Supermanifold

In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.

An informal definition is commonly used in physics textbooks and introductory lectures. It defines a supermanifold as a manifold with both bosonic and fermionic coordinates. Locally, it is composed of coordinate charts that make it look like a "flat", "Euclidean" superspace. These local coordinates are often denoted by

where x is the (real-number-valued) spacetime coordinate, and ${\displaystyle \theta \,}$ and ${\displaystyle {\bar {\theta }}}$ are Grassmann-valued spatial "directions".

The physical interpretation of the Grassmann-valued coordinates are the subject of debate; explicit experimental searches for supersymmetry have not yielded any positive results. However, the use of Grassmann variables allow for the tremendous simplification of a number of important mathematical results. This includes, among other things a compact definition of functional integrals, the proper treatment of ghosts in BRST quantization, the canellcation of infinities in quantum field theory, Witten's work on the Atiyah-Singer index theorem, and more recent applications to mirror symmetry.

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