Non-canonical folding of Dynkin diagrams and reduction of affine Toda theories

TL;DR

This paper explores new non-canonical foldings of affine Dynkin diagrams, leading to novel reductions in affine Toda theories, and establishes general rules for these reductions, expanding the understanding of their symmetry structures.

Contribution

It introduces non-canonical foldings of affine Dynkin diagrams, extending symmetry-based reductions in affine Toda theories beyond traditional methods.

Findings

Most theories reduce to $a_{2n}^{(2)}$

New reduction patterns identified

General rules for non-canonical foldings formulated

Abstract

The equation of motion of affine Toda field theory is a coupled equation for $r$ fields, $r$ is the rank of the underlying Lie algebra. Most of the theories admit reduction, in which the equation is satisfied by fewer than $r$ fields. The reductions in the existing literature are achieved by identifying (folding) the points in the Dynkin diagrams which are connected by symmetry (automorphism). In this paper we present many new reductions. In other words the symmetry of affine Dynkin diagrams could be extended and it leads to non-canonical foldings. We investigate these reductions in detail and formulate general rules for possible reductions. We will show that eventually most of the theories end up in $a_{2n}^{(2)}$ that is the theory cannot have a further dimension $m$ reduction where $m<n$.