N = 3 Conformal Supergravity in Four Dimensions.

In this paper, we derive the action for N = 3 conformal supergravity in four space-time dimensions. We construct a density formula for N = 3 conformal supergravity based on the superform action principle. Finally, we embed the N = 3 Weyl multiplet in the density formula to obtain the invariant action for N = 3 conformal supergravity. We will show that there are two inequivalent embeddings by changing a particular coefficient from real to imaginary. They lead to invariant actions, which will either be the supersymmetrization of the Weyl square term or the Pontryagin density in the eventuality of gauge fixing to Poincar\'e supergravity. We also discuss consistent truncation of N = 4 Weyl multiplet to N = 3 Weyl multiplet and use it for a robust check of our results using the earlier known results in N = 4 conformal supergravity.