Full action for N=2 conformal supergravity in analytic prepotentials

Determine the full action for four-dimensional off-shell \(\mathcal{N}=2\) conformal supergravity, generalizing the square of the Weyl tensor, in terms of the fundamental analytic prepotentials \(\mathcal{H}^{++}\).

Background

The paper formulates off-shell $\mathcal{N}=2$ conformal supergravity in harmonic superspace using unconstrained analytic prepotentials $\mathcal{H}^{++}$ appearing in the covariant harmonic derivative. Although the paper derives the bosonic Wess–Zumino gauge, component transformations, and hypermultiplet couplings, it does not construct the complete conformal-supergravity action in terms of these prepotentials. The authors note that conjectures about the action’s structure exist, but the action itself remains unavailable, making its explicit construction an unresolved problem.