Supersymmetric self-dual Yang–Mills theories from local nilpotent fermionic symmetry

Abstract We present a system of a self-dual vector-spinor and a self-dual Yang–Mills (YM) field with local nilpotent fermionic symmetry (but not supersymmetry) in D = 2 + 2 dimensions that embeds self-dual supersymmetric YM theory as a special set of exact solutions. Our system has local nilpotent fermionic symmetry generator N α I satisfying the algebra { N α I , N β J } = 0 with the adjoint index I of an arbitrary gauge group. Our original field content in D = 2 + 2 is ( A μ I , ψ μ I , χ I ) , where A μ I is the usual YM gauge field, ψ μ I is a Majorana–Weyl vector-spinor gauging N α I , while χ I is a Majorana–Weyl spinor compensator field needed for consistency. This system embeds self-dual supersymmetric YM system with the field content ( A μ I , λ − I ) in D = 2 + 2 . As other examples, we consider similar systems in D = 7 + 0 and D = 8 + 0 embedding respectively N = 1 / 8 + 7 / 8 and N = ( 1 / 8 , 1 ) supersymmetric YM theories with generalized self-dualities, such as F μ ν I = ( 1 / 2 ) f μ ν ρ σ F ρ σ I with a generalized octonionic structure constant f μ ν ρ σ . This result strongly suggests that our local nilpotent fermionic symmetry is more fundamental than the supersymmetric self-dual Yang–Mills systems that are supposed to generate all supersymmetric integrable models in D 4 .