Supermanifolds - Application to Supersymmetry

TL;DR

This paper clarifies the mathematical foundations of supermanifolds using parity and graded algebra, aiming to develop invariant integral definitions and applying these concepts to construct supersymmetric Fock spaces.

Contribution

It provides a unified, invariant framework for supermanifolds and divergence, facilitating the extension of superfunction integrals and their application to supersymmetry.

Findings

Clarified relationships between different supermanifold definitions

Developed invariant divergence complexes for supermanifolds

Constructed supersymmetric Fock spaces using the framework

Abstract

Parity is ubiquitous, but not always identified as a simplifying tool for computations. Using parity, having in mind the example of the bosonic/fermionic Fock space, and the framework of Z_2-graded (super) algebra, we clarify relationships between the different definitions of supermanifolds proposed by various people. In addition, we work with four complexes allowing an invariant definition of divergence: - an ascending complex of forms, and a descending complex of densities on real variables - an ascending complex of forms, and descending complex of densities on Grass mann variables. This study is a step towards an invariant definition of integrals of superfunctions defined on supermanifolds leading to an extension to infinite dimensions. An application is given to a construction of supersymmetric Fock spaces.