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Abstract
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.
Highlights
In a recent paper [1], it was shown that the scalar and vectorial topological Yang–Mills symmetries can be directly constructed, in four and eight dimensions, leading one to a geometrical definition of a closed off-shell twisted sector of Yang–Mills supersymmetric theories, with 8 and 16 generators, respectively
Putting together the results of this paper and of [1], we reach an interesting conclusion for the super Yang–Mills symmetries with 4, 8 and 16 generators, which can be represented as N = 1 theories in 4 and 6 dimensions and N = 2 theories in 4 and 8 dimensions
We found a reverse construction, that clarifies the structure of supersymmetry
Summary
In a recent paper [1], it was shown that the scalar and vectorial topological Yang–Mills symmetries can be directly constructed, in four and eight dimensions, leading one to a geometrical definition of a closed off-shell twisted sector of Yang–Mills supersymmetric theories, with 8 and 16 generators, respectively. To get the N = 1 supersymmetry algebra in a twisted form, the number of independent topological transformations must be reduced by half This leads one to build a TQFT on a Kahler manifold, such that the gauge field can be splitted in holomorphic and antiholomorphic components, A1. The topological construction has the great advantage of purely geometrically determining a closed sector of the supersymmetric algebra, which is large enough to completely determine the theory It determines the Faddeev–Popov ghosts for the supersymmetry algebra, in a way that is relevant for a control of the covariant gauge-fixing of the Yang–Mills symmetry
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