Self-dual Yang-Mills and vector-spinor fields, nilpotent fermionic symmetry, and supersymmetric integrable systems

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 $(A_{\ensuremath{\mu}}{}^{I},\ensuremath{\psi}_{\ensuremath{\mu}}{}^{I},{\ensuremath{\chi}}^{IJ})$, where $I$ is the adjoint index of arbitrary gauge group. The ${\ensuremath{\chi}}^{IJ}$ is a Stueckelberg field for consistency. The system has local nilpotent fermionic symmetry with the algebra ${N_{\ensuremath{\alpha}}{}^{I},N_{\ensuremath{\beta}}{}^{J}}=0$. 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 a self-dual system in four dimensions. Based on our results we conjecture that lower-dimensional supersymmetric integral models can be generated by nonsupersymmetric self-dual systems in higher dimensions only with nilpotent fermionic symmetries.