Calabi-Yau Compactification

When asked about the ten-dimensional nature of superstring theory, Richard Feynman once replied, “The only prediction [string theory] makes is one that has to be explained away because it doesn’t agree with experiment.” [8] In a sense, this skepticism is not unfounded, and if string theory did not have so many other desirable features (being a renormalizable theory containing interacting spin-2 particle, for instance), it seems unlikely that it would have gained the cachet it has today. If one wants to be able to apply the elegance of string theory to explain the observed world, however, one has to come to grips with the fact that the world we observe is most emphatically not ten-dimensional, or at least remains down to scales of about 10 metres (the TeV scale, which has been at least indirectly “observed” at particle colliders.) The usual way to “explain this away”, in Feynman’s words, is to assume that six of the spatial dimensions of the theory are compactified : the spacetime manifold on which the strings move is not an arbitrary ten-dimensional space, but is instead of the form M 4×N6 for some compact sixdimensional manifold N. This compact manifold is usually taken to have a sufficiently small size as to be unobservable with present technology, and thus we would only see the four-dimensional manifold M. There are a horrendously large number of possible six-dimensional compact manifolds one could choose from, though, and one might hope that the requirement that the four-dimensional theory resemble the observed world would limit our choice ofN 6 somewhat. In fact, this turns out to be the case: requiring that supersymmetry is preserved at the compactification scale restricts us to a special class of manifolds known as Calabi-Yau manifolds. Further, the topological properties of these manifolds actually determine some of the particle content observed in the low-energy four-dimensional theory; far from being an ad hoc assumption necessary to “explain away” undesirable features of string theory, compactification might actually provide an elegant explanation of some features of the observed world.