Conformal Field Theory
Introduction
Conformal Field Theory (CFT) is a quantum field theory that is invariant under conformal transformations. These transformations preserve angles but not necessarily distances, making CFTs particularly useful in describing critical phenomena in statistical mechanics, string theory, and various branches of theoretical physics. The study of CFTs has led to significant insights into the structure of quantum field theories and has applications ranging from condensed matter physics to high-energy physics and beyond.
Basic Concepts
Conformal Transformations
Conformal transformations are mappings that locally preserve angles but not necessarily lengths. In \(d\)-dimensional space, these transformations form the conformal group, which includes translations, rotations, dilations (scalings), and special conformal transformations. The conformal group in \(d\) dimensions is isomorphic to the group \(SO(d+1,1)\).
Primary Fields and Operators
In CFT, fields are categorized into primary fields and descendant fields. Primary fields are the basic building blocks, transforming in a simple way under conformal transformations. Descendant fields are derivatives of primary fields and transform in a more complicated manner. The operator product expansion (OPE) is a crucial tool in CFT, expressing the product of two operators as a series involving other operators.
Conformal Algebra
The conformal algebra is the algebra of the generators of the conformal group. In two dimensions, the conformal algebra is infinite-dimensional and is known as the Virasoro algebra. This algebra plays a central role in the study of two-dimensional CFTs, where the central charge \(c\) is a crucial parameter.
Two-Dimensional Conformal Field Theory
Two-dimensional CFTs are particularly rich and well-studied due to their infinite-dimensional symmetry. The Virasoro algebra, with generators \(L_n\) and central charge \(c\), governs the structure of these theories.
Central Charge
The central charge \(c\) is a key parameter in two-dimensional CFTs, appearing in the commutation relations of the Virasoro algebra. It measures the number of degrees of freedom and plays a crucial role in the classification of CFTs.
Minimal Models
Minimal models are a class of exactly solvable two-dimensional CFTs characterized by a finite number of primary fields. These models are labeled by two coprime integers \(p\) and \(q\) and have central charges given by \(c = 1 - 6(p-q)^2/pq\).
Modular Invariance
Modular invariance is a property of two-dimensional CFTs on a torus, requiring that the partition function be invariant under modular transformations. This constraint leads to powerful results, including the classification of possible CFTs and the determination of their spectra.
Higher-Dimensional Conformal Field Theory
While two-dimensional CFTs are the most well-understood, higher-dimensional CFTs also play a crucial role in theoretical physics. These theories are more challenging to study due to the finite-dimensional nature of the conformal group.
Conformal Bootstrap
The conformal bootstrap is a non-perturbative approach to studying CFTs, relying on the consistency conditions imposed by conformal symmetry. By analyzing the crossing symmetry of the four-point correlation functions, one can derive constraints on the spectrum and operator dimensions of the theory.
AdS/CFT Correspondence
The AdS/CFT correspondence is a conjectured duality between a CFT on the boundary of anti-de Sitter (AdS) space and a gravitational theory in the bulk. This correspondence has provided deep insights into both CFTs and quantum gravity, leading to significant developments in string theory and holography.
Applications of Conformal Field Theory
CFTs have a wide range of applications across various fields of physics.
Statistical Mechanics
In statistical mechanics, CFTs describe critical points of second-order phase transitions. The universality class of a critical point is characterized by a CFT, with critical exponents determined by the scaling dimensions of operators.
String Theory
In string theory, CFTs describe the worldsheet dynamics of strings. The consistency of the string theory requires the worldsheet theory to be a CFT with a specific central charge, depending on the type of string theory.
Condensed Matter Physics
CFTs are used to describe various phenomena in condensed matter physics, including quantum Hall effects, topological insulators, and critical behavior in low-dimensional systems.