Quantization of soliton systems and Langlands duality

Abstract

We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to affine Kac--Moody algebras. Our experience with the Gaudin models associated to finite-dimensional simple Lie algebras suggests that the common eigenvalues of the mutually commuting quantum Hamiltonians in a model associated to an affine algebra should be encoded by affine opers associated to the Langlands dual affine algebra. This leads us to some concrete predictions for the spectra of the quantum Hamiltonians of the soliton systems. In particular, for the KdV system the corresponding affine opers may be expressed as Schroedinger operators with spectral parameter, and our predictions in this case match those recently made by Bazhanov, Lukyanov and Zamolodchikov. This suggests that this and other recently found examples of the correspondence between quantum integrals of motion and differential operators may be viewed as special cases of the Langlands duality.

70 pages. Final version, to appear in Proceedings of the conference in honor of A. Tsuchiya (Nagoya, March 2007), published in the series Advanced Studies of Pure Mathematics

Cited by in corpus (7)

• The ODE/IM Correspondence
• Integrability of a family of quantum field theories related to sigma models
• Hofstadter's Butterfly and Langlands Duality
• Quartic Hamiltonians, and higher Hamiltonians at next-to-leading order, for the affine ${\mathfrak{sl}}_2$ Gaudin model
• Cyclotomic Gaudin models with irregular singularities
• Bukhvostov-Lipatov model and quantum-classical duality
• Vacuum energy of the Bukhvostov-Lipatov model