Bohr-Sommerfeld profile surgeries and Disk Potentials

TL;DR

This paper introduces BSP surgery, a new method for modifying Legendrian fillings that preserves certain properties, and applies it to construct exotic monotone Lagrangian tori using wall-crossing formulas.

Contribution

It develops BSP surgery, a novel operation on Legendrians, and demonstrates its utility in producing exotic Lagrangians via disk-potential wall-crossing formulas.

Findings

BSP surgery can preserve monotonicity of Lagrangians.

Wall-crossing formulas describe disk-potential changes under surgery.

Construction of exotic monotone Lagrangian tori in projective space.

Abstract

We construct a new surgery type operation by switching between two exact fillings of Legendrians which we call a BSP surgery. In certain cases, this surgery can preserve monotonicity of Lagrangians. We prove a wall-crossing type formula for the change of the disk-potential under surgery with Bohr-Sommerfeld profiles. As an application, we show that Biran's circle-bundle lifts admit a Bohr-Sommerfeld type surgery. We use the wall-crossing theorem about disk-potentials to construct exotic monotone Lagrangian tori in $\bP^n$.