String Theory and M-Theory

9
String geometry
Since critical superstring theories are ten-dimensional and M-theory is 11dimensional,
something needs to be done to make contact with the fourdimensional
space-time geometry of everyday experience. Two main approaches
are being pursued. 1
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
1 Some mathematical background material is provided in an appendix at the end of this chapter.
Readers not familiar with the basics of topology and geometry may wish to study it first.
354