Hyperkähler manifolds with torsion, supersymmetry and Hodge theory

Let M be a hypercomplex Hermitian manifold, (M, /) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions.The standard linearalgebraic construction produces a canonical nowhere degenerate (2 7 0)-form Cl on (M,/).It is well known that M is hyperkahler if and only if the form Vl is closed.The M is called HKT (hyperkahler with torsion) if Cl is closed with respect to the Dolbeault differential d : A 2 ' 0 (M, /) -> A 3 ' 0 (M, /).Conjecturally, all compact hypercomplex manifolds admit an HKT-metrics.We exploit a remarkable analogy between the de Rham DG-algebra of a Kahler manifold and the Dolbeault DG-algebra of an HKT-manifold.The supersymmetry of a Kahler manifold X is given by an action of an 8dimensional Lie superalgebra g on A*(X), containing the Lefschetz £L(2)-triple, the Laplacian and the de Rham differential.We establish the action of # on the Dolbeault DG-algebra A*' 0 (M, /) of an HKT-manifold.This is used to construct a canonical Lefschetz-type 5L(2)-action on the space of harmonic spinors of M.