Superization of Kaehler and hyper-Kaehler manifolds

Abstract

Let $J$ be an (almost) complex structure on a manifold $\mathcal{M}$; let~ $h$ be a~pseudo-Hermitian with respect to $J$ non-degenerate symmetric form on the tangent bundle over~$\mathcal{M}$. The manifold $\mathcal{M}$ with the triple $(\omega ,h,J)$, where $\omega$ is an almost symplectic differential 2-form defined through $h$ and $J$, is called (almost) Kähler. If $\omega$ and $J$ are \textit{parallel}, i.e., preserved by a~torsion-free connection, the adjective ``almost'' can be dropped; that connection is then unique, but it need not be flat: a~Kähler manifold has curvature, and no coordinates bring the whole triple to a~standard shape. I also describe the local invariants of a pair of bilinear forms and consider what superizing each of the ingredients $(\mathcal{M};\omega ,h,J)$ brings about. The same program is performed with the definition of hyper-Kähler manifolds, where there are three $J$s, and three $\omega$s. Replacing $\omega$ with the dual bivector makes the definition of (almost) Kähler (and hyper-Kähler) supermanifold depend on a supervariety. The Lie superalgebras $\mathfrak{g}$ of invariant operators acting on the spaces of (pseudo)differential and (pseudo)integral forms are described in all cases that arise. Lee's theorem on the two conditions $d\omega =0$ splits into is superized. Applications of $\mathfrak{g}$ to representation theory (analogs of the Hodge-Lepage decomposition) will be given elsewhere, together with interpretation from Howe's duality point of view.

50 pages

References in corpus (3)

• Higher Trace and Berezinian of Matrices over a Clifford Algebra
• Classification of supersymmetries
• Minkowski superspaces and superstrings as almost real-complex supermanifolds