In mathematical physics, the Garnier integrable system, also known as the classical Gaudin model is a classical mechanical system discovered by René Garnier in 1919 by taking the 'Painlevé simplification' or 'autonomous limit' of the Schlesinger equations.
[1][2] It is a classical analogue to the quantum Gaudin model due to Michel Gaudin[3] (similarly, the Schlesinger equations are a classical analogue to the Knizhnik–Zamolodchikov equations).
The classical Gaudin models are integrable.
They are also a specific case of Hitchin integrable systems, when the algebraic curve that the theory is defined on is the Riemann sphere and the system is tamely ramified.
dependence in the denominator by constants
There is a formulation of the Garnier system as a classical mechanical system, the classical Gaudin model, which quantizes to the quantum Gaudin model and whose equations of motion are equivalent to the Garnier system.
This section describes this formulation.
[4] As for any classical system, the Gaudin model is specified by a Poisson manifold
referred to as the phase space, and a smooth function on the manifold called the Hamiltonian.
be a quadratic Lie algebra, that is, a Lie algebra with a non-degenerate invariant bilinear form
is complex and simple, this can be taken to be the Killing form.
, can be made into a linear Poisson structure by the Kirillov–Kostant bracket.
of the classical Gaudin model is then the Cartesian product of
Fixing a basis of the Lie algebra
on the phase space satisfying the Poisson bracket
Next, these are used to define the Lax matrix which is also a
valued function on the phase space which in addition depends meromorphically on a spectral parameter
which is indeed a function on the phase space, which is additionally dependent on a spectral parameter
Poisson commute with all functions on the phase space, but the
These are the conserved charges in involution for the purposes of Arnol'd Liouville integrability.
so the Lax matrix satisfies the Lax equation when time evolution is given by any of the Hamiltonians
The quadratic Casimir gives corresponds to a quadratic Weyl invariant polynomial for the Lie algebra
, but in fact many more commuting conserved charges can be generated using
These invariant polynomials can be found using the Harish-Chandra isomorphism in the case
is complex, simple and finite.
Certain integrable classical field theories can be formulated as classical affine Gaudin models, where
Such classical field theories include the principal chiral model, coset sigma models and affine Toda field theory.
[5] As such, affine Gaudin models can be seen as a 'master theory' for integrable systems, but is most naturally formulated in the Hamiltonian formalism, unlike other master theories like four-dimensional Chern–Simons theory or anti-self-dual Yang–Mills.
A great deal is known about the integrable structure of quantum Gaudin models.
In particular, Feigin, Frenkel and Reshetikhin studied them using the theory of vertex operator algebras, showing the relation of Gaudin models to topics in mathematics including the Knizhnik–Zamolodchikov equations and the geometric Langlands correspondence.