Supergeometry

Supergeometry is a branch of differential geometry that generalizes manifolds by incorporating anticommuting, or odd, coordinates alongside ordinary (even) coordinates. The resulting objects, called supermanifolds or superspaces, are studied using Z2-graded algebras of functions, so that the algebra of functions is split into even and odd parts and signs appear according to parity.

Mathematically, several formalisms are used. In the sheaf-theoretic approach (Berezin–Kostant–Leites), a supermanifold is a locally ringed space (M, O_M) with O_M locally isomorphic to C∞(U) ⊗ Λ•(R^q). Batchelor's theorem gives a concrete realization as a body M together with a vector bundle E such that O_M ≅ Γ(Λ E^*). The functor-of-points viewpoint describes a supermanifold by its S-points for all test supermanifolds S, aligning with Grothendieck’s philosophy. In physics, a more concrete DeWitt approach uses Grassmann-valued coordinates directly.

Key tools include Grassmann algebras, parity grading, and tools of super differential calculus: supervector fields, Lie superalgebras, and integration theory via Berezin integration; the Berezinian generalizes the determinant under coordinate change and plays a central role in superintegration and volumes in superdimensions. Geometry also involves super vector bundles and connections.

Supergeometry provides formal foundations for supersymmetric field theories, supergravity, and superstring theory, and it connects to broader areas such as graded and derived geometry. Ongoing work addresses foundational issues, integration theory, and global constructions.