G. Sardanashvily, Lectures on supergeometry, arXiv: 0910.0092
Elements of supergeometry are an ingredient in many contemporary
classical and quantum field models involving odd fields. For instance, this is
the case of SUSY field theory, BRST theory, supergravity. Addressing to
theoreticians, these Lectures aim to summarize the relevant material on
supergeometry of modules over graded commutative rings, graded manifolds and
supermanifolds.
Contents
1. Graded tensor calculus, 2. Graded differential calculus and connections, 3. Geometry of
graded manifolds, 4. Superfunctions, 5. Supermanifolds, 6. DeWitt supermanifolds,
7. Supervector bundles, 8. Superconnections, 9. Principal superconnections, 10.
Supermetric, 11. Graded principal bundles.
Introduction
Supergeometry is phrased in terms of Z_2-graded modules and sheaves over Z_2-graded commutative algebras. Their algebraic properties
naturally generalize those of modules and sheaves over commutative algebras,
but supergeometry is not a particular case of noncommutative geometry because
of a different definition of
graded erivations.
In these Lectures, we address supergeometry of modules over graded
commutative rings (Lecture 2), graded manifolds (Lectures 3 and 11) and
supermanifolds.
It should be emphasized from the beginning that graded manifolds are not
supermanifolds, though every graded manifold determines a DeWitt
H∞-supermanifold, and vice versa (see Theorem 6.2 below). 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 of
supervector spaces. Note that there are different types of supermanifolds; these
are H∞-, G∞-, GH∞-, G-, and DeWitt supermanifolds. For instance, supervector bundles
are defined in the category of G-supermanifolds.
