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A comment on superspace Bianchi identities and six-dimensional spacetime
Volume 84B, number 2
PHYSICS LETTERS
18 June 1979
A COMMENT ON SUPERSPACE BIANCHI IDENTITIES A N D SIX-DIMENSIONAL SPACETIME S. James GATES Jr. 1 Lyman Laboratory of Phystcs, Harvard University, CambrMge, MA 02138, USA Received 9 April 1979
We investigate the Blanchl Identities for a superspace with a six-dimensional spacetlme, under the assumption of certain superspace kinematic constraints. It is shown that these constraints lead to the supergravity theory recently discussed by Breltenlohner and Kabelschacht However, we conclude that the resulting theory is consistent only if the curvature scalar of the six-dimensional spacetlme vanishes.
The theory of supergravity in higher dimensional spacetimes may play an important role in the systematic study o f extended supergravlty theories. For instance, the six- and ten-dimensional theories should upon dimensional reduction [1] yield 0 ( 2 ) a n d 0 ( 4 ) extended supergravity theories. One nice feature of the six- and ten-dimensional theories is the absence of central charges [2]. Thus, the construction o f these theories should not be complicated by these charges. Recently, the six-dimensional theory has been studied b y Breitenlohner and Kabelschacht [3] who introduce a set o f auxiliary fields to close the algebra of local supersymmetry off mass shell. In this note we shall consider the six-dimensional spacetlme within the context of a fourteen-diinensional superspace. It has been noted by Wess and Zumlno [4], that the analysis of the superspace Bianchi ldentitms is sufficient, in the presence o f kinematic constraints, to gain a complete understanding of supergravity theories in superspace. In a previous paper [5], we have described this procedure in detail and applied it to supergravity theories in spacetimes of two, three, and four dimensions. We introduce a fourteen-dimensional superspace (0 m ' -tim, xU ) consisting of complex eight-component spinors 0 m and ~m together with six bosonlc coordi1 Research supported in part by the National Science Foundation under Grant No. PHY77-22864, and the Harvard Socmty of Fellows.
nates. The splnors satisfy the relations ½ ( 1 - ' ) ' 6 ) m n O n = O,
-om=-(cTO)mnOn* ,
(1)
where C and 3,6 are eight by eight matrices which satlsfy the equations 76 =--(1/6W)ec~eKX3'~TC~@TeTKTX,
(')'~)* = -(CT°')'~c'yO), {V=,V~) = - 2 r ~ ,
e012345 = +1 ,
C')'6 = - 7 6 C ,
(2)
diag (n=e) = ( - 1 , 1 , 1 , 1 , 1 , 1 ) .
Since all o f the spinors are chirally positive, we introduce the following set o f Dirac matrices"
(P+,P_ ¢ ~ , e + o ~ e , P _ o~e~) , P+ = ½(1 + T 6 ) ,
O~t3 - -
1
o ~a6 = -1½ [ ~ r ~ 8 + ~,t~o~ ~ + ;,8 o~t~]
= (i/3 !) v[~./%81
(3)
Within this superspace, we Introduce a spinorial supercovarlant derivative,
C'Da- EaMaM + ~1 WaeSM# ~
(4)
where a M runs over all of the superspace derivatives (am, ~m, 3u) and M~8 denotes the generator o f Lorentz rotations. We also require the charge conJugate of this operator ~a =
-(c~'O)ba % ,
(5)
and a vectorial supercovariant derivative c-/). We de205
note all of these operators collectively by OA" The graded commutator of the supercovariant derivative defines curvature and torsion supertensors: i
[ ~ A , ~ B ) = 2 T A B C ~C+-2RABKXMKx "
(6)
Now let us impose the following kinemahc constraints on the torsion and curvature supertensors:
Tab c = Tab -d = Tab ~ = T.~ ~ + i l (cp_ ")'a)ab = (p+ 7ac)ab Ra ~ gh = 0 .
(7)
These constraints are the six-dimensional analogs of the kinematic constraints which lead to supergravlty with mimmal components m four-dimensional spacetime. By analyzing the Bianchi ldentRies of dimensionality one which follow from [[%,
%},
%)+ [[%,
%}, %}
+ [[ % , % ) , % ) = 0 , [[ % , % ), ~ )
18 June 1979
PHYSICS LETTERS
Volume 84B, number 2
+ [ [ % a ~, - ~ ) , q~b)} = 0 ,
[[q~a, cDb), q)6) + [[q)~, q)(,i}, ~b)) = o, [[%, ~ ) , %} + [[ %, %), %) + [[%, %), % ) = o, [[ %, %), %) + [[ %, ~[e~), % ] } = 0 ,
~.Ga =0,
~aG~ = ½(Ce_o~%~aT~, a
% 63. = ¼ ( C e o ~ % a Y ~ a , rl)a gc~l 6 = _ l (cp_ o ~x °alS)adTKh
,
(P+oKh )a d TKhd = O .
This last equation has great mgnificance whmh we shall discuss shortly. Continuing we note that the final result in eq. (10) implies the following:
(P+"/aC) ab Cl)bG,~ = ( P - ° ~ C) ab ~b G a~.r = c/)aG a =0.
(11)
Thus the isotriplet superfield G a satisfies a conservation equation. By continuing the analysis, we find certain differential constraints which must be satisfied by the superfields. However, we need not consider those here. As has been noted in lower dimensional spacetimes, the vector-vector-spinor component of the torsion supertensor, to lowest order in 0, corresponds to the field strength of the gravitino. Thus, to lowest order in 0 the last result in eq. (10) when hnearized yields
(P+o~)a b aa ff~ = 0 .
(8)
(10)
(12)
This equation is related to but not identical with the equation of motion,
we find the following information:
(e+o~%% a ~ = 0,
Rab~X = - (CP oKxl)ab Gl ,
for the gravitino. Eq. (12) may be obtained from eq. (13) by multiplication by a Dirac gamma matrix. Thus, the kinemanc constraints given by eq. (7) imply that "part" of the equation of motion of the gramino must be satisfied. In other words, the kinematic constraints are too restrictive. In support of this view, we provide an example. As shown m a previous paper [5], the theory of supergravity, with minimal component fields, in a threedimensional spacetlme has the following kinematic constraints:
Tac d= l[(p+ )dc Ga + ¼(P+Ta71 )dc Gill , g~ =G~ + iG~,
Ra-~Kx =(Cp_oKxi)abG3 ,
rac d = l[(p+)dcG2 + ~l.r+TaT) l~. n , d c G l3 6 d
+ ~(p+o~ ) coat8 ] , Tal'r = -G~lV ,
i G~ly = - geol. r 6KXG 6~},,
for some real superfields G a and G=i ~. This set of superfields corresponds to those found in ref. [3]. Furthermore, from the dimensionality three halves Bianchi identities we learn
RalKx = i2(CP_ "/t)ad T~x ~ 1 tSPOtz~-e'~" T -- ~ el~ch ( LI~- OtSvOa )ad e8
206
,
(9)
(13)
Tab d = Tab ~ +1 l ( c T 8 )a b = Taol e = O ,
(14)
Rab 8e = 0 .
(15)
In three dimensions this leads to a supermultiplet (eatS , ~ a a , A) consisting of the graviton, gravitino, and an auxiliary scalar. The algebra of local supersymmetry closes without the use of equations of motion.
Volume 84B, number 2
PHYSICS LETTERS
If these same constraints are applied to four-dimensional O(1) supergravlty, we find a supermultlplet (e~ u, f a s , A s ) which consists of the gravlton, the gravitino, and an auxiliary axial vector. An analysis of the Blanchl identities lmphes that the gravitlno must satisfy eq. (12) and the axial vector must be conserved. As has been known for some time [6], this supermultlplet, although insufficient for ordinary local supersymmetry, is sufficient for the conformal extension of local supersymmetry. In superfield form this has been explicitly demonstrated in ref. [7]. For four-dimensional O(1) supergravity, this situation is alleviated by replacing the constraint In eq. (15) on the curvature supertensor by the weaker condition
(3's (2)ab Rib Kx = 0 .
(16)
Now the supergravlty supermultlplet consists of (esU, ffa s, A s, S, P) and the field strength of the gravltlno is not required to obey eq. (12). Finally, in the lower dimensmnal supergravlty theories we have found that up to terms involving auxiliary fields the curvature scalar is given by
R =--Rote~3 =1%(P+oKX)a b TKa.b .
(17)
This follows quite generally from the constrained Bianchl identities. Thus the final result m eq. (10) also implies that the curvature scalar of the bosonic subspace must vamsh. Therefore, we conclude that the fourteen-dimensional superspace which possesses only the auxiliary superfields Ga and G ~ . r is one which possesses a van-
18 June 1979
lshlng curvature scalar. We also conclude that the 0independent sectors of these super-tensors provide the auxiliary fields for conformal supergravity in six-dimensional spacetlme. However, a knowledge of these fields can provide the starting point for the derivation of the ordinary local supersymmetry. This is precisely the history of four-dimensional O(1) supergravity. From the superspace approach, we must decide how to modify the constraints in eq. (7) to obtain ordinary not conformal, six-dimensional supergravxty. One possible modIfication which suggests itself from our experience with four-dimensional supergravity is the replacement
Ta~=O-" Tas~ +~(~'~'s)aaTdx x = 0 .
(18)
We are presently studying this possibility. We thank Warren Siegel and Ulf h n d s t r o m for interesting conversations.
References [1] J. Scherk and J. Schwarz, Phys. Lett. 57B (1975) 463. [2] L. Brink, J. Schwarz and J. Scherk, Nuel. Phys B121 (1977) 77. [3] P. Breitenlohner and A. Kabelschacht, Nucl. Phys. B148 (1979) 96. [41 J. Wess and B. Zummo, Phys. Lett. 66B (1977) 361, 74B (1978) 51, 79B (1978) 394. [5] M. Brown and S. Gates, Harvard preprmt HUTP-79/A002 (January 1979). [6] S. Ferrara and B. Zumino, Nucl. Phys. B134 (1978) 301. [7] P. Howe and R. Tucker, Phys Lett. 80B (1978) 138.
207
PHYSICS LETTERS
18 June 1979
A COMMENT ON SUPERSPACE BIANCHI IDENTITIES A N D SIX-DIMENSIONAL SPACETIME S. James GATES Jr. 1 Lyman Laboratory of Phystcs, Harvard University, CambrMge, MA 02138, USA Received 9 April 1979
We investigate the Blanchl Identities for a superspace with a six-dimensional spacetlme, under the assumption of certain superspace kinematic constraints. It is shown that these constraints lead to the supergravity theory recently discussed by Breltenlohner and Kabelschacht However, we conclude that the resulting theory is consistent only if the curvature scalar of the six-dimensional spacetlme vanishes.
The theory of supergravity in higher dimensional spacetimes may play an important role in the systematic study o f extended supergravlty theories. For instance, the six- and ten-dimensional theories should upon dimensional reduction [1] yield 0 ( 2 ) a n d 0 ( 4 ) extended supergravity theories. One nice feature of the six- and ten-dimensional theories is the absence of central charges [2]. Thus, the construction o f these theories should not be complicated by these charges. Recently, the six-dimensional theory has been studied b y Breitenlohner and Kabelschacht [3] who introduce a set o f auxiliary fields to close the algebra of local supersymmetry off mass shell. In this note we shall consider the six-dimensional spacetlme within the context of a fourteen-diinensional superspace. It has been noted by Wess and Zumlno [4], that the analysis of the superspace Bianchi ldentitms is sufficient, in the presence o f kinematic constraints, to gain a complete understanding of supergravity theories in superspace. In a previous paper [5], we have described this procedure in detail and applied it to supergravity theories in spacetimes of two, three, and four dimensions. We introduce a fourteen-dimensional superspace (0 m ' -tim, xU ) consisting of complex eight-component spinors 0 m and ~m together with six bosonlc coordi1 Research supported in part by the National Science Foundation under Grant No. PHY77-22864, and the Harvard Socmty of Fellows.
nates. The splnors satisfy the relations ½ ( 1 - ' ) ' 6 ) m n O n = O,
-om=-(cTO)mnOn* ,
(1)
where C and 3,6 are eight by eight matrices which satlsfy the equations 76 =--(1/6W)ec~eKX3'~TC~@TeTKTX,
(')'~)* = -(CT°')'~c'yO), {V=,V~) = - 2 r ~ ,
e012345 = +1 ,
C')'6 = - 7 6 C ,
(2)
diag (n=e) = ( - 1 , 1 , 1 , 1 , 1 , 1 ) .
Since all o f the spinors are chirally positive, we introduce the following set o f Dirac matrices"
(P+,P_ ¢ ~ , e + o ~ e , P _ o~e~) , P+ = ½(1 + T 6 ) ,
O~t3 - -
1
o ~a6 = -1½ [ ~ r ~ 8 + ~,t~o~ ~ + ;,8 o~t~]
= (i/3 !) v[~./%81
(3)
Within this superspace, we Introduce a spinorial supercovarlant derivative,
C'Da- EaMaM + ~1 WaeSM# ~
(4)
where a M runs over all of the superspace derivatives (am, ~m, 3u) and M~8 denotes the generator o f Lorentz rotations. We also require the charge conJugate of this operator ~a =
-(c~'O)ba % ,
(5)
and a vectorial supercovariant derivative c-/). We de205
note all of these operators collectively by OA" The graded commutator of the supercovariant derivative defines curvature and torsion supertensors: i
[ ~ A , ~ B ) = 2 T A B C ~C+-2RABKXMKx "
(6)
Now let us impose the following kinemahc constraints on the torsion and curvature supertensors:
Tab c = Tab -d = Tab ~ = T.~ ~ + i l (cp_ ")'a)ab = (p+ 7ac)ab Ra ~ gh = 0 .
(7)
These constraints are the six-dimensional analogs of the kinematic constraints which lead to supergravlty with mimmal components m four-dimensional spacetime. By analyzing the Bianchi ldentRies of dimensionality one which follow from [[%,
%},
%)+ [[%,
%}, %}
+ [[ % , % ) , % ) = 0 , [[ % , % ), ~ )
18 June 1979
PHYSICS LETTERS
Volume 84B, number 2
+ [ [ % a ~, - ~ ) , q~b)} = 0 ,
[[q~a, cDb), q)6) + [[q)~, q)(,i}, ~b)) = o, [[%, ~ ) , %} + [[ %, %), %) + [[%, %), % ) = o, [[ %, %), %) + [[ %, ~[e~), % ] } = 0 ,
~.Ga =0,
~aG~ = ½(Ce_o~%~aT~, a
% 63. = ¼ ( C e o ~ % a Y ~ a , rl)a gc~l 6 = _ l (cp_ o ~x °alS)adTKh
,
(P+oKh )a d TKhd = O .
This last equation has great mgnificance whmh we shall discuss shortly. Continuing we note that the final result in eq. (10) implies the following:
(P+"/aC) ab Cl)bG,~ = ( P - ° ~ C) ab ~b G a~.r = c/)aG a =0.
(11)
Thus the isotriplet superfield G a satisfies a conservation equation. By continuing the analysis, we find certain differential constraints which must be satisfied by the superfields. However, we need not consider those here. As has been noted in lower dimensional spacetimes, the vector-vector-spinor component of the torsion supertensor, to lowest order in 0, corresponds to the field strength of the gravitino. Thus, to lowest order in 0 the last result in eq. (10) when hnearized yields
(P+o~)a b aa ff~ = 0 .
(8)
(10)
(12)
This equation is related to but not identical with the equation of motion,
we find the following information:
(e+o~%% a ~ = 0,
Rab~X = - (CP oKxl)ab Gl ,
for the gravitino. Eq. (12) may be obtained from eq. (13) by multiplication by a Dirac gamma matrix. Thus, the kinemanc constraints given by eq. (7) imply that "part" of the equation of motion of the gramino must be satisfied. In other words, the kinematic constraints are too restrictive. In support of this view, we provide an example. As shown m a previous paper [5], the theory of supergravity, with minimal component fields, in a threedimensional spacetlme has the following kinematic constraints:
Tac d= l[(p+ )dc Ga + ¼(P+Ta71 )dc Gill , g~ =G~ + iG~,
Ra-~Kx =(Cp_oKxi)abG3 ,
rac d = l[(p+)dcG2 + ~l.r+TaT) l~. n , d c G l3 6 d
+ ~(p+o~ ) coat8 ] , Tal'r = -G~lV ,
i G~ly = - geol. r 6KXG 6~},,
for some real superfields G a and G=i ~. This set of superfields corresponds to those found in ref. [3]. Furthermore, from the dimensionality three halves Bianchi identities we learn
RalKx = i2(CP_ "/t)ad T~x ~ 1 tSPOtz~-e'~" T -- ~ el~ch ( LI~- OtSvOa )ad e8
206
,
(9)
(13)
Tab d = Tab ~ +1 l ( c T 8 )a b = Taol e = O ,
(14)
Rab 8e = 0 .
(15)
In three dimensions this leads to a supermultiplet (eatS , ~ a a , A) consisting of the graviton, gravitino, and an auxiliary scalar. The algebra of local supersymmetry closes without the use of equations of motion.
Volume 84B, number 2
PHYSICS LETTERS
If these same constraints are applied to four-dimensional O(1) supergravlty, we find a supermultlplet (e~ u, f a s , A s ) which consists of the gravlton, the gravitino, and an auxiliary axial vector. An analysis of the Blanchl identities lmphes that the gravitlno must satisfy eq. (12) and the axial vector must be conserved. As has been known for some time [6], this supermultlplet, although insufficient for ordinary local supersymmetry, is sufficient for the conformal extension of local supersymmetry. In superfield form this has been explicitly demonstrated in ref. [7]. For four-dimensional O(1) supergravity, this situation is alleviated by replacing the constraint In eq. (15) on the curvature supertensor by the weaker condition
(3's (2)ab Rib Kx = 0 .
(16)
Now the supergravlty supermultlplet consists of (esU, ffa s, A s, S, P) and the field strength of the gravltlno is not required to obey eq. (12). Finally, in the lower dimensmnal supergravlty theories we have found that up to terms involving auxiliary fields the curvature scalar is given by
R =--Rote~3 =1%(P+oKX)a b TKa.b .
(17)
This follows quite generally from the constrained Bianchl identities. Thus the final result m eq. (10) also implies that the curvature scalar of the bosonic subspace must vamsh. Therefore, we conclude that the fourteen-dimensional superspace which possesses only the auxiliary superfields Ga and G ~ . r is one which possesses a van-
18 June 1979
lshlng curvature scalar. We also conclude that the 0independent sectors of these super-tensors provide the auxiliary fields for conformal supergravity in six-dimensional spacetlme. However, a knowledge of these fields can provide the starting point for the derivation of the ordinary local supersymmetry. This is precisely the history of four-dimensional O(1) supergravity. From the superspace approach, we must decide how to modify the constraints in eq. (7) to obtain ordinary not conformal, six-dimensional supergravxty. One possible modIfication which suggests itself from our experience with four-dimensional supergravity is the replacement
Ta~=O-" Tas~ +~(~'~'s)aaTdx x = 0 .
(18)
We are presently studying this possibility. We thank Warren Siegel and Ulf h n d s t r o m for interesting conversations.
References [1] J. Scherk and J. Schwarz, Phys. Lett. 57B (1975) 463. [2] L. Brink, J. Schwarz and J. Scherk, Nuel. Phys B121 (1977) 77. [3] P. Breitenlohner and A. Kabelschacht, Nucl. Phys. B148 (1979) 96. [41 J. Wess and B. Zummo, Phys. Lett. 66B (1977) 361, 74B (1978) 51, 79B (1978) 394. [5] M. Brown and S. Gates, Harvard preprmt HUTP-79/A002 (January 1979). [6] S. Ferrara and B. Zumino, Nucl. Phys. B134 (1978) 301. [7] P. Howe and R. Tucker, Phys Lett. 80B (1978) 138.
207