Mirror symmetry, Langlands duality, and the Hitchin system

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 mirror partners in one sense will also be mirror partners in the other. This paper concerns a class of...

Mirror Symmetry, Hitchin’s Equations, and Langlands Duality

Geometric Langlands duality can be understood from statements of mirror symmetry that can be formulated in purely topological terms for an oriented two-manifold C. But understanding these statements is extremely difficult without picking a complex structure on C and using Hitchin’s equations. We sketch the essential statements both for the “unramified” case that C is a compact oriented two-mani...

Mirror symmetry, Langlands duality, and commuting elements of Lie groups

By normalizing a component of the space of commuting pairs of elements in a reductive Lie group G, and the corresponding space for the Langlands dual group, we construct pairs of hyperkähler orbifolds which satisfy the conditions to be mirror partners in the sense of Strominger-Yau-Zaslow. The same holds true for commuting quadruples in a compact Lie group. The Hodge numbers of the mirror partn...

2 00 6 Langlands duality for Hitchin systems

Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve

This is a survey of results and conjectures on mirror symmetry phenomena in the nonAbelian Hodge theory of a curve. We start with the conjecture of Hausel–Thaddeus which claims that certain Hodge numbers of moduli spaces of flat SL(n, C) and PGL(n, C)connections on a smooth projective algebraic curve agree. We then change our point of view in the non-Abelian Hodge theory of the curve, and conce...