Mirror symmetry, Langlands duality, and the Hitchin system

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 mirro...

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Bibliographic Details
Published in:Inventiones mathematicae 2003-07, Vol.153 (1), p.197-229
Main Authors: Hausel, Tam s, Thaddeus, Michael
Format: Article
Language:English
Subjects:
Lie groups
Studies
Symmetry
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Summary: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 SL^sub r^-connections on a smooth curve. The mirror is the corresponding space for the Langlands dual group PGL^sub r^. These examples therefore throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program. [PUBLICATION ABSTRACT]
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-003-0286-7