Mirror symmetry, Langlands duality, and the Hitchin system
Mirror symmetry, Langlands duality, and the Hitchin system
复制标题
镜像对称、朗兰兹对偶性和希钦系统
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
M. Thaddeus
中科院分区：
文献类型：
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作者：
Tamás Hausel;M. Thaddeus
镜像对称的主要数学方法是Batyrev-Borisov和Strominger-Yau-Zaslow（SYZ）的方法。第一个是明确的，可以计算，但与物理动机没有明显的关系;第二个是相反的。此外，从某种意义上说，镜像伙伴在另一种意义上也会是镜像伙伴，这一点远不明显。本文讨论一类满足SYZ要求的例子，它们的Hodge数也是相等的。这为支持SYZ提供了重要证据。此外，这些例子本身就很有趣：它们是光滑曲线上的平坦SLr-联络空间。镜像是朗兰兹对偶群PGLR的对应空间。因此，这些例子为李群的对偶理论和更广泛的几何朗兰兹纲领架起了一座桥梁。
Among the major mathematical approaches to mirror symmetry are those of Batyrev-Borisov and Strominger-Yau-Zaslow (SYZ). The first is explicit and amenable to computation but is not clearly related to the physical motivation; the second is the opposite. Furthermore, it is far from obvious that mirror partners in one sense will also be mirror partners in the other. This paper concerns a class of examples that can be shown to satisfy the requirements of SYZ, but whose Hodge numbers are also equal. This provides significant evidence in support of SYZ. Moreover, the examples are of great interest in their own right: they are spaces of flat SLr-connections on a smooth curve. The mirror is the corresponding space for the Langlands dual group PGLr. These examples therefore throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program.