Conformal field theory in four and six dimensions

Abstract

Introduction In this paper, I will be considering conformal field theory (CFT) mainly in four and six dimensions, occasionally recalling facts about two dimensions. The notion of conformal field theory is familiar to physicists. From a mathematical point of view, we can keep in mind Graeme Segal's definition of conformal field theory. Instead of just summarizing the definition here, I will review how physicists actually study examples of quantum field theory, as this will make clear the motivation for the definition. When possible (and we will later consider examples in which this is not possible), physicists make models of quantum field theory using path integrals. This means first of all that, for any n -manifold M n , we are given a space of fields on M n ; let us call the fields Φ. The fields might be, for example, real-valued functions, or gauge fields (connections on a G -bundle over M n for some fixed Lie group G ), or p -forms on M n for some fixed p , or they might be maps Φ : M n → W for some fixed manifold W . Then we are given a local action functional I (Φ). ‘Local’ means that the Euler–Lagrange equations for a critical point of I are partial differential equations. If we are constructing a quantum field theory that is not required to be conformally invariant, I may be defined using a metric on M n . For conformal field theory, I should be defined using only a conformal structure.