14 - Compactification of higher dimensions

Summary

Since superstring theories are necessarily ten-dimensional theories, any discussion of phenomenology must begin with a discussion of how apparent four-dimensional physics is related to underlying ten-dimensional physics. The present chapter is devoted to this question. We will carry out the discussion in the context of field theory, but with an emphasis on properties that depend only on qualitative assumptions, not numerical details, and so can remain valid in string theory. What we will try to accomplish in this chapter is not to develop detailed models of compactification but to set the stage and introduce some of the essential concepts.

Wave Operators in Ten Dimensions

Most of the preceding chapters have been devoted to string propagation in ten-dimensional flat Minkowski space M¹⁰, but henceforth we will consider ten-dimensional space-time to be some more general ten manifold M. We take M to be of the form M⁴ × K, where M⁴ is four-dimensional Minkowski space and K is a compact six manifold which is, unfortunately, as yet unknown. More precisely, we take the vacuum state to be a product M⁴ × K; it must have this form if we wish to maintain four-dimensional Poincaré invariance. Of course, physical fluctuations will not necessarily respect the product form of the vacuum configuration, but as in so many other areas of physics, understanding the ground state is the key to understanding the low-energy excitations.