Part 1. The classification of conformal field theories

Abstract

A Conformal Field Theory is a quantum field theory, usually with a two-dimensional space-time, whose symmetries include the conformal transformations. A category consists of two types of things. One are called objects, and the other are called arrows. Lie algebras appear in numerous places throughout math and mathematical physics. Lie group structure theory can be thought of as a major generalisation of linear algebra. The basic constructions familiar to undergraduates have important analogues valid in many Lie groups. The Lie algebra preserves the local properties of the Lie group, though it loses global topological properties. The representation theory of the simple Lie algebras can perhaps be regarded as an enormous generalisation of trigonometry. The theory of nontwisted affine Kac-Moody algebras is extremely analogous to that of the finite-dimensional simple Lie algebras. Quadratic reciprocity is one of the gems of classical number theory.