Multi-component KdV hierarchy,V-algebra and non-Abelian toda theory

Abstract

We prove the recently conjectured relation between the 2 × 2-matrix differential operatorL = ∂2 −U and a certain nonlinear and nonlocal Poisson bracket algebra (V-algebra), containing a Virasoro subalgebra, which appeared in the study of a non-Abelian Toda field theory. In particular, we show that thisV-algebra is precisely given by the second Gelfand-Dikii bracket associated withL. The Miura transformation that relates the second to the first Gelfand-Dikii bracket is given. 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 -ζ)Ψ = 0 is studied and its coefficientsR l yield an infinite sequence of Hamiltonians with mutually vanishing Poisson brackets. We recall how this leads to a matrix KdV hierarchy, which here are flow equations for the three component fieldsT,V +,V − ofU. ForV ± = 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 ifU is a Hermitiann ×n matrix. Conjectures are made aboutn ×n-matrix,mth-order differential operatorsL and associatedV (n,m)-algebras.