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A note on the symplectic classification of almost-toric systems

Xiudi Tang

Abstract

Since simple semitoric systems were classified about fifteen years ago, and semitoric systems five years ago, we want to move a step forward to almost-toric systems. We give a classification of compact almost-toric systems in dimension four up to fiber-preserving symplectomorphisms, in terms of the base, Taylor series, and twisting indices, analogous to the five invariants for semitoric systems. For convenience, we specify an ordering of focus-focus values and a choice of two cut rays at each of them.

A note on the symplectic classification of almost-toric systems

Abstract

Since simple semitoric systems were classified about fifteen years ago, and semitoric systems five years ago, we want to move a step forward to almost-toric systems. We give a classification of compact almost-toric systems in dimension four up to fiber-preserving symplectomorphisms, in terms of the base, Taylor series, and twisting indices, analogous to the five invariants for semitoric systems. For convenience, we specify an ordering of focus-focus values and a choice of two cut rays at each of them.

Paper Structure

This paper contains 10 sections, 10 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Almost-toric systems
  3. The invariants
  4. Almost-toric bases
  5. The Taylor series invariants
  6. The classification
  7. Almost-toric closed disks
  8. The focus-focus labels
  9. The relative twisting index
  10. Classification in terms of base

Key Result

theorem 10

The base of any almost-toric system $(M, \omega, \mu)$ is an almost-toric closed disk, and any almost-toric closed disk $(B, \mathcal{A}, \mathcal{S})$ is the base of an almost-toric system.

Theorems & Definitions (28)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • definition 7
  • definition 8
  • definition 9
  • theorem 10
  • ...and 18 more