Multi-Component KdV Hierarchy, V-Algebra and Non-Abelian Toda Theory

TL;DR

This paper establishes a deep connection between a matrix differential operator, V-algebra, and non-abelian Toda theory, revealing new integrable structures and hierarchies in mathematical physics.

Contribution

It proves the conjectured relation between a matrix differential operator and V-algebra, linking it to the Gelfand-Dikii brackets and integrable hierarchies.

Findings

V-algebra is given by the second Gelfand-Dikii bracket.

Derived matrix KdV and mKdV hierarchies for multi-component fields.

Extended results to hermitian matrix operators and conjectures for higher-order cases.

Abstract

I prove the recently conjectured relation between the $2\times 2$-matrix differential operator $L=\partial^2-U$, and a certain non-linear and non-local Poisson bracket algebra ($V$-algebra), containing a Virasoro subalgebra, which appeared in the study of a non-abelian Toda field theory. Here, I show that this $V$-algebra is precisely given by the second Gelfand-Dikii bracket associated with $L$. The Miura transformation is given which relates the second to the first Gelfand-Dikii bracket. The two Gelfand-Dikii brackets are also obtained from the associated (integro-) differential equation satisfied by fermion bilinears. The asymptotic expansion of the resolvent of $(L-\xi)\Psi=0$ is studied and its coefficients $R_l$ yield an infinite sequence of hamiltonians with mutually vanishing Poisson brackets. I recall how this leads to a matrix KdV hierarchy which are flow equations for the three component fields $T, V^+, V^-$ of $U$. For $V^\pm=0$ they reduce to the ordinary KdV hierarchy. The corresponding matrix mKdV equations are also given, as well as the relation to the pseudo- differential operator approach. Most of the results continue to hold if $U$ is a hermitian $n\times n$-matrix. Conjectures are made about $n\times n$-matrix $m^{\rm th}$-order differential operators $L$ and associated $V_{(n,m)}$-algebras.