BelavinPolyakovZamolodchikov

BelavinPolyako is a term that often refers to the Belavin-Polyakov-Zamolodchikov (BPZ) equation, a fundamental equation in the study of two-dimensional conformal field theory (CFT). This equation describes the structure of correlation functions in such theories. In essence, it dictates how the values of operators, when placed at different points on a 2D plane, relate to each other. The equation arises from the requirement that these correlation functions must be invariant under conformal transformations, which are transformations that preserve angles but not necessarily lengths.

The derivation of the BPZ equation involves using the properties of conformal symmetry and the operator product expansion (OPE). The OPE states that the product of two local operators at nearby points can be expressed as a sum of other local operators. By applying conformal Ward identities, which are algebraic relations stemming from conformal invariance, one can derive differential equations that the correlation functions must satisfy. These are the BPZ equations.

The BPZ equation has profound implications. It provides a powerful tool for calculating correlation functions in various 2D CFTs, including those that describe critical phenomena in statistical mechanics and string theory. The solutions to the BPZ equation often involve special functions and reveal deep connections between different areas of physics and mathematics. The equation is particularly important in the context of Liouville theory and its applications to quantum gravity.