Vertex Operator Algebra
Introduction
A **Vertex Operator Algebra** (VOA) is a mathematical structure that plays a crucial role in various areas of theoretical physics and mathematics, particularly in conformal field theory, string theory, and representation theory. VOAs are algebraic systems that encapsulate the properties of vertex operators, which are fundamental in the study of two-dimensional conformal field theories. The concept of a vertex operator algebra was developed to provide a rigorous mathematical framework for these operators, which were initially introduced in the context of string theory.
Historical Background
The origins of vertex operator algebras can be traced back to the 1980s, when physicists and mathematicians sought to formalize the algebraic structures underlying string theory and conformal field theory. The pioneering work of Richard E. Borcherds, Igor Frenkel, James Lepowsky, and Arne Meurman laid the foundation for the development of VOAs. Borcherds' introduction of the notion of a vertex algebra, along with the construction of the Monster group using these algebras, highlighted the deep connections between VOAs and other areas of mathematics, such as group theory and modular functions.
Definition and Structure
A vertex operator algebra is a complex vector space \( V \) equipped with a vacuum vector \( |0\rangle \), a conformal vector \( \omega \), and a linear map \( Y: V \to (\text{End}(V))[[z, z^{-1}]] \), where \( z \) is a formal variable. The map \( Y \) assigns to each vector \( v \in V \) a vertex operator \( Y(v, z) \), which is a formal power series in \( z \) with coefficients in \( \text{End}(V) \).
The defining properties of a VOA include:
1. **Vacuum Axiom**: The vacuum vector \( |0\rangle \) satisfies \( Y(|0\rangle, z) = \text{Id}_V \) and \( Y(v, z)|0\rangle \in v + zVz \) for all \( v \in V \).
2. **Translation Axiom**: There exists a derivation operator \( T \) on \( V \) such that \( [T, Y(v, z)] = \frac{d}{dz} Y(v, z) \).
3. **Locality Axiom**: For any \( u, v \in V \), there exists a positive integer \( N \) such that \( (z_1 - z_2)^N [Y(u, z_1), Y(v, z_2)] = 0 \).
4. **Conformal Structure**: The conformal vector \( \omega \) endows \( V \) with a Virasoro algebra structure, where the modes \( L_n \) of \( Y(\omega, z) = \sum_{n \in \mathbb{Z}} L_n z^{-n-2} \) satisfy the Virasoro algebra commutation relations.
Examples of Vertex Operator Algebras
The Heisenberg VOA
The Heisenberg vertex operator algebra is one of the simplest examples of a VOA. It is constructed from a Heisenberg algebra, which is an infinite-dimensional Lie algebra generated by elements \( a_n \) for \( n \in \mathbb{Z} \) and a central element \( c \), satisfying the commutation relations \([a_m, a_n] = m \delta_{m+n, 0} c\). The VOA is built on the Fock space representation of this algebra.
The Lattice VOA
Lattice vertex operator algebras are constructed from even lattices, which are discrete subgroups of Euclidean space equipped with a quadratic form. Given an even lattice \( L \), the lattice VOA \( V_L \) is formed by considering the space of states generated by vertex operators corresponding to lattice elements. These VOAs play a significant role in the theory of modular forms and the moonshine phenomenon.
The Virasoro VOA
The Virasoro vertex operator algebra is associated with the Virasoro algebra, which is a central extension of the Witt algebra. It is characterized by a central charge \( c \) and is constructed on the highest weight representations of the Virasoro algebra. The Virasoro VOA is fundamental in the study of two-dimensional conformal field theories.
Representation Theory of VOAs
The representation theory of vertex operator algebras is a rich and intricate field. A module over a VOA is a vector space equipped with a compatible action of the vertex operators. The study of VOA modules involves understanding the structure and classification of these modules, as well as their interrelations.
Simple and Rational VOAs
A VOA is called simple if it has no nontrivial ideals, and rational if every module is a direct sum of simple modules. Rational VOAs exhibit a finite number of irreducible modules, which are of particular interest in conformal field theory due to their well-behaved modular properties.
Modular Invariance
The modular invariance property of VOAs is a central theme in their representation theory. It refers to the invariance of the characters of VOA modules under the action of the modular group. This property is crucial in the study of modular functions and has deep implications in both mathematics and physics.
Applications in Physics and Mathematics
Vertex operator algebras have profound applications in both theoretical physics and pure mathematics. In physics, they provide the algebraic framework for two-dimensional conformal field theories, which describe critical phenomena in statistical mechanics and string theory. In mathematics, VOAs are instrumental in the study of monstrous moonshine, a mysterious connection between the Monster group and modular functions.
Advanced Topics
Orbifold Theory
Orbifold theory involves the study of VOAs associated with orbifolds, which are spaces obtained by taking the quotient of a manifold by a group action. Orbifold VOAs are constructed by considering the fixed points of the group action on the VOA and have applications in string theory and the study of symmetry in physical systems.
Quantum Groups and VOAs
The relationship between quantum groups and vertex operator algebras is an active area of research. Quantum groups, which are deformations of classical Lie algebras, have connections to VOAs through the study of affine Lie algebras and their representations. This interplay has led to significant developments in the understanding of integrable systems and knot theory.
Higher-Dimensional Generalizations
While VOAs are inherently two-dimensional, there have been efforts to generalize their structure to higher dimensions. These generalizations, known as higher-dimensional vertex algebras, aim to extend the rich algebraic framework of VOAs to more complex geometrical and physical settings.