Garnier integrable system

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. It is a classical analogue to the quantum Gaudin model due to Michel Gaudin (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.

As a limit of the Schlesinger equations

The Schlesinger equations are a system of differential equations for $n+2$ matrix-valued functions $A_{i}:\mathbb {C} ^{n+2}\rightarrow \mathrm {Mat} (m,\mathbb {C} )$, given by

$${\frac {\partial A_{i}}{\partial \lambda _{j}}}={\frac {[A_{i},A_{j}]}{\lambda _{i}-\lambda _{j}}}\qquad \qquad j\neq i$$

$$\sum _{j}{\frac {\partial A_{i}}{\partial \lambda _{j}}}=0.$$

The 'autonomous limit' is given by replacing the $\lambda _{i}$ dependence in the denominator by constants $\alpha _{i}$ with $\alpha _{n+1}=0,\alpha _{n+2}=1$:

$${\frac {\partial A_{i}}{\partial \lambda _{j}}}={\frac {[A_{i},A_{j}]}{\alpha _{i}-\alpha _{j}}}\qquad \qquad j\neq i$$

$$\sum _{j}{\frac {\partial A_{i}}{\partial \lambda _{j}}}=0.$$

This is the Garnier system in the form originally derived by Garnier.

As the classical Gaudin model

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.

As for any classical system, the Gaudin model is specified by a Poisson manifold $M$ referred to as the phase space, and a smooth function on the manifold called the Hamiltonian.

Phase space

Let ${\mathfrak {g}}$ be a quadratic Lie algebra, that is, a Lie algebra with a non-degenerate invariant bilinear form $\kappa$. If ${\mathfrak {g}}$ is complex and simple, this can be taken to be the Killing form.

The dual, denoted ${\mathfrak {g}}^{*}$, can be made into a linear Poisson structure by the Kirillov–Kostant bracket.

The phase space $M$ of the classical Gaudin model is then the Cartesian product of $N$ copies of ${\mathfrak {g}}^{*}$ for $N$ a positive integer.

Sites

Associated to each of these copies is a point in $\mathbb {C}$, denoted $\lambda _{1},\cdots ,\lambda _{N}$, and referred to as sites.

Lax matrix

Fixing a basis of the Lie algebra $\{I^{a}\}$ with structure constants $f_{c}^{ab}$, there are functions $X_{(r)}^{a}$ with $r=1,\cdots ,N$ on the phase space satisfying the Poisson bracket

$$\{X_{(r)}^{a},X_{(s)}^{b}\}=\delta _{rs}f_{c}^{ab}X_{(r)}^{c}.$$

These in turn are used to define ${\mathfrak {g}}$-valued functions

$$X^{(r)}=\kappa _{ab}I^{a}\otimes X_{(r)}^{b}$$

with implicit summation.

Next, these are used to define the Lax matrix which is also a ${\mathfrak {g}}$ valued function on the phase space which in addition depends meromorphically on a spectral parameter $\lambda$,

$${\mathcal {L}}(\lambda )=\sum _{r=1}^{N}{\frac {X^{(r)}}{\lambda -\lambda _{r}}}+\Omega ,$$

and $\Omega$ is a constant element in ${\mathfrak {g}}$, in the sense that it Poisson commutes (has vanishing Poisson bracket) with all functions.

(Quadratic) Hamiltonian

The (quadratic) Hamiltonian is

$${\mathcal {H}}(\lambda )={\frac {1}{2}}\kappa ({\mathcal {L}}(\lambda ),{\mathcal {L}}(\lambda ))$$

which is indeed a function on the phase space, which is additionally dependent on a spectral parameter $\lambda$. This can be written as

$${\mathcal {H}}(\lambda )=\Delta _{\infty }+\sum _{r=1}^{N}\left({\frac {\Delta _{r}}{(\lambda -\lambda _{r})^{2}}}+{\frac {{\mathcal {H}}_{r}}{\lambda -\lambda _{r}}}\right),$$

with

$$\Delta _{r}={\frac {1}{2}}\kappa (X^{(r)},X^{(r)}),\Delta _{\infty }={\frac {1}{2}}\kappa (\Omega ,\Omega )$$

and

$${\mathcal {H}}_{r}=\sum _{s\neq r}{\frac {\kappa (X^{(r)},X^{(s)})}{\lambda _{r}-\lambda _{s}}}+\kappa (X^{(r)},\Omega ).$$

From the Poisson bracket relation

$$\{{\mathcal {H}}(\lambda ),{\mathcal {H}}(\mu )\}=0,\forall \lambda ,\mu \in \mathbb {C} ,$$

by varying $\lambda$ and $\mu$ it must be true that the ${\mathcal {H}}_{r}$'s, the $\Delta _{r}$'s and $\Delta _{\infty }$ are all in involution. It can be shown that the $\Delta _{r}$'s and $\Delta _{\infty }$ Poisson commute with all functions on the phase space, but the ${\mathcal {H}}_{r}$'s do not in general. These are the conserved charges in involution for the purposes of Arnol'd Liouville integrability.

Lax equation

One can show

$$\{{\mathcal {H}}_{r},{\mathcal {L}}(\lambda )\}=\left[{\frac {X^{(r)}}{\lambda -\lambda _{r}}},{\mathcal {L}}(\lambda )\right],$$

so the Lax matrix satisfies the Lax equation when time evolution is given by any of the Hamiltonians ${\mathcal {H}}_{r}$, as well as any linear combination of them.

Higher Hamiltonians

The quadratic Casimir gives corresponds to a quadratic Weyl invariant polynomial for the Lie algebra ${\mathfrak {g}}$, but in fact many more commuting conserved charges can be generated using ${\mathfrak {g}}$-invariant polynomials. These invariant polynomials can be found using the Harish-Chandra isomorphism in the case ${\mathfrak {g}}$ is complex, simple and finite.

Integrable field theories as classical Gaudin models

Certain integrable classical field theories can be formulated as classical affine Gaudin models, where ${\mathfrak {g}}$ is an affine Lie algebra. Such classical field theories include the principal chiral model, coset sigma models and affine Toda field theory. 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.

Quantum Gaudin models

Main article: Gaudin model

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.