Topological symmetry of forms, N=1N=1 supersymmetry and S-duality on special manifolds

We study the quantization of a holomorphic 2-form coupled to a Yang–Mills field on special manifolds in various dimensions, and we show that it yields twisted supersymmetric theories. The construction determines ATQFTs (Almost Topological Quantum Field Theories), that is, theories with observables that are invariant under changes of metrics belonging to restricted classes. For Kähler manifolds in four dimensions, our topological model is related to N=1N=1 Super Yang–Mills theory. Extended supersymmetries are recovered by considering the coupling with chiral multiplets. We also analyse Calabi–Yau manifolds in six and eight dimensions, and seven-dimensional G2G2 manifolds of the kind recently discussed by Hitchin. We argue that the 2-form field could play an interesting rôle in the study of the conjectured S-duality in topological strings. We finally show that in the case of real forms the partition function of our topological model in six dimensions is related to the square of that of the holomorphic Chern–Simons theory, and we discuss the uplift to seven dimensions and its relation to the topological M theory.