Hyperkähler manifolds with torsion, supersymmetry and Hodge theory

Let M be a hypercomplex Hermitian manifold, (M, I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate (2,0)-form Ω on (M, I). It is well known that M is hyperkähler if and only if the form Ω is closed. The M is called HKT (hyperkähler with torsion) if Ω is closed with respect to the Dolbeault differential ∂ : Λ(M, I)−→ Λ(M, I). Conjecturally, all compact hypercomplex manifolds admit an HKT-metrics. We exploit a remarkable analogy between the de Rham DG-algebra of a Kähler manifold and the Dolbeault DG-algebra of an HKT-manifold. The supersymmetry of a Kähler manifold X is given by an action of an 8-dimensional Lie superalgebra g on Λ∗(X), containing the Lefschetz SL(2)-triple, the Laplacian and the de Rham differential. We establish the action of g on the Dolbeault DG-algebra Λ(M, I) of an HKT-manifold. This is used to construct a canonical Lefschetz-type SL(2)-action on the space of harmonic spinors of M .