedding of multlplets into chiral ones here. As a general remark, ~t is important to notme that, in the superconformal case, those embeddings into chiral multiplet are possible only when the multlplets have special values of weights w and n. This comes from the fact that the chiral multlplet has equal chiral and conformal weights, w = n. This point presents a sharp contrast to the usual (Pomcar6) supersymmetry. 2.3.1. Vector--~ chlral. The real vector multiplet V can form a (right-handed) chiral multiplet E(V)=

[½(H-tK),

16~R( D c Z + ~.),

-½(D+E3~C+IDmBm)] (2.53)

only when V has conformal weight w = 2. This is understandable from the fact that the ~ component of the chiral multiplet is S inert while 6sH = ((½w - 1)I75Z and 6 s K = ( ( ½ w - 1 ) Z . This Y~(V) corresponds with D D V and has weights wz~(v)= n~(v) = 3. (Notice that D carries conformal weight ½ and clural weight 3 m the rlgxd case.) S~milarly, for the complex vector multiplet ~ , we find S-inert ~XZand K combinations only when w + n = 2: 6s ( ~ _+ I K ) = 0, Thus the multiplet CV'satlsfylng ~ ( e l f ) = [½(0{~-- t ~ ) ,

w

-

for w _+ n = 2, respectively. n

=

2

can form a chiral multiplet

t@R(DC~+A),

--½(@+[3C~+lD~6j~m)],

(2.54)

T Kugo, S Uehara /

62

N = 1 supergravl(V

which has weights wz{,¢)= n~.(,v)= w + 1. (When C~fhas w + n = 2, we can use this formula for the complex conjugate ~ * . ] Incidentally, the complex hnear multiplet ~ has weights w _+ n = 2, but the chiral multiplet G(~((E)) formed from ~ by the above formula just vamshes (as is ewdent from the c o n s t r a i n t s / 5 / 9 ~ ( E ) = 0 m the rigid case). Hence we cannot form a chlral multiplet from fl linear in ~. In particular, even in the rigid superconformal case,/SE is not a chiral multiplet despite the fact that D ( D ~ ) = 0. /)E does not satisfy the weight condition w = n of a chiral multiplet and hence is not closed under superconformal transformations. 2.3 2 Gauge --* ehtral. The usual field strength W~ Ol~(e-gVOc~e gV) for V having weights w = n = 0 also has a correspondent in the superconformal gravity case, 3 because the (complex) vector multlplet (e-gVD,,e gv) has weights w = ½ and n 2 and satisfies the above condition w - n = 2. Since the gauge multiplet G ~s a V in the Wess-Zumino gauge, W~ becomes linear in c o m p o n e n t s of G and gives a chiral embedding of G as usual: =

W~ --- ~ ( G )

-- (6)R ( 0 " F ~ - ID))B~,

= [ ( ~ R X ) ~,

(2.55) T ~ s chiral multiplet carries weights w = n = 3. 2.3.3. Klnettc multtplet. Finally in this subsection, we give the formula for the kinetic multiplet, which is an analogue of D Drp* in the rigid case. The kinetic multiplet, denoted here by T(Y.), exists only for ]~ having weight w = n = 1 in the conformal case and is given by the formula T ( ~ ) = [oZ.,

6~RD~x,

Hq~*].

(2.56)

This multiplet is right-handed chiral and has weights w = n = 2. 2 4

MULTIPLICATION

LAW AND

INVARIANT

ACTION

FORMULAE

We can combine two supermultiplets to obtain a third one. The multiplication formulae are f o u n d to be quite ~dentical with the usual ones in rigid supersymmetry provided that the derivatives are replaced by conformal covariant derivatives: (a)

% = [C~C2,

% X % = c'~3 ,

W 3 ~ W 1 -k W 2 ,

Gl~;2+ G ~ l ,

C,%2 + C2%~ - ½~Tc~2,

//3 = //1 q- / / 2 "

C~K2 + C2%~ + ½~[C~V5%2, ( G j A 2 + ½(0C, - rysK , + 175~, - D c G 1 ) ~ 2 ) + (1 ~ 2), ~16'~2 + ~26~1 + ~Ct0(~2 + '~,~'~2 - °~°~6~ - DmC~l " Dcm~2 - ~ T C A 2 - ~2TCA , - - ½ ~ T c D c ~ 2 __ ~I 2 CT D

c

~,]

.

(2.57)

63

T Kugo, S Uehara / N = 1 supergravltV

(C: charge conjugation matrix.) (b)

Y'~ × Y'2 = Y'3,

~']'3 = [(~1(~2 '

w3 = n 3 = w~ + wz.

C~R(~IX2+(~2XI)'

~1~6~2 q-~2~6~-~I -/XI6~RX2].

(2 58)

T h e multiplication W(Y.I)XC~t'(Y.~)=cV defines a complex vector multlplet with weights w = w~ + w2 and n = nj - n 2. In the case when n~ = n 2, tins C~'splits into two real vector multlplets: (c)

4 X Re[Cg'(2gl) X ¢V(Y.~)] =-- Y.l ® Y.~ = V

V=[A,A2+BIB2,

(when n, = n2).

- ( ( e l + , v s A , ) x 2 + (1 *-' 2)),

-{(A,F2+B,G2)+(1 ~2)},

(B,F2-A,G2)+(1 ~2),

(BtD,~A2-A,D,~B2 + l2XiTC, YsY,,,X2x) + (1

2),

{(O, + t v s F I ) x 2 + ~9'(B, + lvsA,)'X2 }+ (1 ,-, 2),

2FIF2 + 2G1G 2 - 2D~,,A I . D,"A 2 - 2D,~,B I " D,mB: -xTCI~'X2

-

xTc 5~'XI] ,

(2.59)

where (A, B) and (F, G) are the real c o m p o n e n t fields of 0~ and ~-[see (2.41)]. It is worth r e m e m b e r i n g that the formula ~(~)

X~. 2 = Z ( ~

× c~(~2) )

(2.60)

holds just as (DD6Pl)x ep2 = DD(~lep2 ) m the rigid case. Finally in tins section, we gwe the action formula m v a r i a n t under superconformal transformations First is the well-known F-term formula [1] which gives the m v a n a n t action for the cinral multiplet Y. = [½(A + 1B), @RX, ½ ( F + 1G)] with weights w = n =3:

IF----fd4x[E( ..... 3)]F = fd4xe[F+

½qT"VX + ½ f , , o " " ( A - , v , B ) G

].

(2.61)

T h e second action formula, D - t e r m formula, is derivable from tins formula; indeed, since the real vector multlplet V having conformal weight w = 2 can be e m b e d d e d

64

T Kugo, S Uehara /

N = 1 ~upergraw(v

into the charal multiplet Y.(V) with weights w = n = 3, we apply formula (2 61) to ]g(V) and obtain

Io: f dax[--Z(V)]F = S d4x e[D

+ I~
This formula may be rewritten in a form more similar to the usual Poincar6 supergravlty one by using various equalities, (2.25), (2.47), (2.48), etc.:

= f d4xe[D -

½+.'t,Q,5)t- ~.'y,'l, sZ +

½C(R+ I+-~.R~) (2.63)

where R = ku ~ - 2fmo'~n%,

(2.64a) (2.64b)

For the reader's convemence we give m the appendix the component formula of powers Z~ and V k of the chlral multlplet Z and the vector multiplet V, as well as the one for g(Y.) = Y'~bk~.k and ~(Y~, Z*) = ~ aktC~((Y~k) × ~q-(y,t) k

k,l

3. From conformal to Poincar/~ supergravities As was first shown by Kaku and Townsend [1] and Ferrara, Grisaru and van Nmuwenhuizen [2], the usual Pomcar6 supergravity theory (with "old" minimal auxiliary fields) is understandable as a gauge-equmalent one to a conformal supergravity; that is, the Poincar~ supergravity is obtained from a conformal supergravtty, by fixing the gauges of extraneous local symmetries S, K m, D and A which exist m the conformal supergravity but not in the Poincar6 one. Tlus way of understanding proves very useful; the conformal tensor calculus is simple enough owing to the larger symmetry. Further this umque superconformal calculus derives all the tensor calculi in various Poincar6 theories (with a different set of auxiliary fields) simultaneously.

T Kugo, S Uehara / N = 1 supergravtty

65

The way in which Poincar6 theories are obtained from conformal supergravity is well exemplified in a simple analogous model: the lagranglan = - ~ e R ~ 2 + ½eOrfp" O~'~

(3.1)

is locally scale invariant (6Dg~, ~ = -2?log~, ~, 6Dq~ = )~Dq~). One can obtain the usual Einstein lagrangian from this by fixang thts additional gauge freedom by qffx) = v~-K- 1. In this example, the field qffx) plays the role of fixing the scale unit of the world and may be called the "compensating field" since the Einstein action is made scale lnvariant by virtue of compensation by q,. In conformal supergravity, we need a supermultiplet which plays an analogous role to that of the above compensating field q,. We call such a supermultiplet a "compensating multlplet." Different choices of compensating multiplet(s) leads to different Pomcar6 supergravitles (i.e., with different sets of auxihary fields). The five versions A - E tabulated in (1.1) are known as N = 1 Poincar4 supergravity at present. For the reasons stated in the introducUon, we discuss all four versions from A to D in this section and derive their tensor calculi and acUons in a unffield manner from the superconformal tensor calculus developed in the preceding section. The discussion of versions E is left to the next section. 3 1 OLD MINIMAL POINCARE SUPERGRAVITY This is the most popular form of Poincar6 supergravlty having 6 auxlhary fields: scalar S, pseudoscalar P and axial vector A~~x. This case was explained m detail in ref. [10]. Recapitulating the results as well as adding some new comments, we explain our general procedure. 3.1 1. Gauge f i x i n g and the Q-transformation in P o m c a r k supergravtty. To reproduce this model, we take as a compensating muluplet S a dural multlplet Y~0= [(~o, ~RX0, ~0] = [½(A0 + IBo), @RX0,½(Fo + tGo)] with conformal weight w = 1 ( = n). Gauge fixing for the extraneous gauge freedoms of D, A, S and g m is done by the following con&tions: A o ( x ) = 1,

D gauge,

B 0 = 0,

A gauge,

X0 = 0,

S gauge,

b~ = O,

K m gauge.

These gauge conditions are lnvarlant under general coordinate and local Lorentz transformations, but not under Q-supersymmetry:

(For later convenience, we introduce a notation, 7/-= - ( F 0 + IysGo + ½4h75) [10].) In

66

T Kugo, S Uehara / N = 1 supergravlty

order to preserve X0 = 0, we must take the following combination of the Q and S transformations:

(E) + as

(3.2)

Further, to preserve b, = 0, this should be modified by adding the K,, transformation as

8~( ~) = 80( E) + 8s( ½~le) + 8 K ( ¼~p,,e - ½~m~/E),

(3.3)

which we identify with the Q-transformation m the old mimmal Poincar6 theory. Indeed it reproduces the usual transformation on the gauge fields e~ and ~, in the minimal Polncar6 supergravity: I - rn~., 8 ~ ( e ) e ,m = ~e3'

(3.4a)

8 g ( E ) , . = ( D/, - ~ t 7 5 A . -

IY~,~I) ,

(3.4b)

provided that the auxiliary fields S, P and A~uX are ~dentified with the surviving compensating multlplet components Fo and G o and chiral gauge field A. as follows: S = 3 F 0,

A~. . . . - - 2 ~A ~,

P=-3G0,

(3.5)

then 7/= - -~(S - t 75P - 4lau~t 75 )- The transformation rules of these auxdlary fields also coincide with the usual ones [see eq. (2.7)]:

6 ~ ( E ) S = 32~0'Xo = - ¼g(2y- eg - T"~7+.),

(3.6a)

6 ~ ( E ) P = - 3917sD'Xo = ¼gt75 (2T - q) - 7"~/~b~),

(3.6b)

( ~ ( E ) A ~ ux = -]g(2/"{5T. -- B/"/5~b ) .

(3.6c)

These transformations satisfy the algebra [~(El),~(e2)]

=~GC(~)+~m(__~,O)

....

~IE' mnAl~gkA ..... `

= ~E2]t mE1 ,

+ ,1~20.1-r a n ( j r , _ lqtsP)Ei )

(3.7)

which lS also derivable from the conformal algebra (2.19) and eq. (3.3) directly. 3 1.2 Tensor calculus. We first explain the general strategy used to deduce tensor calculus in Poincar6 supergravity: (i) The matter multiole)~ in conformal

T Kugo, S Uehara / N = 1 supergravt(v

67

supergravity are characterized by conformal weight w and chiral wmght n. But the notion of these weights is an alien concept in Poincar6 supergravlty and thus should be eliminated. This is also accomplished by the help of the compensating multlplet 2go; that is, we can always "normalize" the weight (w, n) of any multiplet to some fixed value (wo, no), the "standard weight," by multiplying suitable powers of 2go or ec((2go). Then the transformanon rule is clearly made independent of (w, n) and is identifiable with that in Polncar6 supergravity. The "standard weight" (wo, no) may be chosen at will for each type of multlplet (chlral, vector, linear, etc.). But it would be s~mplest always to take w0 = n o = 0 (if possible). (ii) One more point to be taken into account is that many formulae in conformal theory are only applicable to multlplets having specific weights; for instance, the embedding formula Y~(eT), (2.54) as vahd only for the vector multlplet W satisfying w - n = 2, and further the constructed multiplet 2g(W) does not necessarily have the standard weights w = n = 0 of Polncare's multiplet. Thus these mismatches of weights are also to be adjusted by the use of the compensating multiplet lifo*. Now let us see these procedures more exphcitly in the old rmnimal case. The chlral multiplet Y'(w) with wmghts w and n ( = w) is redefined as Y,P = N(~) X ( 2 ~ o ) - ~ ,

(3.8)

such that it has the standard weights w = n = 0. Note here that our compensating multiplet Yo has weights w = n = 1. The factor 2 m front of Y'o is just for convenience. ]~e is identified with the chiral multiplet in Pomcar6 supergravlty and eq. (3.8) gives the formula relating the chiral multiplet in conformal supergravlty to that m Pomcar~ supergravlty. Since 2Y~0 = [A o + iBo, 2~RX0, F 0 + lG0] = [1,0, ~ ( S - ~P)] m the present gauge, (22go) -~ is easily calculated to yield = [1, o, -

- ,P)].

(3.9)

Hence eq. (3.8) gives the formula for the (real) component fields as AP=A,

B e= B,

X p = X,

F e = F-

~w(SA + PB),

GP= G-~w(SB-PA),

(3.10)

coinciding with the known result [6]. For vector multiplets also, we can apply just the same procedure. Notice that the vector embeddmgs W(Y.o) and W(Y.~) of our compensating multiplet have weights *A similar observanon was made also by de supergravlty

Wit,

van

Holten and van Proeyen [4] for N = 2

T Kugo, S. Uehara / N = 1 supergravtty

68

w = 1, n = 1 and w = 1, n = - 1 , respectively. Thus the complex vector multiplet ~ w , , ) with weights w and n is reduced to the Poincar6 multiplet by the following formula so as to have w = n = 0: (3.11)

5(v=CV(w,,)× [2c~'(Zo)]-(w+')/z× [2cV(~.;)] - ( w - ' ) / 2 .

In particular for real vector multiplet V~w) with conformal weight w we obtain

v

qw)x v, w/2,

(3.12)

where we have introduced a short-hand notation VI = 4 W ( £ 0 )

x W(~..; ),

(3.131

denoting a real vector multiplet of conformal weight 2 since it will a p p e a r frequently below. In Pomcar6 supergravity, there is no genuine notion of a complex vector multlplet: the vector multiplet W p is in fact reductble into two irreducible vector multiplets Vlp and V2P, the real and imaginary parts of ~ P. Since V { w / 2 = [ 1 , O , ½wS,-½wP,½wA~m~,O,-IIsw(S2+ P2 - (A~,Ux)2)],

(3.14 /

f o r m u l a (3.11) for the real vector multiplet becomes in c o m p o n e n t s C v = C,

(3.15a)

Z v = Z,

(3.15b)

H p = H + ½wCS,

(3.15c)

K p = K-

(3.15d)

½wCP,

B ~ = B , . - ~4- I Wt'~Aaux "~'m ,

Xe = X + ~ w ( S + , y s P + tYsAaUx) Z ,

(3.15e) (3.15f)

D p = D + ~ w ( H S - K P - B . A a~x) + ~ w 2 ( S 2 + p2 _ (Aao~,) 2) _ ¼'w(f,,~l - 2Cg,) y " y , Z .

(3.15g)

Notice also that the "reverse" of this relation is given by the same formula with w replaced by - w. The gauge mulUplet (7 has weights w = n = 0 and hence receives no redefimtion: GP=G.

T Kugo, S Uehara /

69

N = 1 supergrawtl

As for the linear multlplet ~, we identify the Pomcar6 one ~P with E which has weights w = 2 and n = 0, and hence L p = L for the real linear multlplet which has w = 2, n = 0 . Then the general hnear multiplet E(~=~ 2~ with weights w and n = w - 2 IS related to EP through the formula: ~V(~P) = qf( if-(. . . . 2)) X [2c~'(Y~0)] -~, V(LV) = V(L ),

for complex linear multlplet.

for real linear multiplet.

(3.16)

[Notice that this satisfies the hnear multlplet condition D DC~(~ P) = 0 in the rigid case.] Contrary to the case of the complex vector multlplet, there lS a genuine noUon of a complex hnear multiplet here. ~P IS, In fact, an irreducible multlplet. If we use a non-local projection, EP can be decomposed into two real linear multlplets L e and L P and one chlral multlplet YP*. We should notice here that e~/-(Ep) or V(L p) is still a conformal vector multlplet because it has non-zero conformal weight w = 2. The embedding of EP(L r') into the Polncar6 vector multiplet ~ P ( E P ) (VP(LP)) is accomplished by

(vi)', VP(Lp) = V ( L e ) × ( V I )

l,

(3.17a) (3.17b)

which indeed has w = n = 0. A curious point is that such an embedding ~(P(£P) (VP(LP)) does not satisfy the linear multiplet condition D D C ~ / P ( ~ P ) = 0 ( D D VP(L p) = 0) in the rigid case despite the name "linear multiplet", because of the multiplication by the vector multiplet V~ in (3.17); indeed, for instance, VP(L p) has non-vamshmg H and K components! This complication probably explains the reason why the linear multiplet in the old minimal Poincar6 supergravlty has not been discussed in the literature until the recent article of Sohnius and West [1 1]. So we present here the explicit forms of 3~ transformations for each component of L P = [C, Z, Bin] with Dm Bin= O, since they do not seem to have appeared in the literature:

vP(t P) = [c,z,

cs,

--

ce, Bm..~ 2 PAlaux

-D~Z-~(aq-l~5P-l-lY5~aux)z,

_fqc C _ 2 B . Aaux + 2C{$2 + p2 _ (A2,X) 2) _ ½~y,/757Dp" + gl~,~/t, qO],

(3.18) 8g(

)c = ke, ,sz,

(3.19a) (3.19b)

P

-

n

(3.19c)

70

T Kugo, S Uehara /

N = 1 supergrawty

where the conformal covariant derivative D~, is understood as the one for the multiplet with weights w = 2, n = 0. Next consider the other embedding formulae. The embedding of y v into the vector multlplet W P is achieved by the same formulae for W(Nv), eqs. (2.38), as in the conformal case, since W ( y v ) already has the desired weights w = n = 0 Of course, the conformal covariant derivative D;, in eg(Yv) is understood as the one for the multiplet with weights w = n = 0. The converse embedding e v e - + ~ e is not so trivial" since the conformal embedding formula Y.(eV) [eq. (2.54)] is apphcable only to Wsatxsfying the weight relation w - n = 2, we first adjust the wmghts of 'V v by multiplying compensating multiplets cg(Y.0) and e g ( £ ; ) suitably and then apply the formula. Further, since the obtained multiplet £ ( W ) does not necessarily have standard weights w = n = 0, we should adjust its weights again using compensating multlplets. The way of making these two weight adjustments is not unique but yields a unique result as follows:

y v(e,CP) = { y[q(v X (2W(£o)) w-' X (2eg'(Y..g))]} × (2No) -~ = ]~[e~P x (2e"q'(£o))-2 × (2eg'(li]~))].

(3.20)

[Use has been made of eq. (2.60).] This formula, valid also for a real multiplet V P, is lengthy when written in terms of component fields, so we do not give its explicit form here. But one can see that it exactly reproduces the formula obtained by Stelle and West [12]. One should notice that such a complicated formula has been obtained here without any tedious calculations once simple conformal tensor calculus is known. In the same manner we can convert the conformal formulae (2.55) and (2.56) for ehiral embedding of the gauge and kinetic multiplets into the Poincar6 multlplet: w?

=

=

=

Te(N P)=T(Y P×2Y0)X(2Y0)

x

,/2,

(3.21)

2

= [0~* -1- ~1(~*S* , @ R D ~ X , D C ~ * - - 2 S ~ *

(3 22)

- ~6g* ISI 2 ] ,

(3.22)

g=S-tP. The multlphcatlon rules Vlv × V~ = V3v, E1v × I~2P = 1~3 P and I ~ ® Y2p* = g P are exactly the same as those in the conformal case by virtue of the fact that the standard weights of these multiplets V P and y v are chosen as w = n = 0 which is preserved in multiplication. 3.1.3. Actton formula and supergravlty lagranglan. The F-type [D-type] action formula (2.61) [(2.62)] is vahd for chlral multiplet Y. [vector multiplet V] having

71

T Kugo, S Uehara / N = 1 supergravl(v

weights w = n = 3 [w = 2, n = 0]. H e n c e we also need weight adjustment in applying these formulae; thus, /,9 xS, ]3] conformal

IF--fd4x[XPIPF=fd'x[XP×'~--O' lo=-fd4x[V]eo=fd4x[

JF

I./" ] conformal

Vpx ",Jo

'

"

(3.23)

(3.24)

These just reproduce the well-known action formulae of refs. [6, 12] m Polncar6 supergravlty. The usual supergravity lagrangian ~ s c is given as the "kinetic t e r m " of the c o m p e n s a t i n g multlplet [ 1]: -

-2~SO = [ VI ]cDnformal = [ 4 Y 0 × T ( ~ 0 ) ] con F formal "

(3.25)

This can be rewritten by using unit vector and chlral multiplets 1 v and ( = [ 1 , 0 , 0 . . . ] ) as _2~SG= [Iv]P=

[ T e ( l x ) ] P,

Ix

(3.26)

owing to eqs. (3.22)-(3.24). This reflects the fact that the compensating multiplet X 0 becomes the unit multiplet 1 x in Poincar6 gravity by the redefinition formula (3.8) 3 2 NEW MINIMAL POINCARE SUPERGRAVITY

This formulation of Poincar6 supergravity was found by Sohnlus and West [7] and aux contains as auxiliary fields a vector A n and an antisymmetrlc tensor an~. In addition to the usual supergravlty symmetries, it has two local invanances under chiral gauge and vector gauge transformations, the gauge fields for which are A naux and a.~, respectively: 8A (~0)A~ ux= 0no~, 6gaug~(Ap)a,~ = ona ~ - O.A,.

(3.27a) (3.27b)

Thus A~~ has 3 = 4 - 1 (off-shell) degrees of freedom and an, has 3 = 6 - (4 - 1). These comprise 6, the same n u m b e r of auxiliary fields as in the usual minimal formulation. 3.2.1. Compensating multlplet and Pomcar&s Q-supersymmetry. The candidate for the c o m p e n s a t i n g multiplet in this case should therefore be a (4 + 4) multlplet just as in the ordinary minimal case and in particular must contain the antlsymmetric tensor gauge field any. Such a multiplet is unique, a real linear multlplet L0 = [Co, Z0, B °] with weights w -- 2, n = 0. The c o m p o n e n t B ° , being constrained by D ~ B y = 0. indeed represents an antisymmetric tensor gauge field an~ as was seen in (2.50).

72

T Kugo, S Uehara / N = 1 supergravttv

Gauge fixing for extraneous gauge freedoms D, S and K m lS performed by the condmons Co = 1, Z 0 = 0 and b~ = 0, respecttvely. The chlral gauge symmetry A is not fixed here since it ts a symmetry in this new minimal formulation. These gauge conditions are invarlant under the chlral symmetry as well as general coordinate and local Lorentz transformations, but violate Q-supersymmetry. The remaining supersymmetry to be identified with the Polncar6 supersymmetry in this case ~s the following combination: ~ ( e ) = (~Q(~') -1-~s (l/')t5e°e) -~-8 K (/l~m/!' -- ~6~rn/'Y5B°E).

(3.28)

As stated above, B° is written in terms of the a n u s y m m e m c tensor a,~ by eq (2.50), which reads in the present gauge Co - 1 = Z o = 0 ,

B°=e-'e,,,~,e"~°°( O~aoo-~,f~3,p+o).

(3.29)

The auxlhary fields a,~ and A~ux are identifiable with a,~ and the chiral gauge field A,, respectively. For A~ux, however, it would be more convement to identify*" Au. . -.A .u -. e u B

0 m.

(3.30)

The 83 transformation law of the Pomcar6 gauge fields e~, +~ and the auxiliary fields A~ux and a . . (or B° ) is easdy calculated from the conformal formula: ~(E)ep,

1 - m~ , m = ~e'),

(3.31a)

6~2(~)+~,=(D~-3cys(Aa~UX+B°)+¼tJ,5"y~,B°)e, °x =

+

Liy . ~,~0., ~

(~(e)B°m = go;(2t'/,% + ½B°~p.).

(3 31b) (3.31C)

(3.31e)

These transformations satisfy the algebra [8~(e,), 8~(e2)] = (~Gc((" e ") + 6M(--~- ~0"" +

½1Emnkl~kB?)

+(~Q(--~" + ) + (~A(--~" Aaux) + (~gauge( - ½t~o - ~ a . p ) ,

(3.32) which is also derivable from the conformal algebra (2.19) and eq. (3.28), directly. '~ It should be emphasized that eq (3 30) has no essential meaning and is just for convenience to simplify the 8~ transformation law of the charm multlplet

T Kugo, S Uehara / N = 1 supergravtty

73

3.2.2. Tensor calculus.

First consider the redefinition of conformal multlplets into Pomcar6 multiplets. Different from the previous case, this new minimal supergravity has the notion of chlral weight n although the conformal weight w is also an alien concept here. An immediate consequence of this fact is that the chiral multiplet Y. and linear multlplets E and L need not be redefined; that is, Y'~'.) = Y~(w=.), l~n) = ~(w=n+2)'

(3.33) (3.34)

LP = L(w=2) •

This is because the alien conformal weight w is uniquely determined by the chlral weight n for these multiplets by the weight relations w = n for Y., w - n = 2 for E, and w = 2, n = 0 for L. One can, however, adopt an alternative convention for the real linear multiplet L: one may define the Poincar6 multlplet L s w = [C sw, Z sw, Bsw] so as to have vanishing conformal weight w = 0 by using the compensating multiplet L0; v s w ( L s w ) - [cSW, zSW,o,O, BSW,?tSW, DSW] = V ( L ) × [ V ( L 0 ) ] -1

(3.35)

The expression for the extra components Xsw and D sw in terms of C sw, Z sw and Bsw, as well as a constraint on B sw, is derivable from the r.h.s, of this equation. In the present gauge (C o = 1, Z = 0 a n d b. = 0) V(Lo) takes the form

V(Lo) = [ 1 , 0 , 0 , 0 , B°,2tVsy • 99 - ½YmB°~bm, - 1R - f-9~

+

I~ml~5~O+m ] , (3.36)

Hence the extra components Xsw and D sw m (3.35) are given by ~SW = - - D w z S W _ l ~ m ( e s w _~_l . [ 5 D , c S W ) ~ m _}_3/.y5~auxzSW q_ ll.Y5BOzSW '

(3.37a)

Dsw = _D

(DmcSW) ~ - ~,Tinl)tsO~Z sw q - I C m l ~ 5 B S W ~ m - - ~

-- ¼ Z s w y m 4 ~ ° a p m +

0 Z sw ( a m q-Bin)

B °" B sw + 73 Z--SWtY5y • q~.

(3 37b)

Further, B sw can be expressed by using an antisymmetrlc tensor field 3 b~,

B sw = e lem ~e~,~po[o"boo~

_ cSWO~aoo)+ ZSWom.~. "

(3.38)

This defininon of L sw coincides with the convention of Sohnius and West [11], who derived it by direct calculations m new mlmmal supergravlty.

74

T Kugo, S Uehara / N = 1 supergrawty

Poincar6's vector multiplets WP and V P are defined as those having vanishing conformal weight w = 0 just as before:

x [V(to)] -w/2' V p = V(w)× [V(L0) ] -w/2

(3.39a) (3.398)

The gauge multiplet G is not redefined here either: G p = G. By these identifications of Poincar6's multlplets, their Polncar6 supersymmetry transformation 8~ follows from the conformal Q, S and K m gauge transformations. For the reader's convenience, we cxte explicit forms of 8~ for W(,P) and Y.~,). Vector multlplet W{,P)(chiral weight n): 8~(e) C e = ½&~,,~;e,

(3.40a)

6 ~ ( e ) ~ I' = ½(,y,{}t~P - K p - ~3i" + g)cei't75) e - ¼m@I'ty51~°e,

(3.40b)

8~(e)3C 1"= ½gtV5(Dc~; e + A l') + ¼gtvsB°(- ,'/5 + ½m) ~;',

(3.40c)

~ ( e ) KP = - ½e( D C~P + AP) - ¼eB°( -iv5 + ½in) ~2e,

(3.40d)

6~(e)~me= - ½ g ( D ~ ; c P +7,,,AP)_{gBo(tvs+in)./,,,~P

+ ½meP(4~,, - f6,,tysB°)e,

(3.40e)

- ~6nv, y"~;e(aqo,,,e - f m , y S ~ ° e ) ,

(3.40f)

1 P =-m 8~(e)@ p = ½gtysD~Ap + ½mgt~,B°Dc~ e + ¼m&3,sA e - gtn@m(4e p - lffml~/5j~°) E

(3.40g) Chxral multiplet Y.~,) (chiral weight n): 8~(e) ~ P = 'i625RXP, ~(E)@RX

P = @R ( ~ c ( ~ P ..}_6~-~P_1_½mBO(~e)e,

8~(e)~ e = ½gD'@RX e + ¼(1 -- n)g, TsB°@RX P.

(3.41a) (3.418) (3.41c)

In the above formulae covariant derivatwes D~ are understood to be the ones for w = 0 in (3.40) and for w = n in (3.41).

T Kugo, S Uehara / N = 1 supergravtty

75

The vector embedding of Y'~n), ES,) and L P is given by the formulae: c ~ ) ( ~ n ) ) = c~¢'(Z~))× [V(L0)] -~/2,

(3.42a)

% (ee , , , p)

(3.42b)

= ~(eL)×

[V(L0)] -'"+ 2)/2 ,

Ve(L P) = V(L e) X [ V(L0) ] - '

(3.42c)

The last quantity is nothing but vsw(Lsw). Notice here that the vector embedding of the charal multiplet, ~.~,), P (YP(,)j, ~s considerably different from the usual minimal case when the chiral weight n is non-zero*. The coverse embedding of ~ P into YY changes the chlral weight from n to n + 3 and is found just as before: Y[,+ 3, (~(~))= 51.[~ ) ×

( ~ ( e 0 ) ) ' " + 2)/2].

(3.43)

For the reader's convenience, we give explicit form of eqs. (3.42a) and (3.43):

%,(x,o,) =

l

c

P

-- ~

m ~v

~, {(~,"a% - 4trot. ~o) ~" + e°.%xP }, { - g nI + i-m (tYsB 0q % - 4 % , ) + 6 nI R ~^,~, - ½ n ( n + 2 ) ( B ; ) 20 } g

e

-I- ~,,,,,;,u~ 1 ~ 0 ~ m ~<'~P + ¼~Pe~(,~,mBO¢m + 4~'' ~ ) ] , P [ ~P-'~,(n + 3) (c~(P) ) = 1 (

/~P),

t@R(AP+(oc+¼(n+2)leO)~p)

{{E3'G e + ® P - ½ ( n +

'

2)@Se.B°-½n(n+ 2 ) e e ( B ° ) 2+'n'za''~p

+ ½( n + 2)lB° . D~"e p + ¼( n + 2)( f mBOym -- 4lUp.y )@L ~P} ] , where covarlant derivatives DL are understood to be the same as in (3.41) and (3 40). The formula of chlral form W e of the gauge multiptet G is unchanged and has chlral weight n = 3: W~ --=yfa(n = 3/2) (GP'~ ~, ] = Y',~(G) Just as in the conformal case, the kinetic multlplet n = l and h a s n = 2 :

T e

(3.44)

is definable only for y e w~th

Ze,.= 2) (Xf,= ,)) = T(Y.~.=,)) •

(3.45)

The multiphcation rules vie× II2e = V3P, ~ ( nP, ) × P P and ~ P X ~,e ¢,, +,~) are exactly the same as those m the conformal case. The multiplication

= y,P

• To the a u t h o r s ' k n o w l e d g e , this fact has not been notxced before

T Kugo, S Uehara / N = I supergravlty

76

rule ~.1P® (Y.~)* = Vi" valid for the n I = n 2 case should now read ~lP(n) ~ ~ 2 ( n )

P = ( ~ 1 ~ Z~)conformal × [ V ( t o ) ] -n.

(3.46)

In applying these formula, one should take care of the following: the cinral gauge aux field A~ appeanng in the covariant derivative D,~ is not A~ but A~u~ + B°~,, and the conformal weight w in D~ of the vector multiplet component field is zero. The F-type action formula is valid for the cinral multiplet ~P w~th chiral weight n = 3 and has the same form as in the conformal case: p

P

iv=fd,x[~(._~)]r=fd.xr

~

j] conformal F

(3.47)

The following D-type formula holds for the real vector mulnplet V P (hence with n = 0): Io = f d4x [ Vi']~ = f d4x[ Vi' × V ( L 0 ) ] D nf°rmal "

(3.48)

Despite its appearance, this D-type action formula takes the following very simple form written in component fields [Ci', Zi', H t', Ki', B~, ?~i',Di'] = Vi"

=

f d4x[eD i'-

½ef. y,ys?~i'- e~'"°°(B~- ½~Zi')O,aoo ] .

(3.49)

Nonce that the component B~ of V P appears here I n the combmat,on B ~ - ½qT~Zp = B; winch is just the vector component (2.33) of the gauge submultiplet G of V p The components of V p other than those of G, i.e., C p, Z p, H p, K P, and the longitudinal component of B; do not appear in (3.49); indeed eq. (3.49) has an "abelian gauge mvarlance" under the transformation

w +

P

P

(3.50)

with chiral parameter ~,=o). This interesting fact was first noticed by Sohmus and West [11]. In the present context, the origin of this invariance would be clear from expression (3.48); it is nothing but a curved space generalizanon of the following rigid case equahty:

DDDD [¢p~] = DDDD [9~*~] = 0,

(3.51)

valid for real linear mulnplet Cb(DDCb = D D ~ = 0) and cinral field ¢p (/J9~ = 0).

77

T Kugo, S Uehara / N = 1 supergravt O,

Finally in this subsection, we present the supergravlty lagrangian in this new minimal theory. As was seen, the simplest candidate

fd"x

= fd4x[V(Lo)]~ "f°~m~

so

(3.52)

vanishes identically. Since we have only L 0 as a compensating multlplet and V(Lo) already saturates the desired conformal weight w = 2 for the D-term formula, the unique conceivable action would be something like

(3.53)

~SG - [ V(L o )In V( L o )]D"' = [In V( L o )]P.

Here In V(Lo) itself may be regarded as having conformal weight 0, but its power series expansion would be awkward since the conformal weights are different term by term in the senes. To understand this point properly, de Wit and Ro~ek [3] introduced a supplementary chtral multzplet ~o of conformal weight w = 1 ( = n) and changed the lagrangian (3.53) into ~SG 0C [ln{ V( Lo ) / ( Y~o ® N~))]PD.

(3.54)

Then, since this action has the above noticed "abehan gauge invariance" under the transformation (3.50) of V P -- ln(V(Lo)/(Z o ® ~)), V P --) V P + t [c~'(A) - q / ( A * ) ] ,

(3.55)

~sG (3.54) is invariant under ~o ~ e x p ( - t A ) Y , o

(3.56)

with chiral parameter A(,=0). This gauge freedom enables us to "ehmlnate" the supplementary multiplet Y'0: we fix the gauge by 2Z o -- [ 1 , 0 , 0 ] .

(3.57)

It should, however, be noted that the conformal formula (2.59) leads to Y.0 ® ~ = [1,0,0,0, Am,0,

__

(3.58)

1 2 7A,,,] ,

where the chlral gauge field A~ appears. Then the logarithmic multiplet in Eso (3 54) is calculated to be*

ln(V(Lo)/(Y~ o ® Y~ )) = [ 0 , 0 , 0 , 0 , - A aux,

2t~,~, • cp - ½T~B°+m, i

0

2

].

(3.59)

* de Wit and Ro~ek's [3] d e n v a u o n of this multlplet as shghtly more comphcated than ours Eq (3 59) is just what Sohnms and West [11] called the Einstein multlplet.

78

T Kugo, S Uehara / N = 1 supergravtty

By the use of the action formula (2.62), this yields

ESG = 3 [ln( =

leR

V(Lo)/(Y~

o ® Y..;

)}]~

0 2 , ~,,oor l. 32e~,,ooa,uxg ~e ~,'YsT, ItD~,, - ~3 l_r5 A aou x ')v/o+ " ' . ~',aoo + 3e(Bm ) ,

(3.60) reproducing the lagrangian obtained by Sohnius and West [7].

3 3 DE WIT A N D VAN N I E U W E N H U I Z E N M O D E L

We have seen that the old and new minimal supergravity theories utlhze chlral and linear multiplets, respectively, as the compensating multiplet S. The next simple possibility for S is obviously a real vector multiplet Vo. This choice essentially reproduces the supergravity of de Wit and van Nleuwenhuizen, which contains 16 + 16 auxiliary fields. It, however, also turns out to be necessary to introduce a supplementary chiral multiplet Y~0 here, just as in the new minimal case. The reason is as follows. If we have only a real vector multiplet* Vd, we must use the D-term formula for constructing the xnvanant action. After fixing an additional conformal gauge, the Pomcar6 supergravity lagrangian Es~ we can construct is unique up to field redefinitions: E s ~ - [Vo]D= - ½ e R -

½f" R + eD + " " .

(3.61)

But this lagrangian contains the D field only in the term eD, and leads to a contradicting equation of motion e = 0. So we need to introduce another multlplet. We have a way to do so without increasing the number of component fields; that is, we add a chiral multiplet ~'0 but at the same time introduce an additional gauge invariance under the transformation eVo ~ e-,'V(A*)eVoe,'V(A), Y~o~ e-'AY'0 •

(3.62)

So we assign to Vo and Y~o the conformal weights w = 0 and + 1, respectively. The lagrangian ~SG CC[eg'(~)eVo~q'(Y~)] + [W~W"]

(3.63)

has such a gauge invariance and is free from any inconsistency. Here W, is the chlral * In this case the conformal weight of Vd is non-zero m order to be able to apply the action formula for

vd

T Kugo, S Uehara / N = 1 supergravltv

79

multiplet Y.~(G0) of the gauge submultiplet G o of V0. We fix this gauge freedom b y * 2Y. o = [1,0, 0]

(3.64)

just as in the preceding case. We shall see later that the lagrangian (3.63) indeed reproduces Es~ of de Wit and van Nmuwenhuizen. Since the Q, S, A and D t r a n s f o r m a n o n s do not preserve the gauge c o n d m o n (3.64), we first redefine them by combining a suitable gauge transformation so as to preserve it. Taking account of the gauge transformation of 2Y 0 = [1,0, 0], 28gauge(A)Y. o = [ - - l ( ~ A , - - t @ R X A , - - l ~ A ]

,

(3.65)

we find that the transformations

~0 (e) --- ~Q (e) + ~gauge(AQ), AQ = [0, - ~1 , ~ ,

~I-

(~,m~¢ m + 41~, ~ ) ] ,

(3.66a)

~ (if) = ~S (if) -I- ~gauge ( AS),

A s = [ 0 , - 2,@R~',0 ] ,

(3.66b)

~ ( O)~" ~A( O) -F ~gauge(hA),

A A = [10,0,0],

(3.66c)

8D(Ao) - ~.(Ao) + 8gauge(AD),

A D = [--/AD,O,O],

(3.66d)

satisfy the desired condition. The vector compensating multlplet Vo Vo = [C, Z, H, K, B m, A, D ]

(3.67)

transforms under these as ~(~)Vo =

[~e,y,z,

~(,~,H- K - S - ~ + ~cC,~,)~,

-- g { l ( ~ c ' Z "F X) -F I y m ~ 1 -

m -- lY5 y • ~ 9 ) ,

c

½(o" F ~ + ,'ysD)e,

6i(g) vo = [0, - 2z~,,f, - (~T,z. -fz,

½firsD'A],

(3.68a)

f (~_YmZ-- /~'¢km).O,O].

(3.68b)

e~-(O) VO = [0, -- ~,~,ZO, ~OK, -- ~OII. -- O~O, 3,~,XO.O]. 6fi(AD) Vo = [2AD, ½ADZ, At)H, ADK, ADa m, 3 A D A , 2 A D D ] .

(3.68c) (3.68d)

• The gauge-fixangchoice is not umque, of course, for instance we may choose the Wess-Zummo gauge for V0 Any alternauve way, however, leads to the same tensor calculus and the lagranglan £s(~ by field redefimtaons The chome (3 64) as the simplest one

80

T Kugo, S Uehara / N = I ~upergravttv

Notice that the Z component of V0 is usually S inert but is not S inert here. Similarly the C component is D inert but not 17) inert. From here on, the process of reduction to Poincare supergravity is qu~te the same as the preceding examples. We fix the extra gauge freedoms of conformal supergravlty,/3, S and K,,, by the conditions C = 1, Z = 0 and b, = 0, respectively. The chlral gauge ~4 is not fixed and remains as a symmetry in this Poincar6 theory. Then the transformation

8~ ( ~) = 8~( ~) + 8~( ¼( I~ + ,-~K + ,r~B + , v ~ )~) +6K(IUpme--iA6~m(H+lYsK+lY5B+lYsA)e)

(3.69)

preserves these gauge condmons and is identified with the Poincar6 Q-transformation. The gauge fields e~, ~p~ and A, and auxiliary fields transform under 6~ and 87 as 6Q+ Ae;'

= [ 6~(e) + 6A-( 0 )] e~ = ½g./m+~,

(3.70a)

8Q+Aq~, = ( D~ - 3175Aj, + ~y~N)e + 30tysOn,

(3.70b)

6Q+AA, = - gtV5% - ~gNtvs~u + OuO,

(3.70c)

6Q+AH = g ( y ' C p + kymN~brn +

½/y5a) +

3OK,

8Q+AK= gl"ys(V" ep + ~ y ' N + ~ + ~tVsX) - 3OH,

(3.70d) (3.70e)

(3.70f)

8Q+AB m = - g ( l~ + m -~ "~mx ) - OmO , 8O+a~ = ½(o. U + trsD )e + ]Ot),sX,

(3.70g)

8o+aD = ½gl'Cs~'m{D~?~- ½(o" F ~ + ~'/57~)q:m - 3t'~sA,,h ) ,

(3.70h)

F ~ = O~,B.- O.B~, + ½( f~,')',, - ~,,3'~,) X,

(3.70 0

where

N ~ - 3( H + l y s K + t'YsB +

l'y541) -

(3 70J)

The commutation relations [6~(st), 8~(e2) ] = de,c ( ~ . e t') + 8m ( - ~ " com" + ~g2(om'~N + Uo"")el)

+ ~ ( - ~ . ~) + ~-(~. B),

(3.71a) (3 71b)

[~(0,), ~-(02)] =0,

(3.71c)

T Kugo, S Uehara / N ~ 1 supergravtty

81

are derivable from (3.70) or from conformal algebra (2.19) and "abehan" gauge algebra [Sgaug~(Al), 8g~,ge(A 2)] = 0. 3.3.1. Tensor calculus. This de Wit and van Nleuwenhuizen model has chlral invariance and hence is very similar to the preceding new minimal case. Indeed if we define a real vector multiplet (w = 2) as* v, =

=[1,O,H,K, Bm+Am,)~,D+½(HZ+K2-(Bm+Am)~}],

(3.72)

then all the formulae of the preceding subsection concerning relations between Poincar~ and conformal multiplets, embedding and multiplication rules and action formulae remain valid provided that V(Lo) is replaced by V1 (3.72). We now give the supergravity lagrangian. By suitably adjusting the numerical factor, eq. (3.63) becomes £SG = -- ½eR -- ±e"~°V'2'

'f~.f5 . . . . lv

( D ; -~- 31~5B 0 ) ~ o -- ¼e( H 2 + K 2 - (B m + Am) 2)

+3e~.y,ysX-~eD- ~ef,~ 9 2-

9 e~(D'° + 3ly5 A ) X + 9eD2

9~e~.,,m,.,~,r, ( ,*/ v ~.mtF~+fu~),

(3.73)

+64

where

which coincides with the supergravity lagrangian in ref. [8] through the following field redefinitions: S wN= - } H , BwN

=3

pWN= 3K, XwN

= ~X,

A wN= - ~ ( A , + B , ) , D wN

= ~D.

(3.74)

Finally in this subsection we add a comment: this de Wit and van Nieuwenhuizen model is, in fact, an old minimal supergravlty theory with a Fayet-Ihopoulos term of chlral U(1) gauge field. As was pointed out by Stelle and West [12], the FayetIhopoulos term in the old minimal case has to have a form [eV] P, which is

[Y~'~ev® Y'o] D * A n over-all n u m e n c a l factor (the base of the n a t u r a l logarithm) is neglected

(3.75)

T Kugo, S Uehara / N = l supergrawty

82

in conformal notation, Y'0 being the compensating multiplet in the old minimal case. But the lagrangian (3.75) [plus the kinetic term of the gauge multlplet V] is xdentical with £s~ (3.63) in the present de Wit and van Nieuwenhuizen model. So the tensor calculus of the old minimal version in the presence of the Fayet-Iliopoulos term has to be substantially changed into the one in this subsection; that ~s, since Y'0 is no longer gauge lnvariant, it cannot be used to redefine chiral multlplets. Further the compensation of vector multiplets should be performed by using the gauge-invariant multiplet V 1 - ]g~eV® Y'o instead of the gauge-varmnt one Y.~ ® ]go of (3.13).

3 4 BREITENLOHNER MODEL

The only remaining simple possibility for the compensating mulUplet S is a complex hnear multiplet fl0. This choice turns out to reproduce Breitenlohner's Poincar6 theory consisting of 20 + 20 modes. Let us take the weights* of £o to be w = - n = 1 and denote the components of Eo = [C, ~,0C, @m, ?'] by [C~ + lC 2, Z l + tZ 2, H~ + lH 2, B~ + tB2m, X]. After fixing the extraneous gauges D, A, S and Km by the conditions C~ = 1, Ca = 0, Z~ = 0 and b, = 0, respectively, we have 14 boson + 8 fern'non auxiliary fields idenUfiable with Breitenlohner's: two majoranas Z 2 - Z and X, a scalar Hi, and a pseudoscalar H a, one vector B2 and two axial vectors B~ and** A, (chlral gauge field). The Poincar6 Q-transformation compatible with the gauge fixing is given m this case by

6~(e)=SQ(e)+Ss(Me)+SA(glysZ)+Sr(¼~p,,e--

¼fraMe),

(3.76)

where

M =- ½(Hi - I Z Z ) + I ( H 2 + ¼ZtysZ)r'/5 + ½(B1 + ¼Zrys"y~Z)rIfy m, and satisfies the commutation relation [(~(el),

~(e2)]

= (~GC(t~ " eft) + ~ M ( - - ~ . 6 0 rnn- g.2((lmnM + M o r n n ) e l )

+ -

-

. o,,,z).

(3.77)

* Breltenlohner's version in fact exists for ~0 with arbxtrary conformal weight w ( * 0,2, oo) which is related to Slegel's parameter h through w = 2/(3h + 1) [3] Here, however, we have fixed w to 1, for simplicity The extension to general w is strmghtforward * * Precisely speaking, it is A~, = A~,- Zz'ysqJt, instead of A~, that becomes a member of Breltenlohner's auxlhary fields

83

T Kugo, S Uehara / N = 1 supergravtty

Under the 6~ transformation, the gauge and auxiliary fields transform as: m ~.

lg

(3 78a)

m--

6~(e) ~b~,= ( D ; -

~,y,A~, - y~,M) e + ¼tYs~,(Z/'y5E) ,

(3.78b)

6 ~ ( e ) A , = - gt¥scp~ + ~ , , y s M e + O , ( Z I y s e ) ,

8~(e)Z = (2M

-

l(/v5J~l

+ J~2) -

(3.78c)

~'V5z~) E

(3.78d)

-t- 1l')15vm~( ~rnl'~5 Z) -- ) l"[5Z( ~l'~5Z), ~(e)H

l --- l g ( / y 5 ~ . -4- y ' c p - ½ym(H l + tY5H2 +/Y5Sl) ~"rn

+ ½AtysZ + M e ) + 2H2(gtysZ ) ,

(3.78e)

6 ~ ( e ) H , --- - ½g(~k - iysy. ~ + ½tYsym(H, + ,YsH2 + t y s e , ) ~ m

- ½a Z + Mt75 Z) - 2Hl (&Y5 Z ) ,

8~(~)~'., = -½d.,,,,?, + m,~,,,-~,v~(H,

(3.78f)

+'V~&+mS,)q'm

1 2 ½AmZ + MtysymZ } + ~B~,(&ysZ),

-

(3.78g)

8~(e) B~ = g {om"D,~Z - ½Ym( tYs)~ + "Y" cp) + ¼"y,,,y"( H, + lysH 2 + l~15~,) +n

(3.78h)

-1,~,,vmAZ + M.ImZ-'(cp, -M,~m)}-½BL(e, ysZ ),

g(~)x = ['(o . g

+ ,r,
-1-(,~. ,~ + AmCmlr, Z - ½A~ + ~a

- ½&(& + 'r,~ + m&)e,.)m -

('YsH, + H2 + ½B, + ½1"{5J~2- - ~ ) M -

¼ ( +- - , # y s Z ) y

m M

] ,I~

+ ¼/yf,/'Z{ ( ~m - ~m M ) ~) -- ½ ( D e Z

+ ½ym(H l +

tysH 2 +

+ ½,~,,AZ - ~ty,2t)(gt-tsZ),

l"}f5J~l)~ m - .~.~p

(3.780

where the covariant denvative D,~ is understood as that for w = - n = 1. 3 4.1. Tensor calculus. The way to deduce tensor calculus in this case from a conformal one is s~mllar to the old mimmal case. So we confine ourselves to the discussion of the chiral and vector multlplets. From the present compensating

84

T Kugo, S Uehara / N = 1 supergravtty

m u l t l p l e t ~o we c o n s t r u c t the f o l l o w i n g m u l t i p l e t : w i t h w = 1, n = - 1,

c o m p l e x vector: % -= W ( e 0 ) ,

w i t h w = 2 , n = O,

real v e c t o r : V1 ~ W ( ~ o ) > W ( e ~ ) , clural:

~"1 ~- ~(V1),

w i t h w = n = 3.

T h e s e are explicitly

%

=

[1, , z ,

% , - , ~ c , % , (1 -

r~)x -

~cz-

r~cz,,

-~G-

,D~,¢~], (3.79a)

1 + ½Zt.~s.ymZ, 2) ~ V 1 = [ 1 , 0 , 2 H 1 - ½ Z Z , 2 H 2 + ~i Z- t T s Z , 2 B m + {H2 + IYsH, + tYsB1 - ½A + ½ Y m ( f m t Y s Z ) ) Z , c ,,, + 29d~3C* X 2D~mB1

-

, m -Iv ~,A 2m ~2~m6f~

-

Z(Z(tysX+

V

.rp)

+ ]~m(Hl+t'YsIt2+l'Y5J~l)~rn-½~.op-½Am(~mtT5Z)+

(3.79b)

lR],

~']1 = [ ~ * -- / Z @ L / ' I@R{2( ~ ' - l'~'~)-Jr" lVmM~m)

+ (/-/~ + m/-/, + , r , ~ , - ~ a ) z + ½~'z(~7~,-nz)}, - - ~ ( ~ 7 ~ m z - Am- 2 , < ) ~ - ,D;~ m* - ~C%* + ½(<,)~ +Z{,~,x+v

•

] m ,~ -- ~"V'~m-- :,757 z(A,.-Zlv,,;,.)-%5'z)

]

.

(3 79c) T h e n , j u s t s~milar to the old m i n i m a l case, the r e l a t i o n b e t w e e n the P o m c a r 6 a n d c o n f o r m a l m u l t l p l e t s is g w e n b y

:~P = :~=n~ × (:~1)-~/3, vP =

v~.,~x

~=

%~,o, x ('v,) ~°-~'/2 x ("vt)-~w+"'~.

(I1,) w/2,

(3.80a)

(3.SOb) (3.SOc)

T Kugo, S Uehara / N = 1 supergravttv

85

Notice that we are taking the convention of identifying Poincar6 multlplets with conformal ones having vanishing conformal weight. [There is, however, a problem m (3.80a) smce the first component of Y~ is not 1 See below.] Embedding formulae are (3.81a)

= z[ p ×

× (.:,)

2]

(3.81b)

The multiphcatlon rules are the same as in the conformal case. The action formulae are

] conforma] IF- f d'xtXP]%=f d4x[YYx-~-, jF ,

(3.82)

conformal.

(3.83)

Io-fd4x[VP]Po=fd4x[VPxVljo

It is worth noticing here that the F-type action formula (3.82) is, m fact, a D-type formula despite its appearance: indeed, since Y.Px Y.(V~)= Y.(~(Y.P)× V~) by eq. (2.60), f d4x [y.plP = _

f d4x [(c~/'(yp) + h.c.) × I"1]D.

So it seems that no genuine F-type action term can exist in the Breitenlohner case. Tins is true as long as we adhere to the above convention identifying Poincar6's cinral multlplet Y.P with the conformal one having w = 0. But if we take another v as having a conformal weight w0 = 3/m with some integer convention to regard Y'(w0) m, for instance, then the term

[(~..~w0)) m] ; °nf°rmaI

(3.84)

can exist in the action and is really a genuine F-term. It might sound strange that it depends on the "convention" whether the genuine F-term can exist or not, but the problem here is, in fact, not a matter of mere convention. The real problem is what variables we regard as independent. Formally the term (3.84) is rewritten as

x

] conformal

(3.85)

in terms of ~..P with w = O, being related with ~wo) by eq. (3.80a)

~']P = ~"~Wo)X(~']1)-l/m.

(3.86)

T Kugo, S Uehara / N = 1 supergravttv

86

But this relation is, m fact, singular because ~t contains inverse as well as fractional powers of Yq, the first component of whmh is not 1 but a field. So, contrary to the old mimmal case, the two descripuons of using Ee and Y'~w0) are not equivalent. (NoUce, however, that there are no problems in using V~ and cgl for compensation ) In any case we can gwe a F-term w~th a defimte power for each chiral mulUplet, just as in conformal theory. The F-terms w~th different powers for a given chlral multiplet cannot coexist, because a singular compensaUon by Yq would be reqmred in such a case. The supergrawty action is given in this case by P /" J 4 r'~'~ "lconformal f d4xgmo~ f d4x[ | r.]F= -- f d4X[lv]Po= JU Xla,IF .

(3.87)

Rewriting conformal covarmnt derivatives in ~ more exphcitly, we obtain

[(

'-)

eso = e -½ R + 7 + . R

--

-2~C~C*+~*~m-½( Am) " +Z(2,Y5~+T'q0-DPZ)

- z ( H , - ,r,& + ¼,r,a + ~,v,e,)z + ~rn - 0 ,,,,, ( ( H I- _

lYsfft2+ 2 l ~ l ) ~ n + 2( D~_

¼tT5fI,,)Z )

+ Z { ore"( B2 + ,75/}~ ) qJ,~ - ½( B2 - t 75/~ ) ~bm - 2 7"37/+,, } + +roT" ~bB2 -2 q-3( zornnl'~5¢m)( Zl'Y5¢n) At-:(1 Zl'Y5¢rn) --

' "'" ~..)(2,~,~-r~z)-~(z,~,~o

5( Zl'~5~[mz)( Z l ' [ 5 ¢ m )

~ ( g z ) 2], ]

(3.8S)

where rn

D~P = 6oc (( • e ") + 6M( - ~ . oam") + 6 5 ( - ( - ~ P ) ,

(3.89a)

A~, = At, - Zty5 q~,,

(3.89b)

~ = / 4 1 +'/7/2= (HI - ¼ Z Z ) + l(H2 + ¼Z'T5Z),

(3.89c)

~m =

~1 _~ lB 2

_~ ( n~

- ½Am + ¼Z/75Tr,,Z) + tB2 ,

- / ~ / = ½(/7/1 + t75/7/2 + t 7 5 ~ 1 )

•

(3.89d) (3.89e)

T Kugo, S Uehara / N = 1 supergravtty

87

Finally in this subsection we add a comment on the Fayet-Ihopoulos term. Similarly to the old minimal case, it is given by [ ¢V(e)*eVCV(~)] n-

(3.90)

In the presence of this term, the tensor calculus should be modified as before: The compensauon of vector multlplets is to be done by using a gauge-invariant quantity ~ ( ~ ) * e V ~ ( f l ) instead of ~ ( ~ ) * x ~V(¢).

4. Other sets of auxiliary fields As stated before, the "auxiliary fields" of de Wit and van Nleuwenhulzen's supergravity are not all truly auxdtary because there exist propagating fields among them. So the true sets of auxiliary fields discussed in sect. 3 are three: (A) old minimal, (B) new minimal and (D) Breitenlohner. A natural question arises here: " I s there any other irreducible set of auxiliary fields in N = 1 supergravlty?" This question has recently been studied by Rivelles and Taylor at the llneanzed level. They claimed that other than the above three versions, there exist two new irreducible sets of auxihary fields, both of which contain 20 + 20 degrees of freedom (including physical modes). They further argued that those five sets, their two new sets plus the above three, exhaust all possibilities as far as auxiliary field spins are limited to one. We now re-examine these points in our conformal framework. Their new non-minimal sets are easily derivable in our framework and further we automatlcally obtain the full non-linear forms of them. In spite of these authors' claim, however, our result clearly reveals the reduczble character of these two " n e w non-minimal versions". First notice that the auxdlary ferlmons must appear in pairs so as to be eliminated by equations of motion. Thus the compensating multiplets should contain an odd number of fermion components since one fermion is eliminated by S-gauge fixing. Within the multiplets carrying no external Lorentz indices, the irreducible field multiplets satisfying such a condition are only three: chiral Yo, real linear L 0 and complex linear ~o multiplets. These three, corresponding to the cases (A) old minimal, (B) new minimal and (D) BreItenlohner, respectively, therefore exhaust the possibility of utilizing one multlplet as a compensating multlplet. The next posslbdlty is to use two multlplets; the simplest ones are (N0, Vo), (L0, V0) and (E0, V0) [13]. [Notice that the real vector multlplet V0 contains two fermlon components.] The first two choices just reproduce the two new non-minimal sets found by Rwelles and Taylor. The actions

f d,x

= f d4x ([y; ® Y'0J,) + [vg] D),

(4 la)

88

T Kugo, S Uehara / N = 1 supergravm,

and*

f d4xe' Lo'Vo'-f d4x([V(Lo)lnV(Lo)]D+[V#]D)

(4.1b)

give the full n o n - h n e a r versions of their linearlzed actions for the two cases d e n o t e d b y their s u p e r s p i n n o t a t i o n as (0, 0, ~,~ i +) and m ~ + ,2A ~ + }, respectively. In o u r confort ~',2A m a l f r a m e w o r k the r e d u c i b i l i t y of the systems (4.1) is m a n i f e s t since Y'o (or L0) a n d Vo are two mult~plets. Nevertheless, it b e c o m e s quite o b s c u r e in the r e d u c e d Polncar6 theories o b t a i n e d after c o n f o r m a l gauge fixing, which are the forms c o n s t r u c t e d b y Rivelles a n d Taylor. T h e seeming " i r r e d u c i b i l i t y " c o m e s from the following trick of gauge fixing; that is, in the case of (4.1a). W e fix the D a n d A gauges b y * A 0 = 1 a n d B 0 = 0 of Y.0[½(Ao + lBo), ~RX0, ½(Fo + tG0)] Just as before, b u t the S-gauge b y the condiuon Xo = - r t 5 Z ,

(4 2a)

w h e r e Z is the Z - c o m p o n e n t of v e c t o r c o m p e n s a t i n g m u l t i p l e t Vo = [C, Z, H, K, B.,, X, DI. Similarly, in the case of (4 lb), we fix the D gauge b y * * C o = 1 of L o = [C o, Z o, B.°] a n d the S-gauge b y Z 0 = Z.

(4.2b)

T h e K m gauges are fixed b y b~, = 0 as usual. T h e gauge c o n d i t i o n s (4.2) p l a y a key role in achieving the seenung " i r r e d u c i b i l i t y " in the r e s u l t a n t P o m c a r 6 theories; (4.2) causes the m i x i n g of two m u l t i p l e t s Z o ( L 0 ) a n d Vo. I n d e e d the r e m a i n i n g Qt r a n s f o r m a t i o n s 8~ in Poxncar6's stage preserving these gauge c o n d i t i o n s are given b y

1 2(1 + C ) [ ( F - H ) + t V s ( ( G - K ) - ' v s ( B + ½ A )

f(l~ ,

..-

.

..-

-~- 5 ( "~ +m l "Y5 Z -F , "y5 ]l ~ rn Z ) -~1 2(1 + C ) Z ( Z e ) ,

i f(2) = 2(2 - C ) ['YsB° - ( H + 1y, K + ,YsB - 0 C)] e,

(4.3a)

(4.3b)

* In V(Lo) in (4 lb) should be understood in the came way as in subsect 3 2 ** Practically, it would be simpler to fix the D gauge by At) = 1 + C (Co = 1 + C), m ~htch case the Q-transformation of e~' remmns the same as m the usual case

T Kugo, S Uehara / N = 1 supergravltv

89

for the systems (4. l a) and (4.1b), respectively. Under these 8~ transformations the component fields of the two multiplets Y0 (or L0) and V0 are actually mixed up in a completely entangled way. One can easily check that the actions (4.1) and the supersymmetry transformations (4.3) exactly reproduce those of Rivelles and Taylor [5] at the linearized level. (In the original paper of Rivelles and Taylor, there appears a real parameter n which dlscrirmnates three different sectors, n < 0, 0 < n < 1 and n > 1, corresponding to different metric signs of the auxiliary fields. This parameter freedom is also easily incorporated into our case: it is related to the sign of the V02 term in the action (4.1) and the scale freedom of defining component fields of V0 relative to those of Z o (Lo). The details will be reported elsewhere [13].) We have thus seen that the apparent "irreducibility" of the " n e w non-rmnimal versions" comes from the gauge-fixing device (4.2). As was pointed out by de Wit, Philippe and Van Proeyen [14], such "mixing-up" gauges can always be transformed by a field-dependent gauge transformation to such gauges in which the reducible character is manifest, 1.e. the gauges Xo(Zo) = 0. The latter gauges are nothing but those leading to old minimal and new minimal versions, respectively. Thus, if one makes a field redefimtion according to the field-dependent gauge transformation, one can make it exphclt that these " n e w non-minimal versions" reduce to the old and new mammal versions to which an auxiliary vector multlplet V0 is added m a manifestly decoupled form [13]. As for the second question of counting all the possibilities of irreducible sets of auxiliary fields, the arguments of Rivelles and Taylor [5] are essentially correct. So we are led to conclude that all the possible irreducible versions of N = 1 Poincar6 supergravity with auxiliary fields having spin no higher than 1 are already exhausted by the well-known three, 1.e. (A) old minimal, (B) new minimal and (D) Breltenlohner's versions. This conclusion can be confirmed also from our superconformal reasoning but we leave it to the forthcoming paper [13].

Note added. As a practical calculatlonal tool also, the superconformal tensor calculus proves much more flexible and convenient than the usual Polncar6 one. Indeed it has been used to study the N = 1 super Hlggs effect in a recent paper of Cremmer et al. [15]. A more systematic use of superconformal calculus has been made in ref. [16] to simplify such a derivation of the super Hlggs lagranglan, We became aware of a recent paper by M.F. Sohnlus and P.C. West [Nucl Phys. B216 (1983) 100], in which they presented a new tensor calculus based on an off-shell 16 + 16 version of Polncar8 supergravity (which is essentially identical with de Wit and van Nieuwenhulzen's one), and also showed that both old and new minimal versions as well as their tensor calculi are derivable from It We thank P.C. West for informing us of his paper.

T Kugo, S. Uehara / N = 1 supergravtty

90

The authors would like to thank B. de Wit and A. Van Proeyen for pointing out the reducibility of the "new non-mimmal versions" which was misunderstood in the original manuscript of this paper.

Appendix COMPONENT FORMULAE FOR FUNCTIONS OF MULTIPLETS IN CONFORMAL SUPERGRAVITY

The kth power V k of real vector multiplet V = [C, Z, H, K, Bm, X, D] is given by

v =[c

kC k 'z,

kC

-'II-¼k(k -1)c

-2Zz,

kCk-tK + ¼k( k - 1)ck- : z l y s Z , kCk-IBm + ¼k(k - 1 ) c k - Z z I y s y , , Z , kC k t h + ½ k ( k - 1 ) c k - Z ( H - lysK+ i T s B - D c C ) Z - ¼k(k- 1)(k- 2)Ck-SZ(ZZ), kC k

I o -~- I k ( k

-

1 ) C k 2 { H 2 -}- K 2 - 9 2 - ( D m c C ) 2 -

- ¼k(k- 1)(k- 2)Ck-3Z(H+ ~k(k-

2f~Z - Z D ' Z }

trsK+ tv5B)Z

1)(k- 2)(k- 3)C~-4(ZZ)2].

(A 1)

For a complex vector multiplet cv the same formula also holds, provided that the spmor conjugate qT(such as Z and X here) is understood to be +TC but not ++'/0. This, in particular, leads to the power formula for the chiral multiplet Y. = [C~, °2RX, °.3]: ~]k=[@k,

k@k-l@Rx,

k@k ' ~ - l k ( k - 1 ) ~

k 2X~@RX].

(A.2)

This carries, of course, conformal and chiral weights kw and kn when Y, has w and n. For a general function g(Y.), which makes sense if w = n = 0, we obtain from (A.2) the formula g(~..) = [g((~),

g'((~)6~RX ,

g'((~)~-- ¼g"((~)~O~RX ]

(A.3)

By using (A.2) and its hermitian conjugate together with the vector embedding rule c~(~) (2.38), we easily calculate the quantity c~'(~k)× cV(y,t). The result is straight-

T Kugo, S Uehara / N ~ 1 supergravtty

91

forwardly translated into a more general form:

~(X, Z*)= Z a,?V(Z*) X"C(Z*') k,l ~--- [ ~ ( ~ ,

--l(~a@RX--~,a*@LX ) ,

~*),

- ( q , J + ~, ,.~*) + 4'(~ oo~%x + % . ~ . ~ L x ) , 6;-* ,( ~,o~- ~,,o.'~ )-

9,a*

'4,( ,:.o0~%.x m

- epo.o.~%.x).

) + Z~,~*XtYsYmX'

--I {-- ~b.aa.( ,0"~* + j ~ ( ~ * ) + I ~ a a * a * ( X @ L X ) ) @ R X ,

1

--

m

t

*

¢

+ ±q~8, ~ . , , . (~@LX)(~°)RX)],

(A

4)

where we have used abbrevmtlons such as q ~ . = 02q~(~, ~ * ) / 0 ~ 0 ~ * , etc.

(A.5)

References [ 1] M Kaku, P K Townsend and P van Nmuwenhmzen, Phys Rev D 17 (1978) 3179, M Kaku and P K Townsend, Phys Lett 76B (1978) 54, P K Townsend and P van Nmuwenhulzen, Phys Rev DI9 (1979) 3166, 3592 [2] S Ferrara, M T Gnsaru and P van Nmuwenhmzen, Nucl Phys B138 (1978) 430 [3J B de Wit and M Ro~ek, Phys Lett 109B (1982) 439, W Siegel and SJ Gates, Nucl Phys B147 (1979) 77 [41 B de Wit, J W van Holten and A van Proeyen, Nucl Phys B167 (1980) 186, B184 (1981) 77, M de Roo, J W van Holten, B de Wit and A van Proeyen, Nucl Phys BI73 (1980) 175, E Bergshoeff, M de Roo and B de Wit, Nucl Phys B182 (1981) 173 [5] V O Ravelles and J G Taylor, Phys Lett 113B (1982) 467, Nucl Phys B212 (1983) 173 [6] K S S t e l l e a n d P C West, Phys Lett 74B(1978),330, S Ferrara and P van Nmuwenhulzen, Phys Lett 74B (1978) 333 [7] M F Sohnms and P C West, Phys Lett 105B (1981) 353 [8] B de Wit and P van Nmuwenhmzen, Nucl Phys B139 (1978) 216

92

T Kugo, S Uehara / N = 1 supergravtty

[9] P Brextenlohner, Phys Lett 67B (1977)49, Nucl Plays B124 (1977)500 [10] P van Nleuwenhmzen, Phys Reports 68 (1981) 189 [11] M F Sohnms and P C West, m Proc 1981 Nuffleld Workshop, London, Quantum structure of space and time, eds M J Duff and C J Isham (Cambridge Umv Press, Cambridge, UK, 1982) p 187, Nucl Phys B198 (1982) 493 [12] K S Stelle and P C West, Nucl Phys B145 (1978) 175 [13] T Kugo and S Uehara, Nucl Phys B226 (1983) 93 [14] B de Wit, R Phlhppe and A Van Proeyen, Nucl Phys B219 (1983) 143 [15] E Cremmer, S Ferrara, L Glrardello and A Van Proeyen, Nucl Phys B212 (1983) 413 [16] T Kugo and S Uehara, Nucl Phys B222 (1983) 125