Mirror symmetry, Langlands duality, and the Hitchin system

Journal article

We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in two different senses. First, they satisfy the requirements laid down by Strominger-Yau-Zaslow (SYZ), in a suitably general sense involving a B-field or flat unitary gerbe. To show this, we use their hyperkahler structures and Hitchin's integrable systems. Second, their Hodge numbers, again in a suitably general sense, are equal. These spaces provide significant evidence in support of SYZ. Moreover, they throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program.

Authors: Hausel, T; Thaddeus, M

• Publication status: Published
• Journal: INVENTIONES MATHEMATICAE
• Volume: 153
• Issue: 1
• Pages: 197-229
• Publication date: 2002-05-23
• DOI: 10.1007/s00222-003-0286-7
• EISSN: 1432-1297
• ISSN: 0020-9910
• Language: English
• Keywords: 14H60 (Primary); 14D21, 14H40, 14H70, 32S35 (Secondary); math.MP; math.DG; math-ph; hep-th; math.AG