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New Perspectives on Integrable Hamiltonian Systems via the Algebraic Geometry of Twisted Hitchin Moduli Spaces: A Case Study on the Calogero-Franc¸oise Integrable System

Date

2023-04-13

Journal Title

Journal ISSN

Volume Title

Publisher

ORCID

0009-0002-9198-0165

Type

Thesis

Degree Level

Masters

Abstract

Integrable systems are dynamical systems that exhibit very special properties. These systems are exactly solvable, have deep connections with algebraic geometry, and give rise to a maximal set of conserved quantities. Integrable systems have proven to be essential as they arise naturally in various branches of mathematics and physics such as differential and algebraic geometry, partial differential equations, statistical mechanics, quantum field theories, string theory and even more. One of the unique features of integrable systems is that all known integrable systems seem to be inherently related in some sense. This special feature has inspired many mathematicians to attempt and find a single origin of all known integrable systems. One of the most prominent approaches towards this unification process is through realizing different integrable systems as symmetry reductions of the self-dual-Yang-Mills (SDYM) equations. In fact, most known integrable systems (at least in lower dimensions) fit in this paradigm and this thesis is a further step towards this unification. Four integrable systems are in the central attention of this thesis. Namely, these are Euler’s equations for the motion of a rigid body, Nahm’s equations, the Hitchin system, and the the Calogero-Fran¸coise integrable system. Euler’s equations are the most classical example of an integrable system. Moreover, Nahm’s equations are obtained as a dimensional reduction of the SDYM equations to one dimension. Furthermore, the Hitchin system is an algebraically completely integrable system that arises as the space of solutions to Hitchin’s equations. Hitchin’s equations are a coupled system of non-linear partial differential equations that arise as a dimensional reduction of the SDYM equations to two dimensions. Finally, the Calogero-Fran¸coise (CF) integrable system is a finite-dimensional Hamiltonian system that arises as a generalization of the Camassa Holm (CH) dynamics. In this thesis, we show that the dynamics of Euler’s equations and the CF system can be perceived by realizing both systems as twisted Hitchin systems. More specifically, we obtain an explicit solution to Euler’s equations by transforming them to Nahm’s equations and then studying the evolution of the Higgs field of Nahm’s equations. The solution method is different from the classical ones since we used a different formulation than the one usually presented in the literature. More specifically, we formulated the problem on the Lie algebra su(2) rather than formulating it on so(3) as usually done. Furthermore, we study the dynamics of the Calogero-Fran¸coise (CF) integrable system while focusing on the special case of peakon anti-peakon interactions (d = 2). We show explicitly by embedding the CF system into a (twisted) Hitchin system that the CF dynamics is completely governed by the evolution of the corresponding Higgs field. In particular, we show that different singularities in the CF system correspond to very special Higgs fields in the underlying Hitchin system. Furthermore, we show that a periodization (compactification) of the CF dynamics corresponds to a compactification of the underlying Hitchin system. This result is then a direct manifestation of the correspondence between the CF dynamics and the dynamics of the associated Higgs field in the underlying Hitchin system.

Description

Keywords

The Hitchin System, Nahm's equations, Euler's equations, Calogero-Fran¸coise integrable system

Citation

Degree

Master of Science (M.Sc.)

Department

Mathematics and Statistics

Program

Mathematics

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DOI

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