Orbifold Conformal Field Theory and Algebraic Geometry

DMS-0401619Matthew M. SzczesnyThe主要研究者提出了四个项目有关共形场论，顶点代数，代数几何。第一部分是关于手征de Rham复形的轨道折叠及其与Chen-Ruan轨道折叠上同调的关系。更确切地说，PI建议根据orbifold手性de Rham复合物来理解orbifold上同调的产物结构。 第二个（与E. Frenkel和R. Donagi）涉及广义Prym簇上的扭曲共形块和D-模空间。第三个项目涉及的构造和研究的叠上的点G-覆盖的orbifold共形块层。这里的目标是获得一个orbifold CFT推广的KZ方程。最后，最后一个项目（与L。Borisov）涉及手性de Rham络合物的非手性推广。目的是为每个Calabi-Yau流形M构造Kapustin-Orlov意义下的非手征顶点代数层，该层计算目标M的（2，2）-sigma模型。这些项目的目的是将共形场论（CFT）（一种量子场论）的思想应用于代数几何。这在普通共形场论的情况下已经成功地做到了，导致了大量美丽的结果。上述项目中的大多数都与所谓的轨道模型有关，当CFT具有额外的离散对称性时会出现这种模型。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希望将一些在普通CFT的情况下已经知道的几何结果扩展到orbifold设置。上面提出的最后一个项目集中在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，只有“一半”的理论存在的部分建设。

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.