Three-dimensional topological field theory and symplectic algebraic geometry I

Abstract

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 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 we describe a deformation of the B-model and the associated category of branes by forms of arbitrary even degree.

76 pages, AMS-latex. v2: references, acknowledgments, and a discussion of grading ambiguities have been added

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• Truncated affine Rozansky--Witten models as extended defect TQFTs
• BRST-Invariant Deformations of Geometric Structures in Sigma Models
• Formal Global Perturbative Quantization of the Rozansky-Witten Model in the BV-BFV Formalism
• Extended TQFTs and Algebraic Geometry
• A Physical Origin for Singular Support Conditions in Geometric Langlands Theory