supermanifold

A supermanifold is a generalization of a smooth manifold that incorporates both commuting and anticommuting coordinates. Formally, it is a locally ringed space (M, O_M) such that every point has a neighborhood isomorphic to a model (U, C∞(U) ⊗ Λ•R^q), where U ⊂ R^p is open and Λ•R^q denotes the exterior (Grassmann) algebra on q generators. The total dimension is written p|q. The underlying topological space of M, called the reduced manifold M_red, is a smooth manifold of dimension p, obtained by discarding the nilpotent (odd) directions. The structure sheaf O_M contains even functions corresponding to ordinary smooth functions and odd elements generated by Grassmann variables that square to zero.

Local models and charts: A supermanifold is covered by charts mapping open sets of M to open sets of R^{p|q} with coordinates (x^1,…,x^p; θ^1,…,θ^q). Transition maps between charts are morphisms of superalgebras that preserve parity and are smooth in the even coordinates with polynomials in the odd ones. The notion of smooth maps between supermanifolds extends the classical one by requiring compatibility with the sheaf structure.

Morphisms and structures: A morphism of supermanifolds consists of a continuous map on the underlying spaces together with a compatible morphism of sheaves of superalgebras. One can define tangent and cotangent sheaves, vector bundles, and the parity-reversed functor Π for changing the parity of fibers. Supermanifolds support notions such as integration (Berezin integration) and differential forms, with many constructions paralleling ordinary differential geometry.

Foundations and models: Foundational approaches include the sheaf-theoretic Berezin–Leites/Kostant framework and the functor-of-points viewpoint. Batchelor’s theorem provides that, for many purposes, a smooth supermanifold is isomorphic to a model M_red with O_M ≅ C∞(M_red) ⊗ Λ•E for some vector bundle E, linking to practical presentations. The simplest example is the superspace R^{p|q}.