Quantization of soliton systems and Langlands duality
Quantization of soliton systems and Langlands duality
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孤子系统的量化和朗兰兹对偶性
DOI：
10.2969/aspm/06110185
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
E. Frenkel
中科院分区：
文献类型：
--
作者：
B. Feigin;E. Frenkel
我们考虑的问题的量子化的经典孤子可积系统，如KdV的层次，在框架内的一般形式主义的Gaudin模型与仿射Kac-穆迪代数。我们对有限维简单李代数的Gaudin模型的经验表明，在与仿射代数相关联的模型中，相互交换的量子哈密顿算子的公共本征值应该由与Langlands对偶仿射代数相关联的仿射算子编码。这使我们对孤子系统的量子哈密顿量的谱有了一些具体的预言。特别地，对于KdV系统，相应的仿射算子可以表示为具有谱参数的Schroedinger算子，并且在这种情况下，我们的预测与Bazhanov，Lukyanov和Zamolodchikov最近所做的预测相匹配。这表明，这个和其他最近发现的量子运动积分和微分算子之间的对应关系的例子可以被视为朗兰兹对偶的特殊情况。
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.