Three-dimensional topological field theory and symplectic algebraic geometry I

TL;DR

This paper explores boundary conditions and defects in a 3D topological sigma-model with a complex symplectic target, revealing a rich categorical structure related to derived categories and deformation quantization.

Contribution

It establishes a correspondence between boundary conditions and complex Lagrangian submanifolds, and introduces a 2-category framework linking physical defects to advanced mathematical categories.

Findings

Boundary conditions correspond to complex Lagrangian submanifolds with fibrations.

The boundary condition set forms a 2-category categorifying the derived category of X.

Connections to matrix factorizations and deformation quantization are established.

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.