$\mathbb{Z}_2\times \mathbb{Z}_2$-graded supersymmetry: $2$-d sigma models

TL;DR

This paper introduces a new type of supersymmetry based on a $Z_2 imes Z_2$ grading, constructs classical sigma models with this symmetry, and exemplifies it with a double-graded sine-Gordon model.

Contribution

It develops a $Z_2 imes Z_2$-graded supersymmetry framework and constructs sigma models that realize this novel symmetry structure.

Findings

Supercharges close with a commutator instead of an anticommutator.

Constructed classical sigma models with $Z_2 imes Z_2$-graded supersymmetry.

Presented a double-graded supersymmetric sine-Gordon model.

Abstract

We propose a natural $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalisation of $d=2$, $\mathcal{N}=(1,1)$ supersymmetry and construct a $\mathbb{Z}_2^2$-space realisation thereof. Due to the grading, the supercharges close with respect to, in the classical language, a commutator rather than an anticommutator. This is then used to build classical (linear and non-linear) sigma models that exhibit this novel supersymmetry via mimicking standard superspace methods. The fields in our models are bosons, right-handed and left-handed Majorana-Weyl spinors, and exotic bosons. The bosons commute with all the fields, the spinors belong to different sectors that cross commute rather than anticommute, while the exotic boson anticommute with the spinors. As a particular example of one of the models, we present a `double-graded' version of supersymmetric sine-Gordon theory.