9 - Conformal invariance

Summary

It was noted by Polyakov and others in the early seventies that critical models implement a global conformal invariance which goes beyond pure scale invariance. The latter affects relative distances by a constant (i.e. space independent) factor, while other conformal transformations involve a space dependent factor. This invariance property enables one to fix not only the form of two-point but also three-point functions at criticality, when they are nonvanishing. However the conformal group is in general a finite dimensional Lie group, so that the resulting constraints are limited in number. In two dimensions, a new phenomenon arises, well known in the theory of analytic functions, namely there exists a plethora of local conformal transformations. As a result, it was tempting to investigate the possible consequences of local scale invariance in two dimensions. This is what was brilliantly undertaken by Belavin, Polyakov and Zamolodchikov in 1983, launching a new wave of applications in statistical physics. As the subject is still in its development, the present chapter will not be as elementary as previous ones, nor will it presumably remain up to date, especially as it is closely related to string field theory, a promising new approach to the quantum description of extended objects, which attempts to embrace all known interactions including gauge theories and gravity.