Physics:Classical Gaudin model

Short description: Integrable classical system

In mathematical physics, the classical Gaudin model is a classical mechanical system which is a classical analogue to the quantum Gaudin model. The historical development differs from the usual development for physical systems in that the quantum system was defined and studied, by Michel Gaudin, earlier than the classical system. The classical Gaudin models are integrable.

Mathematical formulation

As for any classical system, the Gaudin model is specified by a Poisson manifold [math]\displaystyle{ M }[/math] referred to as the phase space , and a smooth function on the manifold called the Hamiltonian.

Phase space

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

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

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

Sites

Associated to each of these copies is a point in [math]\displaystyle{ \mathbb{C} }[/math], denoted [math]\displaystyle{ \lambda_1, \cdots, \lambda_N }[/math], and referred to as sites.

Hamiltonian

The definition of the Hamiltonian takes considerable set-up. First fixing a basis of the Lie algebra [math]\displaystyle{ \{I^a\} }[/math] with structure constants [math]\displaystyle{ f^{ab}_c }[/math], there are functions [math]\displaystyle{ X^a_{(r)} }[/math] with [math]\displaystyle{ r = 1, \cdots, N }[/math] on the phase space satisfying the Poisson bracket [math]\displaystyle{ \{X^a_{(r)}, X^b_{(s)}\} = \delta_{rs}f^{ab}_c X^c_{(r)}. }[/math]

These in turn are used to define [math]\displaystyle{ \mathfrak{g} }[/math]-valued functions [math]\displaystyle{ X^{(r)} = \kappa_{ab}I^a \otimes X^b_{(r)} }[/math] with implicit summation.

Next, these are used to define the Lax matrix which is also a [math]\displaystyle{ \mathfrak{g} }[/math] valued function on the phase space which in addition depends meromorphically on a spectral parameter [math]\displaystyle{ \lambda }[/math], [math]\displaystyle{ \mathcal{L}(\lambda) = \sum_{r = 1}^N \frac{X^{(r)}}{\lambda - \lambda_r} + \Omega, }[/math] and [math]\displaystyle{ \Omega }[/math] is a constant element in [math]\displaystyle{ \mathfrak{g} }[/math], in the sense that it Poisson commutes (has vanishing Poisson bracket) with all functions.

The Hamiltonian is then [math]\displaystyle{ \mathcal{H}(\lambda) = \frac{1}{2}\kappa(\mathcal{L}(\lambda), \mathcal{L}(\lambda)) }[/math] which is indeed a function on the phase space, which is additionally dependent on a spectral parameter.

Integrable field theories as classical Gaudin models

Certain integrable classical field theories can be formulated as classical affine Gaudin models. Such classical field theories include the principal chiral model, coset sigma models and affine Toda field theory.

Quantum Gaudin models

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.