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A geometric interpretation of local supersymmetry
Volume 67B, n u m b e r 1
PHYSICS LETTERS
14 March 1977
A GEOMETRIC INTERPRETATION OF LOCAL SUPERSYMMETRY P. BREITENLOHNER Max.Planck-Institut for Physik und Astrophysik, Munich, Fed. Rep. Germany Received 7 October 1976 We introduce a set o f fields together with their transformation laws under local s u p e r s y m m e t r y , Lorentz, chiral and internal transformations such that the previously found c o m m u t a t o r algebra holds without any need to use field equations.
Recently several authors [1,2] have constructed a Lagrangian which is (up to a divergence) invariant under supersymmetry transformations with space-time dependent parameter g(x) and have explored the properties of this theory of "supergravity" [3]. Subsequently locally supersymmetric couplings to matter fields have been given for gauge (vector) multiplets [4, 5] and chiral (scalar) multiplets [6]. In all these theories the commutator of two supersymmetry transformations is a gauge-covariant translation plus a field dependent supersymmetry and Lorentz transformation. In order to obtain this result it is, however, necessary to use the field equations. These field equations as well as the transformation laws for the "gravitational" fields vua (vierbein), couab (connexion) and ~u (spin 3/2) depend on the matter fields coupled to supergravity. Consequently the full Lagrangian cannot be separated into a supergravity and a matter Lagrangian, each of them invariant under local supersymmetry. All this seems to indicate that some auxiliary fields have been eliminated by use of their field equations. In the following we introduce a set of fields and their transformation properties such that a commutator algebra holds without any need to use field equations. This will very much simplify the future construction of locally supersymmetric Lagrangians. We start from the observation that the transformation laws of the gravitational fields under (gauge covariant) translation with parameter ~a(x) (expressed in the local Lorentz frame), supersymmetry (parameter ~(x)) and local Lorentz transformation parameter
xab(x))
(1) vua = 3ts~a + .... 6~u= ~ue + .... 6couab=ou~,ab+ ....
suggest their interpretation as gauge fields for these symmetries. We add two more gauge fields Bu5 (parameter a(x)) and Bui (parameter Oi(x)) for chiral and internal symmetry transformations. The corresponding (global) Lie algebra is {Ta, S, Zab , C, Qi}. Using Lie algebra valued parameters
0 = ~aTa + ~S + ½hab Y'ab + ot C + 0 iQi
(2)
,
the commutator 03 = [01,02] is determined by the structure constants of the Lie algebra. Our conventions are
~3a = Xlab ~2 b -- )~2ab ~1 b _ i~ 17ae2 , 1
1
(3 a)
--
-(3 = -- 2 )~lab e2°ab + 2 )~2abel °ab + it~l e2~' 5
-- i°~2eI')'5,
(3b) ~3 ab = ~laC~2c b -- ~2aC~lcb
(3 c)
o~3 = 0 ,
(3d)
o3i = fl.kiO lJ02 k ,
(3 e)
where fii k are the structure constants of the arbitrary internal symmetry group. Due to the form of eqs. (3d) and (3e) we can delete all reference to chiral and/or internal transformations without any change in the results. Next we introduce a Lie-algebra valued connexion (gauge-field)
~u = VuaTa+fu S+l-'" 2 w# abv":"ab+ B u f C + B u t Q i ,
(5)
and curvature (field strength)
Ruv= RuvaTa + ~uvS + 2! O " ' , v ab'~ "ab + FuvSC ~-FuffOi, given by
(5)
Ru, = Ou q/v - av ~Ou- [~Ou, ~v] •
(6)
Except for the special role of supersymmetry the trans49
Volume 67B, number 1
PHYSICS LETTERS
14 March 1977
8 (-~S )x = [gS, X] +½ Xa-&YaX+ iT seD + oab el~ ab , (13b)
formation laws
8(O)~u=OuO+ [0,~Ou]; 8(O)Ruv = [O,R,,~I,
(7)
would be appropriate. We know, however, from global supersymmetry that gauge fields are members of vector multiplets (in the Wess-Zumino gauge) [7,8] and therefore introduce two more Lie-algebra valued fields, a Majorana spinor
X = xaTa + q~S+ ½xab~-,ab+ x5C+ xiQi ,
(8)
and a pseudoscalar
D = DaTa+DS+ ½Dab£ab +D5C +Dial ,
(9)
~u' X and D form a Lie-algebra valued vector multiplet. It should, however, be remembered that the spin content of the components of flu' X and D is determined by the vector multiplet structure and by the spin content of the Lie algebra. In the following we decompose 0 (and similarly flu) in the following way o=~aTa +O'=~aTa+~S +O".
8(~S)D = [gS, D]+(i/2)Da~yaX+½-&r57aZa,
(13c)
with
Ruv=Ruv-(i/2)fJuTvx+(i/2)~vTuX,
(13d)
Zts = OuX - [ VuaTa, X] - 8 (~'u)X + (i/2)4rT~X + kal~au, (13e) are given to a certain extent by the dual role of (~bu, X, D) as Lie algebra valued vector multiplet. The remaining terms are uniquely given by the requirement that [8(~1S), 8(~2S)] depends only on elTae2 and not on g 1oabe2 . This requirement can be motivated by the structure of the Lie algebra (eq. (3)). It can be shown by elementary, although somewhat lengthy calculations, that the following commutator algebra holds on all fields [8(0]), 6(0'2)1 = 6([0' 2, 0'1] ) ,
(14a)
(this includes [8(~1S ), 8(~2S) = 8(~aTa) with ~a = ie 1Tae2 !)
6(~2bTb)] = 8(--~la~2bf~ab ) ,
(14b)
The obvious transformation laws for Lorentz, chiral and internal transformations are
[8(~laTa),
8(0")'I/u = OuO"+ [0", flu]'
(10a)
[8(~aTa), 8(0")] = 6([0", ~aTa])= 8(-~aXabTb).(14d )
8(O")X = [0", X] + ½~kaboabX + i~/5X,
(lOb)
8(0")O = [O",D],
(10c)
An important intermediate result in the derivation of eq. (14a) is the Bianchi identity
and for the gauge covariant translations we require 8(~aG) = 8 ( ~ " G ) - 6 ( ~ " G ) ,
(1 l)
or equivalently 6(~G)
- 8 ( ~ v G ) = O,
(11')
[8(~ara), 8(eS)l = 8((i/2)~a-&YaX ) ,
(14c)
eabed (3bRcd -- (i/2 )~cdfeTb X - [q)b, l~cd ] + i~6"[cZd A
~_.
^
A
- 2 CobceRcd + (i/2)Rcd,YbX + RedReb ) = 0.
(15)
For eqs. (14b-d) we have used eq. (11) and the fact that for field dependent parameters 0'1,0~ eq. (14a) takes the form
for all fields where 8(~vqv) is the variation under a general coordinate transformation xU -+ xU' = xU - ~u
[8(0~), 8(0'2)]= 8(10'2,{J~ ] + 8(0~)0~-8(0~)0'1). (16)
8(~VGv) ~ku = ~vO~u + (Ou~v) ~v ,
(12 a)
8(~VG,)x = ~"8v× ,
If we define a supercovariant derivative (derivative covariant with respect to supersymmetry) through
(12b)
8(~Z'Gv)D = ~VOvD .
(12c)
Eq. (11') once more emphasizes the role of qJu as gauge field (connexion) and establishes the relation between local translations and coordinate transformations. The transformation laws under supersymmetry
8(-~S)tk u = 3ugS + [~S, ~u I +(i/2)vua~yaX, 50
(13a)
~CT) a= 8(~aTa) .
(17)
Eq. (14b) is the Ricci identity for these derivatives. The similarity of this commutator algebra with the results of deWit and Freedman [8] is remarkable but should not be surprising. Expansion of eq. (13 a) yields
Volume 67B, number 1
PHYSICS LETTERS
References
8(gS ) Vua = i ~u'~ e + (i/2)~,uX a , 8(gS) t~u = ~ug + I wlaab~oa b _ iBu5~75 + (i/2)-~Tuq~,
a(-gS)%ab= ( i / 2 ) ~ - r # X
~b ,
6 ('gS)Bu5 = (i/2) gruX 5 , 6(gS)B ui = (i/2) gyuX i.
(18)
This reduces to the transformation laws given in refs. [2, 3] provided the following field equations hold
×a=o, ¢ = 0 , Xab=ieaeCdTs/~ca.
14 March 1977
(19)
It remains to find an invariant Lagrangian which produces these field equations and, on elimination of all the auxiliary fields, reduces to the Lagrangian of refs. [2,31. We would like to acknowledge many useful discussions with Drs. D.Z. Freedman and D. Maison.
[11 D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214. [2] S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335. [3] D.Z. Freedman and P. van Nieuwenhuizen, Stony Brook preprint ITP-SB-76-25 (May 1976). [4] S. Ferrara, J. Scherk and P. van Nieuwenhuizen, Ecole Normale Superieure preprint PTENS 76/17 (August 1976). [5] D.Z. Freedman and J.H. Schwarz: Stony Brook preprint ITP-SB-76-41. [6] S. Ferrara, F. Gliozzi, J. Scherk and P. van Nieuwenhuizen, Ecole Normale Superieure preprint PTENS 76/19 (Sept. 1976); S. Ferrara, D.Z. Freedman, P. van Nieuwenhuizen, P. Breitenlohner, F. Gliozzi and J. Scherk: Stony Brook preprint ITB-SB-76-46 (Sept. 1976). [7] J. Wess and B. Zumino, Nucl. Phys. B78 (1974) 1; S. Ferrara and B. Zumino, Nucl. Phys. B79 (1974) 413; A. Salam and J. Strathdee, Phys, Lett. 51B (1974) 353. [8] B. deWit and D.Z. Freedman, Phys. Rev. D12 (1975) 2286.
.51
PHYSICS LETTERS
14 March 1977
A GEOMETRIC INTERPRETATION OF LOCAL SUPERSYMMETRY P. BREITENLOHNER Max.Planck-Institut for Physik und Astrophysik, Munich, Fed. Rep. Germany Received 7 October 1976 We introduce a set o f fields together with their transformation laws under local s u p e r s y m m e t r y , Lorentz, chiral and internal transformations such that the previously found c o m m u t a t o r algebra holds without any need to use field equations.
Recently several authors [1,2] have constructed a Lagrangian which is (up to a divergence) invariant under supersymmetry transformations with space-time dependent parameter g(x) and have explored the properties of this theory of "supergravity" [3]. Subsequently locally supersymmetric couplings to matter fields have been given for gauge (vector) multiplets [4, 5] and chiral (scalar) multiplets [6]. In all these theories the commutator of two supersymmetry transformations is a gauge-covariant translation plus a field dependent supersymmetry and Lorentz transformation. In order to obtain this result it is, however, necessary to use the field equations. These field equations as well as the transformation laws for the "gravitational" fields vua (vierbein), couab (connexion) and ~u (spin 3/2) depend on the matter fields coupled to supergravity. Consequently the full Lagrangian cannot be separated into a supergravity and a matter Lagrangian, each of them invariant under local supersymmetry. All this seems to indicate that some auxiliary fields have been eliminated by use of their field equations. In the following we introduce a set of fields and their transformation properties such that a commutator algebra holds without any need to use field equations. This will very much simplify the future construction of locally supersymmetric Lagrangians. We start from the observation that the transformation laws of the gravitational fields under (gauge covariant) translation with parameter ~a(x) (expressed in the local Lorentz frame), supersymmetry (parameter ~(x)) and local Lorentz transformation parameter
xab(x))
(1) vua = 3ts~a + .... 6~u= ~ue + .... 6couab=ou~,ab+ ....
suggest their interpretation as gauge fields for these symmetries. We add two more gauge fields Bu5 (parameter a(x)) and Bui (parameter Oi(x)) for chiral and internal symmetry transformations. The corresponding (global) Lie algebra is {Ta, S, Zab , C, Qi}. Using Lie algebra valued parameters
0 = ~aTa + ~S + ½hab Y'ab + ot C + 0 iQi
(2)
,
the commutator 03 = [01,02] is determined by the structure constants of the Lie algebra. Our conventions are
~3a = Xlab ~2 b -- )~2ab ~1 b _ i~ 17ae2 , 1
1
(3 a)
--
-(3 = -- 2 )~lab e2°ab + 2 )~2abel °ab + it~l e2~' 5
-- i°~2eI')'5,
(3b) ~3 ab = ~laC~2c b -- ~2aC~lcb
(3 c)
o~3 = 0 ,
(3d)
o3i = fl.kiO lJ02 k ,
(3 e)
where fii k are the structure constants of the arbitrary internal symmetry group. Due to the form of eqs. (3d) and (3e) we can delete all reference to chiral and/or internal transformations without any change in the results. Next we introduce a Lie-algebra valued connexion (gauge-field)
~u = VuaTa+fu S+l-'" 2 w# abv":"ab+ B u f C + B u t Q i ,
(5)
and curvature (field strength)
Ruv= RuvaTa + ~uvS + 2! O " ' , v ab'~ "ab + FuvSC ~-FuffOi, given by
(5)
Ru, = Ou q/v - av ~Ou- [~Ou, ~v] •
(6)
Except for the special role of supersymmetry the trans49
Volume 67B, number 1
PHYSICS LETTERS
14 March 1977
8 (-~S )x = [gS, X] +½ Xa-&YaX+ iT seD + oab el~ ab , (13b)
formation laws
8(O)~u=OuO+ [0,~Ou]; 8(O)Ruv = [O,R,,~I,
(7)
would be appropriate. We know, however, from global supersymmetry that gauge fields are members of vector multiplets (in the Wess-Zumino gauge) [7,8] and therefore introduce two more Lie-algebra valued fields, a Majorana spinor
X = xaTa + q~S+ ½xab~-,ab+ x5C+ xiQi ,
(8)
and a pseudoscalar
D = DaTa+DS+ ½Dab£ab +D5C +Dial ,
(9)
~u' X and D form a Lie-algebra valued vector multiplet. It should, however, be remembered that the spin content of the components of flu' X and D is determined by the vector multiplet structure and by the spin content of the Lie algebra. In the following we decompose 0 (and similarly flu) in the following way o=~aTa +O'=~aTa+~S +O".
8(~S)D = [gS, D]+(i/2)Da~yaX+½-&r57aZa,
(13c)
with
Ruv=Ruv-(i/2)fJuTvx+(i/2)~vTuX,
(13d)
Zts = OuX - [ VuaTa, X] - 8 (~'u)X + (i/2)4rT~X + kal~au, (13e) are given to a certain extent by the dual role of (~bu, X, D) as Lie algebra valued vector multiplet. The remaining terms are uniquely given by the requirement that [8(~1S), 8(~2S)] depends only on elTae2 and not on g 1oabe2 . This requirement can be motivated by the structure of the Lie algebra (eq. (3)). It can be shown by elementary, although somewhat lengthy calculations, that the following commutator algebra holds on all fields [8(0]), 6(0'2)1 = 6([0' 2, 0'1] ) ,
(14a)
(this includes [8(~1S ), 8(~2S) = 8(~aTa) with ~a = ie 1Tae2 !)
6(~2bTb)] = 8(--~la~2bf~ab ) ,
(14b)
The obvious transformation laws for Lorentz, chiral and internal transformations are
[8(~laTa),
8(0")'I/u = OuO"+ [0", flu]'
(10a)
[8(~aTa), 8(0")] = 6([0", ~aTa])= 8(-~aXabTb).(14d )
8(O")X = [0", X] + ½~kaboabX + i~/5X,
(lOb)
8(0")O = [O",D],
(10c)
An important intermediate result in the derivation of eq. (14a) is the Bianchi identity
and for the gauge covariant translations we require 8(~aG) = 8 ( ~ " G ) - 6 ( ~ " G ) ,
(1 l)
or equivalently 6(~G)
- 8 ( ~ v G ) = O,
(11')
[8(~ara), 8(eS)l = 8((i/2)~a-&YaX ) ,
(14c)
eabed (3bRcd -- (i/2 )~cdfeTb X - [q)b, l~cd ] + i~6"[cZd A
~_.
^
A
- 2 CobceRcd + (i/2)Rcd,YbX + RedReb ) = 0.
(15)
For eqs. (14b-d) we have used eq. (11) and the fact that for field dependent parameters 0'1,0~ eq. (14a) takes the form
for all fields where 8(~vqv) is the variation under a general coordinate transformation xU -+ xU' = xU - ~u
[8(0~), 8(0'2)]= 8(10'2,{J~ ] + 8(0~)0~-8(0~)0'1). (16)
8(~VGv) ~ku = ~vO~u + (Ou~v) ~v ,
(12 a)
8(~VG,)x = ~"8v× ,
If we define a supercovariant derivative (derivative covariant with respect to supersymmetry) through
(12b)
8(~Z'Gv)D = ~VOvD .
(12c)
Eq. (11') once more emphasizes the role of qJu as gauge field (connexion) and establishes the relation between local translations and coordinate transformations. The transformation laws under supersymmetry
8(-~S)tk u = 3ugS + [~S, ~u I +(i/2)vua~yaX, 50
(13a)
~CT) a= 8(~aTa) .
(17)
Eq. (14b) is the Ricci identity for these derivatives. The similarity of this commutator algebra with the results of deWit and Freedman [8] is remarkable but should not be surprising. Expansion of eq. (13 a) yields
Volume 67B, number 1
PHYSICS LETTERS
References
8(gS ) Vua = i ~u'~ e + (i/2)~,uX a , 8(gS) t~u = ~ug + I wlaab~oa b _ iBu5~75 + (i/2)-~Tuq~,
a(-gS)%ab= ( i / 2 ) ~ - r # X
~b ,
6 ('gS)Bu5 = (i/2) gruX 5 , 6(gS)B ui = (i/2) gyuX i.
(18)
This reduces to the transformation laws given in refs. [2, 3] provided the following field equations hold
×a=o, ¢ = 0 , Xab=ieaeCdTs/~ca.
14 March 1977
(19)
It remains to find an invariant Lagrangian which produces these field equations and, on elimination of all the auxiliary fields, reduces to the Lagrangian of refs. [2,31. We would like to acknowledge many useful discussions with Drs. D.Z. Freedman and D. Maison.
[11 D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214. [2] S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335. [3] D.Z. Freedman and P. van Nieuwenhuizen, Stony Brook preprint ITP-SB-76-25 (May 1976). [4] S. Ferrara, J. Scherk and P. van Nieuwenhuizen, Ecole Normale Superieure preprint PTENS 76/17 (August 1976). [5] D.Z. Freedman and J.H. Schwarz: Stony Brook preprint ITP-SB-76-41. [6] S. Ferrara, F. Gliozzi, J. Scherk and P. van Nieuwenhuizen, Ecole Normale Superieure preprint PTENS 76/19 (Sept. 1976); S. Ferrara, D.Z. Freedman, P. van Nieuwenhuizen, P. Breitenlohner, F. Gliozzi and J. Scherk: Stony Brook preprint ITB-SB-76-46 (Sept. 1976). [7] J. Wess and B. Zumino, Nucl. Phys. B78 (1974) 1; S. Ferrara and B. Zumino, Nucl. Phys. B79 (1974) 413; A. Salam and J. Strathdee, Phys, Lett. 51B (1974) 353. [8] B. deWit and D.Z. Freedman, Phys. Rev. D12 (1975) 2286.
.51