The structure of supermanifolds

Abstract

The increasing recognition of Lie superalgebras and their importance in physics inspired a search to find an object, a “supermanifold", which would realize the geometry implicit in Lie superalgebras. This paper analyzes the structure of supermanifolds as defined by B. Kostant. The result is the following structure theorem. The Main Theorem. If E is a real vector bundle over the smooth manifold X, let Λ E \Lambda E be the associated exterior bundle and let Γ ( Λ E ) \Gamma (\Lambda E) be the sheaf of sections of Λ E \Lambda E . Then every supermanifold over X is isomorphic to Γ ( Λ E ) \Gamma (\Lambda E) for some vector bundle E over X. Although the vector bundle E is not unique but is determined only up to isomorphism, and the isomorphism guaranteed is not canonical, the existence of the isomorphism provides a base for a better understanding of geometry in the graded setting.