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The geometrical off-shell structure of pure N = 1; d = 10 supergravity in superspace
Volume 169B, number 4
PHYSICS LETTERS
3 April 1986
THE GEOMETRICAL OFF-SHELL STRUCTURE OF P U R E N = 1; d = 10 S U P E R G R A V I T Y IN S U P E R S P A C E B.E.W. NILSSON 1 CERN, CH-1211 Geneva 23, Switzerland
and A.K. TOLLSTI~N Institute of Theoretical Physics, S-412 96 Gbteborg, Sweden Received 19 November 1985
The N = 1 pure supergravity Bianchi identities are solved in d = 10 superspace off-shell. The solution contains the d = 10 conformal supermultiplet of currents, a necessary condition for consistent coupling to the supersymmetric Yang-Mills theory.
With the recent exciting developments in supersymmetric string theories there has followed a renewed interest in their low-energy limits, in particular d = 0 supergravity [ 1,2] and its coupling to the supersymmetric Yang-MiUs theory [3,4]. The anomaly cancellations of the closed (supersymmetric) heterotic string [5,6] with an E 8 × E 8 or SO(32) Yang-Mills gauge group in its low-energy field theory make this theory seem especially promising. When dealing with supersymmetric field theories of such complexity, it is of course desirable to construct them in superspace. In this paper we will describe a highly non-trivial and, in fact, unexpected property (see below) of pure d = 10 superspace supergravity as it was presented in ref. [2]. This property is essential for a consistent coupling to the supersymmetric Yang-Mills theory [7]. Pure N = 1 supergravity in superspace is quite well understood, and is known to contain an anti-symmetric third-rank tensor in one of its superspace torsion components [2]. In ref. [2], this tensor was constrained to be the field strength of a second-rank tensor potential. This was achieved by an explicit intro-
On leave of absence from the Institute of Theoretical Physics, S-412 96 GOteborg, Sweden.
0370-2693•86•$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
duction of superspace Bianchi identities ensuring this property. When trying to couple pure supergravity to the d = 10 supersymmetric Yang-Mills theory in superspace, one is immediately faced with the following problem. As can be seen from the work of refs. [3,4], the coupled Einstein-Yang-Mills theory formulated in x-space does not contain a Yang-Mills gauge-invariant exact three-form. The three-forms that occur are either exact, i.e. H ' = dB and non-invariant since B transforms under gauge transformations [3,4], or not exact, in which case H = H ' + 6oyM (where 6oyM is the Chern-Simons three-form), and hence gauge-invariant [3,4]. It is of course the gauge-invariant one of these that ultimately has to be related to the three-form occurring in the torsion component mentioned previously. Thus the three-form in the supergravity sector cannot be closed. Now, the curl of this three-form is proportional to one of the components of the conformal supermultiplet of currents [3,8], which also contains a symmetric traceless second-rank tensor and a 7traceless vector vector-spinor. These must of course all emerge when solving the superspace Bianchi identities of pure supergravity. The coupling o f d = 10 supergravity to Yang-Mills in superspace has been described in a previous paper 369
Volume 169B, number 4
by Kallosh and Nilsson [9]. There the Bianchi identities and the constraints of the coupled theory were presented, without, however, explicitly demonstrating the consistency of these equations. The existence of this coupling, using the constraints of refs. [2,9], will be guaranteed once the multiplet of currents is shown to be present in the super torsions of pure supergravity. This will now be shown. (Note that, as explained at the end of this paper, this coupling does not require a fully off-shell solution of the torsion Bianchi identities, and indeed some on-shell structure will emerge from the analysis below.) The chiral superspace introduced in ref. [2] is spanned locally by
z M =(xu,om),
(1)
where xU are bosonic and 0ra are fermionic co-ordinates. By introducing the vielbein E A , which splits into the bosonic zehnbein E a and the fermionic 16bein E a satisfying the Weyl condition E a = Eb(Tll)b a as well as the Majorana condition ff~aCab = E b , and the Lie-algebra-valued spin connection ~A B into superspace, we can construct the torsion tensor T A = DE A = dE A _ E B ^ WBA ,
- E B A RB A ,
TOm = O.
(9)
Note that (7) is related to the corresponding expression in ref. [2] by a Fierz transformation. This new form leads to some useful simplifications. The analysis of (4) proceeds as follows. We will denote by (ABC, D) the tangent space version of (4): (ABC)
(RABC D + DATBc D + TABETEc D) = O, (10)
where the sum contains all cyclic terms. It is convenient to solve the (ABC, D)'s in the following order [21 : (abe, 8);(abT, 8) and (abc,d);(a~% 8) and (abc, d); (a133', 6) and (a~c, d); and finally(a/33,, d). {i) (abc, 8). Since RABC o is valued in the SO(l, 9) Lie algebra it tollows immediately that Ta=L.
(11)
(ii) (abT, 8) and (abc, d). Using (11), these read (ab% 6):
Rab,y 6 + 2irLr6D(a Tb) -- 2i( T')'8 )(a b) = 0,
(12)
• d DbTe) R(abe) d + 16(a
i(Ta)(ab Tlai c)d = 0.
(4)
From (12) and (13) we obtain the following two equations: 72~6D(a Tb) = ( T(.r T6))(ab) ,
(5)
As is well known, (5) follows from (4) [10] and will not be discussed any further• The next step in this approach is to subject T A to constraints and extract the information now contained in (4)• The constraints to be used here are the ones introduced in ref. [2], except for one trivial modification. As asserted in ref. [9], these constraints will also work for the coupled Einstein-Yang-Mills system• This may seem rather surprising but is proved correct by the results found here (see, however, the discussion at the end of the paper). The constraints are
(13)
(14)
l i T .yt6J(ab ,,, ~ ~. tT~6-Jc) ~ d + (Te~)(abTledc)d "i~. = 6 (adD b Tc).
(15)
(14) is just the symmetric part (in 7 and 8) of (12), while (15) is arrived at by eliminating the curvature from (12) and (13). In order to proceed we expand T b C as in ref. [2] and find that (14) and (15) imply T bC = --("fl~la)bCY# + (Ta #'y8 --68#a')"~a)bC Y#3,8(16) where Ya~'r is an arbitrary totally skew-symmetric tensor and
Tab • = -i(,),'V)ab,
(6)
y
TabC = i6(aCTb),
(7)
Furthermore,
370
(8)
(3)
and DRA B = 0.
= i6jq" a,
Obc,d):
These tensors satisfy the Bianchi identities D TA =
rJ
(2)
and the curvature tensor RAB = do)AB - 6oAC A ~C B.
3 April 1986
PHYSICS LETTERS
= - ~6(~/a)abDaTb"
(17)
(7cq ... as )abDa rb = O,
(18)
iTa = D G
(19)
where qbis a scalar superfield. We use the same notation for this field as in ref. [2] since they are put equal onshell [8]. (17) now implies that
r
=
=
(20)
Using these results in (12) we find
(28)
W a = i51 1 "G ~a = ~1( D^b Y ~#v)(T#v)ba '
(29)
Wa ~(3 = -- -~l a. G ,x¢a = ±12 [(fib Ya(3v )(T'r)ba
(30)
1 tD b y y 6 [ ~ JW~ ~t6 aba 1 w#] J J,
-2~
= -2i[(7[.r)lablD~] ~ + 12(7~)ab Y~ ~ 1.
= O.
Wa
(21)
(31)
The curvature componentsRa#3, 6 are Ra#.r~ = ½i[2(Ta[-r7* ])a - (Tv~ 7#)a]
It also follows that 1. o~)ab D ot dp + i(vu#'~)abK(~#v. DaDbO = i~(7
(22)
+ 2inaD(l)~ 1)a b Tb , ([)o)a b = D Sab + Togab.
1. e ~[2T ( v6) e ( b ~"
R[~0~I ~ + i8[~
ra
e
(V~)eO](Vu~)~) a
(23)
where we have defined I)a =- D + i v a .
(iv) (o437, 8) and (a&, d). (a~7, 5) reads
r lar a - - o ,
TO = 0,
(35)
(24)
(25)
+ ¼tr(T( Tol%y~)],
(36)
Da(T #Tv~ en) a = - 2 [ D D tr(T¢] 7~/~en) + tr(TDT~] 7v~ cn)] ,
(37)
where now 6 a = Oa + ~i T a . To facilitate the analysis of (34) to (37) we expand the two spinor indices in DaDbYc~v in terms of'y-matrices and obtain
where Ga#3,# = GuT a = 0, and h a Y#3'a ' DaDb Ycqc~:~a = Qa~az,xa,#( "1
+ 3(Ttav)labWbt~] + (7t~y6)ab Wb ,
(34)
D^ ~ ( r a ~ 8 ) ~ = -8 [R~ay8 + Z1 D¿~ tr(Ta] V78)
It is important to note that the superfield ~ does not enter into (23) (except in Da), and consequently the equation of motion for T a is not derivable (recall that Ta Io=0 is the physical spin-l/2 field in the on-shell theory [2] ). By decomposing the Rarita-Schwinger field strength [2], TaeC = au#e + 2G[ d(3,e] )d c + Gd (,,/a/3)dC,
(33)
and by contracting (a/3c, d) with 8d c , (Tv6)d c and (7v8 cn)d c we find, respectively,
+ ½i(V6)bcTf = l)(b Y~y8 (T~ Bq'8 -- 6 6~'y "~ )c) d,
(32)
where
Note that the three-form Kc~#v defined by (22) is unrelated to Y~t~y" [iii) (a~% 6) and (otbc, d). Here we solve for R a_v~ from (aflT, 6) and plug it into (o~bc, d). We thus ogtain
where
(27)
W a = G a = 0,
Rabv~ = 2i(T[-r7~ ] )(ab)
_
= o,
the solution of (23) can be expressed as follows:
which we will solve by putting [2]
+ (7,r6uvO)ab Y
3 April 1986
PHYSICS LETTERS
Volume 169B, number 4
)ab
(26) + iQa~azaa,O~ ... t3s(7~ "'"as )ab"
(38) 371
Volume 169B, number 4
PHYSICS LETTERS
The Ricci identity then immediately provides us with explicit expressions for Q~x~2~a,# andQ~aa2~,~1 #s in terms of D~-derivatives on ~ and Y~t3~ and)~°r ~'roducts of Y~0./'s: 1
+ ~ [(D[ (p) ra~] n - (Ded~)Yta~erLr I ~ ] -36Y~[~
eY # 3 ' ] e '
+ (1/5!) e~1'''~s ex'"e4 [c Y#q,] el Ye2...e4) ,
(39)
(40)
which is in accord with the Weyl-projected indices appearing in (38). The objective now is to derive from (34) to (37) as much information as possible about Qa1~2~3,01#2#~" In particular, we must verify that the two bosonic components of the conformal supermultiplet of currents are present. [The fermionic partner of these is Go~a ; see (25) and (29).] Qal~2a3,#~# #a split into irreducible tensors under SO(l, 9) as follows [ 11 ] : 120®
S
3 April 1986
portant check of the consistency of this approach. Note that (43) must also be true in the coupled Einstein-Yang-Mills theory and therefore should be viewed, not as a field equation for a two-form potential, but rather as a Bianchi identity for a six-form potential. It is of course only in an off-shell theory that this statement is meaningful and indeed this six-form potential played a crucial role also in the discussion of the linearized off-shell theory in ref. [8]. Finally, we come to the most important terms on the right-hand sides of (41) and (42), namely the representations 54 and 210. As is known from refs. [3,8] the conformal supermultiplet of currents contains a 54 ( m t and 210 (
) and these must be non-zero in or^ der to make a coupling to Yang-Mills possible. Q [~" ~,~3 ~ ~, ,lma~a~ 1~,,.oJ does also appear but is not independent and indeed (37) relates it to the dual of 3'
Q [~1~2,~1#213,' Q[~I ....... c~]
_
1
7-i e~l ...a6
#1 .., #4
X (QT#lth,#3#47 - 6 YTth #2 Y#s#47 )"
(44)
Then from both (36) and (37) we obtain
120=1+54+210+770
e
1
Qe[~#,78] = - ~ [D [a Y#-y8] - 2(D[~b) Y#76] + 1050 + 1050 + 4125,
(41) -
120 ® a 1 2 0 = 45 + 210 + 945 + 5940.
(42)
We will now discuss each of these irreducible tensors in turn. First we note that the 1 and 4125 (with Young tableau ~ ) in (41) are zero due to (28) and (31) respectively. Next we consider 5940 in (42) which appears on the left-hand side of (37). However, there is no such tensor on the right-hand side and it must therefore vanish. 1050 and 1050, on the other hand, will be determined by (37) in terms of y 2 while (36) and (37) imply the same expression for the 945. We now come to the more interesting terms on the right-hand sides of (41) and (42), namely 54,210, 770 and 45. First, Q(770) is found to be related to the Weyl tensor. Secondly, when deriving that Q(45) vanishes, we also discover that D~ Y~t~'~ - 6(D~q~) Y # ' r = 0.
(43)
(43) can be derived in several ways from the Bianchi identities (35) to (37) and therefore provides an im372
12Lt.,Y
1e l
(45)
Similarly, from (36) we find _
144 0 ~/~ =R(~¢) + rl~#l--]¢ + 8D(aD#)¢ 7 ~'r~(c~,~) - 8r/~a(Dr¢) 2 + 8 D ¢ Dag) + 144(r/a#YS'seY ~ e - 7 y 3 ' 8 y/~w6),
(46)
and if we recall that in (41) the singlet vanishes, the trace of (46) becomes R + 18I--1¢ - 72(D ~) 2 + 432 y~#7 Y~¢7 = 0.
(47)
It is important to note that (47) does not imply the rest of the field equations, but does constitute a restriction on possible couplings to matter (see below). This completes the analysis of the Bianchi identities (4) and the constraints (6), (7) and (8). The results are easily summarized as follows. The system has been shown to be off-shell with all torsion and curvature components expressible in terms of one unconstrained scalar superfield q~together with a constrained skew-
Volume 169B, number 4
PHYSICS LETTERS
symmetric third-rank tensor superfield Y~#7" The scalar superfield starts with the physical fields ~(x) and ha(X ) while all components from 02 and up are auxiliary. The superfield Ya#'r has as its lowest component an x-space three-form which is the dual o f the field strength of a six-form potential * t. This follows from (43). At the next two levels in YaB't we find the Weyl tensor C~B.r8 and its on-shell superpartner Gc~Ba together with the conformal supermultiplet of currents consisting of the vector-spinor wa s, the traceless e n e r g y - m o m e n t u m tensor Q76 (~,B)~6 and the fourform current Qe [at~,~ ] e" These currents are related to physical fields by (28), (29) (recall that TaBTaB7 = - 7 2 G 7 + 16G?7), (45) and (46). This should now be compared to the linearized offshell theory obtained by Howe et al. [8]. These authors found the off-shell fields to be a scalar superfield ¢ [identical to the one introduced in (19) above] and the three-form prepotential V~B~. V~B7 satisfies some constraints with the consequence that it is essentially equivalent to an unconstrained scalar superfield T plus the conformal supermultiplet [8]. The auxiliary superfield T does not enter into the theory as presented here. However, this just corresponds to a situation where one of the superspace field equations has been implemented, namely D3'~B~DVaB~ = 0
(48)
in the linearized version o f ref. [8] (48) could have made impossible the coupling to Yang-MiUs, but fortunately this is not the case as we will now explain. In N = 1 supergravity in four dimensions one encounters a similar situation. The d = 4 superspace field equations are two in number, GaB = 0 a n d R = 0, and i f R = 0 (but not GaB = O) is imposed, one can only couple to matter systems which are consistent with zero curvature scalar and zero ?-trace o f the Rarita-Schwinger field equation *=. Similarly in our case, (48) should
3 April 1986
be interpreted as the reason for the constraints (47) and (28). However, these constraints are consistent with the coupling to Yang-Mills in d = 10 as was asserted in ref. [9]. This will be fully demenstrated in ref. [7]. Finally we want to emphasize again that the fact that the torsion constraints used here (and in refs. [2, 9] ) do not lead to the on-shell theory is quite remarkable. This is not the case in for example the chiral N = 2 , d = 10 theory analyzed in ref. [14]. One of us (B.N.) is grateful to L. Mezincescu and E. Witten for stimulating discussions.
References [1] A.H. Chamseddine, Nucl. Phys. B185 (1981) 403. [2] B.E.W. Nilsson, Nucl. Phys. B188 (1981) 176. [3] E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Nucl. Phys. B195 (1982) 97. [4] G.F. Chapline and N.S. Manton, Phys. Lett. 120B (1983) 105. [5] M.B. Green and J.H. Sehwarz, Phys. Lett. 149B (1984) 117. [6] D.J. Gross, J.A. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. 54 (1985) 502; Princeton University preprints. [7] B.E.W. Nilsson and A.K.Tollst6n, CERN-TH 4354/86, Phys. Lett. 171B (1986), to be published. [8] P. Howe, H. Nicolai and A. van Proeyen, Phys. Lett. l12B (1982) 446. [9] R.E. Kallosh and B.E.W. Nilsson, Phys. Lett. 167B (1986) 46. [10] N. Dragon, Z. Phys. C2 (1979) 29. [11] R. Slansky, Phys. Rep. 79 (1981) 1. [12] A.H. Chamseddine, Phys. Rev. D24 (1981) 3065. [13] S.J. Gates Jr. and H. Nishino, Phys. Lett. 157B (1985) 157. [ 14] P. Howe and P. West, Nucl. Phys. B238 (1984) 181.
,1 The formulation of the on-shell theory in terms of a sixform potential is discussed in refs. [12,13]. ,2 We are grateful to K. SteRe for reminding us of this example.
373
PHYSICS LETTERS
3 April 1986
THE GEOMETRICAL OFF-SHELL STRUCTURE OF P U R E N = 1; d = 10 S U P E R G R A V I T Y IN S U P E R S P A C E B.E.W. NILSSON 1 CERN, CH-1211 Geneva 23, Switzerland
and A.K. TOLLSTI~N Institute of Theoretical Physics, S-412 96 Gbteborg, Sweden Received 19 November 1985
The N = 1 pure supergravity Bianchi identities are solved in d = 10 superspace off-shell. The solution contains the d = 10 conformal supermultiplet of currents, a necessary condition for consistent coupling to the supersymmetric Yang-Mills theory.
With the recent exciting developments in supersymmetric string theories there has followed a renewed interest in their low-energy limits, in particular d = 0 supergravity [ 1,2] and its coupling to the supersymmetric Yang-MiUs theory [3,4]. The anomaly cancellations of the closed (supersymmetric) heterotic string [5,6] with an E 8 × E 8 or SO(32) Yang-Mills gauge group in its low-energy field theory make this theory seem especially promising. When dealing with supersymmetric field theories of such complexity, it is of course desirable to construct them in superspace. In this paper we will describe a highly non-trivial and, in fact, unexpected property (see below) of pure d = 10 superspace supergravity as it was presented in ref. [2]. This property is essential for a consistent coupling to the supersymmetric Yang-Mills theory [7]. Pure N = 1 supergravity in superspace is quite well understood, and is known to contain an anti-symmetric third-rank tensor in one of its superspace torsion components [2]. In ref. [2], this tensor was constrained to be the field strength of a second-rank tensor potential. This was achieved by an explicit intro-
On leave of absence from the Institute of Theoretical Physics, S-412 96 GOteborg, Sweden.
0370-2693•86•$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
duction of superspace Bianchi identities ensuring this property. When trying to couple pure supergravity to the d = 10 supersymmetric Yang-Mills theory in superspace, one is immediately faced with the following problem. As can be seen from the work of refs. [3,4], the coupled Einstein-Yang-Mills theory formulated in x-space does not contain a Yang-Mills gauge-invariant exact three-form. The three-forms that occur are either exact, i.e. H ' = dB and non-invariant since B transforms under gauge transformations [3,4], or not exact, in which case H = H ' + 6oyM (where 6oyM is the Chern-Simons three-form), and hence gauge-invariant [3,4]. It is of course the gauge-invariant one of these that ultimately has to be related to the three-form occurring in the torsion component mentioned previously. Thus the three-form in the supergravity sector cannot be closed. Now, the curl of this three-form is proportional to one of the components of the conformal supermultiplet of currents [3,8], which also contains a symmetric traceless second-rank tensor and a 7traceless vector vector-spinor. These must of course all emerge when solving the superspace Bianchi identities of pure supergravity. The coupling o f d = 10 supergravity to Yang-Mills in superspace has been described in a previous paper 369
Volume 169B, number 4
by Kallosh and Nilsson [9]. There the Bianchi identities and the constraints of the coupled theory were presented, without, however, explicitly demonstrating the consistency of these equations. The existence of this coupling, using the constraints of refs. [2,9], will be guaranteed once the multiplet of currents is shown to be present in the super torsions of pure supergravity. This will now be shown. (Note that, as explained at the end of this paper, this coupling does not require a fully off-shell solution of the torsion Bianchi identities, and indeed some on-shell structure will emerge from the analysis below.) The chiral superspace introduced in ref. [2] is spanned locally by
z M =(xu,om),
(1)
where xU are bosonic and 0ra are fermionic co-ordinates. By introducing the vielbein E A , which splits into the bosonic zehnbein E a and the fermionic 16bein E a satisfying the Weyl condition E a = Eb(Tll)b a as well as the Majorana condition ff~aCab = E b , and the Lie-algebra-valued spin connection ~A B into superspace, we can construct the torsion tensor T A = DE A = dE A _ E B ^ WBA ,
- E B A RB A ,
TOm = O.
(9)
Note that (7) is related to the corresponding expression in ref. [2] by a Fierz transformation. This new form leads to some useful simplifications. The analysis of (4) proceeds as follows. We will denote by (ABC, D) the tangent space version of (4): (ABC)
(RABC D + DATBc D + TABETEc D) = O, (10)
where the sum contains all cyclic terms. It is convenient to solve the (ABC, D)'s in the following order [21 : (abe, 8);(abT, 8) and (abc,d);(a~% 8) and (abc, d); (a133', 6) and (a~c, d); and finally(a/33,, d). {i) (abc, 8). Since RABC o is valued in the SO(l, 9) Lie algebra it tollows immediately that Ta=L.
(11)
(ii) (abT, 8) and (abc, d). Using (11), these read (ab% 6):
Rab,y 6 + 2irLr6D(a Tb) -- 2i( T')'8 )(a b) = 0,
(12)
• d DbTe) R(abe) d + 16(a
i(Ta)(ab Tlai c)d = 0.
(4)
From (12) and (13) we obtain the following two equations: 72~6D(a Tb) = ( T(.r T6))(ab) ,
(5)
As is well known, (5) follows from (4) [10] and will not be discussed any further• The next step in this approach is to subject T A to constraints and extract the information now contained in (4)• The constraints to be used here are the ones introduced in ref. [2], except for one trivial modification. As asserted in ref. [9], these constraints will also work for the coupled Einstein-Yang-Mills system• This may seem rather surprising but is proved correct by the results found here (see, however, the discussion at the end of the paper). The constraints are
(13)
(14)
l i T .yt6J(ab ,,, ~ ~. tT~6-Jc) ~ d + (Te~)(abTledc)d "i~. = 6 (adD b Tc).
(15)
(14) is just the symmetric part (in 7 and 8) of (12), while (15) is arrived at by eliminating the curvature from (12) and (13). In order to proceed we expand T b C as in ref. [2] and find that (14) and (15) imply T bC = --("fl~la)bCY# + (Ta #'y8 --68#a')"~a)bC Y#3,8(16) where Ya~'r is an arbitrary totally skew-symmetric tensor and
Tab • = -i(,),'V)ab,
(6)
y
TabC = i6(aCTb),
(7)
Furthermore,
370
(8)
(3)
and DRA B = 0.
= i6jq" a,
Obc,d):
These tensors satisfy the Bianchi identities D TA =
rJ
(2)
and the curvature tensor RAB = do)AB - 6oAC A ~C B.
3 April 1986
PHYSICS LETTERS
= - ~6(~/a)abDaTb"
(17)
(7cq ... as )abDa rb = O,
(18)
iTa = D G
(19)
where qbis a scalar superfield. We use the same notation for this field as in ref. [2] since they are put equal onshell [8]. (17) now implies that
r
=
=
(20)
Using these results in (12) we find
(28)
W a = i51 1 "G ~a = ~1( D^b Y ~#v)(T#v)ba '
(29)
Wa ~(3 = -- -~l a. G ,x¢a = ±12 [(fib Ya(3v )(T'r)ba
(30)
1 tD b y y 6 [ ~ JW~ ~t6 aba 1 w#] J J,
-2~
= -2i[(7[.r)lablD~] ~ + 12(7~)ab Y~ ~ 1.
= O.
Wa
(21)
(31)
The curvature componentsRa#3, 6 are Ra#.r~ = ½i[2(Ta[-r7* ])a - (Tv~ 7#)a]
It also follows that 1. o~)ab D ot dp + i(vu#'~)abK(~#v. DaDbO = i~(7
(22)
+ 2inaD(l)~ 1)a b Tb , ([)o)a b = D Sab + Togab.
1. e ~[2T ( v6) e ( b ~"
R[~0~I ~ + i8[~
ra
e
(V~)eO](Vu~)~) a
(23)
where we have defined I)a =- D + i v a .
(iv) (o437, 8) and (a&, d). (a~7, 5) reads
r lar a - - o ,
TO = 0,
(35)
(24)
(25)
+ ¼tr(T( Tol%y~)],
(36)
Da(T #Tv~ en) a = - 2 [ D D tr(T¢] 7~/~en) + tr(TDT~] 7v~ cn)] ,
(37)
where now 6 a = Oa + ~i T a . To facilitate the analysis of (34) to (37) we expand the two spinor indices in DaDbYc~v in terms of'y-matrices and obtain
where Ga#3,# = GuT a = 0, and h a Y#3'a ' DaDb Ycqc~:~a = Qa~az,xa,#( "1
+ 3(Ttav)labWbt~] + (7t~y6)ab Wb ,
(34)
D^ ~ ( r a ~ 8 ) ~ = -8 [R~ay8 + Z1 D¿~ tr(Ta] V78)
It is important to note that the superfield ~ does not enter into (23) (except in Da), and consequently the equation of motion for T a is not derivable (recall that Ta Io=0 is the physical spin-l/2 field in the on-shell theory [2] ). By decomposing the Rarita-Schwinger field strength [2], TaeC = au#e + 2G[ d(3,e] )d c + Gd (,,/a/3)dC,
(33)
and by contracting (a/3c, d) with 8d c , (Tv6)d c and (7v8 cn)d c we find, respectively,
+ ½i(V6)bcTf = l)(b Y~y8 (T~ Bq'8 -- 6 6~'y "~ )c) d,
(32)
where
Note that the three-form Kc~#v defined by (22) is unrelated to Y~t~y" [iii) (a~% 6) and (otbc, d). Here we solve for R a_v~ from (aflT, 6) and plug it into (o~bc, d). We thus ogtain
where
(27)
W a = G a = 0,
Rabv~ = 2i(T[-r7~ ] )(ab)
_
= o,
the solution of (23) can be expressed as follows:
which we will solve by putting [2]
+ (7,r6uvO)ab Y
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)ab
(26) + iQa~azaa,O~ ... t3s(7~ "'"as )ab"
(38) 371
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The Ricci identity then immediately provides us with explicit expressions for Q~x~2~a,# andQ~aa2~,~1 #s in terms of D~-derivatives on ~ and Y~t3~ and)~°r ~'roducts of Y~0./'s: 1
+ ~ [(D[ (p) ra~] n - (Ded~)Yta~erLr I ~ ] -36Y~[~
eY # 3 ' ] e '
+ (1/5!) e~1'''~s ex'"e4 [c Y#q,] el Ye2...e4) ,
(39)
(40)
which is in accord with the Weyl-projected indices appearing in (38). The objective now is to derive from (34) to (37) as much information as possible about Qa1~2~3,01#2#~" In particular, we must verify that the two bosonic components of the conformal supermultiplet of currents are present. [The fermionic partner of these is Go~a ; see (25) and (29).] Qal~2a3,#~# #a split into irreducible tensors under SO(l, 9) as follows [ 11 ] : 120®
S
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portant check of the consistency of this approach. Note that (43) must also be true in the coupled Einstein-Yang-Mills theory and therefore should be viewed, not as a field equation for a two-form potential, but rather as a Bianchi identity for a six-form potential. It is of course only in an off-shell theory that this statement is meaningful and indeed this six-form potential played a crucial role also in the discussion of the linearized off-shell theory in ref. [8]. Finally, we come to the most important terms on the right-hand sides of (41) and (42), namely the representations 54 and 210. As is known from refs. [3,8] the conformal supermultiplet of currents contains a 54 ( m t and 210 (
) and these must be non-zero in or^ der to make a coupling to Yang-Mills possible. Q [~" ~,~3 ~ ~, ,lma~a~ 1~,,.oJ does also appear but is not independent and indeed (37) relates it to the dual of 3'
Q [~1~2,~1#213,' Q[~I ....... c~]
_
1
7-i e~l ...a6
#1 .., #4
X (QT#lth,#3#47 - 6 YTth #2 Y#s#47 )"
(44)
Then from both (36) and (37) we obtain
120=1+54+210+770
e
1
Qe[~#,78] = - ~ [D [a Y#-y8] - 2(D[~b) Y#76] + 1050 + 1050 + 4125,
(41) -
120 ® a 1 2 0 = 45 + 210 + 945 + 5940.
(42)
We will now discuss each of these irreducible tensors in turn. First we note that the 1 and 4125 (with Young tableau ~ ) in (41) are zero due to (28) and (31) respectively. Next we consider 5940 in (42) which appears on the left-hand side of (37). However, there is no such tensor on the right-hand side and it must therefore vanish. 1050 and 1050, on the other hand, will be determined by (37) in terms of y 2 while (36) and (37) imply the same expression for the 945. We now come to the more interesting terms on the right-hand sides of (41) and (42), namely 54,210, 770 and 45. First, Q(770) is found to be related to the Weyl tensor. Secondly, when deriving that Q(45) vanishes, we also discover that D~ Y~t~'~ - 6(D~q~) Y # ' r = 0.
(43)
(43) can be derived in several ways from the Bianchi identities (35) to (37) and therefore provides an im372
12Lt.,Y
1e l
(45)
Similarly, from (36) we find _
144 0 ~/~ =R(~¢) + rl~#l--]¢ + 8D(aD#)¢ 7 ~'r~(c~,~) - 8r/~a(Dr¢) 2 + 8 D ¢ Dag) + 144(r/a#YS'seY ~ e - 7 y 3 ' 8 y/~w6),
(46)
and if we recall that in (41) the singlet vanishes, the trace of (46) becomes R + 18I--1¢ - 72(D ~) 2 + 432 y~#7 Y~¢7 = 0.
(47)
It is important to note that (47) does not imply the rest of the field equations, but does constitute a restriction on possible couplings to matter (see below). This completes the analysis of the Bianchi identities (4) and the constraints (6), (7) and (8). The results are easily summarized as follows. The system has been shown to be off-shell with all torsion and curvature components expressible in terms of one unconstrained scalar superfield q~together with a constrained skew-
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symmetric third-rank tensor superfield Y~#7" The scalar superfield starts with the physical fields ~(x) and ha(X ) while all components from 02 and up are auxiliary. The superfield Ya#'r has as its lowest component an x-space three-form which is the dual o f the field strength of a six-form potential * t. This follows from (43). At the next two levels in YaB't we find the Weyl tensor C~B.r8 and its on-shell superpartner Gc~Ba together with the conformal supermultiplet of currents consisting of the vector-spinor wa s, the traceless e n e r g y - m o m e n t u m tensor Q76 (~,B)~6 and the fourform current Qe [at~,~ ] e" These currents are related to physical fields by (28), (29) (recall that TaBTaB7 = - 7 2 G 7 + 16G?7), (45) and (46). This should now be compared to the linearized offshell theory obtained by Howe et al. [8]. These authors found the off-shell fields to be a scalar superfield ¢ [identical to the one introduced in (19) above] and the three-form prepotential V~B~. V~B7 satisfies some constraints with the consequence that it is essentially equivalent to an unconstrained scalar superfield T plus the conformal supermultiplet [8]. The auxiliary superfield T does not enter into the theory as presented here. However, this just corresponds to a situation where one of the superspace field equations has been implemented, namely D3'~B~DVaB~ = 0
(48)
in the linearized version o f ref. [8] (48) could have made impossible the coupling to Yang-MiUs, but fortunately this is not the case as we will now explain. In N = 1 supergravity in four dimensions one encounters a similar situation. The d = 4 superspace field equations are two in number, GaB = 0 a n d R = 0, and i f R = 0 (but not GaB = O) is imposed, one can only couple to matter systems which are consistent with zero curvature scalar and zero ?-trace o f the Rarita-Schwinger field equation *=. Similarly in our case, (48) should
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be interpreted as the reason for the constraints (47) and (28). However, these constraints are consistent with the coupling to Yang-Mills in d = 10 as was asserted in ref. [9]. This will be fully demenstrated in ref. [7]. Finally we want to emphasize again that the fact that the torsion constraints used here (and in refs. [2, 9] ) do not lead to the on-shell theory is quite remarkable. This is not the case in for example the chiral N = 2 , d = 10 theory analyzed in ref. [14]. One of us (B.N.) is grateful to L. Mezincescu and E. Witten for stimulating discussions.
References [1] A.H. Chamseddine, Nucl. Phys. B185 (1981) 403. [2] B.E.W. Nilsson, Nucl. Phys. B188 (1981) 176. [3] E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Nucl. Phys. B195 (1982) 97. [4] G.F. Chapline and N.S. Manton, Phys. Lett. 120B (1983) 105. [5] M.B. Green and J.H. Sehwarz, Phys. Lett. 149B (1984) 117. [6] D.J. Gross, J.A. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. 54 (1985) 502; Princeton University preprints. [7] B.E.W. Nilsson and A.K.Tollst6n, CERN-TH 4354/86, Phys. Lett. 171B (1986), to be published. [8] P. Howe, H. Nicolai and A. van Proeyen, Phys. Lett. l12B (1982) 446. [9] R.E. Kallosh and B.E.W. Nilsson, Phys. Lett. 167B (1986) 46. [10] N. Dragon, Z. Phys. C2 (1979) 29. [11] R. Slansky, Phys. Rep. 79 (1981) 1. [12] A.H. Chamseddine, Phys. Rev. D24 (1981) 3065. [13] S.J. Gates Jr. and H. Nishino, Phys. Lett. 157B (1985) 157. [ 14] P. Howe and P. West, Nucl. Phys. B238 (1984) 181.
,1 The formulation of the on-shell theory in terms of a sixform potential is discussed in refs. [12,13]. ,2 We are grateful to K. SteRe for reminding us of this example.
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