# Superfields, nilpotent superfields and superschemes

## TL;DR

This paper develops a functorial mathematical framework for superfields, revising their definition to include algebraic constraints that lead to superschemes, which are generally not regular supermanifolds.

## Contribution

It introduces a functorial interpretation of superfields and explores algebraic constraints resulting in superschemes beyond standard supermanifolds.

## Key findings

- Superschemes can be non-regular, unlike standard supermanifolds.
- Revised definition of superfields accommodates algebraic constraints.
- Functorial formalism clarifies properties of superfields in mathematical terms.

## Abstract

We interpret superfields in a functorial formalism that explains the properties that are assumed for them in the physical applications. The starting point of this research was the need to understand in a sound mathematical framework some algebraic constraints imposed on them, but it lead us to revise the very definition of superfield. The constraints that we investigate in the present work give rise to superschemes that, generically, are not regular, that is, they do not define a standard supermanifold.