A graded manifold is a mathematical structure that generalizes the concept of a smooth manifold by incorporating a grading, typically a $\mathbb{Z}$-grading or a $\mathbb{Z}_2$-grading. In its most common usage, particularly within differential geometry and mathematical physics, a graded manifold refers to a supermanifold, which is a manifold locally modeled on a superdomain.
A superdomain is constructed by taking the product of an open set in $\mathbb{R}^n$ (representing the "even" or bosonic dimensions) and an exterior algebra (or Grassmann algebra) over a finite-dimensional real vector space (representing the "odd" or fermionic dimensions). The coordinates on a supermanifold thus include both commuting (even) and anti-commuting (odd) variables.
More formally, a graded manifold (or supermanifold) can be defined as a pair $(M, \mathcal{A})$, where $M$ is a smooth manifold (the "body" or "reduced manifold") and $\mathcal{A}$ is a sheaf of supercommutative superalgebras over $M$. This sheaf $\mathcal{A}$ locally resembles the tensor product of the sheaf of smooth functions on $M$ with an exterior algebra. The elements of $\mathcal{A}$ are called superfunctions.
The introduction of anti-commuting coordinates allows for the mathematical description of fermionic fields and supersymmetries in theoretical physics, making graded manifolds fundamental in:
- Supersymmetry (SUSY): Graded manifolds provide the geometric framework for supergravity and superstring theories.
- Graded Lie algebras and Lie superalgebras: The tangent bundle of a supermanifold is naturally a graded vector bundle, leading to structures like Lie superalgebras.
- Supergeometry: This broader field encompasses the study of supermanifolds, supervector bundles, super Lie groups, and their associated calculus.
While the term "graded manifold" can, in principle, refer to manifolds endowed with other types of gradings, in the context of differential geometry, it is overwhelmingly used as a synonym for supermanifold, emphasizing the $\mathbb{Z}_2$-grading (even/odd) inherent in supercommutative superalgebras.