Mathematical Sciences: Algebraic Topology of Algebraic Varieties, Conformal Field Theory

9625768 Beilinson该项目研究了朗兰兹通信的一个几何版本，其中自同构形式的作用由复曲线上主丛的模空间上的D-模来扮演，而伽罗瓦表示的作用则由该曲线上的局部系统来扮演。手征代数理论似乎提供了一种强有力的方法来构造与“伽罗瓦表示”相对应的“自同构形式”，这最终可能导致对几何情形下的朗兰兹猜想的完全理解。朗兰兹猜想似乎是现代数论中最深刻的愿景之一；目前似乎没有任何线索来全面理解它们。然而，由于手征代数的机制(起源于数学物理)，几何情况下的并行语句(从技术上讲，数域被复杂曲线上的有理函数域取代)似乎更容易处理。这些平行陈述将在本项目中得到解决。***

9625768 Beilinson The project deals with a geometric version of Langlands' correspondence in which the role of automorphic forms is played by D-modules on the moduli space of principal bundles on a complex curve, and that of Galois representations, by the local systems on the curve. It appears that the theory of chiral algebras provides a powerful method for constructing the 'automorphic forms' corresponding to the 'Galois representations,' which may eventually lead to complete understanding of the Langlands conjectures in the geometric situation. The Langlands conjectures appear to be among the deepest visions of modern number theory; at the moment there seems to be no clue to understanding them in full generality. However, the parallel statements in the geometric situation (technically, with the number field replaced by the field of rational functions on a complex curve) seem to be much more manageable, because of the machinery of chiral algebras (originated in mathematical physics). These parallel statements are to be addressed in this project. ***