My Lectures on Supergeometry

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.

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