Supersymmetries in Calabi - Yau Geometry

We consider a class of Lie superalgebra, called spinc supersym-metry algebras, constructed from spinor representation. They are motivated by supersymmetry algebras used by physicists. On a Riemannian manifold, a KK ahler manifold, and a hyperkk ahler manifold respectively, it is known that some natural operators on the space of diierential forms span spinc (2), spinc(3) and spinc (5) supersymmetry algebras respectively. Motivated by Mirror Symmetry Conjecture, we consider supersymmetries on Calabi-Yau manifolds. The supersymmetry (SUSY) algebra 10] is a special kind of Lie superalgebras 6] which involves spinor representations. It was invented by physicists in the sev-enties to formulate a uniied theory for fermions and bosons. A guiding principle in physics is to examine the symmetries of the Lagrangians. Quite often the classical Lagrangian can be extended to have supersymmetries. For example, Donaldson theory has been interpreted by Witten 12] as a twisted N = 2 supersymmetric quantum eld theory which extends the Lagrangian of the classical Yang-Mills theory. The study of this theory leads to Seiberg-Witten theory 13]. Other examples include supersymmetric extensions of nonlinear sigma models. In this context, it turns out 1] that an N = 1 supersymmetric extension is always possible; to get an N = 2 supersymmetric extension, the target Riemannian manifold has to be KK ahler; to get an N = 4 supersymmetric extension, the target manifold has to be hyperkk ahler. One naturally speculates on whether there is a direct relationship between the manifolds with special holonomy groups and the supersymmetry algebras. This is discussed in a recent paper by Frr ohlich, Grandjean and Recknagel 3]. Their motivation is to nd the analogues of KK ahler manifolds etc. in non-commutative geometry. Their idea is as follows: since the space of exterior forms is a super vector space, the space of linear operators on it is naturally a Lie superalgebra (actually a Poisson superalgebra) under the supercommutators. It is conceivable that a set of operators may generate a nite dimensional Lie (super)algebra. Actually some examples of this type have been well-known. In Riemannian geometry, Witten considered the following very simple Lie superalgebra in his innuential paper on Morse theory 11]: for a Riemannian manifold (M; g), let d : (M) ! (M) be the exterior diierential operator, d its formal adjoint. In KK ahler geometry, the proof of Hard Lefschetz Theorem for KK ahler manifolds, usually attributed to Chern, uses three algebraic operators (see …