TOPOLOGICAL SIGMA MODELS

String theory can be regarded, at the algebraic level, as a two-dimensional conformal field theory coupled to two-dimensional gravity. When the conformal field theory is also a topological field theory (i.e., a theory whose correlation functions do not depend on the metric on the Riemann surface), the resulting string theory turns out to be very simple and in many cases can be completely solved. A string theory that is constructed in this way is called a topological string theory. The starting point for obtaining a topological string theory is therefore a conformal field theory with topological invariance. Such theories are called topological conformal field theories and can be constructed out of N = 2 superconformal field theories in two dimensions by a procedure called twisting. This chapter considers a class of topological string theories in which the topological field theory is taken to be a topological sigma model with target space a Calabi-Yau manifold. The N = 2 supersymmetric sigma model is reviewed, and the twisting procedure is then introduced. The A-type and B-type topological sigma models resulting from two possible twists in two dimensions are examined in detail.