Mirror symmetry, Langlands duality, and the Hitchin system

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 S...

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المؤلفون الرئيسيون: Hausel, T, Thaddeus, M
التنسيق: Journal article
اللغة:English
منشور في: 2002
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author Hausel, T
Thaddeus, M
author_facet Hausel, T
Thaddeus, M
author_sort Hausel, T
collection OXFORD
description 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.
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spelling oxford-uuid:59e7848a-ffcb-41b7-ad5d-f42d5a927d352022-03-26T17:12:24ZMirror symmetry, Langlands duality, and the Hitchin systemJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:59e7848a-ffcb-41b7-ad5d-f42d5a927d35EnglishSymplectic Elements at Oxford2002Hausel, TThaddeus, MWe 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.
spellingShingle Hausel, T
Thaddeus, M
Mirror symmetry, Langlands duality, and the Hitchin system
title Mirror symmetry, Langlands duality, and the Hitchin system
title_full Mirror symmetry, Langlands duality, and the Hitchin system
title_fullStr Mirror symmetry, Langlands duality, and the Hitchin system
title_full_unstemmed Mirror symmetry, Langlands duality, and the Hitchin system
title_short Mirror symmetry, Langlands duality, and the Hitchin system
title_sort mirror symmetry langlands duality and the hitchin system
work_keys_str_mv AT hauselt mirrorsymmetrylanglandsdualityandthehitchinsystem
AT thaddeusm mirrorsymmetrylanglandsdualityandthehitchinsystem