Mirror Symmetry, Hitchin's Equations, And Langlands Duality

Details Mirror Symmetry, Hitchin's Equations, And Langlands Duality Mirror Symmetry, Hitchin's Equations, And Langlands Duality

2008-02-07

arxiv.org/abs/0802.0999v1

Mathematics

Representation Theory

15 pp

Scientific paper

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-manifold without boundary, and the ``ramified'' case that one allows punctures. We also give a few indications of why a more precise description requires a starting point in four-dimensional gauge theory.

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