The God has created a man in order that he creates that the God fails to do

Tuesday, 17 May 2011

What is meant by 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  supermanifold, Lie supergroup and 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. There are graded manifolds, graded Lie groups, and graded bundles. 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.

One usually considers 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.


G.Sardanashvily, Lectures on supergeometry, arXiv: 0910.0092

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