Floer Cohomology and Higher Mutations

TL;DR

This paper generalizes the concept of higher mutation for monotone Lagrangians, showing that Floer cohomology remains invariant under local mutations, which include Lagrangian surgeries, thus extending wall-crossing phenomena.

Contribution

It introduces local higher mutation for a broader class of Lagrangians and proves Floer cohomology invariance under these mutations, generalizing previous wall-crossing results.

Findings

Floer cohomology is invariant under local mutation.

Local higher mutation includes Lagrangian anti-surgery and surgery.

Results unify and extend previous invariance theorems.

Abstract

We extend the construction of higher mutation as introduced in Pascaleff-Tonkonog to local higher mutation, which is applicable to a larger class of monotone Lagrangians. In two-dimensional Lagrangians, local higher mutation is the same as performing a Lagrangian anti-surgery in the sense of Haug followed by a Lagrangian surgery. We prove that up to a change of local systems, the Lagrangian intersection Floer cohomology of a pair of Lagrangians is invariant under local mutation. This result generalizes the wall-crossing formula in Pascaleff-Tonkonog. For two-dimensional Lagrangians, this result agrees with the invariance result in Palmer-Woodward.