The 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.
