Betti Geometric Langlands
Betti Geometric Langlands
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DOI：
10.1090/pspum/097.2/01698
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发表时间：
2016-06
期刊：
影响因子：
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通讯作者：
David Ben-Zvi;D. Nadler
中科院分区：
文献类型：
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作者：
David Ben-Zvi;D. Nadler
我们介绍和调查几何朗兰兹猜想的贝蒂形式，平行于由Beilinson-Drinfeld和Arinkin-Gaitsgory发展的de Rham形式，以及Donagi-Pantev的Dolbeault形式，并受到Kapustin-Witten在超对称规范理论中工作的启发。该猜想提出了一个与紧黎曼曲面X和复约化群G相关联的自守范畴等价于一个与底层拓扑曲面S和朗兰兹对偶群G^相关联的谱范畴。自守范畴由X上G-丛的模栈Bun_G（X）上的合适的C-层组成，而谱范畴由S上G^-局部系统的特征标栈Loc_G^（S）上的合适的O-模组成。该猜想与来自丛和局部系统的修改的两边的自然对称性相容并受其约束。一方面，在德拉姆和贝蒂意义下的尖点赫克本征层预计是重合的，因此人们可以将贝蒂猜想看作是对相同的基本对象提供了不同的“积分测度”。另一方面，贝蒂谱范畴比他们的德拉姆对应更明确，人们可能希望这个猜想不那么具有挑战性。贝蒂纲领也享有来自拓扑场论的对称性：它有望扩展到四维拓扑场论的等价，特别是，闭曲面的猜想有望简化为三次穿孔球面的情况。最后，我们还提出了分歧，量子和积分的猜想变种，并强调连接到其他主题，包括代表性理论的真实的还原群和量子群。
We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of Donagi-Pantev, and inspired by the work of Kapustin-Witten in supersymmetric gauge theory. The conjecture proposes an automorphic category associated to a compact Riemann surface X and complex reductive group G is equivalent to a spectral category associated to the underlying topological surface S and Langlands dual group G^. The automorphic category consists of suitable C-sheaves on the moduli stack Bun_G(X) of G-bundles on X, while the spectral category consists of suitable O-modules on the character stack Loc_G^(S) of G^-local systems on S. The conjecture is compatible with and constrained by the natural symmetries of both sides coming from modifications of bundles and local systems. On the one hand, cuspidal Hecke eigensheaves in the de Rham and Betti sense are expected to coincide, so that one can view the Betti conjecture as offering a different "integration measure" on the same fundamental objects. On the other hand, the Betti spectral categories are more explicit than their de Rham counterparts and one might hope the conjecture is less challenging. The Betti program also enjoys symmetries coming from topological field theory: it is expected to extend to an equivalence of four-dimensional topological field theories, and in particular, the conjecture for closed surfaces is expected to reduce to the case of the thrice-punctured sphere. Finally, we also present ramified, quantum and integral variants of the conjecture, and highlight connections to other topics, including representation theory of real reductive groups and quantum groups.