Conformal field theories (CFTs) have a wide range of applications in theoretical physics, particularly in the study of critical phenomena, quantum field theory, and string theory. Here are some simple applications of CFTs:
Critical Phenomena: CFTs are commonly used to describe the behavior of physical systems at critical points, where phase transitions occur. Examples include the critical behavior of magnets near the Curie temperature or the behavior of fluids near their critical points. CFTs provide a powerful framework to understand and classify critical phenomena by describing the underlying symmetries and scaling properties.
Condensed Matter Physics: CFTs play a significant role in condensed matter physics, especially in two-dimensional systems. They are used to describe the behavior of certain quantum Hall systems, fractional quantum Hall states, and the edge states of topological insulators. CFT techniques, such as conformal mapping, are employed to study the universal properties of these systems.
String Theory: CFTs are a fundamental ingredient in the study of string theory. String theory postulates that fundamental particles are not point-like but rather one-dimensional "strings." CFTs provide a natural description of the worldsheet theory, which describes the dynamics of these strings. They are employed to explore various aspects of string theory, such as string compactification, duality symmetries, and black hole entropy.
Statistical Mechanics: CFTs have applications in the study of statistical mechanics, particularly in two dimensions. They provide a powerful tool to understand the behavior of two-dimensional systems at equilibrium, such as the Ising model, lattice gases, and percolation theory. CFT techniques allow for the calculation of critical exponents, correlation functions, and the classification of universality classes.
AdS/CFT Correspondence: The AdS/CFT correspondence, also known as the gauge/gravity duality, is a remarkable duality between certain CFTs and gravity theories in higher-dimensional Anti-de Sitter (AdS) spacetime. This correspondence has found numerous applications in understanding the dynamics of strongly coupled quantum field theories, such as quark-gluon plasmas and condensed matter systems, by mapping them to classical gravity theories in higher dimensions.
These are just a few examples of the many applications of CFTs. The field is vast and continues to be an active area of research with ongoing developments and discoveries.