Riemannian Geometry of Contact and Symplectic Manifolds

Abstract: We construct supersymmetric field theories on Riemannian three-manifolds $ \mathcal{M} $ , focusing on $ \mathcal{N} $ = 2 theories with a U(1)R symmetry. Our approach is based on the rigid limit of new minimal supergravity in three dimensions, which couples to the flat-space supermultiplet containing the R-current and the energy-momentum tensor. The field theory on $ \mathcal{M} $ possesses a single supercharge if and only if $ \mathcal{M} $ admits an almost contact metric structure that satisfies a certain integrability condition. This may lead to global restrictions on $ \mathcal{M} $ , even though we can always construct one supercharge on any given patch. We also analyze the conditions for the presence of additional supercharges. In particular, two supercharges of opposite R-charge exist on every Seifert manifold. We present general supersymmetric Lagrangians on $ \mathcal{M} $ and discuss their flat-space limit, which can be analyzed using the R-current supermultiplet. As an application, we show how the flat-space two-point function of the energy-momentum tensor in $ \mathcal{N} $ = 2 superconformal theories can be calculated using localization on a squashed sphere.

Abstract: We extend the localization calculation of the 3D Chern-Simons partition func- tion over Seifert manifolds to an analogous calculation in five dimensions. We construct a twisted version of N = 1 supersymmetric Yang-Mills theory defined on a circle bundle over a four dimensional symplectic manifold. The notion of contact geometry plays a crucial role in the construction and we suggest a generalization of the instanton equations to five- dimensional contact manifolds. Our main result is a calculation of the full perturbative partition function on S 5 for the twisted supersymmetric Yang-Mills theory with different Chern-Simons couplings. The final answer is given in terms of a matrix model. Our construction admits generalizations to higher dimensional contact manifolds. This work is inspired by the work of Baulieu-Losev-Nekrasov from the mid 90’s, and in a way it is covariantization of their ideas for a contact manifold.

Abstract: The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $${\mathcal{D}}$$ -homothetic transformation is determined as a special gauge transformation. The η-Einstein manifold are defined, it is proved that their scalar curvature is a constant, and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with $${\mathcal{D}}$$ -homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.

Abstract: We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface Σ). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.

Abstract: Generalized Sasakian-space-forms are introduced and studied. Many examples of these manifolds are presented, by using some different geometric techniques such as Riemannian submersions, warped products or conformal and related transformations. New results on generalized complex-space-forms are also obtained.