The twisting index in semitoric systems

TL;DR

This work clarifies the twisting index in semitoric systems by presenting multiple equivalent formulations—as a difference of momentum maps, actions, Taylor series, and cycles—and by computing it for a concrete two-focus-focus-point family. It derives the first explicit twisting-index values for a system with more than one focus-focus point and provides lower-order terms of the Taylor series invariant, aided by elliptic-integral techniques and Birkhoff normal forms. A key finding is a necessary refinement to the twisting-index group action, introducing an extra term, which has implications for the relationship between polygon data and local models. The results bolster the semitoric classification program and set the stage for extending similar invariants to broader classes of integrable systems, including almost-toric and hypersemitoric settings, and offer insights into quantum inverse problems through invariant recovery.”

Abstract

Semitoric integrable systems were symplectically classified by Pelayo and Vu Ngoc in 2009-2011 in terms of five invariants. Four of these invariants were already well-understood prior to the classification, but the fifth invariant, the so-called twisting index invariant, came as a surprise. Intuitively, the twisting index encodes how the structure in a neighborhood of a focus-focus fiber compares to the large-scale structure of the semitoric system and it was originally defined by comparing certain momentum maps. In the first half of the present paper, we produce several new formulations of the twisting index which give rise to dynamical, geometric, and topological interpretations. More specifically, we describe it in terms of differences of action variables, Taylor series, and homology cycles. In the second half of the paper, we compute the twisting index invariant of a specific family of systems with two focus-focus singular points (the so-called generalized coupled angular momenta), which is the first time that the twisting index has been computed for a system with more than one focus-focus point. Moreover, we also compute the terms of the Taylor series invariant up to second order. Since the other invariants of this family were already computed, this becomes the third family of semitoric systems for which all invariants are known, after the coupled spin oscillators and the coupled angular momenta.

Figures (16)

• Figure 1: Four representatives of the polygon invariant of the system with twisting indices labeled. To prove Theorem \ref{['thm:twist-intro']}, we computed that the twisting index labels for the lower right polygon are $(\kappa_1^s,\kappa_2^s)=(0,0)$, and the others are determined by Equation \ref{['eq:twisact']}.
• Figure 2: The red curve is along the Hamiltonian flow of the ${\mathbb S}^1$-action generated by $L$ and the orange curve is along the Hamiltonian flow of $\Phi_2$. The value of $\tau_1(z)$ is the time spent along the flow of the red curve and $\tau_2(z)$ represents the time spent along the flow of the orange curve.
• Figure 3: Two representatives of the same semitoric polygon related by changing the cut direction at the leftmost focus-focus point.
• Figure 4: A review of the relevant maps for constructing the invariants. We typically use coordinates $(l,h)$ for the image of $F$ in ${\mathbb R}^2$, and coordinates $(l,j)$ for the image of $\Phi_r$ in ${\mathbb R}^2$, or $z = l + ij$ when viewing ${\mathbb R}^2$ as ${\mathbb C}$.
• Figure 5: A lift to the universal cover (displayed on the right) of the concatenated flows of $\mathcal{X}_L$ and $\mathcal{X}_{\Phi_2}$ (displayed on the left). The point $\tilde{x}$ is the corresponding lift of $x$. In the set-up here, the lifted flow of $\mathcal{X}_L$ is moving vertically. For comparison, on the universal cover, we also show $\gamma_\Xi$ (green), which is a straight line.

Theorems & Definitions (54)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Liouville Arnold Mineur theorem Ar
• Remark 2.4
• Definition 2.5