BelavinPolyakov

The Belavin-Polyakov-Zamolodchikov (BPZ) equation is a fundamental relationship in conformal field theory (CFT) that describes the structure of correlation functions of primary fields. It was derived by Alexander Belavin, Alexander Polyakov, and Alexander Zamolodchikov in 1984. The equation provides a powerful tool for analyzing the properties of 2D quantum field theories, particularly those exhibiting conformal symmetry.

At its core, the BPZ equation is a differential equation that relates correlation functions of primary fields to the operators of the theory and their dimensions. It arises from considering the infinitesimal transformations of conformal symmetry and how they act on correlation functions. The equation effectively encodes the constraints imposed by conformal invariance on the possible forms of correlation functions.

The BPZ equation has had profound implications across various fields of physics. In statistical mechanics, it has been crucial for understanding critical phenomena and phase transitions in 2D systems, such as the Ising model and Potts model. It also plays a significant role in string theory, where it is used to study the properties of the worldsheet conformal field theory. Furthermore, the BPZ equation has connections to integrable systems and has found applications in areas like quantum gravity and condensed matter physics. Its derivation and analysis have been a cornerstone of modern theoretical physics.