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Bohr-Sommerfeld profile surgeries and Disk Potentials

Soham Chanda

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 $\bP^n$.

Bohr-Sommerfeld profile surgeries and Disk Potentials

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 .
Paper Structure (32 sections, 34 theorems, 137 equations, 11 figures)

This paper contains 32 sections, 34 theorems, 137 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Applications
  3. Lagrangian Cone Surgeries
  4. Exotic Lagrangian Circle-Bundles
  5. Geometric realization of mutations of polynomials
  6. Exotic tori in $\mathbb{P}^n$
  7. Beginnings of a new surgery
  8. Acknowledgements
  9. Construction of the Bohr-Sommerfeld Profile Surgery
  10. Preliminaries
  11. Bohr-Sommerfeld Profile Lagrangian Handles
  12. Bohr-Sommerfeld Profile Surgery
  13. Examples and relations to other surgeries
  14. Polterovich Surgery
  15. Local Higher Mutation
  16. ...and 17 more sections

Key Result

Proposition 1

Let $L$ be a monotone Lagrangian and $C$ be a Lagrangian cone modeled on a Bohr-Sommerfeld Legendrian $\Lambda$ in the standard contact sphere $S^{2n+1}$. If $C$ intersects $L$ only at its boundary $\partial C$, and the intersection is clean, then $L$ admits a Bohr-Sommerfeld-Profile surgery modeled

Figures (11)

  • Figure 1: Newton polytope of the disk potential of the lifted Vianna tori $\overline{T}_{(1,1,2)}$, and its BSP surgery $\widetilde{T}_{(1,1,2)}$. The red points correspond to the exponents of monomials.
  • Figure 2: Two primitive homotopy classes of embedded paths in $\mathbb{C}^*$ with fixed end-points.
  • Figure 3: $k$-th root of curve $\gamma$
  • Figure 4: Admissible curve $\gamma$
  • Figure 5: $\Lambda_2 = \Lambda \cup e^{i\frac{\pi }{k}}.\Lambda$
  • ...and 6 more figures

Theorems & Definitions (122)

  • Remark 1.1
  • Remark 1.2
  • Proposition
  • Proposition
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.4: Parallel transport along base
  • Lemma 2.5
  • proof
  • ...and 112 more