Strings on Orbifolds*

The most thoroughly understood example of superstring compactification is compactification on a fiat torus [1]. This example is exactly soluble and obeys all of the equations of string theory. It is also clearly a significant example since it is one way to understand the construction of the heterotic string theory [2]. For compact~ficatlon of physical dimensions from ten to four, the fiat torus is not suitable since it leads in many ways to unappealing phenomenology. More realistic compactification schemes involve non-trivial supersymmetric non-linear sigma models, and the realisUc ones seem to be rather complicated mathematically. The purpose of this paper is to descnbe a scheme of compactlfication of extra physical dimensions which is almost as simple as standard toroidal compactxfication and far more reahstic. The main phenomenological question involves the realization of spacetime and gauge symmetry breaking. In the string theory a way to achieve symmetry breaking is to twist the boundary conditions in such a way that there is no net charge corresponding to the broken symmetries on the string world sheet. Such a procedure has been used for supersymmetry [3] and gauge symmetry [4] breaking and it is natural to attempt to do so for Lorentz symmetries as well [5] The idea of using twisted boundary conditions to break Lorentz symmetries is that spacetime coordinates x"(cr) should not be periodic functions of cr but perio&c up to a Lorentz transformation. Modular invarlance puts severe constraints on such attempts, since a string sector with twisted boundary conditions in the o--direction is related by modular transformations to sectors with various boundary conditions in the gand z-&rections. What will be described below is one consistent framework m which the twisted boundary conditions are related to the construction of a new manifold (or at least, a new orbifold, as we will describe later) The method of constructing new manifolds (or orbifolds) that we will employ is a classical geometrical method that we will implement directly in string theory.