Langlands duality for Hitchin systems

Abstract

We show that the Hitchin integrable system for a simple complex Lie group $G$ is dual to the Hitchin system for the Langlands dual group $\lan{G}$. In particular, the general fiber of the connected component $\Higgs_0$ of the Hitchin system for $G$ is an abelian variety which is dual to the corresponding fiber of the connected component of the Hitchin system for $\lan{G}$. The non-neutral connected components $\Higgs_α$ form torsors over $\Higgs_0$. We show that their duals are gerbes over $\Higgs_0$ which are induced by the gerbe of $G$-Higgs bundles $\gHiggs$. More generally, we establish a duality between the gerbe $\gHiggs$ of $G$-Higgs bundles and the gerbe $\lan{\gHiggs}$ of $\lan{G}$-Higgs bundles, which incorporates all the previous dualities. All these results extend immediately to an arbirtary connected complex reductive group $\mathbb{G}$.

75 pages, 1 figure, LaTeX. New version substantially expanded and revised for publication

References in corpus (1)

• Geometric transitions and integrable systems

Cited by in corpus (8)

• S-duality, 't Hooft operators and the operator product expansion
• The Spectral Construction for a (1,8)-Polarized Family of Abelian Varieties
• Autoduality of compactified Jacobians for curves with plane singularities
• Geometry of Character Varieties of Surface Groups
• Hilbert schemes as moduli of Higgs bundles and local systems
• Three lectures on classical integrable systems and gauge field theories
• SYZ Mirror Symmetry of Hitchin's Moduli Spaces Near Singular Fibers I
• A Universal Family of Deformations for the Uniformising Higgs bundle