New results on N = 4 super-Yang-Mills theory

The N=4 super-Yang-Mills theory is covariantly determined by two scalar and one vector BRST topological symmetry operators. This determines an off-shell closed sector of N=4 super-Yang-Mills, with 6 generators, which is big enough to fully determine the theory, in a Lorentz-covariant way. This reduced algebra derives from horizontality conditions in four dimensions. The horizontality conditions only depend on the geometry of the Yang-Mills fields. They also descend from a genuine horizontality condition in eight dimensions. When the four-dimensional manifold is hyper-Kähler, one can perform a twist operation that defines the N=4 supersymmetry and a SL(2,ℍ) intern symmetry (the "Euclidean version" of the SU(4) R-symmetry in Minkowski space). These results extend in a covariant way the light-cone property that the N=4 super-Yang-Mills theory is actually determined by only 8 independent generators, instead of the 16 generators that occur in the physical representation of the super-Poincaré algebra. The topological construction disentangles the off-shell closed sector of the (twisted) maximally supersymmetric theory from the sector that closes only modulo equations of motion. It allows one to escape the question of auxiliary fields in N=4 super-Yang-Mills theory.