UUITP-49/20

Superconformal geometries and local twistors

P.S. Howea,***email<EMAIL_ADDRESS>and U. Lindströmb,c,†††email:
<EMAIL_ADDRESS>
aDepartment of Mathematics, King’s College London

The Strand, London WC2R 2LS, UK

bDepartment of Physics, Faculty of Arts and Sciences,

Middle East Technical University, 06800, Ankara, Turkey

cDepartment of Physics and Astronomy, Theoretical Physics, Uppsala University

SE-751 20 Uppsala, Sweden

Abstract

Superconformal geometries in spacetime dimensions $D=3,4,{5}$ and $6$ are
discussed in terms of local supertwistor bundles over standard superspace.
These natually admit superconformal connections as matrix-valued one-forms. In
order to make contact with the standard superspace formalism it is shown that
one can always choose gauges in which the scale parts of the connection and
curvature vanish, in which case the conformal and $S$-supersymmetry
transformations become subsumed into super-Weyl transformations. The number of
component fields can be reduced to those of the minimal off-shell conformal
supergravity multiplets by imposing constraints which in most cases simply
consists of taking the even covariant torsion two-form to vanish. This must be
supplemented by further dimension-one constraints for the maximal cases in
$D=3,4$. The subject is also discussed from a minimal point of view in which
only the dimension-zero torsion is introduced. Finally, we introduce a new
class of supermanifolds, local super Grassmannians, which provide an
alternative setting for superconformal theories.

Dedicated to this year’s Nobel Laureate in Physics, Sir Roger Penrose,

in appreciation of his many achievements, including the invention of twistor
theory

###### Contents

  1. 1 Introduction
  2. 2 Conformal geometry
  3. 3 Superconformal geometry
  4. 4 $D=3$
  5. 5 $D=4$
  6. 6 $D=6$
  7. 7 $D=5$
  8. 8 Minimal approach
  9. 9 Local Super Grassmannians
  10. 10 Summary

##  1 Introduction

Conformal symmetry has been extensively studied over the years because of its
relevance to various aspects of theoretical physics: two-dimensional conformal
theory models and statistical mechanics; four-dimensional $N=2$ and $4$
superconformal field theories; as an underlying symmetry that may be broken to
Poincaré symmetry in four-dimensional spacetime, and as a tool to construct
off-shell supergravity theories. It played an important role from early on in
studies of quantum gravity [1]. Here we shall be interested in the geometry of
local (super) conformal theories as represented on bundles of supertwistors
over superspace.

The first paper on $D=4,N=1$ conformal supergravity (CSG) [2] used a formalism
in which the entire superconformal group was gauged in a spacetime context.
Although this was not a fully geometrical set-up because supersymmetry does
not act on spacetime itself, but rather on the component fields, this paper
nevertheless introduced the idea that gauging conformal boosts and scale
transformations could be very useful. After Poincaré supergravity had been
constructed in conventional (i.e. Salam-Strathdee [3]) superspace [4],[5], it
was subsequently shown how scale transformations could be incorporated as
super-Weyl transformations [5, 6]. On the other hand, in the completely
different approach to superspace supergravity of [7], super scale
transformations were built in right from the start. However, it has turned out
to be difficult to extend the latter approach to other cases, such as higher
dimensions or higher $N$. The superspace geometry corresponding to all $D=4$
off-shell CSG multiplets was given in [8], [9] using conventional superspace
with an $SL(2,\mbox{\bbbold C})\times U(N)$ group in the tangent spaces
together with real super-Weyl transformations. The superspace geometries
corresponding to most other off-shell CSG multiplets have also been described,
from the conventional point of view in $D=3$ [10, 11, 12] and from conformal
superspace in $D=3,4$ and $5$ [13],[14],[15],[16],[17], and in $D=6$ (the
$(1,0)$ theory) [18], [19], [20]. The $D=4$ $N=2$ theory [21] and the $D=6$
$(1,0)$ theory were also discussed earlier in harmonic superspace in [21] and
[22] respectively. In addition, some years ago, the $D=6$ $(1,0)$ theory was
formulated in in projective superspace [23]. This theory as well as the
$D=6~{}(2,0)$ theory were recently discussed in terms of local supertwistors
in [24].

In the non-supersymmetric case a standard approach is to work with
conventional Riemannian geometry augmented by Weyl transformations of the
metric. The Riemann tensor splits in two parts, the conformal Weyl tensor, and
the Schouten tensor which is a particular linear combination of the Ricci
tensor and the curvature scalar. This object transforms in a connection-type
way under Weyl transformations and can be used to construct a new connection
known as the tractor connection [25, 26, 27]. This takes its values in a
parabolic subalgebra of the conformal algebra and acts naturally on a vector
bundle whose fibres are $\mbox{\bbbold R}^{1,n+1}$ (in the Euclidean case),
thus generalising in some sense the standard conformal embedding of flat
$n$-dimensional Euclidean space. Although this idea does not carry over
straightforwardly to the supersymmetric case, a similar construction, which
does, can be made by replacing the $(n+2)$-dimensional fibre by the relevant
twistor space. This formalism is called the local twistor formalism and was
introduced in $D=4$ in [29]. It has been discussed in the supersymmetric
case111 Global supertwistors and the relation to conformal supersymmetry were
introduced in [28] for $N=1,2$ in $D=4$ by Merkulov, [30, 31, 32]. Such a
formalism depends on the dimension of spacetime because the twistor spaces
also change.

In the following we shall take a slightly different approach to that of
Merkulov in that we start from a connection taking its values in the full
superconformal algebra. The conformal superspace formalism alluded to
previously [13], [14] is a supersymmetric version of the Cartan connection
formalism [33], which was first mentioned in the superspace context in [34].
The formalism we advocate here can be thought of as an associated Cartan
formalism in that the connection acts on a vector bundle rather than a
principal one.

In section 2 we briefly discuss bosonic conformal geometry starting from a
local scale-symmetric formulation from which we recover the standard
formalism; we also review the rôle of the Schouten tensor. We then give a
brief outline of the local twistor point of view in $D=3,4,6$. A connection
for the full superconformal group in the twistor representation is given and
the standard formalism is recovered by a suitable choice of gauge with respect
to local conformal boost transformations. This is then generalised to the
supersymmetric case starting with a quasi-universal discussion in section 3,
and followed up by details of the $D=3,4$ and $6$ cases in sections 4,5 and 6
respectively. In section 7, the $D=5,N=1$ case, which is slightly different,
is discussed. In all cases constraints have to be imposed in order to reduce
the component field content to that of the respective minimal off-shell
conformal supergravity multiplets, the main one being that the even torsion
two-form should be the same as in the flat case. In section 8, we describe the
minimal formalism, in which one introduces only the dimension-zero torsion.
This formalism applies equally well in all cases, including $D=5$ (N=1). In
section 9 we introduce a new class of supermanifolds which we call local super
Grassmannians, which could be useful in constructing alternative approaches to
the subject. In section 10, we make some concluding remarks.

##  2 Conformal geometry

From a mathematical point of view a conformal structure in $n$ dimensions can
be thought of as $G$-structure with $G=CO(n)$, the orthogonal group augmented
by scale transformations, see, for example [26, 27]. This group does not
preserve a particular tensor, but only a metric up to a scale transformation,
so that angles but not lengths are invariant. Similarly, a classical non-
gravitational theory of fields which is locally scale invariant in a
background gravitational field will be conformally invariant with respect to
the standard conformal group of flat spacetime when the metric is taken to be
flat. So conformal transformation are in this sense already present in a
theory which is $CO(n)$ invariant. Nevertheless, it can sometimes be of use to
make conformal boosts manifest. This can be done, for example, in the Cartan
formalism which we shall describe shortly. We shall start from a Weyl
perspective in which the structure group is $CO(n)$, so that, in addition to
the usual curvature, there is also a scale curvature. It can easily be seen
that the torsion is invariant under shifts of the scale connection, and from
there, one can either use this shift symmetry to set the scale connection to
zero, or introduce a new connection in order to make the curvature invariant.
The first method leads back to the conventional approach, whereas the second
leads to the Cartan formalism with the symmetric part of the conformal boost
connection identified with the Schouten tensor, which will be defined below.

Let $e_{a}$ denote a local basis for the tangent space with dual basis forms
denoted by $e^{a}$. So $e_{a}=e_{a}{}^{m}\partial_{m};\
e^{a}=dx^{m}e_{m}{}^{a}$, where $x^{m}$ denotes local coordinates and where
the coordinate and preferred bases are related to each other by the vielbein
$e_{m}{}^{a}$ and its inverse $e_{a}{}^{m}$. Infinitesimal local $\mbox{\goth
c}\mbox{\goth o}(n)$ transformations act on the frames by

$\delta e_{a}=-\hat{L}_{a}{}^{b}e_{b}\qquad:\delta
e^{a}=e^{b}\hat{L}_{b}{}^{a},$ (2.1)

where

$\hat{L}_{a}{}^{b}=L_{a}{}^{b}+\delta_{a}{}^{b}S\ .$ (2.2)

The parameter $L_{a}{}^{b}$ denotes a local $\mbox{\goth o}(n)$ transformation
preserving the flat metric $\eta_{ab}$, and $S$ is a local scale
transformation, although we note that $\eta_{ab}$ is not invariant under the
latter.

The torsion 2-form is defined by

$T^{a}=\hat{D}e^{a}=de^{a}+e^{b}\omega_{b}{}^{a}+e^{a}\omega_{0}:=De^{a}+e^{a}\omega_{0}\
,$ (2.3)

where $\omega_{a}{}^{b}$ is the $\mbox{\goth o}(n)$ connection and
$\omega_{0}:=e^{a}\omega_{a}$ the scale connection. In components

$T_{ab}{}^{c}=f_{ab}{}^{c}+2(\omega_{[a,b]}{}^{c}-\delta_{[a}{}^{c}\omega_{b]})\
,$ (2.4)

where

$f_{ab}{}^{c}=2e_{[a}{}^{m}e_{b]}{}^{n}\partial_{m}e_{n}{}^{c}\ .$ (2.5)

It is clear that the torsion is invariant under shifts of $\omega_{a}$ by a
parameter $X_{a}$, say, provided that the $\mbox{\goth o}(n)$ connection
tranforms by

$\omega_{a,b}{}^{c}\rightarrow\omega_{a,b}{}^{c}+\delta_{a}{}^{c}X_{b}-\eta_{ab}X^{c}\
.$ (2.6)

It is also clear that we can use $\omega_{a,b}{}^{c}$ to set the torsion to
zero while still maintaining this symmetry, and from now on we shall take this
to be the case.

The curvature 2-form is given by
$\hat{R}_{a}{}^{b}=R_{a}{}^{b}+\delta_{a}{}^{b}R_{0}$ where $R_{0}$ is the
scale curvature. Making use of the first Bianchi identity we find

$R_{ab,}{}^{cd}=C_{ab,}{}^{cd}+4\delta_{[a}{}^{[c}Q_{b]}{}^{d]}$ (2.7)

where the trace-free part $C_{ab,cd}$ is the Weyl tensor and where

$Q_{ab}=P_{ab}+\frac{1}{2}(R_{0})_{ab}\ ,$ (2.8)

with the symmetric part $P_{ab}$ being the Schouten tensor (also known as the
rho-tensor). It is given by

$P_{ab}=\frac{1}{n-2}\left(R_{ab}-\frac{1}{2(n-1)}\eta_{ab}R\right)\ .$ (2.9)

Here $R_{ab}$ is the usual symmetric Ricci tensor and R the curvature scalar.
Under a finite shift of $\omega_{a}$ accompanied by a (2.6) transformation we
find

$\Delta Q_{ab}=\hat{D}_{a}X_{b}-X_{a}X_{b}+\frac{1}{2}\eta_{ab}X^{2}\ ,$
(2.10)

where $\Delta$ denotes a finite change.

At this stage one option is to use $X$ to set the scale connection to zero.
This gauge will be preserved by a combined $X$ transformation and a scale
transformation provided that

$X_{a}=Y_{a}:=S^{-1}D_{a}S=e_{a}{}^{m}(S^{-1}\partial_{m}S)=e_{a}{}^{m}Y_{m}\
,$ (2.11)

where $S$ here denotes a finite scale transformation. So at this point we have
regained the conventional formalism: the local scale transformations are no
longer regarded as part of the tangent space group but instead are simply
rescalings of the metric without an additional scale connection. The
transformation of $P_{ab}$ is given by (2.10) but with $X$ replaced by $Y$,
accompanied by an overall factor of $S^{-2}$. This factor then disappears in a
coordinate basis so that the usual formula is recovered.

In the conventional formalism one can define a new connection, the tractor
connection, which takes its values in the Lie algebra of the conformal algebra
$\mbox{\goth s}\mbox{\goth o}(n+1,1)$, and which acts naturally to give a
covariant derivative acting on vector fields in $\mbox{\bbbold R}^{n+1,1}$.
However, the tractor formalism cannot be adapted directly to the
supersymmetric case because the superconformal groups are not simply given by
super Lorentz groups in two higher dimensions, one of which is timelike.
Instead, one should think about supertwistors because they naturally carry the
fundamental representations of superconformal algebras. It is therefore more
relevant to study local twistor connections, introduced in [29] in the non-
supersymmetric case. Then one has to consider different twistor spaces
according to the dimensions of spacetime. In general we can write an element
of the conformal Lie algebra, $h$, in the form

$h=\left(\begin{array}[]{cc}-a^{\alpha}{}_{\beta}&b^{\alpha\beta^{\prime}}\\\
c_{\alpha^{\prime}\beta}&d_{\alpha^{\prime}}{}^{\beta^{\prime}}\end{array}\right)$
(2.12)

where $\alpha,\alpha^{\prime}$ etc denote spinor indices which have two
components for $D=3,4$ and four components for $D=6$. The diagonal elements
$a,d$ are Lorentz and scale transformations while the off-diagonal ones are
translations, $b$, and conformal boosts $c$.

The (Lie algebra valued) connection is

${\cal A}=\left(\begin{array}[]{cc}-\hat{\omega}&e\\\
f&\hat{\omega}^{\prime}\end{array}\right)$ (2.13)

where each entry is a one-form with the index structure given in the previous
equation. The diagonal elements are connections for Lorentz and scale
transformations, while the off-diagonal entries are the vielbein form $e$ and
the conformal connection $f$. The transformation of ${\cal A}$ is

${\cal A}\mapsto g^{-1}{\cal A}g+dg^{-1}g\ ,$ (2.14)

and the curvature ${\cal F}$ is defined by

${\cal F}=d{\cal A}+{\cal A}^{2}\ .$ (2.15)

It transforms covariantly under $g$ without a derivative term. Its components
are given by

${\cal F}=\left(\begin{array}[]{cc}-\hat{\cal R}^{\alpha}{}_{\beta}&{\cal
T}^{\alpha\beta^{\prime}}\\\ {\cal S}_{\alpha^{\prime}\beta}&\hat{\cal
R}_{\alpha^{\prime}}{}^{\beta^{\prime}}\end{array}\right)\ ,$ (2.16)

where the diagonal terms are the covariant Lorentz and scale curvatures,
${\cal T}$ is the torsion and ${\cal S}$ the conformal curvature. In terms of
the standard torsion and curvature one has

${\cal
F}=\left(\begin{array}[]{cc}-\hat{R}^{\alpha}{}_{\beta}-e^{\alpha\gamma^{\prime}}f_{\gamma^{\prime}\beta}&T^{\alpha\beta^{\prime}}\\\
Df_{\alpha^{\prime}\beta}&\hat{R}_{\alpha^{\prime}}{}^{\beta^{\prime}}-f_{\alpha^{\prime}\gamma}e^{\gamma\beta^{\prime}}\end{array}\right)$
(2.17)

The objective now is to construct an element of the conformal group depending
on the scale and conformal parameters, and a connection one-form with values
in the conformal algebra which will transform in the required way provided
that the transformation of the Schouten tensor is as given above in (2.10).
The group element $g$ is given by

$g=\left(\begin{array}[]{cc}S^{-\frac{1}{2}}&0\ ,\\\
S^{-\frac{1}{2}}C&S^{\frac{1}{2}}\end{array}\right)\ ,$ (2.18)

where the diagonal elements involve unit matrices and $C$ is a covector,
$C_{\alpha^{\prime}\beta}$. Under a conformal transformation of this form one
can straightforwardly compute the changes in the components of ${\cal A}$ to
be

$\displaystyle\hat{\omega}$ $\displaystyle\mapsto\hat{\omega}-eC-\frac{1}{2}Y\
,$ (2.19) $\displaystyle\hat{\omega}^{\prime}$
$\displaystyle\mapsto\hat{\omega}^{\prime}-Ce-\frac{1}{2}Y\ ,$ (2.20)
$\displaystyle e$ $\displaystyle\mapsto eS\ ,$ (2.21) $\displaystyle f$
$\displaystyle\mapsto S^{-1}(f-\hat{D}C+CeC)\ ,$ (2.22)

where again the index structure on the various elements follows from the
original definitions, and where $Y=S^{-1}dS$. In order to compare with the
previous discussion we need to eliminate the scale curvature and express the
conformal boost parameter in terms of the scale parameter $S$. In addition, we
use the Lorentz connection to set the torsion to zero as usual. The conformal
connection $f$ is a covector-valued one-form, $f_{b}=e^{a}f_{ab}$, and we can
use the anti-symmetric part of $f_{ab}$ to set $({\cal R}_{0})_{ab}=0$. If we
take the trace of the transformation of $\hat{\omega}$, which is proportional
to the transformation of $\omega_{0}$, we can see that the conformal boost
parameter can be used to set $\omega_{0}=0$, so that $R_{0}=0$ as well. This
gauge will be preserved if $C_{a}\propto Y_{a}$. This gives us the desired
result: the scale curvatures ${\cal R}_{0}$ and $R_{0}$ are both zero as is
the antisymmetric part of $f$ so that we can identify the remaining symmetric
part $f_{(ab)}$ with the Schouten tensor $P_{ab}$.

These transformations agree with the usual ones in the conventional formalism,
expressed in spinor notation and with respect to an orthonormal basis. For the
connection form $\omega_{a}{}^{b}$ this translates to

$\omega_{a}{}^{b}\mapsto e^{b}Y_{a}-e_{a}Y^{b}\ .$ (2.23)

For $P_{ab}$ we recover the standard transformation for the tractor connection
which in an orthonormal basis reads

$P_{ab}\mapsto S^{-2}(D_{a}Y_{b}+Y_{a}Y_{b}-\frac{1}{2}\eta_{ab}Y^{2})\ .$
(2.24)

Now let us return to the Weyl picture with non-zero scale connection but still
with the torsion taken to be zero. If we redefine the Lorentz and scale
curvature two-forms by

$\displaystyle R^{ab}$ $\displaystyle\mapsto
R^{{}^{\prime}ab}=R^{ab}+2e^{[a}\wedge Q^{b]}$ $\displaystyle R_{0}$
$\displaystyle\mapsto R^{\prime}_{0}=R_{0}-e^{a}\wedge Q_{a}\ ,$ (2.25)

where $Q^{b}:=e^{a}Q_{a}{}^{b}$, then we observe that the primed quantities
are invariant under infinitesimal $X$ gauge transformations, for which

$\delta Q^{a}=\hat{D}X^{a}\ .$ (2.26)

We can also define a new curvature two-form

$R^{\prime a}:=\hat{D}Q^{a}\ .$ (2.27)

We can interpret $X$ as a local conformal boost parameter, $Q^{a}$ as the
corresponding gauge field and $R^{\prime a}$ as its curvature. We have thus
arrived at the conformal gauging picture starting from the Weyl perspective.
In fact, we can identify the combined primed curvatures together with the
$\hat{\cal R}$ curvatures, and $Q$ with $f$, in (2.17). Of course, this is
just the converse to deriving the conventional point of view starting from the
conformal perspective as discussed, for example, in [14]

To conclude this outline of non-supersymmetric conformal geometry we briefly
review the theory of Cartan connections of which conformal gauging is an
example. Let $H,G$ be Lie groups, $H\subset G$, with respective Lie algebras
$\mbox{\goth h},\mbox{\goth g}$ let $P$ be a principal $H$-bundle over a base
manifold $M$. A Cartan connection on $P$ is a g-valued form $\omega$
equivariant with respect to $H$, and such that $\forall X\in\mbox{\goth h}$
$\omega(X)=X$ and $\omega$ gives an isomorphism from $T_{p}P$ to g, for any
point $p\in P$.

A simple example is given by an $n$-dimensional manifold $M$ with
$G=SO(n)\ltimes\mbox{\bbbold R}^{n}$, $H=SO(n)$. Then $\mbox{\goth
g}=\mbox{\goth g}_{-1}\oplus\mbox{\goth h}$, where $\mbox{\goth g}_{-1}$
corresponds to translations and h to rotations. The translational part of
$\omega$ is identified with the soldering form, i.e. the vielbein, while the
h-part corresponds to an $\mbox{\goth s}\mbox{\goth o}(n)$ connection. In the
conformal case $\mbox{\goth h}=\mbox{\goth g}_{0}\oplus\mbox{\goth g}_{1}$
where $\mbox{\goth g}_{0}=\mbox{\goth s}\mbox{\goth o}(n)\oplus\mbox{\bbbold
R}$, and $\mbox{\goth g}_{1}=\mbox{\bbbold R}^{n}$. So $\mbox{\goth g}_{o}$
corresponds to rotations and scale transformations while $\mbox{\goth g}_{1}$
corresponds to conformal boosts. The grading of the Lie algebra then
corresponds to the dilatational weights of the various components. The
curvature of $\omega$, ${\cal R}=d\omega+\omega^{2}$, also has components
corresponding to this grading, and it is straightforward to see that they
correspond to the torsion, the curvature and scale curvature, and the
conformal boost field strength respectively.

##  3 Superconformal geometry

In this section we shall present a quasi-universal formalism for
superconformal geometries in $D=3,4$ and $6$, working in complexified
spacetime for the moment. The basic idea is to introduce a connection one-form
${\cal A}$ on superspace taking its values in the appropriate superconformal
algebra acting on the super vector bundle whose fibres are super-twistors.
These have the form

$Z^{\underline{\alpha}}=\left(\begin{array}[]{c}u^{\alpha}\\\
v_{\alpha^{\prime}}\\\ \lambda_{i}\\\ \end{array}\right)\ .$ (3.1)

The spinor indices on $(u,v)$ are two-component for $D=3,4$ and four-component
for $D=6$. The primed spinor indices are dotted indices for $D=4$ and are the
same as the unprimed ones in all other cases. The odd part of a super-twistor
is $\lambda_{i}$, where $i=1,\ldots M$, with $M=N$, the number of
supersymmetries for $D=3,4$ and $M=2N$ for $D=6$.

The connection is given by

$\cal{A}=\left(\begin{array}[]{c c
|c}-\hat{\Omega}^{\alpha}{}_{\beta}&E^{\alpha\beta^{\prime}}&E^{\alpha j}\\\
F_{\alpha^{\prime}\beta}&\hat{\Omega}_{\alpha^{\prime}}{}^{\beta^{\prime}}&F_{\alpha^{\prime}}{}^{j}\\\
\hline\cr\tilde{F}_{i\beta}&\tilde{E}_{i}{}^{\beta^{\prime}}&\Omega_{i}{}^{j}\end{array}\right)\
.$ (3.2)

For $D=3$ the pair ($\alpha\beta)$ on $(E,F)$ are symmetric whereas in $D=6$
they are antisymmetric. The connection components are as follows:
($E^{\alpha\beta^{\prime}},E^{\alpha j}$) correspond to translations and
$Q$-supersymmetry, $(F_{\alpha^{\prime}\beta},F_{\alpha^{\prime}}{}^{j})$ to
conformal and $S$-supersymmetry, while the diagonal $\Omega$s correspond to
Lorentz symmetry above the line and internal symmetry below the line with the
hats indicating that the scale connection is also included.
$(\tilde{E},\tilde{F})$ on the third line are appropriate transpositions of
$(E,F)$. We will occasionally refer to the internal connection as
$\Omega_{I}$.

The curvature two-form is given by ${\cal F}=d{\cal A}+{\cal A}^{2}$; its
components in matrix form are:

$\cal{F}=\left(\begin{array}[]{c c |c}-\hat{\cal R}^{\alpha}{}_{\beta}&{\cal
T}^{\alpha\beta^{\prime}}&{\cal T}^{\alpha j}\\\ {\cal
S}_{\alpha^{\prime}\beta}&\hat{{\cal
R}}_{\alpha^{\prime}}{}^{\beta^{\prime}}&{\cal S}_{\alpha^{\prime}}{}^{j}\\\
\hline\cr\tilde{\cal S}_{i\beta}&\tilde{{\cal T}}_{i}{}^{\beta^{\prime}}&{\cal
R}_{i}{}^{j}\end{array}\right)\ ,$ (3.3)

where

$\displaystyle{\cal T}^{\alpha\beta^{\prime}}$
$\displaystyle=\hat{T}^{\alpha\beta^{\prime}}+E^{\alpha
k}\tilde{E}_{k}{}^{\beta^{\prime}}$ $\displaystyle{\cal T}^{\alpha j}$
$\displaystyle=\hat{T}^{\alpha
j}+E^{\alpha\gamma^{\prime}}F_{\gamma^{\prime}}{}^{j}$ $\displaystyle{\cal
S}_{\alpha^{\prime}\beta}$
$\displaystyle=\hat{D}F_{\alpha^{\prime}\beta}+F_{\alpha^{\prime}}{}^{k}\tilde{F}_{k\beta}$
$\displaystyle{\cal S}_{\alpha^{\prime}}{}^{j}$
$\displaystyle=\hat{D}F_{\alpha^{\prime}}{}^{j}+F_{\alpha^{\prime}\gamma}E^{\gamma
j}$ $\displaystyle\hat{{\cal R}}^{\alpha}{}_{\beta}$
$\displaystyle=\hat{R}^{\alpha}{}_{\beta}+E^{\alpha\gamma^{\prime}}F_{\gamma^{\prime}\beta}+E^{\alpha
k}\tilde{F}_{k\beta}$ $\displaystyle\hat{{\cal
R}}_{\alpha^{\prime}}{}^{\beta^{\prime}}$
$\displaystyle=\hat{R}_{\alpha^{\prime}}{}^{\beta^{\prime}}+F_{\alpha^{\prime}\gamma}E^{\gamma\beta^{\prime}}+F_{\alpha^{\prime}}{}^{k}\tilde{E}_{k}{}^{\beta^{\prime}}$
$\displaystyle{\cal R}_{i}{}^{j}$
$\displaystyle=R_{i}{}^{j}+\tilde{F}_{i\gamma}{E}^{\gamma
j}+\tilde{E}_{i}^{\gamma^{\prime}}F_{\gamma^{\prime}}{}^{j}\ .$ (3.4)

Here $\hat{D}$ denotes the exterior covariant derivative for the Lorentz,
scale and internal parts of the algebra. The first terms on the right of the
first two equations are the standard even and odd superspace torsion tensors
constructed in the usual way from the $\hat{\Omega}$ connections,
$\hat{T}^{\alpha\beta}=\hat{D}E^{\alpha\beta}$, $\hat{T}^{\alpha
j}=\hat{D}E^{\alpha j}$, while the non-calligraphic curvature forms on the
right in the last three lines are the standard curvature tensors constructed
in a similar fashion.

In $D=3,6$ the one-forms $(E^{\alpha\beta^{\prime}},E^{\alpha j})$ can be
identified, after converting pairs of spinor indices to vector indices and, if
necessary, rescaling, with the basis one-forms of conventional superspace,
$E^{A}=(E^{a},E^{\alpha i})$, while in $D=4$,
$\tilde{E}_{i}{}^{\alpha^{\prime}}$, which becomes the complex conjugate of
$E^{\alpha i}$ in real superspace, is also required in order to complete the
basis forms. The forms $E^{A}$ are associated with super-translations, and
play the role of soldering forms in this context. This means that the
translational part of the algebra can be subsumed into super-diffeomorphisms.

In a similar fashion, we can identify the pair
$(F_{\alpha^{\prime}\beta},{F}_{\alpha^{\prime}}{}^{j})$ with a super-
covector-valued one-form $F_{B}=E^{A}F_{AB}$. In $D=4$ the tilded odd forms
are essentially the complex conjugates of the untilded ones (in real
spacetime) and similar identifications can be made in other dimensions.

We remark in passing that, although there is also a conformal supergravity
theory in $D=5$ for $N=1$ [35],[36], which can be described in conventional
superspace [16] and in conformal superspace [17], it does not admit a
straightforward description in the supertwistor formalism. In this case the
spinor indices are four-component, with no distinction between primed and
unprimed, vectors can be represented by skew-symmetric symplectic-traceless
spinors, e.g.
$E^{\alpha\beta}=-E^{\beta\alpha},\,\eta_{\alpha\beta}E^{\alpha\beta}=0$,
where $\eta_{\alpha\beta}$ is the symplectic matrix (charge conjugation
matrix), and where the spacetime part of the curvature is
$\gamma^{a}$-traceless,
$\hat{\Omega}^{\alpha}{}_{\beta}(\gamma^{a})^{\beta}{}_{\alpha}=0$. However,
the bilinear fermion terms in the bosonic curvatures in (3) do not preserve
these contraints. Nevertheless, we shall see in section 7, that the formalism
can be amended to take the $D=5$ case into account.

The Bianchi identity is

${\cal D}{\cal F}:=d{\cal F}+[{\cal F},{\cal A}]=0\ .$ (3.5)

Written out in components this is, for the torsions,

$\displaystyle\hat{D}{\cal
T}^{\alpha\beta^{\prime}}-E^{\alpha\gamma^{\prime}}\hat{\cal
R}_{\gamma^{\prime}}{}^{\beta^{\prime}}-\hat{\cal
R}^{\alpha}{}_{\gamma}E^{\gamma\beta^{\prime}}-iE^{\alpha k}\tilde{{\cal
T}}_{k}{}^{\beta^{\prime}}-i{\cal T}^{\alpha
k}\tilde{E}_{k}{}^{\beta^{\prime}}$ $\displaystyle=0$
$\displaystyle\hat{D}{\cal T}^{\alpha j}+\hat{\cal
R}^{\alpha}{}_{\gamma}E^{\gamma j}-E^{\alpha k}{\cal R}_{k}{}^{j}+i{\cal
T}^{\alpha\gamma^{\prime}}F_{\gamma^{\prime}}{}^{j}-iE^{\alpha\gamma^{\prime}}{\cal
S}_{\gamma^{\prime}}{}^{j}$ $\displaystyle=0\ ,$ (3.6)

for the Lorentz, scale and internal symmetry curvatures,

$\displaystyle D\hat{\cal R}^{\alpha}{}_{\beta}-{\cal
T}^{\alpha\gamma^{\prime}}F_{\gamma^{\prime}\beta}-{\cal T}^{\alpha
k}\tilde{F}_{k\beta}+E^{\alpha\gamma^{\prime}}{\cal
S}_{\gamma^{\prime}\beta}+E^{\alpha k}\tilde{{\cal S}}_{k\beta}$
$\displaystyle=0$ $\displaystyle D{\cal R}_{ij}-2{\cal
T}^{\gamma}_{(i}F_{|\gamma|j)}+2E^{\gamma}_{(i}{\cal S}_{|\gamma|j)}$
$\displaystyle=0\ ,$ (3.7)

and for the superconformal curvatures,

$\displaystyle\hat{D}{\cal S}_{\alpha^{\prime}\beta}+\hat{\cal
R}_{\alpha^{\prime}}{}^{\gamma^{\prime}}F_{\gamma^{\prime}\beta}+F_{\alpha^{\prime}\gamma}{\cal
R}^{\gamma}{}_{\beta}+i{\cal
S}_{\alpha^{\prime}}{}^{k}\tilde{F}_{k\beta}+iF_{\alpha^{\prime}}{}^{k}\tilde{S}_{k\beta}$
$\displaystyle=0$ $\displaystyle\hat{D}{\cal
S}_{\alpha^{\prime}}{}^{j}+\hat{\cal
R}_{\alpha^{\prime}}{}^{\gamma^{\prime}}F_{\gamma^{\prime}}{}^{j}-F_{\alpha^{\prime}}{}^{k}{\cal
R}_{\kappa}{}^{j}+i{\cal S}_{\alpha^{\prime}\gamma}E^{\gamma
j}-iF_{\alpha^{\prime}\gamma}T^{\gamma j}$ $\displaystyle=0\ .$ (3.8)

We now consider the supersymmetric counterpart of the group element (2.18)
given by

$g=\left(\begin{array}[]{c c |c}S^{-\frac{1}{2}}&0&0\\\
S^{-\frac{1}{2}}C&S^{\frac{1}{2}}&\Gamma\\\ \hline\cr
S^{-\frac{1}{2}}\Delta&0&1\end{array}\right)$ (3.9)

with inverse given by

$g^{-1}=\left(\begin{array}[]{c c |c}S^{\frac{1}{2}}&0&0\\\
S^{-\frac{1}{2}}\tilde{C}&S^{-\frac{1}{2}}&-S^{-\frac{1}{2}}\Gamma\\\
\hline\cr-\Delta&0&1\end{array}\right)\ ,$ (3.10)

where

$C+\tilde{C}-\Gamma\Delta=0\ .$ (3.11)

For the moment we can think of $g$ as an element of the complex supergroup
$SL(M_{0}|M_{1})$, where $(M_{0}|M_{1})$ denote the even and odd dimensions of
supertwistor space. In the real cases we will find that $\Delta$ is related to
$\Gamma$ and that $C$ will have symmetry properties depending on the case in
hand. The transformation of the connection ${\cal A}$ is given by (2.14). For
the superspace basis forms we find

$\displaystyle E^{\alpha\beta^{\prime}}$ $\displaystyle\mapsto
SE^{\alpha\beta^{\prime}}$ (3.12) $\displaystyle E^{\alpha j}$
$\displaystyle\mapsto S^{\frac{1}{2}}(E^{\alpha
j}+E^{\alpha\gamma^{\prime}}\Gamma_{\gamma^{\prime}}{}^{j})$
$\displaystyle\tilde{E}_{i}{}^{\beta^{\prime}}$ $\displaystyle\mapsto
S^{\frac{1}{2}}(\tilde{E}_{i}{}^{\beta^{\prime}}-\Delta_{i\gamma}E^{\gamma\beta^{\prime}})\
,$

while for the connections we find

$\displaystyle\hat{\Omega}^{\alpha}{}_{\beta}$
$\displaystyle\mapsto\hat{\Omega}^{\alpha}{}_{\beta}-E^{\alpha\gamma^{\prime}}C_{\gamma^{\prime}\beta}-E^{\alpha
k}\Delta_{k\beta}-\frac{1}{2}\delta^{\alpha}_{\beta}Y$ (3.13)
$\displaystyle\hat{\Omega}_{\alpha^{\prime}}{}^{\beta^{\prime}}$
$\displaystyle\mapsto\hat{\Omega}_{\alpha^{\prime}}{}^{\beta^{\prime}}+\tilde{C}_{\alpha^{\prime}\gamma}E^{\gamma\beta^{\prime}}-\Gamma_{\alpha^{\prime}}{}^{k}\tilde{E}_{k}{}^{\beta^{\prime}}-\frac{1}{2}\delta^{\alpha^{\prime}}_{\beta^{\prime}}Y$
$\displaystyle\Omega_{i}{}^{j}$
$\displaystyle\mapsto\Omega_{i}{}^{j}-\Delta_{i\gamma}E^{\gamma
j}+\tilde{E}_{i}{}^{\gamma^{\prime}}\Gamma_{\gamma^{\prime}}{}^{j}-\Delta_{i\gamma}E^{\gamma\gamma^{\prime}}\Gamma_{\gamma^{\prime}}{}^{j},$

For the $S$\- supersymmetry connections we have

$\displaystyle F_{\alpha^{\prime}}{}^{j}$ $\displaystyle\mapsto
S^{-\frac{1}{2}}\left(F_{\alpha^{\prime}}{}^{j}-\hat{D}\Gamma_{\alpha^{\prime}}{}^{j}+\tilde{C}_{\alpha^{\prime}\beta}(E^{\beta
j}+E^{\alpha\gamma^{\prime}}\Gamma_{\gamma^{\prime}}{}^{j})-\Gamma_{\alpha^{\prime}}{}^{k}\tilde{E}_{k}{}^{\beta^{\prime}}\Gamma_{\beta^{\prime}}{}^{j}\right)$
(3.14) $\displaystyle\tilde{F}_{i\beta}$ $\displaystyle\mapsto
S^{-\frac{1}{2}}\left(\tilde{F}_{i\beta}-\hat{D}\Delta_{i\beta}+(\tilde{E}_{i}{}^{\gamma^{\prime}}-\Delta_{i\gamma}E^{\gamma\gamma^{\prime}})C_{\gamma^{\prime}\beta}-\Delta_{i\gamma}E^{\gamma
j}\Delta_{j\beta}\right)\ ,$

The conformal connection, $F_{\alpha^{\prime}\beta}$, transforms as

$\displaystyle F_{\alpha^{\prime}\beta}$ $\displaystyle\mapsto
S^{-1}(F_{\alpha^{\prime}\beta}-\hat{D}C_{\alpha^{\prime}\beta}+\Gamma_{\alpha^{\prime}}{}^{k}\hat{D}\Delta_{k\beta}+F_{\alpha^{\prime}}{}^{k}\Delta_{k\beta}-\Gamma_{\alpha^{\prime}}{}^{k}\tilde{F}_{k\beta}$
(3.15)
$\displaystyle+\tilde{C}_{\alpha^{\prime}\gamma}E^{\gamma\delta^{\prime}}C_{\delta^{\prime}\beta}+\tilde{C}_{\alpha^{\prime}\gamma}E^{\gamma
k}\Delta_{k\beta}-\Gamma_{\alpha^{\prime}}{}^{k}\tilde{E}_{k}{}^{\delta^{\prime}}C_{\delta^{\prime}\beta})\
.$

The transformations of the field strengths can also be found
straightforwardly. For the torsions we have

$\displaystyle{\cal T}^{\alpha\beta^{\prime}}$ $\displaystyle\mapsto S{\cal
T}^{\alpha\beta^{\prime}}$ (3.16) $\displaystyle{\cal T}^{\alpha j}$
$\displaystyle\mapsto S^{\frac{1}{2}}({\cal T}^{\alpha j}+{\cal
T}^{\alpha\gamma^{\prime}}\Gamma_{\gamma^{\prime}}{}^{j})$
$\displaystyle\tilde{\cal T}_{i}{}^{\beta^{\prime}}$ $\displaystyle\mapsto
S^{\frac{1}{2}}(\tilde{\cal T}_{i}{}^{\beta^{\prime}}-\Delta_{i\gamma}{\cal
T}^{\gamma\beta^{\prime}})\ ,$

while for the Lorentz, scale and internal curvatures we have:

$\displaystyle\hat{\cal R}^{\alpha}{}_{\beta}$ $\displaystyle\mapsto\hat{\cal
R}^{\alpha}{}_{\beta}-{\cal
T}^{\alpha\gamma^{\prime}}C_{\gamma^{\prime}\beta}-{\cal T}^{\alpha
k}\Delta_{k\beta}$ (3.17) $\displaystyle\hat{\cal
R}_{\alpha^{\prime}}{}^{\beta^{\prime}}$ $\displaystyle\mapsto\hat{\cal
R}_{\alpha^{\prime}}{}^{\beta^{\prime}}+\tilde{C}_{\alpha^{\prime}\gamma}{\cal
T}^{\gamma\beta^{\prime}}-\Gamma_{\alpha^{\prime}}{}^{k}\tilde{\cal
T}_{k}{}^{b^{\prime}}$ $\displaystyle{\cal R}_{i}{}^{j}$
$\displaystyle\mapsto{\cal R}_{i}{}^{j}-\Delta_{i\gamma}{\cal T}^{\gamma
j}+\tilde{\cal
T}_{i}{}^{\gamma^{\prime}}\Gamma_{\gamma^{\prime}}{}^{j}-\Delta_{i\gamma}{\cal
T}^{\gamma\gamma^{\prime}}\Gamma_{\gamma^{\prime}}{}^{j},$

The scale curvatures for $D=3,6$ are given by the trace of $\hat{\cal
R}^{\alpha}{}_{\beta}$ given in (3), while for $D=4$ we have to take the sum
of the traces of $\hat{\cal R}^{\alpha}{}_{\beta}$ and its complex conjugate;
in all cases, we can write

$({\cal R}_{0})_{AB}=(d\Omega_{0})_{AB}+kF_{[AB]}\ ,$ (3.18)

where $k$ is a constant depending on the dimension of spacetime, and where
$F_{[AB]}$ is the graded anti-symmetric part of $F_{AB}$, the latter having no
symmetry. Since $F_{A}$ is a connection one-form we are free to add a
tensorial part to it and thereby set the scale curvature ${\cal R}_{0}=0$,
after which $(d\Omega_{0})_{AB}=-F_{[AB]}$. As in the bosonic case
$\Omega_{0}$ transforms by a shift under conformal and $S$-supersymmetry
transformations, as can be seen from the trace of the first line in (3.13),
and can therefore also be set to zero. We are then left with the symmetric
part of $F_{AB}$ which we can be identified as the super Schouten tensor.
There is then a residual local super-Weyl invariance which we will present in
more detail below for each case.

To put more flesh on this general outline we shall now go through the various
cases in turn.

##  4 $D=3$

The superconformal group in $D=3$ is $SpO(2|N)$. This is the same as the
orthosymplectic group, but with the symplectic factor written first to
indicate that it refers to the spacetime part. It acts on the supertwistor
space $\mbox{\bbbold C}^{4|N}$ (in the complex case) and consists of
$(4|N)\times(4|N)$ matrices $g$ which preserve the symplectic-orthogonal form

$J=\left(\begin{array}[]{c c |c}0&1&0\\\ -1&0&0\\\ \hline\cr
0&0&1_{N}\end{array}\right)\ .$ (4.1)

The invariance condition is

$gJg^{st}=J$ (4.2)

where the $st$ superscript denotes the super-transpose of the matrix $g$. This
is the same as the ordinary transpose except for the odd component in the
lower left corner which has an additional minus sign. For a Lie superalgebra
element $h$ we have, correspondingly,

$hJ=-Jh^{st}$ (4.3)

The reality condition needed to restrict to real spacetime (and superspace) is

$gKg^{*}=K$ (4.4)

where $K$ has a similar structure to J but with the minus sign on the second
row replaced by a plus sign. The form of a real superalgebra element $h$ is
therefore

$h=\left(\begin{array}[]{c c |c}a&ib&\gamma\\\ ic&d&\delta\\\
\hline\cr\delta^{t}&-\gamma^{t}&e\end{array}\right)\ ,$ (4.5)

where $d=-a^{t}$, $b$ and $c$ are symmetric, $e$ is anti-symmetric, $\gamma$
is real and $\delta$ is imaginary. Since the connection is a g-valued one-form
it can be written as

$\cal{A}=\left(\begin{array}[]{c c
|c}-\hat{\Omega}^{\alpha}{}_{\beta}&iE^{\alpha\beta}&E^{\alpha j}\\\
iF_{\alpha\beta}&\hat{\Omega}_{\alpha}{}^{\beta}&F_{\alpha}{}^{j}\\\
\hline\cr\tilde{F}_{i\beta}&\tilde{E}_{i}{}^{\beta}&\Omega_{i}{}^{j}\end{array}\right)\
.$ (4.6)

Here $E^{\alpha\beta}$ and $F_{\alpha\beta}$ are real and symmetric,
$E^{\alpha j}$ is real and $F_{\alpha}{}^{j}$ is imaginary (for later
convenience). On the bottom row $\tilde{E}_{i}{}^{\beta}=-E^{\beta}{}_{i}$,
while $\tilde{F}_{i}{}^{\beta}=F^{\beta}{}_{i}$, where the $\mbox{\goth o}(N)$
indices are raised or lowered by the flat Euclidean metric. The Lorentz and
scale connections are

$\hat{\Omega}^{\alpha}{}_{\beta}=\Omega^{\alpha}{}_{\beta}+\frac{1}{2}\delta^{\alpha}_{\beta}\Omega_{0};\qquad\hat{\Omega}_{\alpha}{}^{\beta}=\Omega_{\alpha}{}^{\beta}+\frac{1}{2}\delta^{\beta}_{\alpha}\Omega_{0}\
,$ (4.7)

with $\Omega^{\alpha}{}_{\beta}=\Omega_{\beta}{}^{\alpha}$ as the trace-free
Lorentz connection and $\Omega_{0}$ the scale connection. As usual two-
component spinor indices are raised and lowered with the epsilon tensor, so
$\Omega_{\alpha\beta}$ is symmetric.

It is straightforward to compute the components of the curvature two-form, but
for the moment we shall focus on the scale curvature. It is given by

${\cal R}_{0}=R_{0}+E^{\alpha\beta}F_{\alpha\beta}-E^{\alpha i}F_{\alpha i}\
,$ (4.8)

where $R_{0}=d\Omega_{0}$. The last two terms can be rewritten as

$E^{\alpha\beta}F_{\alpha\beta}-E^{\alpha i}F_{\alpha i}=-E^{a}F_{a}-E^{\alpha
i}F_{\alpha i}:=-E^{A}F_{A}=-E^{B}E^{A}F_{AB}\ ,$ (4.9)

Note that $F_{AB}$ is not necessarily graded-antisymmetric, although it is
when contracted with two sets of basis forms. (We have suppressed the wedge
symbol in the two-forms above). Here we have identified $E^{\alpha i}$ as the
standard odd basis forms of superspace and set
$E^{\alpha\beta}=-E^{a}(\gamma_{a})^{\alpha\beta}$, where $E^{a}$ are the
standard even basis forms. (Note that this involves a rescaling since there
would normally be a factor of $\frac{1}{2}$ when going from bi-spinors to
vectors.) We therefore have

$({\cal R}_{0})_{AB}=(R_{0})_{AB}-2F_{[AB]}\ .$ (4.10)

As discussed briefly above, we can use the freedom to adjust a connection by a
tensorial addition to set $({\cal R}_{0})_{AB}=0$, after which
$(R_{0})_{AB}=2F_{[AB]}$. As we shall see shortly below, superconformal gauge
transformations (i.e. conformal boosts and S-supersymmetry) can be used to set
$\Omega_{0}=0$, after which $F_{AB}$ will become graded symmetric. We can then
identify $F_{AB}$ as the super-Schouten tensor.

We shall now exhibit a finite superconformal transformation which will allow
us to set $\Omega_{0}=0$ and to identify residual superconformal
transformations in terms of scale transformations, or what one could call
(finite) super Weyl transformations in this context. It is given by

$g=\left(\begin{array}[]{c c |c}S^{-\frac{1}{2}}&0&0\\\
C_{0}&S^{\frac{1}{2}}&\Gamma\\\ \hline\cr
S^{-\frac{1}{2}}\Gamma^{t}&0&1\end{array}\right)\ .$ (4.11)

where

$C_{0}=-iS^{-\frac{1}{2}}(C+\frac{i}{2}\Gamma\Gamma^{t})\ .$ (4.12)

The parameters $(S,C,\Gamma)$ are those for scale transformations, conformal
boosts and special supersymmetry respectively. $C$ is symmetric on its spinor
indices (i.e. it is a vector) and real, while $\Gamma$ is taken to be
imaginary. It is straightforward to check that this is indeed an element of
the superconformal group; one can find the effect of such a transformation on
the connection by using the standard formula

${\cal A}\rightarrow dg^{-1}g+g^{-1}{\cal A}g\ .$ (4.13)

For the moment we shall focus on the transformation of the scale connection
$\Omega_{0}$. It is given by

$\displaystyle\Omega_{0}\rightarrow$ $\displaystyle\Omega_{0}$
$\displaystyle-S^{-1}dS-E^{\alpha\beta}C_{\alpha\beta}+E^{\alpha
i}\Gamma_{\alpha i}$ $\displaystyle=$ $\displaystyle\Omega_{0}$
$\displaystyle-Y+E^{a}C_{a}+E^{\alpha i}\Gamma_{\alpha i}\ .$ (4.14)

where the one-form $Y:=S^{-1}dS$. We learn two things from this equation:
first, the parameters $C,\Gamma$ can be used to set the even and odd
components of the one-form $\Omega_{0}$ to zero, and second, we can determine
the residual symmetry transformations in terms of $Y$. In other words, having
set $\Omega_{0}=0$ we have

$C_{a}=Y_{a}=S^{-1}D_{a}S;\qquad\Gamma_{\alpha i}=\Upsilon_{\alpha
i}=S^{-1}D_{\alpha i}S$ (4.15)

where the derivatives are now the standard superspace covariant derivatives,
and where we have denoted the odd component of $Y$ by $\Upsilon$ for later
use. The finite transformations of the basis forms are:

$\displaystyle E^{a}$ $\displaystyle\rightarrow SE^{a}$ (4.16) $\displaystyle
E^{\alpha i}$ $\displaystyle\rightarrow S^{\frac{1}{2}}(E^{\alpha
i}-iE^{a}(\gamma_{a})^{\alpha\beta}\Upsilon_{\beta}{}^{i})$

The finite changes of the Lorentz and $\mbox{\goth o}(N)$ connections are:

$\displaystyle\Delta\Omega^{\alpha\beta}$
$\displaystyle=E^{a}(\gamma_{a})^{(\alpha|\gamma|}(Y_{\gamma}{}^{\beta)}+\frac{i}{2}(\Upsilon^{2})_{\gamma}{}^{\beta)})+E^{(\alpha
i}\Upsilon_{i}{}^{\beta)}$ (4.17) $\displaystyle\Delta\Omega^{ij}$
$\displaystyle=2E^{\alpha[i}\Upsilon_{\alpha}{}^{j]}+iE^{a}\Upsilon_{\alpha}{}^{i}(\gamma_{a})^{\alpha\beta}\Upsilon_{\beta}{}^{j}$

The transformations of the superconformal connections are

$\displaystyle F_{\alpha}{}^{j}$ $\displaystyle\mapsto
S^{-\frac{1}{2}}\left(F_{\alpha}{}^{j}-D\Gamma_{\alpha}{}^{j}+(iC_{\alpha\beta}+\Gamma_{(\alpha}{}^{k}\Gamma_{\beta)k})(E^{\beta
j}+E^{\alpha\gamma}\Gamma_{\gamma}{}^{j})-\Gamma_{\alpha}{}^{k}E^{\beta}{}_{k}{}\Gamma_{\beta}{}^{j}\right)$
(4.18)

and

$\displaystyle F_{\alpha\beta}$ $\displaystyle\mapsto
S^{-1}\left(F_{\alpha\beta}+iDC_{\alpha\beta}-D\Gamma_{(\alpha}{}^{k}\Gamma_{\beta)k}+2F_{(\alpha}{}^{k}\Gamma_{k\beta)}\right.$
(4.19)
$\displaystyle\left.+C_{\alpha\gamma}E^{\gamma\delta}C_{\delta\beta}+2iC_{(\alpha\gamma}E^{\gamma
k}\Gamma_{k\beta)}\right.$
$\displaystyle\left.+iC_{(\alpha\gamma}E^{\gamma\delta}(\Gamma^{2})_{\delta\beta)}+\Gamma^{2}_{(\alpha\gamma}E^{\gamma
k}\Gamma_{\beta)k}+{\textstyle{\frac{1}{4}}}\Gamma^{2}_{\alpha\gamma}E^{\gamma\delta}\Gamma^{2}_{\delta\beta}\right)\
.$

Here $\Gamma^{2}_{\alpha\beta}:=\Gamma_{\alpha}{}^{k}\Gamma_{\beta k}$, is
antisymmetric on $\alpha\,\beta$, while the symmetrisations are on
$\alpha,\beta$ only.

The formalism given above applies quite generally regardless of whether any
constraints have been imposed or not. The basic constraint we shall choose for
$D=3$ (and in fact in all cases) is ${\cal T}_{0}=0$, which is clearly
superconformally invariant, from (3.16). We can now use conventional
constraints, including some for the conformal and superconformal potentials,
as well as the Bianchi identities, to show that we can always choose ${\cal
T}_{1}={\cal R}=0$, remembering that we always choose the scale curvature to
vanish. Thus in $D=3$ the only covariant field strengths that are non-zero are
the internal curvature ${\cal R}_{I}$ and the superconformal curvature ${\cal
S}_{A}=({\cal S}_{a},{\cal S}_{\alpha j})$. From the Bianchi identity (3.6) we
can then see that

$\displaystyle{\cal R}_{\alpha i\beta j,kl}$
$\displaystyle=\varepsilon_{\alpha\beta}W_{ijkl}\ ,$ (4.20)
$\displaystyle{\cal R}_{a\beta j,kl}$
$\displaystyle=(\gamma_{a}\lambda)_{\beta jkl}\ ,$ $\displaystyle{\cal
R}_{ab,kl}$ $\displaystyle=F_{abkl}\ ,$

for $N\geq 2$, where each field is totally antisymmetric on its $SO(N)$
indices. In fact, for $N\geq 2$, the leading component in the Weyl multiplet,
i.e. the conformal supergravity field strength multiplet, is the leading non-
zero component in ${\cal R}_{I}$.

For $N=1$ ${\cal R}_{I}$ is identically zero, and only ${\cal S}_{A}$
survives. From the Bianchi identities it is then easy to show that the only
non-zero components are ${\cal S}_{ab,\gamma}$ and ${\cal S}_{ab,c}$. The
former is equivalent to a gamma-traceless vector-spinor, while the latter is
equivalent to a symmetric traceless tensor. These are the Cottino and Cotton
tensors respectively. It is then straightforward to see that the component
fields in the Weyl multiplet can be arranged diagrammatically as
follows222This multiplet was discussed in the context of the supermembrane in
[38]. :

${{[4,0]}}$${{[5,1]}}$${{[3,1]}}$${{[6,2]}}$${{[4,0]}}$${{[2,2]}}$${{[7,3]}}$${{[1,3]}}$
${{[0,4]}}$

Here, each $[p,q]$ entry denotes a field with $p$ antisymmetrised internal
indices and $q$ symmetrised spinor indices. For $N\leq 4$ clearly only the
right part of the diagram survives, with the last two components being the
Cottino and Cotton tensors respectively. For $N=6$ we note that there is an
extra $U(1)$ gauge field not included in the superconformal group. This field
plays an important role in the BLG formalism for multiple membranes [39, 40],
and was discussed in the superspace context in [12]. For $N=6$, therefore, we
have an additional closed two-form field strength $G$ with components

$\displaystyle G_{\alpha i\beta j}$
$\displaystyle=\varepsilon_{\alpha\beta}W_{ij}$ (4.21) $\displaystyle
G_{a\beta j}$ $\displaystyle=(\gamma_{a}\lambda)_{\beta j}\ ,$

as well as $G_{ab}$, where $W_{ij}$ is the $SO(6)$ dual of $W_{ijkl}$ and
$\lambda_{i}$ is the $SO(6)$ dual of the 5-index fermion on the left of the
diagram above. For $N=8$ an extra constraint is required in order to avoid
having two gravitons; this is achieved by imposing a self-duality constraint
on $W_{ijkl}$, and this in turn implies that the field content of the $N=7$
and $N=8$ Weyl multiplets are the same, so that the left-hand diagonal line
can be terminated at $N=6$.

To conclude this section we translate the above results into conventional
superspace. The main consequence is that the components of the superconformal
potentials now appear explicitly in the torsions and curvatures. These
potentials are graded symmetric and make up the components of the super-
Schouten tensor as we remarked earlier. Making use of equation (3) which
relates the conformally covariant tensors on the left to the standard
superspace ones on the right we find, for the even torsion two-form

$T^{a}=-\frac{i}{2}E^{\alpha i}(\gamma^{a})_{\alpha\beta}E^{\beta}_{i}\ ,$
(4.22)

or, in components,

$\displaystyle T_{\alpha i\beta j}{}^{c}$
$\displaystyle=-i\delta_{ij}(\gamma^{a})_{\alpha\beta}\ ,$ $\displaystyle
T_{\alpha ib}{}^{c}$ $\displaystyle=T_{ab}{}^{c}=0\ ,$ (4.23)

which we recognise as the usual expressions [10]. For the odd torsion we find

$T^{\alpha i}=DE^{\alpha i}=(\gamma_{a})^{\alpha\gamma}E^{a}(E^{\beta
j}F_{\beta j,\gamma}{}^{i}+E^{b}F_{b\gamma}{}^{i})\ .$ (4.24)

This implies for the components,

$\displaystyle T_{\alpha i,\beta j}{}^{\gamma k}$ $\displaystyle=0\ ,$
$\displaystyle T_{a\beta j}{}^{\gamma k}$
$\displaystyle=-(\gamma_{a})^{\gamma\delta}F_{\beta j,\delta}{}^{k}\ ,$
$\displaystyle T_{ab}{}^{\gamma k}$
$\displaystyle=-2\gamma_{[a}^{\gamma\delta}F_{b]\delta}{}^{k},$ (4.25)

where we have written the right-hand sides in terms of the super Schouten
tensor. For the dimension-one component, since $F_{\alpha i,\beta j}$ is
antisymmetric under the interchange of pairs of indices, we have $F_{\alpha
i,\beta j}=\varepsilon_{\alpha\beta}K_{ij}+(\gamma^{a})_{\alpha\beta}L_{aij}$,
where $K$ is symmetric on the internal indices and $L$ antisymmetric, so that

$T_{a\beta j}{}^{\gamma
k}=(\gamma_{a})_{\beta}{}^{\gamma}K_{j}{}^{k}+(\gamma^{b}\gamma_{a})_{\beta}{}^{\gamma}L_{bj}{}^{k}\
.$ (4.26)

This differs slightly from the expression for the dimension one torsion given
in [10, 12], but can be brought into agreement by a further redefinition of
the dimension-one $\mbox{\goth o}(N)$ connection. (On the other hand (4.26)
does agree with the form given in [11].) The dimension three-halves torsion is
the gravitino field strength, which in three dimensions can be dualised to a
vector-spinor $\Psi_{a\gamma k}$. It is given in terms of the super Schouten
tensor by

$T_{ab}{}^{\gamma k}=-2(\gamma_{[a}){}^{\gamma\delta}F_{b]\delta}{}^{k}\ ,$
(4.27)

where

$\Psi_{a}{}^{\alpha i}:=\frac{1}{2}\varepsilon_{a}{}^{bc}\Psi_{bc}{}^{\alpha
i}=\varepsilon_{a}{}^{bc}(\gamma_{b})^{\alpha\beta}F_{c\beta}{}^{i}\ .$ (4.28)

The components of the standard superspace curvature tensors can also be easily
computed from equation (3.4), using the fact that the covariant Lorentz and
scale curvatures are zero.

##  5 $D=4$

The superconformal groups for $D=4$ $N$-extended supersymmetry are
$SU(2,2|N)$, for $N=1,2,3$ and $PSU(2,2|4)$ for $N=4$. Elements of this group
are matrices $g$ with unit superdeterminant which obey

$g^{*}Jg=J$ (5.1)

where

$J=\left(\begin{array}[]{c c |c}0&1_{2}&0\\\ 1_{2}&0&0\\\ \hline\cr
0&0&1_{N}\end{array}\right)\ ,$ (5.2)

and where the numerical subscripts denote the dimenions of the unit matrices.
For an element $h$ of the corresponding Lie superalgebra $\mbox{\goth
s}\mbox{\goth u}(2,2|N)$ we have

$h^{*}J=-Jh$ (5.3)

Such an element can be written

$h=\left(\begin{array}[]{c c |c}-\hat{a}&-ib&-i\varepsilon\\\
-ic&\hat{a}^{*}&\varphi\\\
\hline\cr-\varphi^{*}&-i\varepsilon^{*}&e\end{array}\right)\ ,$ (5.4)

where $b$ and $c$ are hermitian, $e$ is anti-hermitian, and the factors of $i$
are put in for convenience (we follow the conventions of [42] here). The
supertrace condition then implies that

$-a_{0}+\bar{a}_{0}=e_{k}{}^{k}\ ,$ (5.5)

where

$\hat{a}^{\alpha}{}_{\beta}=a^{\alpha}{}_{\beta}+\frac{1}{2}\delta^{\alpha}_{\beta}a_{0}\
.$ (5.6)

The connection in $D=4$ can be written

$\cal{A}=\left(\begin{array}[]{c c
|c}-\hat{\Omega}^{\alpha}{}_{\beta}&-iE^{\alpha\dot{\beta}}&-iE^{\alpha j}\\\
-iF_{\dot{\alpha}\beta}&\hat{\bar{\Omega}}_{\dot{\alpha}}{}^{\dot{\beta}}&\bar{F}_{\dot{\alpha}}{}^{j}\\\
\hline\cr-F^{t}_{i\beta}&-iE^{*}_{i}{}^{\dot{\beta}}&{\Omega}_{i}{}^{j}\end{array}\right)\
.$ (5.7)

and the curvature is

$\cal{F}=\left(\begin{array}[]{c c |c}-\hat{\cal R}^{\alpha}{}_{\beta}&-i{\cal
T}^{\alpha\dot{\beta}}&-i{\cal T}^{\alpha j}\\\ -i{\cal
S}_{\dot{\alpha}\beta}&\hat{\bar{\cal
R}}_{\dot{\alpha}}{}^{\dot{\beta}}&\bar{{\cal S}}_{\dot{\alpha}}{}^{j}\\\
\hline\cr-({\cal S}^{t})_{i\beta}&-i({{\cal T}^{*}})_{i}{}^{\dot{\beta}}&{\cal
R}_{i}{}^{j}\end{array}\right)\ ,$ (5.8)

where the expressions for the various components are given by (3.4) with
appropriate factors of $i$. We set

$\hat{\Omega}^{\alpha}{}_{\beta}=\Omega^{\alpha}{}_{\beta}+\frac{1}{2}\delta^{\alpha}_{\beta}(\Omega_{0}+i\Omega_{1})\
,$ (5.9)

where $\Omega_{0}$ and $\Omega_{1}$ are the scale and $U(1)$ connections
respectively and where $\Omega^{\alpha}{}_{\beta}$ is trace-free.

A group element depending on scale and superconformal transformations is
easily constructed in a similar fashion to $D=3$. It is given by

$g=\left(\begin{array}[]{c c |c}S^{-\frac{1}{2}}&0&0\\\
C_{0}&S^{\frac{1}{2}}&\bar{\Gamma}\\\
\hline\cr-S^{-\frac{1}{2}}\Gamma^{t}&0&1\end{array}\right)\ .$ (5.10)

where

$C_{0}=iS^{-\frac{1}{2}}(C+\frac{i}{2}\bar{\Gamma}\Gamma^{t})$ (5.11)

with $C$ being hermitian. The inverse is given by

$g^{-1}=\left(\begin{array}[]{c c |c}S^{\frac{1}{2}}&0&0\\\
\hat{C}_{0}&S^{-\frac{1}{2}}&-S^{-\frac{1}{2}}\bar{\Gamma}\\\
\hline\cr\Gamma^{t}&0&1\end{array}\right)\ .$ (5.12)

where

$\hat{C}_{0}=-iS^{-\frac{1}{2}}(C-\frac{i}{2}\bar{\Gamma}\Gamma^{t})\ .$
(5.13)

As in $D=3$ we can use the antisymmetric part of the superconformal connection
together with a superconformal transformation to set ${\cal R}_{0}=R_{0}=0$
and leave residual superconformal transformations in terms of the scale
parameter $S$. The transformation of the scale connection $\Omega_{0}$ under
$g$ is given by

$\Delta\Omega_{0}=-Y+E^{\alpha\dot{\alpha}}C_{\dot{\alpha}\alpha}-\frac{i}{2}E^{\alpha
k}\Gamma_{\alpha
k}-\frac{i}{2}\bar{E}^{\dot{\alpha}}_{k}\bar{\Gamma}_{\dot{\alpha}}{}^{k}\ .$
(5.14)

The right-hand side can be rewritten as $-Y+C$, where
$C=E^{A}C_{A}=E^{a}C_{a}+E^{\alpha i}\Gamma_{\alpha
i}-\bar{E}^{\dot{\alpha}}_{i}\Gamma_{\dot{\alpha}}{}^{i}$ (after rescaling
$C_{\alpha\dot{\alpha}}\rightarrow\frac{1}{2}C_{\alpha\dot{\alpha}}$ and
$\Gamma_{\alpha i}\rightarrow 2i\Gamma_{\alpha i}$). We can then use $C$ to
set $\Omega_{0}=0$, leaving residual superconformal transformations with
$Y=S^{-1}dS=C$.

The scale curvature ${\cal R}_{0}$ can be written as

${\cal R}_{0}=R_{0}-E^{A}f_{A}\ ,$ (5.15)

where $R_{0}=d\Omega_{0}$ and where

$f_{a}:=2F_{a};\ \ \ f_{\alpha i}:=-\frac{i}{2}F_{\alpha i};\ \ \
\bar{f}_{\dot{\alpha}}{}^{i}:=\frac{i}{2}\bar{F}_{\dot{\alpha}}{}^{i}\ .$
(5.16)

We therefore have

$({\cal R}_{0})_{AB}=(R_{0})_{AB}-2f_{[AB]}\ .$ (5.17)

Finally, we can use $C_{A}$ to set $(\Omega_{0})_{A}=0$ and the graded
antisymmetric part of the superconformal connection $f_{[AB]}$ to set ${\cal
R}_{0}=0$. After this, we have residual superconformal transformations
determined by $S$, as above, while the remaining part of $f_{AB}$ is graded
symmetric and can be identified as the super-Schouten tensor for $D=4$.

The super-Weyl transformations of the basis forms and the connections are
given by:

$\displaystyle E^{a}$ $\displaystyle\mapsto SE^{a}\ ,$ $\displaystyle
E^{\alpha i}$ $\displaystyle\mapsto S^{\frac{1}{2}}E^{\alpha i}\ ,$
$\displaystyle\Omega^{\alpha}{}_{\beta}$
$\displaystyle\mapsto\Omega^{\alpha}{}_{\beta}-\frac{1}{2}E^{\alpha\dot{\gamma}}(C_{\dot{\gamma}\beta}+4i\bar{\Upsilon}_{\dot{\gamma}}{}^{k}\Upsilon_{\beta
k})+2E^{\alpha k}\Upsilon_{\beta k}\ ,$ $\displaystyle\Omega_{i}{}^{j}$
$\displaystyle\mapsto\Omega_{i}{}^{j}-2(E^{\gamma j}\Upsilon_{\gamma
i}+\bar{E}^{\dot{\gamma}}_{i}\bar{\Upsilon}_{\dot{\gamma}}^{j}+2i\Upsilon_{\gamma
i}E^{\gamma\dot{\gamma}}\bar{\Upsilon}_{\dot{\gamma}}{}^{j})\ ,$

where it is understood that the trace over $\alpha$ and $\beta$ is to be
projected out in the third line. The components of the super-Schouten tensor
at dimension one are $f_{\alpha i,\beta j}$, $f_{\alpha i,\dot{\beta}}^{\ \ \
j}$ plus complex conjugates. Graded symmetry then implies that

$f_{\alpha i,\beta j}=\varepsilon_{\alpha\beta}f_{(ij)}+f_{(\alpha\beta)[ij]}\
,$ (5.19)

while $f_{\alpha i,\dot{\beta}}^{\ \ \ j}$ is skew-hermitian. At dimension
three-halves there is just a complex vector-spinor $f_{a\beta j}$ which can be
identified with that part of the gravitino field-strength tensor which is not
part of the Weyl supermultiplet.

The basic constraint for $D=4$ is ${\cal T}^{a}=0$, as for the other cases.
Using the Bianchi identities and choices for the connections, including the
superconformal ones, one finds that the non-zero components of the covariant
torsion ${\cal T}^{\gamma k}$ are

$\displaystyle{\cal T}_{\dot{\alpha}\dot{\beta}}^{i\,j\,\gamma k}$
$\displaystyle=\varepsilon_{\dot{\alpha}\dot{\beta}}\chi^{\gamma ijk}\ ,$
$\displaystyle{\cal T}_{\alpha\dot{\alpha},\beta,\dot{\gamma}}^{\ \ \ \,j\,k}$
$\displaystyle=i\bar{M}_{\alpha\beta}^{jk}\ ,$ $\displaystyle{\cal
T}_{\alpha\dot{\alpha},\beta,\gamma k}^{\ \ \ j}$
$\displaystyle=-\frac{i}{12}(N-3)\delta_{k}^{j}\bar{\chi}_{\dot{\alpha}}^{lmn}\chi_{\beta
lmn}\ ,$ $\displaystyle{\cal
T}_{\alpha\dot{\alpha},\beta\dot{\beta},\dot{\gamma}}^{\ \ \ k}$
$\displaystyle=\varepsilon_{\alpha\beta}\bar{\Psi}_{\dot{\alpha}\dot{\beta}\dot{\gamma}}^{k}\
.$ (5.20)

Here, vector indices have been converted to pairs of spinor indices using the
sigma-matrices in the usual way:

$v_{a}\rightarrow
v_{\alpha\dot{\alpha}}=(\sigma^{a})_{\alpha\dot{\alpha}}v_{a}\ .$ (5.21)

In (5) multiple internal indices are antisymmetrised, while the spinor indices
on $M$ and $\bar{\Psi}$ are symmetrised. The leading components of the Weyl
multiplets are $\Psi,M$ and $\chi$ for $N=1,2,3$ respectively. For $N=4$ there
is also a complex dimension-zero scalar field that does not appear directly in
the components of the torsion tensor. Instead it parametrises the coset space
$SU(1,1)/U(1)$. The $U(1)$ factor here is part of the superspace internal
connection, but for $N=4$ not part of the gauge group of the superconformal
group which is only $SU(4)$. An additional constraint is required at dimension
one to enforce this. It is

$D_{\alpha i}\chi^{\alpha\,ijk}=0\ .$ (5.22)

The components of the covariant curvature can easily be found; for example, at
dimension one, they are [9]:

$\displaystyle{\cal R}_{\alpha i\beta j,\gamma\delta}$ $\displaystyle=0\ ,$
$\displaystyle{\cal R}_{\alpha i\beta j,\dot{\gamma}\dot{\delta}}$
$\displaystyle=-\varepsilon_{\alpha\beta}\bar{M}_{\dot{\gamma}\dot{\delta}ij}\
,$ $\displaystyle{\cal R}_{\alpha i\beta,k}{}^{jl}$
$\displaystyle=-\varepsilon_{\alpha\beta}\bar{B}_{ijk}{}^{l}\ ,$
$\displaystyle{\cal R}_{\alpha i\dot{\beta},k}^{\hskip 5.60002ptj}{}^{l}$
$\displaystyle=-\bar{\chi}_{\dot{\beta}ikm}\chi_{\alpha}^{jlm}\ ,$ (5.23)

where

$B^{ijk}{}_{l}=\frac{1}{2}D_{\alpha l}\chi^{\alpha ijk}\ .$ (5.24)

Further details of the torsions and curvatures, including the Schouten terms,
can be found in [8, 9].

##  6 $D=6$

The $(1,0)$ theory in $D=6$ has been discussed in $SU(2)$ superspace in [23]
and in conformal superspace [18], [19], [20]. The theory was also discussed
earlier in harmonic superspace in [22] and, some years ago, in projective
superspace [23]. In components, the $(2,0)$ supergravity dates back twenty
years to [43]. Below we summarise the local twistor formulation of $(1,0)$ and
$(2,0)$ $6D$ conformal supergravity recently presented in [24].

The complex conjugate of a four-component $D=6$ spinor $u^{\alpha}$ is denoted
$\bar{u}^{\dot{\alpha}}$ but this representation is equivalent to the undotted
one as there is a matrix $B_{\alpha}{}^{\dot{\alpha}}$ relating the two,
$\bar{u}^{\dot{\alpha}}=\bar{u}^{\alpha}B_{\alpha}{}^{\dot{\alpha}}$. $B$ is
unitary, $B^{*}B=1$, and satisfies $B\bar{B}=-1$.333We use the six-dimensional
conventions of [44]. Similar remarks hold for the inequivalent spinor
representation denoted by a lower index, $v_{\alpha}$ say. So a twistor $z$
consists of a pair of 4-component spinors and can be written

$z=\left(\begin{array}[]{c}u^{\alpha}\\\ v_{\alpha}\end{array}\right)\ .$
(6.1)

A supertwistor in $D=6$ can therefore be written in the form

$Z=\left(\begin{array}[]{c}u^{\alpha}\\\ v_{\alpha}\\\ \lambda_{i}\\\
\end{array}\right)\ .$ (6.2)

where $i=1,\ldots 2N$ for $(N,0)$ supersymmetry, $N=1,2$. Here $(u,v)$ are
commuting objects while $\lambda$ is odd. The superconformal group is
$OSp(8|N)$ in complex superspace and preserves the orthosymplectic metric $K$,
so for an element $g$ of the group we have

$gKg^{st}=K=\left(\begin{array}[]{c c |c}0&1_{4}&0\\\ 1_{4}&0&0\\\ \hline\cr
0&0&J_{0}\end{array}\right)\ .$ (6.3)

where st denotes the supertranspose, which is the same as the ordinary
transpose except for an additional minus sign for each element in the bottom
left (odd) sector. The $2N\times 2N$ matrix $J_{0}$ is the $Sp(N)$ symplectic
invariant. In real spacetime we need to impose the reality constraint

$gRg^{*}=R=\left(\begin{array}[]{c c |c}0&B^{-1}&0\\\ B&0&0\\\ \hline\cr
0&0&1_{2N}\end{array}\right)\ .$ (6.4)

An element of the Lie superalgebra, $h$, has the form

$h=\left(\begin{array}[]{c c
|c}a^{\alpha}{}_{\beta}&b^{\alpha\beta}&\varepsilon^{\alpha j}\\\
c_{\alpha\beta}&d_{\alpha}{}^{\beta}&\varphi_{\alpha}{}^{j}\\\
\hline\cr\lambda_{i\beta}&\rho_{i}{}^{\beta}&e_{i}{}^{j}\end{array}\right)\ .$
(6.5)

The orthosymplectic constraint implies that $b$ and $c$ are skew-symmetric and
$d=-a^{t}$, as before, while

$eJ_{0}=-J_{0}e^{t}\ .$ (6.6)

In indices, setting $(J_{0})_{ij}=\eta_{ij}$, this implies444Indices are
raised and lowered according to the rule: $X^{i}=\eta^{ij}X_{j}\Leftrightarrow
X_{i}=X^{j}\eta_{ji}$ with $\eta^{ik}\eta_{jk}=\delta_{j}{}^{i}$.

$e_{ij}:=e_{i}{}^{k}\eta_{kj}=e_{ji}\ .$ (6.7)

For the odd components we have

$\rho=J_{0}\varepsilon^{t}\qquad\lambda=J_{0}\varphi^{t}\ .$ (6.8)

or, in indices,

$\displaystyle\rho_{i}{}^{\beta}$ $\displaystyle=\eta_{ij}\varepsilon^{\beta
j}\Rightarrow\rho_{i}{}^{\beta}=-\varepsilon^{\beta}{}_{i}\ ,$
$\displaystyle\lambda_{i\beta}$
$\displaystyle=\eta_{ij}\varphi_{\beta}{}^{j}\Rightarrow\lambda_{i\beta}=-\varphi_{\beta
i}\ .$ (6.9)

Next we need to impose reality in order to move to real superspace. This is
done with equation (3.3) but this time with $R$ extended by the unit matrix in
the odd-odd sector, as in (6.4). The result of imposing $gRg^{*}=g$, at the
Lie algebra level is that $a,b,c$ and $d$ obey the same conditions as in the
bosonic case while $e$ satisfies

$e=-e^{*}\ .$ (6.10)

For the independent odd components of $h$ we have:

$\displaystyle\bar{\varepsilon}^{\dot{\alpha}}_{i}$
$\displaystyle=-\eta_{ij}\varepsilon^{\beta j}B_{\beta}{}^{\dot{\alpha}}\ ,$
$\displaystyle\bar{\varphi}_{\dot{\alpha}}^{i}$
$\displaystyle=(B^{-1})_{\dot{\alpha}}{}^{\beta}\eta^{ij}\varphi_{\beta j}\ .$
(6.11)

These constraints simply mean that $\varepsilon^{\alpha i}$ and
$\varphi_{\alpha i}$ are symplectic Majorana-Weyl spinors as one would expect.
They are respectively the parameters for $Q$ and $S$ supersymmetry
transformations.

The connection ${\cal A}$ is

${\cal A}=\left(\begin{array}[]{c c
|c}\hat{\Omega}^{\alpha}{}_{\beta}&iE^{\alpha\beta}&E^{\alpha j}\\\
iF_{\alpha\beta}&\hat{\Omega}_{\alpha}{}^{\beta}&F_{\alpha}{}^{j}\\\
\hline\cr-F_{i\beta}&-E_{i}{}^{\beta}&\Omega_{i}{}^{j}\end{array}\right)\ ,$
(6.12)

where $E^{A}=(E^{a},E^{\alpha i})$, with
$E^{a}=\frac{1}{2}(\gamma^{a})_{\alpha\beta}E^{\alpha\beta}$, will be
identified with the even and odd super-vielbein one-forms of the underlying
superspace, $F_{A}=(F_{a},F_{\alpha i})$ is the connection for superconformal
transformations, i.e. $S$-supersymmetry and standard conformal
transformations, $\hat{\Omega}^{\alpha}{}_{\beta}$
($\hat{\Omega}_{\alpha}{}^{\beta}$) is the Lorentz plus scale connection and
$\Omega_{i}{}^{j}$ the internal $\mbox{\goth s}\mbox{\goth p}(N)$ connection.
On the bottom line, $F_{i\beta}$ and $E_{i}{}^{\beta}$ are transposes of
$F_{\alpha}{}^{j}$ and $E^{\alpha j}$ with the internal index lowered by
$(J_{0})_{ij}=\eta_{ij}$. The curvature two-form, ${\cal F}=d{\cal A}+{\cal
A}^{2}$, has components given in matrix form by:

$\cal{F}=\left(\begin{array}[]{c c |c}\hat{\cal R}^{\alpha}{}_{\beta}&i{\cal
T}^{\alpha\beta}&{\cal T}^{\alpha j}\\\ i{\cal S}_{\alpha\beta}&\hat{{\cal
R}}_{\alpha}{}^{\beta}&{\cal S}_{\alpha}{}^{j}\\\ \hline\cr-{\cal
S}_{i\beta}&-{{\cal T}}_{i}{}^{\beta}&{\cal R}_{i}{}^{j}\end{array}\right)\ ,$
(6.13)

where, from (3.4), with $\alpha^{\prime}\mapsto\alpha$, and with appropriate
factors of $i$,

$\displaystyle{\cal T}^{\alpha\beta}$
$\displaystyle=\hat{D}E^{\alpha\beta}+iE^{\alpha k}E_{k}{}^{\beta}\ ,$
$\displaystyle{\cal T}^{\alpha j}$ $\displaystyle=\hat{D}E^{\alpha
j}+iE^{\alpha\gamma}F_{\gamma}{}^{j}\ ,$ $\displaystyle\hat{{\cal
R}}^{\alpha}{}_{\beta}$
$\displaystyle=\hat{R}^{\alpha}{}_{\beta}-E^{\alpha\gamma}F_{\gamma\beta}-E^{\alpha
k}F_{k\beta}\ ,$ $\displaystyle{\cal R}_{i}{}^{j}$
$\displaystyle=R_{i}{}^{j}-F_{i\gamma}E^{\gamma
j}-E_{i}^{\gamma}F_{\gamma}{}^{j}\ ,$ $\displaystyle{\cal S}_{\alpha\beta}$
$\displaystyle=\hat{D}F_{\alpha\beta}+iF_{\alpha}{}^{k}F_{k\beta}\ ,$
$\displaystyle{\cal S}_{\alpha}{}^{j}$
$\displaystyle=\hat{D}F_{\alpha}{}^{j}+iF_{\alpha\gamma}E^{\gamma j}\ .$
(6.14)

Here, $\hat{D}$ is the superspace covariant exterior derivative with respect
to scale, Lorentz and internal symmetries, while the leading terms on the
right, for the top four lines, are the standard superspace and torsion and
curvature tensors for the corresponding connections (extended by the scale
connection).

The detailed form of the Bianchi identity ${\cal D}{\cal F}=0$ is given,
mutatis mutandis, by (3.6) to (3.8). We shall now repeat the steps carried out
in the non-supersymmetric case to reduce the conformal and superconformal
boost parameters to derivatives of the scale parameter. We introduce a group
element $g(S,C,\Gamma)$ where $S$ is a scale parameter and
$\Gamma_{\alpha}{}^{i}$ is an $S$-supersymmetry parameter. It is given by

$g=\left(\begin{array}[]{c c |c}S^{-\frac{1}{2}}&0&0\\\
iS^{-\frac{1}{2}}\tilde{C}&S^{\frac{1}{2}}&\Gamma\\\ \hline\cr
S^{-\frac{1}{2}}J_{0}\Gamma^{t}&0&1\end{array}\right)\ ,$ (6.15)

where the index structure is as above, in (6.12) for example, where $J_{0}$ is
the $Sp(N)$ invariant discussed previously, and where

$\tilde{C}+\tilde{C}^{t}+i\Gamma J_{0}\Gamma^{t}=0\ .$ (6.16)

If we write

$\tilde{C}=C-\frac{i}{2}\Gamma J_{0}\Gamma^{t}\ ,$ (6.17)

then, from (6.16), $C$ is antisymmetric since $\Gamma J_{0}\Gamma^{t}$ is
symmetric. Reality implies that

$\displaystyle C$ $\displaystyle=BC^{*}B\ ,\ $ $\displaystyle\bar{\Gamma}$
$\displaystyle=-B^{-1}\Gamma J_{0}\ .$ (6.18)

Note also that the latter equation implies that $\Gamma J_{0}\Gamma=B(\Gamma
J_{0}\Gamma)^{*}B$.

Under such a transformation the components of ${\cal A}$ transform as follows:

$\displaystyle E^{\alpha\beta}$ $\displaystyle\mapsto SE^{\alpha\beta}\ ,$
$\displaystyle E^{\alpha j}$ $\displaystyle\mapsto S^{\frac{1}{2}}(E^{\alpha
j}+iE^{\alpha\beta}\Gamma_{\beta}{}^{j})\ ,$ $\displaystyle F_{\alpha\beta}$
$\displaystyle\mapsto
S^{-1}\left(F_{\alpha\beta}-\hat{D}C_{\alpha\beta}-i(\hat{D}\Gamma_{[\alpha}{}^{k})\Gamma_{\beta]k}+2iF_{[\alpha}{}^{k}\Gamma_{\beta]k}+i\tilde{C}_{\gamma\alpha}E^{\gamma\delta}\tilde{C}_{\delta\beta}-2\tilde{C}_{\gamma[\alpha}E^{\gamma
k}\Gamma_{\beta]k}\right)\ ,$ $\displaystyle F_{\alpha}{}^{j}$
$\displaystyle\mapsto
S^{-\frac{1}{2}}\left(F_{\alpha}{}^{j}-\hat{D}\Gamma_{\alpha}{}^{j}+i(E^{\beta
j}+iE^{\beta\gamma}E_{\gamma}{}^{j})\tilde{C}_{\beta\alpha}-E^{\beta
k}\Gamma_{\alpha k}\Gamma_{\beta}{}^{j}\right)\ ,$
$\displaystyle\hat{\Omega}^{\alpha}{}_{\beta}$
$\displaystyle\mapsto\hat{\Omega}^{\alpha}{}_{\beta}-E^{\alpha\gamma}\tilde{C}_{\gamma\beta}-E^{\alpha
k}\Gamma_{\beta k}+\frac{1}{2}\delta^{\alpha}_{\beta}Y\ ,$
$\displaystyle\Omega_{i}{}^{j}$
$\displaystyle\mapsto\Omega_{i}{}^{j}+\Gamma_{\alpha i}E^{\alpha
j}+\Gamma_{\alpha}{}^{j}E^{\alpha}_{\ i}+i\Gamma_{\alpha
i}E^{\alpha\beta}\Gamma_{\beta}{}^{j}\ .$ (6.19)

The curvature transformations are obtained from those for the potentials by
replacing the latter by the former in the equations above. In addition, for
the superconformal curvatures, the derivative terms in (6.19) must be replaced
by curvature terms as follows:

$\displaystyle\hat{D}C_{\alpha\beta}$ $\displaystyle\mapsto
2\hat{R}_{[\alpha}{}^{\gamma}C_{|\gamma|\beta]}\ ,$
$\displaystyle\hat{D}\Gamma_{\alpha}{}^{j}$ $\displaystyle\mapsto
R_{\alpha}{}^{\beta}\Gamma_{\beta}{}^{j}+\Gamma_{\alpha}{}^{k}R_{k}{}^{j}\ .$
(6.20)

If we take the trace of the third equation in (6.14) we find that

$2{\cal R}_{0}=2R_{0}-E^{A}F_{A}\ ,$ (6.21)

where we have defined the super-vector-valued one-form $F_{A}=(F_{a},F_{\alpha
i})$. By adjusting this potential we can choose ${\cal R}_{0}=0$ so that the
(graded) antisymmetric part of $F_{AB}$ is now proportional to $R_{0AB}$.
Taking the trace of the transformation of $\hat{\Omega}^{\alpha}{}_{\beta}$ we
find

$2\Omega_{0}\mapsto 2\Omega_{0}-E^{a}C_{a}-E^{\alpha i}\Gamma_{\alpha i}+2Y\
,$ (6.22)

so that we can use the parameters $C_{a}$ and $\Gamma_{\alpha i}$ to set
$\Omega_{0}=0$. This leaves residual transformations determined by the scale
parameter $S$,

$C_{A}=2Y_{A}=2S^{-1}D_{A}S\ ,$ (6.23)

where $C_{A}=(C_{a},\Gamma_{\alpha i})$. We shall take the components of $Y$
to be given by $Y_{A}=(Y_{a},\Upsilon_{\alpha i})$ in order to clearly
distinguish the even and odd components where necessary.

The basic constraint that we shall choose is to set the even torsion two-form
to zero,

${\cal T}^{a}=0\ ,$ (6.24)

which is clearly covariant. Using this, conventional constraints corresponding
to connection choices (including superconformal ones) and the Bianchi
identities, one finds that the covariant torsions (i.e. the torsion components
of ${\cal F}$) are given by

$\displaystyle{\cal T}_{\alpha i\,\beta j}{}^{\gamma k}$ $\displaystyle=0\ ,$
$\displaystyle{\cal T}_{a\beta j}{}^{\gamma k}$
$\displaystyle=(\gamma_{a})_{\beta\delta}G^{\gamma\delta}_{j}{}^{k}:=(\gamma^{bc})_{\beta}{}^{\gamma}G_{abcj}{}^{k}\
,$ $\displaystyle{\cal T}_{ab}{}^{\gamma k}$
$\displaystyle=\Psi_{ab}{}^{\gamma k}\ ,$ (6.25)

where $G_{abcjk}$ is anti-self-dual on $abc$ (by its definition), anti-
symmetric on $jk$ and symplectic-traceless on $jk$ for $N=2$, and where
$\Psi_{ab}{}^{\gamma k}$ is the gamma-traceless gravitino field strength. For
the curvature tensor components we find

$\displaystyle{\cal R}_{\alpha i\beta j,kl}$ $\displaystyle=0\ ,$
$\displaystyle{\cal R}_{\alpha i\beta j,cd}$
$\displaystyle=4i(\gamma^{a})_{\alpha\beta}G_{acdij}\ ,$ $\displaystyle{\cal
R}_{a\beta j,cd}$
$\displaystyle=-\frac{i}{2}(\gamma_{a}\Psi_{bc}-\gamma_{c}\Psi_{ab}-\gamma_{b}\Psi_{ca})_{\beta
j}\ ,$ $\displaystyle{\cal R}_{a\beta j,kl}$
$\displaystyle=-8(\gamma_{a}\chi)_{\beta(k,l)j}\ ,$ $\displaystyle{\cal
R}_{ab,cd}$ $\displaystyle=C_{ab,cd}\ ,$ $\displaystyle{\cal R}_{ab,kl}$
$\displaystyle=F_{ab,kl}\ .$ (6.26)

The dimension three-halves field $\chi^{\alpha}_{i,jk}$ is antisymmetric on
$jk$ : it is a doublet for $N=1$ while for $N=2$ it is in the 16 of
$\mbox{\goth s}\mbox{\goth p}(2)$, i.e. it is symplectic-traceless on any pair
of indices. The graviton field-strength $C_{ab,cd}$ has the symmetries of the
Weyl tensor, while $F_{ab,kl}$ is the $\mbox{\goth s}\mbox{\goth p}(N)$ field-
strength tensor. We have thus located all of the components of the conformal
supergravity field strength supermultiplets except for the dimension-two
scalars which are given by

$C_{ij,kl}=D_{\alpha i}\chi^{\alpha}{}_{j,kl}\ .$ (6.27)

for the $(1,0)$ case this reduces to a singlet, while for $(2,0)$ $C_{ij,kl}$
is in the $14$-dimensional representation of $\mbox{\goth s}\mbox{\goth
p}(2)$. Fuller details of this multiplet can be found in [24].

The standard superspace torsion tensors differs from the covariant ones by
components of the Schouten tensor $F_{AB}$.

##  7 $D=5$

The $D=5$ superconformal theory, which exists only for $N=1$, where the
algebra is ${\mathfrak{f}}(4)$ [37], has been described in (conformal)
superspace in [16], [17]. However,it does not fit into the supertwistor
formalism as well as the other cases due to the constraints that one has to
impose on the top left quadrant of ${\cal A}$ (starting from the $D=6,(1,0)$
case). These are

$\displaystyle(\gamma^{a})^{\alpha}{}_{\beta}\hat{\Omega}^{\beta}{}_{\alpha}$
$\displaystyle=0\ ,$ $\displaystyle\eta_{\alpha\beta}E^{\alpha\beta}$
$\displaystyle=0\ ,$ $\displaystyle\eta^{\alpha\beta}F_{\alpha\beta}$
$\displaystyle=0\ ,$
$\displaystyle(\gamma^{a})_{\alpha}{}^{\beta}\hat{\Omega}_{\beta}{}^{\alpha}$
$\displaystyle=0\ ,$ (7.1)

where $\alpha=1,\ldots 4$, $i=1,2$ and where the vector indices are now five-
dimensional. Similar constraints should hold for the curvature ${\cal F}$, but
the fermion bilinears in the ${\cal A}^{2}$ term do not preserve them. Thus
with the same definition of ${\cal F}$ as before, ${\cal F}=d{\cal A}+{\cal
A}^{2}$, the curvatures in the top left quadrant should be defined by

$\displaystyle\hat{\cal R}^{\alpha}{}_{\beta}$ $\displaystyle={\cal
F}^{\alpha}{}_{\beta}-\frac{1}{4}(\gamma^{a})^{\alpha}{}_{\beta}(\gamma_{a})^{\gamma}{}_{\delta}{\cal
F}^{\delta}{}_{\gamma}\ ,$ $\displaystyle{\cal T}^{\alpha\beta}$
$\displaystyle={\cal
F}^{\alpha\beta}-\frac{1}{4}\eta^{\alpha\beta}\eta_{\gamma\delta}{\cal
F}^{\gamma\delta}\ ,$ $\displaystyle{\cal S}_{\alpha\beta}$
$\displaystyle={\cal
F}_{\alpha\beta}-\frac{1}{4}\eta_{\alpha\beta}\eta^{\gamma\delta}{\cal
F}_{\gamma\delta}\ ,$ $\displaystyle\hat{\cal R}_{\alpha}{}^{\beta}$
$\displaystyle={\cal
F}_{\alpha}{}^{\beta}-\frac{1}{4}(\gamma^{a})_{\alpha}{}^{\beta}(\gamma_{a})_{\gamma}{}^{\delta}{\cal
F}_{\delta}{}^{\gamma}\ ,$ (7.2)

where the ${\cal F}$s on the right-hand side are the top left quadrant entries
in ${\cal F}$, and where $\eta_{\alpha\beta}$ is the $4\times 4$ symplectic
matrix, (the charge-conjugation matrix in $D=5$); the odd torsion,
S-supersymmetry curvature and internal curvature are defined as before. The
relations between the covariant curvatures and the standard superspace ones is
the same as in (6.14) with the difference that the unwanted terms have to be
projected out as in the preceding equation. For example, we have

$\displaystyle{\cal T}^{\alpha\beta}$ $\displaystyle=T^{\alpha\beta}-i\langle
E^{\alpha k}E^{\beta}{}_{k}\rangle\ ,$ $\displaystyle\hat{{\cal
R}}^{\alpha}{}_{\beta}$
$\displaystyle=\hat{R}^{\alpha}{}_{\beta}-E^{\alpha\gamma}F_{\gamma\beta}-\langle
E^{\alpha k}F_{k\beta}\rangle\ ,$ (7.3)

where the angle brackets indicate that the symplectic trace terms on the first
line and the $(\gamma^{a})$-trace terms on the second line have been projected
out. Note that the middle term on the right in the second equation is
automatically $(\gamma^{a})$-traceless because $E^{\alpha\gamma}$ and
$F_{\gamma\beta}$ are now five-dimensional vectors, so that multiplying by
$\gamma^{a}$ and taking the trace is identically zero.

We now briefly describe the constraints and the geometry that follows from
them. As in the other cases we set ${\cal T}^{\alpha\beta}=0$ from which the
even torsion two-form $T^{\alpha\beta}$ takes its usual $\gamma$-matrix form,
as can be seen from the first line in (7) on setting the left-hand-side to
zero. The only non-zero torsion components are at dimension one and three-
halves. The former is

${\cal T}_{a\beta j}{}^{\gamma
k}=\delta_{j}{}^{k}\left((\gamma^{bc})_{\beta}{}^{\gamma}G_{abc}+2(\gamma^{b})_{\beta}{}^{\gamma}G_{ab}\right)\
,$ (7.4)

where $G_{abc}$ is the dual of $G_{ab}$,

$G_{abc}=\frac{1}{2}\varepsilon_{abcde}G^{de}\ .$ (7.5)

The field $G_{ab}$ can be identified as the leading component of the $D=5,N=1$
Weyl multiplet. The covariant dimension three-halves torsion is the gamma-
traceless gravitino field strength. Note that (7.4) can be obtained by
dimensional truncation from the $D=6,N=(1,0)$ expression. The remaining
components are a dimension three-halves spinor, $\chi_{\alpha i}$, and, at
dimension two, the Weyl tensor $C_{ab,cd}$, the $\mbox{\goth s}\mbox{\goth
p}(1)$ field strength $F_{ab,kl}$ and a scalar field $C$. The curvature
components are formally the same as the $D=6,(1,0)$ case given in (6.26),
except that

${\cal R}_{\alpha i\beta
j,cd}=4i\varepsilon_{ij}\left(\eta_{\alpha\beta}G_{cd}+(\gamma^{a})_{\alpha\beta}G_{acd}\right)\
,$ (7.6)

while the dimension three-halves spinor $\chi_{\alpha
i,jk}\rightarrow\varepsilon_{i(j}\chi_{\alpha k)}$. Finally, the dimension two
scalar can be defined to be

$C=D_{\alpha i}\chi^{\alpha i}\ .$ (7.7)

The standard superspace torsions and curvatures can be obtained from the
covariant ones in the same way as in $D=6,(1,0)$, using (6.14). They are the
leading terms on the right-hand sides of the first four equations, but with
the symplectic and $\gamma^{a}$ traces projected out. The Schouten terms come
from the equation for the odd torsion in (6.14).

##  8 Minimal approach

We shall now describe the superspace geometry corresponding to these conformal
supergravity multiplets from a minimal perspective. Since the details coincide
with those derived previously we will be brief. We define a superconformal
structure on a supermanifold with $(\rm{even}|\rm{odd})$ dimension
$(D|D^{\prime})$ to be a choice of odd tangent bundle $T_{1}$ (of dimension
$(0|D^{\prime})$ which is maximally non-integrable, so that the even tangent
bundle $T_{0}$ is generated by commutators of sections of $T_{1}$, and such
that the Frobenius tensor, 𝔽, defined below, is invariant under $\mbox{\bbbold
R}\oplus\mbox{\goth s}\mbox{\goth p}\mbox{\goth i}\mbox{\goth
n}(1,D-1)\oplus\mbox{\goth g}$, where g is the internal symmetry algebra for
the case in hand: $\mbox{\goth g}=\mbox{\goth s}\mbox{\goth o}(N)$ for $D=3$,
$\mbox{\goth u}(N)$ for $D=4$, $\mbox{\goth s}\mbox{\goth p}(N)$ for $D=6$,
with $N=1,2$, and for $D=5$, with $N=1$. The components of 𝔽 with respect to
local bases $E_{\alpha i}$ for $T_{1}$ and $E^{a}$ for $T_{0}^{*}$ are given
by

$\mbox{\bbbold F}_{\alpha i\beta j}{}^{c}=\langle[E_{\alpha i},E_{\beta
j}],E^{c}\rangle=-ik_{ij}(\gamma^{c})_{\alpha\beta}\ ,$ (8.1)

where $k_{ij}=\delta_{ij}$ for $D=3$ and $\eta_{ij}$ for $D=5,6$ respectively
and where $\langle,\rangle$ denotes the pairing between vectors and forms. The
ℝ factor denotes an infinitesimal scale transformation, $\delta E^{a}=SE^{a}$,
$\delta E_{\alpha i}=-\frac{1}{2}SE_{\alpha i}$, while the spin and symplectic
algebras act in the natural way on the spacetime and internal indices. For
$D=4$, $T_{1}$ is the sum of two complex conjugate bundles of dimension $2N$,
$T_{1}={\cal T}_{1}\oplus\bar{{\cal T}}_{1}$, and we have

$\displaystyle\mbox{\bbbold F}_{\alpha i\beta j}{}^{c}$
$\displaystyle=\langle[E_{\alpha i},E_{\beta j}],E^{c}\rangle=0,$ (8.2)
$\displaystyle\mbox{\bbbold F}_{\dot{\alpha}\,\dot{\beta}}^{\,i\ j\,c}$
$\displaystyle=\langle[\bar{E}_{\dot{\alpha}}^{i},\bar{E}_{\dot{\beta}}^{j}],E^{c}\rangle=0,$
$\displaystyle\mbox{\bbbold F}_{\alpha i\dot{\beta}}^{\ \ \,j\,c}$
$\displaystyle=\langle[E_{\alpha
i},\bar{E}_{\dot{\beta}}^{j}],E^{c}\rangle=-i\delta_{i}{}^{j}(\sigma^{c})_{\alpha\dot{\beta}}\
.$

We now introduce connections for $\mbox{\goth s}\mbox{\goth p}(N)$ and
$\mbox{\goth s}\mbox{\goth p}\mbox{\goth i}\mbox{\goth n}(1,D-1)$ and define
the torsion and curvatures in the usual way. Note that this procedure involves
the complementary basis $E_{\alpha i}$ for $T_{1}^{*}$ which is only
determined modulo $T_{0}^{*}$, i.e. shifts of the form

$E_{\alpha i}\mapsto E_{\alpha i}+L_{\alpha i}{}^{b}E_{b}\ .$ (8.3)

We could include this in the structure group, along with a corresponding
connection, but we shall instead follow the standard procedure of using this
freedom to impose some additional constraints at dimension one-half. In
addition we shall not include a scale connection so that we have the standard
superspace geometrical set-up.

It is clear that the Frobenius tensor is invariant under scale transformations
of the form $E^{a}\mapsto SE^{a},\ E_{\alpha i}\mapsto
S^{-\frac{1}{2}}E_{\alpha i}$ (with the same transformation for
$\bar{E}_{\dot{\alpha}}^{i}$ in $D=4$). If we impose constraints to determine
$E_{a}$ and $E^{\alpha i}$ at dimension one-half we can then determine their
transformations as well. A convenient one to consider, as it does not involve
any connection terms, is

$(\langle[E_{\alpha i},E_{b}],E^{c}\rangle)_{(bc)}=0\ ,$ (8.4)

where the symmetrisation is understood to include lowering the $c$ index on
the left-hand side. Making finite super-Weyl (scale) transformations, we find
that this constraint will be preserved if

$\displaystyle E_{a}$ $\displaystyle\mapsto
S^{-1}\left(E_{a}+i(\gamma_{a})^{\gamma\delta}\Upsilon_{\gamma}{}^{k}E_{\delta
k}\right)\ ,$ $\displaystyle E^{ai}$ $\displaystyle\mapsto
S^{\frac{1}{2}}\left(E^{\alpha
i}-iE^{a}(\gamma_{c})^{\alpha\beta}\Upsilon_{\beta}{}^{i}\right)\ ,$ (8.5)

where $\Upsilon_{\alpha i}=S^{-1}D_{\alpha i}S$. The super-Weyl
transformations for the Lorentz and internal connections are given in (4.17)
for $D=3$, (5) for $D=4$ and (6.19) for $D=6$. The $D=5$ case can be derived
from $D=6$ by dimensional truncation.

Identifying $\mbox{\bbbold F}_{\alpha i\beta j}{}^{c}$ with the dimension-zero
torsion $T_{\alpha i\beta j}{}^{c}$, imposing suitable constraints on various
components of the torsion corresponding to fixing the odd basis $E_{\alpha i}$
using (8.3) and making appropriate choices for the $\mbox{\goth s}\mbox{\goth
p}\mbox{\goth i}\mbox{\goth n}(1,D-1)$ and g connections, one can show, with
the aid of the usual superspace Bianchi identities and some algebra, that the
components of the torsion and curvature tensors can be chosen to agree with
those derived previously. The finite super-Weyl transformations in the general
case were given in (5.23), from which it is straightforward to obtain
particular cases.

In addition to the fields of the conformal supergravity multiplet, this
geometry will also contain the components of the super Schouten tensor
$F_{AB}$, whose transformations can be found in (6.19). We can recover the
covariant forms for the torsions and curvatures by reversing the steps made
earlier.

##  9 Local Super Grassmannians

In the previous sections we have considered superconformal geometries starting
from local supertwistors in various dimensions. In all cases we have used the
standard split of supertwistor spaces into even twistors together with
additional odd components. However, we can choose different $(even|odd)$
splittings which naturally give rise to formulations of superconformal
geometry on different supermanifolds. These include chiral, projective [46],
[47],[49] and harmonic superspaces [48],and can be thought of from the
viewpoint of super flag manifolds as discussed from a pure mathematical
perspective in [50],[51] and presented in a more accessible form in [42]. More
recently the present authors have used them to discuss super-Laplacians and
their symmetries in the context of rigid supersymmetry [52, 53, 54]. In this
section we shall give a brief discussion of some cases relevant to four-
dimensional complex spacetime, deferring a fuller discussion to a later
publication.

Grassmannians are spaces of $p$-dimensional planes in $\mbox{\bbbold C}^{n}$,
with $p<n$, and super Grassmannians are spaces of $(p|q)$ planes in
$\mbox{\bbbold C}^{m|n}$, with $p\leq m,\ q\leq n$ (but with $p+q<m+n$). For
example, complex Minkowski space in $D=4$ can be thought of as an open set in
the Grassmannian of 2-planes in $\mbox{\bbbold C}^{4}$,. In the super case,
consider $D=4$-dimensional spacetime with $N=2$ supersymmetries. The
corresponding supertwistor space is then $\mbox{\bbbold C}^{4|2}$, and the
Grassmannian of $(2|1)$ planes in $\mbox{\bbbold C}^{4|2}$ is known as
analytic superspace. A supertwistor, i.e. an element of $\mbox{\bbbold
C}^{4|2}$ can be split into two halves $u^{A}$ and $v_{A^{\prime}}$, where
$A=(\alpha,1)$ and $A^{\prime}=(\alpha^{\prime},1^{\prime})$, where we have
used a prime to denote a dotted two-component spinor index. This allows us to
rewrite a supertwistor as

$Z^{\underline{A}}=\left(\begin{array}[]{c}u^{A}\\\ \ \ v_{A^{\prime}}\
.\end{array}\right)\ .$ (9.1)

We can then define a superconformal connection ${\cal A}$ acting on
supertwistors in this basis in the form

${\cal A}=\left(\begin{array}[]{cc}-\hat{\Omega}^{A}{}_{B}&E^{AB^{\prime}}\\\
F_{A^{\prime}B}&\hat{\Omega}_{A^{\prime}}{}^{B^{\prime}}\end{array}\right)\ ,$
(9.2)

with corresponding curvature

${\cal F}=d{\cal A}+{\cal A}^{2}=\left(\begin{array}[]{cc}-\hat{\cal
R}^{A}{}_{B}&{\cal T}^{AB^{\prime}}\\\ {\cal S}_{A^{\prime}B}&\hat{\cal
R}_{A^{\prime}}{}^{B^{\prime}}\end{array}\right)\ .$ (9.3)

The action of the superconformal group on ${\cal A}$ and ${\cal F}$ is
formally the same as in the bosonic case, equation (2.14), but note that the
diagonal transformations are not simply scale and Lorentz transformations,
because the diagonal subgroups are also supergroups. This is therefore a
somewhat different way of looking at superconformal transformations which we
propose to investigate further in a forthcoming paper.

Finally, we note that such geometries can also be thought of super versions of
paraconformal geometries [55].

##  10 Summary

In this article we have discussed the supergeometries describing off-shell
conformal supergravity multiplets in $D=3$ for $N=1$ to $8$, in $D=4$ for
$N=1$ to $4$, and in $D=6$ for $(N,0)$ supersymmetries with $N=1,2$, from the
perspective of local supertwistors555 We note that there is a related
construction in four dimensions described in [56]. . In this formalism one
introduces connections taking their values in the superconformal algebras in
the twistor representation, which can be thought of as an associated version
of the Cartan connection formalism. Similarly, the conformal superspace
construction [13] is an adaption of the Cartan formalism to superspace, and is
expected to be equivalent to our local supertwistor approach.. From this
starting point one can then derive the standard superspace formalism in a
systematic fashion. In order to specialise to the minimal off-shell conformal
supergravity multiplets one then has to impose constraints. In addition, we
discussed the $D=5,N=1$ case for which a slight amendment of the formalism is
necessary. We also showed that the same results can be obtained from the
minimal formalism in which only the dimension-zero torsion, or Frobenius
tensor, is specified. This minimal formalism was previously applied to $D=3$
[10, 12], where an additional self-duality constraint at dimension one is
required, while it was also previously shown in the $D=4$ case that the
supergeometries also follow from the dimension-zero torsion constraint [8]. In
this case an additional dimension one constraint is required to ensure the
correct number of component fields. In section 10 we briefly introduced the
idea of local super Grassmannians. Such superspaces are best viewed in terms
of different $(even|odd)$ splittings of supertwistor space.

Acknowledgement:
We are grateful to Sergei Kuzenko and Gabriele Tartaglino-Mazzucchelli for
careful reading and comments on the manuscript. The research of U.L. is
supported in part by the 2236 Co-Funded Circulation Scheme2 (CoCirculation2)
of TÜBITAK (Project No:120C067)666However the entire responsibility of the
publication is the authors’. The financial support received from TÜBITAK does
not mean that the content of the publication is approved in a scientific sense
by TÜBITAK. as well as by Längmanska Kulturfonden.

## References

  * [1] D. M. Capper and M. J. Duff, “Conformal Anomalies and the Renormalizability Problem in Quantum Gravity,” Phys. Lett. A 53 (1975) 361. doi:10.1016/0375-9601(75)90030-4
  * [2] M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, “Properties of Conformal Supergravity,” Phys. Rev. D 17 (1978) 3179. doi:10.1103/PhysRevD.17.3179
  * [3] A. Salam and J. A. Strathdee, “On Superfields and Fermi-Bose Symmetry,” Phys. Rev. D 11 (1975) 1521. doi:10.1103/PhysRevD.11.1521
  * [4] J. Wess and B. Zumino, “Superspace Formulation of Supergravity,” Phys. Lett. 66B (1977) 361. doi:10.1016/0370-2693(77)90015-6
  * [5] W. Siegel and S. J. Gates, Jr., “Superfield Supergravity,” Nucl. Phys. B 147 (1979) 77. doi:10.1016/0550-3213(79)90416-4
  * [6] P. S. Howe and R. W. Tucker, “Scale Invariance in Superspace,” Phys. Lett. 80B (1978) 138. doi:10.1016/0370-2693(78)90327-1
  * [7] V. I. Ogievetsky and E. S. Sokatchev, “The Gravitational Axial Superfield and the Formalism of Differential Geometry,” Sov. J. Nucl. Phys. 31 (1980) 424 [Yad. Fiz. 31 (1980) 821].
  * [8] P. S. Howe, “A Superspace Approach To Extended Conformal Supergravity,” Phys. Lett. 100B (1981) 389. doi:10.1016/0370-2693(81)90143-X
  * [9] P. S. Howe, “Supergravity in Superspace,” Nucl. Phys. B 199 (1982) 309. doi:10.1016/0550-3213(82)90349-2
  * [10] P.S. Howe, J.M. Izquierdo, G. Papadopoulos and P.K. Townsend, “New supergravities with central charges and Killing spinors in (2+1) dimensions,” Nucl. Phys. B467(1996) 183 [arXiv:hep-th/9505032] doi:10.1016/0550-3213(96)00091-0
  * [11] S. M. Kuzenko, U. Lindström and G. Tartaglino-Mazzucchelli, “Off-shell supergravity-matter couplings in three dimensions,” JHEP 03 (2011), 120 doi:10.1007/JHEP03(2011)120 [arXiv:1101.4013 [hep-th]].
  * [12] U. Gran, J. Greitz, P.S Howe and B.E.W. Nilsson, “Topologically gauged superconformal Chern-Simons matter theories,” JHEP 12 (2012) 046 [arXiv:1204.2521] doi:10.1007/JHEP12(2012)046
  * [13] D. Butter, “N=1 Conformal Superspace in Four Dimensions,” Annals Phys. 325 (2010), 1026-1080 doi:10.1016/j.aop.2009.09.010 [arXiv:0906.4399 [hep-th]].
  * [14] D. Butter, S. M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, “Conformal supergravity in three dimensions: New off-shell formulation,” JHEP 09 (2013), 072 doi:10.1007/JHEP09(2013)072 [arXiv:1305.3132 [hep-th]].
  * [15] D. Butter, S. M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, “Off-shell actions for conformal supergravity in three dimensions,” Phys. Part. Nucl. Lett. 11 (2014) no.7, 927. doi:10.1134/S1547477114070085; D. Butter, S. M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, “Conformal supergravity in three dimensions: Off-shell actions,” JHEP 1310 (2013) 073 doi:10.1007/JHEP10(2013)073 [arXiv:1306.1205 [hep-th]].; D. Butter, “N=2 Conformal Superspace in Four Dimensions,” JHEP 10 (2011), 030 doi:10.1007/JHEP10(2011)030 [arXiv:1103.5914 [hep-th]].
  * [16] S. M. Kuzenko and G. Tartaglino-Mazzucchelli, “Super-Weyl invariance in 5D supergravity,” JHEP 04 (2008), 032 doi:10.1088/1126-6708/2008/04/032 [arXiv:0802.3953 [hep-th]].
  * [17] D. Butter, S. M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, “Conformal supergravity in five dimensions: New approach and applications,” JHEP 1502 (2015) 111 doi:10.1007/JHEP02(2015)111 [arXiv:1410.8682 [hep-th]].;
  * [18] D. Butter, S. M. Kuzenko, J. Novak and S. Theisen, “Invariants for minimal conformal supergravity in six dimensions,” JHEP 1612 (2016) 072 doi:10.1007/JHEP12(2016)072 [arXiv:1606.02921 [hep-th]].
  * [19] D. Butter, J. Novak and G. Tartaglino-Mazzucchelli, “The component structure of conformal supergravity invariants in six dimensions,” JHEP 1705 (2017) 133 doi:10.1007/JHEP05(2017)133 [arXiv:1701.08163 [hep-th]].
  * [20] D. Butter, J. Novak, M. Ozkan, Y. Pang and G. Tartaglino-Mazzucchelli, “Curvature squared invariants in six-dimensional ${\cal N}=(1,0)$ supergravity,” JHEP 1904 (2019) 013 doi:10.1007/JHEP04(2019)013 [arXiv:1808.00459 [hep-th]].
  * [21] A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky and E. Sokatchev, “$N=2$ Supergravity in Superspace: Different Versions and Matter Couplings,” Class. Quant. Grav. 4 (1987), 1255 doi:10.1088/0264-9381/4/5/023
  * [22] E. Sokatchev, ”Off-shell Six-dimensional Supergravity in Harmonic Superspace,” Class. Quant. Grav.”, 5 (1988) 1459-1471. reportNumber = ”CERN-TH-5015-88”, doi = ”10.1088/0264-9381/5/11/009”,
  * [23] W. D. Linch, III and G. Tartaglino-Mazzucchelli, “Six-dimensional Supergravity and Projective Superfields,” JHEP 08 (2012), 075 doi:10.1007/JHEP08(2012)075 [arXiv:1204.4195 [hep-th]].
  * [24] P. S. Howe and U. Lindström, “Local supertwistors and conformal supergravity in six dimensions,” [arXiv:2008.10302 [hep-th]].
  * [25] Thomas, T. Y., ”On conformal differential geometry”, Proc. N.A.S. 12 (1926), 352; ”Conformal tensors”, Proc. N.A.S. 18 (1931), 103
  * [26] Bailey, T. N.; Eastwood, M. G.; Gover, A. R., ”Thomas’s structure bundle for conformal, projective and related structures”, Rocky Mountain J. 24 (1994), 1191.
  * [27] A. R. Gover, A. Shaukat and A. Waldron, “Tractors, Mass and Weyl Invariance,” Nucl. Phys. B 812 (2009) 424 doi:10.1016/j.nuclphysb.2008.11.026 [arXiv:0810.2867 [hep-th]].
  * [28] A. Ferber, “Supertwistors and Conformal Supersymmetry,” Nucl. Phys. B 132 (1978), 55-64 doi:10.1016/0550-3213(78)90257-2
  * [29] R. Penrose and W. Rindler, “Spinors And Space-time. Vol. 2: Spinor And Twistor Methods In Space-time Geometry,” doi:10.1017/CBO9780511524486
  * [30] S. A. Merkulov, “Supertwistor Connection And Conformal Supergravity,” Theor. Math. Phys. 64 (1985) 933 [Teor. Mat. Fiz. 64 (1985) 426]. doi:10.1007/BF01018355
  * [31] S. A. Merkulov, “N=2 Local Supertwistors,” Class. Quant. Grav. 5 (1988) 1373. doi:10.1088/0264-9381/5/10/016
  * [32] S. A. Merkulov, “Local supertwistors and N=2 conformal supergravity,” Theor. Math. Phys. 79 (1989) 517 [Teor. Mat. Fiz. 79 (1989) 253]. doi:10.1007/BF01016533
  * [33] Cartan, Élie, ”Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre”, Annales Scientifiques de l’ École Normale Supérieure, 27: (1910) 109.
  * [34] J. Lott, “The Geometry of supergravity torsion constraints,” math/0108125 [math-dg].
  * [35] T. Kugo and K. Ohashi, “Gauge and nongauge tensor multiplets in 5-D conformal supergravity,” Prog. Theor. Phys. 108 (2003), 1143-1164 doi:10.1143/PTP.108.1143 [arXiv:hep-th/0208082 [hep-th]].
  * [36] E. Bergshoeff, S. Cucu, T. De Wit, J. Gheerardyn, R. Halbersma, S. Vandoren and A. Van Proeyen, “Superconformal N=2, D = 5 matter with and without actions,” JHEP 10 (2002), 045 doi:10.1088/1126-6708/2002/10/045 [arXiv:hep-th/0205230 [hep-th]].
  * [37] W. Nahm, “Supersymmetries and their Representations,” Nucl. Phys. B 135 (1978), 149 doi:10.1016/0550-3213(78)90218-3
  * [38] P. S. Howe and E. Sezgin, “The Supermembrane revisited,” Class. Quant. Grav. 22 (2005), 2167-2200 doi:10.1088/0264-9381/22/11/017 [arXiv:hep-th/0412245 [hep-th]].
  * [39] J. Bagger and N. Lambert, “Gauge symmetry and supersymmetry of multiple M2-branes,” Phys. Rev. D 77 (2008), 065008 doi:10.1103/PhysRevD.77.065008 [arXiv:0711.0955 [hep-th]].
  * [40] A. Gustavsson, “Algebraic structures on parallel M2-branes,” Nucl. Phys. B 811 (2009), 66-76 doi:10.1016/j.nuclphysb.2008.11.014 [arXiv:0709.1260 [hep-th]].
  * [41] A. A. Rosly, “Gauge Fields in Superspace and Twistors,” Class. Quant. Grav. 2 (1985), 693-699 doi:10.1088/0264-9381/2/5/011
  * [42] P. S. Howe and G. G. Hartwell, “A Superspace survey,” Class. Quant. Grav. 12 (1995), 1823-1880 doi:10.1088/0264-9381/12/8/005
  * [43] E. Bergshoeff, E. Sezgin and A. Van Proeyen, “(2,0) tensor multiplets and conformal supergravity in D = 6,” Class. Quant. Grav. 16 (1999) 3193 doi:10.1088/0264-9381/16/10/311 [hep-th/9904085].
  * [44] P. S. Howe, G. Sierra and P. K. Townsend, “Supersymmetry in Six-Dimensions,” Nucl. Phys. B 221 (1983) 331. doi:10.1016/0550-3213(83)90582-5
  * [45] S. M. Kuzenko, U. Lindström, E. S. N. Raptakis, and G. Tartaglino-Mazzucchelli, “Symmetries of N = (1, 0) supergravity backgrounds in six dimensions” , Work in progress.
  * [46] A. Karlhede, U. Lindström and M. Roček, “Selfinteracting Tensor Multiplets in $N=2$ Superspace,” Phys. Lett. B 147 (1984), 297-300 doi:10.1016/0370-2693(84)90120-5
  * [47] J. Grundberg and U. Lindström, “Actions for Linear Multiplets in Six-dimensions,” Class. Quant. Grav. 2 (1985), L33 doi:10.1088/0264-9381/2/2/005
  * [48] A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky and E. S. Sokatchev, “Harmonic superspace,” doi:10.1017/CBO9780511535109
  * [49] S. M. Kuzenko, U. Lindström, M. Roček and G. Tartaglino-Mazzucchelli, “On conformal supergravity and projective superspace,” JHEP 0908 (2009) 023 doi:10.1088/1126-6708/2009/08/023 [arXiv:0905.0063 [hep-th]].
  * [50] A. A. Rosly, “Gauge Fields in Superspace and Twistors,” Class. Quant. Grav. 2 (1985), 693-699 doi:10.1088/0264-9381/2/5/011
  * [51] Yu. I. Manin, “Holomorphic supergeometry and Yang-Mills superfields,” J. Sov. Math. 30, 1927 (1985); Gauge Field Theory and Complex Geometry, Springer, Berlin, 1988 (Second Edition: 1997).
  * [52] P. S. Howe and U. Lindström, “Notes on Super Killing Tensors,” JHEP 03 (2016), 078 doi:10.1007/JHEP03(2016)078 [arXiv:1511.04575 [hep-th]].
  * [53] P. S. Howe and U. Lindström, “Super-Laplacians and their symmetries,” JHEP 05 (2017), 119 doi:10.1007/JHEP05(2017)119 [arXiv:1612.06787 [hep-th]].
  * [54] P. S. Howe and U. Lindström, “Some remarks on (super)-conformal Killing-Yano tensors,” JHEP 11 (2018), 049 doi:10.1007/JHEP11(2018)049 [arXiv:1808.00583 [hep-th]].
  * [55] T. N. Bailey and M. G. Eastwood, “Complex Paraconformal Manifolds-their Differential Geometry and Twistor Theory” Forum Math. 3 (1991), 061.
  * [56] M. Wolf, “Self-Dual Supergravity and Twistor Theory,” Class. Quant. Grav. 24 (2007), 6287-6328 doi:10.1088/0264-9381/24/24/010 [arXiv:0705.1422 [hep-th]].