N=1 Superconformal Tensor Calculus: Multiplets with External Lorentz Indices and Derivative Operations

In N=1 conformal supergravity, we completely determine the transformation laws of the most general multiplets with arbitrary external Lorentz indices, and then define the spinor derivative operations. Unconstrained (general type) multiplets exist for arbitrary external indices while chiral and linear multiplets exist only for purely undotted spinor indices. This comes from the particular fact that the superconformal spinor derivative is covariant only on multiplets satisfying some restrictive conditions. We define also a new class of spinor derivative operators each of which depends on the choice of a multiplet u playing the role of covariantization but is applicable covariantly to any multiplets. New chiral and linear multiplets defined through these spinor derivatives exist for arbitrary external Lorentz indices when ū is a (real or complex) linear multiplet. The connections of these spinor derivatives and constrained multiplets to those in superspace formulations of Poincaré supergravities are clarified.