A convenient method for dealing with off-shell formulations of supergravity theories is provided by the superconformal multiplet calculus. This calculus was originally constructed for 4d ${\cal N}=2$ supergravity by de Wit et al. For 5d supergravity, conformal supergravity approach was developed relatively recently by several groups of Bergshoeff et al, Ohashi et al,Hanaki et al.
http://xxx.lanl.gov/abs/hep-th/0104113 , http://xxx.lanl.gov/abs/hep-th/0104130 , http://xxx.lanl.gov/abs/hep-th/0611329 .
The main idea is that the Poincare algebra is extended to the superconformal algebra to obtain an off-shell version of Poincare supergravity. It turns out that extending conformal supergravity to a gauge theory of superconformal algebra provides an irreducible off-shell realization of the gravity and matter multiplets. Then, by imposing constraints, the gauge theory is identified as a gravity theory. Upon gauge fixing the extra superconformal symmetries, one obtains the Poincare supergravity. In this formalism, one has a systematic way to construct invariant actions since the transformation laws are simpler and completely fixed by the superconformal algebra. Following this approach, one gets an off-shell formulation of supergravity coupled to vector multiplets.
As far as I know, there is no off-shell formulation of 6d supergravity by superconformal multiplet calculas. Why is there no conformal supergravity in 6D? Is there any obstruction to the formulation?
