Advanced Studies in Pure Mathematics

Quantization of soliton systems and Langlands duality

Boris Feigin and Edward Frenkel

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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 $\widehat{\mathfrak{g}}$ should be encoded by affine opers associated to the Langlands dual affine algebra ${}^L\widehat{\mathfrak{g}}$. 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 Schrödinger 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.

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Exploring New Structures and Natural Constructions in Mathematical Physics, K. Hasegawa, T. Hayashi, S. Hosono and Y. Yamada, eds. (Tokyo: Mathematical Society of Japan, 2011), 185-274

Received: 11 November 2007
Revised: 6 April 2008
First available in Project Euclid: 24 November 2018

Permanent link to this document euclid.aspm/1543085348

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Feigin, Boris; Frenkel, Edward. Quantization of soliton systems and Langlands duality. Exploring New Structures and Natural Constructions in Mathematical Physics, 185--274, Mathematical Society of Japan, Tokyo, Japan, 2011. doi:10.2969/aspm/06110185.

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