In the first introductory part of the talk (that may be skipped by the request of the audience), we remind the classical Witten's result that the de Rham complex is isomorphic to the Hilbert space of wave functions in a certain supersymmetric quantum mechanical system. There is an ``industrial'' way to construct supersymmetric systems which is based on the superspace formalism, which we also describe. We then discuss Kahler manifolds and show how the classical result that any Kahler metric can be derived from the Kahler potential, $h_{m \bar n} = \partial_m \bar \partial_n \, K(z^p, \bar z^p)$ can be easily derived in the {\it extended} superspace formalism.
The hyper-Kahler models enjoy ${\cal N} = 8$ supersymmetry. We do not know how to fully implement the latter in the conventional superspace approach, it is only possible in the harmonic superspace formalism that we briefly outline. Any hyper-Kahler metric can be derived from a certain
function (prepotential) depending on the harmonic superfields and harmonics. In contrast to the Kahler case, the relation of the metric to the prepotential is not so simple, one has to solve a system of certain differential equations.
Finally we go over to the so-called HKT manifolds. These are the manifolds admitting three quaternionic complex structures that are covariantly constant with respect to a certain torsionful connection. (HK manifolds is a subclass of HKT manifolds where this connection is torsionless). We describe them in the harmonic superspace framework. In constrast to the HK metric, a general HKT metric needs {\it two} different functions for its description. We show that the space of all HKT metrics is divided in families such that the Obata curvatures of all members of one family coincide.
Vasily Pestun
