An action of N = 8 self - dual supergravity in ultra - hyperbolic harmonic superspace 1

The N-extended self-dual supergravity in the ultra-hyperbolic four-dimensional space-time of kleinian signature (2+2) is given in the N-extended harmonic superspace. We reformulate the on-shell N-extended self-dual supergravity constraints of Siegel to a 'zero-curvature' representation, and solve all of them but one in terms of a single superfield prepotential, by using a covariant Frobenius gauge. An off-shell super-space action, whose equation of motion yields the remaining constraint, is found. Our manifestly Lorentz-covariant action in harmonic superspace is very similar to the non-covariant Chern-Simons-type action, which was proposed earlier by Siegel in the light-cone N = 8 superspace. Our action is also invariant under the residual superdiffeomorphisms and the residual local OSp(8|2) super-Lorentz rotations, which are left after imposing the Frobenius gauge. The infinitesimal superfield parameters of the residual symmetries are expressed in terms of independent analytic superfields.