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摘要翻译：
在与仿射KAC-Moody代数相关的Gaudin模型的一般形式框架下，我们考虑了经典孤子可积系统的量子化问题，如KdV族。我们对与有限维简单李代数相关的Gaudin模型的经验表明，在与仿射代数相关的模型中，相互交换的量子哈密顿量的公共特征值应该由与Langlands对偶仿射代数相关的仿射子编码。这使我们对孤子系统的量子哈密顿谱有了一些具体的预测。特别地，对于KdV系统，相应的仿射算子可用带谱参数的Schroedinger算子表示，我们在这种情况下的预测与Bazhanov、Lukyanov和Zamolodchikov最近的预测相吻合。这表明，运动的量子积分和微分算符之间的对应关系的这个例子和其他最近发现的例子可以看作是朗兰对偶的特例。
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英文标题：
《Quantization of soliton systems and Langlands duality》
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作者：
Boris Feigin and Edward Frenkel
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最新提交年份：
2009
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分类信息：
一级分类：Mathematics 数学
二级分类：Quantum Algebra 量子代数
分类描述：Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
量子群，skein理论，运算代数和图解代数，量子场论
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一级分类：Physics 物理学
二级分类：High Energy Physics - Theory 高能物理-理论
分类描述：Formal aspects of quantum field theory. String theory, supersymmetry and supergravity.
量子场论的形式方面。弦理论，超对称性和超引力。
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
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.
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PDF链接：
https://arxiv.org/pdf/0705.2486
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