Self-Dual Yang-Mills and Vector-Spinor Fields, Nilpotent Fermionic Symmetry, and Supersymmetric Integrable Systems

TL;DR

This paper introduces a self-dual Yang-Mills and vector-spinor field system with nilpotent fermionic symmetry in 2+2 dimensions, which generates supersymmetric integrable systems in lower dimensions, revealing a new approach to supersymmetric models.

Contribution

It presents a novel self-dual field system with nilpotent fermionic symmetry that produces supersymmetric integrable equations via dimensional reduction.

Findings

Generates supersymmetric KP equations in 2+1 dimensions.

Produces supersymmetric KdV equations in 1+1 dimensions.

Shows higher-dimensional self-dual systems can generate lower-dimensional supersymmetric models.

Abstract

We present a system of a self-dual Yang-Mills field and a self-dual vector-spinor field with nilpotent fermionic symmetry (but not supersymmetry) in 2+2 dimensions, that generates supersymmetric integrable systems in lower dimensions. Our field content is (A_\mu{}^I, \psi_\mu{}^I, \chi^{I J}), where I and J are the adjoint indices of arbitrary gauge group. The \chi^{I J} is a Stueckelberg field for consistency. The system has local nilpotent fermionic symmetry with the algebra \{N_\alpha{}^I, N_\beta{}^J \} = 0. This system generates supersymmetric Kadomtsev-Petviashvili equations in D=2+1, and supersymmetric Korteweg-de Vries equations in D=1+1 after appropriate dimensional reductions. We also show that a similar self-dual system in seven dimensions generates self-dual system in four dimensions. Based on our results we conjecture that lower-dimensional supersymmetric integral models can be generated by non-supersymmetric self-dual systems in higher dimensions only with nilpotent fermionic symmetries.