Reconstruction of N=1 supersymmetry from topological symmetry

The scalar and vector topological Yang-Mills symmetries on Calabi-Yau manifolds geometrically define consistent sectors of Yang-Mills D=4,6, N=1 supersymmetry, which fully determine the supersymmetric actions up to twist. For a CY2 manifold, both N=1, D=4 Wess and Zumino and super-Yang-Mills theory can be reconstructed in this way. A superpotential can be introduced for the matter sector, as well as the Fayet-Iliopoulos mechanism. For a CY3 manifold, the N=1, D=6 Yang-Mills theory is also obtained, in a twisted form. Putting these results together with those already known for the D=4,8, N=2 cases, we conclude that all Yang-Mills supersymmetries with 4, 8 and 16 generators are determined from topological symmetry on special manifolds.