Conformal Field Theory In Four And Six Dimensions
Edward Witten
TL;DR
The notes examine the link between electric-magnetic duality in four dimensions and a set of non-Lagrangian, six-dimensional conformal theories. They begin with linear abelian theories and p-forms, showing how modular invariance arises in 4d and how a self-dual theory in 6d encodes duality via compactification on tori. They then discuss nonlinear, supersymmetric theories, notably 4d N=4 SYM with PSU(2,2|4) symmetry and its six-dimensional origin from an ADE-related theory whose reduction to 4d reproduces Montonen-Olive duality. The material connects modular properties, theta-function constructions, and high-dimensional gerbe theories to explain dualities that lack a conventional Lagrangian. It highlights the significance of six-dimensional theories in understanding the structure of four-dimensional conformal and supersymmetric field theories.
Abstract
The goal of these notes is to give a brief explanation of how electric-magnetic duality in four dimensions is related to the existence of an unusual conformal field theory in six dimensions.