Holomorphic disks and the disk potential for a fibered Lagrangian

TL;DR

This paper develops a method to lift holomorphic disks from the base to the total space in fibered symplectic manifolds, providing formulas for the disk potential and criteria for unobstructedness, with explicit computations.

Contribution

It introduces a strategy for lifting holomorphic disks in fibered Lagrangians and derives a formula for the disk potential in the product case, along with unobstructedness criteria.

Findings

Derived a formula for the leading order potential of fibered Lagrangians.

Established criteria for unobstructedness of the associated $A_ abla$ algebra.

Constructed explicit examples involving Floer-non-trivial tori in fiber bundles.

Abstract

We consider a fibered Lagrangian $L$ in a compact symplectic fibration with small monotone fibers, and develop a strategy for lifting $J$-holomorphic disks with Lagrangian boundary from the base to the total space. In case $L$ is a product, we use this machinery to give a formula for the leading order potential and formulate an unobstructedness criteria for the $A_\infty$ algebra. We provide some explicit computations, one of which involves finding an embedded 2n+k dimensional submanifold of Floer-non-trivial tori in a 2n+2k dimensional fiber bundle.