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

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_μ^I, ψ_μ^I, χ^{I J}), where I and J are the adjoint indices of arbitrary gauge group. The χ^{I J} is a Stueckelberg field for consistency. The system has local nilpotent fermionic symmetry with the algebra \{N_α^I, N_β^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.