Langlands duality for Hitchin systems
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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 $\Higgs_{\alpha}$ 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 $\mathbb{G}$.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/73be86bdddb1a5ff18c1e8d9aaf506b760e63a6d
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Holomorphic reduction of N = 2 gauge theories, Wilson-'t Hooft operators, and S-duality (2006, background): “The duality of Hitchin fibrations for G and G is known as classical geometric Langlands duality and has been proved for nonsingular fibers in [30, 31, 32, 33].”
- Twisted wild character varieties (2015, background): “This duality is known to be related to Langlands duality in some cases [29, 25], but apparently suffers from “missing orbits” in certain untwisted cases [28].”
- Electric-Magnetic Duality And The Geometric Langlands Program (2006, methodology): “The question has also been analyzed by Donagi and Pantev for any semi-simple Lie group using an abstract approach to spectral covers [82].”
- Langlands duality and G2 spectral curves (2006, result): “Rather than appeal to the Lie-theoretical approach of [ 11 ], or the earlier work of [22], we choose to describe G2 as the identity component of the stabilizer of a three-form in seven dimensions, drawing on the papers [19],[20] dealing with G2 in a different context.”