Vertex algebras and geometry of manifolds

这个项目的目的是进一步了解顶点代数是如何与流形的几何相关的。虽然顶点代数是代数中一个相对较新的概念，一直被理解为共形场理论的重要组成部分，但它们与高维几何的关系，即使隐含在弦理论中，也有一段时间是相当模糊的。1998年，Malikov，Scheck htman和Vaintrob提出了手征微分算子的概念，作为直接的这种联系。后来的工作发现了这一新理论的几个应用，最近，Witten和Kapustin用弦理论中的所谓Witten半扭曲模型确定了手征微分算子的代数。这一提议的一个特征是它真正属于几个不同学科的接口，如代数、几何和理论物理。事实上，顶点代数是纯粹的代数对象，直到最近才被发现与流形的几何密切相关。另一方面，这一研究领域本质上是弦物理学的数学对应物。因此，预计该提案的结果将对研究界产生广泛影响。

The aim of this project is to further the understanding of how vertex algebras are related to geometry of manifolds. While vertex algebras, a relatively new concept in algebra, have always been understood as an essential part of conformal field theory, their relation to higher dimensional geometry, even though implicit in string theory, had been rather obscure for a while. In 1998 Malikov, Schechtman and Vaintrob proposed the notion of chiral differential operators as a direct such link. Later work has found several applications of this new theory and, rather recently, Witten and Kapustin identified algebras of chiral differential operators with the so-called Witten half-twisted model in string theory.One characteristic feature of this proposal is that it truly belongs to the interface of several disparate disciplines, such as algebra, geometry, and theoretical physics. Indeed, vertex algebras are purely algebraic objects, which only relatively recently were found to be closely related to geometry of manifolds. On the other hand, this area of research is essentially a mathematical counterpart of the physics of strings. The results of the proposal are therefore expected to have a broad impact on research community.