Orbifold Conformal Field Theory and Algebraic Geometry

主要研究人员提出了四个项目，涉及共形场理论、顶点代数和代数几何。第一个问题是关于手性de Rham络合物的上同调结构及其与Chen-Ruan上同调关系的关系。更准确地说，PI建议通过Orbiold手性De Rham络合物来理解Orbiold上同调的产物结构。第二部分(与E.Frenkel和R.Donagi)讨论了广义Prym簇上的扭曲共形块空间和D-模。第三个项目涉及在尖点G-覆盖的堆叠上构造和研究奥布里奥德共形块层。这里的目的是得到KZ方程的一个或多个CFT推广。最后，最后一个项目(与L·鲍里索夫)涉及手性德罗姆络合物的非手性推广。其目的是为每个Calabi-Yau流形M构造一个Kapustin-Orlov意义下的非手性顶点代数，它计算与目标M有关的(2，2)-sigma模型。这个构造应该具有复杂的Kahler依赖。这些项目的目的是将共形场论(CFT)的思想应用到代数几何中。这已经在普通的保形场理论的情况下成功地完成了，导致了大量美丽的结果。大多数上述项目都涉及所谓的奥比福尔德模型，即当CFT具有额外的离散对称性时出现的模型。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想要将在普通CFT的情况下已知的一些几何结果推广到OBORBORLD设置。上面提出的最后一个项目集中于严格定义和构建一个重要的量子场论，称为西格玛模型。到目前为止，这一理论只存在“一半”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.