By F. J. Flaherty
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3. Superxeld actions and equations of motion Invariant actions in superspace supergravity are obtained upon integrating superspace densities over the commuting and anticommuting directions of superspace. Densities, in this case, are constructed with the help of E, the superdeterminant of E . As we have already alluded to above, + the supergravity action in standard superspace geometry is just the volume of superspace. In our present situation where both supergravity and matter occur together in a generalized superspace geometry, the volume element corresponding to this superspace geometry yields the complete kinetic actions for the supergravity/matter system.
A@? 2 @ A L K? 12 A K ? e K ) M M ! 32) Plrep=1020=EM=VVC P. Bine& truy et al. / Physics Reports 343 (2001) 255}462 313 for the supercovariant gravitino "eld strength. The contributions of the matter and Yang}Mills sector are hidden in the covariant derivatives through the de"nitions given above. @". # A ? " is given in terms of ?. ". Fortunately enough, in the present formulation, the LK@ projection technique takes care of these complications automatically in a concise way. @" . e. @P" P @ ?
2(2 I ? 25) These terms will appear explicitly in various places of component "eld expressions later on as well. " for the A@ gravitino. A and its conjugate ¹ . The @ ? 28) L K L K K L@ K L D M "R M # M Q @Q ! M A . 29) L K? L K? K@ L ? K? " , A@ A@ A @ and similarly for ¹ . Using the explicit form of the torsion coe$cients appearing here, and ? b? A@ L K K L @ K ? A 2 @ A 12 A K ? @ i ! e K M ) b? D M )! (e K M K L? @ K ? A@? 2 @ A L K? 12 A K ? e K ) M M ! 32) Plrep=1020=EM=VVC P.
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