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# Communications in Number Theory and Physics

## Volume 4 (2010)

### Number 3

### Three-dimensional topological field theory and symplectic algebraic geometry II

Pages: 463 – 549

DOI: http://dx.doi.org/10.4310/CNTP.2010.v4.n3.a1

#### Authors

#### Abstract

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.