Orbifold Conformal Field Theory and Algebraic Geometry

主要研究者提出了四个项目，涉及共形场论、顶点代数和代数几何。第一个是关于手性de Rham配合物的轨道及其与陈朗的轨道上同调的关系。更准确地说，PI提出了从轨道手性de Rham配合物的角度来理解在轨道上同调上的产物结构。第二部分（与E. Frenkel和R. Donagi合作）讨论了广义Prym变异上的扭曲共形块和d模的空间。第三个项目涉及在一堆尖G型盖上建造和研究轨道共形块。这里的目标是得到KZ方程的一个轨道CFT推广。最后，最后一个项目（与L. Borisov）是关于手性de Rham配合物的非手性推广。目的是为每个Calabi-Yau流形M构造一组Kapustin-Orlov意义上的非手性顶点代数，用于计算目标M的(2,2)-sigma模型。这种构造应该使Kahler依赖变得复杂。这些项目的目的是将共形场论（CFT）（量子场论的一种）中的思想应用于代数几何。这已经在普通共形场理论中成功地做到了，并导致了大量美丽的结果。上述大多数项目都与所谓的轨道模型有关，当CFT具有额外的离散对称性时，就会出现这种模型。在这种情况下，与代数几何的联系还没有得到充分的探索，PI希望将普通CFT情况下已知的一些几何结果扩展到轨道设置。上面提出的最后一个项目侧重于严格定义和构建一个重要的量子场论，称为sigma模型。到目前为止，只存在“一半”理论的部分构建。

DMS-0401619Matthew M. SzczesnyThe principal investigator proposes four projects relating conformal field theory, vertex algebras, and algebraic geometry. The first one concerns orbifolds of the chiral de Rham complex and their relation to Chen-Ruan's orbifold cohomology. More precisely, the PI proposes to understand the product structure on orbifold cohomology in terms of the orbifold chiral de Rham complex. The second (with E. Frenkel and R. Donagi) relates spaces of twisted conformal blocks and D-modules on generalized Prym varieties. The third project involves the construction and study of sheaves of orbifold conformal blocks over the stack of pointed G--covers. The objective here is to obtain an orbifold CFT generalization of the KZ equations. Finally, the last project (with L. Borisov) is concerned with a non-chiral generalization of the chiral de Rham complex. The aim is to construct, for each Calabi-Yau manifold M, a sheaf of non-chiral vertex algebras in the sense of Kapustin-Orlov, which computes the (2,2)-sigma model with target M. This construction should have complexified Kahler dependence.The purpose of these projects is to apply ideas in conformal field theory (CFT) ( a type of quantum field theory) to algebraic geometry. This has already been done successfully in the case of ordinary conformal field theories, leading to a large number of beautiful results. Most of the above projects are concerned with so called orbifold models, which arise when the CFT has additional discrete symmetries. In this case, the connections with algebraic geometry have not been fully explored, and the PI wants to extend some of the geometric results already known in the case of ordinary CFT's to the orbifold setting. The last of the projects proposed above focuses on rigorously defining and constructing an important quantum field theory, called the sigma model. So far, only a partial construction of "half" the theory exists.