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

该项目涉及朗兰兹对应的几何版本，其中自同构形式的角色由复杂曲线上主束模空间上的d模扮演，而伽罗瓦表示的角色则由曲线上的局部系统扮演。手性代数理论似乎为构造与伽罗瓦表示相对应的“自同构形式”提供了一种强有力的方法，这可能最终导致对几何情况下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见解之一；目前，似乎没有任何线索可以全面地理解它们。然而，由于手性代数的机制（起源于数学物理），几何情况下的并行陈述（技术上，用复杂曲线上的有理函数域取代数域）似乎更易于管理。这些并行语句将在本项目中解决。＊＊＊

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. ***