Geometry and Topology of Orbifolds

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

DMS-4724 Ernesto Lupercio This proposal consists of a number of research projects on the topic of cohomology theories for orbifolds and related subjects. The suggested methods combine algebraic topology with techniques coming from stack theory in algebraic geometry and tools arising in symplectic geometry. It proposes the use of the theory of groupoids as a unifying tool in the solution these questions; a methodology that has proved successful before in our study of orbifolds. The following is a brief summary of some aspects of the project. The first project is program whose objective is the study of the cohomological invariants that can be naturally obtained from several natural spaces associated to an orbifold. A second project consist of a study the orbifold elliptic genus, its modularity and rigidity properties from the homotopy theory point of view. The third project describes a program to define and study Orbifold Deligne cohomologies and their relation to gerbes defined over the orbifold. The fourth is a project to study the relation between the Hodge-Deligne numbers of an orbifold resolution of singularities of an orbifold and the original orbifold. Orbifolds are geometric spaces in which it is very important to keep track of the local symmetries of the particular situation. So in general the points of an orbifold are classified by the amount of local symmetry. In ordinary spaces all points are equal, while in an orbifold points carry different weights corresponding to the amount of local symmetry. Orbifolds have been used in the study of structural crystallography allowing for a convenient book-keeping device by reducing the redundant information in the space group diagrams of the crystallographic groups. Orbifolds have also become extremely important in the field of theoretical physics referred to as Superstring theory. In String Theory, the multitude of particle types of classical high energy physics is replaced by a single fundamental building block, a `string'. As the string moves through time it traces out a tube or a sheet, according to whether it is closed or open. Moreover, the string can vibrate, and different vibrational modes of the string represent the different particle types, since different modes are seen as different masses or spins. In an attempt to make these theories describe the physical universe, physicists have been motivated to let the string move on an orbifold space. In these cases the theory acquires properties that are very desirable for it to model reality. The work proposed in this project involves the development of rigorous mathematical methods to study the geometry of these spaces.