Gaudin model

When ${\displaystyle {\mathfrak {g}}}$ has rank greater than 1, the commuting algebra spanned by the Gaudin Hamiltonians and the identity can be expanded to a larger commuting algebra, known as the Gaudin algebra. Similarly to the Harish-Chandra isomorphism, these commuting elements have associated degrees, and in particular the Gaudin Hamiltonians form the degree 2 part of the algebra. For ${\displaystyle {\mathfrak {g}}={\mathfrak {sl}}_{2}}$, the Gaudin Hamiltonians and the identity span the Gaudin algebra. There is another commuting algebra which is 'universal', underlying the Gaudin algebra for any choice of sites and weights, called the Feigin–Frenkel center. See here.

Then eigenvectors of the Gaudin algebra define linear functionals on the algebra. If ${\displaystyle X}$ is an element of the Gaudin algebra ${\displaystyle {\mathfrak {G}}}$, and ${\displaystyle v}$ an eigenvector of the Gaudin algebra, one obtains a linear functional ${\displaystyle \chi _{v}:{\mathfrak {G}}\rightarrow \mathbb {C} }$ given by

$${\displaystyle Xv=\chi _{v}(X)v.}$$

The linear functional ${\displaystyle \chi _{v}}$ is called a character of the Gaudin algebra. The spectral problem, that is, determining eigenvalues and simultaneous eigenvectors of the Gaudin algebra, then becomes a matter of determining characters on the Gaudin algebra.

For ${\displaystyle {\mathfrak {g}}={\mathfrak {sl}}_{2}}$, let ${\displaystyle \{E,H,F\}}$ be the standard basis. For any ${\displaystyle X\in {\mathfrak {g}}}$, one can define the operator-valued meromorphic function

$${\displaystyle X(z)=\sum _{i=1}^{N}{\frac {X^{(i)}}{z-z_{i}}}.}$$

Its residue at ${\displaystyle z=z_{i}}$ is ${\displaystyle X^{(i)}}$, while ${\displaystyle \lim _{z\rightarrow \infty }zX(z)=\sum _{i=1}^{N}X^{(i)}=:X^{(\infty )},}$ the 'full' tensor representation.

The ${\displaystyle X(z)}$ and ${\displaystyle X^{(\infty )}}$ satisfy several useful properties

• ${\displaystyle [X(z),Y^{(\infty )}]=[X,Y](z)}$
• ${\displaystyle S(u)={\frac {1}{2}}\sum _{a}I_{a}(z)I^{a}(z)}$
• ${\displaystyle [H_{i},X^{(\infty )}]=0}$

but the ${\displaystyle X(z)}$ do not form a representation: ${\displaystyle [X(z),Y(z)]=-[X,Y]'(z)}$. The third property is useful as it allows us to also diagonalize with respect to ${\displaystyle H^{\infty }}$, for which a diagonal (but degenerate) basis is known.

For an ${\displaystyle {\mathfrak {sl}}_{2}}$ Gaudin model specified by sites ${\displaystyle z_{1},\cdots ,z_{N}\in \mathbb {C} }$ and weights ${\displaystyle \lambda _{1},\cdots ,\lambda _{N}\in \mathbb {N} }$, define the vacuum vector to be the tensor product of the highest weight states from each representation: ${\displaystyle v_{0}:=v_{\lambda _{1}}\otimes \cdots \otimes v_{\lambda _{N}}}$.

A Bethe vector (of spin deviation ${\displaystyle m}$) is a vector of the form

$${\displaystyle F(w_{1})\cdots F(w_{m})v_{0}}$$

for ${\displaystyle w_{i}\in \mathbb {C} }$. Guessing eigenvectors of the form of Bethe vectors is the Bethe ansatz. It can be shown that a Bethe vector is an eigenvector of the Gaudin Hamiltonians if the set of equations

$${\displaystyle \sum _{i=1}^{N}{\frac {\lambda _{i}}{w_{k}-z_{i}}}-2\sum _{j\neq k}{\frac {1}{w_{k}-w_{j}}}=0}$$

holds for each ${\displaystyle k}$ between 1 and ${\displaystyle m}$. These are the Bethe ansatz equations for spin deviation ${\displaystyle m}$. For ${\displaystyle m=1}$, this reduces to

$${\displaystyle {\boldsymbol {\lambda }}(w):=\sum _{i=1}^{N}{\frac {\lambda _{i}}{w-z_{i}}}=0.}$$

Completeness

In theory, the Bethe ansatz equations can be solved to give the eigenvectors and eigenvalues of the Gaudin Hamiltonian. In practice, if the equations are to completely solve the spectral problem, one must also check

• The number of solutions predicted by the Bethe equations
• The multiplicity of solutions

If, for a specific configuration of sites and weights, the Bethe ansatz generates all eigenvectors, then it is said to be complete for that configuration of Gaudin model. It is possible to construct examples of Gaudin models which are incomplete. One problem in the theory of Gaudin models is then to determine when a given configuration is complete or not, or at least characterize the 'space of models' for which the Bethe ansatz is complete.

Analogues of the Bethe ansatz equation can be derived for Lie algebras of higher rank.[2] However, these are much more difficult to derive and solve than the ${\displaystyle {\mathfrak {sl}}_{2}}$ case. Furthermore, for ${\displaystyle {\mathfrak {g}}}$ of rank greater than 1, that is, all others besides ${\displaystyle {\mathfrak {sl}}_{2}}$, there are higher Gaudin Hamiltonians, for which it is unknown how to generalize the Bethe ansatz.

There is an ODE/IM isomorphism between the Gaudin algebra (or the universal Feigin–Frenkel center), which are the 'integrals of motion' for the theory, and opers, which are ordinary differential operators, in this case on ${\displaystyle \mathbb {P} ^{1}}$.