Foundations of modern physics 5: Supergeometry

SUSY extension of field theory including supergravity is greatly motivated by grand-unification models and contemporary string and brane theories. However, there are different notions of a Lie supergroup, a supermanifold and a superbundle.

Let us mention a definition of a super Lie group as a Harish -- Chandra pair of a Lie group and a super Lie algebra, a graded Lie group and a Lie supergroup.

It should be emphasized that graded manifolds are not supermanifolds in a strict sense. However, every graded manifold can be associated to a DeWitt infinity-smooth H-supermanifold, and vice versa.

Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves on supervector spaces. In field theory, treating odd fields on a smooth manifold X, one can follow the well-known Serre--Swan theorem extended to graded manifolds. It states that, if a graded commutative ring over a ring C(X) of smooth functions on X is generated by a projective C(X)-module of finite rank, it is isomorphic to a ring of graded functions on a graded manifold whose body is X. Therefore, odd fields in field theory are described in terms of graded manifolds.

One usually consider supermanifolds over Grassmann algebras of finite rank. This is the case of infinity-smooth GH-, H-, G-supermanifolds and G-supermanifolds. By analogy with familiar smooth manifolds, infinity-smooth supermanifolds are constructed by gluing of open subsets of supervector spaces endowed with the Euclidean topology. However, if a supervector space is provided with the non-Hausdorff DeWitt topology, we are in the case of DeWitt supermanifolds.

In a more general setting, one considers supermanifolds over the so called Arens--Michael algebras of Grassmann origin. They are R-supermanifolds obeying a certain set of axioms. In the case of a finite Grassmann algebra, the category of R-supermanifolds is equivalent to the category of G-supermanifolds.

The most of theoreticians however ignore these mathematical details.

References

G. Sardanashvily, Lectures on supergeometry arXiv: 0910.0092

WikipediA  Supergeometry

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