Higher index focus-focus singularities in the Jayne-Cummings-Gaudin model : symplectic invariants and monodromy

TL;DR

This paper investigates the symplectic structure of the Jaynes-Cummings-Gaudin model with an odd number of spins, identifying focus-focus singularities, constructing normal flows, and computing symplectic invariants and monodromy.

Contribution

It provides a detailed analysis of higher index focus-focus singularities, including explicit calculations of symplectic invariants and monodromy in the model.

Findings

Identification of maximal Williamson type focus-focus singularities

Construction of linearized normal flows near singularities

Calculation of leading terms of symplectic invariants and monodromy

Abstract

We study the symplectic geometry of the Jaynes-Cummings-Gaudin model with $n=2m-1$ spins. We show that there are focus-focus singularities of maximal Williamson type $(0,0,m)$. We construct the linearized normal flows in the vicinity of such a point and show that soliton type solutions extend them globally on the critical torus. This allows us to compute the leading term in the Taylor expansion of the symplectic invariants and the monodromy associated to this singularity.