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Geometric endoscopy and mirror symmetry. (English) Zbl 1223.14014

This paper gives considerable detail about the interpretation of geometric Langlands correspondence in terms of mirror symmetry. A full description of its contents would go beyond the scope of a review (and indeed beyond the knowledge of the reviewer), and would require an account of the work of Ngô as well as a discussion of mirror symmetry, Hitchin fibrations and geometric Langlands in general. However, the introduction (thirteen pages long) summarises the paper, so let us summarise the introduction.
The geometric Langlands duality can be interpreted in terms of mirror symmetry for Hitchin fibrations for moduli of Higgs \(G\)-bundles over a curve \(C\): this is fairly straightforward, relatively speaking, at the general fibre, which is a torus, but further interesting phenomena arise when the Hitchin fibres acquire orbifold singularities. This paper handles the case \(G={\mathrm{SL}}_2\) in detail. The \(T\)-duality interchanges \(B\)-branes (e.g.skyscraper sheaves) on the Hitchin fibre of \({\mathcal M}_H({}^L G)\) (moduli of Higgs bundles: \({}^L{\mathrm{SL}}_2\) is \(\mathrm{SO}_3\) but at the singular points we can work with \(\text{O}_2\)) and \(A\)-branes on \({\mathcal M}_H(G)\). At the most general orbifold point the inertia group is \({\mathbb Z}/2\), the centre of \(\text{O}_2\), so the \(B\)-branes are the representations of that. Thus there are two irreducible objects and one should try to find out what they are. These \(A\)-branes are in any case a substitute for \({\mathcal D}\)-modules, which in turn correspond over \({\mathbb C}\) to Hecke eigensheaves, and a decomposition of one should decompose the other.
The authors “make another leap of faith” and move to curves over \({\mathbb F}_q\) instead of \({\mathbb C}\). Here Hecke eigensheaves correspond to automorphic functions on the adelic group \(G({\mathbb A}_F)\), where \(F\) is the function field of \(C\). Here they encounter endoscopy, which they describe in the introduction by referring to its appearance in the Langlands correspondence for \({\mathrm{GL}}_n\). Much of what is stated here remains conjectural, though much less so after Ngô’s work, but there is sufficient confidence for the authors to feel able to offer interpretations and predictions about how the results (or conjectures) relating to endoscopy should translate, via the correspondences above, into statements about \({\mathcal D}\)-modules and Hecke eigensheaves, Hitchin fibrations, etc. The term “geometric endoscopy” refers to this process and its outcome. It turns out that the \(A\)-branes are convenient for expressing this, and that the endoscopy groups arise (on the \(B\)-model side) in a more natural and transparent way than they do in the classical context.
The introduction concludes with a description of the relation with Ngô’s work, a brief allusion to aspects relating to QFT, and an outline of the structure of the paper itself. This last explains, which for reasons of space we cannot here, what it is exactly that the authors do. One thing they do not do, by and large, is prove theorems: this is in the nature of a survey article.

MSC:

14D24 Geometric Langlands program (algebro-geometric aspects)
11F23 Relations with algebraic geometry and topology
14J33 Mirror symmetry (algebro-geometric aspects)
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings

Citations:

Zbl 1098.14023
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