## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- 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

### Quantization of soliton systems and Langlands duality

Boris Feigin and Edward Frenkel

#### 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 $\widehat{\mathfrak{g}}$ 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.

#### Article information

**Dates**

Received: 11 November 2007

Revised: 6 April 2008

First available in Project Euclid:
24 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543085348

**Digital Object Identifier**

doi:10.2969/aspm/06110185

**Mathematical Reviews number (MathSciNet)**

MR2867148

**Zentralblatt MATH identifier**

1245.81062

#### Citation

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. https://projecteuclid.org/euclid.aspm/1543085348