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

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 $m^{\mathrm{th}}$-order differential operators $L$ and associated $V_{(n,m)}$-algebras.

20 pages, revised: several references to earlier papers on multi-component KdV equations are added

References in corpus (1)

• Non Abelian Toda Theory : A Completely Integrable Model for Strings on a Black Hole Background

Cited by in corpus (9)

• Higher Grading Generalisations of the Toda Systems
• Non-Local Matrix Generalizations of W-Algebras
• Frobenius manifolds and Frobenius algebra-valued integrable systems
• W-algebras for non-abelian Toda systems
• Heat Trace and Functional Determinant in One Dimension
• NON-LOCAL EXTENSIONS OF THE CONFORMAL ALGEBRA : MATRIX $W$-ALGEBRAS, MATRIX KdV-HIERARCHIES AND NON-ABELIAN TODA THEORIES,
• V-Algebras and Their Free Field Realizations
• On the Classical $W_4^{(2)}$ Algebra
• Nonlocal Matrix Generalizations of N=2 Super Virasoro Algebra