Three-dimensional topological field theory and symplectic algebraic geometry. I. (English) Zbl 1194.81224
Summary: We study boundary conditions and defects in a three-dimensional topological sigma-model with a complex symplectic target space \(X\) (the Rozansky-Witten model). We show that boundary conditions correspond to complex Lagrangian submanifolds in \(X\) equipped with complex fibrations. The set of boundary conditions has the structure of a 2-category; morphisms in this 2-category are interpreted physically as one-dimensional defect lines separating parts of the boundary with different boundary conditions. This 2-category is a categorification of the \({\mathbb Z}_2\)-graded derived category of \(X\); it is also related to categories of matrix factorizations and a categorification of deformation quantization (quantization of symmetric monoidal categories). In the Appendix B we describe a deformation of the B-model and the associated category of branes by forms of arbitrary even degree.
MSC:
| 81T45 | Topological field theories in quantum mechanics |
| 81T10 | Model quantum field theories |
| 81S10 | Geometry and quantization, symplectic methods |
| 53D55 | Deformation quantization, star products |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
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