Principal connection
In
mathematics, an Oper is a
principal connection, or in more elementary terms a type of
differential operator. They were first defined and used by
Vladimir Drinfeld and Vladimir Sokolov
[1] to study how the
KdV equation and related integrable PDEs correspond to algebraic structures known as
Kac–Moody algebras. Their modern formulation is due to Drinfeld and
Alexander Beilinson.
[2]
History
Opers were first defined, although not named, in a 1981 Russian paper by Drinfeld and Sokolov on Equations of Korteweg–de Vries type, and simple Lie algebras. They were later generalized by Drinfeld and Beilinson in 1993, later published as an e-print in 2005.
Formulation
Abstract
Let be a
connected
reductive group over the
complex plane , with a distinguished
Borel subgroup . Set , so that is the
Cartan group.
Denote by and the corresponding
Lie algebras. There is an open -orbit consisting of vectors stabilized by the radical such that all of their negative simple-root components are non-zero.
Let be a smooth curve.
A G-oper on is a triple where is a principal -bundle, is a connection on and is a -
reduction of , such that the
one-form takes values in .
Example
Fix the
Riemann sphere. Working at the level of the algebras, fix , which can be identified with the space of traceless complex matrices. Since has only one (complex) dimension, a one-form has only one component, and so an -valued one form is locally described by a matrix of functions
where
are allowed to be
meromorphic functions.
Denote by the space of valued meromorphic functions together with an action by , meromorphic functions valued in the associated
Lie group . The action is by a formal
gauge transformation:
Then opers are defined in terms of a subspace of these connections. Denote by
the space of connections with . Denote by the subgroup of meromorphic functions valued in of the form
with meromorphic.
Then for it holds that . 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 have been used by
Boris Feigin,
Edward Frenkel and
Nicolai Reshetikhin to characterize the
spectrum of the
Gaudin model.
[3]
Specifically, for a -Gaudin model, and defining 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 opers.
References