Extensions of 2d Gravity * †

Conformally invariant theories in two dimensions play an important role in the study of string theories, second order phase transitions and integrable systems. The Virasoro algebra and its extensions form the cornerstone of conformal field theories. More recently a lot of attention was devoted to the study of a particular class of conformal field theories: gravity in two dimensions. As I will show in the next section, gravity in two dimensions is a purely quantum mechanical artefact. Contrary to the case of higher dimensions, gravity in two dimensions allows for infinitely many extensions such as higher spin fields, supersymmetry, Yang-Mills symmetries, etc. These extensions are in one to one correspondence with the extensions of the Virasoro algebra. The interest in 2D gravity arose from its close relation to non-critical string theories. In these theories, the matter sector is a minimal model. A propagating gravity sector contributes to the conformal anomaly in such a way that the conformal anomaly of the combined matter-gravity-ghost sectors vanishes. Noncritical string theories allow for a non-perturbative treatment thus providing a testing ground for techniques and ideas which might be applicable to more “realistic” string theories. More recently 2D gravity gave rise to models which made the study of quantum aspects of black hole evaporation possible.