Fukaya A_\infty-structures associated to Lefschetz fibrations. III

TL;DR

This paper explores the structure of Floer cohomology groups in the context of Lefschetz fibrations, introducing connections that lead to differential equations and linking symplectic cohomology with enumerative geometry.

Contribution

It extends the theory of Floer cohomology by incorporating connections over Novikov fields in the setting of Lefschetz fibrations, revealing new relations with enumerative geometry.

Findings

Floer cohomology groups can be equipped with connections under certain conditions.

Differential equations for Floer classes are derived using these connections.

A novel relation between symplectic cohomology and enumerative geometry is established.

Abstract

Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschetz fibrations, and obtain a relation with enumerative geometry.