Floer cohomology and higher mutations

We extend the construction of higher mutation as introduced in [42] to local higher mutation, which is applicable to a larger class of monotone Lagrangians. For two-dimensional Lagrangians, local higher mutation is the same as performing a Lagrangian anti-surgery in the sense of [24] 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 [42]. For two-dimensional Lagrangians, this result agrees with the invariance result in [43].