Introduction to Vertex Algebras, Borcherds Algebras, and the Monster Lie Algebra

TL;DR

This paper provides a comprehensive introduction to vertex algebras, their connection to conformal field theory, and their role in constructing Borcherds and Monster Lie algebras, highlighting explicit examples and fundamental properties.

Contribution

It offers a pedagogical overview of vertex algebras, illustrating their axiomatic foundations, connections to conformal field theory, and their application in constructing Borcherds and Monster Lie algebras.

Findings

Explicit construction of vertex algebras from even lattices

Derivation of properties of Borcherds algebras as generalized Kac-Moody algebras

Sketch of the construction of the Monster Lie algebra

Abstract

The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory. In this context Borcherds algebras arise as certain ``physical'' subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction into this rapidly-developing area of mathemat% ics. Based on the machinery of formal calculus we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake Monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analysed from the point of view of symmetry in quantum theory and the construction of the Monster Lie algebra is sketched.