Lagrangian and covariant field equations for supersymmetric Yang–Mills theory in 12D

Abstract

We present a lagrangian formulation for recently-proposed supersymmetric Yang-Mills theory in twelve dimensions. The field content of our multiplet has an additional auxiliary vector field in the adjoint representation. The usual Yang-Mills field strength is modified by a Chern-Simons form containing this auxiliary vector field. This formulation needs no constraint imposed on the component field from outside, and a constraint on the Yang-Mills field is generated as the field equation of the auxiliary vector field. The invariance check of the action is also performed without any reference to constraints by hand. Even though the total lagrangian takes a simple form, it has several highly non-trivial extra symmetries. We couple this twelve-dimensional supersymmetric Yang-Mills background to Green-Schwarz superstring, and confirm fermionic kappa-invariance. As another improvement of this theory, we present a set of fully Lorentz-covariant and supercovariant field equations with no use of null-vectors. This system has an additional scalar field, whose gradient plays a role of the null-vector. This system exhibits spontaneous breaking of the original Lorentz symmetry SO(10,2) for twelve-dimensions down to SO(9,1) for ten-dimensions.