# Dimensional reduction of Rozansky-Witten theory

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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

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