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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?

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Satoshi, I know of one paper which does this: "Nucl Phys B 264, Superconformal tensor calculus and matter couplings in six dimensions, Pages 653-686, E. Bergshoeff, E. Sezgin, A. Van Proeyen"

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Thank you very much for the information. How about different supersymmetry? Namely ${\cal N}=(2,0)$ and ${\cal N}=(1,1)$? –  Satoshi Nawata Dec 2 '11 at 17:48

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