#
On a ℤ_{2}^{n}-Graded Version of Supersymmetry

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. ${\mathbb{Z}}_{2}^{n}$-Manifolds and Their Basic Geometry

**Definition**

**1**

**.**A locally ${\mathbb{Z}}_{2}^{n}$- ringed space , $n\in \mathbb{N}\backslash \left\{0\right\}$, is a pair $X:=\left(\right|X|,{\mathcal{O}}_{X})$ where $\left|X\right|$ is a second-countable Hausdorff topological space and a ${\mathcal{O}}_{X}$ is a sheaf of ${\mathbb{Z}}_{2}^{n}$-graded, ${\mathbb{Z}}_{2}^{n}$-commutative associative unital $\mathbb{R}$-algebras, such that the stalks ${\mathcal{O}}_{X,p}$, $p\in \left|X\right|$ are local rings.

**Definition**

**2**

**.**A (smooth) ${\mathbb{Z}}_{2}^{n}$-manifold of dimension $p|\mathbf{q}$ is a locally ${\mathbb{Z}}_{2}^{n}$-ringed space $M:=\left(\left|M\right|,{\mathcal{O}}_{M}\right)$, which is locally isomorphic to the ${\mathbb{Z}}_{2}^{n}$-ringed space ${\mathbb{R}}^{p|\mathbf{q}}:=\left({\mathbb{R}}^{p},{C}_{{\mathbb{R}}^{p}}^{\infty}\left[\left[\xi \right]\right]\right)$. Here ${C}_{{\mathbb{R}}^{p}}^{\infty}$ is the structure sheaf on the Euclidean space ${\mathbb{R}}^{p}$. Local sections of ${\mathbb{R}}^{p|\mathbf{q}}$ are formal power series in the ${\mathbb{Z}}_{2}^{n}$-graded variables ξ with smooth coefficients, i.e.,

**Example**

**1**(The local model)

**.**

**Proposition**

**1**

**.**Let M be a ${\mathbb{Z}}_{2}^{n}$-manifold. Then ${\mathcal{O}}_{M}$ is $\mathcal{J}$-adically Hausdorff complete as a sheaf of ${\mathbb{Z}}_{2}^{n}$-commutative rings, i.e., the morphism

**Remark**

**1.**

**Example**

**2**(${\mathbb{Z}}_{2}^{n}$-graded Cartesian spaces)

**.**

**Example**

**3**(Manifolds and supermanifolds)

**.**

**Example**

**4**(Double vector bundles)

**.**

**Remark**

**2.**

#### 2.2. A Toy ${\mathbb{Z}}_{2}^{2}$-Superspace

#### 2.3. Majorana Spinors

**Remark**

**3.**

**Remark**

**4.**

#### 2.4. ${\mathbb{Z}}_{2}^{n}$-Graded Majorana Spinors

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

## 3. ${\mathbb{Z}}_{2}^{n}$-Extended Supersymmetry

#### 3.1. A ${\mathbb{Z}}_{2}^{n}$-Extended Poincaré Algebra

**Remark**

**5.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

**Remark**

**6.**

#### 3.2. Direct Consequences of the Algebra

- Positivity of energy: a direct computation shows $E:={P}_{0}=\frac{1}{2}{\sum}_{\alpha}[{Q}_{I}^{\alpha},{Q}_{I}^{\alpha}]$ (no sum over I). Then, passing to the representation as Hermitian operators allow us to write $\widehat{E}={\sum}_{\alpha}{\left({Q}_{I}^{\alpha}\right)}^{\u2020}{Q}_{I}^{\alpha}$ and thus for any state$$\langle \psi |\widehat{E}|\psi \rangle =\langle \psi |\sum _{\alpha}{\left({Q}_{I}^{\alpha}\right)}^{\u2020}{Q}_{I}^{\alpha}|\psi \rangle =\left|\right|\sum _{\alpha}{Q}_{I}^{\alpha}{|\psi \rangle ||}^{2}\ge 0.$$
- Irreducible representations of supersymmetry carry the same value of ${P}^{\mu}{P}_{\mu}=-{m}^{2}$: this follows from $[P,Q]=0$, which implies that ${P}^{2}$ is a Casimir.
- The spin of each state in a multiplet varies in steps of 1/2: this follows from $[Q,J]\sim Q$.

#### 3.3. ${\mathbb{Z}}_{2}^{n}$-Minkowski Space-Time

**Definition**

**3.**

**Remark**

**7.**

#### 3.4. Invariant Differential Forms

#### 3.5. Superfields

**Remark**

**8.**

**Remark**

**9.**

#### 3.6. ${\mathbb{Z}}_{2}^{2}$-Minkowski Space-Time

**Remark**

**10.**

## 4. Closing Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

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Bruce, A.J.
On a ℤ_{2}^{n}-Graded Version of Supersymmetry

. *Symmetry* **2019**, *11*, 116.
https://doi.org/10.3390/sym11010116

**AMA Style**

Bruce AJ.
On a ℤ_{2}^{n}-Graded Version of Supersymmetry

. *Symmetry*. 2019; 11(1):116.
https://doi.org/10.3390/sym11010116

**Chicago/Turabian Style**

Bruce, Andrew James.
2019. "On a ℤ_{2}^{n}-Graded Version of Supersymmetry

" *Symmetry* 11, no. 1: 116.
https://doi.org/10.3390/sym11010116