Mirror Symmetry, Hitchin's Equations, And Langlands Duality

Abstract

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.

15 pp

References in corpus (4)

• Generalized complex geometry
• Gauge Theory And Wild Ramification
• Geometric Endoscopy and Mirror Symmetry
• Local geometric Langlands correspondence and affine Kac-Moody algebras

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• Landau's Last Paper and its Impact on Developments in Mathematics, Physics and Other Disciplines in New Millennium
• AP Theory IV: Intrinsic Topological Quantum Langlands Theory