摘要翻译：
几何Langlands对应被解释为两对偶还原基团Hitchin纤维的镜像对称性。这种镜像对称性反过来又在一般的Hitchin纤维上减少到T-对偶性，这些纤维是光滑的圆环。本文研究了B型侧Hitchin光纤产生轨道奇点时的情况。这些奇点对应于具有有限群自同构的局部系统。在经典的Langlands程序中，这种类型的局部系统被称为内窥镜。它们在自守表示理论中起着重要作用，特别是在迹公式的稳定性方面。我们的目标是利用Hitchin纤维的镜像对称性来揭示这些局域系统在几何理论中所起的特殊作用。通过对双Hitchin纤维上A-膜种类的研究，我们可以发现一些与内窥镜有关的几何Langlands对应的有趣现象。然后我们沿着我们的预言回到经典的自同构函数理论。这使我们能够测试和确认它们。我们使用的几何学类似于B.C.在最近的工作中所利用的几何学。Ngo,这一事实对理解迹公式有重要意义。
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英文标题：
《Geometric Endoscopy and Mirror Symmetry》
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作者：
Edward Frenkel and Edward Witten
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Physics        物理学
二级分类：High Energy Physics - Theory        高能物理-理论
分类描述：Formal aspects of quantum field theory. String theory, supersymmetry and supergravity.
量子场论的形式方面。弦理论，超对称性和超引力。
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一级分类：Mathematics        数学
二级分类：Quantum Algebra        量子代数
分类描述：Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
量子群，skein理论，运算代数和图解代数，量子场论
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一级分类：Mathematics        数学
二级分类：Representation Theory        表象理论
分类描述：Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra
代数和群的线性表示，李理论，结合代数，多重线性代数
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英文摘要：
  The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry, in turn, reduces to T-duality on the generic Hitchin fibers, which are smooth tori. In this paper we study what happens when the Hitchin fibers on the B-model side develop orbifold singularities. These singularities correspond to local systems with finite groups of automorphisms. In the classical Langlands Program local systems of this type are called endoscopic. They play an important role in the theory of automorphic representations, in particular, in the stabilization of the trace formula. Our goal is to use the mirror symmetry of the Hitchin fibrations to expose the special role played by these local systems in the geometric theory. The study of the categories of A-branes on the dual Hitchin fibers allows us to uncover some interesting phenomena associated with the endoscopy in the geometric Langlands correspondence. We then follow our predictions back to the classical theory of automorphic functions. This enables us to test and confirm them. The geometry we use is similar to that which is exploited in recent work by B.-C. Ngo, a fact which could be significant for understanding the trace formula.
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PDF链接：
https://arxiv.org/pdf/0710.5939