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

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.

23 pages