Symplectic bifurcation theory for integrable systems

TL;DR

This paper establishes a symplectic bifurcation theory for four-dimensional integrable systems, linking the structure of their bifurcation diagrams to the topology of their fibers, and providing a framework for quantum spectral analysis.

Contribution

It introduces conditions under which the fibers are connected and characterizes the bifurcation diagram as a planar region with specific singularities, advancing the understanding of integrable systems.

Findings

Fibers are connected if no hyperbolic singularities and no vertical tangencies.

Bifurcation diagram is a planar region bounded by continuous graphs.

Contains a countable set of interior rank zero singularities.

Abstract

This paper develops a symplectic bifurcation theory for integrable systems in dimension four. We prove that if an integrable system has no hyperbolic singularities and its bifurcation diagram has no vertical tangencies, then the fibers of the induced singular Lagrangian fibration are connected. The image of this singular Lagrangian fibration is, up to smooth deformations, a planar region bounded by the graphs of two continuous functions. The bifurcation diagram consists of the boundary points in this image plus a countable collection of rank zero singularities, which are contained in the interior of the image. Because it recently has become clear to the mathematics and mathematical physics communities that the bifurcation diagram of an integrable system provides the best framework to study symplectic invariants, this paper provides a setting for studying quantization questions, and spectral theory of quantum integrable systems.