\(\mathscr {N}\,{=}\,2\) Supergravities in Harmonic Superspace

Basics of \(\mathscr {N}\,{=}\,2, 4D\) conformal and Einstein supergravities in the harmonic superspace approach are outlined. The crucial merit of this formulation is that the relevant off-shell supermultiplets, in particular \(\mathscr {N}\,{=}\,2, 4D\) superconformal Weyl multiplet, are accommodated by the harmonic-analytic unconstrained prepotentials with a clear geometric meaning, like in the analogous formulation of \(\mathscr {N}\,{=}\,2, 4D\) 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 \(\mathscr {N}\,{=}\,2\) Einstein supergravity are obtained as the actions of the appropriate harmonic analytic compensators in the background of conformal \(\mathscr {N}\,{=}\,2\) supergravity. The version admitting the most general couplings to quaternion-Kähler matter is the “principal” version with the unconstrained harmonic analytic hypermultiplet superfield as a compensator. It involves an infinite number of auxiliary fields.