On Contact Manifolds

A new construction of symplectic manifolds

then M is said to be a differentiable manifold with almost complex structure. (Tensor fields of the form given above may exist only for some manifolds with even dimension.) We shall call φ the fundamental collineation of the almost complex structure. The set of differentiable manifolds with almost complex structure is wider than the set of complex manifolds. Every differentiable manifold with almost complex structure φ admits a poistive definite Riemannian metric g such that

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On differentiable manifolds with certain structures which are closely related to almost contact structure i

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.

methods

Twisted supersymmetric 5D Yang-Mills theory and contact geometry

/pdf/the-theory-of-quasi-sasakian-structures-3jxtjvnjby.pdf

The theory of quasi-Sasakian structures

We introduce a multiplication ⋆ (we call it a join) on the space of all compact Sasakian-Einstein orbifolds and show that has the structure of a commutative associative topological monoid. The set of all compact regular Sasakian–Einstein manifolds is then a submonoid. The set of smooth manifolds in is not closed under this multiplication; however, the join of two Sasakian–Einstein manifolds is smooth under some additional conditions which we specify. We use this construction to obtain many old and new examples of Sasakain–Einstein manifolds. In particular, in every odd dimension greater that five we obtain spaces with arbitrary second Betti number.

On Contact Manifolds

Citations

A new construction of symplectic manifolds

On differentiable manifolds with certain structures which are closely related to almost contact structure i

Twisted supersymmetric 5D Yang-Mills theory and contact geometry

The theory of quasi-Sasakian structures

On sasakian–einstein geometry

Related Papers (5)

On differentiable manifolds with certain structures which are closely related to almost contact structure i

Riemannian Geometry of Contact and Symplectic Manifolds

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