Holomorphic Aspects of String Theory

Abstract

A string is a piecewise smooth map of the interval to a manifold M. A closed string is a map of the circle S ¹ into M. In string theory the strings replace the points of the manifold M as fundamental objects. The enormous amount of work done on quantized string models in physics has been motivated by the hope that the quantum string theory would produce a finite quantized theory of gravity, free of the divergences of the ordinary quantized Einstein theory of gravitation. So far the proof is missing but work is continuing. It has been proposed that some kind of string theory would be the unified theory of all fundamental interactions in physics. However, the fundamental principles of string theory have not been clearly formulated and up to now there are no predictions of string theory which could be experimentally tested. The great interest in string models cannot be explained only by the (rather meager) physical resusts, but there are very interesting mathematical problems which require an exciting blend of algebraic and geometric methods.