So far, we have focussed on rather simple models, involving toroidal compactifications and their orbifold generalizations. But while by far the simplest, these turn out to be only a tiny subset of the possible manifolds on which to compactify string theories. A particularly interesting and rich set of geometries is provided by the Calabi–Yau manifolds. These are manifolds which are Ricci flat, RMN = 0. Their interest arises in large part because these compactifications can preserve some subset of the full ten-dimensional supersymmetry. This is significant if one believes that low-energy supersymmetry has something to do with nature. It is also important at a purely theoretical level, since, as usual, supersymmetry provides a great deal of control over any analysis; at the same time, there is less supersymmetry than in the toroidal case, so a richer set of phenomena are possible.

This chapter is intended to provide an introduction to this subject. In the first section, we will provide some mathematical preliminaries. Unlike the toroidal or orbifold compactifications, it is not possible, in most instances, to provide explicit formulas for the underlying metric on the manifold and other quantities of interest. The six-dimensional Calabi–Yau spaces, for example, have no continuous isometries (symmetries), so at best one can construct the metrics by numerical methods. But it turns out to be possible to extract much important information without detailed knowledge of the metric from topological considerations. The machinery required to define these spaces and to extract at least some of this information includes algebraic geometry and cohomology theory, subjects not part of the training of most physicists.