J. Mod. Phys., 2010, 1, 59-66
doi:10.4236/jmp.2010.11007 Published Online April 2010 (http://www.scirp.org/journal/jmp)
Supertranslations to All Orders*
Rainer Dick
Department of Physi cs a nd Engineering Phys ic s, Universi t y of Sa skat chewan, Sask ato o n, C a na d a
Received December 29, 2009; revised February 25, 2010; accepted March 18, 2010
Abstract
We calculate the transformation laws of the general linear superfield


,,Vx

 and chiral superfields under
1N supertranslations

exp iQQ





to all orders in the translation parameters

, . We use the
superfield formalism with complete expansions of the component fields in the coordinate shifts

xi



  

. The results show in particular how a general supertranslation transforms each
component field of a supermultiplet into a complete superfield. The results also provide complete parametri-
zations of orbits of component fields under supertranslations.
Keywords: Supersymmetry, Supermultiplets, Supertranslations
1. Introduction
Quantum field theories with exact correspondences be-
tween bosonic and fermionic helicity states are not only
basic ingredients for superstring theories, but have
dominated both theoretical investigations and experi-
mental searches for particle physics beyond the current
“Standard Model’’ of particle physics for over three dec-
ades now.
The minimal version of supersymmetric extensions of
the Standard Model extends the generators
M

, p

of the Poincaré group by a set of fermionic generators
Q and Q
 in the (1/2,0) and (0 ,1/2) represen tations of
the proper orthochronous Lorentz group in four dimen-
sions. It has been recognized early on that this extension
of the Poincaré algebra can be represented linearly (and
in a reducible, but not fully reducible manner) on a set
comprising 4 complex spin-0 fields, 4 Weyl spinors and
one complex spin-1 field. This set constitutes the so
called general linear multiplet or general linear superfield
V and its irreducible subsets had also been identified.
It is sufficient to know the action of the supertransla-
tion generators Q and Q
on the components of V,
or equivalently the action of the supertranslation

exp iQQ





to first order in the parameters


,


, to construct supersymmetric action principles and
the related supercurrents. Therefore the first order trans-
formation laws for the components of V have been
calculated a long time ago and can be found in many
books and review articles on supersymmetry and with
our current understanding this is all that is needed to
discuss the physical implications of supersymmetry. Re-
cent research in supersymmetry focuses on important
applications like new solutions and structure of super-
gravity theories [1-5], impact of supersymmetry on per-
turbative calculations [6], cosmological implications of
supersymmetry [7-9] and in particular the interesting
problem how supersymmetry can be reconciled with a
cosmological constant and help to explain it [10,11]. The
structure of superpotentials in theories with broken su-
persymmetry is also an active area of research with phe-
nomenological relevance [12-14].
From a mathematical point of view it is clearly desir-
able to also have the full tr ansformation properties of the
general linear multiplet readily available for reference.
To provide such a reference is the purpose of this paper.
To make these results also easily accessible for beginners
in supersymmetry, the super-Poincaré algebra and the
basic techniques of superspace calculations are also re-
viewed. Therefore the outline of the paper is as follows.
Our conventions for spinor representations of the
Lorentz group and the super-Poincaré algebra are intro-
duced in Section 2. Superspace is reviewed in Section 3
and the full supertrans lation properties of the component
fields of the general linear multiplet are calculated in
Section 4.
*This work was supported by NSERC Canada.

R. DIC K 60
Chiral superfields provide a particular irreducible re-
presentation within the reducible linear multiplet. Due
to their practical relevance for the supersymmetrization
of matter fields, the resulting supertranslation properties
of the components of chiral superfields are listed in
Section 5.
Appendix 1 contains a translation of our results into
the conventions of Wess and Bagger [15]. The relevant
spinor indices are reviewed in Appendix 2.
Our conventions for spinor representations and super-
space calculations differ from Wess and Bagger only
with regard to the definition of superderivatives and the
definition of the 2nd order epsilon spinors with lower
indices. Sections 2 and 3 are included to make the paper
self-contained and easily accessible and to clarify con-
ventions. However, the new results in Sections 4 and 5
are not affected by the different definitions. The cogno-
scenti should therefore go straight to Section 4.
2. The Super-Poincaré Algebra
The basic methodology for calculations with linear su-
per-multiplets in four dimensions was developed some
35 years ago by Wess, Zumino, Salam, Strathdee and
Ferrara [16-19]. This section and the following section
provide a brief but concise introduction to the calcula-
tional techniques of supersymmetry and its linear repre-
sentations in four dimensions.
We use 001 for the Minkowski metric and
standard notation   with
 
 
00 11
22 33
10 01
,,
01 10
010
,
101
i
i
 
 
 
 
 
 
  
 
 

 


for the Pauli matrices.
Complex conjugation turns undotted indices into dot-
ted indices and vice versa,

,
a






and hermiticity of the Pauli matrices implies for the
complex conjugate matrices
.

 


(1)
We pull spinor indices with the two-dimensional epsi-
lon spinors
12 12
12 12
1, 1,
,,
,.
 
 
 
 

 
 



 
 

(2)
The Equations (1) then imply that the conjugate Pauli
matrices with upper spinor indices are
.
 

 


 (3)
Numerically, we have with the upper index positions
for the barred matrices and lower index positions for the
unbarred matrices
00
,.
ii

 
Although not formally required, use of upper indices
for barred Pauli matrices and lower indices for un-
barredPauli matrices is a useful and very common con-
vention.
Relations for Pauli matrices are meticulously compiled
in [15]. For convenience, we recall those relations which
are directly relevant for the derivation of supertransla-
tions to all orders, 2,
 


 




(4)


2,

 




 (5)


2,

 



 




 (6)


2,Tr 

 

  


 (7)
and 0123 .i
   


 (8)
The factor 01231

 was included to allow for ready
use of both conventions for the four-dimensional epsilon
tensor.
We will briefly recall below that pulling spinor indices
with the 2nd order epsilon spinors is motivated by the fact
that this yields Lorentz invariant spinor products
,
 














(9)
where the anti-commutation property of spinors was used.
Conjugation also implies re-ordering of spinor quantities,
such that conjugation of (9) yields
.



















 (10)
The vector representation matrices of the Lorentz al-
gebra,


,L






 
appear as structure constants in the Poincaré algebra. The
spinor representations of a proper orthochronous Lorentz
transformation
1
exp exp
22
i
M
L
 



   


are given by

exp 2
i
US



 


and

exp ,
2
i
US







with generators






,
4
.
4
i
S
i
S


  


 

   
   

 

R. DICK 61
The relations

0123
1,
2
Tr SSi
  
 (11)


0123
1.
2
Tr SSi
  
 (12)
are used in the deriv ation of supersymmetric Maxwell or
Yang-Mills actions.
The spinor products (9,10) are invariant because the
matrices

U and

U
 are (2, )SL  matrices,
 
,UU

 

 
 
.UU

  
 


 


 (13)
Stated differently, the epsilon spinors are Lorentz in-
variant.
We can now write down the sup er-Poincaré algebra in
the form




,,MMiLMiLM

 






,,
M
piL p

 






,
,
M
QSQ

 

 



,,
M
QSQ


 





 (14)

 
,22,
,0,,0,
,0,,0.
QQ p p
QQ QQ
pQ pQ

 
 
 


 


 



The (2, )SL  property (13) reads in first order



SS
 





and implies that Equation (14) can also be written as

,.MQ QS
 





 
The super-Poincaré algebra satisfies all the pertinent
super-Jacobi identities as a consequence of the represen-
tation properties of the vector and spinor representations
of the Lorentz algebra. The particular super-Jacobi iden-
tity
  
,,,
,,,
M
QQMQQMQ Q
  
 


 


holds as a consequence of the fact that the Pauli matrices
have the same form in every inertial frame,




1
41.
4
L 
  


 



 
 

  
  




(15)
This can be verified from Equations (5,6) by commut-
ing the


matrices into the middle positions in the
products on the right hand side. It can also be verified as
a direct consequence of Equation (8).
3. 1N

Superspace
The Poincaré algebra is realized on spacetime coordi-
nates
x

through derivative operators


.
,
M
ixxp i



  (16)
In a nutshell, superspace is based on the observation
that this construction can be extended to the su-
per-Poincaré algebra by supplementing Minkowski
spacetime with fermionic coordinates   and


 and
correspondin g fermionic derivat i ves
,.



 




(17)
The super-Poincaré algebra is then realized on the su-
perspace coordinates


,,x



 by amending the
representations (16) of the bosonic operators with the
realizations
,,Qi Qi
 

 

 


(18)
for the fermionic operators and complementing the Lor-
entz generators to include the action on Q and Q

,
 
.Mixx SS
 

  








A superfield


,,Vx

 maps a spacetime point
x
into the algebra over  which is generated by the five
elements


1, ,

, subject to the relations (35,36) in the
Appendix A.2. Note that this definition explicitly refers
to the fermionic arguments of V. Supertranslations are
based on the concept that there are infinitely many in-
carnations of the four fermionic generators

 and



and that we can freely move between these infinitely
many copies of the same algebra. In particular, if


1, ,

 and


1, ,

generate the same algebra, we
require that


1, ,





also generates the same
algebra. This requirement is equivalent to an-
ti-commutation properties
,,











and the corresponding conjugate equations, i.e. the alge-
bra generated by


1, ,

 is a subalgebra of a corre-
sponding infinite-dimensional algebra.
The relations in Appendix A.2 imply that the expan-
sion of every superfield with respect to the fermionic

R. DICK 62
elements   and   can be written in terms of four
scalars
 


,,,,
x
Mx Nx Dx four Weyl fermions
  
,,,,
x
xxx and a vector field


A
x
,

 
   
222 222
,,
.
VxxxxA x
M
xNxxxDx


 
  
The commutation relations

,,,,
,
iQ iQ
iQQx i
 

 
 
 






  



imply for unitary supertranslations


,, exp,,
,, ,
xiQQx
xi


 







 





and therefore
 
 
,,,, exp
,,.
Vxxi QQV
Vxi


 


  (19)
We can calculate the transformation properties of the
component fields by comparing










 

222 2
22
,,Vxxx xAx
Mx Nxxx
Dx


 
 
 
 
  
with the expansion of the right hand side of (19) with
respect to the fermionic variables   and  .
4. Supertranslations of the General Linear
Multiplet
Equation (19) implies in particular that supertranslations
shift the argument
x
of component fields to

.Xxi
 


  

(20)
We can calculate the transformation properties of the
components of the supermultiplet V to all ord ers in the
translation parameters

,

, by expanding the right
hand side





,,VX xi
xi




 

 

  
of Equation (19) to all orders in  and .
The first step requires the expansion of the component
fields with respect to the coordinate shifts


,xi




  

e.g.



















1
2
6
1
24
Xxi x
x
i
x
x
xi x













 
   

   
 
  








22 222
22 222
222222
12
4
1
2
4
1,
16
x
x
ix
x


 



  
    
  
 
and corresponding expansions for combinations of the
other eight component fields with various factors, which
are different in each case due to the presence of fer-
mionic variables in the extra factors. Altogether, this
includes 35 more relations, e.g.
 
 




2222
22
2
2222
21
24
1
4
1
4
.
8
i
X
xx
ixx
x
x
ix










   

 
  
 
Substitution of all the expansions in terms of standard
words in the Grassmann variables into Equation (19)
yields the full supertranslation properties of the compo-
nent fields, which are reported in Equations (21-29). The
transformation equations of the component fields are
organized by contributions from the nine component
fields


,
x



,
x



,
x


,
A
x


,
M
x

,Nx


,
x



,
x
 and


Dx, instead of organization by expan-
sion in the supertranslation parameters

and

. In
this way, supertranslations act on the component fields
like matrices which have Grassmann valued differential

R. DICK 63
operators as entries. The reader can easily re-organize the
transformation equations in terms of supertranslation
parameters. The supertranslation equations are
 
 
  
22
2222
,
xx xx
A
xMxNx
x
xDx




 
  
(21)
 

 
 
 
 

2
2
22
22
2
2
2
2
2
2
2,
xix x
ix
ixAx
iAx Mx
iMxx
i
x
x
Dx
















 

 

(22)
 
 
 
 
 
 
2
2
2
22
22
2
2
2
2
2
2,
i
xix x
i
xx
i
Ax Ax
Nx iNx
i
xx
xDx














 

 
  
 
 

  
(23)
 
  
 
  
22
22 2
222
22
1
42
1
42
1
4
,
2
i
Mxx x
i
x
Ax Mx
iMx Mx
i
x
xDx














 

 
 
  
(24)
  
 
 
  
22
22 2
222
22
1
42
1
42
1
4
,
2
i
Nx xx
i
x
Ax Nx
iNx Nx
i
x
xDx








 
 
 
 
(25)
 
 
 
 
 
 

 

 
2
222
22
2
22 2
22
2
1
2
2
11
42
1
24
1
2
2
12
4
Ax x
i
x
x
xx
i
x
x
xAx
iAx Ax
Ax Ax
iMxiNx x
i


 



 

 




 

   

 



 

 
 
  
 


 
 
 
2
2
2
2
2.
2
i
xx
xi x
ixDx




 

   
  
(26)
For conversion of the last equation into standard
words in the Grassmann variables

and

, note that
from Equation (8)
 

0123
0123
1
22
.
iFx Fx
Fx
 



 


The remaining transformation equations are
 




 
  
 



 

22 22
2
22 2
22
22
2
1
44
1
4
1
2
82
12
41
2
2
2
2,
i
xxx
x
x
ii
x
Ax
Ax Ax
iNx Nxx
ix
i
ix x
Dx



















  
 
 
 

 

(27)

R. DICK 64
 

 

 
 
 
 



 
22
2
22 222
22
2
22 2
4
12
41
84
1
24
1
2
1
22
2
2,
i
xx
xx
ixx
iAx Ax
Ax iMx
i
Mxx x
ix
ixDx






 
 
 
 










 

  
 
 
 
 
 
 
(28)
 
 
 

 
 
2222
222 2
2
22 22
1
16
88
12
4
11
442
.
2
Dx x
ii
x
x
Ax Ax
i
Mx Nxx
ixDx









  
 






 
(29)
These transformation laws are compatible with the re-
ality constraints

†
,, ,,VxV x  which define
the vector multiplet,



††
,, ,
x
xx xAxAx

 
 




††
,, .
M
xNx xxDxDx
5. Supertranslations of the Chiral Mu lti ple t
Besides the superderivatives (18) one can also define
supercovariant de rivatives [18,19]
,,Di Di
 
 
 


(30)
such that





,0,,0,,2,DD DD DDi

 




and








, 0,, 0,,0,,0.QD QD QDQD
  




The condition for chiral sup erfields

,, 0ix


 

is therefore invariant under supertranslations .
The basic solut i ons


0, 0,DDxi





imply [15]





2
,, ,
.
xxi xi
xi Fxi

  

The relation

222
1
2



  
yields


 
 
 
222
22
,,
1
4
.
2
x
xi x
xx
ixFx




 
 
 
(31)
The chiral superfield corresponds to the following
substitutions in the general superfield V,






0,, ,
x
Ax ixMxFx


  
0, 0,,
2
i
Nx xxx


 
 
2
1.
4
Dx x
It is clear from the construction, but can also be
checked explicitly that these constraints are compatible
with the transformation laws (21-29) of the full linear
multiplet.
We find the following supertranslations of the com-
ponents of the chiral multiplet,
  
  
222
22
1
4
,
2
x
xi xx
i
xxFx





 
  
(32)







 
 
22
222
2
2
1
24
2,
x
ix xx
Sxx
Fx iFx








  
 
(33)




 
 
22
22
222
1
21.
4
Fxx ix
xFx
iFx Fx






 

 

(34)
Please note that this presentation does not involve the
usual rescaling


2
x
x of the spinor compo-
nent of the chiral superfield, which is required for ca-
nonically normalized kinetic terms in supersymmetric
Lagrangians.

R. DICK 65
6. Conclusions
The supertranslation properties of the component fields of
a general linear supermultiplet and of a chiral multiplet
were reported to all orders in the translation parameters

and

in (21-29) and (32-34), respectively. On the
one hand, one can think of these results as explicit pa-
rametrizations of orbits of supertranslations in the space
of component fields of a supersymmetric theory. On the
other hand, one can consider the transformed fields as
superfields in the variables


,,x

, because e.g.



 



0,
0
,,
,,
,,,, ,
xx
Vxi
iVxiDVx








 





 





 





and higher order derivatives with respect to the

and
 variables at 0 , 0 can also be expressed as
supercovariant derivatives with respect to

and

.
For example, the transformed vector field is



0,
0
,, ,,
,,.
Ax Ax
Vx i

 
 





 

 



We have







,
,
,
,
,,
,,
,, ,
Xxi
Xxi
Vxi
iVX
X
iX
iVX
X



   


 



  

  
 

 
 


   





 





 


 










and therefore

 

1
,,,, .
2
Ax DDVx





 



From this point of view, (21-29) and (32-34) tell us
explicitly how supertranslation of the components of V
induces corresponding superfields.
7. References
[1] R. Clarkson, A. M. Ghezelbash and R. B. Mann, “New
Reducible Membrane Solutions in D=11 Supergravity,”
Journal of High Energy Physics, 2004.
[2] A. M. Ghezelbash, “Supergravity Solutions without
Tri-Holomorphic U (1) Isometries,” Physical Review D,
Vol. 78, 2008, p. 126002.
[3] de B. Wit, H. Nicolai and H. Samtleben “Gauged Super-
gravities, Tensor Hierarchies and M-Theory,” Journal of
High Energy Physics, Vol. 2, 2008, p. 044.
[4] G. Compère, K. Copsey, de S. Buyl and R. B. Mann,
“Solitons in Five-Dimensional Minimal Supergravity:
Local Charge, Exotic Ergoregions and Violations of the
BPS Bound,” Journal of High Energy Physics, Vol. 12,
No. 47, 2009, p. 46.
[5] U. Gran, J. Gutowski and G. Papadopoulos, “M-Theory
Backgrounds with 30 Killing Spinors are Maximally Su-
persymmetric,” ARXIV, 2010.
[6] R. Britto, E. Buchbinder, F. Cachazo and B. Feng,
“One-Loop Amplitudes of Gluons in SQCD,” Physical
Review D, Vol. 72, 2005, p. 27.
[7] R. Dick, “Implications of a Dilaton in Gauge Theory and
Cosmology,” Fortschr der Physik, Vol. 45, 1997, pp.
537-587.
[8] R. Dick, “Heterotic String Dark Matter from the Graviton
Multiplet,” Physics Letters B, Vol. 510, 2001, pp. 10-16.
[9] J. McDonald and N. Sahu, “Kev Warm Dark Matter Via
the Supersymmetric Higgs Portal,” Physical Review D,
Vol. 79, 2009, p. 6.
[10] A. A. Bytsenko, S. D. Odintsov and S. Zerbini, “The
Effective Action in Gauged Supergravity on Hyperbolic
Background and Induced Cosmological Constant,” Phys-
ics Letters B, Vol. 336, 1994, pp. 355-361.
[11] E. Elizalde, S. Jhingan, S. Nojiri, S. D.Odintsov, M. Sami
and I. Thongkool, “Dark Energy Generated From a (Su-
per) String Effective Action with Higher Order Curvature
Corrections and a Dynamical Dilaton,” European Physi-
cal Journal C, Vol. 53, 2008, pp. 447-457.
[12] A. E. Nelson and N. Seiberg, “R-symmetry Breaking
Versus Supersymmetry Breaking,” Nuclear Physics B,
Vol. 416, 1994, 46-62.
[13] K. A. Intriligator, N. Seiberg and D. Shih, Supersym-
metry Breaking, R-symmetry Breaking and Metastable
Vacua,” Journal of High Energy Physics, Vol. 7, 2007, p.
19.
[14] M. Dine, G. Festuccia and Z. Komargodski, “A Bound on
the Superpotential,” ARXIV, 2009, p. 2527.
[15] J. Wess and J. Bagger, “Supersymmetry and Supergravity,
2nd Edition,” Princeton University Press, 1992.
[16] J. Wess and B. Zumino, “Supergauge Transformations in
four Dimensio ns,” Nu clear Ph ysic s B, Vol. 70, pp. 39-50; “A
Lagrangian Model Invariant under Supergauge Transforma-
tions,” Phys ics Le tters B, Vol. 49, 1974, pp. 52-54.
[17] A. Salam and J. Strathdee, “Supergauge Transforma-
tions,” Nuclear Physics B, Vol. 76, 1974, pp. 477-482.
[18] S. Ferrara, J. Wess and B. Zumino, “Supergauge Multi-
plets and Superfields,” Physics Letters B, Vol. 51, 1974,
pp. 239-241.
[19] A. Salam and J. Strathdee, “On Superfields and Fermi-Bose
Symmetry,” Physical Review D, Vol. 11, 1975, pp. 1521-1535.

R. DICK 66
Appendix
A.1. Translation of Our Results Into the
Conventions of Wessand Bagger
Our superspace realizations (18) and (30) are related to
the realizations in Wess and Bagger [15] according to
 
,,
WB WB
QiQQiQ

 

 
,.
WB WB
DiDDiD

 

With these conventions the component field expan-
sions for chiral superfields agree and the generators for
supertranslations are also the same (cf. (4.11,4.12) in
[15]),

 
,
WB WB
iQQQQ







i.e. our results for supertranslations to all orders also di-
rectly apply as generalizations of the first order trans-
formation laws reported in [15].
Note that Wess and Bagger use an operator represen-
tation of the super-Poincaré algebra with the same sig-
nature of the Minkowski metric but

WB
pip

 .
This comes from the familiar sign difference between
field theoretic and quantum mechanical operator realiza-
tions of Noether charges. If


x
x



is a field op-
erator, the momentum operators

30
0
Pdx









L
L
satisfy
 
,Pxi x




and generate translations according to
 

 



exp exp
expexp .
WB
xiPxiPx
xxipxip

 
 
Similar relations hold for Lorentz and gauge charges.
Another way to look at the sign difference is through
Jacobi identities. If the generators a
X
satisfy the Lie
algebra,


,,
c
ab abc
X
XiCX
then the adjoint matrix representation is given by

.
cc
aab
b
XiC
A.2. Spinor Identities
There are several useful identities for products o f spinors
which are used in the determination of the general linear
multiplet and its transformation laws.
The following identities are a direct consequence of
the anti-commutation properties of spinors and the defi-
nitions (9) and (10) of spinor products,

121 2
12
2
1
2
11
,
22
 
 


 
  
(35)
12 2
1,
2
 



 
(36)
22
11
,.
22
 



 

Please note that 2

in the first line in Equation (35)
denotes the 2



component of the spinor , but in the
second line it is 2

. In every equation in super-
symmetry it is clear from the context what 2

means.
In all equations in previous sections of this paper 2

always refers to 2

.
The following relations also use the prop erties (3-7) of
the Pauli matrices,
,


 



 

(37)
,
 

     (38)
1,
2
 


 
 (39)
22
1.
2
 

 

  

 (40)