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 NαI satisfying the algebra {NαI,NβJ}=0 with the adjoint index I of an arbitrary gauge group. Our original field content in D=2+2 is (AμI,ψμI,χI), where AμI is the usual YM gauge field, ψμI is a Majorana–Weyl vector-spinor gauging NαI, while χI is a Majorana–Weyl spinor compensator field needed for consistency. This system embeds self-dual supersymmetric YM system with the field content (AμI,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 FμνI=(1/2)fμνρσFρσI with a generalized octonionic structure constant fμνρσ. 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.