An In-Depth Mathematical Exploration of Conformal Field Theory- Principles and Applications

A Mathematical Introduction to Conformal Field Theory

Conformal Field Theory (CFT) is a branch of theoretical physics that deals with the study of physical systems that are invariant under conformal transformations. It has found applications in various areas of physics, including string theory, statistical mechanics, and condensed matter physics. This article aims to provide a mathematical introduction to conformal field theory, focusing on the fundamental concepts and techniques used in its study.

In the first section, we will discuss the basics of conformal transformations and their properties. Conformal transformations are continuous transformations that preserve angles and ratios of lengths. They are characterized by a conformal factor, which is a scalar function that scales the coordinates of the space. The group of conformal transformations in two dimensions is known as the conformal group, denoted by SO(2,1).

The Conformal Group and Representation Theory

The conformal group plays a crucial role in conformal field theory. It acts on the space of fields, and the representations of the conformal group describe the symmetries of the physical system. Representation theory is a branch of mathematics that studies the representations of groups and algebras. In the context of conformal field theory, representation theory provides a framework for classifying and understanding the possible symmetries of the fields.

We will introduce the concept of a representation of a group and discuss some of the key representations of the conformal group. These representations will be used to construct the conformal algebra, which is a Lie algebra that encodes the generators of the conformal group. The conformal algebra is a fundamental object in conformal field theory, as it governs the equations of motion for the fields.

Vertex Algebras and Conformal Field Theories

Vertex algebras are algebraic structures that are used to construct conformal field theories. A vertex algebra is a vector space equipped with a collection of associative products called vertices. These vertices allow us to build field theories by gluing together local operators, which are functions that transform according to the representations of the conformal group.

We will discuss the construction of vertex algebras and their associated conformal field theories. The key idea is to define a vertex operator, which is a linear map from the vertex algebra to the space of functions on the complex plane. The vertex operator allows us to express the fields in terms of the local operators, and it also encodes the conformal symmetry of the theory.

Conformal Blocks and Correlation Functions

Conformal blocks are functions that encode the correlation functions of the fields in a conformal field theory. Correlation functions are statistical measures that describe the interactions between the fields. They are essential for understanding the physical properties of the system.

We will introduce the concept of conformal blocks and discuss their properties. Conformal blocks can be used to compute the correlation functions of the fields, and they are also related to the conformal invariance of the theory. We will present some examples of conformal blocks and illustrate their applications in computing correlation functions.

Conclusion

In this article, we have provided a mathematical introduction to conformal field theory. We have discussed the basics of conformal transformations, representation theory, vertex algebras, and conformal blocks. These concepts are essential for understanding the structure and properties of conformal field theories. By studying conformal field theory, we can gain insights into the fundamental principles of physics and explore the connections between different areas of theoretical physics.