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Semitoric Families on Pentagon Spaces

Yichen Liu, Aerim Si

TL;DR

The paper constructs explicit semitoric transition families on pentagon (5-gon) spaces by interpolating two toric-type semitoric systems via a parameter $t$. A unique rank-zero transition point $P$ is shown to switch from elliptic-elliptic to focus-focus and back as $t$ crosses the interval $(t^-,t^+)$, with transition times given by a quadratic whose coefficients depend on the pentagon edge lengths. All other singularities are proven non-degenerate and of elliptic-elliptic or elliptic-regular type, ensuring the family is semitoric away from $t^-$ and $t^+$. The construction employs polygon-space reductions, local normal form analyses, and reduction-by-stages arguments to establish the transition and toric limits, providing new explicit semitoric examples on 4D manifolds and illustrating the effect of varying symplectic form on transition data.

Abstract

Semitoric systems are a special type of 4-dimensional integrable system where one of the functions is the moment map of a Hamiltonian $S^1$-action. These systems were classified by Pelayo and V{ũ} Ng\d{o}c, but there are not many explicit examples. Recently, Le Floch and Palmer introduced semitoric transition families in which a singular point transitions between elliptic-elliptic type and focus-focus type as the parameter varies. In this paper, we construct semitoric transition families on pentagon spaces by interpolating two semitoric systems of toric type. More specifically, we show that there exists a unique singular point that changes from elliptic-elliptic type to focus-focus type and back to elliptic-elliptic type as the parameter varies.

Semitoric Families on Pentagon Spaces

TL;DR

The paper constructs explicit semitoric transition families on pentagon (5-gon) spaces by interpolating two toric-type semitoric systems via a parameter . A unique rank-zero transition point is shown to switch from elliptic-elliptic to focus-focus and back as crosses the interval , with transition times given by a quadratic whose coefficients depend on the pentagon edge lengths. All other singularities are proven non-degenerate and of elliptic-elliptic or elliptic-regular type, ensuring the family is semitoric away from and . The construction employs polygon-space reductions, local normal form analyses, and reduction-by-stages arguments to establish the transition and toric limits, providing new explicit semitoric examples on 4D manifolds and illustrating the effect of varying symplectic form on transition data.

Abstract

Semitoric systems are a special type of 4-dimensional integrable system where one of the functions is the moment map of a Hamiltonian -action. These systems were classified by Pelayo and V{ũ} Ng\d{o}c, but there are not many explicit examples. Recently, Le Floch and Palmer introduced semitoric transition families in which a singular point transitions between elliptic-elliptic type and focus-focus type as the parameter varies. In this paper, we construct semitoric transition families on pentagon spaces by interpolating two semitoric systems of toric type. More specifically, we show that there exists a unique singular point that changes from elliptic-elliptic type to focus-focus type and back to elliptic-elliptic type as the parameter varies.

Paper Structure

This paper contains 10 sections, 15 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Polygon spaces
  3. Semitoric Systems
  4. Singularities of $4$-dimensional integrable systems
  5. Local normal forms of singularities of a semitoric system
  6. Rank zero points
  7. Rank one points
  8. Non-degeneracy of non-transition points
  9. The transition point $P$
  10. Proof of Theorem \ref{['thm:new']}

Key Result

Theorem 1.1

Let $r_1,\ldots,r_5$ be positive numbers such that Then the family $F_t:=(\ell_{12},t\ell_{34}^2 +(1-t)\ell_{45}^2)$ is a semitoric transition family on $\mathcal{P}(r_1,r_2,r_3,r_4,r_5)$, with transition point $P$ represented by the planar pentagon, whose five edges are given by and transition times Here $j = r_3+r_4-r_5$.

Figures (1)

  • Figure 1: In both figures, $J_{\min} = |r_1-r_2|, J_{\max} = r_1+r_2, j=r_3+r_4-r_5$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 19 more