Three-dimensional topological field theory and symplectic algebraic geometry II

TL;DR

This paper develops a 2-category framework for holomorphic symplectic manifolds inspired by topological sigma-model boundary conditions, revealing new algebraic structures related to derived categories and deformations.

Contribution

It defines a novel 2-category associated with holomorphic symplectic manifolds and explores its properties, especially for cotangent bundles and their deformations.

Findings

The 2-category's objects are holomorphic lagrangian submanifolds.

Endomorphism categories for cotangent bundles are A-infinity deformations.

The framework connects topological field theory with symplectic algebraic geometry.

Abstract

Motivated by the path integral analysis of boundary conditions in a 3-dimensional topological sigma-model, we suggest a definition of the 2-category associated with a holomorphic symplectic manifold X and study its properties. The simplest objects of this 2-category are holomorphic lagrangian submanifolds of 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-infinity deformation of the 2-periodic derived category of U.