A new construction of homogeneous quaternionic manifolds and related geometric structures
Abstract
Description
Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a module over the even Clifford algebra Cl^0 (V). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V)-equivariant linear map Π: \wedge^2 W \to V. If the skew symmetric vector valued bilinear form Πis nondegenerate then (M,Q) is endowed with a canonical pseudo-Riemannian metric g such that (M,Q,g) is a homogeneous quaternionic pseudo-Kähler manifold.
The construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any spin(V)-equivariant linear map Π: Sym^2 W \to V a homogeneous quaternionic supermanifold (M,Q) is constructed and, moreover, a homogeneous quaternionic pseudo-Kähler supermanifold (M,Q,g) if the symmetric vector valued bilinear form Πis nondegenerate.
to appear in the Memoirs of the AMS, 81 pages, Latex source file
to appear in the Memoirs of the AMS, 81 pages, Latex source file