Wiki. “几何 Langlands 对应” [几何Langlands对应]

Recently, in a groundbreaking advance, Kapustin and Witten have linked the geometric Langlands correspondence and the S-duality of 4D supersymmetric gauge theories. This and subsequent works have already led to striking new insights into the geometric Langlands Program, which in particular involve the Homological Mirror Symmetry of the Hitchin moduli spaces of Higgs bundles on algebraic curves associated to two Langlands dual Lie groups.

Hitchin 模叠

Galois 表示理解为 l-进层.

非分歧自守表示给出 $\mathrm{GL}_n(F) \backslash \mathrm{GL}_n(\mathbb A) / \mathrm{GL}_n(\mathcal O)$ 上的函数 $f_\pi$ (差一个常数), 它是 Hecke 代数 $\mathcal H_x, x\in X$ 的特征函数. 使用 Grothendieck 函数–层字典,

范畴式几何 Langlands 对应

范畴式几何 Langlands 对应的主旨是两个 DG 范畴的比较:

• 几何 (“自守”) 侧, 即 $X$ 上 $G$-丛的模空间上的 D-模;
• 谱 (“Galois”) 侧, 即 $X$ 上 $\check G$-局部系统的叠上的拟凝聚层.