Mirror symmetry, Langlands duality, and the Hitchin system
Abstract
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.
Keywords
Cite
@article{arxiv.math/0205236,
title = {Mirror symmetry, Langlands duality, and the Hitchin system},
author = {Tamas Hausel and Michael Thaddeus},
journal= {arXiv preprint arXiv:math/0205236},
year = {2009}
}
Comments
31 pages, LaTeX with packages amsfonts, latexsym, [dvips]graphicx, [dvips]color, one embedded postscript figure