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

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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

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Abstract

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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

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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

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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

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  1. ^ Drinfeld, Vladimir; Sokolov, Vladimir (1985). "Lie algebras and equations of Korteweg-de Vries type". Journal of Soviet Mathematics. 30 (2): 1975–2036. doi:10.1007/BF02105860. S2CID 125066120. Retrieved 10 October 2022.
  2. ^ Beilinson, Alexander; Drinfeld, Vladimir (2005). "Opers". arXiv:math/0501398.
  3. ^ Feigin, Boris; Frenkel, Edward; Reshetikhin, Nikolai (1994). "Gaudin Model, Bethe Ansatz and Critical Level". Commun. Math. Phys. 166 (1): 27–62. arXiv:hep-th/9402022. Bibcode:1994CMaPh.166...27F. doi:10.1007/BF02099300. S2CID 17099900.