#
Superfields, Nilpotent Superfields and Superschemes^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Superspaces and Scalar Superfields

#### 2.1. Some Mathematical Definitions

**Definition**

**1.**

**Example**

**1.**

**Definition**

**2.**

**Remark**

**1.**

**Definition**

**3.**

**Definition**

**4.**

#### 2.2. Superfields

**Scalar superfields on ${A}^{1|1}$**

**Spacetime is a point**

**Definition**

**5.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 3. The Even Rules Principle

**Remark**

**2.**

**Definition**

**6.**

**Theorem**

**1.**

**Proof.**

- ${v}_{1},{v}_{2}$ even. Then, we may take ${b}_{1}={b}_{2}=1$ (in an arbitrary algebra $\mathcal{B}$) and:$$f({v}_{1}\otimes {v}_{2}):={f}_{\mathcal{B}}({v}_{1}\otimes {v}_{2})\phantom{\rule{0.166667em}{0ex}}.$$
- ${v}_{1}$ even, ${v}_{2}$ odd. Then, we take for example $\mathcal{B}=\mathsf{\Lambda}\left[\xi \right]$, ${b}_{1}=1$, and ${b}_{2}=\xi $. The equality:$${f}_{\mathcal{B}}({v}_{1}\otimes \xi {v}_{2})=\xi f({v}_{1}\otimes {v}_{2})$$
- ${v}_{1}$ odd, ${v}_{2}$ odd. It is enough to consider $\mathcal{B}=\mathsf{\Lambda}[{\xi}^{1},{\xi}^{2}]$, and the equality:$${f}_{\mathcal{B}}({\xi}^{1}{v}_{1}\otimes {\xi}^{2}{v}_{2})=-{\xi}^{1}{\xi}^{2}f({v}_{1}\otimes {v}_{2})$$

**Example**

**5.**

**Remark**

**3**

## 4. How to Deal with Algebraic Constraints

#### 4.1. Schemes and Superschemes

**Example**

**6.**

**Definition**

**7.**

**Definition**

**8.**

#### 4.2. Constraint ${\mathsf{\Phi}}^{2}=0$

#### 4.3. General Constraint $f\left(\mathsf{\Phi}\right)=0$

**Remark**

**4.**

**Remark**

**5.**

#### 4.4. Cubic Constraint with Two Superfields

#### 4.5. Cubic Constraint with an Arbitrary Number of Superfields

## 5. A Non-Algebraic Constraint

## 6. Observables and Nilpotent Variables

**Example**

**7.**

- 1.
- We consider the ring of polynomials in one variable $F=\mathbb{C}\left[x\right]$. The prime ideals of F are of the form ${\mathfrak{p}}_{a}=(x-a)$, $a\in \mathbb{C}$ or the ideal $\left(0\right)$. It is not difficult to see that the residue field at ${\mathfrak{p}}_{a}$ is $\kappa \left({\mathfrak{p}}_{a}\right)\cong \mathbb{C}$ and $\kappa \left(\right)open="("\; close=")">\left(0\right)$ is the field of rational functions.
- 2.
- If the ground field is $\mathbb{R}$, we have that every irreducible polynomial in $\mathbb{R}\left[x\right]$ generates a prime ideal. We still have the maximal ideals as ${\mathfrak{p}}_{a}=(x-a)$ that give all the points of $\mathbb{R}$. At them, the residue field is $\kappa \left({\mathfrak{p}}_{a}\right)\cong \mathbb{R}$. However, for example, the irreducible polynomial ${x}^{2}+1$ also generates a prime ideal. It is not difficult to realize that the elements of the residue field are of the form $a+xb$, with $a,b\in \mathbb{R}$ and x such that ${x}^{2}+1=0$, so $\kappa \left(({x}^{2}+1)\right)\cong \mathbb{C}$.

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Some Basic Definitions

**Definition**

**A1.**

**Definition**

**A2.**

- 1.
- ${\mathrm{res}}_{U,U}=\mathrm{id}$.
- 2.
- ${\mathrm{res}}_{V,U}\circ {\mathrm{res}}_{W,V}={\mathrm{res}}_{W,U}$ for all $U\subset V\subset W\subset \left|X\right|$.

**Example**

**A1.**

- 1.
- Continuous, differentiable, real analytic, or complex analytic functions on a topological space are all sheaves of algebras.
- 2.
- Sections of a vector bundle over a topological space are a sheaf of modules over some algebra of functions.
- 3.
- Constant functions over a topological space are, generically, only a presheaf. If the space is connected, then the sheaf condition is satisfied. Furthermore, on a not necessarily connected space, one can define the sheaf of locally constant functions, that is functions that are constant on an open neighborhood of each point.

**Definition**

**A3.**

**Definition**

**A4.**

## Appendix B. Notation for Spinors

## Appendix C. Supersymmetry Transformations

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Lledó, M.A.
Superfields, Nilpotent Superfields and Superschemes. *Symmetry* **2020**, *12*, 1024.
https://doi.org/10.3390/sym12061024

**AMA Style**

Lledó MA.
Superfields, Nilpotent Superfields and Superschemes. *Symmetry*. 2020; 12(6):1024.
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**Chicago/Turabian Style**

Lledó, María Antonia.
2020. "Superfields, Nilpotent Superfields and Superschemes" *Symmetry* 12, no. 6: 1024.
https://doi.org/10.3390/sym12061024