Higher Grading Generalisations of the Toda Systems

TL;DR

This paper introduces new integrable generalizations of the Toda system using higher Z-grading subspaces of Lie algebras, providing their solutions and exploring their relation to nonabelian Toda theories and Hamiltonian reduction.

Contribution

It develops integrable Toda system generalizations based on higher Z-gradings and constructs their solutions, linking them to nonabelian Toda theories and Hamiltonian reduction.

Findings

Constructed integrable generalizations of Toda systems.

Derived solutions for the generalized systems.

Explored relations between abelian and nonabelian Toda systems.

Abstract

In the present paper we obtain some integrable generalisations of the Toda system generated by flat connection forms taking values in higher ${\bf Z}$--grading subspaces of a simple Lie algebra, and construct their general solutions. One may think of our systems as describing some new fields of the matter type coupled to the standard Toda systems. This is of special interest in nonabelian Toda theories where the latter involve black hole target space metrics. We also give a derivation of our conformal system on the base of the Hamiltonian reduction of the WZNW model; and discuss a relation between abelian and nonabelian systems generated by a gauge transformation that maps the first grading description to the second. The latter involves grades larger than one.