Topics in the Differential Geometry of Supermanifolds: Super Holonomy Theorem

The book combines the algebraic and differential geometric approaches to supermanifold theory. It begins with a superspace, constructed by taking the Cartesian product of copies of the even and odd parts of an infinite-dimensional Banach Grassmann algebra. The superspace serves as a model space of G-infinite supermanifolds. The theory of super Lie groups and super Lie algebras, (in particular the existence theorems on super Lie groups) is given. Super principal fiber bundles equipped with connections and the action of super Lie groups and supervector fields are introduced. The concept of parallel transport along smooth curves is introduced in the same way as in conventional differential geometry. It is shown that a smooth curve in the base space can be lifted uniquely into the bundle and that parallel displacements along closed smooth curves form a group, the ¿super holonomy group¿. It is further shown that the group is a sub-super Lie group of the structure group of the bundle. The result follows from the super analogues of the Freudenthal theorem and the reduction theorem of bundles. The book concludes with a proof of a super version of the Ambrose-Singer holonomy theorem.