Hard Lefschetz actions in Riemannian geometry with special holonomy

Abstract

It is known that the hard Lefschetz action, together with Kähler identities for Kähler (resp. hyperkähler) manifolds, determines a $\mathfrak{su}(1,1{)}_{sup}$ (resp. $\mathfrak{sp}(1,1{)}_{sup}$) Lie superalgebra action on differential forms. In this paper, we explain the geometric origin of this action, and we also generalize it to manifolds with other holonomy groups. For semi-flat Calabi-Yau (resp. hyperkähler) manifolds, these symmetries can be enlarged to a $\mathfrak{so}(2,2{)}_{sup}$ (resp. $\mathfrak{su}(2,2{)}_{sup}$) action.

16 pages