25 - Compactification of string theory I. Tori and orbifolds

Summary

We don't live in a ten-dimensional world, and certainly not in a twenty-sixdimensional world without fermions. But if we don't insist on Lorentz invariance in all directions, there are other possible ways to construct consistent string theories. In this chapter we will uncover many consistent string theories in four dimensions (and in others). If anything, our problem will shortly be an embarrassment of riches: we will see that there are vast numbers of possible string constructions. The connection of these various constructions to one another is not always clear. Many of these can be obtained from one another by varying expectation values of light fields (moduli). One might imagine that others could be obtained by exciting massive fields as well. In general, though, this is not known, and, in any case, the meaning of such connections in a theory of gravity is obscure. But before exploring these deep and difficult questions, we need to acquire some experience with constructing strings in different dimensions.

Compactification in field theory: the Kaluza–Klein program

The idea that space-time might be more than four-dimensional was first put forward by Kaluza and Klein shortly after Einstein published his general theory of relativity. They argued that, in this case, five-dimensional general coordinate invariance would give rise to both four-dimensional general coordinate invariance and a U(1) gauge invariance, unifying electromagnetism and gravity. In modern language, they considered the possibility that space-time is five-dimensional, with the structure M⁴ × S¹. This is, on first exposure, a bizarre concept, but its implications are readily understood by considering a toy model. Take a single scalar field, Φ, in five dimensions.