Rozansky-Witten theory is a 3d topological sigma model which is used to study topological invariants of 3-manifolds. In what follows, X will denote its target space.
In a question posted here - https://www.physicsoverflow.org/21310/is-there-a-mirror-of-the-rozansky-witten-theory, it is written that "One can show that if we take X=T∗Y for Y a complex manifold (T∗Y is naturally holomorphic symplectic), then the dimensional reduction over a circle of the 3d Rozansky-Witten TQFT of target X is the B-model 2d TQFT of target Y."
(Here, T∗Y refers to the total space of the cotangent bundle of Y. The B-model is a 2d topological sigma model first studied by Witten.)
My question is how does one show this explicitly? Why is it that the target space of the 3d theory is T∗Y but the target space of its dimensional reduction to 2d is Y? It seems that some scalar fields which parametrize the T∗Y target space should be set to zero during the dimensional reduction, but I cannot see why.
References would be appreciated.
This post imported from StackExchange Physics at 2017-06-10 16:47 (UTC), posted by SE-user Mtheorist