- Email: [email protected]
Superspace bianchi identities and the supercovariant derivative
ANNALS
OF PHYSICS
Superspace
122, 443-462 (1979)
Bianchi
Identities
and the Supercovariant
Derivative*
M. BROWN Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 AND
S. JAMESGATES, JR.+ Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138 Received January 31, 1979
We analyze the theory of supergravity, with minimal component fields, in spacetimes of two, three, and four dimensions. The analyses begins by investigating the superspace Bianchi identities. We show that in two and three dimensions, the supergravity multiplet consists of the gravition, gravitino, and a single auxiliary field. In four dimensions, we analyze the Bianchi identities in order to investigate the structure of the vielbein and connection superfields. Utilizing this analysis, we explicitly construct these superfields to quadratic order in 8 and give a prescription for determining them completely.
I. INTR~DUC~ON
A complete superspace formulation of the theory of supergravity now exists [I, 2,3]. The theory which hasemerged is extremely elegant showing a very remarkable interplay between two different superspacegroups, the “chiral” gauge group and the “vector” gauge group [2, 4, 51. By starting with an inverse vielbein superfield, EAM, and a connection superfield, WiA, (transforming only under the “vector” gauge group) which are constrained it has been shown [I, 21 that the solutions to these constraints transform in an entirely different manner. In fact, it is only by solving the constraints that we find the existence of the “chiral” gauge group. For it is the gauge phasefactor WM and the density superfields (@ or Y) which transform as somerepresentation of the “chiral” gauge group [2]. As an alternative to solving the constraints, Wess and Zumino [3] have noted that the superspaceBianchi identities may also be used to gain knowledge about the * Research supported in part by the National + Junior Fellow, Harvard Society of Fellows.
Science Foundation
under Grant PHY77-22864.
443 OOO3-4916/79/110443-2O$OXOO/O Copyright @ 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
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structure of the vielbein and connection superfields. By working only with the vielbein and connection superfields, we are forced to consider only the “vector” gauge group. We were thus motivated to investigate the equivalence of these two very different approaches to understanding superspace geometry. This is the main purpose of this paper. As a secondary purpose we are striving to understand how one goes about deriving a theory of superspace geometry, for an arbitrary superspace, which has the minimal number of component fields. This is of crucial importance for extended supergravity theories. Since in an arbitrary superspace one is not guaranteed that it is always possible to solve a set of contraints in superfield form, we thought the Bianchi identities might serve as an alternative. This may be particulary true of a spacetime of ten dimensions (O(4) supergravity after dimensional reduction) where spinors exist that are simultaneously Weyl and Majorana. Thus, one is not even guaranteed the existence of UM [l] in this theory. In section two, we review the structure of supergravity in spacetimes of two and three dimensions [6]. We start by introducing the most general possible supercovariant derivative and specialize to a particular subclass. Within this subclass, we give the form of the spinorial supercovariant derivative in Wess-Zumino gauge [7]. We also present the superfield actions for pure supergravity in these spacetimes. In section three, we restrict these geometries even further to find theories which have the minimal number of component fields. In each of these superspaces it is shown, by analyzing the Bianchi identities, that the minimal supergravity multiplets consist of the graviton, gravitino, and a single auxiliary scalar field. In section four, we start with a very restrictive superspace geometry for ordinary supergravity. In this superspace which possesses a four dimensional spacetime we repeat the type of analysis which is carried out in section three. We show, in detail, how the Bianchi identities may be used to express all of torsion and curvature supertensors in terms of Taadand TaBa. We note the existence of a recussion formula which allows the evaluation of any number of spinorial derivatives of Tuodand T,,“. Finally, we make a brief comparison with the nonminimal theory [4,8]. In section five, we show how the information gained from the Bianchi identities may be combined with a previously derived procedure [6] for finding a generalized W-Z gauge to reconstruct the 0 expansion for the spinorial supercovariant derivative. We explicitly construct this operator to quadratic order and give a prescription for determining the remainder. We thus demonstrate precisely how the Bianchi identities determine the vielbein and connection superfields up to a supergauge transformation. This constitutes a proof that a complete analysis of the Bianchi identities is equivalent to solving the constraints on the vielbein and connection superfields as has been done previously [l, 21. We give a discussion and summary in the final section of the paper. We include three appendices which give our conventions and representations, the equations of motion for superspace, and the 0 independent sector of all of the geometric supertensors.
BIANCHI IDENTITIES AND THE SUPERCOVARIANT
IT. SUPERGRAVITY
445
DERIVATIVE
IN Two AND THREE DIMENSIONAL
SPACETIME
In a previous paper [6] we have discussed the most general possible theories of local sypersymmetry in two and three dimensional spacetimes. Here we briefly review those results. Tn a two dimensional spacetime embedded within a four dimensional superspace, we defined a pinorial supercovariant derivative D, = EaMaM + &W,M
(2.0
and a vectorial supercovariant derivative D, = (27’)‘, [-i
&&I’)~* {D, , Db) + l’?iDd + &K,w
(2.2)
where det 2 = 1. We believe that this is the most general form a vectorial connection can have in superspace. Specializing to a geometry where Za8 - 8; = I$ = KB = 0
(2.3)
we see that it is possible to define the vectorial supercovariant derivative in terms of the spinorial one. We have found that in a Wess-Zumino gauge the spinorial supercovariant derivative can assume the form, D, = a, - i j+y”~a@)aV, -
i ~$(y”y~0)~V + @L~(~~V),
V, = ‘Iy,“a,,, + eNPaU+ +w,M
v = Pa, + ~9,
-+- *BM
?a = V” - $((Y + ul,Da) + y5yalYa)” V + %+ya(Y + YlgW + r8Vul,>P Va - W
+ 03”
+ Py”wJ
2)
Va = ($y”)am a,,, + x”“V, + ;g”M
(2.4)
Even though there are no dynamics associated with two dimensional there is a supergravity action. 64,, =
s
d2x d28 E-l[-i(y~)“c
Taoc]
For the three dimensional spacetime embedded in a five dimensional we also introduce a ferminonic supercovariant derivative D, = EaMaMf 4 WauhMKA
supergravity,
(2.5) superspace, (2.6)
and a bosonic supercovariant derivative of the same general form as (2.2). Once again we can specialize the geometry so that 2, fl, and K are determined as in (2.3). 595/122/2-16
446
BROWN
AND
GATES
In this superspace, the fermionic supercovariant derivative can assume the following form in a W-Z gauge D, = a, - i &(y”y@,
v, + tt%(~~?‘)~
(2.7)
Finally, the action for pure supergravity in the five dimensional superspace is practically the same as (2.5). We need only integrate over an additional bose coordinate and multiply by a factor of inverse kappa, the gravitational coupling constant. Before ending this section, it is amusing to note that the action for pure supergravity in four dimensional spacetime (in chiral representation) can be written as LZ& = &
1 d4x d20+ +3[i(P_rs)“c
Taoc]+ h.c.
(2.8)
where p is the chiral density measure superfield discovered by Siegel [l]. So very roughly speaking, the action for supergravity in two, three, and four dimensional spacetimes is the same. If this pattern were to continue, then in a six dimensional spacetime (assuming an action exists) we would expect the (chiral representation) action to be of the same form as the vector representation action of four dimensional supergravity. III.
LOCAL SUPERSYMMETRY AND MINIMAL
COMPONENT
FIELDS
The superspaces discussed in the previous section provide the geometric starting point for models of spinning strings (four dimensional superspace) and spinning membranes (five dimensional superspace). In this section we will consider, in component form, the theories with the minimal component fields. The simplest way to do this is to use the method of Wess and Zumino [3] and investigate the superspace Bianchi identities. We will consider first, the four dimensional superspace; then the five dimensional superspace. The constraints for the four dimensional superspace imply that the anticommutator of two spinorial derivatives takes the form,
Pa, Dd = -i(y”y%
0, - iiW”y5),~ M,
(3.1)
where R is a scalar superfield. Now using equation (2.4) we compute the above to zero order in 8. We immediately find that the component fields Y and D are set equal
BIANCHI
IDENTITIES
AND
THE
SUPERCOVARIANT
to zero. The component field B is the lowest component tuting the above equation into the Bianchi identiy,
then gives the following
of the superfield R. Sudsti-
equations --i2(y”y9,, Talcs+ permutations R,, + ii(y0y5)ab(DcR) + permutations
-i$(yOya)ab -i2(y”y”),b
TEcd+ $R(y”y5),b(y5)de
+ permutations
= 0 = 0
Tab = TBec+ i&& Now we use the constraints identity.
R = Rbrr + (y5ya)%(D,R)
(3.3)
= 0.
After a little algebra, we learn that these equations imply further constraints components of the torsion and curvature supertensors.
Thus, rewriting
447
DERIVATIVE
on certain
= 0.
in equations (3.1) and (3.4) in the following
(3.4) Bianchi
this identity we find i2(y”y6)bc Tmsy = 0
From the first and third of these equations we find,
Taay= l Taad
+ i$(yeyo)dc R,, = 0.
Now we compute, to lowest order in 8, the component (3.4) and (3.7). In this way, we learn,
Thus, we conclude that the constraints
(3.7)
fields which enter equations
implied by equation (3.1) lead to a geometry
448
BROWN
AND
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in which erru, yl,m , and B are the only supergravity fields. The transformation of these fields under local supersymmetry are:
laws
The transformation of B is found from the last relation in equation (3.4). This equation tells us that the linear B term in R is proportional to the 8 independent term in Rba . Since the supersmmetry generator, to lowest order, is &I, the transformation law follows. The constraints for the five dimensional superspace allow us to write,
I& , Dbl = -i(r”r%
D, (3.10)
[D, , DB] = 2TasdDd + h,““M,,
The second equation, when computed to lowest order, implies that the auxiliary field xQ” is zero. From the Bianchi identity in equation (3.2) we now can prove, (3.11).
P&t , &I + Q(Y~YJ”JDs 2 Da> = 0,
and this is the only information contained in the first Bianchi identity. Thus, we turn to the Bianchi identity in equation (3.5). This leads to the following set of equations
-i
&“f),
Rapx” -
+Dp,Rc)sxA = 0
(3.12) --i 2(y”fhc
TEDa- ~D(,T~J~” + i ~R,H~I~(Y~~I~) = 0 8--= R AB %,.‘. RAB
The first of these is particularly Taab
=
useful. We find that it and equation (3.11) give, ht)“b
Tab
-
which implies that the T,,” component the form, T,b’
&3(~8~b
Taab
=
0,
(3.13)
of the torsion supertensor can be expressed in =
i%%>cb
for some scalar superfield G. Furthermore,
Td - &JG
G,
(3.14)
we also find the following relation, = 0.
(3.15)
BIANCHI
IDENTITIES
Now by using equations equations
AND
THE
SUPERCOVARIANT
DERIVATIVE
449
(3.11), (3.14), (3.15) and the last of (3.12) we derive two D,G = &(y”y&
Rb: = 0
Tusd - i ~(~“y”)“”
RbhoD= 0.
(3.16)
As before, we compute to lowest order in B the component fields which enter equations (3.14), (3.15), and the last of (3.16). We find to lowest order,
(3.17)
So, once again we find the supergravity multiplet consists of e,“, !l-‘i’r, , and A. The transformation laws of these fields are given by (3.9) for eau and y”I, . However, the transformation law of A is here given by; &A = i&“~~(+&,).
(3.18)
It should be noted, that the transformation law of the scalar field A is determined by the first relation in equation (3.16). Thus, we have explicitly shown that for the five-dimensional superspace, the constraints given by
T6ab = -i j$(y”y”)ab ; Tarn” = 0 9
(y~‘)‘~
RnbK’ = 0
(3.20)
imply a supergravity theory with minimal component fields. It is amusing to note that the constraints in (3.20) when applied to eight dimensional superspace also lead to a theory of supergravity with minimal auxiliary fields. Similarly the constraints of Wess and Zumino [3] also lead to minimal component supergravity theories in the four and five dimensional superspaces. The reader should note that these theories with minimal component fields are second order formalisms. This is not an accident. In order to construct a theory which has minimal component fields, we must start with a theory which has the minimal number of superfields. Thus, in the minimal component supergravity theories, the Lorentz connections, IV, KA, are determined in terms of the inverse vielbein EaM. We already know from section II that both EaM and WmKAmay also be constructed from EnM and WaKA. The fact that Z is nonzero allows this construction. Thus, the nonzero torsion decreu.ses the number of component fields in the supergravity multi-
450
BROWN AND GATES
plet. Therefore, in minimal component supergravity theories all supergravity fields are contained EaM. Furthermore, this superfield is constrained. This situation should persist in all superspaces. It is interesting to note the differences in these lower dimensional theories and four dimensional supergravity. In two dimensions we find the superfield R but there is no analog of the four dimensional G, . In three dimensions, there is no R but the analog of G, exists in form of a scalar, G.
IV. MINIMAL
COMPONENTS
AND SUPERGRAVITY
In this section we review the constraints which lead to minimal component supergravity in four dimensional spacetime. We may choose a set of constraints which imply the following equations. (4.1) (4.2) (4.3) Now using the equations above and the Bianchi identities,
[Pa , &I, Dc>+ [[& >&>, Da> + Wc , QJ, Db>= 0 ND, , D& D,eJ - [[Do , &h Da> + W, , Da>, Dd = 0
(4.4) (4.5)
we must first find all algebraic constraints which are implied for the Taabcomponent of the torsion supertensor. This component of the torsion supertensor appears purely algebraically in equations (4.4) and (4.5) as, -ii2(y”y”)&
Tacd + ifRabKA(uJdc + permutations
i%‘0y8)daTb)sd
-
i2h0yahb
Using both of these equations simultaneously, on TaBc.
T,B’
+
&b’D
= 0 =
0
(4.6) (4.7)
we can derive the following constraints
T Mla = (o~,)~, TBab = (Y")"~ Taba - +~~ac8(~~8)0b Toab = 0
(4.8)
Equation (4.7) and the first relation in equation (4.8) may be used to solve for Tap8 and RabKAin terms of TaBb. Tao8 = i&(~"oI)"~ Tsab &a88
Next we multiply
=
ifd~odcA)ab[(uMAy8)od
Tacdl
equation (4.10) by the chiral projection
(4.9) (4.10) operators, +(I f y5), and
BIANCHI
IDENTITIES
AND
THE
SUF’ERCOVARIANT
symmetrize in the 6 and /3 indices. In this way, we derive the following
two equations,
TG,d = 0
(4.12)
+ (Y~u,AY,$~ Tcad = 0
(4.12)
tu,nya)Gd Td + (QY$~ ty5u,n)cd r,,’
451
DERIVATIVE
as results. In these two equations we contract the 01and h indices, and this leads us to conclude that the equations immediately above are equivalent to, t~a)‘d Tcad - hi4y6)e~ (y5yu)ed Tcod - hot~~y”)~~
Tcsd = 0 Tcsd = 0
Finally, equation (4.9) implies one more simple constraint
(4.13) (4.14)
on Tasc.
t~sa)al, Gab + (crag)% T-2 = 0
(4.15)
The constraints in equations (4.8) (4.13) (4.14) and (4.15) imply that the superfield T,$ must be of the form, Tas’ = iHyOy@yO)ac;
@ = -M
+ y5y”G, + iy5N
(4.16)
for some superfields M, N, and G, . In terms of these superfields the supertorsion component TEByand the supercurvature component RabKA are found from equations (4.9) and (4.10) to be, YSGs Ta8’ = - +E,~ RabKA= -i
$(y’i%b
[6~&jM
(4.17) f
E,B~‘NI
(4.18)
To lowest order in 8, the superfieldsM, N, and G, correspond precisely to the auxiliary fields found for minimal component supergravity [9]. In the presenceof matter superfields, the superfields M, N, and G, are set equal to certain matter dependent superfields. With these substitutions, equations (4.16) (4.17), and (4.18) are equations of motion. In particular, equation (4.17) has been obtained previously [lo] as an equation of motion in the presenceof the (1, l/2) matter supermultiplet. We may define a complex scalar superfield by the equation, R- = ift(P-o,,yO)* &bKA
(4.19)
and from equations (4.4) and (4.18) we are able to deduce the results below. Da-R- = 0;
R- = 3(M - iN)
(4.20)
Thus R- is the chiral scalar superfield introduced by Wessand Zumino [3]. In a similar fashion, we find that the superfield Gab of Wess and Zumino is related to G, by the equation, Gob = i(yOP+yaLb 6
(4.21)
452
BROWN
AND
GATES
The reason this superfield appears on the right hand side of equation (4.17) in place of zero as in reference [3] is that we have replaced the constraint TaBy = 0 by the constraint (yayopb R,, %A= 0. When coupling supergravity to either the (l/2,0) or (1, l/2) matter supermultiplets, these constraints are equivalent. This is not surprising since both of these supermultiplets are described by scalar (chiral and real) superfields. Now we turn to the problem of calculating the spinorial derivative of the matrix super-field @. To this end, we note the spinorial derivative of this superfield enters the Bianchi identities (4.4) and (4.5) as, i(yOyol)& REcKA+ ~~~~~~~~ + permutations -2&To$
+ WCmKA(uKAh
- i2(y”ya),,
Also (4.17) suggests that we should consider the additional
= 0
(4.22) (4.23)
7’2 = 0
Bianchi identity,
[ED, , 44, Dc>+ [[Do, Da>,Q4 + [PB , Do>,QJ = 0
(4.24)
which gives the following information, Q~~D,G,
-
iXy”y%
Cod
+
&,‘a
=
(4.25)
0
Equation (4.23) we use to express parts of the curvature supertensor in terms of the torsion supertensor and its first spinorial derivative.
R~a-~~(~+~x~)~c = --i W+)bd [~wTI,+)B~ + i(r”~-r% T,B~I R Ba+KA(~oP+%A>bc
=
+ -
i
4[(y”p+)ad (y”P+)cd
&b+ tTI c+)/
&+I
Tb+)sd
-
h’P+)bd
&+I
Tlc+)~dl
(4.26)
In these expressions, the plus or minus sign following a spinorial index indicates multiplication by (1 + ys)/2 or (1 - y6)/2. By contracting the b and c indices, in the first of these results, we learn 2D,M
+ (y6ym)ba DbG, - i2(y5)b, D&
= -i2(y00as)ab
T,;.
(4.27)
Furthermore, by adding the two formulae in (4.26) together with the conjugate formulae we can express the curvature RaaxA as, R aBxA=
-(Y~Y~%A),~ +
TaBa+ i(udbd hTmd
i[(~+~~~~“)““(~“~+)da
+
(~-GO’“)bch’o~-)da~
Db
T,B~
(4.28)
Similarly, we may use (4.25) to derive an alternative expression for RaBKA. By equating the two expressions we find another equation which relates the spinorial .derivative of Cp to the torsion supertensor. Finally, we observe that equation (4.20) implies as a consequence, D,M -/- i(y5)ba D,N
= 0
(4.29)
BIANCHI
IDENTITIES
AND
THE
StJPERCOVARIANT
453
DERIVATIVE
We have thus derived a set of three linear equations which may be used to express the spinorial derivatives of M, N, and G, in terms of a component of the torsion supertensor.
This last result we substitute R cal3v=
into (4.25) to express Raaa,, in terms of TmBd. iW”yJcd
T,Yd
-
i&aBys(yoy5~~af”)~d
Note that (4.30) implies an algebraic constraint N, and G, . (y5y”)ab D,G, = D&l This can be seen to be equivalent superfield P,, .
(4.31)
T,,id
on the spinorial
derivatives
(4.32)
- i(y5pb D,N
to an equation
of M,
of conservation
for the matrix
Da@, = 0
(4.33)
So far we have been able to express most of the nontrivial geometric supertensor in terms of Tadc or Tag. The only remaining exception is RmoxA. There are two ways in which we may proceed. First, we note the Bianchi identity in (4.5) implies, - Dt, RbjBKA- i (YO~&)~~RnaKh- DBRnbKA= 0.
(4.34)
This combined with (4.31) yields an equation for RasKn , R UBKh= -i(uauB)ab D,TKAb- i&,BK~DaDJ(P+@P+ - P-@P-)
y”)ab
(4.35)
and a constraint on the spinorial derivative of reDa. DaTEDa= 0
(4.36)
Alternately, we find from the Bianchi identity in (4.24), D,TNBd
=
@,I
TclBld
+
~~~~~~~~~~
+
jWaOPA(u,Jdc
+
TcdTo~d
(4.37)
+
(4-W
This we solve for ReaMA and recalling (4.16) and (4.17), we find, RaBlc,a = -j(cA)cd DJ’k
+ ~~~~~~~~~~ G&,
+
4
‘WuKA@m@m)
GA~~$IGs
Thus, we have found two alternative ways to write RasK,,in terms of ToaCand Tnsc. Compatibility of these expressionssimply give more differential constraints on TnRc and Tagc.
454
BROWN
AND
GATES
Finally, we note the result
WhKa
=
&,I
&I~I
Ka +
G~‘G&VK~
-
i(@yr,#,
Rdlofa
follows from the Bianchi identity in (4.24). At this point we can end our investigation of the Bianchi identities. We note that if the 8 = 0 values of all the curvature and torsion tensors were known, then equations (4.30), (4.31), (4.37), and (4.39) imply that all higher spinorial derivatives of these quantities can be expressed in terms of the 0 = 0 values. But as we shall demonstrate in the next section, the required 0 = 0 values can be determined (see appendix A). At this point we can make some comparison between the minimal component theory and the nonminimal component theory. As shown in reference [2], there are many forms of the nonminimal theory. For simplicity we consider the II = ---I version of the nonminimal theory. The constraints for this theory are Tabd = 0;
Tabs = -i
&(y0y6)ab ;
RaeKh = 0 (4.40)
By repeating arguments analogous to those given earlier in this section we find there exists a matrix superfield @ such that Tao” = i~(y”y@yo)ac;
Furthermore,
@
X
-A4
+ iy”V, + y5ynG, + iy5N.
(4.41)
it can be shown that T,aB
=
(~“~af)ob
(4.42)
Tb
for some spinor superfield Tb. Also, we find that the 0 independent following quantities
sector of the
(4.43)
are not algebraically related to any of the 19independent sectors of the geometric supertensors. All of these new quantities imply that with the constraints in (4.40) the supergravity multiplet includes, in addition to the minimal set of fields, a vector, a second axial vector, and two spinors. These fields correspond to the 0 independent sectors of V019
(y5y”yo)ab DaTbl’,
Tb,
(y5y”yo)ab DaTbm’
This corresponds to the set of fields found in reference [ll].
(4.44)
BIANCHI
IDENTlTlES
V. THE 0 EXPANSION
AND
OF
THE
SUPERCOVARIANT
THE SPINORIAL
DERIVATIVE
SUPERCOVARIANT
455
DERIVATIVE
In this section, we discusshow the Bianchi identities in the previous section allow us to reconstruct the 0 expansion of the spinorial component of the supercovariant derivative. In general, we know that the spinorial component of the supercovariant derivative can be written in for form,
for some operators Vu) (suppressing the lower indices). Each of these operators is expanded over the algebra generated by the O-derivative, x-derivative, and the Lorentz generator, Q(i)
=
[&iqm
a,
+
[@)]u
3,
+
$.[&Ciqd
M,,
(5.2)
where the coefficients A, 2, and e are functions of x alone. The transformation law for the spinorial component of the supercovariant derivative is
SD, = FL D,l
(5.3)
where Q, the parameter of supergauge transformations, has a form similar to that of D, . sz = ~(0) + ebip
+ BefP) + i0peiP
+ f7+fec2~)
+ Beeb@) + (Oe)z ~2~6)(5.4)
Each of the component operators J2ci) has the samegeneral form as the component operators VCi). At this point, we find it is essential to work in a generalized WessZumino gauge [6]. Without loss of generality, we make the modification
on the lowest component in D, . To find a W-Z gauge, we need only retain the following term from the transformation law. SD, = -d&'
(5.6)
It is immediately apparent that sZj,l’may be used to “gauge away” 6:). Furthermore, Q@), Q3), and !2?) may be used to “gauge away” the antisymmetric part of V$. Thus, by only using the transformation, we find that in some W-Z gauge, D, can assume the form
for some operators V, and V,, . By considering the B independent sector of (4.1)
456
BROWN
AND
GATES
and (4.18) we determine these operators. Continuing, we note that equation (5.6) also implies that the operators Qi2) and eA3’ satisfy the relation $7(3) a
=
-Q(2)
(5.8)
a
in some W-Z gauge. Therefore, the quadratic 9 terms in D, can take the form,
Given this fact, we can compute the I3 independent sectors of Tnas, Taab, and RnaKA, using (4.2) to define D, . When this is done, we find that
The terms not esplicitly given above (the deltas) are functions which only depend on .4(l), B(l), C(r). It is clear that by making the definitions A(J)b aa
= --
~(4)b at-c
_
A(db. aa
,54kA aoL
= -
3 &4hA an
@4)8 no
_ -
_
A (4h an
B(4)8 au
_
A(4)” aa
(5.11)
we find, KA oc &4)KA R acx
(5.12)
The deltas are, in general, noncovariantly transforming functions of the component fields. A covariant quantity is one which has a transformation law which does not depend on the spacetime derivative of the gauge parameters in +Qco).Since the torsion and curvature super-tensors are by definition covariant quantities, the “shifts” which take us from the component quantities with carats to those without are necessary. We have encountered these sort of shifts in several of our previous works [2, 41. The fact that Taas is identically zero determines BAtj8 to be zero. Similarly, equation (4.16) determines A,,“)b . Finally equation (3.31) determines CLt)K’. The spin connection, OJ,~~Y, is then defined by (4.17). Now a convenient feature of the gauge in (5.9) is that TRLIB,ToLab,and RaaK* are all independent of 6,“) . In order to determine this operator we note that is contained in the B independent sectors of (5.13)
The first of these vanishes identically
from (4.1). The second quantity can be found
BlANCHI
IDENTITIES
AND
THE
SUPERCOVARIANT
DERIVATIVE
457
from (4.18) and (4.30). When all of this is done, we find that to quadratic order in 6’ the spinorial covariant derivative is given by, V, - i&“u”“e), BasKAMK. + g80(y”(s - iy5P) 2)a + gt7y5e(yoyys - iy5P) 2)@ + i@~5y~~tyof%), + om
D, = a, - i&OyaS),
(5.14)
where the following definitions are to be understood.
The result in (5.14) can be used to compute the 6’ independent sector of all of the superspace curvature and torsion tensors. These are given in an appendix. Arguments given previously 161indicate that the generator of local supersymmetry must be of the form,
and thus the commutator
of two supersymmetry generators is [12],
E ‘a =
- i(~lym~2) ‘y,” (5.16)
hlnR = -i(&fr,) 5’” = -i(ZIya~2)
w:’
- i )(+#‘E~)
BeTa
e,”
where QE and Qn, are the generators of local Lorentz and general coordinate transformations, respectively. To calculate the transformation laws for the various compo-
458
BROWN
AND
GATES
ponent fields, we may simply utilize (4.30) and the result for TBBcgiven in Appendix A. The supersymmetry generator in equation (5.14) may be rewritten as, ii?&) = E”&
+ i(+“e) D, - $@ebRabKAMKA+ o(e’)
(5.17)
Thus, we see that this supersymmetry generator differs from that given in the last of reference [3] by an E dependent Lorentz transformation. This can be understood by noting this supersymmetry generator preserves the W-Z gauge of D, ; whereas the generator given in reference [3] preserves the W-Z gauge only up to a Lorentz transformation. It remains for us to determine the cubic and quartic 6 sectors of D, . Before discussing these, we note that by using the remaining freedom, sZL6),we can choose a gauge where (#y
9%) = 0
(5.18)
At this point we have used all of the gauge freedom in Q, except that in Q(O). These remaining degrees of freedom correspond to local supersymmetry, general coordinate, and local Lorentz transformations which we must retain in the theory. Although we shall not completely present the two remaining 8 sectors of D, , we will describe the procedure which can be used to determine these remaining sectors. First using (5.1) and (5.14) we can compute the supertensors TaED and RaaKAto quadratic order in 8. The 0 independent sector of 2(DraTblsD)
DD
+
~(kz&~i?‘?
M,A
(5.18)
will contain the as yet undetermined operator 6 bi). But the results in (4.30), (4.31) and (4.37) may also be used to evaluate the 8 independent part of this expression. Thus, we can write an equation which expresses 6%’ in terms of previously determined quantiries. In a similar manner, we note that the 6’ independent sector of
2(&zD,]T,,D) D, + ;-(Dd’b&,F”) M,A
(5.19)
can be computed from (5.1) and (5.14) with ej$ now determined. It is easily seen that the 0 inedependent sector of (5.19) will contain the only remaining undetermined operator, eA6’. Once again, however, we can evaluate the t9 independent sector of (5.19) by using (4.30), (4.31), (4.37) and (4.39). This would give the final equation determine 6L6). Thus in this section we have proven that by using information derived from the Bianchi identities, we are able to construct the vielbein and connection superfields. In some previous works [I, 1,4] these superfields have been constructed by actually solving the constraints on the torsion and curvature supertensors. So we have demonstrated the equivalence of these two different approaches.
BIANCHI IDENTITIES AND THE SUPERCOVARIANT
DERIVATIVE
459
VI. DISCUSSION We have explicitly demonstrated that an analysis of the superspaceBianchi identities is completely equivalent to solving the constraints on vielbein and connection superfields as was done in references [l, 21.We have also shown the existence of somesimilarities in the theory of supergravity, with minimal component fields, in two, three, and four dimensional spacetimes.For instance, the constraints for all of thesetheories are the same as given in (3.20). Furthermore, in each of these theories the auxiliary fields form the samerepresentation of the Lorentz group as the product
eaeb
(6.1)
In two and three dimensions,this is a scalar under the Lorentz group. In four dimensions, this is a reducible representation transforming as a scalar, pseudoscalar, and axial vector. If this were to continue for the extended supergravity theories, we would expect the auxiliary, bosonic fields to fall into the samerepresentation as
where i and j are the O(N) labels of the Majorana spinor. For instance, for O(2) supergravity this would suggestthat the auxiliary, bosonic fields in addition to those of O(1) supergravity are isovectors (scalar, pseudoscalar, and axialvector) and isoscalars(vector and antisymmetric tensor). Another interesting feature of the minimal component theories is that the required auxiliary fields all appear at the linear 0 sector of the spinorial supercovariant derivative. As a consequenceall of auxiliary fields appear in the commutator of two supersymmetry transformations. This is in sharp contrast with the nonminimal theories where additional, independent, auxiliary fields appear beyond the linear B sector of D, . Thus, we think it is a very important feature to have all of the auxiliary fields in the linear 0 sector of D, . In closing, we note that the analysis of the superspaceBianchi identities and the subsequent construction of the vielbein and connection superfields can be carried out for an arbitrary superspace.
APPENDIX
A: LOWEST ORDER TERMS IN GEOMETRIC
SUPERTENSORS
Here we give the 8 independetn sector of all of the geometric tensors of the eight dimensional superspacewith minimal component fields. We define the curvature and torsion supertensor by the equation, P, , &I) = 2T/mDDD + +RAPMKA.
460
BROWN AND GATES
From equation (5.14) we compute these geometric tensors to lowest order in 8.
These expressions may be compared with those which are obtained in the nonminimal component approach [4, lo]. In computing these results, we have made use of the fact that the vectorial component of the supercovariant derivative to first order in B is given by
APPENDIX
B: EQUATIONS OF MOTION
FOR SU~ERSPACE
Here we give the equations of motion for superfield supergravity coupled to (1, l/2) or (l/2,0) matter supermultiplets described by a bispinor +b . Tabs = -i$(y”y6)aa Tabd = 0
~,BY”(yoy5ys>~b TV: = -iDt,((p+Tp+ &6aL3
+ p-Tp-)
y.)‘,
+ i(fjba
DbJa
= Wmwyd - 2(rlmJaJb- JsJa) + UW~~P+TP+ + p-Tp-) y,~ysysy’)~~ - @U?d~~sye~~~*~ - i%~~~s~~)c* DcDdJ,s - w?~&JA
BIANCHI
IDENTITIES
AND
THE
SUPERCOVARIANT
461
DERIVATIVE
In these equations, the quantity J, is defined by, J, = Tr(y5y,T)
and has been referred to as the “supercurrent” for some time [13]. The superfield ~a,, is the superspace analog of the ordinary energy-momentum tensor. Additionally, the superspace energy-momentum bispinor, +b , is subject to the constraints (Y”T)ab = -(Y”%z Dana,,
0
=
in analogy with the symmetry and conservation requirements on the ordinary energymomentum tensor. Recently, Ogievetsky and Sokatchev have given an algorithm for the derivation of Tab for an aribtrary superfield action [14]. APPENDIX In the four dimensional
C:
NOTATION
AND
CONVENTIONS
superspace, we use a representation
where
y” 5%(02, iu3) y5
3
i&,y"y~
=
iyoyl
=
-ial
The full set of Dirac matrices is 1, y”, and y5. y”yP = -.q%Q + i&y5 diag(T@) = (-1,
1);
--
co1
-@l
= 1
In this representation y” is antisymmetric, while y”y” and y”y6 are symmetric. In the five dimensional superspace, we define, y” = (u2, icr3, id) y~y6 = -+S _ ieaS8y-
diag(q*B) = (-1, For the four dimensional we define
1, 1);
6012
=
-eo12
=
1
spacetime embedded in the eight dimensional
ye = (CT” @ u2, iI @ d, ic? @ u2, iI @ u3)
1
We therefore have, yUy.8 = -.qd - iuCX6 diag(q”fl) = (-1,
1, 1, 1);
l 0123= -co123 = 1
superspace,
rt62
BROWN AND GATES
and the matrices y”, y”y5y”, and y”y5 are antisymmetric, while y”y” and y”&’ are symmetric. All of these representations are Majorana representations so that the charge conjugation of a spinor is equivalent to ordinary complex conjugation. We use capital latin indices (A, B,...) to denote a supervector label. Lower case latin indices (a, b,...) denotes fermionic labels. Lower case greek indices (01,p,...) denote bosonic labels. Letters from the beginning to middle of the alphabet denote components of a tensor with respect to an anholonomic coordinate system (ie “flat” labels), while those from the latter part of the alphabet denote holonomic components (“curvy” labels). The distinction between flat and curvy spinors, however, is immaterial.
MKh z Generator of Lorentz transformations
ACKNOWLEDGMENT We wish to thank Warren Siegel and Lee Smolin for interesting and stimulating discussions.
REFERBNCE.3 1. W. SIEGEL, Nucl. Whys. B 142 (1978), 301. 2. W. SIEGEL AND S.. J. GATES, JR., Harvard preprint HUTP-78/A019 (1978). 3. J. MESS AND B. ZUMINO, Phys. Lett. B 66 (1977), 361; Phys. Lett. B 74 (1978), 51; Phys. Left. B 79 (1978), 394. 4. S. J. GATES, JR., AND J. A. SHAPIRO, Phys. Rev. D 18 (1978), 2768. 5. J. WESS,Supersymmetry-Supergravity, Lectures given at the VIII G.I.F.T. Seminar, Salamanca, 1977, to be published in Springer Tracts. 6. S. J. GATES, JR., Harvard preprint HUTP-78/A028 (1978). 7. J. Wsss AND B. ZUMINO, Nucl. Phys. B 78 (1974), 1. 8. P. BREITENWHNER, Phys. Lett. B 67 (1977), 49; Nucl. Phys. B 124 (1977), 500. 9. IS. ‘s. STELLE AND P. C. WEST, Phys. Lett. B 74 (1978), 330; S. FERRARA ANI, P. VAN NIETJWENHLJIZEN, Phys. Lett. B 74 (1978), 333. 10. L. BRINK, M. GELL-MANN, P. RAMOND, AND J. H. SCHWARZ, Phys. Lett. B 74 (1978), 336. 11. B. DE WITT AND M. GRISARU, Phys. Lett. B 74 (1978), 57. 12. Equation (5.15) is not exact as written. Since the supersymmetry generator 8, is field dependent, there are correction terms which must be included in this equation. See Ref. [6] for a complete discussion. 13. S. FERRARA AND B. ZUMINO, Nucl. Phys. B 87 (1975), 207. 14. V. OGIEVETSKY AND E. SOKATCHEV, JINR preprint E2-11528, Dubna (1978).
OF PHYSICS
Superspace
122, 443-462 (1979)
Bianchi
Identities
and the Supercovariant
Derivative*
M. BROWN Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 AND
S. JAMESGATES, JR.+ Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138 Received January 31, 1979
We analyze the theory of supergravity, with minimal component fields, in spacetimes of two, three, and four dimensions. The analyses begins by investigating the superspace Bianchi identities. We show that in two and three dimensions, the supergravity multiplet consists of the gravition, gravitino, and a single auxiliary field. In four dimensions, we analyze the Bianchi identities in order to investigate the structure of the vielbein and connection superfields. Utilizing this analysis, we explicitly construct these superfields to quadratic order in 8 and give a prescription for determining them completely.
I. INTR~DUC~ON
A complete superspace formulation of the theory of supergravity now exists [I, 2,3]. The theory which hasemerged is extremely elegant showing a very remarkable interplay between two different superspacegroups, the “chiral” gauge group and the “vector” gauge group [2, 4, 51. By starting with an inverse vielbein superfield, EAM, and a connection superfield, WiA, (transforming only under the “vector” gauge group) which are constrained it has been shown [I, 21 that the solutions to these constraints transform in an entirely different manner. In fact, it is only by solving the constraints that we find the existence of the “chiral” gauge group. For it is the gauge phasefactor WM and the density superfields (@ or Y) which transform as somerepresentation of the “chiral” gauge group [2]. As an alternative to solving the constraints, Wess and Zumino [3] have noted that the superspaceBianchi identities may also be used to gain knowledge about the * Research supported in part by the National + Junior Fellow, Harvard Society of Fellows.
Science Foundation
under Grant PHY77-22864.
443 OOO3-4916/79/110443-2O$OXOO/O Copyright @ 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
444
BROWN
AND
GATES
structure of the vielbein and connection superfields. By working only with the vielbein and connection superfields, we are forced to consider only the “vector” gauge group. We were thus motivated to investigate the equivalence of these two very different approaches to understanding superspace geometry. This is the main purpose of this paper. As a secondary purpose we are striving to understand how one goes about deriving a theory of superspace geometry, for an arbitrary superspace, which has the minimal number of component fields. This is of crucial importance for extended supergravity theories. Since in an arbitrary superspace one is not guaranteed that it is always possible to solve a set of contraints in superfield form, we thought the Bianchi identities might serve as an alternative. This may be particulary true of a spacetime of ten dimensions (O(4) supergravity after dimensional reduction) where spinors exist that are simultaneously Weyl and Majorana. Thus, one is not even guaranteed the existence of UM [l] in this theory. In section two, we review the structure of supergravity in spacetimes of two and three dimensions [6]. We start by introducing the most general possible supercovariant derivative and specialize to a particular subclass. Within this subclass, we give the form of the spinorial supercovariant derivative in Wess-Zumino gauge [7]. We also present the superfield actions for pure supergravity in these spacetimes. In section three, we restrict these geometries even further to find theories which have the minimal number of component fields. In each of these superspaces it is shown, by analyzing the Bianchi identities, that the minimal supergravity multiplets consist of the graviton, gravitino, and a single auxiliary scalar field. In section four, we start with a very restrictive superspace geometry for ordinary supergravity. In this superspace which possesses a four dimensional spacetime we repeat the type of analysis which is carried out in section three. We show, in detail, how the Bianchi identities may be used to express all of torsion and curvature supertensors in terms of Taadand TaBa. We note the existence of a recussion formula which allows the evaluation of any number of spinorial derivatives of Tuodand T,,“. Finally, we make a brief comparison with the nonminimal theory [4,8]. In section five, we show how the information gained from the Bianchi identities may be combined with a previously derived procedure [6] for finding a generalized W-Z gauge to reconstruct the 0 expansion for the spinorial supercovariant derivative. We explicitly construct this operator to quadratic order and give a prescription for determining the remainder. We thus demonstrate precisely how the Bianchi identities determine the vielbein and connection superfields up to a supergauge transformation. This constitutes a proof that a complete analysis of the Bianchi identities is equivalent to solving the constraints on the vielbein and connection superfields as has been done previously [l, 21. We give a discussion and summary in the final section of the paper. We include three appendices which give our conventions and representations, the equations of motion for superspace, and the 0 independent sector of all of the geometric supertensors.
BIANCHI IDENTITIES AND THE SUPERCOVARIANT
IT. SUPERGRAVITY
445
DERIVATIVE
IN Two AND THREE DIMENSIONAL
SPACETIME
In a previous paper [6] we have discussed the most general possible theories of local sypersymmetry in two and three dimensional spacetimes. Here we briefly review those results. Tn a two dimensional spacetime embedded within a four dimensional superspace, we defined a pinorial supercovariant derivative D, = EaMaM + &W,M
(2.0
and a vectorial supercovariant derivative D, = (27’)‘, [-i
&&I’)~* {D, , Db) + l’?iDd + &K,w
(2.2)
where det 2 = 1. We believe that this is the most general form a vectorial connection can have in superspace. Specializing to a geometry where Za8 - 8; = I$ = KB = 0
(2.3)
we see that it is possible to define the vectorial supercovariant derivative in terms of the spinorial one. We have found that in a Wess-Zumino gauge the spinorial supercovariant derivative can assume the form, D, = a, - i j+y”~a@)aV, -
i ~$(y”y~0)~V + @L~(~~V),
V, = ‘Iy,“a,,, + eNPaU+ +w,M
v = Pa, + ~9,
-+- *BM
?a = V” - $((Y + ul,Da) + y5yalYa)” V + %+ya(Y + YlgW + r8Vul,>P Va - W
+ 03”
+ Py”wJ
2)
Va = ($y”)am a,,, + x”“V, + ;g”M
(2.4)
Even though there are no dynamics associated with two dimensional there is a supergravity action. 64,, =
s
d2x d28 E-l[-i(y~)“c
Taoc]
For the three dimensional spacetime embedded in a five dimensional we also introduce a ferminonic supercovariant derivative D, = EaMaMf 4 WauhMKA
supergravity,
(2.5) superspace, (2.6)
and a bosonic supercovariant derivative of the same general form as (2.2). Once again we can specialize the geometry so that 2, fl, and K are determined as in (2.3). 595/122/2-16
446
BROWN
AND
GATES
In this superspace, the fermionic supercovariant derivative can assume the following form in a W-Z gauge D, = a, - i &(y”y@,
v, + tt%(~~?‘)~
(2.7)
Finally, the action for pure supergravity in the five dimensional superspace is practically the same as (2.5). We need only integrate over an additional bose coordinate and multiply by a factor of inverse kappa, the gravitational coupling constant. Before ending this section, it is amusing to note that the action for pure supergravity in four dimensional spacetime (in chiral representation) can be written as LZ& = &
1 d4x d20+ +3[i(P_rs)“c
Taoc]+ h.c.
(2.8)
where p is the chiral density measure superfield discovered by Siegel [l]. So very roughly speaking, the action for supergravity in two, three, and four dimensional spacetimes is the same. If this pattern were to continue, then in a six dimensional spacetime (assuming an action exists) we would expect the (chiral representation) action to be of the same form as the vector representation action of four dimensional supergravity. III.
LOCAL SUPERSYMMETRY AND MINIMAL
COMPONENT
FIELDS
The superspaces discussed in the previous section provide the geometric starting point for models of spinning strings (four dimensional superspace) and spinning membranes (five dimensional superspace). In this section we will consider, in component form, the theories with the minimal component fields. The simplest way to do this is to use the method of Wess and Zumino [3] and investigate the superspace Bianchi identities. We will consider first, the four dimensional superspace; then the five dimensional superspace. The constraints for the four dimensional superspace imply that the anticommutator of two spinorial derivatives takes the form,
Pa, Dd = -i(y”y%
0, - iiW”y5),~ M,
(3.1)
where R is a scalar superfield. Now using equation (2.4) we compute the above to zero order in 8. We immediately find that the component fields Y and D are set equal
BIANCHI
IDENTITIES
AND
THE
SUPERCOVARIANT
to zero. The component field B is the lowest component tuting the above equation into the Bianchi identiy,
then gives the following
of the superfield R. Sudsti-
equations --i2(y”y9,, Talcs+ permutations R,, + ii(y0y5)ab(DcR) + permutations
-i$(yOya)ab -i2(y”y”),b
TEcd+ $R(y”y5),b(y5)de
+ permutations
= 0 = 0
Tab = TBec+ i&& Now we use the constraints identity.
R = Rbrr + (y5ya)%(D,R)
(3.3)
= 0.
After a little algebra, we learn that these equations imply further constraints components of the torsion and curvature supertensors.
Thus, rewriting
447
DERIVATIVE
on certain
= 0.
in equations (3.1) and (3.4) in the following
(3.4) Bianchi
this identity we find i2(y”y6)bc Tmsy = 0
From the first and third of these equations we find,
Taay= l Taad
+ i$(yeyo)dc R,, = 0.
Now we compute, to lowest order in 8, the component (3.4) and (3.7). In this way, we learn,
Thus, we conclude that the constraints
(3.7)
fields which enter equations
implied by equation (3.1) lead to a geometry
448
BROWN
AND
GATES
in which erru, yl,m , and B are the only supergravity fields. The transformation of these fields under local supersymmetry are:
laws
The transformation of B is found from the last relation in equation (3.4). This equation tells us that the linear B term in R is proportional to the 8 independent term in Rba . Since the supersmmetry generator, to lowest order, is &I, the transformation law follows. The constraints for the five dimensional superspace allow us to write,
I& , Dbl = -i(r”r%
D, (3.10)
[D, , DB] = 2TasdDd + h,““M,,
The second equation, when computed to lowest order, implies that the auxiliary field xQ” is zero. From the Bianchi identity in equation (3.2) we now can prove, (3.11).
P&t , &I + Q(Y~YJ”JDs 2 Da> = 0,
and this is the only information contained in the first Bianchi identity. Thus, we turn to the Bianchi identity in equation (3.5). This leads to the following set of equations
-i
&“f),
Rapx” -
+Dp,Rc)sxA = 0
(3.12) --i 2(y”fhc
TEDa- ~D(,T~J~” + i ~R,H~I~(Y~~I~) = 0 8--= R AB %,.‘. RAB
The first of these is particularly Taab
=
useful. We find that it and equation (3.11) give, ht)“b
Tab
-
which implies that the T,,” component the form, T,b’
&3(~8~b
Taab
=
0,
(3.13)
of the torsion supertensor can be expressed in =
i%%>cb
for some scalar superfield G. Furthermore,
Td - &JG
G,
(3.14)
we also find the following relation, = 0.
(3.15)
BIANCHI
IDENTITIES
Now by using equations equations
AND
THE
SUPERCOVARIANT
DERIVATIVE
449
(3.11), (3.14), (3.15) and the last of (3.12) we derive two D,G = &(y”y&
Rb: = 0
Tusd - i ~(~“y”)“”
RbhoD= 0.
(3.16)
As before, we compute to lowest order in B the component fields which enter equations (3.14), (3.15), and the last of (3.16). We find to lowest order,
(3.17)
So, once again we find the supergravity multiplet consists of e,“, !l-‘i’r, , and A. The transformation laws of these fields are given by (3.9) for eau and y”I, . However, the transformation law of A is here given by; &A = i&“~~(+&,).
(3.18)
It should be noted, that the transformation law of the scalar field A is determined by the first relation in equation (3.16). Thus, we have explicitly shown that for the five-dimensional superspace, the constraints given by
T6ab = -i j$(y”y”)ab ; Tarn” = 0 9
(y~‘)‘~
RnbK’ = 0
(3.20)
imply a supergravity theory with minimal component fields. It is amusing to note that the constraints in (3.20) when applied to eight dimensional superspace also lead to a theory of supergravity with minimal auxiliary fields. Similarly the constraints of Wess and Zumino [3] also lead to minimal component supergravity theories in the four and five dimensional superspaces. The reader should note that these theories with minimal component fields are second order formalisms. This is not an accident. In order to construct a theory which has minimal component fields, we must start with a theory which has the minimal number of superfields. Thus, in the minimal component supergravity theories, the Lorentz connections, IV, KA, are determined in terms of the inverse vielbein EaM. We already know from section II that both EaM and WmKAmay also be constructed from EnM and WaKA. The fact that Z is nonzero allows this construction. Thus, the nonzero torsion decreu.ses the number of component fields in the supergravity multi-
450
BROWN AND GATES
plet. Therefore, in minimal component supergravity theories all supergravity fields are contained EaM. Furthermore, this superfield is constrained. This situation should persist in all superspaces. It is interesting to note the differences in these lower dimensional theories and four dimensional supergravity. In two dimensions we find the superfield R but there is no analog of the four dimensional G, . In three dimensions, there is no R but the analog of G, exists in form of a scalar, G.
IV. MINIMAL
COMPONENTS
AND SUPERGRAVITY
In this section we review the constraints which lead to minimal component supergravity in four dimensional spacetime. We may choose a set of constraints which imply the following equations. (4.1) (4.2) (4.3) Now using the equations above and the Bianchi identities,
[Pa , &I, Dc>+ [[& >&>, Da> + Wc , QJ, Db>= 0 ND, , D& D,eJ - [[Do , &h Da> + W, , Da>, Dd = 0
(4.4) (4.5)
we must first find all algebraic constraints which are implied for the Taabcomponent of the torsion supertensor. This component of the torsion supertensor appears purely algebraically in equations (4.4) and (4.5) as, -ii2(y”y”)&
Tacd + ifRabKA(uJdc + permutations
i%‘0y8)daTb)sd
-
i2h0yahb
Using both of these equations simultaneously, on TaBc.
T,B’
+
&b’D
= 0 =
0
(4.6) (4.7)
we can derive the following constraints
T Mla = (o~,)~, TBab = (Y")"~ Taba - +~~ac8(~~8)0b Toab = 0
(4.8)
Equation (4.7) and the first relation in equation (4.8) may be used to solve for Tap8 and RabKAin terms of TaBb. Tao8 = i&(~"oI)"~ Tsab &a88
Next we multiply
=
ifd~odcA)ab[(uMAy8)od
Tacdl
equation (4.10) by the chiral projection
(4.9) (4.10) operators, +(I f y5), and
BIANCHI
IDENTITIES
AND
THE
SUF’ERCOVARIANT
symmetrize in the 6 and /3 indices. In this way, we derive the following
two equations,
TG,d = 0
(4.12)
+ (Y~u,AY,$~ Tcad = 0
(4.12)
tu,nya)Gd Td + (QY$~ ty5u,n)cd r,,’
451
DERIVATIVE
as results. In these two equations we contract the 01and h indices, and this leads us to conclude that the equations immediately above are equivalent to, t~a)‘d Tcad - hi4y6)e~ (y5yu)ed Tcod - hot~~y”)~~
Tcsd = 0 Tcsd = 0
Finally, equation (4.9) implies one more simple constraint
(4.13) (4.14)
on Tasc.
t~sa)al, Gab + (crag)% T-2 = 0
(4.15)
The constraints in equations (4.8) (4.13) (4.14) and (4.15) imply that the superfield T,$ must be of the form, Tas’ = iHyOy@yO)ac;
@ = -M
+ y5y”G, + iy5N
(4.16)
for some superfields M, N, and G, . In terms of these superfields the supertorsion component TEByand the supercurvature component RabKA are found from equations (4.9) and (4.10) to be, YSGs Ta8’ = - +E,~ RabKA= -i
$(y’i%b
[6~&jM
(4.17) f
E,B~‘NI
(4.18)
To lowest order in 8, the superfieldsM, N, and G, correspond precisely to the auxiliary fields found for minimal component supergravity [9]. In the presenceof matter superfields, the superfields M, N, and G, are set equal to certain matter dependent superfields. With these substitutions, equations (4.16) (4.17), and (4.18) are equations of motion. In particular, equation (4.17) has been obtained previously [lo] as an equation of motion in the presenceof the (1, l/2) matter supermultiplet. We may define a complex scalar superfield by the equation, R- = ift(P-o,,yO)* &bKA
(4.19)
and from equations (4.4) and (4.18) we are able to deduce the results below. Da-R- = 0;
R- = 3(M - iN)
(4.20)
Thus R- is the chiral scalar superfield introduced by Wessand Zumino [3]. In a similar fashion, we find that the superfield Gab of Wess and Zumino is related to G, by the equation, Gob = i(yOP+yaLb 6
(4.21)
452
BROWN
AND
GATES
The reason this superfield appears on the right hand side of equation (4.17) in place of zero as in reference [3] is that we have replaced the constraint TaBy = 0 by the constraint (yayopb R,, %A= 0. When coupling supergravity to either the (l/2,0) or (1, l/2) matter supermultiplets, these constraints are equivalent. This is not surprising since both of these supermultiplets are described by scalar (chiral and real) superfields. Now we turn to the problem of calculating the spinorial derivative of the matrix super-field @. To this end, we note the spinorial derivative of this superfield enters the Bianchi identities (4.4) and (4.5) as, i(yOyol)& REcKA+ ~~~~~~~~ + permutations -2&To$
+ WCmKA(uKAh
- i2(y”ya),,
Also (4.17) suggests that we should consider the additional
= 0
(4.22) (4.23)
7’2 = 0
Bianchi identity,
[ED, , 44, Dc>+ [[Do, Da>,Q4 + [PB , Do>,QJ = 0
(4.24)
which gives the following information, Q~~D,G,
-
iXy”y%
Cod
+
&,‘a
=
(4.25)
0
Equation (4.23) we use to express parts of the curvature supertensor in terms of the torsion supertensor and its first spinorial derivative.
R~a-~~(~+~x~)~c = --i W+)bd [~wTI,+)B~ + i(r”~-r% T,B~I R Ba+KA(~oP+%A>bc
=
+ -
i
4[(y”p+)ad (y”P+)cd
&b+ tTI c+)/
&+I
Tb+)sd
-
h’P+)bd
&+I
Tlc+)~dl
(4.26)
In these expressions, the plus or minus sign following a spinorial index indicates multiplication by (1 + ys)/2 or (1 - y6)/2. By contracting the b and c indices, in the first of these results, we learn 2D,M
+ (y6ym)ba DbG, - i2(y5)b, D&
= -i2(y00as)ab
T,;.
(4.27)
Furthermore, by adding the two formulae in (4.26) together with the conjugate formulae we can express the curvature RaaxA as, R aBxA=
-(Y~Y~%A),~ +
TaBa+ i(udbd hTmd
i[(~+~~~~“)““(~“~+)da
+
(~-GO’“)bch’o~-)da~
Db
T,B~
(4.28)
Similarly, we may use (4.25) to derive an alternative expression for RaBKA. By equating the two expressions we find another equation which relates the spinorial .derivative of Cp to the torsion supertensor. Finally, we observe that equation (4.20) implies as a consequence, D,M -/- i(y5)ba D,N
= 0
(4.29)
BIANCHI
IDENTITIES
AND
THE
StJPERCOVARIANT
453
DERIVATIVE
We have thus derived a set of three linear equations which may be used to express the spinorial derivatives of M, N, and G, in terms of a component of the torsion supertensor.
This last result we substitute R cal3v=
into (4.25) to express Raaa,, in terms of TmBd. iW”yJcd
T,Yd
-
i&aBys(yoy5~~af”)~d
Note that (4.30) implies an algebraic constraint N, and G, . (y5y”)ab D,G, = D&l This can be seen to be equivalent superfield P,, .
(4.31)
T,,id
on the spinorial
derivatives
(4.32)
- i(y5pb D,N
to an equation
of M,
of conservation
for the matrix
Da@, = 0
(4.33)
So far we have been able to express most of the nontrivial geometric supertensor in terms of Tadc or Tag. The only remaining exception is RmoxA. There are two ways in which we may proceed. First, we note the Bianchi identity in (4.5) implies, - Dt, RbjBKA- i (YO~&)~~RnaKh- DBRnbKA= 0.
(4.34)
This combined with (4.31) yields an equation for RasKn , R UBKh= -i(uauB)ab D,TKAb- i&,BK~DaDJ(P+@P+ - P-@P-)
y”)ab
(4.35)
and a constraint on the spinorial derivative of reDa. DaTEDa= 0
(4.36)
Alternately, we find from the Bianchi identity in (4.24), D,TNBd
=
@,I
TclBld
+
~~~~~~~~~~
+
jWaOPA(u,Jdc
+
TcdTo~d
(4.37)
+
(4-W
This we solve for ReaMA and recalling (4.16) and (4.17), we find, RaBlc,a = -j(cA)cd DJ’k
+ ~~~~~~~~~~ G&,
+
4
‘WuKA@m@m)
GA~~$IGs
Thus, we have found two alternative ways to write RasK,,in terms of ToaCand Tnsc. Compatibility of these expressionssimply give more differential constraints on TnRc and Tagc.
454
BROWN
AND
GATES
Finally, we note the result
WhKa
=
&,I
&I~I
Ka +
G~‘G&VK~
-
i(@yr,#,
Rdlofa
follows from the Bianchi identity in (4.24). At this point we can end our investigation of the Bianchi identities. We note that if the 8 = 0 values of all the curvature and torsion tensors were known, then equations (4.30), (4.31), (4.37), and (4.39) imply that all higher spinorial derivatives of these quantities can be expressed in terms of the 0 = 0 values. But as we shall demonstrate in the next section, the required 0 = 0 values can be determined (see appendix A). At this point we can make some comparison between the minimal component theory and the nonminimal component theory. As shown in reference [2], there are many forms of the nonminimal theory. For simplicity we consider the II = ---I version of the nonminimal theory. The constraints for this theory are Tabd = 0;
Tabs = -i
&(y0y6)ab ;
RaeKh = 0 (4.40)
By repeating arguments analogous to those given earlier in this section we find there exists a matrix superfield @ such that Tao” = i~(y”y@yo)ac;
Furthermore,
@
X
-A4
+ iy”V, + y5ynG, + iy5N.
(4.41)
it can be shown that T,aB
=
(~“~af)ob
(4.42)
Tb
for some spinor superfield Tb. Also, we find that the 0 independent following quantities
sector of the
(4.43)
are not algebraically related to any of the 19independent sectors of the geometric supertensors. All of these new quantities imply that with the constraints in (4.40) the supergravity multiplet includes, in addition to the minimal set of fields, a vector, a second axial vector, and two spinors. These fields correspond to the 0 independent sectors of V019
(y5y”yo)ab DaTbl’,
Tb,
(y5y”yo)ab DaTbm’
This corresponds to the set of fields found in reference [ll].
(4.44)
BIANCHI
IDENTlTlES
V. THE 0 EXPANSION
AND
OF
THE
SUPERCOVARIANT
THE SPINORIAL
DERIVATIVE
SUPERCOVARIANT
455
DERIVATIVE
In this section, we discusshow the Bianchi identities in the previous section allow us to reconstruct the 0 expansion of the spinorial component of the supercovariant derivative. In general, we know that the spinorial component of the supercovariant derivative can be written in for form,
for some operators Vu) (suppressing the lower indices). Each of these operators is expanded over the algebra generated by the O-derivative, x-derivative, and the Lorentz generator, Q(i)
=
[&iqm
a,
+
[@)]u
3,
+
$.[&Ciqd
M,,
(5.2)
where the coefficients A, 2, and e are functions of x alone. The transformation law for the spinorial component of the supercovariant derivative is
SD, = FL D,l
(5.3)
where Q, the parameter of supergauge transformations, has a form similar to that of D, . sz = ~(0) + ebip
+ BefP) + i0peiP
+ f7+fec2~)
+ Beeb@) + (Oe)z ~2~6)(5.4)
Each of the component operators J2ci) has the samegeneral form as the component operators VCi). At this point, we find it is essential to work in a generalized WessZumino gauge [6]. Without loss of generality, we make the modification
on the lowest component in D, . To find a W-Z gauge, we need only retain the following term from the transformation law. SD, = -d&'
(5.6)
It is immediately apparent that sZj,l’may be used to “gauge away” 6:). Furthermore, Q@), Q3), and !2?) may be used to “gauge away” the antisymmetric part of V$. Thus, by only using the transformation, we find that in some W-Z gauge, D, can assume the form
for some operators V, and V,, . By considering the B independent sector of (4.1)
456
BROWN
AND
GATES
and (4.18) we determine these operators. Continuing, we note that equation (5.6) also implies that the operators Qi2) and eA3’ satisfy the relation $7(3) a
=
-Q(2)
(5.8)
a
in some W-Z gauge. Therefore, the quadratic 9 terms in D, can take the form,
Given this fact, we can compute the I3 independent sectors of Tnas, Taab, and RnaKA, using (4.2) to define D, . When this is done, we find that
The terms not esplicitly given above (the deltas) are functions which only depend on .4(l), B(l), C(r). It is clear that by making the definitions A(J)b aa
= --
~(4)b at-c
_
A(db. aa
,54kA aoL
= -
3 &4hA an
@4)8 no
_ -
_
A (4h an
B(4)8 au
_
A(4)” aa
(5.11)
we find, KA oc &4)KA R acx
(5.12)
The deltas are, in general, noncovariantly transforming functions of the component fields. A covariant quantity is one which has a transformation law which does not depend on the spacetime derivative of the gauge parameters in +Qco).Since the torsion and curvature super-tensors are by definition covariant quantities, the “shifts” which take us from the component quantities with carats to those without are necessary. We have encountered these sort of shifts in several of our previous works [2, 41. The fact that Taas is identically zero determines BAtj8 to be zero. Similarly, equation (4.16) determines A,,“)b . Finally equation (3.31) determines CLt)K’. The spin connection, OJ,~~Y, is then defined by (4.17). Now a convenient feature of the gauge in (5.9) is that TRLIB,ToLab,and RaaK* are all independent of 6,“) . In order to determine this operator we note that is contained in the B independent sectors of (5.13)
The first of these vanishes identically
from (4.1). The second quantity can be found
BlANCHI
IDENTITIES
AND
THE
SUPERCOVARIANT
DERIVATIVE
457
from (4.18) and (4.30). When all of this is done, we find that to quadratic order in 6’ the spinorial covariant derivative is given by, V, - i&“u”“e), BasKAMK. + g80(y”(s - iy5P) 2)a + gt7y5e(yoyys - iy5P) 2)@ + i@~5y~~tyof%), + om
D, = a, - i&OyaS),
(5.14)
where the following definitions are to be understood.
The result in (5.14) can be used to compute the 6’ independent sector of all of the superspace curvature and torsion tensors. These are given in an appendix. Arguments given previously 161indicate that the generator of local supersymmetry must be of the form,
and thus the commutator
of two supersymmetry generators is [12],
E ‘a =
- i(~lym~2) ‘y,” (5.16)
hlnR = -i(&fr,) 5’” = -i(ZIya~2)
w:’
- i )(+#‘E~)
BeTa
e,”
where QE and Qn, are the generators of local Lorentz and general coordinate transformations, respectively. To calculate the transformation laws for the various compo-
458
BROWN
AND
GATES
ponent fields, we may simply utilize (4.30) and the result for TBBcgiven in Appendix A. The supersymmetry generator in equation (5.14) may be rewritten as, ii?&) = E”&
+ i(+“e) D, - $@ebRabKAMKA+ o(e’)
(5.17)
Thus, we see that this supersymmetry generator differs from that given in the last of reference [3] by an E dependent Lorentz transformation. This can be understood by noting this supersymmetry generator preserves the W-Z gauge of D, ; whereas the generator given in reference [3] preserves the W-Z gauge only up to a Lorentz transformation. It remains for us to determine the cubic and quartic 6 sectors of D, . Before discussing these, we note that by using the remaining freedom, sZL6),we can choose a gauge where (#y
9%) = 0
(5.18)
At this point we have used all of the gauge freedom in Q, except that in Q(O). These remaining degrees of freedom correspond to local supersymmetry, general coordinate, and local Lorentz transformations which we must retain in the theory. Although we shall not completely present the two remaining 8 sectors of D, , we will describe the procedure which can be used to determine these remaining sectors. First using (5.1) and (5.14) we can compute the supertensors TaED and RaaKAto quadratic order in 8. The 0 independent sector of 2(DraTblsD)
DD
+
~(kz&~i?‘?
M,A
(5.18)
will contain the as yet undetermined operator 6 bi). But the results in (4.30), (4.31) and (4.37) may also be used to evaluate the 8 independent part of this expression. Thus, we can write an equation which expresses 6%’ in terms of previously determined quantiries. In a similar manner, we note that the 6’ independent sector of
2(&zD,]T,,D) D, + ;-(Dd’b&,F”) M,A
(5.19)
can be computed from (5.1) and (5.14) with ej$ now determined. It is easily seen that the 0 inedependent sector of (5.19) will contain the only remaining undetermined operator, eA6’. Once again, however, we can evaluate the t9 independent sector of (5.19) by using (4.30), (4.31), (4.37) and (4.39). This would give the final equation determine 6L6). Thus in this section we have proven that by using information derived from the Bianchi identities, we are able to construct the vielbein and connection superfields. In some previous works [I, 1,4] these superfields have been constructed by actually solving the constraints on the torsion and curvature supertensors. So we have demonstrated the equivalence of these two different approaches.
BIANCHI IDENTITIES AND THE SUPERCOVARIANT
DERIVATIVE
459
VI. DISCUSSION We have explicitly demonstrated that an analysis of the superspaceBianchi identities is completely equivalent to solving the constraints on vielbein and connection superfields as was done in references [l, 21.We have also shown the existence of somesimilarities in the theory of supergravity, with minimal component fields, in two, three, and four dimensional spacetimes.For instance, the constraints for all of thesetheories are the same as given in (3.20). Furthermore, in each of these theories the auxiliary fields form the samerepresentation of the Lorentz group as the product
eaeb
(6.1)
In two and three dimensions,this is a scalar under the Lorentz group. In four dimensions, this is a reducible representation transforming as a scalar, pseudoscalar, and axial vector. If this were to continue for the extended supergravity theories, we would expect the auxiliary, bosonic fields to fall into the samerepresentation as
where i and j are the O(N) labels of the Majorana spinor. For instance, for O(2) supergravity this would suggestthat the auxiliary, bosonic fields in addition to those of O(1) supergravity are isovectors (scalar, pseudoscalar, and axialvector) and isoscalars(vector and antisymmetric tensor). Another interesting feature of the minimal component theories is that the required auxiliary fields all appear at the linear 0 sector of the spinorial supercovariant derivative. As a consequenceall of auxiliary fields appear in the commutator of two supersymmetry transformations. This is in sharp contrast with the nonminimal theories where additional, independent, auxiliary fields appear beyond the linear B sector of D, . Thus, we think it is a very important feature to have all of the auxiliary fields in the linear 0 sector of D, . In closing, we note that the analysis of the superspaceBianchi identities and the subsequent construction of the vielbein and connection superfields can be carried out for an arbitrary superspace.
APPENDIX
A: LOWEST ORDER TERMS IN GEOMETRIC
SUPERTENSORS
Here we give the 8 independetn sector of all of the geometric tensors of the eight dimensional superspacewith minimal component fields. We define the curvature and torsion supertensor by the equation, P, , &I) = 2T/mDDD + +RAPMKA.
460
BROWN AND GATES
From equation (5.14) we compute these geometric tensors to lowest order in 8.
These expressions may be compared with those which are obtained in the nonminimal component approach [4, lo]. In computing these results, we have made use of the fact that the vectorial component of the supercovariant derivative to first order in B is given by
APPENDIX
B: EQUATIONS OF MOTION
FOR SU~ERSPACE
Here we give the equations of motion for superfield supergravity coupled to (1, l/2) or (l/2,0) matter supermultiplets described by a bispinor +b . Tabs = -i$(y”y6)aa Tabd = 0
~,BY”(yoy5ys>~b TV: = -iDt,((p+Tp+ &6aL3
+ p-Tp-)
y.)‘,
+ i(fjba
DbJa
= Wmwyd - 2(rlmJaJb- JsJa) + UW~~P+TP+ + p-Tp-) y,~ysysy’)~~ - @U?d~~sye~~~*~ - i%~~~s~~)c* DcDdJ,s - w?~&JA
BIANCHI
IDENTITIES
AND
THE
SUPERCOVARIANT
461
DERIVATIVE
In these equations, the quantity J, is defined by, J, = Tr(y5y,T)
and has been referred to as the “supercurrent” for some time [13]. The superfield ~a,, is the superspace analog of the ordinary energy-momentum tensor. Additionally, the superspace energy-momentum bispinor, +b , is subject to the constraints (Y”T)ab = -(Y”%z Dana,,
0
=
in analogy with the symmetry and conservation requirements on the ordinary energymomentum tensor. Recently, Ogievetsky and Sokatchev have given an algorithm for the derivation of Tab for an aribtrary superfield action [14]. APPENDIX In the four dimensional
C:
NOTATION
AND
CONVENTIONS
superspace, we use a representation
where
y” 5%(02, iu3) y5
3
i&,y"y~
=
iyoyl
=
-ial
The full set of Dirac matrices is 1, y”, and y5. y”yP = -.q%Q + i&y5 diag(T@) = (-1,
1);
--
co1
-@l
= 1
In this representation y” is antisymmetric, while y”y” and y”y6 are symmetric. In the five dimensional superspace, we define, y” = (u2, icr3, id) y~y6 = -+S _ ieaS8y-
diag(q*B) = (-1, For the four dimensional we define
1, 1);
6012
=
-eo12
=
1
spacetime embedded in the eight dimensional
ye = (CT” @ u2, iI @ d, ic? @ u2, iI @ u3)
1
We therefore have, yUy.8 = -.qd - iuCX6 diag(q”fl) = (-1,
1, 1, 1);
l 0123= -co123 = 1
superspace,
rt62
BROWN AND GATES
and the matrices y”, y”y5y”, and y”y5 are antisymmetric, while y”y” and y”&’ are symmetric. All of these representations are Majorana representations so that the charge conjugation of a spinor is equivalent to ordinary complex conjugation. We use capital latin indices (A, B,...) to denote a supervector label. Lower case latin indices (a, b,...) denotes fermionic labels. Lower case greek indices (01,p,...) denote bosonic labels. Letters from the beginning to middle of the alphabet denote components of a tensor with respect to an anholonomic coordinate system (ie “flat” labels), while those from the latter part of the alphabet denote holonomic components (“curvy” labels). The distinction between flat and curvy spinors, however, is immaterial.
MKh z Generator of Lorentz transformations
ACKNOWLEDGMENT We wish to thank Warren Siegel and Lee Smolin for interesting and stimulating discussions.
REFERBNCE.3 1. W. SIEGEL, Nucl. Whys. B 142 (1978), 301. 2. W. SIEGEL AND S.. J. GATES, JR., Harvard preprint HUTP-78/A019 (1978). 3. J. MESS AND B. ZUMINO, Phys. Lett. B 66 (1977), 361; Phys. Lett. B 74 (1978), 51; Phys. Left. B 79 (1978), 394. 4. S. J. GATES, JR., AND J. A. SHAPIRO, Phys. Rev. D 18 (1978), 2768. 5. J. WESS,Supersymmetry-Supergravity, Lectures given at the VIII G.I.F.T. Seminar, Salamanca, 1977, to be published in Springer Tracts. 6. S. J. GATES, JR., Harvard preprint HUTP-78/A028 (1978). 7. J. Wsss AND B. ZUMINO, Nucl. Phys. B 78 (1974), 1. 8. P. BREITENWHNER, Phys. Lett. B 67 (1977), 49; Nucl. Phys. B 124 (1977), 500. 9. IS. ‘s. STELLE AND P. C. WEST, Phys. Lett. B 74 (1978), 330; S. FERRARA ANI, P. VAN NIETJWENHLJIZEN, Phys. Lett. B 74 (1978), 333. 10. L. BRINK, M. GELL-MANN, P. RAMOND, AND J. H. SCHWARZ, Phys. Lett. B 74 (1978), 336. 11. B. DE WITT AND M. GRISARU, Phys. Lett. B 74 (1978), 57. 12. Equation (5.15) is not exact as written. Since the supersymmetry generator 8, is field dependent, there are correction terms which must be included in this equation. See Ref. [6] for a complete discussion. 13. S. FERRARA AND B. ZUMINO, Nucl. Phys. B 87 (1975), 207. 14. V. OGIEVETSKY AND E. SOKATCHEV, JINR preprint E2-11528, Dubna (1978).