Toda Theory From Six Dimensions

TL;DR

This paper demonstrates how compactifying the six-dimensional (2,0) theory on a four-sphere yields a two-dimensional Toda theory, revealing deep connections between higher-dimensional theories, edge modes, and W-algebras.

Contribution

It provides a novel geometric construction linking the (2,0) theory to Toda field theory via compactification and edge modes, elucidating their relationship through conformal anomalies.

Findings

Realization of chiral Toda fields as edge modes near sphere poles

Connection between (2,0) theory operators and W-algebras

Equality of conformal anomalies between theories

Abstract

We describe a compactification of the six-dimensional (2,0) theory on a four-sphere which gives rise to a two-dimensional Toda theory at long distances. This construction realizes chiral Toda fields as edge modes trapped near the poles of the sphere. We relate our setup to compactifications of the (2,0) theory on the five and six-sphere. In this way, we explain a connection between half-BPS operators of the (2,0) theory and two-dimensional W-algebras, and derive an equality between their conformal anomalies. As we explain, all such relationships between the six-dimensional (2,0) theory and Toda field theory can be interpreted as statements about the edge modes of complex Chern-Simons on various three-manifolds with boundary.