The Analytic Geometry of Two-dimensional Conformal Field Theory*

Some years ago, Polyakov [1] proposed constructing all conformally invariant quan tum field theories by using the constraint of conformal invariance to make concrete the fundamental principles of quantum field theory. This is the conformal boots t rap program. Conformal field theories describe the universality classes of critical phenomena, or equivalently the short distance limits of quantum field theories, so the conformal bootstrap program is an attempt to find all possible critical phenomena, and all possible quantum field theories, and to describe explicitly their short-distance behavior. The subject of two-dimensional conformal field theory originated simultaneously in the theory of critical phenomena [21] and in string theory [3]. In recent years there has been progress in the two-dimensional conformal bootstrap program, based on investigation of the two-dimensional conformal anomaly and the Virasoro algebra [4-17]. There has also been progress in the two-dimensional super-conformal boots t rap program, leading to the discovery of supersymmetric critical phenomena [8,18-22]. Two-dimensional conformal field theory also has several applications in mathematics. Modifications of the Ricci-flat Calabi-Yau spaces [23], and certain generalizations [24] are thought to provide examples of two-dimensional superconformal