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N = 2 conformal supergravity in N = 1 superspace
Nuclear PhysicsB278 (1986) 851-880 North-Holland, Amsterdam
N - - 2 C O N F O R M A L SUPERGRAVITY IN N - - 1 S U P E R S P A C E J.M.F. LABASTIDA1 The Institute for Advanced Study, Princeton, NJ 08540, USA E. S,~NCHEZ-VELASCO 2' * and Peter WILLS2"**
Institute for Theoretical Physics, S U N Y at Stony Brook, N Y 11794, USA
Received 18 April 1986
We apply the method of projecting extended supersymmetric theories into N = 1 superspace to the case of N = 2 conformal supergravity. We solve the constraints in terms of unconstrained matter prepotentials and N = 1 conformalsupergravity fields.
1. I n t r o d u c t i o n
Supergravity theories, besides being mathematically interesting, are thought to be part of the low-energy limit of superstring theories, which are attractive candidates for a unified theory involving gravity. A complete understanding of supergravity theories is thus desirable. One of the nice properties of supergravity theories is that they have better ultraviolet behavior than their non-supersymmetric counterparts. However, in general, computations are very difficult, and consequently issues that involve quantum corrections, like the finiteness question, have not been resolved. The simplest and geometrically most natural way to define supersymmetric theories is in superspace. Also, in superspace, one can define the Feynman rules of the theory using supergraph techniques, which greatly simplify the quantum computations. However, to apply supergraph techniques one needs a formulation of the theory in terms of unconstrained prepotentials. For extended supergravity theories such formulations are not known in their corresponding extended superspaces (except for one of the N = 2 Poincar6 supergravities at the linear level as presented in [1]). Two possible solutions to this problem are the modification of extended 1Research supported by US Department of EnergyContract No. DE-AC02-76ER02220. 2Research supported in part by NSF grant PHY 85-07627. * Address after Sep. 1, 1986: Newman Laboratory for Nuclear Studies, Cornel1University, Ithaca, NY 14853, USA. ** Address after Sep. 1, 1986: Physics Department, University of California at San Diego, La Jolla, CA 92093, USA. 0550-3213/86/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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superspace [2], and a light-cone superspace [3]. These approaches are under investigation and they still possess technical difficulties. Another approach consists of projecting extended theories into N = 1 superspace. This method has the disadvantage that only one of the supesymmetries is manifest; however, it has the advantage that if one is able to solve the constraints in terms of unconstrained propotentials, the well-developed machinery of N = 1 supergraphs [4] would facilitate quantum calculations enormously. The method of the projection to N = 1 superspace has been successfully utilized for N = 2 super-Yang-Mills [5]. This method has started to be applied to extended supergravities in recent years. For the version of N = 2 Poincar~ supergravity of [1], the theory was projected to N = 1 superspace and the constraints were solved in terms of unconstrained prepotentials [6, 7]. The corresponding action was constructed in [8]. Furthermore, the coupling of this version of N = 2 Poincar6 supergravity to an abelian N = 2 vector multiplet was analyzed in this context in [9]. There, the theory was solved in terms of unconstrained prepotentials and the action was constructed. The version of N = 2 Poincar6 supergravity of [1] is at present formulated in the adequate form to perform quantum calculations. However, this is not the simplest form of N = 2 Poincar6 supergravity to carry out a quantum analysis because of the tower of ghosts problem [10]. The reason for this is that the formulation of this theory in N = 1 superspace is made in terms of non-minimal N = 1 Poincar6 supergravity. A formulation based on minimal N = 1 Poincar~ supergravity would be preferred. A possible way to obtain such a formulation could be the study of dual forms of the N = 2 Poincar6 supergravity theory of [6-8]. Another way is to analyze N = 2 conformal supergravity and then, through compensator techniques, generate the different versions of N = 2 Poincar6 supergravity in N = 1 superspace. This paper constitutes the first step towards the analysis of this last approach to the problem. Conformal supergravity theories were introduced because, besides being interesting by themselves [11], they provide a mechanism to analyze the structure of Poincar~ supergravity theories. The utility of this mechanism has been demonstrated successfully for N = 1 supergravity (in ordinary space [12] and in superspace [4]) and for N = 2 (in ordinary space [13] and (in part) in N = 2 superspace [14]). In the cases where it has been applied, it has clarified the structure of Poincar6 supergravity and greatly facilitated the construction of invariant actions. In this paper we present the projection to N = 1 superspace of N = 2 conformal supergravity and the solution of the corresponding N = 1 constraints. We start with the N = 2 superspace formulation of [14,15] and we define N = 1 components of N = 2 superspace objects by covariantly projecting the 8 2~ (and 8~) components. In doing this we use part of the symmetry of the theory to gauge away algebraically some of the N = 1 components. This gives an N = 1 superspace formulation in a Wess-Zumino gauge in a similar way as the one for N = 2 Poincar6 supergravity [6, 7]. However, in the case of N = 2 conformal supergravity one possesses a bigger
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853
symmetry which provides the possibility of a further gauge fixing. We call this gauge superconforrnal Wess-Zumino gauge because we use certain components of the superconformal transformation parameter to gauge away algebraically some of the superfields. The next step is to formulate N = 2 conformal supergravity as N = 1 conformal supergravity coupled to a set of matter fields in such a way that the constraints for these matter fields can be solved in terms of unconstrained prepotentials. To carry out this program we have found very convenient to make a further gauge fixing, the so-called chiral gauge [7]. The N = 1 superspace formulation that we present in this paper shows that N = 2 conformal supergravity can be expressed as N = 1 conformal supergravity plus some matter couplings and that the constraints satisfied by the matter superfields can be solved in terms of unconstrained prepotentials. The paper is organized as follows. In sect. 2, we describe the formulation of N = 2 conformal supergravity in N = 2 superspace. In sect. 3, we project the theory into N = 1 superspace. In sect. 4, we perform the superconformal Wess-Zumino gauge fixing. In sect. 5, we present the solution to the constraints in terms N = 1 conformal supergravity and unconstrained prepotentials. Finally, in sect. 6, we state our conclusion. We also include one appendix which contains a summary of our notation.
2. N = 2 conformal supergravity in N - - 2 superspace
In this section we review the formulation of N = 2 supergravity in N = 2 superspace as presented in [15,14] and rewrite the equations we will need in a form convenient for our purposes. This approach is based on the fact that ordinary N = 2 superspace [4] is perfectly suitable for the description of conformal supergravity. Although ordinary N superspace has as tangent space group SL(2, C) × U ( N ) and the superconformal group is SU(2, 2IN), the additional parameters, i.e. conformal and superconformal boosts, can be absorbed into general coordinate transformations, while Weyl and S supersymmetry transformations appear as extra invariances of the constraints that define the theory. This approach is a superspace generalization of the work done in components [13,16,17]. We will use the N = 2 index notation of [4] which is summarized in the appendix. We denote SL(2,C) generators by M r . We introduce curved N = 2 superspace covariant derivatives V_A (V~_,V~, V ~ ) which contain SU(2) connections I',~b, and U(1) connections I'A, (2.1) where
rab
and Y are the SU(2) and U(1) generators respectively. The SU(2)
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generators Y~b satisfy the reality condition ~,,o= C,cCOUYca,
(2.2)
where C "h is the SU(2) invariant tensor (see the appendix), and their action on an N = 2 superfield is: [Yah, T~,] = - 1Cc(aTb) Y .
(2.3)
The U(1) generator Y is hermitian and its action on covariant derivatives is: [Y, X7~] = XT_~,
(2.4a)
[Y, v a ] = -V_~,
(2.4b)
[r,v~a] =0.
(2.4c)
Under gauge transformations, the covariant derivatives transform as 8V a = [iK, V_A1,
(2.5)
where, K=i(K~va+K~OM#~+Kj~M/~+KabY,
b+kY ) .
(2.6)
The covariant derivatives possess the following graded commutator: [VA, VB } = TABBCVc + RABvSMn v + RA2÷~M~* + F~AdYcd + FA2Y,
(2.7)
where we have used the standard notation [4] to define torsions and field strengths. The theory is defined by demanding the following constraints [15,14]:
T.__f= O, = o,
T~_,a#~' = O,
T._~y,=.t ~ ab S~~ , = o,
T,~,, #O"v~'- 0 -
•
(2.8)
Introducing these constraints into the Bianchi identities one can compute the rest of the torsions. Then, the Bianchi identities are solved for field strengths and curvatures in terms of them. The solution for all the field strengths up to the vector-spinor one and for all the curvatures up to the spinor-dotted spinor one, which are enough
J.M.F. Labastida et aL / Conformal supergravity
855
for our work, can be summarized in:
{ Va_, ~ } = i[ 8(a°Cs)vSab + CasCabN, ° ] Mav + iCaBCabW~'iM~9 + i [ 3(,,d3b)qV~tj + G~qbs4cecc/u] Ycu,
(2.9a)
( Va_,~_B } = i~b~TalJ + iS(ao<)tJbaMov + iS~'iGSt)abM6 ~ + i
+
cec~:Gal~e b] Ycd,
(2.9b)
[Voa, V 0] = - 23:GoatfV z + [ C, oCb~Wa~ + 3:( C~1~$b¢+ ChiN.o)] ~,_ -t- 1 R ( a l " c ) d [ ) -- 1 - ~ ( e 3 "b "" ~Tea~rfl&(ff 2v.~
X(
1
+
Co S ,e +
o) or" V°°GO,i(b")] Y + Roa,gvaM8 v + Roa,t~,SM~ " v," (2.9c)
°r'~[V(bfl~Trof a
where Noa, Sob and WoO are symmetric N = 2 superfields and G~aob satisfies the reality condition Gaaa b = - CJa&ba . (2.10) These superfields are not all independent. From the analysis of the Bianchi identities one can obtain:
V_~W,,, = 0,
(2.11a)
V,,#N~,o + Vo(oNv)p = 0,
(2.11b)
Co(~Vt,),N~,,
CB(~V.~oS~= O,
-
(2.11c)
V ( b f l S c ) a "Jr V a f l S b c = O,
(2.11d)
Cbcv~s,~ + V,rNvo = O,
(2.11e)
b
+ v-~auo, fl
V:g,2)
(2.11f)
= o,
g"~ab,¢.z ~ C__ - ,-Vb(p',-'o)/~ -- O,
~-c 3 ~ W a IJ + C ab (VaSbc -- 5Vb,,G°a~ ¢ -- 4V:Goab ~) = 0.
(2.11g) (2.11h) (2.11i)
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The constraints (2.9) and (2.11) are invariant under the following N = 2 superconformal transformations:
8LE~_ = ½LE~_,
(2.12a)
SLY,, = ½LV. + 2(V,#~L ) M. B + 2(V.~L ) ccbYab -- ½(vail ) Y,
(2.12b)
8LS,, b = LS,, h - ½iV(.aVo)°L,
(2.12c)
~. ,,bW.O,Vba)L, 8LN,,I~ = LN.a + ~tC
(2.12d)
8LW.a = L W,q~,
(2.12e)
6LGd~f = L G.B f - l i [Va., ~ blj] L ,
(2.12f)
~Li~7.B= Li~7.~ + 2 ( v . . L )V"B + 2(v/~L ) V . . +i(~TaB L ) M r + i((7.aL )MBa,
(2.12g)
where L is a real scalar N = 2 superfield. In reducing the constraints (2.9) to N = 1 superspace it is convenient to use a shifted N = 2 covariant vector derivative defined by i V . a - { V 2 . , V 2 a } . This breaks manifest SU(2) covariance. In terms of this new derivative the constraints (2.9) become:
{ V~_, V_lj) = i[8(.actj)vS~b + C.I~C~bN, a ] 34ov + iC.13C~bW~,°Moi' + i[ 6(.aSb)CN.l~ + C.¢C~bSeICecC/d] r,. a , { V2,,,~ 2B } = iV. B ,
{v
..v'B ) = voB + i[ 8B,., o+,11 -
(2.13a) (2.13b)
M.,
+i [~.(aGy,B11 - ~.(°G--y)B221Ma YAt-i(~TetB11- ~.B22) Y - 4 i ( G . B ix + G.B22 )Y12,
(2.13c)
{ ~1., "~ 29 } = iS.(nl~v)BZaMav + iSB(r'Gag)xZMa* + ia.Ba2Y + 2i((7.B11 + G.B22) Yn,
(2.13d)
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J.M. E Labastidaet aL / Conformal supergravity
{ V2~,,~',~ } = i~,,(°Gv)B'2Mo ' + i~(dG~,)2'Ma ~ + iGa~2'Y
(2.13e)
- 2 / ( c ~ ? + ~o~?) r~:,
C,~,sS,'~la
[V,,/,,VlO] = -2(t70/,1'+ G#a22)Vl,, - 2C,,#GV,u2W2v+ + [C,,,o(W,~ i' + 8e,i'S,2 ) + ~a*N~]~ 2, 1
1
1
2
1
"t
2
2 1 4 1 °t- [-- 3~2a((Tfl&l -- (Tfl&22) "F ~71a(Tfl&2 -- 2~71fl(Ta&21
+v,~o, + c o ~ ( v , , < , -
}(v'~Sx= + v ~s=~))] rx,
+ V 2&Na,R+ Carl('~ 2~,~'V&~ + { ( V I&Sll + V 2&S12))1 Y12 + 2V,~G~,,EyE2,
(2.130
Iv.a, V2~] = -- 2CaBGV~ElVl v - 4CaBGVi, E2V2 v +
Ca~S2~
28
+ [coa(,2s,~- ~ ? ) - ,?N.~]~', 1
1
+ 2[3V2(fl(l~a)&l
1
1
-- ~a)&22) "Jr -~Vl,a(78,a.2
1
+ C~#V,vG va2' - C,,/~V2v ( G~' - GaY22)] Y - 2 V2/~G,,a2' Y1, 2 1 4 1 + [ - ~v~o(a,~. - a,~?) - 4v~ao~? + ~v.oa,~
..[_~ 1Nail + Carl( ~ li, w&.9 _ I ( V 1&S12 + V 2822 ))] Y12 _ 2
1_
3V2aGflal
+
2V2fGaal 2
+ v 2 a N ~ + C~(~2,Wa * + }(vlas n + v2aS~2))] Y22. (2.13h)
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J.M.F. Labastida et al. / Conformal supergravity
3. Projection to N - - 1 superspace
In this section we derive the N = 1 form of constraints (2.11) and (2.13), and the second supersymmetry, gauge and superconformal transformations of the N = 1 superfields. We apply the method described in [6, 7]. The N = 1 component superfields of an N = 2 superfield are obtained by acting with spinor covariant derivatives respect to the second 0 and then taking the second-O independent part. The 0 2~ (and 02a) dependent parts of K in (2.6) are used to algebraically gauge away components of the vielbien and the connections. In this Wess-Zumino gauge the N = 1 form of the constraints simplifies notably. As we will describe below the superconformal invariance of the theory allows us to use the 0 2~ (and /~2~) dependent parts of L to algebraically gauge away additional N = 1 superfields. It is in this superconformal Wess-Zumino gauge (plus another gauge fixing that we call chiral gauge and that we describe below) where we are able to solve the N = 1 form of the constraints in terms of the unconstrained prepotentials. The N = 1 superspace second supersymmetry transformations and the gauge transformations are generated by the 0 2a (and 02a) independent part of K. Superconformal transformations are generated by the 0 2~ (and t~2a) independent part of L. The 0 2~ (and 02~) independent part of any superfield quantity X is denoted by XI. For an operator, X-~ xMiOMJr X"M.,
(3.1)
X I - xMIiaM + X ~ l i g ~ ,
(3.2)
we have
where M~ represents any generator of tangent space transformations. We define V,~ I - VA + ~/A"02, + ~PA~O-'2~+ I'AbcYbc, ~7A = EAMOM Jr dOAvBMBv Jr eOA~tJMIj~+ FAY ,
(3.3a) (3.3b)
where V,~ - (Vx,,Vla, V,s) and the rest of the fields are N = 1 superfields. We now describe the fixing of the Wess-Zumino gauge for the spinor derivatives V2, and V 2a. Since from (2.5) 3V2.1 =
-i02,,K I + ...
(3.4)
we use the 0 2" (and 02s) parts of K to go to a gauge where v2.1
= a..o2. - 02.,
=
(3.5)
Using higher second derivative components of K we fix [V2.,V2~]I = 0 ,
(3.6)
J.M.F. Labastida et at / Conformal supergravity
859
so, in the Wess-Zumino gauge, after using (2.13a),
Vz,,VelJI =
½{ V2,, V2t~ )l = - iS22 IM, a +
iN, l~lY22-
(3.7)
In the same way we find, ~ 2 V 2 . I = i1{ V--2 a , V 2 , , } l = ½iV,~al •
(3.8)
Finally, we complete the projection of all the first components of the covariant derivatives. After using (3.3), (3.5), (3.7) and (3.8) we find, V 2 a ~ B I = [V2,~,VB}I "at*( - - ) B [ / " l l ( v a
"}-~)a"/V2y] "[-~ofiV2~l "}'-Fa¢cd Ycd) -
-
" 3'(~8(~ 1 oC.)oS=IMo 0 + /~5,a I Y22 ) +tq'n
+½iq, B*V.,I + (8.~(FB+ F~2)+OBJ)V2~I].
(3.9)
The commutators of this expression are evaluated using (2.13). We define the following N = 1 superfields:
w~- W~l, S- Snl, =
$121,
X=S221
,
A . , i = iG,~,i221 ,
wo,~ -= v::vB~I, N . , ~ - V2.NB~ I , s~ - v 2 . S n l ,
_--2
~ a --~ ~72aS12[ ,
(~a,/~fi-= iv2aG$/hll , ~k~ --- v2aS221 , s~ - ~ S l ~ l ,
Xa,BB ~ iv2aG~/~211 , ~:~ --
~S~l, A a , ~ = iV2aG~221 .
2% - v2S221 ,
(3.10)
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J.M.F. Labastida et a L / Conformal supergravity
Higher components of VA can be determined extending this procedure. Using all the 02a (and 02a) dependent components of K to gauge away part of them one finds that the rest can be expressed in terms of the fields in (3.3) and components of the field strengths. The definitions made in (3.10) are enough for our analysis. Higher components can be expressed in terms of these ones. In fact, the ones defined in (3.10) are not independent. They satisfy constraints as a consequence of (2.9) and (2.11). The gauge transformations that remains in the Wess-Zumino gauge are the 0 2° (and 02~) independent parts of K,
iKI = - e , '02, - e-a'02e, - K'~bYas + iK,
(3.11)
where e° and K ~h are N = 1 superfields generating second supersymmetry and SU(2) transformations respectively, and K is the parameter of a general N = 1 gauge transformation,
K = i( KAVA + K flMB" + K a % a
+kY).
(3.12)
The supersymmetry generated by K is manifest since the theory is written in terms of N = 1 superfields. However, the other symmetries transform superfields into each other. This implies that in order to keep the Wess-Zumino gauge we have to add gauge restoring compensating transformations using the higher components of K. This also applies for the superconformal transformations of the N = 1 superfields. We now determine the N = 1 superspace geometry. As in (2.7), we introduce torsions and field strengths:
[ VA, VB} = TABCVc+ RAB~,SM~V + R A ~ M ~
+ FAB Y.
(3.13)
The content of (3.13) is determined by taking the 0 2° (and 02a) independent part of (2.9), and specializing to the part proportional to V~, M,# and Y. In addition, we obtain more constraints on the N = 1 superfields from the coefficient of V2vl and the coefficient of Y~bAt this point we make a further gauge fixing. We adopt the so-called chiral gauge in which q,f i = ~pfl = 0. Here we assume its existence and we will justify it later. Projecting the constraints (2.9) into N = 1 superspace in the chiral Wess-Zumino gauge we find:
+
+ %N;) + ! 2a~, ' ( a va, oa,c" xl ruro (3.14a) ' r B ) v ( o ~ v ) 0 1 ~'~r ,
J.M.F. Labastida et al. / Conformal supergravity
( V.,~9} = i(1+
861
.,,,i
+( ~jrj2 + j/JrJ')% - t ".*rl2,~. ,. - % *rd22)v,-I- [ - - i g
(a~
--
. ..a
.i'
o
+2+'%+,] M:
+
+ (aG-B - +:¢XaB - +B+X~+)Y,
(3.14b)
[W:a, Ua] = i#+:a+q+¢v~7"r+ + [iS OABa+ 8~o r2~ 12 + + . d r ~ 1 + 2,+~ox • --o ~ + ,L l ] vo ~ . o r l o,j --
+
1•
.
glX(a.B)a
2-
1_.
11
O --
E22V
y
1 e a - ±/-'n~" 2 a ~ & -7.X"s-qJo°Xo°a-2F~. + ~i4',8,,[~-G,,
+ R,~,,I3~aMav + R:s,~ +t M~f ,
~ 7 ( a ~ )12 __--f l i p 2 2
+ ,t,
~ ' ( a f f l ) 22 = "'<::B) "IF'12F '22
+
ypllp12
,L
yp11T,22
,L
+
(3.15a)
ypllrll
;,l,
(3.15b)
YA[
_
(3.15c)
_
i~b(:'@ °N. -
V'k~b.v i(1 + X)~k ~b°, +
2~b VFJ2 +
=
12--o X a]}Y
(3.14c)
V<~@B)Y-- :<~vp)rn'"or,roy+ 8< v r ~ + 2 @~. . vr12 :~), 4
]
(3.15d) ,
(3.15e)
Wa/-'/~ 1 + W k [ 11 = i ( 1 + a)aBv'~. a a r l oa l t + ~k -tF'll ~ r'121a a--/'111-,12j_/) "1- ~t
- , ~ :4:a ~- - ' rda gTaFl} 2 + --~7]~F~12=
i(1 + X )
flT'.IIP II
~fl "B
+
tb / ~ F 2 2 F II
-a
-a
~5' '
(3.15f)
afl'°6r12106 + lapllr221]~ -- FatllF~22 + 2E.12r12B *a
3v~ B ~ rl* l r~ 12
F12 - A ~ a , + , tb~ -./~~ F22 *~
(3.15g)
J. M.F. Labastida et al. / Conformal supergravity
862
22R y v'..% ' - v'~+.~ "= i~/l~P,~a~b,,~r + qJ,,a°I'~lq%~ + 2iC~l~X~a + ~F ' ,,a"t~ 12
• .a- ,/, a,I, Y F '11
(3.15h)
+ 2iq~.aXOaq%" ,
FOz2%a+++°- i%'q~.a+q~,+° - (1 + X)¢aV°N.,
+(1 + x)~:.+~? + (1 - x ) ~ d z + +odr~ ~ - ~.oX3a ~ - C.a~a~S, ~Ta&F~ 1
- v ~ r 11~ -_ Fa21p11
11
(3.15i)
21
2t 11
½iG.,l~a + C~t~Flals
3'uOal a 2 : ' ~r n + iGaaF/~l _ iAaaF/~l _ 3'ABa 1 8 ' V r12.
+ t.4oa[, 2 . 11 _
+ 4 i L , r ~ 2 - 4i(17.~a + , . 02 o,#a) + 2i( v'a x.a + q~zP2p,.a) + VaN~a + q~a°~rb,.,
+ co~(g',wd + q~?w~d) 1
--
ll
11
-,o:(~c..x-a
11 r;i1 ) +~o2[-AB~+r~ 22 rX11 + , -, . /~r;~]
+~/Oo[_2iC.oY,aF~l - C.o(3a, X _ - - ~w.)r¢ , 11 11 11 + N.oF ~11 - 2i2o,.a + F'.sF; ],
vo.r~ ~- v~r2 = F a l 2 p
12 q- ,L
?'B
* a&*'r
ypllp12
q.. ,I,
;,I,
-i,p 2 2 p 1 2
"i',L Y I ~'12
12
-t-
(3.15j)
12
__
,L
Y~[
p12
-+ ~ i ( ~ . a ~ - v.&~) + 2;v~&~- ~(r: 11 x~. + r:22 x.~)
+ 2i(F~X.a + Fff2X.a)- ~iq~¢V(G.,va- A . , , a ) - 2iqq~,~AV,~,a + -1~ c . ~-[- v ~ s + % * s , - 2 ( r 212s + r s
22
z) +z.]
- ~,k o . (-% 1 : + G ~. ~ . + r l~xs - rd22x + M ) 1
+
+
--
"i' ..~
4;,L v p l Z V
5'~'/~
".
"'va
½Na,/~"
±
-
d
9
--
½~VuaN,-
pllp22
+
~iX.
22
,
11
Bs
~,_ 1_,1, :,,~,.~,~r
~i~#v(W.~v a + ~o o-xo,,,) +
~z ., ~
y
r2l l ( G ,
-
Ay&),
(3.15k)
J.M.F. Labastida et al. / Conformal supergravity
vSd
~ - v ~ r ~ = ,11-,21/-22 ~ ~
~
~/-,22 ~ /-'12 _j _ co~r.~22s + ,•A S ~
+ i(A.a - aJrd
863 22
2 - 2i( W#X.a + ~pBo)(o,.a)
+ 4iX~aFJ 2 + i~b.a, aS + 2iqJo~X°aF22
+~JO.(3a*Z - W ? ) F ¢ 2 + 2 i q J f ~ , . a
4"
~l ^
-- ~fNa,. r + +B.W,, a~' - ½i+f(Vr.Gva + +.°Go, ra ) y
22
4ij ,
prlly
++~ N.,r.~ + i~.avO/B°Nov + 3'v/~ - o "'oa 4" p 22-, 0/-11/',22 + s,q,~ r ; x , ~ + J~'a *.a-. ,
(3.151)
where G~a = 2(G,a - A,a),
(3.16a)
A,a -= 2(G,a + A~a),
(3.16b)
G~,t~# = 2( G-,aB - A~,#~ ),
(3.16c)
A~,a~ = 2(6~,a~ + A-,aB),
(3.16d)
(1 ++_X ) . f 8
=- 3f3a B _+~f~pa 8.
(3.16e)
To complete the projection of the constraints we have to take into account also the ones in (2.11). We consider only the N = 1 constraints generated by taking the 0 2~ (and 02a) independent part of (2.11). We find, ~aW, a = 0,
(3.17a)
Xo = 0,
(3.17b)
2o + (r2' + rl=x + ~ v . x ) = o ,
(3.17c)
S. + 2 ( 7 . 2 + r n s - r227,)- 24~f(rJ~s + FJ2?~ + ½va?, ) = 0 ,
(3.17d)
+24,.'% a (r~n + r ~ x + ½VaX) 2 ( q 2 S + F d = 2 ) = 0 ,
(3.17e)
J.M.F. Labastida et al. / Conformal supergraoity
864
N., tJv - Z(I~Cy). = O, 0
(3.17 0
1
__
x7.N~, + ~ No,~, - ~S(~C.~ - O,
(3.17g)
2t'~_ 12 _ W(.Xa)# + CJX~,a)~ +z_ 1 / ~ 2(~ ~a)a. _ 2F(~Xa)~O,
(3.17h)
x~.,a)~ = 0, X(.,O)
#
(3.17i)
1 la . - ~V'(~Ga) # - ±2d/(~ v Gy,a)B - 2F(~Xl3)a
-
2Y(.
22_ X~)B - 0 ,
(3.17j) (3.17k)
iS a + graxa a + ~at~X#" aa + tp22~a2.~ ~ a-4- 2F~ X . = O, i(V~A + ~kaaA# + 2(F21,~
+
r i g a ) ) + x ,o~ = o,
i(~r~S +@a~S~- 2(ra12S + r 2 2 z ) -
(3.171) (3.17m)
2za) + ½v~o~
+ 2!,f,~ar:_ v '-'#,,,a - 2F~IX~ - 2 F2:X a a a - X , a~~ , = 0, 11
i
22
(3.17n)
a
- 2-~ ~ a - 2F~ X~ = O,
(3.17o)
VaN~l~ + ~JJJN[~,~1~+ ~tA(~,l~)~ ~" = 0,
(3.17p)
N~,~a - ½i( Vr(~Aa)~ + $ J A r , a ) ~ ) = 0,
(3.17q)
3(V/~W~~ + ~¢~*W,f) + ~'~Z + , fZ/~ + -
-,r'ao
n _2r~
x
~)
3 a +i(G~,~a + iA~, ~) = 0 ,
(3.17r)
3WB~ + 2( e i'=s + Fd2Z) - Z a + 4i(r2~x= + r : 22-,, x ~ - x ° o~) -- -v & S - q J & % q" l(• IT.G - & q-- @aflGfl, at& - 32\[ r~r,~ - & + ~ k f A o ~ ) ) = 0 .
(3.17s)
We now compute the symmetry transformations of the N = 1 superfields. The second supersymmetry and SU(2) gauge transformations of the N = 1 covariant derivatives, and ~b~t~ and F; a are obtained by evaluating the 0 2~ (and 02a)
865
J.M.F. Labastida et al. / Conformal supergravity
independent part of the N -- 2 transformation (2.5). Similarly for the N = 1 superfields defined in (3.10) one has, for example, for W,a:
[iK, W,~,]I,
(3.18)
etc. For the computation of the superconformal transformations of the N = 1 fields we take the 0 2" (and 02a) independent part of (2.12). In computing all these transformations we have to take into account that to restore the Wess-Zumino gauge we have to perform compensating transformations using the higher components of K. These components are easily obtained by the requirement that the Wess-Zumino gauge is preserved under all transformations, e.g., from (3.5):
V2,~iKI = 6LW2,1 + iKV2,d.
(3.19)
Then, using this relation and (3.11) we find:
8~7A1 = 8LVAI + [iK, vA]I = ~LVA I -- IpA'SC~LV2,81 -- (EflV2,8 q- ~ F 2 / } "~- g c d Y c d ) V A I
-'}-~:TA(gflV2/3 -Jr-~ / ~ 2/} q_ gcdYcd) ] ..}_I~Ay(EBV2 fl .~_ ~ "q-~,3'(E/~V2/~ "~ ~eF 2/~ "~ g c d r c d ) V 2 , ]
2/~ q_ gcdYcd)V2y] (3.20)
-~- l~dycdgefy_e f .
On the other hand, from (3.3a) we have,
8Val =SVA + (&ka~) 02, + (&Pae)a2 + (8FAbc)Ybc.
(3.21)
Computing (3.20) by using (2.12b) and (3.9) and comparing to (3.21) we find, BY,, = ½Lv,~ + 2(wilL + 4,~vLv)M,~~ - 2q~.13LvMis ~'- ½(F'.L) Y
-elJ [ i( 8(13°C.)vX - CI~.NvO) MoV - iC~,~W~,OM~f- r2ilv/3] -~[-8(.°Xv)¢Mo +
v + 89(°X.))M. "~+ X . B Y - F:2F'#] +
+
+ Kn
:)
(3.22a)
" 8+,~B = W.eB+ ey ( F 2126vB + F~11+v13) + i~x+-x+v~B + 2 ~p,¢Kx2 + 6 j~p v/~K11 + 6 J K 22,
(3.22b)
J. M, F. Labastida et al. / Conformal supergravity
866
1 11 ~ r ~ ' = ~LC~ - 2 G
+ Eft( iC.B~ + FallF/} 1 ) + ~/~( Fd2F~ 1 + i~.BFJ~ -- h./~)
+ (V. + 2r]2) K 1' + ~b,,l~r~'K n - f'~1K12, 8~ 2 =
(3.22c)
½LFa12 + VaL + 24,,,BLB
+~( ~qoS- iNo~+ r11r~~) + ~(r~r~2 + i+o~r~) + ( V',, + F~2)K12 + ~ , ? F ~ 2 K n - F ~ X K 2 2 + Fa2K n ,
(3.22d)
aF~22 = ½Lr~ 2 - 2~bJ(~7/~L + +~YLv)
+~,(~q.s + r~rj2 + 2i,:N~) +~(r~2ry + ;,o, rJJ) + ( V , - 2F~2) K 22 + 3Fd2K 12 + +,/~Fd2K n ,
(3.22e)
where we have used the following definitions for the components of L: L=LI,
(3.23a)
L,, - V2.LI ,
(3.23b)
1 . ~72.L1 M - ~V2
,
(3.23C)
L,,,~ - ½[ ~ 2a, V2.] L I,
(3.23d)
m . ~ ~l~t''l v 2 a v~ 2 ~
(3.23e)
v
2aLl,
M ' = x~V z V ~ 2 & -V- 2 aV2,LI.
(3.23f)
Symmetry transformations of the N = 1 superfields defined in (3.10) are determined as in (3.18) plus the 0 2" (and 02~) independent parts of (2.12c-f). We reproduce here some of these transformations: 6 W,#~ = L W.I ~- e v Wv, ,~/~,
(3.24a)
-2i~,"13~7.L1~- i( V',,d/"1~+ FI~"~p~ #) L B + i~"Y~,wM - e " S . - ~*S, + 2(K'2S + K22X),
(3.24b)
J.M.F. Labastida et aL / Conformal supergravity 8 ~ = L~, + i~7~V'~L
-
iI'lllF'at
-iF~.2L " - i( V.+ "a + -e"~, - ~a
-
867
i~aV~L,
F.x~"~)La
+ (K22X + K l l S ) ,
8X = L ~ + 2 i M - ~ah a - 2(KXl~ + K l E X ) ,
(3.24c) (3.24d)
6N,~ a = LN,~ a + iV(,,L#) + i F ~ ( V'~)L + ~b#)VLv) + iF~ZLa) -i~b(,~a)M-
eVNv, ,~a - ~'N~, ,~a '
(3.24e)
8G,,a = LG,,a - 2L,~a - [V,,,-Va] L + 2q~,fq~a~i,~/~ + [ ( F a + ~ a ) L a + +.aF~I( V a L + +aVLv)
+ 2 oaF L a -
F L- h.c.] + 4(KllX~a + K22X~a),
(3.24f)
- e a X a , , a - ~/3X/3,~ a - 7I K 1 2 ~~ a + 2K12X~a,
(3.24g)
-eaGa,,a-~BGB,,a
-[(v.+~) Za + 2+~avoZ~- +oar2~(vat, + +a'L,) - F21L,i - 4',~aF]2L~ + I22( Va L + q',i%) - h.c.] -
eaAa, ,~a - i~AB, ,~,~,
(3.24h)
where use have been made of constraint (3.17a). The parameter of the second supersymmetry transformations is constrained due to the fact that the chiral gauge has been chosen. This constraint is easily computed by considering the second supersymmetry transformation of +,~. We find, 17~ a + (Fx,~81~~ - F22q,Ba - i~pJ+aB a ) ~ = o.
(3.25)
Integrability of this equation is necessary for the consistency of the chiral gauge. We will discuss this question in sect. 5.
J.M.F. Labastida et aL / Conformal supergravity
868
4. Supereonformal gauge So far we have obtained N = 2 superconformal supergravity in N = 1 superspace with full N = 2 superconformal symmetry. The theory is invariant under all the N = 1 superconformal parameters defined in (3.23). Most of these parameters can be used to gauge away algebraically some of the N = 1 superfields of the theory. The residual superconformal symmetry is the N = 1 one and it is represented by L (see (3.23a)). In this form, the theory contains the following symmetries: N = 1 local supersymmetry (manifest), N = 1 local superconformal (L), local second supersymmetry (%, which satisfies constraint (3.25)), and local SU(2) (K"b). This additional gauge fixing implies new compensating transformations to restore the Wess-Zumino gauge. This is carried out in this case by obtaining the expression of the higher components of L in terms of L, % and K ~b. Now we describe the gauge fixing. From the variation of F2 ~ (3.22c) we observe that it can be gauged away by using the L-component: L~. Then, we fix the gauge by demanding F l 1 = 0,
(4.1)
and so we obtain, 2~ t ~ ,/3 +i~b,~/3FJ~) + ~(~7~ + 2F~12 ) K L,,=-½ihe,~+ ±zMr'z2r1l
11.
(4.2)
Similarly, from (3.24d) we observe that h can be gauged away using the next L-component: M. We have, ?,=0,
(4.3)
M = _ 2l(e a. -a Xa + 2 K U ~ ) .
(4.4)
The next component, L~a, appears in the variations of G ~ and A~. We choose to gauge away A~a. We have, A~a = 0,
(4.5)
L~a= ½(1 + X);2/3~{[~Y/3, ~t~] L - [ ( ~ a ~ / 3 ) L / 3 + +. /3F~I i (v/3L + ~bfLv)
+ 2~/3~L/3 - F22L/3 -F~t2--~TaL- h.c.] + eYAr./3p + ~A,,/3B}.
(4.6)
It is now clear by looking at (2.12c) and the definition of Xa in (3.10) that this field can be gauged away algebraically using Ma. We then have,
(4.7)
J.M.F. Labastida et al. / Conformal supergravity
869
The expression for M, in terms of L, G and K ab can be obtained by computing the full transformation of 2,~. Finally, the last L-component M ' can be used to gauge away some of the higher field components that we have not defined explicitly. In this gauge, constraints (3.14), (3.15) and (3.17) simplify,
{vo, v~} = ,2
+
+i[6(,,rc#>oS-~b(,,v(Sv'~C#) o + 8 # , ' C v o ) ~ . ] M , ° ,
(4.8a)
{ Vr~, ~k} = i(1 + X ) , # Vr,e
+ [ _ 1j~ ( o ~
. u-~o ,,,>~_ G<°x,>~ + +~.) a<° x.+] Mo,
+ [ ½6k<°G:+,- ~/k<~X.+)- J/#~k
(4.8b)
+ (1G,,B - +,~#2#~ - ~bk+X,~+)Y,
[vo~, v~] = i+o2+/v,+
+[8. o
12
11 21~k#,X- ° a + ~k#a r2a] Vo
+ [ +o:ry + (c+s + __ 1 "
q_
gtX(~,,B)e, t"
< o
2 :p22V
- Z2 '•C , # X , , va - ~a+X#+ + ~i~b#'G(r,,)a
+ ~i+eo[~o°°~- VoyO~- G oxo - o ~-2r.' ,~-o x ~]}r + R :a. fl,'~M,~' + R :a, +8-p<~M,+, lT(a@fl)Y
=
,R
Vp22
"(~' *B}
+
"l,t,
~,p12
" ~ ' ( " *fl) '
N~# = 0,
~" l*p 1( a2 p 2*#)2
(4.9a) (4.98)
~(,,FJ~ = - ig/<~,VNl~)v - i~<,#3)Z, ~7'
(4.8c)
+ t. ~b<,v~b#)°N,, - i~b<,rG)vS,
(4.9c) (4.9d)
870
J . M . F . L a b a s t i d a et al. /
W ~ f = i(1 + X ) . ¢ --
~7aFJI i(1 =
v,~FJ2 +
-12 = W~q
i(1
.06
Conformal supergravity
}oo y + 2 + / r f
+ + . % f r J ~,
(4.9e)
~ a BF£22F~11
(4.9f)
. 0 6 1r l0l6 -~+ Av ,} Ot]~
3Fj11 C 12 +
-t" ~ V ~) ~ . 0 6 p~o~+ 12
2F~12 F~12 + ~.r, Cr.nr12 + ~a ~F~22F~12 ,
(4.9g)
P',~,i+f - V'#~,a v = i+f~,~a~'~,,-i, v + 2 i C , FXVa 4- F 22 v + ~')p12"t'aa,fB"y
+,,,"t"B..r, ~r"-+./pjz_ "t" o a & v.~
~ = ry+.~%
x714//3, G s + 2 i * . ¢ X O a * o v,
(4.9h)
~ - i + / ~ o ~ % ~ ~ - (1 + X ) B ~ N . ~
+ (1 + X)Ba,~ ~--~ W+ +
(1 - x ) ~ j ~ + ~ ° ) r 7 (4.9i)
- C,,B+,~ S, v . ~ F J ' - v~F-J~ = + ½iG.,Ba + C~Bfla 1S - iA.fl'J ~ - 5"Ba'.':v ,.~2
- ~,( vox~, + +. x,.~,) + 6
+¢~,.,
j,
1
. /~ 11 + 4iL~FJ' + ¢.s°[F~22 F;11 + ,+, F~o]
+,~o[ - c . o ( 6 j x - ~j)F~ ~- 2i~o..a ] , vo.rJ' - v~rJ1-- rLr~"12 +
(4.9j)
vda&'L* ~p 21 2 1±x]*~ -t- *~afiiJ'Y,t,~yflyp12 J - d , ~__ , . ) ~flY p12±a&.7 pll
+2,Gt~aF'~12 + C~¢SF;i2 + , +' ¢ . ( 2 X--o a/';12 + [ 8 2 Z •
w.*]r.~ 2)
~. _ 52 i F :~: ~ ~a + 2iFff22.a - 3,+~ + Stw~G~a 1. v (O,~,va-A,~.vs)
o 11 22
4 , j , TF12"y
_ ±.,, 2 ~ B v.,, "rB.,~ "'9,
Va
~,~, ~ W 2.
5Ne,,t
~ fla + 1
+ ~iqJf( w,~Xva + t),/'X,, va) ,
'
6~ W~,~ ) ":/
(4.9k)
871
J. M. F. Labastida et al. / Conformal supergravity
V,,,fff 2 - V'/3F,~za2 = ap2~p22o. ,,a*B + C,,/3U,~22S+ i ( a , a -
G.a)F/2-Zi(vt~X.a+tk¢°do,as)
+ 4 i X , aF~ 2 + i#.e,,l~S + q,~so( - Fff2F22 - i+¢° F 22)
+ ~4,,,~ [ V,~S + q,,]'S+ - 2(F22S +/'222;) + 2a] + 2iq,¢~X°aF 22
1' g ++l,o( ~ d z - w d )r? ~ - ~'q'l, ( voG~ + G°G, ~)
(4.91)
+ 4_i, L 0 p l l y ..J- ~i,I, # F ' 2 2 y ,L p p l l r '22 3"'Ffl * a .-Xp&-- 3,~t.,fl *-a "xO&"{- 5vfl *~&J-,o ,
v.s-
V'~W~ = O,
(4.10a)
)% = O,
(4.10b)
N, = 0,
(4.10c)
X7~,~ = O,
(4.10d)
2(rJ~s + r:.~y.)=o,
(4.10e)
N,,, ~, = 0,
(4.10f)
S,~ -- 0,
(4.10g)
1 F22g%"
") F ' 1 2 V
(4.10h)
~
X(:,~>k = 0, X ( a , f l ) fi - - ~l 7 ( a a f l )
(4.10i)
~ - - 12~/b(=G.y,fl)fl ~, . - - 2 - r ( ,2, 2X-e- ~ a = 0 ,
(4.10j)
v<:-~t~,a + +<,,'-x,.~> e. 0 2r::.g~,a , ++ ~G(o.~,e 1 = 2G' x ~
(4.10k) 0,
2iFlalN + X -,~ ,,,a = 0,
,(+:s + !,l,aflg7
+ 2v
"B, ~
2r2:s__
(4.101) (4.10m)
lvo<
2r; 22 x- -~ -- Xa, ,~a = o ,
(4.10n)
872
J.M.F. Labastidaet al. / Conformalsupergravity
2i(Wa~ + ~ba/~Z/)+ F]IS) + ½G~,~a + 17"-~a + ~b"BXo,~a- 2F22-~"4 = 0, (4.10o)
~ba/~NK ~ + ½iA(~,/~)a = 0,
(4.10p)
" Y Na,.~ - l2,qq. Av,B) ~ = 0
(4.10q)
3(~Wd + ~,~w,~) + G z + ~ ' z , + r~'s + i( G."~ + ~A."~) + 4i( V'"X.a + ~k"I~xI~,.~- 2F22X"a) =O, 3WBaB - ~aS - tka~ + 2/'128 - Z,i +
4i(F2a2%-
(4.10r)
x L..)
+ i(W.G" a + tk./~G, "a - -32~b.BAt<"a)= 0.
(4.10s)
Finally, we give the symmetry transformations of the N = 1 covariant derivatives and superfields in the superconformal chiral Wess-Zumino gauge:
3V'. = ½Lv~ + 2 ( V B L ) M . a - ½(V.L)Y -eO[i3ffC,~,vX - M. ~ - iCt3,~M~ ~'] -~[-3(.°X~,)~Mo v + ~/;(°X.~,)Mo* + X . ~ Y - F22~
+ i~b,'~/~V,t~ + (( Vv + 2F12)Kn)( ~kI~VM~--~J~MI~v) + (K123~ v + Kn@~v) Wv,
~ a fl = ~Taefl+ eflF12 + i~aY~y~ fl + 2 ~ J K 12 + tkJq, v/~Kn + ($JK 22 ,
(4.11a) (4.11b)
~r 1~= ~Lr2 + vo~ + ~oZ + ~(rd~r~ + ~o~(r~ + tyrO' + + , r ~ ) )
+(vo + r2=)~ ~ + ~o~(v. + r ~ ) r 11+ +o~r~r" + rd=r~', (4.nc) a r f = ½Lrd ~ - 2<~( V,L + +e~L,) + i~.S
+ ~(,<~r~,:,- +o%,(rSr~l+ ;
(4.11d)
X I
+
%
~~,~.I~
II t~
o.<11
I
-I-
~.~
~~' ~ .
~.
+
~,.
X
~
,
#
°+
"~
~
~'~.
+
I
~
o.~'
#
+
+
o~
•
•
~.~
~.~'~
+
I
~
o~
+
<
X
I
r.
'%'
"~"
"~
~ - ~
+
~-~
~.
~.
#
+ ~,
~
+ +
~b~ =
o.~l
I
I
b
b
~
e~
°
-F
i ~
+
%
b~
"1~
+
I
b~
II
I
~
+ X
+
+
I
II
~, o~ ~I
I ~, •
I
+ ~
il
il
o.~l
'
,~
+
+ i i
i
J.M.F. I.abastida et al. / Conformal supergravity
874
5. Solution to the constraints
In this section we solve constraints (4.8), (4.9) and (4.10) in terms of unconstrained matter prepotentials and N = 1 conformal supergravity, whose solution is known. The method is to make redefinitions of our covariant derivatives until the commutator constraints (4.8a-c) take the form of those of N = 1 conformal supergravity [4]:
( d~, d# } = - 2isM~/~,
(5.1a)
{ d~, da } = idea,
(5.1b)
=
(5.1c
-
where d~, da and d~a are the covariant derivatives of N = 1 conformal supergravity, and where s satisfies:
d~s = 0.
(5.2)
The first step is to postulate a rather general relation between the covariant derivative V~ and the N = 1 covariant derivative d~:
~7,= d, + Q,o o Mop + Q~ a Mo + Q~Y.
(5.3)
Note that we do not include a vielbein redefinition. This is because the commutator (4.8a) does not contain already vector and dotted spinor torsion terms, and this property would be spoiled by a vielbein redefinition. Next, we choose the Q's in (5.3) in such a way that when the ~7~ and ~7a are replaced in terms of the d~'s and da's the constraints (4.9a-c) take the same form as those in (5.1). Then we can identify some of our fields in terms of the pure N = 1 conformal supergravity fields. The rest of the fields represent matter fields and must be solved for in terms of prepotentials using the rest of eq. (4.8), and eq. (4.9-10). To determine the Q's, we note that the N = 1 conformal supergravity constraints can be expressed succinctly in the form:
d~a = - i ( d ~ , d a } , ~,~=~,~ffo)=0, ~v
(5.4a) (5.4b)
= 0,
(5.4c)
~¢v* = ~ ¢ * = 0,
(5.4d)
where the Y ' s are the torsions for pure N = 1 conformal supergravity. The form of
J.M.F. Labastida et al. / Conformal supergravitv
875
the constraints (5.1) is then completely fixed by Bianchi identities. This implies that if we can arrange the Q's such that (5.4) are satisfied, then (5.1) will be automatically satisfied. Next, we will show how this method will give a linear set of equations for the Q's. As we explained before, eq. (5.4d) are automatically satisfied, given that, in the chiral gauge the constraints (4.8a-c) do not have such torsions and that our redefinitions (5.3) are chosen not to produce them. Eq. (5.4c) implies that the Q's satisfy:
= (rS-
(5.5)
Eq. (5.4a) is solved trivially by a redefinition of the vector derivative. Finally, eq. (5.4b) implies: Q.
=
- ±4~~ . ~ . . ~ ~ ,
~) , Q,k# = 114o 4 = ' o /x(&
(5.6a) (5.6b)
where H¢."a~' = (1 + X).ao~q~t~°+oaO(1 + X ) 2 " ~ __ Fal2(~&-i, b
22
(VB(1 + X).~a°~)(1 + X ) 2 "~
~,
These equations can be solved for the Q's once the objects that appear in them are known in terms of unconstrained prepotentials. Progress is made by noting that the Y field strengths in both (5.1a) and (4.8a) are zero. This implies, using (5.5), that:
d(,~Qt3) = 0,
(5.8)
Q~ = d~2,
(5.9)
giving
where ~2 is an unconstrained prepotential. Next, from (4.9a), we find, using (5.5): d(~bt~)v - 2Q(,~t~)v = 3(,,~' [Yff~ + Q~)o°tk, ° + F ~ x - Q ~ ) x ] ,
(5.10)
where we have used the notation x = qJ~'. By symmetrizing and anti-symmetrizing in fl and y in eq. (5.10), we find: d(~q~t~v) - 2Q(.+~y) = O,
(5.11a) (5.11b)
J.M.F. Labastida et al. / Conformal supergravity
876
F r o m eq. (5.11a) we find the solution of ~b~¢ in terms of two new unconstrained prepotentials q~a and ~: ~,~ = ( d e - 2Q~)ff¢ + C . ¢ ~ .
(5.12)
O n the other hand, from eq. (4.9e) we obtain:
(5.13) Eqs. (5.5), (6), (9), (llb), (12), (13) are a closed set of linear equations which allow us to solve for Q~, Q~aV, Q , ~ , ~ B , ~b~v, F~2, F~ 2 in terms of the unconstrained prepotentials ~2, ~, and q~a, and the N = 1 conformal supergravity covariant derivatives of these prepotentials. Eqs. (5.9) and (5.12) are two of these solutions. The additional expressions are long and not enlightening and we do not include them here. The rest of the constraints (4.8)-(4.10) give expressions for the remaining fields in terms of the unconstrained prepotentials I2, ~, and q'B, and N = 1 conformal supergravity objects. The process is tedious but straightforward (all the equations are algebraic and linear). We will give a representative example by obtaining the explicit expression of S and ~. Comparing the M~B terms of (4.8a) and (5.1) we find: S = s - ~i.ga - Z x ,
(5.14)
£p= d,~Qff~B + Q,~Qff~/3+ Q,ffQ ,~¢.
(5.15)
where
In expression (5.14), s is an N = 1 conformal supergravity object, and **.coand x = qJ~ are already solved. We now need another equation which involves 2J, and this is provided by eq. (4.9i): Z = 3F1 - S.Ze~2 ,
(5.16)
where ,~F1 = ~ ( 1 + X)s~I/~aK/~aa ~ ,
(5.17a)
~2 =Jg(1 +
,
(5.17b)
K¢, ~a~ = V'~b~s~ - Fff2+,a*~p,~ - ~,a°FJ 2 + i+av~p,s+~kv,o,
(5.17c)
-,~o
-~
Jg'= (Tr[(1 + X ) - I ( 1 - X)]) -1 ~,
(5.17d)
JU1 and OF2 are already solved in terms of prepotentials. Eqs. (5.14) and (5.16)
J.M.F. Labastida et al. / Conformalsupergravity
877
are a system of two linear equations with two unknowns, which is trivially solved. We find for S:
S=(s-
½iAq- x , ~ l ) / ( 1 + x,gU2).
(5.18)
It is easy to check, using constraints (4.8)-(4.10), that for all the fields one can find an appropriate system of linear equations to solve for them, similarly to the example above. Actually, one finds more equations than are necessary to solve for the unknown fields. The constraints which do not solve for unknown fields should be automatically satisfied, without giving constraints on the prepotentials. We have checked this for all equations at the linearized level, and some at the full level, and we found no such constraints on the prepotentials. Finally, we comment on the existence of the chiral gauge. We note that all of the constraints are consistent with its existence, and that the constraint imposed on the second supersymmetry parameter e~ is integrable at the linearized level. To see this, note that at the linearized level, constraint (3.25) becomes, (d~ + 2/-~2)~/~=0.
(5.19)
To solve for e~ we observe first that from (5.1a) and (5.13b) we have, { d,~ + 2/-~ z, d/~ + 2F/}2 } = - 2isM~13,
(5.20)
so the solution to (5.19) is
~ = [½(d" + 2/'12~)(d. + 2/"12,~) +
is]~ ~ ,
(5.21)
where q~ is an arbitrary superfield. It would be desirable to prove the integrability at the full level, but we have not been able to do this. To summarize this section, we have solved all the fields in terms of the N = 1 unconstrained prepotentials I2, ~, and q~, and pure N = 1 conformal supergravity fields. 6. Conclusions In this paper, we have studied N = 2 conformal supergravity in N = 1 superspace. In particular we have treated the following two aspects: first we have applied the method of the projection of extended supersymmetric theories into N = 1 superspace; second, we have solved the resulting constraints in terms of unconstrained prepotentials. We have found that for the case of conformal extended theories, besides the usual Wess-Zumino gauge fixing, one can perform an additional gauge fixing by using some of the N = 1 superspace components of the superconformal transformation
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J.M.F. Labastida et al. / Conformal supergravity
parameter L. This provides a formulation of the theory with a minimum set of matter superfields. Also we have been working in the already familiar chiral gauge [6, 7]. We argued that in this case this gauge facilitates the expression of N = 2 conformal supergravity as N = 1 conformal supergravity coupled to matter superfields. The constraints satisfied by these matter fields have been solved in terms of N = 1 conformal supergravity plus the matter prepotentials ~ , (b and ~2. This work provides the adequate framework in N = 1 superspace to construct the different formulationss of N = 2 Poincar6 supergravity by using compensator techniques. The first step will consists of the coupling of the vector multiplet and the appropriate choice of gauge to obtain the so-called minimal fieM representation of N = 2 Poincar6 supergravity, which is not superscale invariant. At that stage, one will have still the SU(2) gauge invariance in the theory which is not part of the symmetries of N = 2 Poincar6 supergravity. One will need further compensating fields to compensate for this gauge invariance. This will require the coupling of N = 2 matter supermultiplets. Different formulations of Poincar6 supergravity will show up after the SU(2) gauge symmetry is then compensated. We thank Martin Ro~ek for many useful and stimulating conversations. One of us (J.M.F.L.) would like to thank the Institute for Theoretical Physics at SUNY at Stony Brook for support during the summer of 1985, where part of this work was carried out.
Appendix In this appendix we summarize our conventions. Our notation is based in [4]. We denote SL(2, C) indices by lower case Greek letters, simple superspace indices by upper case Latin letters A = (a, &, a&), and N = 2 superspace indices by underlined Latin letters A = (a, a_',a&), a_ = aa, where a is an SU(2) isospin index. To raise and lower SL(2, C) indices, we use the tensor,
° -i) i
0 "
(A.1)
For SU(2) indices, we use 0 Cab = c a b = ( --1
Symmetrization of isospin indices is defined to mean tracelessness: X(aY b) =_ 2X, Y b - 8abXc Yc"
(A.3)
According to our conventions, the hermitian conjugation of an N = 1 superspace equation is obtained from the following rule: change dotted (undotted) indices into
J. M.F. Labastida et al. / Conformal supergravity
879
undotted (dotted) without changing the order of indices (except for vector index a& which goes into a&), add a minus sign for each vector index and for each i, change the upper SU(2) indices of Fn (F j2) into F j2 (FJX), and add a minus sign for each F 12. For example, the hermitian conjugate of (5.13) is,
i~b~/~= (1 + X)2°~( V~q:f - 2~fF~ 2 - q:f+oBF¢I).
(A.4)
The action of our SU(2) generators on superfields is defined in (2.3). Accordingly the commutation relations for the generators are: (A.5) Y-charges of the N = 2 superfields are easily derived by using (2.4) and the commutation relations (2.9). One finds, [Y, W~/~] = - 2W~e,
[Y, sob] = 2sob, [ Y, N~a ] = 2 N~/3 ,
=o.
(A.6)
Then, using (3.3) and (3.10) (or (3.15)) one can easily compute the Y-charges of the N = 1 superfields. We list here some of them: [Y, X] = 2X,
[ r , v ; ] = v,o, [Y, ~ J ] = 0 ,
[Y,~] =22,,
[Y, 4',f] = 24',f,
[ Y, N~I~] = 2 N~I~ ,
[r,r:l=r:,
[ Y. Wo~] = - 2 w o , .
[Y, SI = 2 s ,
It. oo~] =o.
(A.7)
where ~ represents any object with only a vector index. To compute the Y-charges of the prepotentials and the parameter of the second supersymmetry transformation one needs to use (2.6), (5.6) and (5.7). We find, [Y, q~] = - q ~ ,
[Y, ~2] = 0 ,
[Y,*I =0,
[Y,d
=
-%.
(A.8)
References [1] S.J. Gates and W. Siegel, Nucl. Phys. B195 (1982) 39
[2] A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetskyand E. Sokatchev, Class. Quant. Grav. 2 (1986) 601,617
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J. M.F. Labastida et al. / Conformal supergravity
[3] W. Siegel and S.J. Gates, Nucl. Phys. B189 (1981) 295; S. Mandelstan, Nuct. Phys. B213 (1983) 149; L. Brink, O. Lindgren and B.E.W. Nilsson, Nucl. Phys. B212 (1983) 401; Phys. Lett. 123B (1983) 323 [4] S.L. Gates, M.T. Grisaru, M. Ro~ek and W. Siegel, Superspace (Benjamin Cummings, 1983) [5] M.F. Sohnius, Nucl. Phys. B136 (1978) 135; E. Witten, Phys. Lett. 77B (1978) 394; M.T. Grisaru, W. Siegel and M. Ro~ek, Nucl. Phys. B159 (1979) 429 [6] S.J. Gates, A. Karlhede, U. Lindstr~Sm and M. Ro~ek, Class. Quant. Grav. 1 (1984) 277 [7] S.J. Gates, A. Karlhede, U. Lindstr6m and M. Rogek, Nucl. Phys. B243 (1984) 221 [8] J.M.F. Labastida, M. Ro~ek, E. S~nchez-Velasco and P. Wills, Phys. Lett. 151B (1985) 111 [9] J.M.F. Labastida, E. Sfinchez-Velasco and P. Wills, Nucl. Phys. B256 (1985) 394 [10] M.T. Grisaru and D. Zanon, Nucl. Phys. B252 (1985) 578, 591 [11] E.S. Fradkin and A.A. Tseytlin, Phys. Reports 119 (1985) 233 [12] S. Ferrara, L. Girardello, T. Kugo and A. van Proeyen, Nucl. Phys. B223 (1983) 191; T. Kugo and S. Uehara, Nucl. Phys. B226 (1983) 49 [13] B. de Wit, J.W. van Holten and A. van Proeyen, Nucl. Phys. B184 (1981) 77, (E) B222 (1983) 516 [14] P. Howe, Nucl. Phys, B199 (1982) 309 [15] P. Howe, Phys. Lett. 100B (1981) 389 [16] M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Phys. Lett. 69B (1977) 304; Phys. Rev. D17 (1978) 3179 [17] E. Bergshoeff, M. de Roo and B. de Wit, Nucl. Phys. B182 (1981) 173
N - - 2 C O N F O R M A L SUPERGRAVITY IN N - - 1 S U P E R S P A C E J.M.F. LABASTIDA1 The Institute for Advanced Study, Princeton, NJ 08540, USA E. S,~NCHEZ-VELASCO 2' * and Peter WILLS2"**
Institute for Theoretical Physics, S U N Y at Stony Brook, N Y 11794, USA
Received 18 April 1986
We apply the method of projecting extended supersymmetric theories into N = 1 superspace to the case of N = 2 conformal supergravity. We solve the constraints in terms of unconstrained matter prepotentials and N = 1 conformalsupergravity fields.
1. I n t r o d u c t i o n
Supergravity theories, besides being mathematically interesting, are thought to be part of the low-energy limit of superstring theories, which are attractive candidates for a unified theory involving gravity. A complete understanding of supergravity theories is thus desirable. One of the nice properties of supergravity theories is that they have better ultraviolet behavior than their non-supersymmetric counterparts. However, in general, computations are very difficult, and consequently issues that involve quantum corrections, like the finiteness question, have not been resolved. The simplest and geometrically most natural way to define supersymmetric theories is in superspace. Also, in superspace, one can define the Feynman rules of the theory using supergraph techniques, which greatly simplify the quantum computations. However, to apply supergraph techniques one needs a formulation of the theory in terms of unconstrained prepotentials. For extended supergravity theories such formulations are not known in their corresponding extended superspaces (except for one of the N = 2 Poincar6 supergravities at the linear level as presented in [1]). Two possible solutions to this problem are the modification of extended 1Research supported by US Department of EnergyContract No. DE-AC02-76ER02220. 2Research supported in part by NSF grant PHY 85-07627. * Address after Sep. 1, 1986: Newman Laboratory for Nuclear Studies, Cornel1University, Ithaca, NY 14853, USA. ** Address after Sep. 1, 1986: Physics Department, University of California at San Diego, La Jolla, CA 92093, USA. 0550-3213/86/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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J. M.F. Labastida et al. / Conformal supergravity
superspace [2], and a light-cone superspace [3]. These approaches are under investigation and they still possess technical difficulties. Another approach consists of projecting extended theories into N = 1 superspace. This method has the disadvantage that only one of the supesymmetries is manifest; however, it has the advantage that if one is able to solve the constraints in terms of unconstrained propotentials, the well-developed machinery of N = 1 supergraphs [4] would facilitate quantum calculations enormously. The method of the projection to N = 1 superspace has been successfully utilized for N = 2 super-Yang-Mills [5]. This method has started to be applied to extended supergravities in recent years. For the version of N = 2 Poincar~ supergravity of [1], the theory was projected to N = 1 superspace and the constraints were solved in terms of unconstrained prepotentials [6, 7]. The corresponding action was constructed in [8]. Furthermore, the coupling of this version of N = 2 Poincar6 supergravity to an abelian N = 2 vector multiplet was analyzed in this context in [9]. There, the theory was solved in terms of unconstrained prepotentials and the action was constructed. The version of N = 2 Poincar6 supergravity of [1] is at present formulated in the adequate form to perform quantum calculations. However, this is not the simplest form of N = 2 Poincar6 supergravity to carry out a quantum analysis because of the tower of ghosts problem [10]. The reason for this is that the formulation of this theory in N = 1 superspace is made in terms of non-minimal N = 1 Poincar6 supergravity. A formulation based on minimal N = 1 Poincar~ supergravity would be preferred. A possible way to obtain such a formulation could be the study of dual forms of the N = 2 Poincar6 supergravity theory of [6-8]. Another way is to analyze N = 2 conformal supergravity and then, through compensator techniques, generate the different versions of N = 2 Poincar6 supergravity in N = 1 superspace. This paper constitutes the first step towards the analysis of this last approach to the problem. Conformal supergravity theories were introduced because, besides being interesting by themselves [11], they provide a mechanism to analyze the structure of Poincar~ supergravity theories. The utility of this mechanism has been demonstrated successfully for N = 1 supergravity (in ordinary space [12] and in superspace [4]) and for N = 2 (in ordinary space [13] and (in part) in N = 2 superspace [14]). In the cases where it has been applied, it has clarified the structure of Poincar6 supergravity and greatly facilitated the construction of invariant actions. In this paper we present the projection to N = 1 superspace of N = 2 conformal supergravity and the solution of the corresponding N = 1 constraints. We start with the N = 2 superspace formulation of [14,15] and we define N = 1 components of N = 2 superspace objects by covariantly projecting the 8 2~ (and 8~) components. In doing this we use part of the symmetry of the theory to gauge away algebraically some of the N = 1 components. This gives an N = 1 superspace formulation in a Wess-Zumino gauge in a similar way as the one for N = 2 Poincar6 supergravity [6, 7]. However, in the case of N = 2 conformal supergravity one possesses a bigger
J.M.F. Labastida et al. / Conformal supergravity
853
symmetry which provides the possibility of a further gauge fixing. We call this gauge superconforrnal Wess-Zumino gauge because we use certain components of the superconformal transformation parameter to gauge away algebraically some of the superfields. The next step is to formulate N = 2 conformal supergravity as N = 1 conformal supergravity coupled to a set of matter fields in such a way that the constraints for these matter fields can be solved in terms of unconstrained prepotentials. To carry out this program we have found very convenient to make a further gauge fixing, the so-called chiral gauge [7]. The N = 1 superspace formulation that we present in this paper shows that N = 2 conformal supergravity can be expressed as N = 1 conformal supergravity plus some matter couplings and that the constraints satisfied by the matter superfields can be solved in terms of unconstrained prepotentials. The paper is organized as follows. In sect. 2, we describe the formulation of N = 2 conformal supergravity in N = 2 superspace. In sect. 3, we project the theory into N = 1 superspace. In sect. 4, we perform the superconformal Wess-Zumino gauge fixing. In sect. 5, we present the solution to the constraints in terms N = 1 conformal supergravity and unconstrained prepotentials. Finally, in sect. 6, we state our conclusion. We also include one appendix which contains a summary of our notation.
2. N = 2 conformal supergravity in N - - 2 superspace
In this section we review the formulation of N = 2 supergravity in N = 2 superspace as presented in [15,14] and rewrite the equations we will need in a form convenient for our purposes. This approach is based on the fact that ordinary N = 2 superspace [4] is perfectly suitable for the description of conformal supergravity. Although ordinary N superspace has as tangent space group SL(2, C) × U ( N ) and the superconformal group is SU(2, 2IN), the additional parameters, i.e. conformal and superconformal boosts, can be absorbed into general coordinate transformations, while Weyl and S supersymmetry transformations appear as extra invariances of the constraints that define the theory. This approach is a superspace generalization of the work done in components [13,16,17]. We will use the N = 2 index notation of [4] which is summarized in the appendix. We denote SL(2,C) generators by M r . We introduce curved N = 2 superspace covariant derivatives V_A (V~_,V~, V ~ ) which contain SU(2) connections I',~b, and U(1) connections I'A, (2.1) where
rab
and Y are the SU(2) and U(1) generators respectively. The SU(2)
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J.M.F. Labastida et al. / Conformal supergravity
generators Y~b satisfy the reality condition ~,,o= C,cCOUYca,
(2.2)
where C "h is the SU(2) invariant tensor (see the appendix), and their action on an N = 2 superfield is: [Yah, T~,] = - 1Cc(aTb) Y .
(2.3)
The U(1) generator Y is hermitian and its action on covariant derivatives is: [Y, X7~] = XT_~,
(2.4a)
[Y, v a ] = -V_~,
(2.4b)
[r,v~a] =0.
(2.4c)
Under gauge transformations, the covariant derivatives transform as 8V a = [iK, V_A1,
(2.5)
where, K=i(K~va+K~OM#~+Kj~M/~+KabY,
b+kY ) .
(2.6)
The covariant derivatives possess the following graded commutator: [VA, VB } = TABBCVc + RABvSMn v + RA2÷~M~* + F~AdYcd + FA2Y,
(2.7)
where we have used the standard notation [4] to define torsions and field strengths. The theory is defined by demanding the following constraints [15,14]:
T.__f= O, = o,
T~_,a#~' = O,
T._~y,=.t ~ ab S~~ , = o,
T,~,, #O"v~'- 0 -
•
(2.8)
Introducing these constraints into the Bianchi identities one can compute the rest of the torsions. Then, the Bianchi identities are solved for field strengths and curvatures in terms of them. The solution for all the field strengths up to the vector-spinor one and for all the curvatures up to the spinor-dotted spinor one, which are enough
J.M.F. Labastida et aL / Conformal supergravity
855
for our work, can be summarized in:
{ Va_, ~ } = i[ 8(a°Cs)vSab + CasCabN, ° ] Mav + iCaBCabW~'iM~9 + i [ 3(,,d3b)qV~tj + G~qbs4cecc/u] Ycu,
(2.9a)
( Va_,~_B } = i~b~TalJ + iS(ao<)tJbaMov + iS~'iGSt)abM6 ~ + i
+
cec~:Gal~e b] Ycd,
(2.9b)
[Voa, V 0] = - 23:GoatfV z + [ C, oCb~Wa~ + 3:( C~1~$b¢+ ChiN.o)] ~,_ -t- 1 R ( a l " c ) d [ ) -- 1 - ~ ( e 3 "b "" ~Tea~rfl&(ff 2v.~
X(
1
+
Co S ,e +
o) or" V°°GO,i(b")] Y + Roa,gvaM8 v + Roa,t~,SM~ " v," (2.9c)
°r'~[V(bfl~Trof a
where Noa, Sob and WoO are symmetric N = 2 superfields and G~aob satisfies the reality condition Gaaa b = - CJa&ba . (2.10) These superfields are not all independent. From the analysis of the Bianchi identities one can obtain:
V_~W,,, = 0,
(2.11a)
V,,#N~,o + Vo(oNv)p = 0,
(2.11b)
Co(~Vt,),N~,,
CB(~V.~oS~= O,
-
(2.11c)
V ( b f l S c ) a "Jr V a f l S b c = O,
(2.11d)
Cbcv~s,~ + V,rNvo = O,
(2.11e)
b
+ v-~auo, fl
V:g,2)
(2.11f)
= o,
g"~ab,¢.z ~ C__ - ,-Vb(p',-'o)/~ -- O,
~-c 3 ~ W a IJ + C ab (VaSbc -- 5Vb,,G°a~ ¢ -- 4V:Goab ~) = 0.
(2.11g) (2.11h) (2.11i)
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J.M.F. Labastida et al. / Canformal supergravity
The constraints (2.9) and (2.11) are invariant under the following N = 2 superconformal transformations:
8LE~_ = ½LE~_,
(2.12a)
SLY,, = ½LV. + 2(V,#~L ) M. B + 2(V.~L ) ccbYab -- ½(vail ) Y,
(2.12b)
8LS,, b = LS,, h - ½iV(.aVo)°L,
(2.12c)
~. ,,bW.O,Vba)L, 8LN,,I~ = LN.a + ~tC
(2.12d)
8LW.a = L W,q~,
(2.12e)
6LGd~f = L G.B f - l i [Va., ~ blj] L ,
(2.12f)
~Li~7.B= Li~7.~ + 2 ( v . . L )V"B + 2(v/~L ) V . . +i(~TaB L ) M r + i((7.aL )MBa,
(2.12g)
where L is a real scalar N = 2 superfield. In reducing the constraints (2.9) to N = 1 superspace it is convenient to use a shifted N = 2 covariant vector derivative defined by i V . a - { V 2 . , V 2 a } . This breaks manifest SU(2) covariance. In terms of this new derivative the constraints (2.9) become:
{ V~_, V_lj) = i[8(.actj)vS~b + C.I~C~bN, a ] 34ov + iC.13C~bW~,°Moi' + i[ 6(.aSb)CN.l~ + C.¢C~bSeICecC/d] r,. a , { V2,,,~ 2B } = iV. B ,
{v
..v'B ) = voB + i[ 8B,., o+,11 -
(2.13a) (2.13b)
M.,
+i [~.(aGy,B11 - ~.(°G--y)B221Ma YAt-i(~TetB11- ~.B22) Y - 4 i ( G . B ix + G.B22 )Y12,
(2.13c)
{ ~1., "~ 29 } = iS.(nl~v)BZaMav + iSB(r'Gag)xZMa* + ia.Ba2Y + 2i((7.B11 + G.B22) Yn,
(2.13d)
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J.M. E Labastidaet aL / Conformal supergravity
{ V2~,,~',~ } = i~,,(°Gv)B'2Mo ' + i~(dG~,)2'Ma ~ + iGa~2'Y
(2.13e)
- 2 / ( c ~ ? + ~o~?) r~:,
C,~,sS,'~la
[V,,/,,VlO] = -2(t70/,1'+ G#a22)Vl,, - 2C,,#GV,u2W2v+ + [C,,,o(W,~ i' + 8e,i'S,2 ) + ~a*N~]~ 2, 1
1
1
2
1
"t
2
2 1 4 1 °t- [-- 3~2a((Tfl&l -- (Tfl&22) "F ~71a(Tfl&2 -- 2~71fl(Ta&21
+v,~o, + c o ~ ( v , , < , -
}(v'~Sx= + v ~s=~))] rx,
+ V 2&Na,R+ Carl('~ 2~,~'V&~ + { ( V I&Sll + V 2&S12))1 Y12 + 2V,~G~,,EyE2,
(2.130
Iv.a, V2~] = -- 2CaBGV~ElVl v - 4CaBGVi, E2V2 v +
Ca~S2~
28
+ [coa(,2s,~- ~ ? ) - ,?N.~]~', 1
1
+ 2[3V2(fl(l~a)&l
1
1
-- ~a)&22) "Jr -~Vl,a(78,a.2
1
+ C~#V,vG va2' - C,,/~V2v ( G~' - GaY22)] Y - 2 V2/~G,,a2' Y1, 2 1 4 1 + [ - ~v~o(a,~. - a,~?) - 4v~ao~? + ~v.oa,~
..[_~ 1Nail + Carl( ~ li, w&.9 _ I ( V 1&S12 + V 2822 ))] Y12 _ 2
1_
3V2aGflal
+
2V2fGaal 2
+ v 2 a N ~ + C~(~2,Wa * + }(vlas n + v2aS~2))] Y22. (2.13h)
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J.M.F. Labastida et al. / Conformal supergravity
3. Projection to N - - 1 superspace
In this section we derive the N = 1 form of constraints (2.11) and (2.13), and the second supersymmetry, gauge and superconformal transformations of the N = 1 superfields. We apply the method described in [6, 7]. The N = 1 component superfields of an N = 2 superfield are obtained by acting with spinor covariant derivatives respect to the second 0 and then taking the second-O independent part. The 0 2~ (and 02a) dependent parts of K in (2.6) are used to algebraically gauge away components of the vielbien and the connections. In this Wess-Zumino gauge the N = 1 form of the constraints simplifies notably. As we will describe below the superconformal invariance of the theory allows us to use the 0 2~ (and /~2~) dependent parts of L to algebraically gauge away additional N = 1 superfields. It is in this superconformal Wess-Zumino gauge (plus another gauge fixing that we call chiral gauge and that we describe below) where we are able to solve the N = 1 form of the constraints in terms of the unconstrained prepotentials. The N = 1 superspace second supersymmetry transformations and the gauge transformations are generated by the 0 2a (and 02a) independent part of K. Superconformal transformations are generated by the 0 2~ (and t~2a) independent part of L. The 0 2~ (and 02~) independent part of any superfield quantity X is denoted by XI. For an operator, X-~ xMiOMJr X"M.,
(3.1)
X I - xMIiaM + X ~ l i g ~ ,
(3.2)
we have
where M~ represents any generator of tangent space transformations. We define V,~ I - VA + ~/A"02, + ~PA~O-'2~+ I'AbcYbc, ~7A = EAMOM Jr dOAvBMBv Jr eOA~tJMIj~+ FAY ,
(3.3a) (3.3b)
where V,~ - (Vx,,Vla, V,s) and the rest of the fields are N = 1 superfields. We now describe the fixing of the Wess-Zumino gauge for the spinor derivatives V2, and V 2a. Since from (2.5) 3V2.1 =
-i02,,K I + ...
(3.4)
we use the 0 2" (and 02s) parts of K to go to a gauge where v2.1
= a..o2. - 02.,
=
(3.5)
Using higher second derivative components of K we fix [V2.,V2~]I = 0 ,
(3.6)
J.M.F. Labastida et at / Conformal supergravity
859
so, in the Wess-Zumino gauge, after using (2.13a),
Vz,,VelJI =
½{ V2,, V2t~ )l = - iS22 IM, a +
iN, l~lY22-
(3.7)
In the same way we find, ~ 2 V 2 . I = i1{ V--2 a , V 2 , , } l = ½iV,~al •
(3.8)
Finally, we complete the projection of all the first components of the covariant derivatives. After using (3.3), (3.5), (3.7) and (3.8) we find, V 2 a ~ B I = [V2,~,VB}I "at*( - - ) B [ / " l l ( v a
"}-~)a"/V2y] "[-~ofiV2~l "}'-Fa¢cd Ycd) -
-
" 3'(~8(~ 1 oC.)oS=IMo 0 + /~5,a I Y22 ) +tq'n
+½iq, B*V.,I + (8.~(FB+ F~2)+OBJ)V2~I].
(3.9)
The commutators of this expression are evaluated using (2.13). We define the following N = 1 superfields:
w~- W~l, S- Snl, =
$121,
X=S221
,
A . , i = iG,~,i221 ,
wo,~ -= v::vB~I, N . , ~ - V2.NB~ I , s~ - v 2 . S n l ,
_--2
~ a --~ ~72aS12[ ,
(~a,/~fi-= iv2aG$/hll , ~k~ --- v2aS221 , s~ - ~ S l ~ l ,
Xa,BB ~ iv2aG~/~211 , ~:~ --
~S~l, A a , ~ = iV2aG~221 .
2% - v2S221 ,
(3.10)
860
J.M.F. Labastida et a L / Conformal supergravity
Higher components of VA can be determined extending this procedure. Using all the 02a (and 02a) dependent components of K to gauge away part of them one finds that the rest can be expressed in terms of the fields in (3.3) and components of the field strengths. The definitions made in (3.10) are enough for our analysis. Higher components can be expressed in terms of these ones. In fact, the ones defined in (3.10) are not independent. They satisfy constraints as a consequence of (2.9) and (2.11). The gauge transformations that remains in the Wess-Zumino gauge are the 0 2° (and 02~) independent parts of K,
iKI = - e , '02, - e-a'02e, - K'~bYas + iK,
(3.11)
where e° and K ~h are N = 1 superfields generating second supersymmetry and SU(2) transformations respectively, and K is the parameter of a general N = 1 gauge transformation,
K = i( KAVA + K flMB" + K a % a
+kY).
(3.12)
The supersymmetry generated by K is manifest since the theory is written in terms of N = 1 superfields. However, the other symmetries transform superfields into each other. This implies that in order to keep the Wess-Zumino gauge we have to add gauge restoring compensating transformations using the higher components of K. This also applies for the superconformal transformations of the N = 1 superfields. We now determine the N = 1 superspace geometry. As in (2.7), we introduce torsions and field strengths:
[ VA, VB} = TABCVc+ RAB~,SM~V + R A ~ M ~
+ FAB Y.
(3.13)
The content of (3.13) is determined by taking the 0 2° (and 02a) independent part of (2.9), and specializing to the part proportional to V~, M,# and Y. In addition, we obtain more constraints on the N = 1 superfields from the coefficient of V2vl and the coefficient of Y~bAt this point we make a further gauge fixing. We adopt the so-called chiral gauge in which q,f i = ~pfl = 0. Here we assume its existence and we will justify it later. Projecting the constraints (2.9) into N = 1 superspace in the chiral Wess-Zumino gauge we find:
+
+ %N;) + ! 2a~, ' ( a va, oa,c" xl ruro (3.14a) ' r B ) v ( o ~ v ) 0 1 ~'~r ,
J.M.F. Labastida et al. / Conformal supergravity
( V.,~9} = i(1+
861
.,,,i
+( ~jrj2 + j/JrJ')% - t ".*rl2,~. ,. - % *rd22)v,-I- [ - - i g
(a~
--
. ..a
.i'
o
+2+'%+,] M:
+
+ (aG-B - +:¢XaB - +B+X~+)Y,
(3.14b)
[W:a, Ua] = i#+:a+q+¢v~7"r+ + [iS OABa+ 8~o r2~ 12 + + . d r ~ 1 + 2,+~ox • --o ~ + ,L l ] vo ~ . o r l o,j --
+
1•
.
glX(a.B)a
2-
1_.
11
O --
E22V
y
1 e a - ±/-'n~" 2 a ~ & -7.X"s-qJo°Xo°a-2F~. + ~i4',8,,[~-G,,
+ R,~,,I3~aMav + R:s,~ +t M~f ,
~ 7 ( a ~ )12 __--f l i p 2 2
+ ,t,
~ ' ( a f f l ) 22 = "'<::B) "IF'12F '22
+
ypllp12
,L
yp11T,22
,L
+
(3.15a)
ypllrll
;,l,
(3.15b)
YA[
_
(3.15c)
_
i~b(:'@ °N. -
V'k~b.v i(1 + X)~k ~b°, +
2~b VFJ2 +
=
12--o X a]}Y
(3.14c)
V<~@B)Y-- :<~vp)rn'"or,roy+ 8< v r ~ + 2 @~. . vr12 :~), 4
]
(3.15d) ,
(3.15e)
Wa/-'/~ 1 + W k [ 11 = i ( 1 + a)aBv'~. a a r l oa l t + ~k -tF'll ~ r'121a a--/'111-,12j_/) "1- ~t
- , ~ :4:a ~- - ' rda gTaFl} 2 + --~7]~F~12=
i(1 + X )
flT'.IIP II
~fl "B
+
tb / ~ F 2 2 F II
-a
-a
~5' '
(3.15f)
afl'°6r12106 + lapllr221]~ -- FatllF~22 + 2E.12r12B *a
3v~ B ~ rl* l r~ 12
F12 - A ~ a , + , tb~ -./~~ F22 *~
(3.15g)
J. M.F. Labastida et al. / Conformal supergravity
862
22R y v'..% ' - v'~+.~ "= i~/l~P,~a~b,,~r + qJ,,a°I'~lq%~ + 2iC~l~X~a + ~F ' ,,a"t~ 12
• .a- ,/, a,I, Y F '11
(3.15h)
+ 2iq~.aXOaq%" ,
FOz2%a+++°- i%'q~.a+q~,+° - (1 + X)¢aV°N.,
+(1 + x)~:.+~? + (1 - x ) ~ d z + +odr~ ~ - ~.oX3a ~ - C.a~a~S, ~Ta&F~ 1
- v ~ r 11~ -_ Fa21p11
11
(3.15i)
21
2t 11
½iG.,l~a + C~t~Flals
3'uOal a 2 : ' ~r n + iGaaF/~l _ iAaaF/~l _ 3'ABa 1 8 ' V r12.
+ t.4oa[, 2 . 11 _
+ 4 i L , r ~ 2 - 4i(17.~a + , . 02 o,#a) + 2i( v'a x.a + q~zP2p,.a) + VaN~a + q~a°~rb,.,
+ co~(g',wd + q~?w~d) 1
--
ll
11
-,o:(~c..x-a
11 r;i1 ) +~o2[-AB~+r~ 22 rX11 + , -, . /~r;~]
+~/Oo[_2iC.oY,aF~l - C.o(3a, X _ - - ~w.)r¢ , 11 11 11 + N.oF ~11 - 2i2o,.a + F'.sF; ],
vo.r~ ~- v~r2 = F a l 2 p
12 q- ,L
?'B
* a&*'r
ypllp12
q.. ,I,
;,I,
-i,p 2 2 p 1 2
"i',L Y I ~'12
12
-t-
(3.15j)
12
__
,L
Y~[
p12
-+ ~ i ( ~ . a ~ - v.&~) + 2;v~&~- ~(r: 11 x~. + r:22 x.~)
+ 2i(F~X.a + Fff2X.a)- ~iq~¢V(G.,va- A . , , a ) - 2iqq~,~AV,~,a + -1~ c . ~-[- v ~ s + % * s , - 2 ( r 212s + r s
22
z) +z.]
- ~,k o . (-% 1 : + G ~. ~ . + r l~xs - rd22x + M ) 1
+
+
--
"i' ..~
4;,L v p l Z V
5'~'/~
".
"'va
½Na,/~"
±
-
d
9
--
½~VuaN,-
pllp22
+
~iX.
22
,
11
Bs
~,_ 1_,1, :,,~,.~,~r
~i~#v(W.~v a + ~o o-xo,,,) +
~z ., ~
y
r2l l ( G ,
-
Ay&),
(3.15k)
J.M.F. Labastida et al. / Conformal supergravity
vSd
~ - v ~ r ~ = ,11-,21/-22 ~ ~
~
~/-,22 ~ /-'12 _j _ co~r.~22s + ,•A S ~
+ i(A.a - aJrd
863 22
2 - 2i( W#X.a + ~pBo)(o,.a)
+ 4iX~aFJ 2 + i~b.a, aS + 2iqJo~X°aF22
+~JO.(3a*Z - W ? ) F ¢ 2 + 2 i q J f ~ , . a
4"
~l ^
-- ~fNa,. r + +B.W,, a~' - ½i+f(Vr.Gva + +.°Go, ra ) y
22
4ij ,
prlly
++~ N.,r.~ + i~.avO/B°Nov + 3'v/~ - o "'oa 4" p 22-, 0/-11/',22 + s,q,~ r ; x , ~ + J~'a *.a-. ,
(3.151)
where G~a = 2(G,a - A,a),
(3.16a)
A,a -= 2(G,a + A~a),
(3.16b)
G~,t~# = 2( G-,aB - A~,#~ ),
(3.16c)
A~,a~ = 2(6~,a~ + A-,aB),
(3.16d)
(1 ++_X ) . f 8
=- 3f3a B _+~f~pa 8.
(3.16e)
To complete the projection of the constraints we have to take into account also the ones in (2.11). We consider only the N = 1 constraints generated by taking the 0 2~ (and 02a) independent part of (2.11). We find, ~aW, a = 0,
(3.17a)
Xo = 0,
(3.17b)
2o + (r2' + rl=x + ~ v . x ) = o ,
(3.17c)
S. + 2 ( 7 . 2 + r n s - r227,)- 24~f(rJ~s + FJ2?~ + ½va?, ) = 0 ,
(3.17d)
+24,.'% a (r~n + r ~ x + ½VaX) 2 ( q 2 S + F d = 2 ) = 0 ,
(3.17e)
J.M.F. Labastida et al. / Conformal supergraoity
864
N., tJv - Z(I~Cy). = O, 0
(3.17 0
1
__
x7.N~, + ~ No,~, - ~S(~C.~ - O,
(3.17g)
2t'~_ 12 _ W(.Xa)# + CJX~,a)~ +z_ 1 / ~ 2(~ ~a)a. _ 2F(~Xa)~O,
(3.17h)
x~.,a)~ = 0, X(.,O)
#
(3.17i)
1 la . - ~V'(~Ga) # - ±2d/(~ v Gy,a)B - 2F(~Xl3)a
-
2Y(.
22_ X~)B - 0 ,
(3.17j) (3.17k)
iS a + graxa a + ~at~X#" aa + tp22~a2.~ ~ a-4- 2F~ X . = O, i(V~A + ~kaaA# + 2(F21,~
+
r i g a ) ) + x ,o~ = o,
i(~r~S +@a~S~- 2(ra12S + r 2 2 z ) -
(3.171) (3.17m)
2za) + ½v~o~
+ 2!,f,~ar:_ v '-'#,,,a - 2F~IX~ - 2 F2:X a a a - X , a~~ , = 0, 11
i
22
(3.17n)
a
- 2-~ ~ a - 2F~ X~ = O,
(3.17o)
VaN~l~ + ~JJJN[~,~1~+ ~tA(~,l~)~ ~" = 0,
(3.17p)
N~,~a - ½i( Vr(~Aa)~ + $ J A r , a ) ~ ) = 0,
(3.17q)
3(V/~W~~ + ~¢~*W,f) + ~'~Z + , fZ/~ + -
-,r'ao
n _2r~
x
~)
3 a +i(G~,~a + iA~, ~) = 0 ,
(3.17r)
3WB~ + 2( e i'=s + Fd2Z) - Z a + 4i(r2~x= + r : 22-,, x ~ - x ° o~) -- -v & S - q J & % q" l(• IT.G - & q-- @aflGfl, at& - 32\[ r~r,~ - & + ~ k f A o ~ ) ) = 0 .
(3.17s)
We now compute the symmetry transformations of the N = 1 superfields. The second supersymmetry and SU(2) gauge transformations of the N = 1 covariant derivatives, and ~b~t~ and F; a are obtained by evaluating the 0 2~ (and 02a)
865
J.M.F. Labastida et al. / Conformal supergravity
independent part of the N -- 2 transformation (2.5). Similarly for the N = 1 superfields defined in (3.10) one has, for example, for W,a:
[iK, W,~,]I,
(3.18)
etc. For the computation of the superconformal transformations of the N = 1 fields we take the 0 2" (and 02a) independent part of (2.12). In computing all these transformations we have to take into account that to restore the Wess-Zumino gauge we have to perform compensating transformations using the higher components of K. These components are easily obtained by the requirement that the Wess-Zumino gauge is preserved under all transformations, e.g., from (3.5):
V2,~iKI = 6LW2,1 + iKV2,d.
(3.19)
Then, using this relation and (3.11) we find:
8~7A1 = 8LVAI + [iK, vA]I = ~LVA I -- IpA'SC~LV2,81 -- (EflV2,8 q- ~ F 2 / } "~- g c d Y c d ) V A I
-'}-~:TA(gflV2/3 -Jr-~ / ~ 2/} q_ gcdYcd) ] ..}_I~Ay(EBV2 fl .~_ ~ "q-~,3'(E/~V2/~ "~ ~eF 2/~ "~ g c d r c d ) V 2 , ]
2/~ q_ gcdYcd)V2y] (3.20)
-~- l~dycdgefy_e f .
On the other hand, from (3.3a) we have,
8Val =SVA + (&ka~) 02, + (&Pae)a2 + (8FAbc)Ybc.
(3.21)
Computing (3.20) by using (2.12b) and (3.9) and comparing to (3.21) we find, BY,, = ½Lv,~ + 2(wilL + 4,~vLv)M,~~ - 2q~.13LvMis ~'- ½(F'.L) Y
-elJ [ i( 8(13°C.)vX - CI~.NvO) MoV - iC~,~W~,OM~f- r2ilv/3] -~[-8(.°Xv)¢Mo +
v + 89(°X.))M. "~+ X . B Y - F:2F'#] +
+
+ Kn
:)
(3.22a)
" 8+,~B = W.eB+ ey ( F 2126vB + F~11+v13) + i~x+-x+v~B + 2 ~p,¢Kx2 + 6 j~p v/~K11 + 6 J K 22,
(3.22b)
J. M, F. Labastida et al. / Conformal supergravity
866
1 11 ~ r ~ ' = ~LC~ - 2 G
+ Eft( iC.B~ + FallF/} 1 ) + ~/~( Fd2F~ 1 + i~.BFJ~ -- h./~)
+ (V. + 2r]2) K 1' + ~b,,l~r~'K n - f'~1K12, 8~ 2 =
(3.22c)
½LFa12 + VaL + 24,,,BLB
+~( ~qoS- iNo~+ r11r~~) + ~(r~r~2 + i+o~r~) + ( V',, + F~2)K12 + ~ , ? F ~ 2 K n - F ~ X K 2 2 + Fa2K n ,
(3.22d)
aF~22 = ½Lr~ 2 - 2~bJ(~7/~L + +~YLv)
+~,(~q.s + r~rj2 + 2i,:N~) +~(r~2ry + ;,o, rJJ) + ( V , - 2F~2) K 22 + 3Fd2K 12 + +,/~Fd2K n ,
(3.22e)
where we have used the following definitions for the components of L: L=LI,
(3.23a)
L,, - V2.LI ,
(3.23b)
1 . ~72.L1 M - ~V2
,
(3.23C)
L,,,~ - ½[ ~ 2a, V2.] L I,
(3.23d)
m . ~ ~l~t''l v 2 a v~ 2 ~
(3.23e)
v
2aLl,
M ' = x~V z V ~ 2 & -V- 2 aV2,LI.
(3.23f)
Symmetry transformations of the N = 1 superfields defined in (3.10) are determined as in (3.18) plus the 0 2" (and 02~) independent parts of (2.12c-f). We reproduce here some of these transformations: 6 W,#~ = L W.I ~- e v Wv, ,~/~,
(3.24a)
-2i~,"13~7.L1~- i( V',,d/"1~+ FI~"~p~ #) L B + i~"Y~,wM - e " S . - ~*S, + 2(K'2S + K22X),
(3.24b)
J.M.F. Labastida et aL / Conformal supergravity 8 ~ = L~, + i~7~V'~L
-
iI'lllF'at
-iF~.2L " - i( V.+ "a + -e"~, - ~a
-
867
i~aV~L,
F.x~"~)La
+ (K22X + K l l S ) ,
8X = L ~ + 2 i M - ~ah a - 2(KXl~ + K l E X ) ,
(3.24c) (3.24d)
6N,~ a = LN,~ a + iV(,,L#) + i F ~ ( V'~)L + ~b#)VLv) + iF~ZLa) -i~b(,~a)M-
eVNv, ,~a - ~'N~, ,~a '
(3.24e)
8G,,a = LG,,a - 2L,~a - [V,,,-Va] L + 2q~,fq~a~i,~/~ + [ ( F a + ~ a ) L a + +.aF~I( V a L + +aVLv)
+ 2 oaF L a -
F L- h.c.] + 4(KllX~a + K22X~a),
(3.24f)
- e a X a , , a - ~/3X/3,~ a - 7I K 1 2 ~~ a + 2K12X~a,
(3.24g)
-eaGa,,a-~BGB,,a
-[(v.+~) Za + 2+~avoZ~- +oar2~(vat, + +a'L,) - F21L,i - 4',~aF]2L~ + I22( Va L + q',i%) - h.c.] -
eaAa, ,~a - i~AB, ,~,~,
(3.24h)
where use have been made of constraint (3.17a). The parameter of the second supersymmetry transformations is constrained due to the fact that the chiral gauge has been chosen. This constraint is easily computed by considering the second supersymmetry transformation of +,~. We find, 17~ a + (Fx,~81~~ - F22q,Ba - i~pJ+aB a ) ~ = o.
(3.25)
Integrability of this equation is necessary for the consistency of the chiral gauge. We will discuss this question in sect. 5.
J.M.F. Labastida et aL / Conformal supergravity
868
4. Supereonformal gauge So far we have obtained N = 2 superconformal supergravity in N = 1 superspace with full N = 2 superconformal symmetry. The theory is invariant under all the N = 1 superconformal parameters defined in (3.23). Most of these parameters can be used to gauge away algebraically some of the N = 1 superfields of the theory. The residual superconformal symmetry is the N = 1 one and it is represented by L (see (3.23a)). In this form, the theory contains the following symmetries: N = 1 local supersymmetry (manifest), N = 1 local superconformal (L), local second supersymmetry (%, which satisfies constraint (3.25)), and local SU(2) (K"b). This additional gauge fixing implies new compensating transformations to restore the Wess-Zumino gauge. This is carried out in this case by obtaining the expression of the higher components of L in terms of L, % and K ~b. Now we describe the gauge fixing. From the variation of F2 ~ (3.22c) we observe that it can be gauged away by using the L-component: L~. Then, we fix the gauge by demanding F l 1 = 0,
(4.1)
and so we obtain, 2~ t ~ ,/3 +i~b,~/3FJ~) + ~(~7~ + 2F~12 ) K L,,=-½ihe,~+ ±zMr'z2r1l
11.
(4.2)
Similarly, from (3.24d) we observe that h can be gauged away using the next L-component: M. We have, ?,=0,
(4.3)
M = _ 2l(e a. -a Xa + 2 K U ~ ) .
(4.4)
The next component, L~a, appears in the variations of G ~ and A~. We choose to gauge away A~a. We have, A~a = 0,
(4.5)
L~a= ½(1 + X);2/3~{[~Y/3, ~t~] L - [ ( ~ a ~ / 3 ) L / 3 + +. /3F~I i (v/3L + ~bfLv)
+ 2~/3~L/3 - F22L/3 -F~t2--~TaL- h.c.] + eYAr./3p + ~A,,/3B}.
(4.6)
It is now clear by looking at (2.12c) and the definition of Xa in (3.10) that this field can be gauged away algebraically using Ma. We then have,
(4.7)
J.M.F. Labastida et al. / Conformal supergravity
869
The expression for M, in terms of L, G and K ab can be obtained by computing the full transformation of 2,~. Finally, the last L-component M ' can be used to gauge away some of the higher field components that we have not defined explicitly. In this gauge, constraints (3.14), (3.15) and (3.17) simplify,
{vo, v~} = ,2
+
+i[6(,,rc#>oS-~b(,,v(Sv'~C#) o + 8 # , ' C v o ) ~ . ] M , ° ,
(4.8a)
{ Vr~, ~k} = i(1 + X ) , # Vr,e
+ [ _ 1j~ ( o ~
. u-~o ,,,>~_ G<°x,>~ + +~.) a<° x.+] Mo,
+ [ ½6k<°G:+,- ~/k<~X.+)- J/#~k
(4.8b)
+ (1G,,B - +,~#2#~ - ~bk+X,~+)Y,
[vo~, v~] = i+o2+/v,+
+[8. o
12
11 21~k#,X- ° a + ~k#a r2a] Vo
+ [ +o:ry + (c+s + __ 1 "
q_
gtX(~,,B)e, t"
< o
2 :p22V
- Z2 '•C , # X , , va - ~a+X#+ + ~i~b#'G(r,,)a
+ ~i+eo[~o°°~- VoyO~- G oxo - o ~-2r.' ,~-o x ~]}r + R :a. fl,'~M,~' + R :a, +8-p<~M,+, lT(a@fl)Y
=
,R
Vp22
"(~' *B}
+
"l,t,
~,p12
" ~ ' ( " *fl) '
N~# = 0,
~" l*p 1( a2 p 2*#)2
(4.9a) (4.98)
~(,,FJ~ = - ig/<~,VNl~)v - i~<,#3)Z, ~7'
(4.8c)
+ t. ~b<,v~b#)°N,, - i~b<,rG)vS,
(4.9c) (4.9d)
870
J . M . F . L a b a s t i d a et al. /
W ~ f = i(1 + X ) . ¢ --
~7aFJI i(1 =
v,~FJ2 +
-12 = W~q
i(1
.06
Conformal supergravity
}oo y + 2 + / r f
+ + . % f r J ~,
(4.9e)
~ a BF£22F~11
(4.9f)
. 0 6 1r l0l6 -~+ Av ,} Ot]~
3Fj11 C 12 +
-t" ~ V ~) ~ . 0 6 p~o~+ 12
2F~12 F~12 + ~.r, Cr.nr12 + ~a ~F~22F~12 ,
(4.9g)
P',~,i+f - V'#~,a v = i+f~,~a~'~,,-i, v + 2 i C , FXVa 4- F 22 v + ~')p12"t'aa,fB"y
+,,,"t"B..r, ~r"-+./pjz_ "t" o a & v.~
~ = ry+.~%
x714//3, G s + 2 i * . ¢ X O a * o v,
(4.9h)
~ - i + / ~ o ~ % ~ ~ - (1 + X ) B ~ N . ~
+ (1 + X)Ba,~ ~--~ W+ +
(1 - x ) ~ j ~ + ~ ° ) r 7 (4.9i)
- C,,B+,~ S, v . ~ F J ' - v~F-J~ = + ½iG.,Ba + C~Bfla 1S - iA.fl'J ~ - 5"Ba'.':v ,.~2
- ~,( vox~, + +. x,.~,) + 6
+¢~,.,
j,
1
. /~ 11 + 4iL~FJ' + ¢.s°[F~22 F;11 + ,+, F~o]
+,~o[ - c . o ( 6 j x - ~j)F~ ~- 2i~o..a ] , vo.rJ' - v~rJ1-- rLr~"12 +
(4.9j)
vda&'L* ~p 21 2 1±x]*~ -t- *~afiiJ'Y,t,~yflyp12 J - d , ~__ , . ) ~flY p12±a&.7 pll
+2,Gt~aF'~12 + C~¢SF;i2 + , +' ¢ . ( 2 X--o a/';12 + [ 8 2 Z •
w.*]r.~ 2)
~. _ 52 i F :~: ~ ~a + 2iFff22.a - 3,+~ + Stw~G~a 1. v (O,~,va-A,~.vs)
o 11 22
4 , j , TF12"y
_ ±.,, 2 ~ B v.,, "rB.,~ "'9,
Va
~,~, ~ W 2.
5Ne,,t
~ fla + 1
+ ~iqJf( w,~Xva + t),/'X,, va) ,
'
6~ W~,~ ) ":/
(4.9k)
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J. M. F. Labastida et al. / Conformal supergravity
V,,,fff 2 - V'/3F,~za2 = ap2~p22o. ,,a*B + C,,/3U,~22S+ i ( a , a -
G.a)F/2-Zi(vt~X.a+tk¢°do,as)
+ 4 i X , aF~ 2 + i#.e,,l~S + q,~so( - Fff2F22 - i+¢° F 22)
+ ~4,,,~ [ V,~S + q,,]'S+ - 2(F22S +/'222;) + 2a] + 2iq,¢~X°aF 22
1' g ++l,o( ~ d z - w d )r? ~ - ~'q'l, ( voG~ + G°G, ~)
(4.91)
+ 4_i, L 0 p l l y ..J- ~i,I, # F ' 2 2 y ,L p p l l r '22 3"'Ffl * a .-Xp&-- 3,~t.,fl *-a "xO&"{- 5vfl *~&J-,o ,
v.s-
V'~W~ = O,
(4.10a)
)% = O,
(4.10b)
N, = 0,
(4.10c)
X7~,~ = O,
(4.10d)
2(rJ~s + r:.~y.)=o,
(4.10e)
N,,, ~, = 0,
(4.10f)
S,~ -- 0,
(4.10g)
1 F22g%"
") F ' 1 2 V
(4.10h)
~
X(:,~>k = 0, X ( a , f l ) fi - - ~l 7 ( a a f l )
(4.10i)
~ - - 12~/b(=G.y,fl)fl ~, . - - 2 - r ( ,2, 2X-e- ~ a = 0 ,
(4.10j)
v<:-~t~,a + +<,,'-x,.~> e. 0 2r::.g~,a , ++ ~G(o.~,e 1 = 2G' x ~
(4.10k) 0,
2iFlalN + X -,~ ,,,a = 0,
,(+:s + !,l,aflg7
+ 2v
"B, ~
2r2:s__
(4.101) (4.10m)
lvo<
2r; 22 x- -~ -- Xa, ,~a = o ,
(4.10n)
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J.M.F. Labastidaet al. / Conformalsupergravity
2i(Wa~ + ~ba/~Z/)+ F]IS) + ½G~,~a + 17"-~a + ~b"BXo,~a- 2F22-~"4 = 0, (4.10o)
~ba/~NK ~ + ½iA(~,/~)a = 0,
(4.10p)
" Y Na,.~ - l2,qq. Av,B) ~ = 0
(4.10q)
3(~Wd + ~,~w,~) + G z + ~ ' z , + r~'s + i( G."~ + ~A."~) + 4i( V'"X.a + ~k"I~xI~,.~- 2F22X"a) =O, 3WBaB - ~aS - tka~ + 2/'128 - Z,i +
4i(F2a2%-
(4.10r)
x L..)
+ i(W.G" a + tk./~G, "a - -32~b.BAt<"a)= 0.
(4.10s)
Finally, we give the symmetry transformations of the N = 1 covariant derivatives and superfields in the superconformal chiral Wess-Zumino gauge:
3V'. = ½Lv~ + 2 ( V B L ) M . a - ½(V.L)Y -eO[i3ffC,~,vX - M. ~ - iCt3,~M~ ~'] -~[-3(.°X~,)~Mo v + ~/;(°X.~,)Mo* + X . ~ Y - F22~
+ i~b,'~/~V,t~ + (( Vv + 2F12)Kn)( ~kI~VM~--~J~MI~v) + (K123~ v + Kn@~v) Wv,
~ a fl = ~Taefl+ eflF12 + i~aY~y~ fl + 2 ~ J K 12 + tkJq, v/~Kn + ($JK 22 ,
(4.11a) (4.11b)
~r 1~= ~Lr2 + vo~ + ~oZ + ~(rd~r~ + ~o~(r~ + tyrO' + + , r ~ ) )
+(vo + r2=)~ ~ + ~o~(v. + r ~ ) r 11+ +o~r~r" + rd=r~', (4.nc) a r f = ½Lrd ~ - 2<~( V,L + +e~L,) + i~.S
+ ~(,<~r~,:,- +o%,(rSr~l+ ;
(4.11d)
X I
+
%
~~,~.I~
II t~
o.<11
I
-I-
~.~
~~' ~ .
~.
+
~,.
X
~
,
#
°+
"~
~
~'~.
+
I
~
o.~'
#
+
+
o~
•
•
~.~
~.~'~
+
I
~
o~
+
<
X
I
r.
'%'
"~"
"~
~ - ~
+
~-~
~.
~.
#
+ ~,
~
+ +
~b~ =
o.~l
I
I
b
b
~
e~
°
-F
i ~
+
%
b~
"1~
+
I
b~
II
I
~
+ X
+
+
I
II
~, o~ ~I
I ~, •
I
+ ~
il
il
o.~l
'
,~
+
+ i i
i
J.M.F. I.abastida et al. / Conformal supergravity
874
5. Solution to the constraints
In this section we solve constraints (4.8), (4.9) and (4.10) in terms of unconstrained matter prepotentials and N = 1 conformal supergravity, whose solution is known. The method is to make redefinitions of our covariant derivatives until the commutator constraints (4.8a-c) take the form of those of N = 1 conformal supergravity [4]:
( d~, d# } = - 2isM~/~,
(5.1a)
{ d~, da } = idea,
(5.1b)
=
(5.1c
-
where d~, da and d~a are the covariant derivatives of N = 1 conformal supergravity, and where s satisfies:
d~s = 0.
(5.2)
The first step is to postulate a rather general relation between the covariant derivative V~ and the N = 1 covariant derivative d~:
~7,= d, + Q,o o Mop + Q~ a Mo + Q~Y.
(5.3)
Note that we do not include a vielbein redefinition. This is because the commutator (4.8a) does not contain already vector and dotted spinor torsion terms, and this property would be spoiled by a vielbein redefinition. Next, we choose the Q's in (5.3) in such a way that when the ~7~ and ~7a are replaced in terms of the d~'s and da's the constraints (4.9a-c) take the same form as those in (5.1). Then we can identify some of our fields in terms of the pure N = 1 conformal supergravity fields. The rest of the fields represent matter fields and must be solved for in terms of prepotentials using the rest of eq. (4.8), and eq. (4.9-10). To determine the Q's, we note that the N = 1 conformal supergravity constraints can be expressed succinctly in the form:
d~a = - i ( d ~ , d a } , ~,~=~,~ffo)=0, ~v
(5.4a) (5.4b)
= 0,
(5.4c)
~¢v* = ~ ¢ * = 0,
(5.4d)
where the Y ' s are the torsions for pure N = 1 conformal supergravity. The form of
J.M.F. Labastida et al. / Conformal supergravitv
875
the constraints (5.1) is then completely fixed by Bianchi identities. This implies that if we can arrange the Q's such that (5.4) are satisfied, then (5.1) will be automatically satisfied. Next, we will show how this method will give a linear set of equations for the Q's. As we explained before, eq. (5.4d) are automatically satisfied, given that, in the chiral gauge the constraints (4.8a-c) do not have such torsions and that our redefinitions (5.3) are chosen not to produce them. Eq. (5.4c) implies that the Q's satisfy:
= (rS-
(5.5)
Eq. (5.4a) is solved trivially by a redefinition of the vector derivative. Finally, eq. (5.4b) implies: Q.
=
- ±4~~ . ~ . . ~ ~ ,
~) , Q,k# = 114o 4 = ' o /x(&
(5.6a) (5.6b)
where H¢."a~' = (1 + X).ao~q~t~°+oaO(1 + X ) 2 " ~ __ Fal2(~&-i, b
22
(VB(1 + X).~a°~)(1 + X ) 2 "~
~,
These equations can be solved for the Q's once the objects that appear in them are known in terms of unconstrained prepotentials. Progress is made by noting that the Y field strengths in both (5.1a) and (4.8a) are zero. This implies, using (5.5), that:
d(,~Qt3) = 0,
(5.8)
Q~ = d~2,
(5.9)
giving
where ~2 is an unconstrained prepotential. Next, from (4.9a), we find, using (5.5): d(~bt~)v - 2Q(,~t~)v = 3(,,~' [Yff~ + Q~)o°tk, ° + F ~ x - Q ~ ) x ] ,
(5.10)
where we have used the notation x = qJ~'. By symmetrizing and anti-symmetrizing in fl and y in eq. (5.10), we find: d(~q~t~v) - 2Q(.+~y) = O,
(5.11a) (5.11b)
J.M.F. Labastida et al. / Conformal supergravity
876
F r o m eq. (5.11a) we find the solution of ~b~¢ in terms of two new unconstrained prepotentials q~a and ~: ~,~ = ( d e - 2Q~)ff¢ + C . ¢ ~ .
(5.12)
O n the other hand, from eq. (4.9e) we obtain:
(5.13) Eqs. (5.5), (6), (9), (llb), (12), (13) are a closed set of linear equations which allow us to solve for Q~, Q~aV, Q , ~ , ~ B , ~b~v, F~2, F~ 2 in terms of the unconstrained prepotentials ~2, ~, and q~a, and the N = 1 conformal supergravity covariant derivatives of these prepotentials. Eqs. (5.9) and (5.12) are two of these solutions. The additional expressions are long and not enlightening and we do not include them here. The rest of the constraints (4.8)-(4.10) give expressions for the remaining fields in terms of the unconstrained prepotentials I2, ~, and q'B, and N = 1 conformal supergravity objects. The process is tedious but straightforward (all the equations are algebraic and linear). We will give a representative example by obtaining the explicit expression of S and ~. Comparing the M~B terms of (4.8a) and (5.1) we find: S = s - ~i.ga - Z x ,
(5.14)
£p= d,~Qff~B + Q,~Qff~/3+ Q,ffQ ,~¢.
(5.15)
where
In expression (5.14), s is an N = 1 conformal supergravity object, and **.coand x = qJ~ are already solved. We now need another equation which involves 2J, and this is provided by eq. (4.9i): Z = 3F1 - S.Ze~2 ,
(5.16)
where ,~F1 = ~ ( 1 + X)s~I/~aK/~aa ~ ,
(5.17a)
~2 =Jg(1 +
,
(5.17b)
K¢, ~a~ = V'~b~s~ - Fff2+,a*~p,~ - ~,a°FJ 2 + i+av~p,s+~kv,o,
(5.17c)
-,~o
-~
Jg'= (Tr[(1 + X ) - I ( 1 - X)]) -1 ~,
(5.17d)
JU1 and OF2 are already solved in terms of prepotentials. Eqs. (5.14) and (5.16)
J.M.F. Labastida et al. / Conformalsupergravity
877
are a system of two linear equations with two unknowns, which is trivially solved. We find for S:
S=(s-
½iAq- x , ~ l ) / ( 1 + x,gU2).
(5.18)
It is easy to check, using constraints (4.8)-(4.10), that for all the fields one can find an appropriate system of linear equations to solve for them, similarly to the example above. Actually, one finds more equations than are necessary to solve for the unknown fields. The constraints which do not solve for unknown fields should be automatically satisfied, without giving constraints on the prepotentials. We have checked this for all equations at the linearized level, and some at the full level, and we found no such constraints on the prepotentials. Finally, we comment on the existence of the chiral gauge. We note that all of the constraints are consistent with its existence, and that the constraint imposed on the second supersymmetry parameter e~ is integrable at the linearized level. To see this, note that at the linearized level, constraint (3.25) becomes, (d~ + 2/-~2)~/~=0.
(5.19)
To solve for e~ we observe first that from (5.1a) and (5.13b) we have, { d,~ + 2/-~ z, d/~ + 2F/}2 } = - 2isM~13,
(5.20)
so the solution to (5.19) is
~ = [½(d" + 2/'12~)(d. + 2/"12,~) +
is]~ ~ ,
(5.21)
where q~ is an arbitrary superfield. It would be desirable to prove the integrability at the full level, but we have not been able to do this. To summarize this section, we have solved all the fields in terms of the N = 1 unconstrained prepotentials I2, ~, and q~, and pure N = 1 conformal supergravity fields. 6. Conclusions In this paper, we have studied N = 2 conformal supergravity in N = 1 superspace. In particular we have treated the following two aspects: first we have applied the method of the projection of extended supersymmetric theories into N = 1 superspace; second, we have solved the resulting constraints in terms of unconstrained prepotentials. We have found that for the case of conformal extended theories, besides the usual Wess-Zumino gauge fixing, one can perform an additional gauge fixing by using some of the N = 1 superspace components of the superconformal transformation
878
J.M.F. Labastida et al. / Conformal supergravity
parameter L. This provides a formulation of the theory with a minimum set of matter superfields. Also we have been working in the already familiar chiral gauge [6, 7]. We argued that in this case this gauge facilitates the expression of N = 2 conformal supergravity as N = 1 conformal supergravity coupled to matter superfields. The constraints satisfied by these matter fields have been solved in terms of N = 1 conformal supergravity plus the matter prepotentials ~ , (b and ~2. This work provides the adequate framework in N = 1 superspace to construct the different formulationss of N = 2 Poincar6 supergravity by using compensator techniques. The first step will consists of the coupling of the vector multiplet and the appropriate choice of gauge to obtain the so-called minimal fieM representation of N = 2 Poincar6 supergravity, which is not superscale invariant. At that stage, one will have still the SU(2) gauge invariance in the theory which is not part of the symmetries of N = 2 Poincar6 supergravity. One will need further compensating fields to compensate for this gauge invariance. This will require the coupling of N = 2 matter supermultiplets. Different formulations of Poincar6 supergravity will show up after the SU(2) gauge symmetry is then compensated. We thank Martin Ro~ek for many useful and stimulating conversations. One of us (J.M.F.L.) would like to thank the Institute for Theoretical Physics at SUNY at Stony Brook for support during the summer of 1985, where part of this work was carried out.
Appendix In this appendix we summarize our conventions. Our notation is based in [4]. We denote SL(2, C) indices by lower case Greek letters, simple superspace indices by upper case Latin letters A = (a, &, a&), and N = 2 superspace indices by underlined Latin letters A = (a, a_',a&), a_ = aa, where a is an SU(2) isospin index. To raise and lower SL(2, C) indices, we use the tensor,
° -i) i
0 "
(A.1)
For SU(2) indices, we use 0 Cab = c a b = ( --1
Symmetrization of isospin indices is defined to mean tracelessness: X(aY b) =_ 2X, Y b - 8abXc Yc"
(A.3)
According to our conventions, the hermitian conjugation of an N = 1 superspace equation is obtained from the following rule: change dotted (undotted) indices into
J. M.F. Labastida et al. / Conformal supergravity
879
undotted (dotted) without changing the order of indices (except for vector index a& which goes into a&), add a minus sign for each vector index and for each i, change the upper SU(2) indices of Fn (F j2) into F j2 (FJX), and add a minus sign for each F 12. For example, the hermitian conjugate of (5.13) is,
i~b~/~= (1 + X)2°~( V~q:f - 2~fF~ 2 - q:f+oBF¢I).
(A.4)
The action of our SU(2) generators on superfields is defined in (2.3). Accordingly the commutation relations for the generators are: (A.5) Y-charges of the N = 2 superfields are easily derived by using (2.4) and the commutation relations (2.9). One finds, [Y, W~/~] = - 2W~e,
[Y, sob] = 2sob, [ Y, N~a ] = 2 N~/3 ,
=o.
(A.6)
Then, using (3.3) and (3.10) (or (3.15)) one can easily compute the Y-charges of the N = 1 superfields. We list here some of them: [Y, X] = 2X,
[ r , v ; ] = v,o, [Y, ~ J ] = 0 ,
[Y,~] =22,,
[Y, 4',f] = 24',f,
[ Y, N~I~] = 2 N~I~ ,
[r,r:l=r:,
[ Y. Wo~] = - 2 w o , .
[Y, SI = 2 s ,
It. oo~] =o.
(A.7)
where ~ represents any object with only a vector index. To compute the Y-charges of the prepotentials and the parameter of the second supersymmetry transformation one needs to use (2.6), (5.6) and (5.7). We find, [Y, q~] = - q ~ ,
[Y, ~2] = 0 ,
[Y,*I =0,
[Y,d
=
-%.
(A.8)
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J. M.F. Labastida et al. / Conformal supergravity
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