Three-dimensional topological field theory anda categorification of the derived category of coherent sheaves
The Rozansky-Witten model is a 3d topological sigma-model whose target space X is a complex symplectic manifold. I will describe the 2-category structure on the set of its boundary conditions and show that it is a categorification of the derived category of coherent sheaves on X. In the special case when X is a cotangent bundle to a complex manifold Y, this 2-category is closely related to the 2-category of derived categorical sheaves over Y introduced by Toen and Vezzosi. I will also explain a surprising connection between a categorification of deformation quantization and complex symplectic geometry.