Since critical superstring theories are ten-dimensional and M-theory is 11-dimensional, something needs to be done to make contact with the four-dimensional space-time geometry of everyday experience. Two main approaches are being pursued.

Kaluza–Klein compactification

The approach with a much longer history is Kaluza–Klein compactification. In this approach the extra dimensions form a compact manifold of size lc. Such dimensions are essentially invisible for observations carried out at energy E ≪ 1/lc. Nonetheless, the details of their topology have a profound influence on the spectrum and symmetries that are present at low energies in the effective four-dimensional theory. This chapter explores promising geometries for these extra dimensions. The main emphasis is on Calabi–Yau manifolds, but there is also some discussion of other manifolds of special holonomy. While compact Calabi–Yau manifolds are the most straightforward possibility, modern developments in nonperturbative string theory have shown that noncompact Calabi–Yau manifolds are also important. An example of a noncompact Calabi–Yau manifold, specifically the conifold, is discussed in this chapter as well as in Chapter 10.

Brane-world scenario

A second way to deal with the extra dimensions is the brane-world scenario. In this approach the four dimensions of everyday experience are identified with a defect embedded in a higher-dimensional space-time. This defect is typically given by a collection of coincident or intersecting branes.