Oper (mathematics)

Formulation

Abstract

Let ${\displaystyle G}$ be a connected reductive group over the complex plane ${\displaystyle \mathbb{C}}$, with a distinguished Borel subgroup ${\displaystyle B=B_G\subset G}$. Set ${\displaystyle N=[B,B]}$, so that ${\displaystyle H=B/N}$ is the Cartan group.

Denote by ${\displaystyle \mathfrak{n}<\mathfrak{b}<\mathfrak{g}}$ and ${\displaystyle \mathfrak{h}=\mathfrak{b}/\mathfrak{n}}$ the corresponding Lie algebras. There is an open ${\displaystyle B}$-orbit ${\displaystyle \mathbf{O}}$ consisting of vectors stabilized by the radical ${\displaystyle N\subset B}$ such that all of their negative simple-root components are non-zero.

Let ${\displaystyle X}$ be a smooth curve.

A G-oper on ${\displaystyle X}$ is a triple ${\displaystyle (\mathfrak{F},\mathrm{\nabla },{\mathfrak{F}}_B)}$ where ${\displaystyle \mathfrak{F}}$ is a principal ${\displaystyle G}$-bundle, ${\displaystyle \mathrm{\nabla }}$ is a connection on ${\displaystyle \mathfrak{F}}$ and ${\displaystyle {\mathfrak{F}}_B}$ is a ${\displaystyle B}$-reduction of ${\displaystyle \mathfrak{F}}$, such that the one-form ${\displaystyle \mathrm{\nabla }/{\mathfrak{F}}_B}$ takes values in ${\displaystyle {\mathbf{O}}_{{\mathfrak{F}}_B}}$.

Example

Fix ${\displaystyle X={\mathbb{P}}^1={\mathbb{C}\mathbb{P}}^1}$ the Riemann sphere. Working at the level of the algebras, fix ${\displaystyle \mathfrak{g}=\mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$, which can be identified with the space of traceless ${\displaystyle 2\times 2}$ complex matrices. Since ${\displaystyle {\mathbb{P}}^1}$ has only one (complex) dimension, a one-form has only one component, and so an ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$-valued one form is locally described by a matrix of functions $${\displaystyle A(z)=\left(\begin{array}{cc}a(z)& b(z)\\ c(z)& -a(z)\end{array}\right)}$$ where ${\displaystyle a,b,c}$ are allowed to be meromorphic functions.

Denote by ${\displaystyle {\text{Conn}}_{\mathfrak{s}\mathfrak{l}(2,\mathbb{C})}({\mathbb{P}}^1)}$ the space of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ valued meromorphic functions together with an action by ${\displaystyle g(z)}$, meromorphic functions valued in the associated Lie group ${\displaystyle G=SL(2,\mathbb{C})}$. The action is by a formal gauge transformation: $${\displaystyle g(z)\ast A(z)=g(z)A(z)g(z{)}^{-1}-g^{\prime }(z)g(z{)}^{-1}.}$$

Then opers are defined in terms of a subspace of these connections. Denote by ${\displaystyle {\text{op}}_{\mathfrak{s}\mathfrak{l}(2,\mathbb{C})}({\mathbb{P}}^1)}$ the space of connections with ${\displaystyle c(z)\equiv 1}$. Denote by ${\displaystyle N}$ the subgroup of meromorphic functions valued in ${\displaystyle SL(2,\mathbb{C})}$ of the form ${\displaystyle \left(\begin{array}{cc}1& f(z)\\ 0& 1\end{array}\right)}$ with ${\displaystyle f(z)}$ meromorphic.

Then for ${\displaystyle g(z)\in N,A(z)\in {\text{op}}_{\mathfrak{s}\mathfrak{l}(2,\mathbb{C})}({\mathbb{P}}^1),}$ it holds that ${\displaystyle g(z)\ast A(z)\in {\text{op}}_{\mathfrak{s}\mathfrak{l}(2,\mathbb{C})}({\mathbb{P}}^1)}$. It therefore defines an action. The orbits of this action concretely characterize opers. However, generally this description only holds locally and not necessarily globally.

Gaudin model

Opers on ${\displaystyle {\mathbb{P}}^1}$ have been used by Boris Feigin, Edward Frenkel and Nicolai Reshetikhin to characterize the spectrum of the Gaudin model.

Specifically, for a ${\displaystyle \mathfrak{g}}$-Gaudin model, and defining ${\displaystyle {}^L\mathfrak{g}}$ as the Langlands dual algebra, there is a bijection between the spectrum of the Gaudin algebra generated by operators defined in the Gaudin model and an algebraic variety of ${\displaystyle {}^L\mathfrak{g}}$ opers.