On Generalized Statistics and Stability in $\mathbb{Z}_2^2$-Graded Supersymmetric Yang-Mills Theory

TL;DR

This paper constructs a classical $ ext{Z}_2^2$-graded supersymmetric Yang-Mills theory, demonstrating its stability and positive Hamiltonian, thus showing generalized statistics can be realized in stable interacting models.

Contribution

It provides the first explicit classical $ ext{Z}_2^2$-graded supersymmetric gauge theory with stable dynamics and positive energy, extending the understanding of graded symmetries in quantum field theory.

Findings

All kinetic terms have correct signs, indicating no classical ghost instabilities.

The Hamiltonian's positivity follows from the $ ext{Z}_2^2$-graded supersymmetry algebra.

The theory realizes $ ext{Z}_2^2$-graded generalized statistics at the classical level.

Abstract

In the standard formulation of relativistic quantum field theory, a $\mathbb{Z}_2$-graded structure is assumed to realize locality and the boson-fermion dichotomy. While $\mathbb{Z}_2^n$-graded extensions are known to be allowed at the level of symmetry, their realization in interacting quantum field theories remains unclear. In this paper, we construct a classical minimal $\mathbb{Z}_2^2$-graded supersymmetric Yang-Mills theory. We derive the invariant action and show that all kinetic terms have the correct sign, indicating the absence of classical ghost-like instabilities. Moreover, the positivity of the Hamiltonian follows from the $\mathbb{Z}_2^2$-graded supersymmetry algebra. As a result, we show that $\mathbb{Z}_2^2$-graded generalized statistics can be realized at the classical level in a stable interacting supersymmetric gauge theory.