Some algebraic geometry

In chapters 12 and 14 we developed some simple tools in differential geometry and used them to gain some insight concerning the compactification of hidden dimensions as well as some insight concerning phenomena on the string world sheet. We now turn our attention to some more specialized mathematical tools involving complex manifolds and algebraic geometry. Again, the motivation is twofold. The world sheet of a string is a complex manifold – a Riemann surface, in fact – and as string theory develops, the deeper study of world-sheet phenomena is likely to involve deeper aspects of algebraic geometry, which have already begun to enter in recent works on multiloop diagrams. Also, algebraic geometry has been a tool in recent attempts to formulate more realistic models of string compactification.