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
Authors
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