${\cal N}=2\,$ Supergravities in Harmonic Superspace

Basics of ${\cal N}=2, 4D$ conformal and Einstein supergravities in the harmonic superspace approach are outlined. The crucial merit of this formulation consists in that the relevant off-shell supermultiplets, in particular ${\cal N}=2, 4D$ superconformal Weyl multiplet, are accommodated by the harmonic-analytic unconstrained prepotentials with a clear geometric meaning, like in the analogous formulation of ${\cal N}=2, 4D$ supersymmetric gauge theory. The fundamental gauge group of conformal supergravity is constituted by the analyticity-preserving diffeomorphisms of harmonic superspace. The superfield actions of various off-shell versions of ${\cal N}=2$ Einstein supergravity are obtained as the actions of the appropriate harmonic analytic compensators in the background of conformal ${\cal N}=2$ supergravity. The version admitting the most general couplings to quaternion-K\"ahler matter is the ``principal'' version with the unconstrained harmonic analytic hypermultiplet superfield as a compensator. It involves an infinite number of auxiliary fields.