- Email: [email protected]
Components of superspace
Volume
79B, number
3
PHYSICS
20 November
LETTERS
1978
COMPONENTS OF SUPERSPACE Martin ROCEK ’ Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA
and Ulf LINDSTRijM Institute for Theoretical Physics, University of Stockholm, S-l 13 46 Stockholm, Sweden Received
2 August
1978
We give the explicit
superspace
form of the newly
discovered
Although the original formulation of supergravity used explicit components, elegant reformulations have been given in superspace [2]. Recently a tensor calculus for the chiral multiplet [l] has been given, again in component form. In this brief letter we use this formulation to find a precise relation between the component approach and superspace. We begin by deriving the superspace transformation laws that correspond to the local supersymmetry transformations of ref. [ 11. Since we are working in a specific gauge, we can expand the chiral superfield in purely left (right) handed 0’s. A left handed multiplet takes the form: @(z, 0) =a@) +2~,&(z)+~A&+(z).
(1)
(We use two-component spinors throughout l).) Combining the standard transformation of scalars,
(see table property
sq~,e)=-(6eA(a/ae~)~-sz~Jf'(a/a~~~')~), with the explicit form of 6@ in ref. [ 11, we find: +$t=eA
+,/%+B’$ABB’-@&~A
+ ;++A’
for the chiral multiplet
in part by DOE contract
No. EY-76-S-02-
[ 11.
BAe A’ - eA’tiB tBA,G/B’
Although the formulae (3) appear complicated, their interpretation is simple: the leading terms are the same as the rigid transformations of global supersymmetry, the $ terms are their obvious supercovariantizations, and the terms with w, J * and A provide the appropriate Lorentz rotations in the commutator of two transformations. Applications of these results will be presented in a later paper currently in preparation; we merely note that our approach, though superficially similar to that of Ogievetsky and Sokatchev [3], is distinct in that, by construction, it gives the algebra of ref. [l]. In ref. [4], the kinetic multiplet for the chiral super field was presented in component form. We use this result to construct several important quantities in superspace. Recently [5], a chiral gauge was constructed in superspace. In this gauge, the spinor (right chiral) derivative has the simple form:
(3)
and the chiral projector d(a/aeA’)(a/aeAf)(P2(z,8,
where
3227.
calculus
QA’ =cp(Z,e,8,)(a/aeA'),
,
-~zMM'=~(~~M~'-~,~C~~,~LM'MB'),
’ Work supported
(2)
tensor
(4)
[6] takes the form [5]: 8).
(5)
Using this information, a straightforward calculation yields both the shift from z to z* (note that the transformation law (3) of z shows that z must be complex), 217
Volume
79B, number
3
PHYSICS
LETTERS
20 November
Table 1
vMM’
Our conventions are the standard conventions of the NewmannPenrose formalism, except that our metric has the opposite signature, and hence: nab = -E~BEA’B’; the relation with other notations is summarized below.
~M’=~MB,B’M’+fiAMM’_~~BMB’~MRB,,
Refs. -_
[ 1,4]
Ref. [6]
v-2= i +,/tie*
4 R
in the notation
$2i) (F - F*) z(cl+J*) (l/W (J -J *)
of refs.
?p_G
!?+O =
X/?*,
sip,‘Y&
= dXAV,J’,
XP-.yas =
_
*MM’ _ _
-b, -R,, - i~clvpo ___._~~__
WJ EwP~ ______~_..
=
t
i?n Y5 -G -M -N
[l] and [4] : Pk = (1 f -ys)/2,
XA's*' , xP++,
= -JZ,*?ff,
etc.
&xA’~*,
The spin 3/2 field is represented by: P+$ = $AMMg. The chirally supercovariant spin 3/2 fieltequation is written: P+R,,= P+Epvpoy5’Yv(ap+ $wpabdb+ The spin connection
has 3/2 torsion
Wllab= t (~K~~*B~A’B’
&&Ys)$JL==*M,W.
and is written:
+ ~MA~A’B’EAB)
.
And finally pi’ = gfipr * = - PNN In our conventions, complex conjugation does not effect the order of spinors; this is inconvenient for quantization, but makes algebra simpler.
the unknown function p(z). The procedure goes as follows: we take the scalar field @(z, 0), and complex conjugate it; we then shift z* by an unknown function Az(z, 0, e), and multiply by an unknown function (p2(z, 0, f). Finally, we apply 4 a*‘a, I by taking the BAl B* component, and we set the result equal to the kinetic multiplet. Terms without derivatives determine (p2(z, 0, 3) unambiguously, and terms with derivatives determine Az(z, 8, e). We find:
-
$ iANN’
_. f/Q,NLMc~NMM’,
and
_~___
xrl ir5 G s P
Further,
This paper
= -rNN’
1978
QA,**‘- f eAe*(ILA,MM,~*‘MM’t~6*)
eAeAj,/fipAA’eAeAeAf@‘t eA’eA’(iR),
where
&
-&/ijj,**‘_ -
wAA’B’
+BwM’_
~~BB,~‘BB’+DBB,~‘BB;6) ;&,#BB’~B,CC’~Bcc,,
and R is precisely the Ricci scalar multiplet called R in ref. [2] and called U in ref. [ 1). Since q2 has leading term 1, cpcan easily be found. The appearance of R as the coefficient of eA t tl* ’ was predicted by Zumino [S], and served as a gratifying check on our algebra. A further check was made by observing that x = f (z +z*) = z + g AZ is a real quantity; we then found that when AZ is expressed in terms of x, it is indeed a purely imaginary quantity. Although the variables x, 0, and e form a real symmetric basis in superspace, they are not convenient coordinates; this is because their transformation laws, when expressed in terms of x, 0, and 8, contain derivatives of E, the parameter of supersymmetry transformations. A more convenient symmetric basis can be found by adding a real shift to x and corresponding shifts to 8 and fi. We are currently investigating this and related problems.
and
nzMM’
= -2.JZeMeM’
+ 2(oCoc)oAj$
t (e,ec)
M’MA’_
(e,,eCj
x (vM”‘-~MM’-~~M*M’~~,), where
218
2(eC,eC’)e,
$MAM’
We would like to thank the Institut d’Etudes Scientifiques de Cargese for its hospitality, the many participants of the Summer School on “Recent developments in gravitation” and in particular Prof. Zumino for fruitful discussions. One of us (M.R.) would like to thank Prof. S.W. Hawking for his hospitality at D.A.M.P.T., where this investigation started. [l 1 S. Ferrara and P. van Nieuwenhuizen,
Phys. Lett. 76B (1978) 404. P:I E.g.: J. Wess and B. Zumino, Phys. Lett. 74B (1978) 51, and references therein. Joint Inst. for Nucl. [3 I V. Ogievetsky and E. Sokatchev, Research Dubna (1978). preprint Lab. Phys. ]41 S. Ferrara and P. van Nieuwenhuizen, Theo. Ecole Normale Superieure LPTENS 78/14. ]51 B. Zumino, Lectures given at Cargese summer school (1978), to be published in the proc. [61 K.S. Stelle and P.C. West, Imperial College preprint ICTP77-78124.
79B, number
3
PHYSICS
20 November
LETTERS
1978
COMPONENTS OF SUPERSPACE Martin ROCEK ’ Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA
and Ulf LINDSTRijM Institute for Theoretical Physics, University of Stockholm, S-l 13 46 Stockholm, Sweden Received
2 August
1978
We give the explicit
superspace
form of the newly
discovered
Although the original formulation of supergravity used explicit components, elegant reformulations have been given in superspace [2]. Recently a tensor calculus for the chiral multiplet [l] has been given, again in component form. In this brief letter we use this formulation to find a precise relation between the component approach and superspace. We begin by deriving the superspace transformation laws that correspond to the local supersymmetry transformations of ref. [ 11. Since we are working in a specific gauge, we can expand the chiral superfield in purely left (right) handed 0’s. A left handed multiplet takes the form: @(z, 0) =a@) +2~,&(z)+~A&+(z).
(1)
(We use two-component spinors throughout l).) Combining the standard transformation of scalars,
(see table property
sq~,e)=-(6eA(a/ae~)~-sz~Jf'(a/a~~~')~), with the explicit form of 6@ in ref. [ 11, we find: +$t=eA
+,/%+B’$ABB’-@&~A
+ ;++A’
for the chiral multiplet
in part by DOE contract
No. EY-76-S-02-
[ 11.
BAe A’ - eA’tiB tBA,G/B’
Although the formulae (3) appear complicated, their interpretation is simple: the leading terms are the same as the rigid transformations of global supersymmetry, the $ terms are their obvious supercovariantizations, and the terms with w, J * and A provide the appropriate Lorentz rotations in the commutator of two transformations. Applications of these results will be presented in a later paper currently in preparation; we merely note that our approach, though superficially similar to that of Ogievetsky and Sokatchev [3], is distinct in that, by construction, it gives the algebra of ref. [l]. In ref. [4], the kinetic multiplet for the chiral super field was presented in component form. We use this result to construct several important quantities in superspace. Recently [5], a chiral gauge was constructed in superspace. In this gauge, the spinor (right chiral) derivative has the simple form:
(3)
and the chiral projector d(a/aeA’)(a/aeAf)(P2(z,8,
where
3227.
calculus
QA’ =cp(Z,e,8,)(a/aeA'),
,
-~zMM'=~(~~M~'-~,~C~~,~LM'MB'),
’ Work supported
(2)
tensor
(4)
[6] takes the form [5]: 8).
(5)
Using this information, a straightforward calculation yields both the shift from z to z* (note that the transformation law (3) of z shows that z must be complex), 217
Volume
79B, number
3
PHYSICS
LETTERS
20 November
Table 1
vMM’
Our conventions are the standard conventions of the NewmannPenrose formalism, except that our metric has the opposite signature, and hence: nab = -E~BEA’B’; the relation with other notations is summarized below.
~M’=~MB,B’M’+fiAMM’_~~BMB’~MRB,,
Refs. -_
[ 1,4]
Ref. [6]
v-2= i +,/tie*
4 R
in the notation
$2i) (F - F*) z(cl+J*) (l/W (J -J *)
of refs.
?p_G
!?+O =
X/?*,
sip,‘Y&
= dXAV,J’,
XP-.yas =
_
*MM’ _ _
-b, -R,, - i~clvpo ___._~~__
WJ EwP~ ______~_..
=
t
i?n Y5 -G -M -N
[l] and [4] : Pk = (1 f -ys)/2,
XA's*' , xP++,
= -JZ,*?ff,
etc.
&xA’~*,
The spin 3/2 field is represented by: P+$ = $AMMg. The chirally supercovariant spin 3/2 fieltequation is written: P+R,,= P+Epvpoy5’Yv(ap+ $wpabdb+ The spin connection
has 3/2 torsion
Wllab= t (~K~~*B~A’B’
&&Ys)$JL==*M,W.
and is written:
+ ~MA~A’B’EAB)
.
And finally pi’ = gfipr * = - PNN In our conventions, complex conjugation does not effect the order of spinors; this is inconvenient for quantization, but makes algebra simpler.
the unknown function p(z). The procedure goes as follows: we take the scalar field @(z, 0), and complex conjugate it; we then shift z* by an unknown function Az(z, 0, e), and multiply by an unknown function (p2(z, 0, f). Finally, we apply 4 a*‘a, I by taking the BAl B* component, and we set the result equal to the kinetic multiplet. Terms without derivatives determine (p2(z, 0, 3) unambiguously, and terms with derivatives determine Az(z, 8, e). We find:
-
$ iANN’
_. f/Q,NLMc~NMM’,
and
_~___
xrl ir5 G s P
Further,
This paper
= -rNN’
1978
QA,**‘- f eAe*(ILA,MM,~*‘MM’t~6*)
eAeAj,/fipAA’eAeAeAf@‘t eA’eA’(iR),
where
&
-&/ijj,**‘_ -
wAA’B’
+BwM’_
~~BB,~‘BB’+DBB,~‘BB;6) ;&,#BB’~B,CC’~Bcc,,
and R is precisely the Ricci scalar multiplet called R in ref. [2] and called U in ref. [ 1). Since q2 has leading term 1, cpcan easily be found. The appearance of R as the coefficient of eA t tl* ’ was predicted by Zumino [S], and served as a gratifying check on our algebra. A further check was made by observing that x = f (z +z*) = z + g AZ is a real quantity; we then found that when AZ is expressed in terms of x, it is indeed a purely imaginary quantity. Although the variables x, 0, and e form a real symmetric basis in superspace, they are not convenient coordinates; this is because their transformation laws, when expressed in terms of x, 0, and 8, contain derivatives of E, the parameter of supersymmetry transformations. A more convenient symmetric basis can be found by adding a real shift to x and corresponding shifts to 8 and fi. We are currently investigating this and related problems.
and
nzMM’
= -2.JZeMeM’
+ 2(oCoc)oAj$
t (e,ec)
M’MA’_
(e,,eCj
x (vM”‘-~MM’-~~M*M’~~,), where
218
2(eC,eC’)e,
$MAM’
We would like to thank the Institut d’Etudes Scientifiques de Cargese for its hospitality, the many participants of the Summer School on “Recent developments in gravitation” and in particular Prof. Zumino for fruitful discussions. One of us (M.R.) would like to thank Prof. S.W. Hawking for his hospitality at D.A.M.P.T., where this investigation started. [l 1 S. Ferrara and P. van Nieuwenhuizen,
Phys. Lett. 76B (1978) 404. P:I E.g.: J. Wess and B. Zumino, Phys. Lett. 74B (1978) 51, and references therein. Joint Inst. for Nucl. [3 I V. Ogievetsky and E. Sokatchev, Research Dubna (1978). preprint Lab. Phys. ]41 S. Ferrara and P. van Nieuwenhuizen, Theo. Ecole Normale Superieure LPTENS 78/14. ]51 B. Zumino, Lectures given at Cargese summer school (1978), to be published in the proc. [61 K.S. Stelle and P.C. West, Imperial College preprint ICTP77-78124.