Three-dimensional topological field theory and symplectic algebraic geometry II

Motivated by the path-integral analysis [6] of boundary conditions in a three-dimensional topological sigma model, we suggest a definition of the two-category ¨L(X) associated with a holomorphic symplectic manifold X and study its properties. The simplest objects of ¨L(X) are holomorphic lagrangian submanifolds Y ⊂ X. We pay special attention to the case when X is the total space of the cotangent bundle of a complex manifold U or a deformation thereof. In the latter case, the endomorphism category of the zero section is a monoidal category which is an A_∞ deformation of the two-periodic derived category of U.