A short introduction to conformal invariance

Abstract

An easily readable introduction to the main concepts and techniques of conformal invariance is provided. Starting from the global scale-invariance at a critical point, it is argued, through the local conformal Ward identities, that under mild conditions an extension to a local form of scale-invariance, namely conformally invariance, is in general possible. In two space dimensions, the particular role of the infinite-dimensional Lie algebra of conformal transformations is outlined and the main concepts, namely those of a primary scaling operator, the conformal energy-momentum tensor, the Virasoro algebra and the central charge and the main facts of their unitary and/or minimal representations will be presented. Some simple applications for the explicit calculation of two-point functions will be given. The free boson will be used as a paradigmatic illustration and we shall close with an outline of surface critical phenomena and their description in terms of boundary conformal field-theory. © 2012 Springer-Verlag Berlin Heidelberg.